diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzlixz" "b/data_all_eng_slimpj/shuffled/split2/finalzzlixz" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzlixz" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nErgodicity of many-body systems and its breaking is one of the central research areas in modern statistical mechanics.\nThe basic definition is well understood: the physical system can be called ergodic, if during its time evolution all accessible microstates are visited.\nHowever, the detailed division whether the ergodicity is broken or not, especially, when applied to a large variety of quantum systems is yet to be established~\\cite{DAlessio2016}.\nA few general examples of ergodicity breaking in these are quantum scars \\cite{Bernien2017, Serbyn2021}, Bethe ansatz integrable systems~\\cite{QuantumCradle}, lattice gauge theories \\cite{LatticeGaugeTheory}, fractons and confinement~\\cite{Fractons_confinement}, and Hilbert space fragmentation \\cite{Sala2020, PhysRevB.101.174204}.\n\nAll these examples of ergodicity breaking and lack of thermalization in closed systems provoke a number of questions on the microscopic characterization of ergodicity in quantum systems. \nIn particular, there are interesting connections between thermalization and properties of eigenstates of microscopic Hamiltonian, which are nicely summarized in the form of the eigenstate thermalization hypothesis \\cite{SrednickiETH, DeutschETH}, and also with the quasiclassical limit, where these notions overlap with classical chaos \\cite{haake1991quantum}.\nRather generally, the notions of quantum chaos and thermalization in closed systems can be used interchangeably, and we thereafter use different criteria of quantum chaoticity, as, in particular, spectral statistics, to detect absence of thermalization. \n\nAs for experimental verification of the mentioned model studies and uncovering new related phenomena,\nover the past two decades, the range of accessible quantum many-body systems has been sufficiently extended. \nThis progress is largely due to an impressive development of experimental techniques for cooling and loading atoms into optical lattices \\cite{Bloch2008RMP, Esslinger2010ARCMP}.\nIn these artificial systems, many relevant parameters can be controlled and tuned with a high degree of freedom: the external potential (with additional disorder, linear, or any specific), the interaction (both the amplitude and the range), the initial lattice filling, particle statistics, symmetries, etc.\nAs a natural consequence of this freedom, in particular, the celebrated Anderson localization phenomenon is now viewed as only a member of the wider class of many-body localization (MBL) transitions in interacting systems \\cite{Nandkishore2015, Abanin2019RMP}.\n\nFrom the success of the Anderson localization, the natural platforms for MBL were initially the systems with disorder.\nThese platforms were successfully realized in experiments, see, e.g., Ref.~\\cite{ExperimentMBL}, where one required artificial disorder produced by quasiperiodic potentials or other means.\nRecently, it was shown that analogous systems with only short-range interactions and disorder-free (linear or harmonic) potentials can exhibit localized behavior in a wide range of Hamiltonian parameters, which was named as the Stark (or Bloch) localization \\cite{Nieuwenburg2019, StarkLocalization, Yao2020} and was also observed experimentally \\cite{Scherg2021}.\n\nThe disorder-free potentials with a linear tilt are common in the field of cold atoms in optical lattices \\cite{Raizen1997}.\nTypically, the interactions between cold atoms are short-range, however, there are many cases, where these become sufficiently nonlocal, as in gases of atomic isotopes possessing the dipole moment in the ground state~\\cite{DipolarGasesOpticalLattice} and atoms in the metastable excited Rydberg states~\\cite{Rydberg_atoms_review}. The latter are especially attractive in the context of ergodicity breaking due to quantum scar effects \\cite{Bernien2017}.\nFurthermore, there is a number of both experimental and theoretical studies on many-body regimes in atomic gases with cavity-mediated interactions \\cite{Dogra2016,Landig2016,Sierant2019} (see also the review \\cite{CavityQED}). Coupling to the cavity modes in these systems sufficiently extends the effective range of interactions between atoms.\n\nFrom the theoretical point of view, the combination of the above realizations, namely, the tilted lattice systems, where atoms or quantum spins interact nonlocally, was not studied in detail. \nThis motivates us to focus on a wide class of model Hamiltonians, with various types of experimentally available long-range interactions or long-range hopping processes and analyze the fate of many-body localization in these systems. As we show below, long-range interactions also impact the spectrum in the same regularizing way as an additional disordered or harmonic potential for short-range interacting systems. It turns out that it is sufficient to employ the disorder-free linear external potential with moderate-range interactions between particles to observe and study MBL transitions. \n\n\n\nIt should be noted that aspects of Stark localization in similar context attracted much interest recently. In particular, there are studies of the tilted Heisenberg spin chain with the next-nearest couplings \\cite{vernek2021robustness}, tilted lattice systems with long-range hopping \\cite{PhysRevB.102.085133} and with cavity-mediated interactions \\cite{Chanda2021manybody}.\n\n\n\\section{Models and methods}\n\\subsection{Lattice models}\nIn this section, we introduce one-dimensional theoretical models in the order of increasing complexity. Starting from the noninteracting limit with only (long-range) hopping and external linear potential, we discuss the localized wave functions and influence of hopping. \nWe extend further our description by including power-law interactions, various representations and symmetries of these models. Furthermore, we study a model of localization in all-to-all potentials, which can be realized in cavities \\cite{Landig2016}.\n\n\n\\subsubsection{Noninteracting model}\nThe Hamiltonian of noninteracting lattice model consists of the external linear potential and the hopping term, which describes long-range tunneling processes. Here, we start from an infinite system, i.e., neglect the boundary effects for simplicity,\n\\begin{equation}\\label{eq:Hnonint}\n \\hat{H} = - \\sum_{j=1}^{m}J_{j}\\sum_{k} \n (\\hat{a}^{\\dagger}_{k} \\hat{a}_{k+j} +{\\rm H.c.}) \n + F \\sum_{k} k \\hat{a}^{\\dagger}_{k} \\hat{a}_{k},\n\\end{equation}\nwhere $\\hat{a}^{\\dagger}_{k}$ and $\\hat{a}_{k}$ are bosonic or fermionic creation and annihilation operators on site $k$, respectively. The quantity $F$ characterizes the amplitude of external linear potential and $J_{j}$ are the hopping amplitudes, which depend on the distance~$j$ between the lattice sites. The upper limit~$m$ in the sum denotes the maximal range of hopping. This maximal range can be both finite or infinite in the case of power-law hopping $J_{j} \\propto 1\/j^\\beta$.\n\nThe introduced model is quadratic in creation and annihilation operators, thus it is sufficient to solve its one-particle sector. \nHence, the wave function can be written in the form\n\\begin{equation}\n |\\psi\\rangle = \\sum_{k} c_{k} \\hat{a}^{\\dagger}_{k} |0\\rangle,\n\\end{equation}\nwhere $c_{k}$ are the coefficients and $|0\\rangle$ is the vacuum state.\nWe can map the Hilbert space built on the basis states $\\hat{a}^{\\dagger}_{k}|0\\rangle$ onto the Hilbert space of functions on the circle according to the rule \\cite{Hartmann_2004}\n\\begin{equation}\n \\hat{a}^{\\dagger}_{k} |0\\rangle \\rightarrow \n \\frac{\\exp{\\left(i k \\phi\\right)}}{\\sqrt{2 \\pi}},\n\\end{equation}\nwhere $\\phi$ is the polar angle. This results in the mapping\n\\begin{equation}\n |\\psi\\rangle \n \\rightarrow \n \\psi(\\phi)=\\sum_{k} \\frac{c_{k} \\exp{\\left(i k \\phi\\right)}}{\\sqrt{2 \\pi}}.\n\\end{equation}\n\nWithin the introduced procedure, it is possible to map operators entering the Hamiltonian~\\eqref{eq:Hnonint} to differential operators on the circle. For the linear potential term, the corresponding mapping can be written as follows:\n\\begin{multline}\\label{eq:map1}\n \\sum_{m} m a^{\\dagger}_{m} a_{m} \\sum_{k} c_{k} a^{\\dagger}_{k} |0\\rangle = \\sum_{k} k c_{k} a^{\\dagger}_{k} |0\\rangle \\rightarrow \n \\\\ \n \\rightarrow \\sum_{k} \\frac{ k c_{k} \\exp{\\left(i k \\phi\\right)}}{\\sqrt{2 \\pi}} \n =\n -i \\frac{d}{d\\phi} \\psi(\\phi).\n\\end{multline}\nWe see that the external linear potential is mapped to the derivative. \nFinally, let us perform analogous mapping for the hopping terms,\n\\begin{multline}\\label{eq:map2}\n \\sum_{m} \\hat{a}^{\\dagger}_{m} \\hat{a}_{m+j} \n \\sum_{k} c_{k} \\hat{a}^{\\dagger}_{k} |0\\rangle \n = \\sum_{k} c_{k+j} \\hat{a}^{\\dagger}_{k} |0\\rangle \\rightarrow \n \\\\ \n \\rightarrow \\sum_{k} \\frac{ c_{k+j} \\exp{\\left(i k \\phi\\right)}}{\\sqrt{2 \\pi}} \n = \n \\exp{\\left(-i j \\phi \\right)} \\psi(\\phi).\n\\end{multline}\nThese terms are mapped to the basis functions multiplied by the range-dependent phase factors. \n\nBy means of the obtained mapping rules \\eqref{eq:map1} and \\eqref{eq:map2}, the Hamiltonian~\\eqref{eq:Hnonint} can be expressed as \n\\begin{equation}\n {H} = -2\\sum_{j=1}^{m} J_{j} \\cos{\\left(j\\phi\\right)} -i F \\frac{d}{d\\phi}.\n\\end{equation}\nThe eigenstates of this Hamiltonian can be determined by solving the first-order differential equation, while the eigenvalues are obtained by the condition that the eigenstates must be periodic functions on the circle.\nAs a result, we obtain the eigenstates,\n\\begin{equation}\\label{eq:psi_n}\n \\psi_{n}(\\phi) = \\frac{\\exp{\\left(i n \\phi + 2 i \\sum_{j=1}^{m} \\frac{J_{j} \\sin{(j \\phi)}}{j F}\\right)}}{\\sqrt{2 \\pi}} ,\n\\end{equation}\nand the eigenvalues\n\\begin{equation}\n E_{n} = F n, \\quad n \\in \\mathbb{Z}.\n\\end{equation}\n\nThe spectrum of the introduced model is independent of the hopping amplitudes and it is the same as of the Hamiltonian with only a potential term. \nIt is natural to suggest that the wave functions in the presence of hopping are continuously connected to the wave functions in the atomic limit (the latter are completely localized on one site). The nonzero hopping processes lead to broadening of the wave functions around that center site with a corresponding exponential decay of the density distribution. \nBelow, we also show it more directly by expressing the coefficients~$c_k$ that determine the wave function in the initial basis~$|\\psi\\rangle$. \n\nFrom the form of the wave function~\\eqref{eq:psi_n} it is clear that eigenfunctions with $n\\neq0$ can be obtained from the eigenstate with $n=0$ simply by translation. \nIt can also be deduced from the fact that a commutator of the shift operator with the Hamiltonian results in the shift operator itself. \nHence, for the models considered below, the shift operator can be viewed as a raising operator: all eigenstates can be obtained by repeated action of the shift operator on a particular eigenstate. For $n=0$, we obtain the following coefficients $c_{k}$ in the initial basis:\n\\begin{equation}\n c_{k} = \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} \\exp{\\left(-i k \\phi + 2 i \\sum_{j=1}^{m} \\frac{J_{j} \\sin{(j \\phi)}}{j F}\\right)} d\\phi\n\\end{equation}\n\nIn the simplest case of only the nearest-neighbor hopping, these coefficients are determined in terms of the Bessel functions~${\\cal J}_k(x)$ as $c_{k} = {\\cal J}_{k}\\left({2J_{1}}\/{F}\\right)$. \nIn a similar case of hopping only between the next-nearest neighbors ($J_{2}\\neq0$, while $J_j=0$ for $j\\neq2$), the wave functions vanish for odd $k$, while for even $k$ they are given by $c_{2k} = {\\cal J}_{k}\\left({J_{2}}\/{F}\\right)$.\nNote that in the case of nearest-neighbor hopping, the localization is generally stable to interactions if $F>2J_{1}$, or if the argument of the Bessel functions is smaller than one.\nNote that in this case, the exponential vanishing of the wave functions is clear from the expansion of the Bessel functions into series over small argument $x=2J_1\/F$, which gives ${\\cal J}_{k}(x) \\propto x^{k}(1+O(x^2))$. \nIn case of the next-nearest-neighbor hopping, we can conjecture analogously that the localization is stable to interactions if the argument of the Bessel function for the noninteracting wave function is smaller than one, or if $J_{2} < F$. \nMore generally, we can conclude that the single particle Stark localization is stable if $2J_{m}1$.\n\n\n\\subsubsection{Models with power-law long-range interactions}\nAs one can see from the preceding results, the many-body states of the noninteracting model are localized for all values of parameters.\nBut the localization may not be stable with respect to interactions between particles. \nNow, we introduce the long-range many-body interactions and study the possibility of localization in this system.\nThe issue of MBL in the presence of long-range interactions is also of conceptual value, since for a long time it was accepted that systems with long-range interactions described by the power-law dependence cannot demonstrate localization features.\n\nFor definiteness, let us introduce the interacting one-dimensional system consisting of spinless fermions on the finite lattice with $L$ sites [see also Fig.~\\subfigref{models}{a}].\nIt is described by the Hamiltonian\n\\begin{equation}{\\label{H_fermionic}}\n \\hat{H} \n = -\\sum_{k=1}^{m} J_{k} \\sum_{i=1}^{L-k} \n (\\hat{f}^{\\dagger}_{i} \\hat{f}_{i+k} +{\\rm H.c.}) \n +F\\sum_{i=1}^{L} i \\hat{n}_{i} + U \\sum_{1\\leq i < j}^L\n \\frac{\\hat{n}_{i} \\hat{n}_{j}}{|i-j|^{\\alpha}},\n\\end{equation}\nwhere $\\hat{f}^{\\dagger}_{i}$ and $\\hat{f}_{i}$ are the fermionic creation and annihilation operators on site $i$, respectively, and $\\hat{n}_{i}=\\hat{f}^{\\dagger}_{i}\\hat{f}^{\\phantom{\\dagger}}_{i}$ is the corresponding number operator on site $i$. $F$ determines the strength of the external linear potential, as before, $U$ corresponds to the magnitude of interactions between particles, and $\\alpha$ is the exponent characterizing the power-law decay of interactions. \n$J_{k}$ are the hopping amplitudes, while $m$ determines the maximal range of hopping as in the noninteracting model. \n\\begin{figure}[t]\n \\includegraphics[width=\\linewidth]{fig1_models.pdf}\n \\caption{\\label{fig:models}%\n Schematic illustration of many-body systems under study and relevant couplings.\n }\n\\end{figure}\nBelow, we focus on small values of the exponent $\\alpha$, in particular, $\\alpha \\in [0.5,3]$, since at larger values of $\\alpha$ the system behaves as the one with short-range interactions. \nThe case $\\alpha = 3$ is especially relevant, as it can be realized experimentally with dipolar ultracold gases \\cite{DipolarGasesOpticalLattice}. \n\nIn case of bosonic system, the Hamiltonian is analogous to Eq.~\\eqref{H_fermionic}, except of the additional possibility of the on-site interaction, controlled by the parameter $V$ [see also Fig.~\\subfigref{models}{b}]. For the completeness, we specify the explicit form as follows:\n\\begin{multline}{\\label{H_bosonic}}\n \\hat{H} = -\\sum_{k=1}^{m} J_{k} \\sum_{i=1}^{L-k} \n (\\hat{a}^{\\dagger}_{i} \\hat{a}_{i+k} + {\\rm H.c.}) \n \\\\ \n +V\\sum_{i=1}^{L} \\hat{n}_{i}(\\hat{n}_{i}-1)\n +F\\sum_{i=1}^{L} i \\hat{n}_{i} + U \\sum_{1\\leq i < j \\leq L} \n \\frac{\\hat{n}_{i} \\hat{n}_{j}}{|i-j|^{\\alpha}},\n\\end{multline}\nwhere $\\hat{a}^{\\dagger}_{i}$ and $\\hat{a}_{i}$ are the bosonic creation and annihilation operators on site $i$, respectively. \nAll other quantities have the same meaning, as in the fermionic case. In numerical calculations, the dimension of the local bosonic Hilbert space has to be restricted to a finite value. We set the maximal number of bosons on the same site equal to three, which is sufficient at moderate values of $V$.\n\nNote that for the purpose of succeeding analysis in the framework of the time-dependent variational principle (TDVP) and the Shrieffer-Wolff transformation, it is necessary to reformulate the fermionic Hamiltonian in the bosonic language. \nFor this purpose, we employ the Jordan-Wigner transformation to map the fermionic system onto the spin-1\/2 chain. \nIn this procedure, the creation and annihilation operators are mapped onto the Jordan-Wigner chains according to the rules: $\\hat{f}^{\\dagger}_{i} \\to \\prod_{j=1}^{i-1} (-\\hat{\\sigma}^{z}_{j}) \\hat{S}^{+}_{i}$ \nand $\\hat{f}_{i} \\to \\prod_{j=1}^{i-1} (-\\hat{\\sigma}^{z}_{j}) \\hat{S}^{-}_{i}$, where $\\hat{\\sigma}^{z}$ is the Pauli matrix with a conventional correspondence to the spin projection operator to the $z$ axis, $\\hat{S}^z=\\hat{\\sigma}^z\/2$, while $\\hat{S}^{+}=(\\hat{\\sigma}^x+i\\hat{\\sigma}^y)\/2$ and $\\hat{S}^{-}=(\\hat{\\sigma}^x-i\\hat{\\sigma}^y)\/2$ are the spin-raising and spin-lowering operators, respectively. \nThe particle number operator $\\hat{n}_i$ is mapped to the local projection operator as $\\hat{n}_i \\to 1\/2 + \\hat{S}^{z}_{i}$.\nUsing these rules, it is possible to map the fermionic Hamiltonian with only the nearest-neighbor hopping ($m=1$) to the following spin-chain Hamiltonian:\n\\begin{multline}{\\label{H_spin}}\n \\hat{H} = - J_{1}\\sum_{i=1}^{L-1} (\\hat{S}^{+}_{i}\\hat{S}^{-}_{i+1} + \\hat{S}^{-}_{i}\\hat{S}^{+}_{i+1})\n + U \\sum_{1\\leq i < j \\leq L} \n \\frac{\\hat{S}^{z}_{i} \\hat{S}^{z}_{j}}{|i-j|^{\\alpha}}\n \\\\\n + \\sum_{i=1}^{L} (F i +w_{i}) \\hat{S}^{z}_{i}\n , \n \\quad\n w_{i} = \\frac{U}{2}\\sum_{j=1, j\\neq i}^{L} \\frac{1}{|i-j|^{\\alpha}}.\n\\end{multline}\nIn the given form, this model describes the $XXZ$ spin chain in the external inhomogeneous magnetic field $h_i^z\\equiv(Fi+w_i)$, which has additional Ising-type couplings between the spins located farther from each other than nearest neighbors [see also Fig.~\\subfigref{models}{c}].\nThe additional on-site potential term $w_{i}$ becomes constant in the limit of infinite $L$. On the finite lattice, this term is almost constant in the bulk and decreases only at the boundaries. \n\n\n\\subsubsection{Model in a cavity}\\label{subsec:cavity}\nIn addition to the model with power-law long-range interactions, let us also introduce the model of localization in the all-to-all potential. \nThis kind of interatomic potential can be realized with a system in a cavity, where presence of the cavity modes, strongly interacting with particles, can induce completely nonlocal interaction patterns. Here, we study a simple model, which captures some basic characteristics of real cavities. In this model [see also Fig.~\\subfigref{models}{d}], we introduce an additional term \\cite{Dogra2016, Landig2016, Sierant2019, CavityQED, BoseHubbardQuenches}, which is added to the above-specified Hubbard-type Hamiltonians \\eqref{H_fermionic} or \\eqref{H_bosonic},\n\\begin{equation}{\\label{H_cavity}}\n \\hat{H}_{\\rm cav} = - \\frac{U_{\\rm cav}}{L} \\left( \\sum_{i=1}^{L} (-1)^{i+1} \\hat{n}_{i} \\right)^{2}.\n\\end{equation}\n\nThis cavity term leads to all-to-all interactions with the same strength between particles irrespectively of the separating distance. \nThe interaction amplitude $U_{\\rm cav}$ is normalized by the lattice size $L$ to be meaningful in the infinite-lattice limit. \nIt is interesting to investigate whether the localization induced by a linear potential is stable to these long-range interactions. \n\n\n\\subsection{Methods}\nTo distinguish between chaotic and MBL phases, we employ several methods, which are commonly used in the literature on many-body localization. For small system sizes (up to $L = 18$ for spinless fermionic or spin-1\/2 systems), it is possible to obtain full spectrum using exact diagonalization. \nSince chaotic and localized systems have different level statistics \\cite{Alet2018ManybodyLA}, characteristics of the spectrum can be used as probes of localization.\nAs the system size grows, exact diagonalization quickly becomes infeasible due to exponential growth of the Hilbert space with the number of the latticce sites~$L$. \nTherefore, below we also employ methods based on the matrix product states to access dynamics of much larger bosonic and fermionic systems after quenches. \nIn these simulations, MBL manifests itself as a lack of thermalization of local observables and slow logarithmic growth of the entanglement entropy, similar to observations given, e.g., in Refs.~\\cite{Alet2018ManybodyLA, EntanglementGrowth1,EntanglementGrowth2, EntanglementGrowth3, EntanglementGrowth4}. \nFor a large linear tilt $F$, we apply the Schrieffer-Wolff transformation to obtain effective Hamiltonians. Within the effective models, we also analyze limiting cases of spectral characteristics and evolution of relevant physical observables.\n\n\n\\subsubsection{Exact diagonalization and level statistics}\\label{subsec:ED}\nFor a small system size and moderate local Hilbert space dimensions (or in a dilute limit, not studied in this work), it is possible to determine full spectrum of the system Hamiltonian in the fixed symmetry sector. The exponential growth of the Hilbert space with a number of lattice sites~$L$ limits these calculations to $L \\approx 18$ for spinless fermions, spins or hard-core bosons, or even smaller numbers for bosons with moderate on-site interactions or spinful fermions. \n\nChaotic and MBL spectra have different level statistics: Poissonian for MBL phase and Wigner-Dyson for chaotic systems \\cite{Alet2018ManybodyLA}. It is connected to the fact that the MBL phase has an extensive set of quasilocal integrals of motion. \nThe eigenstates with different eigenvalues of these integrals have uncorrelated energy eigenvalues that leads to the Poisson distribution. However, Hamiltonians of chaotic systems can be represented as random matrices. Eigenvalues of random matrices are distributed according to the Dyson-Wigner ensembles \\cite{Mehta2004}. \nDue to the fact that the full spectrum statistics contains an immense amount of information, it is more reasonable to employ a simple quantity, which distinguishes chaotic and MBL systems. \nSuch a commonly-used criterion is the gap ratio, usually denoted by $r$. \nTo evaluate it, we sort the energy spectrum and calculate the quantity\n\\begin{equation}{\\label{r}}\n {r_i} = \\frac{\\min(E_{i}-E_{i-1}, E_{i+1} - E_{i})}{\\max(E_{i}-E_{i-1}, E_{i+1} - E_{i})}.\n\\end{equation}\nNext, this ratio is averaged over all neighbor triples $\\{E_{i-1}, E_{i}, E_{i+1}\\}$ in the sorted spectrum, $r=\\langle r_i\\rangle$.\nIt is established that $r\\approx0.38$ for the MBL systems and $r\\approx0.53$ for the chaotic ones \\cite{PhysRevLett.110.084101,PhysRevB.75.155111}. \nBy analyzing this criterion as a function of system parameters, i.e., by constructing an effective ``phase diagram'', we can determine boundaries between chaotic and localized behavior \\cite{PhysRevA.92.041601, PhysRevB.82.174411}.\nIn this study, we perform the corresponding numerical analysis by means of the \\textsc{QuSpin} open-source package \\cite{SciPostPhys.2.1.003, 10.21468\/SciPostPhys.7.2.020}.\n\nThe exact diagonalization (ED) technique can also be employed for studying time dynamics after quenches. \nWithin this method, we directly obtain all Hamiltonian eigenstates and project the initial wave function onto them.\nAlthough ED is feasible for small system sizes, its substantial benefit is that the evolution of physical observables can be analyzed on exponentially large timescales (in contrast to the TDVP approach, where the complexity scales linearly with time). \nAccess to quantities in this regime allows us to study asymptotic behavior of relevant observables and their fluctuations. \n\n\n\\subsubsection{Ensemble-based analysis}\\label{subsec:ensembles}\nThe physical observables obtained within the ED approach can also be compared with those from the diagonal and microcanonical ensembles \\cite{DAlessio2016}. \nThis comparison is an explicit test for thermalization or its absence, since for the thermalized system the local observables must stay in agreement with those provided by the microcanonical ensemble. \nThe observables from the microcanonical or diagonal ensembles can be evaluated from the available data obtained in the simulation of system dynamics.\n\nFor the microcanonical ensemble, we calculate observables in the following way. First, we calculate the expectation value~$E^{(0)}=\\langle\\psi^{(0)}| \\hat{H} |\\psi^{(0)} \\rangle$ of the Hamiltonian in the initial state~$|\\psi^{(0)} \\rangle$ before the quench.\nNext, we specify the range of energies $\\Delta E$ and determine all eigenstates ${\\psi_i}$ with the energies $E_i\\in[E^{(0)} - \\Delta E, E^{(0)} + \\Delta E]$.\nWe choose $\\Delta E$ in the way that the number $N_{\\rm st}$ of the available eigenstates in the interval is about $N_{\\rm st}=50$. \nFinally, we evaluate the expectation values of the operator $\\hat{\\cal O}$ in the microcanonical ensemble (ME) according to the standard formula, \n\\begin{equation}\\label{eq:obs_ME}\n \\langle \\hat{\\cal O}\\rangle_{\\rm ME}\n = {N_{\\rm st}}^{-1}\\sum_{i} \n {\\langle \\psi_{i}|\\hat{\\cal O}|\\psi_{i} \\rangle}, \n\\end{equation}\nwhere the summation is performed over all eigenstates in the specified energy range. \n\nThe diagonal ensemble describes long-time asymptotics of expectation values \\cite{DAlessio2016}. To access it, we calculate all eigenstates $|v_{j}\\rangle$ in the symmetry sector (e.g., the block with the fixed total number of particles) of the initial state $|\\psi^{(0)}\\rangle$.\nNext, we calculate the projection coefficients of the initial state onto the eigenstates $\\langle v_{j}|\\psi^{(0)}\\rangle$.\nThe average values in the diagonal ensemble (DE) are calculated according to the formula\n\\begin{equation}\\label{eq:obs_DE}\n \\langle \\hat{\\cal O}\\rangle_{\\rm DE}\n =\\sum_{j} |\\langle v_{j}|\\psi^{(0)}\\rangle|^{2} \\langle v_{j}|\\hat{\\cal O}|v_{j} \\rangle,\n\\end{equation}\nwhere the summation is performed over all energy eigenstates~$|v_{j}\\rangle$. \nSince the long-time asymptotes of physical observables after quench are equal to expectation values in the diagonal ensemble, we compare $\\langle \\hat{\\cal O}\\rangle_{\\rm DE}$ with $\\langle \\hat{\\cal O}\\rangle_{\\rm ME}$ to determine whether the system is thermalized. \n\nAs an additional important observable, we also calculate the entanglement entropy. To this end, we consider a state~$|\\psi\\rangle \\equiv|\\psi\\rangle_{AB}$ and a bipartition of the system $AB$ into two parts: $A$ and $B$ with the respective sizes $L_{A}$ and $L_{B}$.\nThen, we can define the entanglement entropy of the subsystem $A$ as the von Neumann entropy of the reduced density matrix $\\rho_{A}$ characterizing the subsystem $A$,\n\\begin{equation}\\label{eq:ent_entropy}\n S=-\\operatorname{Tr}(\\rho_{A}\\ln{\\rho_{A}}).\n\\end{equation}\nWe calculate the reduced density matrix according to the standard formula $\\rho_{A} = \\operatorname{Tr}_{B}\\rho$, where $\\rho = |\\psi\\rangle \\langle \\psi|$ is the density matrix of the full system under study and the trace is taken over degrees of freedom in the subsystem $B$. \n\n\\subsubsection{Matrix-product state approaches}\nAs we mentioned above, for large systems the ED approach is not feasible.\nHowever, it is still possible to employ algorithms based on matrix product states (MPS). \nIn these methods, it is assumed that the targeted state can be represented as an MPS of a relatively small bond dimension~$D$ (typically, $D\\lesssim 100$). There are several classes of such algorithms applicable to MBL systems. \nIn particular, the density-matrix renormalization group (DMRG) approach with the corresponding generalization (DMRG-X) can be used to determine eigenstates in the middle of the spectrum of MBL systems \\cite{PhysRevLett.116.247204}. \nUnitary matrix-product operator algorithm \\cite{PhysRevB.94.041116} finds the full unitary matrix that diagonalizes the Hamiltonian of the localized system.\nTDVP \\cite{PhysRevB.94.165116} and time evolving block decimation (TEBD) \\cite{PhysRevLett.93.040502} can be used to determine dynamics of wave functions after quenches or time dynamics of operators in the Heisenberg picture. \nBelow, we restrain ourselves to studying only time dynamics of wave functions. \nAlthough both TEBD and TDVP approaches can be employed for this purpose, TEBD is restricted to Hamiltonians with short-range interactions. \nAs the Hamiltonians of our models contain the long-range terms, TDVP becomes more beneficial for the simulation of quenches. \nIn this study, we perform the corresponding tensor-network calculations by means of the \\textsc{ITensor} numerical package~\\cite{itensor}.\n\nMPS-based approaches are powerful in representing the states with low entanglement entropy. They have a control parameter, the bond dimension $D$, with the maximal entanglement of the representable states, which scales as $\\log(D)$. \nOne can use unentangled states as initial wave functions, which can be exactly represented as MPS. \nThen, we propagate this state in time within the TDVP approach. \nNaturally, the entanglement entropy increases during the time evolution.\nAs soon as the entropy reaches approximately the same value as the maximal entanglement entropy for the given bond dimension, results from TDVP become unreliable \\footnote{In fact, as results for the system with short-range interactions show, TDVP may become unreliable significantly earlier, see, e.g., the analysis of TDVP convergence in Ref.~\\cite{Sierant2021}}. \nFor this reason, TDVP is effective only on finite time intervals. However, it can be effectively used for a detection of the MBL regime in which the dynamics significantly slows down. It is much more difficult to unambiguously confirm MBL phase without investigating much longer timescales \\cite{Sierant2021}, and there are clear differences between disordered systems and systems with quasiperiodic potentials. The question of how Stark localized systems fit in this scheme needs further investigation, which is beyond the scope of the current study.\n\nFor chaotic systems, the entropy increases linearly in time after quenches. Due to this fact, TDVP is applicable on relatively small timescales. In contrast, for MBL systems, the entropy grows logarithmically in time, thus numerical simulations can cover significantly larger time intervals at moderate bond dimensions. \nMoreover, the entropy evolution can be used by itself as a criterion of localization in numerical algorithms. \nThereafter, the growth of the entropy is used both as an indicator of reliability of the obtained results and as one of representative quantities, which are sensitive to transition between chaotic and localized behavior.\n\nIn the subsequent analysis, we use the following quench protocol: we initialize the wave function in the product state, where all even sites of the lattice are filled with one particle and all odd sites are empty (in the case of spin chain, even and odd sites are occupied by spin-up and spin-down particles, respectively); then, the time dynamics of this state for the given model Hamiltonian is studied. In case of fermionic system, we perform the Jordan-Wigner transformation to map the system to the spin chain [see also Eqs.~\\eqref{H_fermionic} and \\eqref{H_spin}], where one can apply the TDVP approach in a straightforward manner.\n\n\nWhile analyzing time evolution of the system, we measure several quantities characterizing the many-body wave function. \nOne of them is the above-mentioned entanglement entropy \\eqref{eq:ent_entropy}. \nIt is also possible to compute expectation values of operators, which can characterize ergodicity breaking. \nSuch an experimentally-relevant observable (see, e.g., Ref.~\\cite{Scherg2021}), which is especially convenient for our quench protocol, is the even-odd site occupation imbalance~$I$ (or the so-called amplitude of the charge-density wave). It is defined as a difference between the number of particles on even and odd sites of the lattice ($N_{e}$ and $N_{o}$, respectively), normalized by the total number of particles $N$,\n\\begin{equation}\\label{imbalance}\n I(t)\n = \\frac{1}{N} \\sum_{i=1}^{L} (-1)^{i} \n \\langle\\psi(t)|\\hat{n}_{i}|\\psi(t)\\rangle\n = ({N_{e} - N_{o}})\/{N}.\n\\end{equation}\n\nFor the above-specified initialization of the wave function, the initial state yields $I(0)=1$ and this is its maximal value, i.e., $|I(t)|\\leq1$. \nDuring the time evolution, this observable typically decreases to a certain constant value and then oscillates around this value with a small amplitude. \nIn the chaotic phase, this constant value is close to zero. In contrast, in the MBL phase this value remains relatively large. \nThis shows that MBL phase contains some memory of the initial state and its inhomogeneities, which are partly measured by the parameter~$I$. \nTherefore, the asymptotic behavior of imbalance at long times can be used as reliable indicator of localization.\n\n\n\\subsubsection{Schrieffer-Wolff transformation}\\label{subsec:SWT}\nLet us briefly discuss the case of large external potential in the Hamiltonian.\nNote that the corresponding amplitude~$F$ is proportional to the dipole moment of particles or spins. \nThe spectrum of the dipole operator entering the Hamiltonian is highly degenerate. Therefore, it seems natural to employ the degenerate perturbation theory based on the Schrieffer-Wolff transformation to effectively describe the system under study. \n\nThe traditional Schrieffer-Wolff transformation (SWT) \\cite{sw66} relies on the following procedure: we divide the Hamiltonian into the leading term $\\hat{H}_{0}$, which determines the largest energy scale of the full Hamiltonian, and the residual part. \nThe latter can be additionally divided into parts $\\hat{T}$ and $\\hat{V}$ containing operators that do not commute and commute with $\\hat{H}_{0}$, respectively.\nFor definiteness, the spin Hamiltonian~\\eqref{H_spin} can be written as\n\\begin{equation}\\label{SW-decomposition}\n \\hat{H} = \\hat{H}_{0} + \\hat{T} +\\hat{V},\n \\qquad\n \\hat{H}_{0} = F\\sum_{i=1}^{L} i \\hat{S}^{z}_{i},\n\\end{equation}\nwhere the part commuting with $\\hat{H}_0$ is given by\n\\begin{equation}\n \\hat{V} = \\sum_{i=1}^{L} w_{i} \\hat{S}^{z}_{i} + U \\sum_{1\\leq i < j \\leq L} \n \\frac{\\hat{S}^{z}_{i} \\hat{S}^{z}_{j}}{|i-j|^{\\alpha}}\n\\end{equation}\nand the noncommuting perturbation has the form\n\\begin{equation}\n \\hat{T} = - J_{1}\\sum_{i=1}^{L-1} (\\hat{S}^{+}_{i}\\hat{S}^{-}_{i+1} + \\hat{S}^{-}_{i}\\hat{S}^{+}_{i+1}).\n\\end{equation}\n\nUpon this (or a similar) division, we apply the unitary transformation $\\hat{\\cal U}$ to the Hamiltonians \\eqref{H_fermionic}--\\eqref{H_spin}. This unitary transformation is represented in the form $\\hat{\\cal U} = \\exp{\\hat{\\cal S}}$, where $\\hat{\\cal S}$ is anti-hermitian operator. \nThe operator $\\hat{\\cal S}$ is expressed in terms of a series in the expansion parameter (${1}\/{F}$ in our case) in the way to cancel terms in the Hamiltonian that do not commute with $\\hat{H}_{0}$.\nThis transformation yields an effective Hamiltonian, which is block-diagonal (up to small higher-order corrections in the expansion parameter), with the size of blocks determined by the degeneracy of $\\hat{H}_{0}$. \nFor the models with the linear potential we obtain the dipole-conserving Hamiltonians. Note that in the limit of infinite system, the resulting effective Hamiltonian is translationally invariant. In this sense, the systems in linear or quadratic external potentials are close to translational invariance. \n\n\n\nLet us now discuss the form of the effective Hamiltonians for the above-specified models. For the spin Hamiltonian~\\eqref{H_spin}, we obtain\n\\begin{multline}\\label{H_effective_spin}\n \\hat{H}_{\\rm eff} = F\\sum_{i=1}^{L} i \\hat{S}^{z}_{i} + \n + \\sum_{i=1}^{L} w_{i} \\hat{S}^{z}_{i} + U \\sum_{1\\leq i < j \\leq L} \n \\frac{\\hat{S}^{z}_{i} \\hat{S}^{z}_{j}}{|i-j|^{\\alpha}} \n \\\\\n + \\frac{J_{1}^{2}}{F} \\left( \\hat{S}_{L}^{z} - \\hat{S}_{1}^{z} \\right) \n + \\hat{H}_{\\rm eff}^{(2)},\n\\end{multline}\nwhere the explicit form of the second-order terms $\\hat{H}_{\\rm eff}^{(2)}$ is given in Appendix~\\ref{App1} for the sake of compactness.\nAll terms in the effective model commute with the dipole operator $\\sum_{i=1}^{L} i \\hat{S}^{z}_{i}$. If only short-range interactions are present, this Hamiltonian is additionally fragmented into noninteracting sectors, as described in Ref.~\\cite{Sala2020}.\n\nEither from the effective Hamiltonian \\eqref{H_effective_spin} with the inverse Jordan-Wigner transformation, or directly from the Fermi-Hubbard model \\eqref{H_fermionic}, the effective Hamiltonian can be written as\n\\begin{multline}\\label{H_effective_fermionic}\n \\hat{H}_{\\rm eff} \n = F \\sum_{i=1}^{L} i \\hat{n}_{i} \n + U \\sum_{1\\leq i < j \\leq L} \n \\frac{\\hat{n}_{i} \\hat{n}_{j}}{|i-j|^{\\alpha}} \n \\\\\n + \\frac{J_{1}^{2}}{F} \\left( \\hat{n}_{L} - \\hat{n}_{1} \\right)\n + \\hat{H}_{\\rm eff}^{(2)}.\n\\end{multline}\n\nUp to quadratic terms in the expansion series over ${1}\/{F}$, the generator $\\hat{\\cal S}$ for the spin model~\\eqref{H_spin} has the following form:\n\\begin{equation}\\label{SW_spin}\n \\hat{\\cal S} \n = -\\frac{J_{1}}{F} \\sum_{i=1}^{L-1} (\\hat{S}_{i}^{-} \\hat{S}_{i+1}^{+} - \\hat{S}_{i}^{+} \\hat{S}_{i+1}^{-}) \n + \\hat{\\cal S}^{(2)},\n\\end{equation}\nsee also Appendix~\\ref{App1} for the explicit form of $\\hat{\\cal S}^{(2)}$.\nNote that in the fermionic system, the transformation has a similar form except for the absence of terms with $w_{i}$ in the operator $\\hat{\\cal S}$. All other terms can be obtained from Eq.~\\eqref{SW_spin} by applying the Jordan-Wigner fermionization rules. \n\nThe bosonic model \\eqref{H_bosonic} contains an additional on-site interaction term with the coupling $V$, thus the effective Hamiltonian up to linear terms in $1\/F$ differs from Eq.~\\eqref{H_effective_fermionic} only by the term $V\\sum_{i=1}^{L} \\hat{n}_{i}(\\hat{n}_{i}-1)$. \nAt the same time, the explicit forms of the quadratic corrections $\\hat{H}_{\\rm eff}^{(2)}$ and $\\hat{\\cal S}^{(2)}$ are substantially different for cases of fermions and bosons; these are given separately in Appendix~\\ref{App1}.\n\n\n\\section{Results}\\label{sec:results}\n\\subsection{Spectral characteristics}\nIn this section, we discuss results for the above-introduced ergodicity criterion $r$ [see Eq.~\\eqref{r}], which we evaluate by means of the exact diagonalization of various Hamiltonians with long-range deformations of the Hubbard model in the presence of a linear potential (see Subsec.~\\ref{subsec:ED}).\nWe begin our analysis from the Fermi-Hubbard model~\\eqref{H_fermionic} with the next-nearest hopping processes ($m=1$).\nIn Fig.~\\ref{fig:fermionic_spectrum} we show the characteristic diagrams of the parameter $r$ in the fermionic system with long-range interactions described by different values of exponents $\\alpha$ ranging from $\\alpha=0.5$ to $\\alpha = 3.0$.\nLet us emphasize that the former ($\\alpha=0.5$) is far beyond the predicted boundary values of $\\alpha$, where localization can occur according to the perturbation theory \\cite{Yao2014, Burin2015a, Burin2015b}. Note that there are also ED studies of the MBL persistence in the presence of similar long-range interactions and aperiodic potentials revealing similar behavior \\cite{Nag2019, Prasad2021}.\n\\begin{figure}[t]\n \\includegraphics[width=\\linewidth]{fig2_fermions.pdf}\n \\caption{\\label{fig:fermionic_spectrum}\n Dependencies of the parameter $r$ on the strength of the external linear tilt $F$ and the interaction strength $U$ at different $\\alpha=\\{0.5,1,2,3\\}$ for the fermionic long-range interacting model~\\eqref{H_fermionic} with $L=16$, $N=8$, $J_1=1$, and $m=1$. \n }\n\\end{figure}\n\n\nOne of central observations of our study is that the systems with small but nonzero long-range interaction typically remain localized at $F>2$.\nThis holds in a wide range of the employed parameters $\\alpha$ and $U$. \nWe attribute it to the fact that long-range interactions completely lift all degeneracies in the spectrum yielding completely regular spectrum statistics with no need for further introduction of the on-site disorder or harmonic potential. \n\nAt large amplitudes of the interaction potential $U$, the systems under study are localized for almost every value of $F$, but this effect is caused rather by conventional Mott-like localization, than by the external linear tilt. \nSince these systems are spinless and do not have internal degrees of freedom, their dynamics is trivial in the strong-coupling limit (in contrast to the effective Heisenberg chains for systems with internal degrees of freedom).\n\nAs one can see from Fig.~\\ref{fig:fermionic_spectrum}, the chaotic phase is the most pronounced in the interval $U \\in (2 ,5)$. To further clarify the influence of the exponent $\\alpha$, we fix $U=3.5$ and study the dependence of the gap ratio $r(F, \\alpha)$ as shown in Fig.~\\ref{fig:Alpha_dependency}. At large values of $\\alpha$, the system is effectively short-range and the localization boundary only moderately depends on the exponent~$\\alpha$. \nAt small $\\alpha$, the system becomes additionally localized due to long-range interactions. We use finite-size scaling analysis as described in Appendix~\\ref{App2} to extract the critical value $F_{c}$ at different $\\alpha$ ~\\cite{StarkMobilityEdge, StarkSuperconductingCircuits}. The obtained critical values are indicated by circles in Fig.~\\ref{fig:Alpha_dependency}.\n\\begin{figure}[t]\n \\includegraphics[width=\\linewidth]{fig3_alphadep.pdf}\n \n \\caption{\\label{fig:Alpha_dependency}\n Dependencies of the parameter $r$ in the fermionic system on the strength of the external linear tilt $F$ and the exponent $\\alpha$ at $L=16$, $N=8$, and $U=3.5$. Red points correspond to the critical amplitude~$F_{c}$ obtained from the finite-size scaling analysis.\n }\n\\end{figure}\n\n\n\nUp to this moment, we analyzed stability of localization with respect to introduction of long-range interactions with different power-law dependencies. \nLet us also discuss how long-range hopping can influence MBL. \nTo this end, we introduce the next-nearest neighbor (nnn) hopping term with the amplitude $J_{2}$ [see also Eqs.~\\eqref{H_fermionic} and \\eqref{H_bosonic}] and study its influence on the many-body localization. Note, that the influence of long-range hopping was studied in Ref.~\\cite{vernek2021robustness} for the $J_{1}$-$J_{2}$ spin chain in external linear field. Our results agree well with the observations of that study. \n\n\\begin{figure}[t]\n \\includegraphics[width=\\linewidth]{fig4_fermions_lr-L16.pdf}\n \\caption{\\label{fig:lr_hopping}%\n Dependencies of the parameter $r$ on the strength of the external linear tilt $F$ and the next-nearest-neighbor hopping amplitude $J_{2}$ at $\\alpha=1$ and two different $U=\\{3,6\\}$ for the fermionic long-range interacting model~\\eqref{H_fermionic} with $L=16$, $N=8$, $J_1=1$, and $m=2$. \n }\n\\end{figure}\nFigure~\\ref{fig:lr_hopping} shows the dependence of $r$ on the external tilt $F$ and the hopping amplitude $J_{2}$ at two values of long-range interaction strength: $U=3$ and $U=6$. \nIn the regime of small $J_{2}$ ($J_{2} \\lesssim J_{1}$), the nnn hopping does not impact the localization transition in a visible way. \nOnly at $J_{2} > J_{1}$ the transition becomes substantially determined by the amplitude~$J_{2}$. \nIn particular, the transition curve exhibits approximately linear dependence of the critical tilt $F_{\\rm loc}$ on $J_{2}$, as one can also conclude from the noninteracting model. \nSince in experimental realizations the longer-range hopping terms are usually smaller than the nearest-neighbor terms, the former become largely irrelevant to the issue of stability of Stark localization. We should note that at large interaction strength $U$ the dependence of the gap ratio $r$ on $J_2$ can be more complex, as the system shows localized value of $r$ for all $F$ at $J_{2} =0$. In this case, the nonzero $J_{2}$ can drive the system into the chaotic phase. This behavior is partly shown in Fig.~\\ref{fig:lr_hopping} at $U=6$, where the system has intermediate values of $r$ in the limit $J_{2} = 0$.\n\nWe have also checked the case of more general power-law hopping $J_{m} = {J_1}\/{m^{\\beta}}$. The hopping processes parameterized in this way do not destroy localization even at small values of the parameter~$\\beta$ ($1< \\beta < 2$). \nNote that the stability of MBL was theoretically studied for the lattice model with long-range interactions and the same parametrization of long-range hopping in Ref.~\\cite{Nag2019}, but with aperiodic potentials. The given results agree with our observations.\n\n\nThe observed robustness of MBL even upon inclusion of the long-range hopping can be partly explained by arguments based on resonances, which were used to predict breaking of MBL in the case of disordered potentials \\cite{Yao2014,Burin2015a,Burin2015b}. \nResonances are generally present if the difference of energies $|\\tilde E_i-\\tilde E_{j}|$\nbetween the two eigenstates $|\\tilde{\\psi}_{i}\\rangle$ and $|\\tilde{\\psi}_{j}\\rangle$ of the Hamiltonian without hopping (which includes both many-body interactions and external potential) are smaller than the hopping matrix element between these two respective states. \nIn case of an external disorder potential with a randomly distributed amplitude $\\epsilon_{n} \\in [-W,W]$, there is a nonzero probability that $|\\tilde E_i-\\tilde E_{j}|$ is very small, and resonances are present. \nIf the number of such resonances diverges with distance between resonating sites, MBL is not stable.\nFrom this, one can derive that for stability of MBL in one-dimensional case with random external potential and local interactions, the condition $\\beta > 1$ must be fulfilled.\nIf more general interactions between the resonances are considered, even more strong restrictions on $\\alpha$ and $\\beta$ can be obtained. \nFor Stark localization, generally, if states $|\\psi_{i}\\rangle$ and $|\\psi_{j}\\rangle$ are coupled by a single hopping process between, for example, the sites $m$ and $n$, then the difference between the respective energies $| E_i- E_{j}|$ will be mainly determined by the external tilt, $| E_i- E_{j}| \\approx F|m-n|$. \nResonance will be present only if ${J_1}\/{|m-n|^{\\beta}} > F|m-n|$, which is generally not the case for large enough $F$ and $|m-n|$ with $\\beta > 0$. \nHence, the natural mechanism of MBL destabilization by long-range hopping is significantly suppressed by the nature of potential, which largely inhibits the possibility of resonances between the states coupled by a single hopping process. \n\nThe above analysis reveals stability of the Stark many-body localization upon inclusion of long-range interactions and long-range hopping processes. The natural question arises on experimentally realistic interaction terms that are able to make Stark MBL unstable or, at least, to shift the localization boundary to the larger values of $F$.\nThese interactions must contain even the longer-range coupling than in the power-law dependence.\nThe obvious type of interactions to examine are the cavity-mediated interactions (see Subsec.~\\ref{subsec:cavity}), which have already been studied in the context of MBL \\cite{Sierant2019,Chanda2021manybody}.\n\\begin{figure}[t]\n \\includegraphics[width=\\linewidth]{fig5_fermions_cav.pdf}\n \\caption{\\label{fig:cavity_spectrum}%\n Dependence of the parameter $r$ on the amplitude of the cavity-mediated interaction $U_{\\rm cav}$ and the external linear tilt $F$. The fixed parameters are $L=16$, $N=8$, $U=3$, and $\\alpha = 3$. \n }\n\\end{figure}\nIn Fig.~\\ref{fig:cavity_spectrum} we analyze the ergodicity parameter~$r$ as a function of $F$ and $U_{\\rm cav}$, while the amplitudes $U$ and $\\alpha$ are kept fixed. \nWe observe that the localization boundary shifts to the larger values of $F$ compared to the case of $U_{\\rm cav}=0$. \nAt larger values of the external tilt $F$ (in particular, $F \\gtrsim 5$ for the chosen set of parameters), this system remains localized.\n\n\n\nNote that in the limit of large tilt $F$, we also verified the obtained ED results for spectral characteristics by means of the SWT-based calculations [see Subsec.~\\ref{subsec:SWT}].\nThe corresponding analysis confirms, in particular, that the spectral characteristics of the effective Hamiltonians \\eqref{H_effective_spin} and \\eqref{H_effective_fermionic} are the same as of the full models \\eqref{H_fermionic}--\\eqref{H_spin} up to corrections proportional to $1\/F^{3}$.\n\n\n\\subsection{Dynamics: Imbalance and entropy}\nThe localized and chaotic regimes can be identified by clear signatures in the dynamics of physical observables, such as the particle imbalance $I$ [see Eq.~\\eqref{imbalance}].\nCharacteristic examples of this dynamics are shown in Fig.~\\ref{fig:Imbalance_dyn} for the fermionic system in the chaotic ($F=0.5$) and localized ($F=3.0$) regimes. \n\\begin{figure}[t]\n \\includegraphics[width=\\linewidth]{fig6_dynamics-ED_corr2.pdf}\n \n \\caption{\\label{fig:Imbalance_dyn}%\n Time evolution of the imbalance $I$ and entanglement entropy~$S$ after quench in the fermionic model at two different values of the external tilt, $F=0.5$ and $F=3.0$. \n For sake of visibility, the entropy growth is shown on two timescales: linear (c) and logarithmic (d).\n Other parameters are $L=18$, $U=2$, and $\\alpha = 1$. Entanglement entropy is computed for the bipartition of the system on two equal parts.\n }\n\\end{figure}\nNote the difference in timescales used in the corresponding subfigures. \nIn the chaotic regime [see Fig.~\\subfigref{Imbalance_dyn}{a}], the expectation values in microcanonical and diagonal ensembles are nearly the same and close to zero. \nThe initial imbalance relaxes to this expectation value on the timescale of the order of $1\/J$ (here and below $\\hbar=1$, $J\\equiv J_1$, {$N=L\/2$}, and $m=1$). After this relaxation, the fluctuations of the imbalance become negligibly small. \nIn contrast, in the localized regime with much larger $F$ [see Fig.~\\subfigref{Imbalance_dyn}{b}], the diagonal and microcanonical ensembles provide us with different expectation values of the imbalance. \nThis observable oscillates around the respective expectation value in the diagonal ensemble for a significantly larger period of time. \n\nClear indications of localization can also be observed in the dynamics of the entanglement entropy $S$ [see Eq.~\\eqref{eq:ent_entropy}], as we show in Figs.~\\subfigref{Imbalance_dyn}{c} and \\subfigref{Imbalance_dyn}{d}. \nIn the chaotic regime with $F=0.5$, the entanglement entropy grows linearly for a short period of time and then saturates to a constant value. \nThe period of the linear growth is approximately the same as a period of relaxation of the imbalance~$I$, see also Fig.~\\subfigref{Imbalance_dyn}{a}. \nIn contrast, in the localized regime with $F=3$, the entropy~$S$ grows only logarithmically in time and demonstrates characteristic oscillations. \nAt much longer times it also saturates, but to a smaller value than in the chaotic regime. A more general discussion of the entanglement growth in long-range interacting localized systems can be found in Ref.~\\cite{NonAlgebraicEntanglementGrowth}.\n\nWe can use these observations on the behavior of the imbalance~$I(t)$ to study localization transition in more detail. \nBelow, we obtain full spectrum of the Hamiltonians under study and calculate the imbalance both from the diagonal and microcanonical ensembles, as discussed in Subsec.~\\ref{subsec:ensembles}.\nFor the purpose of quantifying the observed differences in system dynamics, we introduce an auxiliary ergodicity parameter~$\\rho$,\n\\begin{equation}\n \\rho=-\\log|\\langle I \\rangle_{\\rm DE}-\\langle I \\rangle_{\\rm ME}|.\n\\end{equation}\n\nIn terms of $\\rho$, first, we compare predictions given by these two ensembles for the fermionic model~\\eqref{H_fermionic} in Fig.~\\ref{fig:Dynamics_fermions}.\n\\begin{figure}[t]\n \\includegraphics[width=\\linewidth]{fig7_all.pdf}\n \\caption{\\label{fig:Dynamics_fermions}%\n Dependence of the auxiliary ergodicity parameter $\\rho$ on the strength of linear tilt $F$ and strength of long-range interactions $U$ for the fermionic (left, $L=16$, $\\alpha = 1$) and bosonic (right, $L=12$, $\\alpha=2$, $V=4$) systems.\n }\n\\end{figure}\nHere, the boundary between the chaotic and localized behavior at $F\\approx2$ and moderate $U$ can be seen much more clearly. \nThis boundary starts to shift to larger values of $F$ at higher interaction strength $U$, but it is necessary to note that in this regime both microcanonical and diagonal ensembles predict large values of the final imbalance. \nThis shift of the localization boundary to larger interaction strengths is also confirmed by calculations of the entanglement entropy, which exhibits a linear growth to large values even at $F=2.4$ and $U=9$. \n\nFor small values of the interaction strength~$U$, the results from the dynamics and level statistics show some discrepancies (cf. Figs.~\\ref{fig:fermionic_spectrum} and \\ref{fig:Dynamics_fermions}). \nWe further checked the behavior of the entanglement entropy~$S$ in the region of parameter space, where dynamics and level statistics suggest different results. \nThe entanglement entropy shows a logarithmic growth to the values typical for chaotic systems, while the imbalance~$I$ fluctuates as in the localized system, but, at the same time, predictions for the mean values $\\langle I\\rangle_{\\rm ME}$ and $\\langle I\\rangle_{\\rm DE}$ agree. \nFurthermore, the microcanonical ensemble predictions become sensitive to the energy range~$\\Delta E$ (equivalently, to the number $N_{\\rm st}$) used in the definition~\\eqref{eq:obs_ME} of the corresponding observables more strongly than in the case of large interaction strength.\nIn Appendix~\\ref{App3} we discuss how the impact of the mentioned discrepancy can be further reduced by analyzing temporal fluctuations of main observables.\n\n\nNext, for the bosonic model~\\eqref{H_bosonic}, we obtain results for the imbalance dynamics. \nLet us note that, according to additional analysis, the results for the spectrum statistics are different at larger values of $V$, as there are states in the spectrum with double or triple occupancies on some sites and these states have energies uncorrelated with other states. \nFor bosonic system we restrict ourselves to half-filling to compare with the fermionic case. For larger densities chaotic behavior can survive to higher values of $F$, as effective hopping is enhanced by bosonic statistics.\nTherefore, the results from the dynamics become more relevant.\nWe show the characteristic phase diagram in Fig.~\\ref{fig:Dynamics_fermions}. At small and intermediate interaction strength, $\\langle I\\rangle_{\\rm ME}$ and $\\langle I\\rangle_{\\rm DE}$ agree at $F<2$. At larger interactions, as in the fermionic case, microcanonical and diagonal ensembles show similar results only at relatively large tilts $F$. \n\nFor larger systems ($L\\geq20$), we employ the TDVP approach to study the imbalance and entanglement entropy behavior. \nIn Fig.~\\ref{fig:ImbEntTDVP}, we show the results for imbalance dynamics in the bosonic and fermionic systems ($L=50$) at different values of the tilt $F$. In Fig.~\\ref{fig:EntTDVPLogScale}, we additionally analyze the dynamics of entanglement entropy on logarithmic timescale to ensure logarithmic growth of entropy in the localized phase. \n\\begin{figure}[t]\n \\includegraphics[width=\\linewidth]{fig8_tdvp_corr.pdf}\n \\caption{\\label{fig:ImbEntTDVP}%\n Time dynamics of the imbalance $I$ and entanglement entropy~$S$ for fermionic (left) and bosonic (right) systems obtained by TDVP. The parameters are $\\alpha = 1$, $U = 3$, $V = 5$ (for bosons), $L = 50$, and $D = 100$. Entanglement entropy is computed for the bipartition of the system on two equal parts.\n }\n\\end{figure}\n\n\\begin{figure}[t]\n \\includegraphics[width=\\linewidth]{fig9_entr_logscale.pdf}\n \n \\caption{\\label{fig:EntTDVPLogScale}%\n Time dynamics of the entanglement entropy on logarithmic timescale in fermionic system for different values of the tilt $F$ and system sizes $L$. The parameters are $\\alpha = 1$, $U=3$, and $D=50$. Entanglement entropy is computed for the bipartition of the system on two equal parts.\n }\n\\end{figure}\n\nIt is clear that the dynamics of imbalance changes for both statistics at $F\\approx2$. \nAt larger $F$, the imbalance~$I$ exhibits oscillations and relaxes to a nonzero value, while at $F\\approx1$ it quickly approaches zero. Behavior is qualitatively the same as was observed in small systems with the exact diagonalization. \nIn the TDVP analysis, we employ a relatively small value of the bond dimension ($D=100$), which restricts our calculations at small tilts $F$ to short times, since for larger times the employed MPS approach is not able to accurately represent the amount of entanglement in the wave function. \nThis effect can also be seen in Fig.~\\ref{fig:ImbEntTDVP}, which shows the growth of the entanglement entropy with time. At small $F$, the entropy reaches the maximum value allowed by the bond dimension~$D$ at several $Jt$. \nThis invalidates our results at larger values of $t$, but also heralds the chaoticity of the system.\n\n\n\\section{Conclusion}\\label{sec:conclusion}\nWe theoretically studied the many-body localization in the case of one-dimensional lattice systems with long-range interaction between particles and linear external potential.\nThe obtained results reveal that the systems with small but nonzero long-range interaction typically remain localized at moderate and large amplitude of the external linear potential.\nThese observations hold in a wide range of parameters characterizing long-range interaction potential including the cases of cavity-mediated interactions and long-range tunneling. This makes inclusion of the additional on-site disorder or harmonic potential unnecessary. \n\n\nIn addition to quantitative analysis of spectral characteristics of systems in wide ranges of parameters, we analyzed the dynamical evolution of relevant physical observables: even-odd site occupation imbalance and entanglement entropy.\nThe dynamics of both quantities clearly indicates differences between the chaotic and localized many-body regimes in lattice systems with the external linear tilt.\n\nUpon calculation of the imbalance within the microcanonical and diagonal ensembles, we introduced an auxiliary (ensemble-based) ergodicity parameter.\nFor the fermionic systems, we observe qualitative agreement in structures of phase diagrams constructed by means of the ergodicity paramenters from different (spectrum- and ensemble-based) approaches, whereas for the bosonic system, the ensemble-based ergodicity parameter becomes more accurate in certain regimes of the on-site interaction strengths.\nDepending on the system size, we applied both ED and TDVP approaches, which agree well in determining localization transitions.\nWe also confirmed the obtained numerical results in the limit of the applicability of the effective models, where we derived the effective Hamiltonians for the systems under study.\n\nIn general, our findings significantly extend the class of systems, where the transitions between the localized and chaotic many-body regimes can be studied in detail by accessing relevant observables in cold-atom experiments \\cite{Scherg2021}.\nThe systems under study are completely disorder free and quasi translationally invariant in the sense that the shift operator commutes with the Hamiltonians up to a constant.\nThis makes the system identical at different spatial positions and allows one to study it in a kind of thermodynamic limit.\nThus, the approach becomes efficient for the Wegner-flow and Schrieffer-Wolff studies relying on translational invariance \\cite{Pekker2017, LongRangeWegnerFlow}.\n \n\n\n\\begin{acknowledgments}\nThe authors acknowledge support from \nthe National Research Foundation of Ukraine, Grant No.~0120U104963,\nthe Ministry of Education and Science of Ukraine, Research Grant No.~0120U102252, and the National Academy of Sciences of Ukraine, Project No. 0121U108722.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\\label{sec:introduction}\n\nThe \\gls{iot} makes it possible to remotely monitor and control a wide set of\nheterogeneous objects through an Internet connection. This paradigm foresees\nmultiple applications in a large variety of scenarios: from fleet tracking and\nprocess monitoring in industrial scenarios to smarter garbage collection and\nintelligent light control in cities; from monitoring of soil moisture in\nagriculture to home temperature control and personal health monitoring\n\\cite{zanella2014internet, yuehong2016internet, dlodlo2015internet,\n chiariotti2018symbiocity}. Also, the \\gls{iot} paradigm can be applied to\nsurveillance-related applications~\\cite{bovenzi2018iot}, as event detectors and\nalarms~\\cite{dos2020performance}.\n\nThe presence of several use cases spawned an ample market, and encouraged the\ndevelopment of multiple technologies meeting the need for low-cost ubiquitous\nconnectivity. A large part of \\gls{iot} nodes will consist in sensors that\ngenerate sporadic traffic, without strict constraints in terms of latency and\nthroughput. This calls for new wireless solutions able to support a massive\nnumber of devices, with an affordable cost for both user equipment and network\ninfrastructure. Therefore, high energy efficiency, extended coverage, and\ninfrastructure simplicity are aspects of primary importance.\n\nSuch requirements motivated the creation of a new family of wireless\ntechnologies collectively called \\glspl{lpwan}, characterized by long\ncoverage range and low power consumption. A prominent \\gls{lpwan}\ntechnology is LoRaWAN, which claims up to 10~years of battery lifetime for\ndevices, and a transmission range between 1.5~km in urban scenarios and\n30~km in rural areas~\\cite{centenaro2016long}.\n\n\n\nSince the deployment of a dense \\gls{iot} network is expensive and time\nconsuming, performance assessments using simulations and mathematical models\nbecome essential to gauge the effect of network parameters and estimate the\nperformance at a reduced cost. In this work, we propose an analytical model of\nthe performance of a LoRaWAN network, accounting for the most relevant features\nof the LoRaWAN standard. This model considers network-layer performance,\nassuming perfect orthogonality between signals modulated with different\n\\glspl{sf}. However, compared to previous models in the literature (discussed in\nSec. III), our model takes into account a wider range of aspects of the PHY and\nMAC layers, such as the possibility of transmitting multiple times both\nconfirmed and unconfirmed packets, the limitations on the channel occupancy time\nimposed by the different national regulations, the interference produced by\nmultiple overlapping transmissions, the capture effect, and the limited number\nof demodulators available at the \\gls{gw}. Furthermore, the model formulation\noffers great flexibility in setting some system parameters, thus making it\npossible to analyze the system performance under different conditions and to\nshed light on possible trade-offs. We consider as performance\nmetrics the packet success probability, average delays, and fairness, from which\nit is possible to derive other measures of interest, such as energy consumption,\nsystem's reliability and the achievable \\gls{qos} in multiple scenario.\nThe proposed model is validated by comparing the results with those obtained\nthrough detailed ns-3 simulations. The analysis shows how the model can be used\nto maximize different performance metrics, proving a very powerful and\nconvenient tool to determine the best network configuration.\n\n\nThe rest of this work is structured as follows. To make the paper\nself-contained, in Sec.~\\ref{sec:technology} we present the main features of the\nLoRaWAN standard, while in Sec.~\\ref{sec:soa} we give an overview of the current\nstate of the art in the performance modeling of this technology.\nSec.~\\ref{sec:model} introduces the proposed model and describes how some of its\nparameters can be tuned to explore different behaviors of the network, while\nSec.~\\ref{sec:simulation} briefly describes the simulation framework used for\nvalidation. Sec.~\\ref{sec:results}, then, compares the output of the analytical\nand simulation models, also showing how they can be used to provide different\ninsights of the network behavior. Finally, Sec.~\\ref{sec:conclusion} draws the\nconclusions and discusses possible future developments.\n\n\n\n\n\n\\section{Technology overview}\n\\label{sec:technology}\n\nThis section describes the key LoRaWAN features, dwelling upon the elements and\nproperties that have a significant impact on the system-level performance, which\nwill then be considered in the model formulation.\n\n\\subsection{The LoRa modulation}\n\\glsreset{sf}\n\nLoRa is a modulation technique based on \\gls{css}, patented by Semtech. Bitrate\nand coverage range depend on the \\gls{sf} parameter that can vary from 7 to 12. Lower \\gls{sf}\nvalues achieve higher data rates and shorter transmission times, but require\nhigher signal powers at the receiver for correct decoding, which implies shorter\ncoverage ranges. On the other hand, signals transmitted using higher \\gls{sf}\nvalues are more robust to channel impairments and can thus achieve longer\ntransmission distances, at the price of an increased transmission time due to\ntheir lower data rates. Furthermore, signals modulated with different \\glspl{sf}\nare almost orthogonal: even if overlapping in time and frequency, two or more\nsignals transmitted with different \\glspl{sf} can be simultaneously decoded,\nprovided that their received powers satisfy some\nconditions~\\cite{croce2018impact}.\n\n\nWhen multiple packets transmitted with the same \\gls{sf} overlap in time and\nfrequency, instead, they may generate destructive mutual interference,\ndisrupting each other's reception and resulting in what is called a\n\\textit{packets collision} event. However, if one signal is significantly\nstronger than the others, by a power margin greater than the so-called\n``co-channel rejection parameter'' $CR_{\\rm dB}$, then it can be received\ncorrectly despite the interference, giving rise to a \\emph{capture} phenomenon.\n\nThe value of $CR_{dB}$ has been estimated to be around 6~dB\nin~\\cite{goursaud2015dedicated}. In order to take advantage of these features,\nthe SX1301 LoRa PHY chipset, typically employed in \\glspl{gw}~\\cite{sx1301},\nprovides 8 parallel demodulation chains, which allow the chip to demodulate up\nto 8 different signals simultaneously, irrespective of their \\glspl{sf} and\nfrequency. We also remark that the \\glspl{gw} do not support full-duplex\ntransmission and reception: in order to send a \\gls{dl} packet they have to\ninterrupt any ongoing reception, regardless of the frequency channels in which\ntransmission and reception occur.\n\n\n\n\\subsection{The LoRaWAN standard}\n\n\\glsreset{ns} \\glsreset{ed} \\glsreset{gw} The LoRaWAN standard~\\cite{lorawan}\ndefines a star-of-stars topology, as represented in Fig.~\\ref{fig:infrastruc},\nwith three kinds of devices: the \\textit{\\gls{ns}}, which is the central network\ncontroller and can be located anywhere in the Internet; the \\textit{\\glspl{ed}},\nperipheral nodes (usually sensors or actuators) that transmit using the LoRa\nmodulation; and the \\textit{\\glspl{gw}}, relay nodes that collect messages from\nthe \\glspl{ed} through the LoRa interface and forward them to the \\gls{ns} using\na reliable IP connection, and \\textit{vice versa}.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=\\figurescaling\\linewidth]{figures\/lorawanInfrastructureBN.png}\n \\caption{LoRaWAN network infrastructure. Dotted lines represent LoRa links,\n while solid lines represent IP connections.}\n \\label{fig:infrastruc}\n \\vspace{-1em}\n\\end{figure}\nThe standard also defines three classes of \\glspl{ed}, which differ for the time\nthey spend in reception mode. This article considers \\textit{Class A} devices,\nwhich have the strictest requirements in terms of energy consumption. In order\nto save battery, these devices stay in sleep mode most of the time, opening two\nreception windows only 1 and 2 seconds after the end of an \\gls{ul} packet\ntransmission. Fig.~\\ref{fig:cycle} depicts the different operational phases of a\nClass-A device when transmitting a \\gls{ul} packet and, then, receiving a\n\\gls{dl} packet in the second receive window.\n\nThe \\glspl{ed} have the possibility of transmitting \\textit{unconfirmed} or\n\\textit{confirmed} packets. In the latter case, an \\gls{ack} is expected in one\nof the two reception opportunities after the transmission to confirm the correct\nreception of the packet by the \\gls{ns}.\\footnote{Although in this paper we\n focus on ACK transmissions, the model and the analysis equally hold for any\n \\gls{dl} packet returned by the \\gls{ns} to the ED after the reception of a\n \\gls{ul} packet by the NS.} If the \\gls{ack} is not received in either of the\ntwo reception windows, a re-transmission can be performed at least\nRETRANSMIT\\_TIMEOUT seconds after the end of the second receive window. The\nstandard recommends to randomly pick the value for RETRANSMIT\\_TIMEOUT\nuniformly between 1 and 3 seconds~\\cite{regional}.\nIf the \\gls{ack} is not received, the same confirmed message can be\nre-transmitted up to $m-1$ times, after which the packet is dropped. The value\nof $m$ can be configured by the \\gls{ns}.\\footnote{This behavior holds for the\n LoRaWAN 1.1 standard~\\cite{lorawan, regional}: other versions of the standard\n may differ.} To increase the robustness of unconfirmed transmissions, instead,\nthe \\gls{ed} can transmit each packet $h$ times. Once again, the value of $h$\ncan be set by the \\gls{ns}. It is worth noting that the reception windows are\nopened after every UL transmission, irrespective of whether or not an \\gls{ack}\nis expected, in order to give the \\gls{ns} the opportunity to send a \\gls{dl}\npacket to the \\glspl{ed}, if needed. The \\gls{ul} messages transmitted by an\n\\gls{ed} are collected by all the \\glspl{gw} in the coverage range of the\ntransmitter, and forwarded to the \\gls{ns}. If the \\gls{ed} requires a reply,\nthe \\gls{ns} can pick any of these \\glspl{gw} to transmit the \\gls{dl} message.\n\n\\begin{figure}[t]\n \\centering\n \\input{figures\/cycle.tex}\n \\caption{Example of operational phases for a Class-A \\gls{ed}. The device,\n initially in sleep mode, wakes up to transmit from time\n $t_0$ to $t_1$. Then, the node remains in the idle state for 1 second, and\n at time $t_2=t_1+1~s$ opens the \\acrfull{rx1}. If no packets are\n received, the device remains idle until the \\acrfull{rx2} is opened at\n time $t_3=t_1+2~s$. }\n \\label{fig:cycle}\n \\vspace{-1em}\n\\end{figure}\n\nLoRaWAN operates in the ISM unlicensed spectrum, the use of which is subject to\nnational regulations that define the maximum transmit power, and\nthe \\gls{dc} limit, i.e., the maximum percentage of time a node can actively\ntransmit on a certain frequency band. The frequency bands, power and \\gls{dc}\nrestrictions that apply to different regions are reported in the\nstandard~\\cite{regional}. In particular, Table~\\ref{tab:channels}\nshows the configuration mandated for the European region, which entails three\nbidirectional channels and a fourth channel reserved to \\gls{dl} transmissions\nonly. The 868.1, 868.3, 868.5~MHz channels belong to the same regulatory\nsub-band (\\gls{sb1}), and have to share a \\gls{dc} limitation of 1\\%, while the\nchannel reserved for \\gls{dl}, located in the 869~MHz sub-band (\\gls{sb2}), can\nbenefit from a more lenient \\gls{dc} of 10\\% and a higher transmission power.\n\n\\begin{table}[h]\n \\footnotesize\n \\centering\n \\caption{Available LoRaWAN channels in the two sub-bands.}\n \\label{tab:channels}\n \\begin{tabular}{llrr}\n \\toprule\n Sub-band & Frequency [MHz] & Use & Duty Cycle \\\\\n \\midrule\n \\multirow{3}{*}{SB1} & 868.1 & UL\/DL & 1\\%, shared \\\\\n & 868.3 & UL\/DL & 1\\%, shared \\\\\n & 868.5 & UL\/DL & 1\\%, shared \\\\\n \\arrayrulecolor{black!70}\\midrule\n SB2 & 869.525 & DL & 10\\%, dedicated \\\\\n \\arrayrulecolor{black}\\bottomrule\n \\end{tabular}\n\\end{table}\n\nThe \\gls{sf} used for a device's transmission is configured by the device itself\nor set by the \\gls{ns} according to some network management policies. By\ndefault, \\glspl{ed} open the \\gls{rx1} on the same frequency channel of the\n\\gls{ul} transmission, and expect a signal modulated with the same \\gls{sf}. The\n\\gls{rx2}, instead, is opened on the 869.525~MHz channel and the incoming signal is\nassumed to use \\gls{sf} 12, to maximize the coverage rate. The standard allows\nthe \\gls{ns} to modify this pre-defined configuration by communicating the new\nsettings to the \\gls{ed} through appropriate \\gls{mac} commands, allowing for\nthe use of any \\gls{sf} in the second window.\n\n\n\\subsection{Packet life cycle}\n\\label{sec:lifecycle}\n\nMessages transmitted by \\glspl{ed} to the \\gls{gw} are subject to multiple\ncauses of losses:\n\n\\begin{itemize}\n\\item \\textit{Interference}: packets sent in the same frequency channel and\n with the same \\gls{sf} collide. A transmission can survive a collision\n event if its received power is sufficiently higher than that of the other\n overlapping signals (capture effect).\n\\item \\textit{\\gls{gw} already in transmission}: the \\gls{gw} can not lock\n on a \\gls{ul} packet while performing a \\gls{dl} transmission.\n\\item \\textit{\\gls{gw} starting a transmission}: an ongoing packet\n reception may be interrupted if the \\gls{gw} needs to send a \\gls{dl}\n packet.\n\\item \\textit{No available demodulation chains at the \\gls{gw}}: all demodulators\n are already busy decoding incoming signals.\n\\end{itemize}\n\nMoreover, confirmed \\gls{ul} messages cause the \\gls{ns} to generate\n\\glspl{ack} that need to be transmitted by the \\gls{gw}. Such DL transmissions\nmay as well be impaired by a number of events:\n\\begin{itemize}\n\\item \\textit{Unavailability of receive windows}: this event occurs when\n all available \\glspl{gw} are prevented from transmitting in both the\n receive windows because of the \\gls{dc} constraint or other ongoing\n transmissions.\n\\item \\textit{Interference}: \\gls{dl} packets transmitted in \\gls{rx1} can\n collide with \\gls{ul} packets transmitted by other \\glspl{ed} in the same\n channel and with the same \\gls{sf}.\n\\end{itemize}\n\nIn this work, we provide a network model that accounts for all these\nevents.\n\n\n\n\\section{State of the art in LoRaWAN modeling}\n\\label{sec:soa}\n\nIn the last years,\nmathematical modeling has been applied to assess the network\nperformance with respect to various metrics. In~\\cite{adelantado2017under}, the\nauthors address high-level questions about LoRaWAN's suitability for a range of\nsmart city applications, from metering to video surveillance, by modeling the\nsystem as a superposition of different Aloha networks. They conclude that, even if\nthe long coverage range of a single \\gls{gw} makes the infrastructure able to\nserve several devices, the network must be carefully dimensioned to meet the\napplication requirements. The work presented in~\\cite{georgiou2017low} is one of\nthe first to address the issue of scalability, using stochastic geometry to\nmodel interference in a LoRaWAN network. However, the study considers scenarios\nwith only \\gls{ul} traffic.\nIn~\\cite{sorensen2017analy} instead, queueing theory is applied to model latency\nand throughput of an \\gls{ed} subject to \\gls{dc} constraints, again focusing on\n\\gls{ul} communication only. The authors of~\\cite{bankov2017mathem,\n bankov2019lorawan} provide a model based on Poisson arrival processes which\ntakes \\gls{dl} communications, re-transmissions and capture effect into account.\nHowever, the analysis holds only in limited-size networks, where nodes can\nemploy any transmission rate and their received powers are similar. The authors\nof~\\cite{croce2019lora} consider the features of the technology at the \\gls{phy}\nlayer, by focusing on the capture effect and imperfect orthogonality between\n\\glspl{sf}: after performing empirical measurements, they model these effects\nand derive the throughput achieved by the network for different cell\nconfigurations and number of \\glspl{gw}. In~\\cite{heusse2020capacity}, the\nproblem of network scalability is faced through mathematical modeling and\nPython-based simulations, taking into account also the capture effect, and\nevaluating the impact of \\gls{sf} allocation and power control. In all these\nworks, however, the main focus is on the \\gls{phy} layer, and downlink traffic\nand re-transmissions are not considered. Finally, the work presented\nin~\\cite{khan2019model} proposes a model to calculate energy consumption and\ndelay for reliable \\gls{ul} traffic in a LoRaWAN network. The results for a\nlimited number of devices are compared to real test-bed measurements and to the\noutcome of ns-3 simulations. The analysis, based on Markov-chain theory,\nneglects the \\gls{dc} constraints in the different sub-bands, and assumes that\n\\glspl{ack} are always sent in one specific receive window (either \\gls{rx1} or\n\\gls{rx2}). Markov chains are also proposed in~\\cite{delgado2021batteryless} to\ncharacterize the performance of a LoRaWAN battery-less device; however, the\nstudy considers a single device, and the network analysis is left for future\nwork.\n\nThe work presented in this paper is an extension of our previous conference\npaper~\\cite{capuzzo2018mathematical}, where we modeled a wide network with\nPoisson packet arrivals, considering the \\gls{dc} limitations and a set of\nnetwork parameters. Here we revise the model by developing a novel approach to\naccurately consider the limited availability of reception chains at the \\gls{gw},\nthe peculiarities of the two receive windows, and the \\gls{dc} constraints.\nAdditionally, we include packet re-transmissions and the capture\neffect. Compared to the state of the art, our model includes the ability to take\ninto account the coexistence of unconfirmed and confirmed traffic and, at the\nsame time, maintains the possibility of estimating the network behavior under\nseveral network configurations with minimal effort. The results obtained through\nthis model are compared with those given by a state-of-the-art and open source\nLoRaWAN simulator, presented in~\\cite{magrin2017performance}, further attesting\nthe accuracy of the proposed approach and exploring the impact of common\nassumptions. Finally, we also show some possible usages of the model to evaluate\na wide variety of network configurations with limited effort.\n\n\n\n\n\\section{Model}\n\\label{sec:model}\n\nThe aim of the model proposed in this paper is to characterize the behavior of a\nLoRaWAN network with a single \\gls{gw}, which receives packets from a set of\n\\glspl{ed} and needs to reply in one of the two receive windows when an \\gls{ed}\nrequires confirmation. The system performance is assessed in terms of packet\nsuccess probability, following the approach used\nin~\\cite{capuzzo2018mathematical} and extending it with a more accurate\ncharacterization of the \\gls{gw} behavior. This performance metric is proxy to\nother fundamental metrics, such as throughput and network capacity, which can be\nstraightforwardly derived from it. The following sub-sections are structured as\nfollows. The reference scenario, model assumptions, system parameters and their\neffects are described in Sec.~\\ref{sec:scenario}, together with a brief\npresentation of the structure of the model and its underlying rationale;\nSec.~\\ref{sec:quantities}, then, describes some relevant quantities and\nparameters of the proposed model. We then delve into the analytical formulation\nby decoupling the analysis of the \\gls{ul} traffic\n(Sec.~\\ref{sec:ultrafficrates} and~\\ref{sec:phy-probs}) and \\gls{dl} messages\n(Sec.~\\ref{sec:acktransmission}), and derive the formulas for \\gls{dl} success\nprobabilities in Sec.~\\ref{sec:succprobs}. Finally, Sec.~\\ref{sec:metrics},\ndescribes different performance metrics and their computation. Note that,\nbecause of the mutual dependency of some values, some terms may be described and\nintroduced before the corresponding equation can be derived, in which case\nreferences are provided in the text.\n\n\\subsection{Scenario and assumptions}\n\\label{sec:scenario}\n\nWe consider a scenario where the \\glspl{ed} are randomly and uniformly\ndistributed around a single \\gls{gw}. Application-layer packets are generated\naccording to a Poisson Process with aggregate packet generation rate $\\lambda$\n[pck\/s], and can be either confirmed or unconfirmed.\n\nFor tractability, we assume perfect orthogonality between different \\glspl{sf},\ni.e., only packets employing the same \\gls{sf} can collide. In this case, one of\nthe two packets can survive if its received power is sufficiently higher than\nthat of the colliding packet (collisions with more than two packets happen with\nnegligible probability and are not considered). While the orthogonality\nassumption has been shown to have an impact on the \\gls{phy}-layer performance\nof \\gls{ul} only traffic~\\cite{croce2018impact}, the results discussed in\nSec.~\\ref{sec:results} show that the effect is much more limited in the presence\nof confirmed traffic, where the performance is severely limited by other\nfactors.\n\n\\begin{figure}[t]\n \\centering \\input{figures\/diagram}\n \\caption{Representation of the model's packet filtering structure. $R^{phy}$ is\n the rate of \\gls{ul} traffic (see~\\eqref{eq:rphytot}), while $r^1$ and\n $r^2$ represent the rate of \\glspl{ack} sent in \\gls{sb1} and \\gls{sb2},\n respectively (see~\\eqref{eq:r1},~\\eqref{eq:r2}).}\n \\label{fig:diagram}\n \\vspace{-1em}\n\\end{figure}\n\nFig.~\\ref{fig:diagram} shows the structure of the packet reception model,\nconsisting in successive filtering of Poisson processes. At the base of the\nfigure, arrows are used to represent the \\gls{ul} traffic generated by the\n\\glspl{ed}, including both new packet transmissions and re-transmissions of\nfailed packets. This process is assumed to be Poisson for tractability,\nignoring the fact that re-transmissions of a certain packet are correlated\nin time because of \\gls{dc} limitations. An initial filtering of this\nprocess excludes some arrivals, modeling packet losses due to interference\nfrom other \\glspl{ed}, unavailability of \\gls{gw} demodulators, or ongoing\n\\gls{dl} transmissions from the \\gls{gw}. This yields a process with a\nreduced rate, which now represents the packets that are correctly received\nby the \\gls{gw}.\n\nWhen the received \\gls{ul} message requires confirmation, an \\gls{ack} must be\nsent by the \\gls{gw} during one of the two receive windows of the target\n\\gls{ed}. The ability of the \\gls{gw} to perform such a transmission is modeled\nthrough two independent alternating renewal processes, in which the system\nalternates between the ON and OFF states. The two processes represent the\nopportunity of sending the \\gls{ack} in \\gls{rx1} or \\gls{rx2}, respectively,\nwhich are opened on \\gls{sb1} or \\gls{sb2}, i.e., on the shared or dedicated\nsub-band. If a confirmed packet finds a process in the ON state, it means that\nthe \\gls{gw} will be able to send an \\gls{ack} in that sub-band. In this case,\nthe process will switch to the OFF state to model the unavailability of that\nsub-channel for a certain period of time following the ACK transmission, due to\nthe DC restrictions.\n\nSince the sub-bands are disjoint, we assume that the two processes are\nuncorrelated, neglecting the fact that the very packets that need to be served\nin \\gls{sb2} are those that found \\gls{sb1} in the OFF state. If\nthe \\gls{dl} packet finds at least one of the two processes in the ON state, an\n\\gls{ack} is sent. If the \\gls{ack} is sent on \\gls{sb1} (hence, using frequencies\nshared by \\gls{ul} and \\gls{dl} traffic), it can be destroyed by the interference created by other \\glspl{ed}.\nIf the \\gls{ack} is sent on \\gls{sb2}, instead, it is assumed to be\nalways successful.\n\nFor the sake of clarity, the following list describes some examples of the\nlife cycle of the packets in Fig.~\\ref{fig:diagram}:\n\\begin{enumerate}[(A)]\n\\item This packet is lost because of interference or \\gls{gw} transmission\n or unavailability of demodulators. Hence, it does not pass the first\n filter.\n\\item This is an unconfirmed \\gls{ul} packet, which is successfully\n received by the \\gls{gw}. It does not generate any \\gls{ack}.\n\\item This is a confirmed packet successfully received by the \\gls{gw}. It\n generates an \\gls{ack}, which finds the \\gls{sb1} process in the ON state. The\n \\gls{ack} is successfully sent, and the \\gls{sb1} process switches to the\n OFF state.\n\\item This is another confirmed packet which is successfully received by\n the \\gls{gw}. Since the \\gls{gw} has just sent an \\gls{ack} for packet\n (C), it cannot reply in \\gls{sb1} due to \\gls{dc} constraints; \\gls{sb2}\n is however in the ON state, and the \\gls{gw} can thus reply to the\n \\gls{ed}, making the second process switch to the OFF state.\n\\item This is another confirmed packet, which gets a treatment similar to\n that of packet (D). However, since the \\gls{gw} has transmitted the\n \\gls{ack} for packet (D) and is still under the \\gls{dc} constraints, it\n cannot reply to packet (E) in either of the two receive windows (both\n \\gls{sb1} and \\gls{sb2} processes are in the OFF state). The \\gls{dl}\n packet is hence discarded, and the \\gls{ed} will re-transmit the \\gls{ul}\n message at a later time.\n\\end{enumerate}\n\n\\subsection{Model Quantities}\\label{sec:quantities}\n\nOur model offers some tunable parameters to increase its flexibility,\nenabling the evaluation of the network performance in various\nconfigurations with minimal effort. The model makes it possible to specify\nthe following values:\n\\begin{itemize}\n \\item $\\mathcal{SF}=\\{7,\\ldots,12\\}$ indicates the set of all SFs.\n \\item $\\alpha$: fraction of application-layer traffic requiring confirmation;\n \\item $p^u_{i}, p^c_{i}$: fraction of devices generating unconfirmed and\n confirmed traffic with a specific SF $i \\in \\mathcal{SF}$, respectively.\n Note that $\\sum_{i \\in \\mathcal{SF}} p_i^u = \\sum_{i \\in \\mathcal{SF}} p_i^c\n = 1$;\n \\item $h$: number of times an application-layer unconfirmed packet is\n transmitted;\n \\item $m$: maximum number of transmission attempts for confirmed packets;\n \\item $\\delta_{SB1}$ and $\\delta_{SB2}$: ratio between silent time and\n transmission time in SB$k$, corresponding to the \\gls{dc} constraint.\n For instance, in Europe, we have $\\delta_{SB1}=99$ and $\\delta_{SB2}=9$\n corresponding to a \\gls{dc} of 1\\% in SB1 and 10\\% in SB2. In general,\n when $\\delta_{SBk}>0$ the \\gls{dc} constraint applies to all devices\n transmitting in subchannel SB$k$. Instead, the setting $\\delta_{SB_k}=0$\n corresponds to a DC constraint of 100\\%, which means that there is no\n limitation on the transmission time\\footnote{This setting is not allowed\n by current RF recommendations but is considered in this study to gain\n insights on the impact of \\gls{dc} limitations in the considered\n scenarios.};\n \\item $\\tau_1$ and $\\tau_2$: prioritization flags. If $\\tau_k=0$, the\n \\gls{gw} prioritizes reception operations over transmission during the\n $k$-th receive window, with $k=1, 2$. In this case, the \\gls{gw} will\n drop any DL message that needs to be transmitted while a \\gls{ul}\n reception is ongoing. Instead, if TX is prioritized ($\\tau_k = 1$), the\n reception of any incoming packet will be interrupted in order to send\n the \\gls{ack};\n \\item $C$: number of \\gls{ul} frequency channels. Note that each \\gls{ul}\n channel can also be used for \\gls{dl} transmissions. Instead, the\n channel in \\gls{sb2} is \\gls{dl} only;\n \\item $T_{i}^{ack_2}$: duration of the transmission of the \\gls{ack} in\n \\gls{rx2} when using \\gls{sf} $i$. (The standard requires the use of SF\n 12 in \\gls{rx2} as a pre-configured setting, corresponding to\n $T_{12}^{ack_2}$. Note that this default setting can be changed by the\n NS, and accordingly in our model.)\n \\item $T_i^{data}$ and $T_i^{ack_1}$ indicate the time durations of a data\n packet and of an \\gls{ack} transmitted in \\gls{sb1} with \\gls{sf} $i$,\n respectively. If \\glspl{ack} transmitted in \\gls{sb2} use \\gls{sf}12,\n irrespective of the \\gls{sf} employed in the \\gls{ul} transmission, then\n $T_i^{ack_2} = T_{12}^{ack_1}, \\: \\forall i \\in \\mathcal{SF}$.\n \n \n \n \n \n \n\\end{itemize}\n\n\nIn the formulas, the notation generally respects the following scheme. The\nprobability is indicated with $S$ or $F$ if it corresponds to a ``success''\nor ``failure'' event, respectively; if this rule does not apply, the probability\nis denoted simply as $P$. The superscript indicates the considered event, while\nthe subscript the \\gls{sf}. For example, in~\\eqref{eq:Sint}, the symbol\n$S_i^{INT}$ represents the probability of successfully surviving interference\nwhen using \\gls{sf}~$i$. Different uses of the notation are specified in the\ntext. The following sections provide a mathematical formulation for some\nrelevant quantities in this model.\n\n\\subsection{Uplink traffic rates}\n\\label{sec:ultrafficrates}\n\nThe assumption of perfect orthogonality between different \\glspl{sf} makes it\npossible to split the network traffic in different logical channels that do not\ninterfere with each other. The traffic load for each \\gls{sf} $i$ is split\nuniformly over the given $C$ frequency channels (since \\glspl{ed} pick a random\n\\gls{ul} frequency for each transmission attempt). Thus, the traffic generated\nat the application layer by the \\glspl{ed} using confirmed and unconfirmed\nmessages is, respectively, given by:\n\\begin{align}\n \\label{eq:RateApp}\n R_i^{c, app} &= \\frac{p_i^c \\cdot \\lambda}{C} \\cdot \\alpha, \\\\\n R_i^{u, app} &= \\frac{p_i^u \\cdot \\lambda}{C} \\cdot (1 - \\alpha).\n\\end{align}\n\nSince \\glspl{ed} using unconfirmed traffic will perform $h$ transmissions of\neach application-layer packet, the PHY rate of these devices can be computed as\n$R_i^{u, phy} = R_i^{u, app} \\cdot h$. For \\glspl{ed} transmitting confirmed\nmessages, instead, the number of re-transmitted packets depends on the success\nof both the \\gls{ul} transmission and the corresponding \\gls{ack}. We indicate\nas $P_{i,j}^{DL}$ the probability that a confirmed \\gls{ul} packet sent with\n\\gls{sf} $i$ is successfully received and acknowledged at the $j$-th\ntransmission attempt, which will be derived in~\\eqref{eq:psucc_dl}.\nTherefore, we have that the rate of confirmed packets transmitted at \\gls{sf}\n$i$, $R_i^{c,phy}$, is given by the product of the application-level rate,\n$R_i^{c,app}$, and the average number of times a confirmed packet is transmitted\nat the \\gls{phy} layer.\n\n\\begin{align}\n \\label{eq:Rc}\n \\begin{split}\n R_i^{c, phy} = R_i^{c, app} \\Bigg[ &\\sum_{j = 1}^{m-1} j \\cdot\n P_{i,j}^{DL}\n + m \\left( 1 - \\sum_{j = 1}^{m-1}P_{i,j}^{DL} \\right) \\Bigg].\n \\end{split}\n\\end{align}\nThe first summation in the square brackets of~\\eqref{eq:Rc} takes into account\ntransmissions that are successfully received before the $m$th attempt, while the\nsecond term considers the case when the packet is transmitted $m$ times\n(irrespective of whether the last transmission is successful or not).\n\nThe total traffic for a single frequency channel and for \\gls{sf}~$i$ is\ntherefore given by\n\\begin{equation}\n \\label{eq:rphytot}\n R_i^{phy} = R_i^{u, phy} + R_i^{c, phy}.\n\\end{equation}\n\nIn general, the distribution of the \\glspl{sf} for the transmitted packets at the\n\\gls{phy} layer will differ from the native distribution of \\glspl{sf} among the\ndevices, \\{$p_i^u, p_i^c$\\}, because of re-transmissions. Thus, we\ndefine\n\\begin{equation}\n \\label{eq:d}\n d_i = \\frac{R_i^{phy}}{\\sum_j R_j^{phy}},\n\\end{equation}\nas the ratio of \\gls{phy} layer packets that are transmitted at \\gls{sf} $i \\in\n\\mathcal{SF}$.\n\n\n\n\n\\subsection{PHY layer probabilities}\n\\label{sec:phy-probs}\n\nA \\gls{ul} packet is successfully received by the \\gls{gw} if all the following\nconditions are met: (i) it does not overlap with another \\gls{ul} transmission\nusing the same \\gls{sf} on the same frequency, or it overlaps with another\n\\gls{ul} packet, but the received power is sufficiently large to allow for\ncorrect decoding despite the interference (capture), (ii) it does not overlap\nwith a \\gls{gw} \\gls{dl} transmission in any channel, and (iii) it finds an\navailable demodulator. These conditions are represented by the first filter in\nFig.~\\ref{fig:diagram}.\n\nSince packets are generated following a Poisson process, the probability of\nevent (i) is given by two components. The first is the probability that there\nare no other arrivals during the $2T_i^{data}$ vulnerability period across the\npacket arrival instant. The second, considers a collision with one packet, and\nthe fact that the receiver successfully captures the frame. For the \\gls{ul}, we\nconsider the capture probability $ \\mathbb{W}^{GW}$ as computed\nin~\\cite{bankov2017mathem}. Since these two events are disjoint, the probability\nof surviving interference (event (i)) is given by the sum of the two components,\nwhich results in\n\\begin{equation}\n \\label{eq:Sint}\n S_i^{INT} = e^{-2T_i^{data}R_i^{phy}} +\n 2T_i^{data}R_i^{phy}e^{-2T_i^{data}R_i^{phy}} \\cdot \\mathbb{W}^{GW},\n\\end{equation}\nwhere, in the right-most term, we computed the probability that either of the\ntwo colliding packets is captured (collision events with more than two packets\nare neglected).\n\n\n\n\n\n\nTo compute the probability of event (ii), we observe that a \\gls{ul} message is\nalways lost when it arrives at the \\gls{gw} during the transmission of an\n\\gls{ack}. Otherwise, the \\gls{gw} will start the reception of the \\gls{ul}\nmessage, which will take a time $T_i^{data}$. If reception on SB$k$ is\nprioritized (i.e., $\\tau_k=0$), this process cannot be interrupted, and the\n\\gls{ul} message will be successfully delivered to the \\gls{ns}. Conversely, if\n$\\tau_k=1$, i.e., we prioritize transmission on SB$k$, the reception of the\n\\gls{ul} packet may be aborted at any time during the period $T_i^{data}$, in\norder to give priority on the \\gls{ack} transmission. Therefore, the\nvulnerability period is given by the \\gls{ack} transmission time\n$T_{s}^{ack_k}$, to which we need to add the interval $T_i^{data}$ only if\n$\\tau_k=1$. Denoting by $b_s^k$ the probability that an \\gls{ack} is transmitted\non SB$k$ with \\gls{sf} $s\\in \\mathcal{SF}$ (which will be derived later\nin~\\eqref{eq:bk}), the average vulnerability period is then given by\n$\\overline{T_k} = \\sum_{s\\in \\mathcal{SF}} b_s^k T_s^{ack_k} + T_i^{data}\\cdot\n\\tau_k$. Now, according to the Poisson Arrivals See Time Averages (PASTA)\nproperty, the probability that a \\gls{ul} packet arrival falls in the\nvulnerability period of channel SB$k$, with $k=1,2$, can be expressed as\n\\begin{equation}\n \\label{eq:f}\n F_{i}^{TXk}=\\frac{\\sum_{s\\in \\mathcal{SF}}{b_s^k T_s^{ack_k}}+ T_i^{data} \\cdot \\tau_k}{E_{ON}^k+E_{OFF}^k},\n\\end{equation}\nwhere the denominator is the mean renewal time of the SB$k$ process, given\nby the sum of $E_{ON}^k$ and $E_{OFF}^k$, i.e., the expected times the SB$k$\nprocess spends in the ON and OFF states during a renewal period (ON-OFF cycle),\nwhich will be computed in~\\eqref{eq:eon} and~\\eqref{eq:eoff}. Then, assuming\n(for ease of analysis) that events in \\gls{sb1} and \\gls{sb2} are independent,\nthe probability that a \\gls{ul} packet is successfully received (event (ii)) is\ngiven by\n\\begin{equation}\n \\label{eq:Sitx}\n S_i^{TX} =(1-F_i^{TX1})(1-F_i^{TX2}).\n\\end{equation}\n\nNext, we compute the probability of event (iii), i.e., that at least one\ndemodulator out of 8 is available. Each demodulator chain is modeled through an\nalternating renewal process, where the demodulator can be in an ``available''\nstate $A$, when idle or in a ``locked'' state $L$, when occupied with the\nreception of another signal. We assume that the different demodulators are\nactivated in succession: if all are available, an incoming signal will be\nreceived by the first demodulator; if the first demodulator is in the $L$ state,\nthe packet will be handled by the second demodulator, and so on. Let $E^L$ be\nthe expected time a demodulator will be locked on a incoming signal. Since the\noccupation will last for the duration of \\gls{ul} LoRa packets at the \\gls{phy}\nlayer, we have:\n\\begin{equation}\n \\label{eq:el}\n E^L = \\sum_{i \\in \\mathcal{SF}}d_i \\cdot T_i^{data}.\n\\end{equation}\nThe average time the first demodulator is in the $A$ state, instead, is\ncomputed as the average inter-arrival time of \\gls{ul} packets, regardless\nof their \\gls{sf} and selected frequency:\n\\begin{equation}\n E^{A, 1} = \\frac{1}{C \\cdot \\sum_{i\\in \\mathcal{SF}} R_i^{phy}}.\n\\end{equation}\nThen, the process of packets that require the second demodulator is filtered by\nthe probability of finding the first demodulator occupied. Thus, the expected\ntime the second demodulator is available is given by\n\\begin{equation}\n E^{A, 2} = \\frac{E^{A,1}}{P^{L,1}} = \\frac{1}{P^{L, 1} \\cdot C \\cdot \\sum_{i\\in \\mathcal{SF}} R_i^{phy}},\n\\end{equation}\nwhere $P^{L,1}$ is the probability that the first modulator is in the $L$ state\n(see~\\eqref{eq:pl1}).\nWith a similar reasoning, we compute the expected time for which the $j$-th\ndemodulator is available as\n\\begin{equation}\n E^{A, j} = \\frac{E^{A,j-1}}{P^{L,j-1}} = \\frac{1}{\\prod_{\\ell=1}^{j-1}P^{L, \\ell} \\cdot C \\cdot \\sum_{i\\in \\mathcal{SF}} R_i^{phy}}.\n\\end{equation}\nThe probability $P^{L,\\ell}$ of finding the $\\ell$-th demodulator in the $L$ state, in turn, can be expressed as\n\\begin{equation}\n \\label{eq:pl1}\n P^{L, \\ell} = \\frac{E^{L}}{E^{A, \\ell} + E^{L}}.\n\\end{equation}\n\nThen, a packet finds an available demodulator (event (iii)) with probability:\n\\begin{equation}\n \\label{eq:Sdemod}\n S^{demod} = 1 - \\prod_{j=1}^8 P^{L, j}.\n\\end{equation}\n\nThe overall \\gls{ul} packet success probability, considering events (i), (ii)\nand (iii) described above, is finally expressed as\n\\begin{equation}\n \\label{eq:Sul} S_i^{UL} = S_i^{INT} \\cdot S_i^{TX} \\cdot S^{demod} .\n\\end{equation}\n\n\\subsection{\\gls{ack} transmission}\n\\label{sec:acktransmission}\nOnce a confirmed packet is correctly received by the \\gls{gw}, an \\gls{ack}\nneeds to be transmitted back to the \\gls{ed}. Eq.~\\eqref{eq:Sul} gives the\nprobability of successful packet reception at the \\gls{gw}. Therefore, the rate\nof \\gls{ack} messages that the \\gls{gw} will try to send in \\gls{sb1} is:\n\\begin{equation}\n \\label{eq:r1}\n r_i^1 = R_i^{c, phy} \\cdot S_i^{UL}.\n\\end{equation}\n\\begin{figure}[t]\n \\centering\n \\input{figures\/dldiagram.tex}\n \\caption{Diagram for successful \\gls{ack} reception.}\n \\label{fig:dldiagram}\n \\vspace{-1em}\n\\end{figure}\nA visual representation of the possible \\gls{ack} life cycles considered in the\nmodel is shown in Fig.~\\ref{fig:dldiagram}. Labels refer to the probabilities of\nthe different events, which we derive next. In general, an \\gls{ack} is\ntransmitted in SB$k$ if both the following conditions hold: (i) $\\tau_k=1$ (TX\nis prioritized) or $\\tau_k=0$ and the GW is idle; (ii) SB$k$ is available (i.e.,\nnot blocked by DC constraints). If either condition is not satisfied, the\n\\gls{ack} is dropped.\n\nLet $T$ denote the event ``the \\gls{gw} \\textit{may} transmit,''\nwhich depends on the TX\/RX prioritization policy. If $\\tau_k=1$, the\n\\gls{gw} can transmit the \\gls{dl} packet whenever it needs to; otherwise,\nif $\\tau_k=0$, the \\gls{gw} can transmit in SB$k$ only if no reception is\nongoing. We denote by $P^{T,k}$ the probability of $T$, which can be computed as\n\\begin{equation}\n \\label{eq:Pnorx}\n P^{T, k} =\n \\begin{cases}\n 1, & \\textrm{if $\\tau_k$ = 1}; \\\\\n e^{-\\sum_{i \\in \\mathcal{SF}} C \\cdot R_i^{phy} T_i^{data}}, &\n \\textrm{if $\\tau_k$ = 0};\n \\end{cases}\n\\end{equation}\nwhere the second expression is the probability that no \\gls{ul} packet was\ngenerated in the last $T^{data}_i$ seconds.\n\nIf \\gls{sb1} is not available, the \\gls{gw} will try to process the\n\\gls{ack} in \\gls{sb2}. Such packets form a process with rate\n\\begin{equation}\n \\label{eq:r2}\n r_i^2= r_i^1 [P^{OFF, 1} + P^{ON, 1}(1 - P^{T, 1})],\n\\end{equation}\nwhere $P^{ON, 1}$ and $P^{OFF, 1}$ are the probabilities of finding \\gls{sb1}\nin the ON and OFF state, respectively, and $(1 - P^{T, 1})$ is the\nprobability that the \\gls{gw} is not available for \\gls{dl} transmission.\nThe ON and OFF probabilities for the SB$k$ process, with $k=1,2$, are given\nby\n\\begin{align}\n \\label{eq:Ponoff}\n P^{ON, k} &= \\frac{E^{ON, k}}{E^{ON, k} + E^{OFF ,k}}, \\\\\n P^{OFF, k} &= \\frac{E^{OFF, k}}{E^{ON, k} + E^{OFF, k}},\n\\end{align}\nwhere $E^{ON,k}$ and $E^{OFF,k}$ are the mean sojourn times in ON and OFF\nstates, respectively, which are computed as follows.\nBy considering the arrival rate of successful \\gls{ul} packets in the\n$k$-th sub-band, we have:\n\\begin{align}\n \\label{eq:eon}\n \\begin{split}\n E^{\\mathrm{ON}k} &= \\frac{1}{\\sum_{i \\in \\mathcal{SF}} C \\cdot r_i^k}.\n \\end{split}\n\\end{align}\nNote that the switch from the ON to the OFF state will be caused by a packet\nsent in any of the $C$ \\gls{ul} channels: therefore, we need to multiply the\nrates $r_i^k$ of arrivals to SB$k$ with SF $i$ by the number of available\nchannels.\n\nIn order to compute the expected duration of the OFF periods, we first need\nto derive the probability distribution $b_i^k$ of the \\glspl{sf} used for\n\\gls{ack} transmissions, which is given by\n\\begin{equation}\n \\label{eq:bk}\n b_i^k = \\frac{r_i^k}{\\sum\\limits_{s \\in \\mathcal{SF}}r_s^k}.\n\\end{equation}\nIn our model, the OFF period accounts for the time the \\gls{gw} is\nprevented from transmitting a new data packet, which includes the time to\nsend the ACK using the given \\gls{sf}, plus the waiting time imposed by the\n\\gls{dc} limitations. We hence have\n\\begin{align}\n \\label{eq:eoff}\n \\begin{split}\n E^{\\rm OFF, 1} &= \\sum_{s \\in \\mathcal{SF}} b_s^1 (T_s^{\\rm ack_1} + \\delta_{SB1} \\cdot T_s^{\\rm ack_1}), \\\\\n E^{\\rm OFF, 2} &= \\sum_{s \\in \\mathcal{SF}} b_s^2 (T_s^{\\rm ack_2} + \\delta_{SB2} \\cdot\n T_s^{\\rm ack_2}).\n \\end{split}\n\\end{align}\n(Note that, by including the parameter $\\delta_{SBk}$ as defined in\nSec.~\\ref{sec:scenario}, we can change the \\gls{dc} limitations in the $k$-th\nsub-band, thus making it possible to analyze its impact.)\n\n\n\n\nFinally, we remark that \\gls{dl} packets sent by the \\gls{gw} in \\gls{sb1} also\nhave to avoid interference from other \\glspl{ed}. In the absence of collisions,\nthe vulnerability period is given by the sum of two terms. The first term\ncorresponds to the case of no \\gls{ul} transmissions starting while the \\gls{dl}\npacket is being sent ($T^{ack_1}$); the second term represents the event where\nno \\gls{ul} transmissions started before the \\gls{ack} is sent. Note that if\n$\\tau_1=0$ the second term is not present, since in that case the \\gls{ack}\nwould not be generated at all. Furthermore, an \\gls{ack} can survive an\ninterfering packet sent by another \\gls{ed} in case of capture, which happens\nwith probability $\\mathbb{W}^{ED}$ (equivalent to the $\\mathbb{W}^{Mote}$ as\nderived in~\\cite{bankov2017mathem}). Therefore, the probability that the\n\\gls{ack} does not collide with a \\gls{ul} packet in \\gls{sb1}, or is captured\ndespite the collision, is equal to\n\\begin{equation}\n \\label{eq:SintAck}\n S_i^{INT, ack_1} =\n e^{-R_i^{phy} (T_i^{ack_1} + \\tau_1 \\cdot T_i^{data})} +\n R_i^{phy} (T_i^{ack_1} + T_i^{data}) \\cdot e^{-R_i^{phy} (T_i^{ack_1} + T_i^{data})} \\cdot \\mathbb{W}^{ED}.\n\\end{equation}\nFor packets sent in \\gls{sb2}, instead, the reception is assumed to\nbe always successful, since the 869.525~MHz channel is dedicated to \\gls{dl}\ncommunication and the \\gls{gw} only transmits one packet at a time (note\nthat this assumption does not hold in the case of multiple \\glspl{gw}).\n\n\\subsection{\\gls{dl} success probability}\n\\label{sec:succprobs}\nGiven that a confirmed \\gls{ul} packet sent with \\gls{sf} $i$ has been\nsuccessfully received by the \\gls{gw}, the probability that the corresponding\n\\gls{ack} is also successfully returned to the \\gls{ed} is expressed as\n\\begin{equation}\n \\label{eq:Sdl1ack} S_i^{\\rm DL} = S_{i}^{\\textrm{SB1}} + S^{\\textrm{SB2}},\n\\end{equation}\nwhere $S_{i}^{\\textrm{SB1}} $ describes the probability of a successful\n\\gls{ack} transmission in \\gls{sb1} with \\gls{sf} $i$, while $S^{\\textrm{SB2}}$\naccounts for the probability that \\gls{sb1} is not available, and the \\gls{ack}\nis successfully sent in \\gls{sb2}. These probabilities, in turn, can be\nexpressed as follows:\n\\begin{align}\n \\label{eq:SB1B2}\n S_{i}^{\\textrm{SB1}} &=\\; P^{ON, 1} \\cdot P^{T, 1} \\cdot S_i^{INT, ack_1}, \\\\\n S^{\\textrm{SB2}} &=\\;[P^{OFF, 1} + P^{ON, 1} \\cdot (1 - P^{T, 1}) ]\\cdot P^{ON, 2} \\cdot P^{T, 2}.\n\\end{align}\n\nFig.~\\ref{fig:dldiagram} can be used as a reference for the computation of\nthis quantity.\n\n\nFinally, we can compute the success probabilities over $m$ transmissions. We\nrecall that, for the sake of simplicity, we neglect the time correlation of\npacket re-transmissions due to \\gls{dc} constraints, (the impact of this\napproximation will be analyzed by simulation). We recall that $P_{i,j}^{UL}$\nindicates the probability that a \\gls{ul} packet with \\gls{sf} $i$ is\nsuccessfully received at the \\gls{gw} at exactly the $j$-th transmission\nattempt, which can be computed as:\n\\begin{equation}\n \\label{eq:psucc_ul}\n P_{i,j}^{UL} = S_i^{UL} \\left(1 - S_i^{UL}\\right)^{j - 1} .\n\\end{equation}\nThen, the \\gls{ed} successfully receives the \\gls{ack} at exactly the $j$-th\nattempt if both the \\gls{ul} and the \\gls{dl} transmissions succeed. The\nprobability $P_{i,j}^{DL}$ of this event is hence given by:\n\\begin{equation}\n \\label{eq:psucc_dl}\n P_{i,j}^{DL} = \\left[1 - (S_i^{UL}S_i^{DL})\\right]^{j - 1} \\cdot (S_i^{UL}S_i^{DL}).\n\\end{equation}\n\nOnce all intermediate quantities are computed, the model can be summarized\nby two inter-dependent equations:\n\\begin{equation*}\n \\begin{cases*}\n S^{UL} = f(S^{UL}, S^{DL}),\\\\\n S^{DL} = g(S^{UL}, S^{DL}).\n \\end{cases*}\n\\end{equation*}\nwhere $S^{UL}=[S^{UL}_7,\\ldots,S^{UL}_{12}]$ and\n$S^{DL}=[S^{DL}_7,\\ldots,S^{DL}_{12}]$, while $f()$ and $g()$ are implicit\nfunctions given by the chaining of the sequence of operations that\nyield~\\eqref{eq:Sul} and~\\eqref{eq:Sdl1ack}, respectively.\n\nThis system admits a fixed-point solution, which can be found through\nfixed-point iteration. From a practical perspective, when initialized with the\nstates $S^{UL} = S^{DL} = [1, 1, 1, 1, 1, 1]$, the iterative process has always\nreached convergence to the stable fixed point after a few iterations (order of\nfew units) for all the parameter combinations considered in this work. The proof\nof the system's convergence is provided in~\\cite{magrin2021proof}. An\nimplementation of the model, allowing the interested readers to easily replicate\nthe results shown in this paper, is publicly available\nat~\\cite{publishedmodelcode}.\n\n\n\n\n\n\\subsection{Performance metrics}\n\\label{sec:metrics}\nTo evaluate the system performance, we consider three classes of key performance\nSndicators, namely: reliability, delay, and fairness metrics which are better\ndetailed in the remainder of this section together with the methodology to\ndetermine their value using the proposed model. Once a set of parameters is\nfixed, the model can be solved and the performance metrics can be estimated\nstarting from $S^{UL}$ and $S^{DL}$. Conversely, it is possible to employ the\nmodel to optimize a given performance metric, finding the parameter setting that\nmaximizes it, as shown in Sec.~\\ref{sec:results}.\n\n\\subsubsection{Reliability Metrics}\n\nWe consider three \\gls{pdr} indexes, namely:\n\\begin{itemize}\n\\item \\textit{\\gls{uu}}: fraction of (application-layer) unconfirmed\n packets that are successfully received by the \\gls{gw};\n\\item \\textit{\\gls{cu}}: fraction of (application-layer) confirmed packets that\n are successfully received by the \\gls{gw}, irrespective of whether or not the\n corresponding \\gls{ack} is successfully returned to the \\gls{ed};\n\\item \\textit{\\gls{cd}}: fraction of (application-layer) confirmed packets\n that are successfully acknowledged by the NS.\n\\end{itemize}\nClearly, CD $\\leq$ CU, since a packet needs to be successfully received by\nthe \\gls{gw} in order to be acknowledged. Note that the \\gls{cu} metric\ncaptures the performance of applications for which it is important to\ndeliver packets to the \\gls{ns} and \\glspl{ack} are only used to stop\nre-transmissions (and thus avoid a useless increase in traffic), while\n\\gls{cd} is more interesting for applications that require the \\glspl{ed}\nto get explicit feedback from the \\gls{ns}, for instance containing control\ninformation addressed to the \\gls{ed}.\n\n\n\nWe obtain the \\gls{uu} and \\gls{cu} values by averaging the \\gls{ul} success\nprobability ($UU_i$ and $CU_i$ for unconfirmed and confirmed packets,\nrespectively) for each \\gls{sf} $i$ over the \\gls{sf} distribution, i.e.,\n\\begin{equation}\n \\label{eq:uu}\n {\\rm UU} = \\sum_{i \\in \\mathcal{SF}} \\left(p_i^u \\cdot \\mathrm{UU}_i\\right) = \\sum_{i \\in \\mathcal{SF}} \\left(p_i^u \\cdot \\sum\\limits_{j=1}^h P_{i, j}^{UL}\\right),\n\\end{equation}\n\\begin{equation}\n \\label{eq:cu}\n {\\rm CU} = \\sum_{i \\in \\mathcal{SF}} \\left(p_i^c \\cdot \\mathrm{CU}_i \\right) = \\sum_{i \\in \\mathcal{SF}} \\left(p_i^c \\cdot \\sum\\limits_{j=1}^mP_{i, j}^{UL}\\right).\n\\end{equation}\n\nSimilarly, \\gls{cd} is computed as the probability of success for a\nconfirmed packet within the available re-transmission attempts\n\\begin{equation}\n \\label{eq:cd}\n {\\rm CD} = \\sum\\limits_{i \\in \\mathcal{SF}} \\left( p_i^c \\cdot \\sum\\limits_{j=1}^mP^{DL}_{i,j}\\right).\n\\end{equation}\n\n\\subsubsection{Delay Metrics}\nWe define two delay metrics, considering confirmed traffic only: $\\Delta^{\\rm\n UL}$ measures the time from the first transmission attempt to the successful\ndelivery to the \\gls{gw} of an \\gls{ul} confirmed packet, while $\\Delta^{\\rm\n DL}$ accounts for the time from the first transmission of a confirmed packet\nto the successful reception of the corresponding reply. Delays are computed for\nsuccessful packets only, and the propagation delay is assumed to be negligible.\nTo compute these metrics with our model, we assume the RETRANSMIT\\_TIMEOUT value\nto be a uniformly distributed random variable with mean $\\mu$, and consider that\n\\glspl{ed} employ the shared sub-band with $\\delta_{SB1}$ \\gls{dc} limitations.\nTherefore, the average time between two transmissions of the same MAC-layer\npacket by a device is given by:\n\\begin{equation}\n \\label{eq:intertranmissionTime}\n \\gamma_i = (\\delta_{SB1} + 1) \\cdot T_i^{data} + \\mu.\n\\end{equation}\n\nThe average delay from the successful reception of a packet at the \\gls{gw}\nto the transmission of the \\gls{ack} is given by:\n\\begin{equation}\n \\label{eq:avgAckTransmissionTime}\n \\phi_i = S_i^{\\rm SB1} \\cdot (1 + T_i^{ack_1}) + S^{\\rm SB2} \\cdot (2 + T_i^{ack_2}),\n\\end{equation}\nwhere we take into account that the \\gls{ack} will be served in SB1 (opened\nafter 1 second) with probability $S_i^{\\rm SB1}$, and in SB2 (opened after 2\nseconds) with probability $S^{\\rm SB2}$.\n\nIf a packet is re-transmitted $m$ times, each re-transmission $j$ is\nassociated with a certain \\gls{ul} success probability $P_{i,j}^{\\rm UL}$.\nThe average delay at each \\gls{sf} $i \\in \\mathcal{SF}$ can be computed as:\n\\begin{equation}\n \\label{eq:uldelay}\n \\Delta^{\\rm UL} = \\sum_{i\\in\\mathcal{SF}} p_i^c \\cdot \\left( \\sum_{j=1}^m \\bar{P}_{i,j}^{\\rm UL} \\left(T_i^{data} + (j-1) \\cdot \\gamma_i\\right)\\right),\n\\end{equation}\nwhere we define $\\bar{P}_{i,j}^{\\rm UL} = P_{i,j}^{\\rm\n UL}\/\\sum_jP_{i,j}^{\\rm UL}$ to obtain the distribution of successful\n\\gls{ul} packet transmissions.\n\nSimilarly, we can compute the average \\gls{ack} delay:\n\\begin{equation}\n \\label{eq:dldelay}\n \\Delta^{\\rm DL} = \\sum_{i\\in\\mathcal{SF}} p_i^c \\cdot \\left( \\sum_{j=1}^m \\bar{P}_{i,j}^{\\rm DL} \\left(T_i^{data} + (j-1) \\cdot \\gamma_i + j \\cdot \\phi_i\\right) \\right),\n\\end{equation}\nwhere, in addition to the inter-transmission time between two packets, we\nalso account for the time to perform the \\gls{ack} transmission.\n\n\n\n\n\n\\subsubsection{Fairness}\nFinally, we consider the fairness of the system in different scenarios. Indeed,\n\\glspl{ed} employing confirmed traffic or higher \\glspl{sf} will use more\nsystem resources (e.g., channel occupancy), possibly affecting the application\nperformance of devices that employ different settings.\nTo this aim, we use Jain's fairness index, defined as\n\\begin{equation}\n \\label{eq:j}\n J(\\mathbf{x}) = \\frac{\\Big(\\sum_{i=1}^{n}x_i\\Big)^2}{n \\cdot \\sum_{i=1}^{n}x_i^2},\n\\end{equation}\nwhere $n$ is the total number of user categories, each with throughput\n$x_i$. Note that $1\/n \\leq J(\\mathbf{x}) \\leq 1$, and the system is perfectly fair if\n$J(\\mathbf{x}) = 1$. In particular, in the following section, we will consider\nthe fairness among devices employing different \\glspl{sf}. Furthermore, since\nall the devices have equal packet generation rate, and transmit packets with the same length, instead of the throughput\nwe can simply consider the \\gls{ul} success probability, i.e., \\gls{uu} for\nnodes employing unconfirmed traffic and \\gls{cu} for devices transmitting confirmed\nmessages. Therefore, the fairness is computed by taking\n$\\mathbf{x} = [\\mathbf{x^u}, \\mathbf{x^c}]$, where the elements correspond to\n$x_i^u = UU_i$, and $x_i^c = CU_i$, as defined in~\\eqref{eq:uu},~\\eqref{eq:cu}.\n\n\\section{Network Simulations}\n\\label{sec:simulation}\n\nIn order to validate our model, we compared the performance estimates obtained\nfrom the model with those observed in more realistic simulations, in which most\nof the simplifying assumptions of the model are removed.\n\nThis section describes how we employ the LoRaWAN ns-3 module described\nin~\\cite{magrin2020thorough} to perform such a validation. To be noted that the\nmore accurate modeling of the LoRaWAN standard considered in the simulator comes\nat the cost of a much larger computational time to assess the system\nperformance. Indeed, for the same parameter set, the performance evaluation is\nbasically instantaneous when employing the theoretical model, while each ns-3\nsimulation run takes in the order of tens of seconds, with execution times\nrapidly increasing when the traffic load, the number of devices and the number\nof required randomized runs grow.\n\nThe merit of the simulator is that it strives to be as realistic as possible,\nalso taking into account some factors that are overlooked by the model for\ntractability reasons. For instance, the assumption of perfect orthogonality\nbetween transmissions employing different \\glspl{sf} is removed, and the\nsimulator relies on the link-level model provided\nin~\\cite{goursaud2015dedicated} to determine the actual reception probability in\ncase of overlapping transmissions, which also accounts for the capture effect.\n\n\\begin{table}[t]\n \\footnotesize\n \\centering\n \\caption{Values of $T^{data}$, $T^{ack}$ and SF distributions $p$.\n Payload of data packets is 10 bytes; \\glspl{ack} have no\n payload.}\n \\begin{tabular}[c]{ccccc}\n \n \n \n \n \n \n \n \n \n \n \n \\toprule\n SF & $T^{data}$ [s] & $T^{ack}$ [s] & $p_{\\rm equal}$ & $p_{\\rm EXPLoRa}$\\\\\n \\midrule\n 7 & 0.051 & 0.041 & 0.166 & 0.487 \\\\\n 8 & 0.102 & 0.072 & 0.166 & 0.243 \\\\\n 9 & 0.185 & 0.144 & 0.166 & 0.135 \\\\\n 10 & 0.329 & 0.247 & 0.166 & 0.076 \\\\\n 11 & 0.659 & 0.495 & 0.166 & 0.038 \\\\\n 12 & 1.318 & 0.991 & 0.166 & 0.019 \\\\\n \\bottomrule\n \\end{tabular}\n \\label{tab:resparams}\n\\end{table}\n\nThe simulation setting is as follows.\n\\begin{itemize}\n %\n\\item \\textit{Traffic load} -- The number of \\glspl{ed} is fixed to 1200, and the \\glspl{ed}' application layer is set to periodically generate packets to be transmitted by the MAC layer. The traffic load in the network is modified by varying the packet generation\n period. It is to be noted that this periodic traffic generation pattern is likely more realistic than the Poisson traffic assumed in the model. Nonetheless, the good match of simulation and analytical results confirms that the Poisson assumption is valid when the number of nodes is sufficiently large.\n\n %\n\\item \\textit{Channel allocation} -- We consider the typical frequency\n allocation scheme for Europe, as reported in Tab.~\\ref{tab:channels}.\n Therefore, the number of different frequency channels for \\gls{ul} is $C\n = 3$.\n %\n\\item \\textit{Duty cycle} -- The simulator considers the \\gls{dc} limitations applied in the\n European region~\\cite{regional}, which corresponds to setting $\\delta_{SB1} = 99$ and $\\delta_{SB2} = 9$ in the model.\n\\item \\textit{Channel model} -- Differently from the model, simulated LoRaWAN\n nodes experience a log-distance propagation path loss,\n as for an open-air scenario. Thus, farther devices will suffer increased\n loss, and their performance will be penalized with respect to \\glspl{ed} that\n are close to the \\gls{gw}. Note that we do not include fast-fading components,\n which are supposed to be averaged out by the LoRa modulation, nor\n time-dependent variations in the channel, which remains constant throughout the\n entire simulation. Also, the channel is assumed to be symmetric, and\n \\gls{dl} transmissions will suffer the same impairments as in the \\gls{ul}.\n %\n \\item \\textit{\\gls{sf} distribution} -- \\glspl{ed} are located around the\n single \\gls{gw} in a circular area of radius 2500~m, which allows for\n communications with any SFs with negligible channel error probability\n (in the absence of interference). Instead, the positions of the nodes\n are randomly picked at each simulation run. \\glspl{sf} are assigned\n uniformly (see Tab.~\\ref{tab:resparams}, $p_{\\rm equal}$). A different\n \\gls{sf} distribution ($p_{\\rm EXPLoRa}$) is considered in some\n scenarios, to evaluate the impact of this parameter on the different\n metrics.\n %\n \\item \\textit{Interference and capture effect} -- To model interference, in\n the simulator we consider the collision matrix provided\n in~\\cite{goursaud2015dedicated} and the overlapping time between\n packets, as described in~\\cite{magrin2017performance}.\\footnote{Note\n that, in the simulator, the capture event is determined also considering\n the partial overlapping of the colliding packets.} A packet survives\n interference from a signal modulated with the same \\gls{sf} if its power\n is at least $CR_{dB} = 6$~dB higher than the colliding one. In order to\n provide a comparison with this scenario, in the analytical model we\n leverage the assumption of uniformly distributed \\glspl{ed} around the\n \\gls{gw} to compute the capture probabilities as\n in~\\cite{bankov2017mathem}, which results in $\\mathbb{W}^{GW} = 0.1796$,\n and $\\mathbb{W}^{ED} = 0.5682 $. We remark that different distributions\n of \\glspl{ed} around the \\gls{gw} can be modeled by adapting this\n derivation.\n\\end{itemize}\n\n\nSince the \\gls{gw} implementation in the simulator attempts to emulate the\nbehavior of a real device, a \\gls{ul} packet is successfully received when all the\nfollowing conditions are satisfied:\n\\begin{enumerate}\n %\n\\item The packet finds an available demodulator;\n %\n\\item The packet's reception is not interrupted by \\gls{dl} transmissions;\n %\n\\item Once the reception is finished, the packet was not corrupted by\n interference.\n %\n\\end{enumerate}\n\nTo count packets at the \\gls{phy} layer coherently with the simulator\nimplementation, the model's packet loss probabilities due to lack of\ndemodulators ($F_{NMD}$), \\gls{gw} transmission ($F_{GWTX}$) and interference\n($F_{INT}$) are plotted in the following section using, respectively, the\nfollowing expressions:\n\\begin{enumerate}\n\\item $F_{\\rm NMD}= 1 - S^{demod}$;\n\\item $F_{\\rm GWTX} = E_{i}\\left[S^{demod} \\cdot (1 - S_i^{TX})\\right]$;\n\\item $F_{\\rm INT} = E_{i}\\left[S^{demod} \\cdot S_i^{TX} \\cdot (1 - S_i^{INT})\\right]$;\n\\end{enumerate}\nby exploiting~\\eqref{eq:Sint},~\\eqref{eq:Sitx}, and~\\eqref{eq:Sdemod}, and where\n$E_i\\left[\\cdot\\right]$ indicates the expectation over the distribution of\n\\glspl{sf} and $S^{demod}$ the probability that, in the simulations, a packet\ncan lock on an available demodulator.\n\n\n\n\\section{Results}\n\\label{sec:results}\n\nThis section provides a comparison between the performance estimated with the\nproposed model and by the ns-3 simulator. Results are presented for both\n\\gls{phy} and \\gls{mac} layer, and the impact of the model's assumptions is\nshown to be mostly negligible, or at least acceptable. Finally, some results\nwill showcase how the model can be used to gain insight on the behavior of the\nLoRaWAN technology in a quick and effortless way, analyzing the effects of\nvarious parameters on the performance of the network. In the plots of this\nsection the analytical results are represented by lines, while markers\ncorrespond to simulation outcomes.\n\n\\begin{figure}[t]\n\t\\centering\n \\includegraphics[width=\\figurescaling\\linewidth]{modelFigures\/phy-confirmed.pdf}\n\t\\caption{PHY-level performance with $m=8$, $\\alpha=1$.}\n\t\\label{fig:phy}\n \\vspace{-1em}\n\\end{figure}\nFig.~\\ref{fig:phy} shows the packet outcome probabilities at the \\gls{phy}\nlayer in a network employing confirmed traffic. Although obtained with\ndifferent approaches, such probabilities are overall consistent, proving\nthe effectiveness of the model.\n\n\\begin{figure}[t]\n\t\\centering\n \\begin{subfigure}[t]{\\figurescaling\\linewidth}\n \\centering\n \\includegraphics[width=\\linewidth]{modelFigures\/cu-confirmed.pdf}\n \\vspace{-0.5cm}\n \\caption{CU for different values of $m$, $\\alpha=1$}\n \\label{fig:cu}\n \\end{subfigure}%\n \\begin{subfigure}[t]{\\figurescaling\\linewidth}\n \\centering\n \\includegraphics[width=\\linewidth]{modelFigures\/cd-confirmed.pdf}\n \\vspace{-0.5cm}\n \\caption{CD for different values of $m$, $\\alpha=1$}\n \\label{fig:cd}\n \\end{subfigure}%\n\t\\caption{Comparison of model and simulation results in terms of CU and\n CD.}\n\t\\label{fig:cucd}\n \\vspace{-1em}\n\\end{figure}\n\nThe good match between model and simulation is also reflected in\nFig.~\\ref{fig:cucd}, which shows the \\gls{cu} and \\gls{cd} metrics for a network\nin which all \\glspl{ed} generate confirmed traffic ($\\alpha=1$), and for\ndifferent values of $m$. Also in this case, the model results are quite close to\nthose given by the simulations. Fig.~\\ref{fig:cu} shows that the number of\navailable transmissions helps the correct delivery of the message at the\n\\gls{mac} layer, providing performance above 0.9 also for relatively high\ntraffic levels, when an average of one packet per second is generated by the\nnetwork at the application layer. The \\gls{cd} performance shown in\nFig.~\\ref{fig:cd} exhibits a similar behavior, but reaches much lower values\nmostly because the rate of \\gls{dl} messages that the \\gls{gw} can generate is\nlimited by the \\gls{dc} restrictions. The fact that this loss in performance is\ncaused by the \\gls{gw}'s \\gls{dc} is confirmed by the lilac dash-dotted line in\nFig.~\\ref{fig:cd}: to obtain these results, the \\gls{dc} restrictions were\nlifted by setting $\\delta_{SB1} = \\delta_{SB2} = 0$ in the model, producing\nmarkedly better results when compared to the corresponding green curve, where\n\\gls{dc} is enabled. Another example of the model's flexibility in considering\nalso non-standard settings is given by the densely dash-dotted brown line, which\nrepresents the \\gls{cd} metric when $\\delta_{SB1} = \\delta_{SB2} = 9$, i.e.,\nwhen transmissions in both sub-bands are subject to a \\gls{dc} of 10\\%. Although\nbeing an ideal setting, this case shows that even a small increase in the\n\\gls{dc} allowance in SB1 can yield considerable performance gains.\n\n\\begin{figure}[t]\n\t\\centering\n \\includegraphics[width=\\figurescaling\\linewidth]{modelFigures\/alpha.pdf}\n\t\\caption{Performance when varying the fraction of confirmed traffic, with\n $\\lambda=1, m=8, h=1$.}\n\t\\label{fig:alpha}\n \\vspace{-1em}\n\\end{figure}\nFig.~\\ref{fig:alpha} compares simulation and theoretical results, in terms of\n\\gls{uu}, \\gls{cu} and \\gls{cd}, when different fractions of confirmed traffic\nare employed in the network. For this comparison, we set the network application\nlayer packet arrival rate to $\\lambda=1$ pck\/s, the maximum number of\ntransmissions for confirmed traffic to $m=8$, and the number of repetitions for\nunconfirmed traffic to $h=1$. As the fraction of \\glspl{ed} employing confirmed\ntraffic increases, the data delivery performance decreases for all the\n\\glspl{ed}, in particular for nodes employing unconfirmed traffic which do not\nhave the chance of re-transmitting their packets. The match between the\nsimulator and the model is confirmed to be excellent for all values of $\\alpha$.\n\n\\begin{figure}[t]\n\t\\centering\n \\begin{floatrow}\n \\ffigbox{\\includegraphics[width=0.95\\linewidth]{modelFigures\/delays-confirmed.pdf}}{\\caption{Delays for a confirmed traffic network, $m=8$.}\\label{fig:delays}}\n \\ffigbox{\\includegraphics[width=0.95\\linewidth]{modelFigures\/fairness.pdf}}{\\caption{Fairness for different SF distributions when $m=8, h=8, \\tau=1, \\alpha=0.3$.} \\label{fig:fairness}}\n \\end{floatrow}\n\\end{figure}\nThe final metric that we evaluate through both model and simulation is the\ndelay, as described in Sec.~\\ref{sec:metrics}. Fig.~\\ref{fig:delays} shows\nhow delays generally increase with the traffic load, since more\nre-transmissions are needed to successfully deliver a packet. Note that for\nhigh values of $\\lambda$ the average \\gls{ack} delay $\\Delta^{\\textrm{DL}}$\ndecreases: this is explained by the fact that devices employing higher\n\\glspl{sf}, (which may increase the average delay due to their longer\ninter-packet transmission times) heavily suffer from interference and are\noften dropped (unsuccessful packets are not considered in the delay\ncomputation). Although not shown here, it is worth noting that the model\nformulation makes it easy to extract per-\\gls{sf} metrics that can help\ntroubleshoot the network configuration under study.\n\nWe now analyze how the fairness varies with the traffic load for different\nconfigurations of $\\alpha$, $p^u$ and $p^c$. We consider the \\gls{sf}\ndistributions $p_{\\rm equal}$ and $p_{\\rm EXPLoRa}$ as defined in\nTab.~\\ref{tab:config}. The $p_{\\rm EXPLoRa}$ distribution, first presented\nin~\\cite{cuomo2017explora}, aims at equalizing the aggregate time on air of each\ngroup of devices employing the same \\gls{sf} to minimize the collision\nprobability. In Fig.~\\ref{fig:fairness} we can observe that, when the \\glspl{sf}\nare uniformly allocated independently of the traffic type (i.e.,\n$p^u = p^c = p_{\\rm equal}$), the fairness decreases for an increasing traffic\nintensity. Indeed, as the traffic grows, nodes employing lower \\glspl{sf} will\nsuffer less from interference because of the shorter transmission times. The\nfairness grows when $\\alpha=0.3$ and $p^c = p_{\\rm EXPLoRa}$, since with this\nconfiguration 30\\% of the generated packets will use lower \\glspl{sf} with\nhigher probability, diminishing the channel and \\gls{gw} occupancy. However,\nsince the traffic load is high and the fairness is measured on the uplink\nperformance (\\gls{uu} and \\gls{cu}), the beneficial effect of allocating\n\\glspl{sf} according to the $p_{\\rm EXPLoRa}$ distribution are more evident when\nit is used for most of the devices, i.e., the 70\\% of nodes employing\nunconfirmed traffic. Finally, the maximum fairness is achieved when the\n\\glspl{sf} are allocated using $p_{\\rm EXPLoRa}$ both for $p^u$ and $p^c$\n(dotted line in\nFig.~\\ref{fig:fairness}).\nNote that, when $\\lambda \\leq 1$, the load in the network is low enough to have\n$J=1$ for every $p^u, p^c$, since the collision probability is low and the \\gls{gw} is not busy with\n\\gls{ack} transmissions.\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=\\figurescaling\\linewidth]{modelFigures\/retxdistribution.pdf}\n \\caption{Distribution of re-transmissions, $m=4$, $\\alpha=1$.}\n \\label{fig:retxdistribution}\n \\vspace{-1em}\n\\end{figure}\n\nAn example of insight that the analytical model can offer is presented in\nFig.~\\ref{fig:retxdistribution}, which shows the fraction of traffic that\nachieves success after a certain number of re-transmission attempt for different\ntraffic loads, derived from $P^{DL}_{i,j}$. This data, for instance, can be used\nto estimate the power consumption at the nodes: for low traffic loads the vast\nmajority of \\gls{mac} layer packet transmissions succed with just one \\gls{phy}\nlayer transmission attempt. As the traffic load increases, the fraction of\ndevices needing multiple re-transmissions to correctly receive an \\gls{ack}\ncorrespondingly increases. After a certain point, packet reception fails with\nsuch a high rate that most \\glspl{ed} need to employ the maximum number of\ntransmissions and, despite the high energy expenditure, still fail to receive an\n\\gls{ack} from the \\gls{gw}.\n\n\\begin{table}\n \\footnotesize\n \\centering\n \\caption{Configurations employed in Fig.~\\ref{fig:improvements}}\n \\label{tab:config}\n \\begin{tabular}{lccccc}\n \\toprule\n Configuration & $\\tau_1$ & $\\tau_2$ & $m$ & $h$ & $p^u = p^c$ \\\\\n \\midrule\n C1 & 1 & 1 & 1 & 1 & $p_{\\rm equal}$ \\\\\n C2 & 0 & 1 & 1 & 4 & $p_{\\rm EXPLoRa}$ \\\\\n C3 & 0 & 1 & 4 & 4 & $p_{\\rm EXPLoRa}$ \\\\\n \n \n \\bottomrule\n \\end{tabular}\n\\end{table}\n\\begin{figure}[t]\n\t\\centering\n \\includegraphics[width=\\figurescaling\\linewidth]{modelFigures\/improvements.pdf}\n\t\\caption{UU and CD performance for different network configurations,\n $\\alpha=0.3$.}\n\t\\label{fig:improvements}\n \\vspace{-1em}\n\\end{figure}\nFinally, we show how the model can be applied to investigate the impact of\ndifferent network parameters on the performance. In the example of\nFig.~\\ref{fig:improvements}, 30\\% of the \\glspl{ed} employ confirmed traffic,\nand we show results obtained with the proposed mathematical model. The parameter\nconfigurations are summarized in Table~\\ref{tab:config}. Configuration C1\nprovides a baseline: priority is given to \\gls{dl} transmission in both windows,\ndevices employ a single transmission attempt for both confirmed and unconfirmed\ntraffic, and \\glspl{sf} are uniformly distributed. In this case the curves have\na shape similar to those shown in Fig.~\\ref{fig:cucd} for $m=1$, but, since\nfewer devices require \\glspl{ack}, the \\gls{gw} is able to receive more packets\nand profitably send replies, leading to better performance. To improve \\gls{uu}\na second configuration (C2) considers the prioritization of \\gls{ack}\ntransmissions in \\gls{rx2}, where their reception suffers less interference.\nMoreover, unconfirmed packets are sent multiple times and we use $p^u = p^c =\np_{\\rm EXPLoRa}$. This configuration provides a considerable\nimprovement with respect to the \\gls{uu} metric, and some gains are also\nachieved in the \\gls{cu} performance. To improve also the results for confirmed\ntraffic, a further step (configuration C3) is to set $m=4$. This provides a\nsignificant improvement of \\gls{cu}, at the cost of a (minimal) decrease in\n\\gls{uu} performance. As a final step, we fully leverage the analytical model to\nidentify the optimal parameter configuration (i.e., $m$, $h$, $p_{u}$ and\n$p_{c}$) for each plotted traffic load, with the objective of maximizing the\naverage of \\gls{uu} and \\gls{cu}. The red curves of this setting (C4) show how\nthis optimization process enabled by the model can significantly improve the\nglobal performance of the network, significantly improving the \\gls{cd}\nperformance at the price of a very small reduction in packet success rate for\nunconfirmed devices.\n\nThe optimization problem that is solved to obtain configuration C4 is defined\nas:\n\\begin{equation}\n\\begin{aligned}\n\\max_{p_{u}, p_{c}} \\quad & \\textrm{UU} + \\textrm{CD} \\\\\n\\textrm{s.t.} \\quad & 0 \\le p_{i}^{u} \\le 1 \\\\\n & 0 \\le p_{i}^{c} \\le 1 \\\\\n & \\sum_{i} p_{i}^{u} = 1 \\\\\n & \\sum_{i} p_{i}^{c} = 1 \\\\\n\\end{aligned}\n\\label{eq:optimization}\n\\end{equation}\nwhere we explore the entire space defined by $m$, $h$ and $\\lambda$, by solving~\\eqref{eq:optimization}\nto find the best $p_{u}$ and $p_{c}$, and finally pick the best solution for\neach $\\lambda$. The search is performed using the trust region method as\nimplemented by the \\texttt{scipy} library, and we always set $p_{i}^{u} =\np_{i}^{c} = 1\/6$ as the initial parameter value for the algorithm.\n\n\\begin{figure}[t]\n\t\\centering\n \\includegraphics[width=0.6\\linewidth]{modelFigures\/distributions.pdf}\n\t\\caption{Optimal values of $p_{u}$, $p_{c}$, $m$ and $h$ as computed through model-driven optimization, for various values of $\\lambda$.}\n\t\\label{fig:optimal-parameters}\n \\vspace{-1em}\n\\end{figure}\nFigure~\\ref{fig:optimal-parameters} displays the parameters of configuration C4\nfor some representative values of $\\lambda$, showing $p_{u}$ in the first row,\n$p_{c}$ in the second row, and a combination of the two weighed on $\\alpha$ on\nthe third row. For a low value of generated traffic ($\\lambda = 0.1$, first\ncolumn), we see that the optimization stops almost immediately, yielding a\ndistribution that is very similar to the initial value of $p_{u}$ and $p_{c}$.\nIn this case, as can also be seen in Figure~\\ref{fig:improvements}, since the\ntraffic load is low the performance is indeed very good for high values of $m$\nand $h$, and needs little optimization of the \\gls{sf} distributions. For\n$\\lambda = 1$, instead, the optimization process yields a more distinctive value\nof $p_{c}$, setting almost all devices to use \\gls{sf}7. This is motivated by\nthe fact that, \\gls{rx1} is set to employ the same \\gls{sf} used in the\n\\gls{ul}. Therefore, having most of the confirmed devices employ an \\gls{sf} as\nlow as possible is advantageous, since it guarantees faster \\gls{ack}\ntransmissions in the \\gls{dl} and, as a consequence, shorter silent times\nimposed by the \\gls{dc}, and a larger set of devices can thus be served. Devices\nemploying unconfirmed traffic, instead, are set to use a variety of \\gls{sf}\nvalues. Notably, the selected values are such that the aggregated distribution\nconsidering both unconfirmed and confirmed traffic (visible in the third row)\ntakes a shape that is very similar to that of $p_{\\rm EXPLoRa}$. This behavior\nis even more marked when $\\lambda = 10$, with the notable difference that higher\n\\gls{sf} values are not used in the optimized network: this is because of the\nlimited number of demodulators at the \\gls{gw} (a factor which is accounted for\nin our model). Indeed, although using all \\gls{sf} values would bring an\nadditional gain, a packet with high \\gls{sf} value occupies a demodulator for\nquite a long time, increasing the probability that other incoming packets are\ndropped because of unavailability of reception chains at the \\gls{gw}. Finally,\nwe note that $m$ and $h$ are consistently set to their maximum values (8 here)\nup to $\\lambda = 1$. After this value, instead, it pays off to reduce the number\nof repetitions employed by both unconfirmed and confirmed \\glspl{ed}.\n\nAlthough this analysis showcases the potential of the mathematical model to\nidentify the optimal settings, an evaluation of the trade-offs associated to\nparameter configurations and their effect on other metrics of interest, such as\ndelays and energy consumption, needs a deeper investigation, which we leave for\nfuture work.\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nIn this work, we presented a model for the performance evaluation of a\nLoRaWAN network in the presence of both confirmed and unconfirmed traffic,\ntaking into account the influence of different settings of multiple network\nconfiguration parameters.\n\nThe model is able to capture both the \\gls{phy} layer and \\gls{mac} layer\nperformance, and describes the multiple events that affect both \\gls{ul} packet\nreception and \\gls{dl} transmission: interference, capture effect, availability\nof demodulator, \\gls{dc} constraints, ongoing transmissions and receptions.\nWe validated the model results with ns-3 simulations, showing the consistency\namong the two sets of results. Finally, we presented some examples of how the\nmodel can be employed to analyze the effects of possible changes to the standard\nparameter settings, and to identify optimal configurations with minimum effort.\n\nSeveral extensions of this work are possible. A first improvement to the model\nis the inclusion of multi-\\gls{gw} scenarios, where \\gls{ul} packets are\npotentially received by several \\glspl{gw}, and the network \\gls{dl} capacity is\nincreased. A second aspect of interest is to leverage the proposed model to\nbetter investigate trade-offs among different network parameters in various\nscenarios, or when specific performance requirements are provided. A third\npossible improvement would involve characterizing the capture effect for\nnon-uniform spatial distribution of the devices. Finally, a fourth direction is\nto employ the proposed model to identify optimal network settings when different\nmetrics of interest are used as optimization functions, as we showed in the\nresults section with some simple cases. We point out that the target of the\nmodel was to explore the capabilities of LoRaWAN networks, thus, in this work,\nwe neglected some features of LoRa, such as the interference between overlapping\npackets modulated with different \\glspl{sf}. The model can be extended by\nincluding this, as well as other specific features of the LoRa technology. Such\nextensions are left for future work.\n\nWe remark that all figures contained in this paper, covering both\nmodel evaluations and simulation results, can be easily reproduced using\nthe tool available at~\\cite{publishedmodelcode}.\n\n\\section*{Acknowledgment}\n\nPart of this work was supported by MIUR (Italian Ministry for Education and\nResearch) under the initiative \"Departments of Excellence\" (Law 232\/2016).\n\n\n\\bibliographystyle{IEEEtran} ","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\\label{Sec:intro}\n\nThe conduction electron Wannier orbitals in transition-metal compounds are generally \nfairly localized in space so that electronic correlations, i.e. all effects that deviate from the \nindependent-particle picture, are sometimes strong enough to give rise to metal-insulator \ntransitions in particular temperature and pressure conditions. The correlation-driven \nmetal-insulator transition, known as Mott transition,\\cite{Mott_original, Mott} is \noften accompanied by rather spectacular phenomena that appear in its proximity, high-temperature \nsuperconductivity being the most popular example. This makes $3d$ metal \nelements and \ncompounds a natural laboratory for intriguing many-body physics, which despite \na rich history and many studies is worth exploring further. \n\nElectronic structure methods (sometimes referred to as \"first principles\" methods) that rely on independent \nparticle descriptions, such as \\ac{hf} or \\ac{dft} within \\ac{lda}, are by construction \nincapable of capturing the Mott transition, \nwhich has no counterpart in a one electron picture.\nFor this reason, \\ac{hf} and \\ac{lda}, while \ngenerally quite successful for many materials,\nmay sometimes fail in the description of solids involving transition metals. In fact, most of our knowledge \nabout Mott electron localization has been attained by means of simplified lattice models, the best known \nbeing the Hubbard model,\\cite{Hubbard} which are accessible by methods better suited to deal with correlations, \nsuch as quantum Monte Carlo,\\cite{QMC_review} density-matrix renormalization group\\cite{Schwollock_DMRG} \nand dynamical mean-field theory.\\cite{DMFT} \n\nClearly, for the purpose of a quantitative understanding of real materials, it is of key importance to \nsew the two worlds together, bringing in particular the many body\nexpertise gained on lattice models over to realistic, off-lattice, first-principles calculations of solids.\n\nThis has historically been attempted through ad-hoc improvements of \\ac{dft}. \nFor instance, the inclusion (in fact, the addition and subtraction) of an intra-site Coulomb repulsion $U$ (the \"Hubbard $U$\") \nin the Kohn-Sham Hamiltonian permits a decrease of the so-called self-interaction error, \na severe flaw of \\ac{lda} for partially or fully occupied localized orbitals -- just the case of transition metals. \nWhen added to \\ac{lda}, this procedure, the so called \n\\ac{ldau},\\cite{LDAU_Anisimov_Andersen, LDAU_jphys} often improves results, and can for example stabilize magnetic phases which \nstraight \\ac{lda} would miss. Yet, \\ac{ldau} remains basically a mean-field, independent particle approach that cannot \ndescribe Mott localization. The problem can be overcome if, for instance, the Kohn-Sham Hamiltonian of \\ac{lda} \nsupplemented by $U$ is solved through \\ac{dmft}, by the so-called \\ac{ldadmft}.\\cite{LDADMFT_theory} \nAlternatively, variational Quantum Monte Carlo \\ac{vqmc} approaches\\cite{Ceperley_VMC,Sorella_SR} have been successfully \napplied to the electronic \nproperties of atoms and simple molecules,\\cite{Sorella_dimers} and its development appears to be promising \nfor more ambitious applications.\n\nAt present, both \\ac{ldadmft} and \\ac{vqmc} are numerically much more involved and far more demanding \nthan conventional \\ac{lda} or even \\ac{ldau}, which owe much of their success \nto simplicity. The desirability of approaches joining together the simplicity of \\ac{lda} and the description of\ncorrelations typical of many body methods is therefore still very high. \nIn the context of lattice models, a simple approach to \nstrong correlations was proposed long ago by Martin C. Gutzwiller.\\cite{Gutzwiller1,Gutzwiller2} This method,\nprojecting out of a trial Slater determinant an adjustable proportion of costly configurations and \nevaluating average values by approximate formulas, is strictly variational in the limit of \ninfinite lattice-coordination\\cite{Gebhard} -- the same limit where \\ac{dmft} is exact -- providing much more accurate \nresults than \\ac{hf}. That success invites the use of the Gutzwiller method even when \nthe lattice space dimension, and thus the site coordination, is finite, as people do with \\ac{dmft}. \n\\ac{ga} electronic structure calculations \nhave the great advantage \nto couple extreme LDA-level simplicity with qualitatively, often quantitatively, increased accuracy in the\ndescription of correlations. For example, \\ac{ga} has been able to describe conducting materials that are \ninsulators ``in disguise'',\\cite{Fazekas} i.e. whose properties depend on correlations that are already present \nin their Mott insulating phase, and that continue to play an important role even in the nearby metallic phases. \nA well known example is the RVB scenario for high-temperature superconductors,\\cite{Anderson_RVB_Science} \nwhere Cooper pairing is explained as a byproduct of doping a parent state of resonating valence bonds, \nwhich is the remnant of antiferromagnetism when N\\`eel long range order disappears. Another famous \nresult of the \\ac{ga} is the Brinkman-Rice description of the Mott transition in vanadium sesquioxide, originally \nderived by the \\ac{ga} solution to the Hubbard model.\\cite{brinkman&rice}. \n\nBecause of its simplicity, a great deal of effort has therefore been \ndevoted in recent years to extend \\ac{ga} from simple lattice models to more realistic off-lattice \ncases.\\cite{Ho_LDAG_condmat, ZhongFang_LDAG, Andersen_Gebhard_Gutzwiller, ZhongFang_LDAG_app1, Lanata_efficient, Ho,Lanata-Kotliar} \nHere we implement a density self-consistent algorithm that exploits the Gutzwiller variational wave \nfunction together with the conventional \\ac{lda} for the density functional. The Levy-Lieb constrained-search \nformulation of \\ac{dft} provides a solid theoretical framework for the introduction of Gutzwiller variational \nparameters in the density functional, while a localized atomic basis set (we use in particular the Siesta electronic structure code) makes \nthe definition of the Gutzwiller-projected states straightforward.\n\nWe test the power of the \\ac{ldag} functional by calculating the electronic structure of nonmagnetic\nand ferromagnetic $bcc$ Fe, motivated by long standing basic \nquestions about the electronic origin of magnetic order, \n\\cite{vollhardt2001} including \na recent \\ac{ldadmft} study \nby Anisimov and coworkers\\cite{anisimov} \nsuggesting that $bcc$ iron might be an orbital-selective Mott insulator. According to that picture, the \npoorly dispersive $e_{g}$-type electrons of metallic Fe may be fully localized due to interactions, \nso that conduction phenomena are restricted within the $t_{2g}$ manifold (besides of course \nthe $s$ electrons). In that picture\\cite{stollhoff_ironloc, Goodenough_ironloc, stearns_ironloc} \nferromagnetic alignment \nwould not be due to inter-site Coulomb exchange, as is \nordinarily assumed, but rather to double-exchange, as in colossal magnetoresistance manganites.\\cite{manganites}\nThe Mott localized $e_g$ electrons \nform spin-1 moments that couple \nferromagnetically via intra-atomic Hund's exchange to the electrons in the \nnearly-full itinerant $t_{2g}$ bands. In order to preserve coherent $t_{2g}$ hole motion, \nthe local $e_g$ moments order ferromagnetically. As in the manganites, ferromagnetism is \nthus driven by a kinetic energy gain rather than a potential energy one. \nEven though our \\ac{ldag} approach is still mean-field and thus cannot address dynamical \nphenomena such as orbital selective Mott transitions --\nespecially so in a delicate case where the two sets of orbitals, $e_{g}$ and $t_{2g}$, hybridize with each other \nin the Brillouin zone -- we find that calculation of the total energy and a detailed analysis of its \nseparate kinetic and potential energy contributions actually supports double-exchange \nas the driving mechanism of \nferromagnetism in iron, rather than the conventional Stoner instability. On the whole, this work \nmay also be of general use as a very detailed\nexample of {\\sl ab-initio} application of Gutzwiller correlations to a realistic electronic structure problem.\n\nThe plan of this article is as follows: in \\sect{Sect1} we introduce the formalism of \\ac{ldag} starting \nfrom the constrained-search formulation of Density Functional Theory,\ndemonstrating\nhow the Gutzwiller \nwavefunction can be used to generalize \\ac{ldau} by allowing the expectation value of the atomic \nHamiltonian to be computed on a multi-determinant wavefunction. In \\sect{Sect:Gutz_expect} and\n\\sect{Sec:Gutz_practice} we then show how the different terms of the \\ac{ldag} density functional can be computed \nby means of \\ac{ga}, and how the total energy of a correlated electronic system can be minimized by a three-step \niterative procedure. In \\sect{Sec:results} we finally present and comment on the physical results for paramagnetic and \nferromagnetic $bcc$ Fe, and connect back to the basics questions about the origin of ferromagnetic order.\n\n\n\\section{Constrained-search formulation of a Gutzwiller Density Functional Theory}\\label{Sect1}\n\nA convenient way to introduce a Gutzwiller density functional is through the formalism \nindependently proposed by Levy~\\cite{Levy1,Levy2} and Lieb~\\cite{Lieb1}.\nStarting from the Rayleigh-Ritz definition for the ground state energy $E_{\\rm GS}$ of a system\n\\begin{align}\\label{Eq_min_GSenergy}\nE_{\\rm GS} = \\min_{\\Psi} \\expect{\\Psi}{\\hat{H}}\\,,\n\\end{align}\nwhere the electron Hamiltonian $\\hat{H}$ includes the kinetic energy $\\hat{T}$, the electron-electron interaction $\\hat{V}_{\\rm ee}$, \nand a local external potential $\\hat{V}_{\\rm ext}$, Levy and Lieb converted the variational principle for \nthe ground state wavefunction into a variational principle for the ground state density through a constrained minimization at fixed density $n({\\bm r})$\n\\begin{eqnarray}\n\\label{Eq_LL_DF}\nE_{\\rm GS}[V_{\\rm ext}({\\bm r})] &=& \\min_{n({\\bm r})} \\Bigg\\{ \\min_{\\Psi\\rightarrow n({\\bm r})} \\expect{\\Psi}{\\hat{T}+\\hat{V}_{\\rm ee}} \\nonumber \\\\\n&& ~~~~~~~~~~+ \\int V_{\\rm ext}({\\bm r}) n({\\bm r})\\Bigg\\}\\,.\n\\end{eqnarray}\nThe first term on the right-hand side of (\\ref{Eq_LL_DF}) is nothing but the constrained-search definition of \nthe Hohenberg-Kohn functional\\cite{DFT_HKtheorem}, i.e.\n\\begin{align}\\label{Eq_LL_HKfunc_def}\nF_{\\rm HK}[n({\\bm r})] = \\min_{\\Psi\\rightarrow n({\\bm r})} \\expect{\\Psi}{\\hat{T}+\\hat{V}_{\\rm ee}}\\,,\n\\end{align}\nwhich is independent of the external potential $V_{\\rm ext}$.\nThe wavefunction $\\Psi$ in the definition~\\eqn{Eq_LL_HKfunc_def} should span the whole many-body Hilbert space, \ngenerally too large \nto allow a \nstraightforward numerical evaluation of $F_{\\rm HK}[n({\\bm r})]$.\nWithin the Kohn-Sham scheme, the generality of \\eqn{Eq_LL_HKfunc_def} is abandoned in \nfavor of a more practical definition of the Hohenberg and Kohn functional, in which the latter is split \ninto kinetic, Hartree, and exchange-correlation terms, namely\n\\begin{align}\\label{Eq_HK_semilocal}\nF_{\\rm HK}[n({\\bm r})] = T_{s}[n({\\bm r})] +E_{\\rm H}[n({\\bm r})] + E_{\\rm xc}[n({\\bm r})],\n\\end{align}\nwhere $E_{\\rm H}[n({\\bm r})]$ is simply the electrostatic energy of the electron density regarded as a classical charge distribution.\nA constrained search is then retained only for the kinetic contribution\n\\begin{align}\\label{Eq_HK_nonint}\nT_{s}[n({\\bm r})] = \\min_{\\Psi\\rightarrow n({\\bm r})} \\expect{\\Psi}{\\hat{T}}\\,,\n\\end{align}\nwhich, because $\\hat{T}$ is a one-body operator, has a solution within the class of Slater determinants, \na relatively simple task to accomplish through \nauxiliary non-interacting electron Hamiltonians whose ground state local density $n({\\bm r})$ coincides \nwith that of the physical interacting model. The insurmountable difficulties of the original many-body problem \nhave thus been hidden in the unknown exchange-correlation functional $E_{\\rm xc}[n({\\bm r})]$. All DFT approximation \nschemes correspond just to different guesses of a physically sensible functional form of $E_{\\rm xc}[n({\\bm r})]$ \nin terms of the local density. \n\nThe main problem that arises from the density-dependent parametrization \\eqn{Eq_HK_nonint} \nis that $E_{\\rm H}[n({\\bm r})]$ contains a spurious \\ac{si} \nterm -- finite even when $n({\\bm r})$ is the density of a single electron! -- a term which should be identically \ncancelled in the exact $E_{\\rm xc}[n({\\bm r})]$. Unfortunately, all semi-local approximations to $E_{\\rm xc}[n({\\bm r})]$, \nsuch as \\ac{lda} and \\ac{gga}, fail to fully subtract such a \\ac{si} term from the density functional, which \nbrings about results that by construction contain a certain level of \nself-interaction \nerror. \n\nThe spurious SI one-electron energy is larger for spatially localized electronic wavefunctions. \nFor instance, a single electron with a simple gaussian wavefunction feels an \\ac{si} that is\ninversely proportional to the standard deviation of the gaussian, only $70\\%$ of which is \nsubtracted by the \\ac{lda} exchange functional. The improvements attained by \nbetter functionals do not seem major.\\cite{korzdorfer_SIC}\nAll density-functional calculations are affected to some extent by the \\ac{si} error, more important \nwhen the real-space density matrix is more localized. That is especially \nthe case for most transition metals \nand transition-metal oxides. In a density functional calculation with semi-local functionals, the spurious \\ac{si} term acts effectively as a penalty term \npreventing electronic localization, thus often spoiling \nagreement with experimental data for band gaps, \nmagnetization, and other physical observables such as lattice constant and bulk modulus.\n\n\\subsection{LDA+U}\nA popular way to reduce the \\ac{si} \nwhile \nstill remaining in the context of local or semi-local density functionals \nis by including in the kinetic functional \\eqn{Eq_HK_nonint} also part of the electron-electron interaction, \nspecifically the projection $\\hat{H}_{\\rm at}$ of $\\hat{V}_{\\rm ee}$ on atomic-like orbital (se below).\nThe common choice is \nto consider only orbitals that are partially occupied within standard LDA, hence which suffer more from the \\ac{si}. \nThe non-interacting kinetic functional $T_{s}[n({\\bm r})]$ is thus turned into a modified kinetic functional $T_i[n({\\bm r})]$:\n\\begin{align}\\label{Eq_HK_modified}\nT_{s}[n({\\bm r})] \\rightarrow T_i[n({\\bm r})] = \\min_{\\Psi_{0} \\rightarrow n({\\bm r})} \\expect{\\Psi_{0}}{\\hat{T}+\\hat{H}_{\\rm at}}\\,,\n\\end{align}\nand the Hohenberg and Kohn functional changes into \n\\begin{align}\nF_{\\rm HK}[n({\\bm r})] &= T_i[n({\\bm r})] +E_{\\rm H}[n({\\bm r})] \\nonumber\\\\\n& + E_{\\rm xc}[n({\\bm r})] - {E_{\\rm dc}}[n({\\bm r})]\\,,\\label{Eq_HK_plusU}\n\\end{align}\nwhere ${E_{\\rm dc}}[n({\\bm r})]$ is a double-counting energy which must cancel\nthe contribution of $\\hat{H}_{\\rm at}$ already included within LDA.\n\nIn \\eqn{Eq_HK_modified} the constrained-search is still restricted to the space of Slater-determinants \n$\\Psi_{0}$, so that the modified kinetic functional can be dealt with within an independent-particle picture, \nand therefore included in the Kohn-Sham scheme. Essentially, the interaction $\\hat{H}_{\\rm at}$ is treated by Hartree-Fock, \nwhich is devoid of SI -- while still unable to capture the Mott localization phenomenon, \na correlation effect. \nIn section \\ref{Sect:gutz} we shall discuss how to improve the functional $T_i$ so as to make Mott physics \naccessible. Here in addition we \nbriefly discuss how to define properly $\\hat{H}_{\\rm at}$.\nTypically $\\hat{H}_{\\rm at}=\\sum_{{\\bm R}} \\hat{H}_{\\rm at}^{({\\bm R})}$, with $\\hat{H}_{\\rm at}^{({\\bm R})}$ accounting for the leading order \nmultipolar expansion of the Coulomb interaction projected onto a selected set of \natomic-like orbitals $\\ket{\\phi^{(l)}_{{\\bm R},m}}$ with angular momentum $l$ \nat atomic site ${\\bm R}$ in the lattice, \n\\begin{eqnarray}\n\\hat{H}^{({\\bm R})}_{\\text{at}} &=& \\frac{F_0}{2}\\,\\hat{N}_{\\bm R}\\left(\\hat{N}_{\\bm R}-1\\right)\n+ \\frac{1}{2}\\,\\sum_{L>0}^{2l}F_L\\,\\left(C^{l0}_{l0\\,L0}\\right)^2\\nonumber\\\\\n&& ~~~~~~\\sum_{M=-L}^L\\,(-1)^M C^{lm}_{lm'\\,LM}\nC^{lm_1}_{lm'_1\\,L-M}\\nonumber\\\\\n&& ~~~~~~~~~~~~~~c^\\dagger_{{\\bm R},m\\sigma}c^\\dagger_{{\\bm R},m_1\\sigma_1}\nc^{\\phantom{\\dagger}}_{{\\bm R},m'_1\\sigma_1}c^{\\phantom{\\dagger}}_{{\\bm R},m'\\sigma}\n\\label{Eq:hHat_exact}\n\\end{eqnarray}\nwhere $\\hat{N}_{\\bm R}$ is the total electron number operator at site ${\\bm R}$ projected onto \nthe selected set of atomic orbitals, $L = 2n$ with $n=1,\\dots,l$, and $C^{lm}_{lm'\\,LM}$ are \nthe Clebsch-Gordan coefficients. The parameters $F_L$ are commonly \nknown as Slater integrals. The first term on the right-hand side of \\eqn{Eq:hHat_exact}, \nwhich we shall denote hereafter as $\\hat{H}_{\\rm Hub}^{({\\bm R})}$, is a pure charge repulsion usually referred \nto as the Hubbard term, its coupling constant $F_0$ generally called the \"Hubbard $U$\". The remaining terms instead \nenforce Hund's first and second rules, hence they may be referred to as \nthe Hund's rule exchange ($\\hat{H}_{\\rm Hund}$).\nIn fact, in the case of $p$ orbitals ($l=1$), the exact multipolar expansion can be rewritten solely in terms \nof the number operator $\\hat{N}_{\\bm R}$, the total spin $\\mathbf{S}_{\\bm R}$ and total angular momentum $\n\\mathbf{L}_{\\bm R}$ operators projected on the set $\\ket{\\phi^{(1)}_{{\\bm R},m}}$:\n\\begin{align}\n\\label{Eq_Coulomb_on_p}\n\\hat{H}_{\\rm at} &= \\frac{F_0}{2} \\left[\\hat{N}_{\\bm R}\\left(\\hat{N}_{\\bm R}-\\mathds{1}\\right)\\right] \\\\\n&+ \\frac{F_2}{2}\\left[\\frac{4}{5}\\hat{N}_{\\bm R}-\\frac{\\hat{N}_{\\bm R}^2}{5}-\n\\frac{3}{25}\\left(4 \\hat{\\mathbf{S}}_{\\bm R}\\cdot\\hat{\\mathbf{S}}_{\\bm R} +\\hat{\\mathbf{L}}_{\\bm R}\\cdot\\hat{\\mathbf{L}}_{\\bm R}\\right)\\right]\\,,\\nonumber\n\\end{align}\nexplicitly \nshowing the content of the first two Hund rules. For $l>1$, it is no longer\npossible to \nrewrite \\eqn{Eq:hHat_exact} in terms of simple operators like spin and angular momentum.\n\nThe well-known \\ac{ldau} method truncates the multipolar expansion of the Coulomb operator, \n\\eqn{Eq:hHat_exact}, to the zeroth-order term, therefore setting $\\hat{H}_{\\rm at}=\\hat{H}_{\\rm Hub}$.\nWith this recipe, the density dependence of the expectation value $\\expect{\\Psi_{0}}{\\hat{H}_{\\rm at}}$ can be written \nin terms of the matrix elements \n$n^{(0)}_{lm\\sigma{\\bm R},lm'\\sigma'{\\bm R}} = \\expect{\\Psi_{0}}{c^\\dagger_{{\\bm R}, lm\\sigma}\nc^{\\phantom{\\dagger}}_{{\\bm R},lm'\\sigma'}}$ \nof the local single-particle density matrix $\\hat{n}^{(0)}_{\\bm R}$ , which is an implicit function of the density $n({\\bm r})$. \nIf lattice periodicity is unbroken and the set of correlated orbitals is characterized by a single \nvalue of the angular momentum, we can drop both indices $l$ and ${\\bm R}$ in any local operator, \nand write $\\hat{n}^{(0)}_{\\bm R} = \\hat{n}^{(0)}$, $\\forall {\\bm R}$.\nThe double-counting correction ${E_{\\rm dc}}[n({\\bm r})]$ in \\ac{ldau} is commonly chosen \nso as to cancel $\\expect{\\Psi_{0}}{\\hat{H}_{\\rm at}}$ in the limiting case \nof an idempotent single-particle density matrix $\\hat{n}^{(0)}$,\\cite{mazin} which corresponds to assuming \nthat, within straight \\ac{lda}, $\\langle \\hat{n}^{(0)}_{\\bm R} \\hat{n}^{(0)}_{\\bm R} \\rangle = \n\\langle \\hat{n}^{(0)}_{\\bm R} \\rangle\\langle \\hat{n}^{(0)}_{\\bm R} \\rangle $.\nWith this assumption, the $U$-dependent part of Kohn-Sham Hamiltonian is equal to \nthe positive definite contribution\n\\begin{align}\n\\label{Eq:LDAU_total_ham}\n\\expect{\\Psi_{0}}{\\hat{H}_{\\rm at}}-{E_{\\rm dc}}[n({\\bm r})] = \\frac{U}{2} \\text{Tr}\\Big[\\hat{n}^{(0)}\\big(\n1-\\hat{n}^{(0)}\\big)\\Big].\n\\end{align}\nAn optimal value of $U$ can be estimated by linear response calculations \\cite{Cococcioni_DeGironcoli_LinLDAU, \nAnisimov_Zaanen}, or \nempirically determined by agreement with experimental data. \n\nThe advantage of using \\eqn{Eq:LDAU_total_ham} to improve the description of systems with \nstrongly localized electrons is both its simplicity, involving no further computational effort than \nthat needed to solve the Kohn-Sham equations, and its success in removing the self-interaction \nwhenever $U$ is sensibly chosen.\nHowever, there are of course situations in which the empirical \\ac{ldau} functional will \nnot be adequate.\nWe previously mentioned that Mott localization \nbecause of its genuinely many-body, collective nature, \nis not accessible by \\ac{ldau} \nnor by any other technique that relies on a single-particle description.\nMoreover, it is well known that only the spherically-averaged strength of the exchange-correlation \nhole is correctly accounted for by the LDA functional, but not its angular dependence. For these reasons \none cannot expect that\n\\ac{ldau} will be apt to describe systems that display strongly orbital-dependent correlations, as was shown \nto be the case of body-centered cubic iron.\\cite{anisimov} Indeed recent studies\non iron pnictides and chalcogenides\\cite{Kotliar_coherence_incoherence, Kotliar_ironcalco, magnetism_chargedyn_pnictides} \nsuggest that the orbital selectivity displayed by these iron compounds crucially depends on atomic Hund's rules. These \nobservations indicate that a way to further improve LDA beyond \\ac{ldau} will not only be the inclusion \nof correlations in the modified kinetic functional so as to make Mott localization accessible, \nbut also \nthe introduction of an appropriate expression for Hund's interaction $\\hat{H}_{\\text{Hund}}$ in \nthe atomic Hamiltonian $\\hat{H}_{\\text{at}}$, so as to account for orbital selectivity. However, \nwhen Hund's rule exchange, the second term in \nthe r.h.s. of \\eqn{Eq:hHat_exact}, is taken into account, one faces the problem of finding a proper expression \nfor electron double counting. The latter should by definition be equivalent to the LDA approximation \nto the atomic interaction energy, \\eqn{Eq:hHat_exact}. However, that average depends in principle on the \nspecific point symmetry of the system, and one cannot find a general expression valid for every case. \nThe conventional way to proceed is to dismiss the hope of including within \\ac{ldau} the whole \natomic interaction \\eqn{Eq:hHat_exact}, and instead be content with only terms that depend \non angular-averaged local operators, specifically the total number operator $\\hat{N}_{\\bm R}$ and total spin \n$\\hat{\\mathbf{S}}_{\\bm R}$. These terms are identified by noting that, using the \nre-coupling formula\n\\begin{eqnarray*}\n&&\\sum_M\\, (-1)^M\\, C^{lm}_{lm'\\, LM} C^{lm_1}_{lm'_1\\,L-M} = \n\\sum_{\\Lambda\\lambda}\\, (2\\Lambda+1)\\, \n\\begin{Bmatrix}\nL & l & l\\\\\n\\Lambda & l & l\n\\end{Bmatrix} \\\\\n&& ~~~~~~~~~~~~~~~~(-1)^{L+\\Lambda}\\, (-1)^{\\lambda}\\,C^{lm}_{lm'_1\\,\\Lambda\\lambda} \n\\, C^{lm_1}_{lm'\\,\\Lambda-\\lambda}\\, ,\n\\end{eqnarray*}\nthe $L>0$ contribution of \\eqn{Eq:hHat_exact} can be also written as \n\\begin{eqnarray*}\n\\hat{H}^{({\\bm R})}_{\\text{at}\\, L>0} &=& \n- \\frac{1}{2}\\sum_{mm_1m'm'_1}\\sum_{\\sigma\\sigma_1}\\sum_{L>0}^{2l}F_L\\,\\left(C^{l0}_{l0\\,L0}\\right)^2\\nonumber\\\\\n&& ~~\\sum_{\\Lambda}\\sum_{\\lambda=-\\Lambda}^\\Lambda \n(-1)^{\\Lambda+\\lambda} (2\\Lambda+1) \\begin{Bmatrix}\nL & l & l\\\\\n\\Lambda & l &l \n\\end{Bmatrix}\\nonumber\\\\\n&& C^{lm}_{lm_1'\\,\\Lambda\\lambda}C^{lm_1}_{lm'\\,\\Lambda-\\lambda}\\,\nc^\\dagger_{lm\\sigma}c^{\\phantom{\\dagger}}_{lm'_1\\sigma_1}c^\\dagger_{lm_1\\sigma_1}c^{\\phantom{\\dagger}}_{lm'\\sigma}\\,,\n\\end{eqnarray*}\nwhere $\\{\\dots\\}$ denote the Wigner $6j$-symbols. We can then select out the term with $\\Lambda=0$, \nwhich depends on rotationally invariant densities, \nre-couple back $m$ with $m'$ and $m_1$ with $m'_1$ in the remaining terms, and iterate the procedure.\nAt the end, we obtain a term that involves rotationally invariant densities, plus another interaction that \ncannot be expressed by any means in terms of those densities. The former \ntogether with the Hubbard $U$ define the part of the atomic interaction \\eqn{Eq:hHat_exact} \neasier to implement within \\ac{ldau}, namely \n\\begin{align}\n\\hat{H}^{({\\bm R})}_{\\text{at}} &\\simeq& \\frac{U}{2} \\hat{N}_{\\bm R}\\Big(\\hat{N}_{\\bm R}-1\\Big)\n- \\frac{2l+1}{2l+2}\\,J \\,\\Bigg[\\hat{\\mathbf{S}}_{{\\bm R}}\\cdot \\hat{\\mathbf{S}}_{{\\bm R}}\n-\\frac{3}{4}\\,\\hat{N}_{\\bm R} \\nonumber\\\\\n&& + \\fract{\\hat{N}_{\\bm R}\\left(\\hat{N}_{\\bm R}-1\\right)}{4} \n+\\fract{\\hat{N}_{\\bm R}\\left(\\hat{N}_{\\bm R}-1\\right)}{2(2l+1)} \n\\Bigg],\\label{Eq:hHat_approssimata}\n\\end{align}\nwhere $J$ is conventionally defined as \\cite{Cococcioni_DeGironcoli_LinLDAU, Anisimov_Zaanen} \n\\begin{equation}\nJ = \\frac{1}{2l}\\sum_{L>0}^{2l} \\, \\left(C_{l0\\, L0}^{l0}\\right)^2\\, F_L,\\label{def:J}\n\\end{equation}\nwhich, for $d$-orbitals, i.e. $l=2$, is $J=(F_0+F_4)\/14$. The double counting term associated with \n\\eqn{Eq:hHat_approssimata} is obtained analogously as before and reads in the general case of a \nspin-polarized calculation \n\\begin{align}\nE_{\\text{dc}} &=& \\frac{U}{2} N\\big(N-1\\big) - \\frac{2l+1}{4l+4}\\,J \\,\\bigg[ N_\\uparrow\\left(N_\\uparrow-1\\right) \n\\nonumber\\\\\n&& ~~~~~~~~~~+ \n N_\\downarrow\\left(N_\\downarrow-1\\right) + \\fract{N(N-1)}{2l+2}\\bigg].\\label{double-counting-Hund}\n\\end{align}\nThe expression \\eqref{Eq:hHat_approssimata} can be further simplified to get rid of the $l$ dependence, \nby readsorbing the $l$ in the definition of $J$, and by adopting a simplified version\nof the last term in square brackets, leading to the following results\n\\begin{align}\\label{Eq:recast_J}\n\\hat{H}_{\\rm Hund}=-J \\left\\{ \\hat{S}^2 - \\frac{3}{4} \\hat{N} + \\frac{\\hat{N} (\\hat{N}-1)}{4} +\\sum_{m} \\hat{n}_{m\\uparrow}\\hat{n}_{m{\\downarrow}} \\right\\}\\,.\n\\end{align}\nfor which we choose a double-counting \nenergy \nof the type\n\\begin{align}\\label{Eq:Edc_ours}\n&{E_{\\rm dc}}^{\\rm Hund}[n({\\bm r})] = \\nonumber \\\\\n&= -J \\sqbra{\\frac{N_{\\up} (N_{\\up}-1)}{2} +\\frac{N_{\\dw}(N_{\\dw}-1)}{2} + \\frac{N_{\\up}N_{\\dw}}{2l+1} }\\,.\n\\end{align}\n\\subsection{Extending LDA+U to LDA+Gutzwiller}\\label{Sect:gutz}\nThe key difference between \\ac{ldag} and \\ac{ldau} resides in the definition of the modified kinetic \nfunctional $T_{\\rm i}$. Within \\ac{ldag}, the definition \\eqn{Eq_HK_modified} changes to\n\\begin{align}\\label{Eq_GW_kinetic}\nT_{\\rm i}[n({\\bm r})] \\rightarrow T_{\\rm G}[n({\\bm r})] = \\min_{{\\Psi_{\\rm G}} \\rightarrow n({\\bm r})} \\expect{{\\Psi_{\\rm G}}}{\\hat{T}+\\hat{H}_{\\rm at}}\\,,\n\\end{align}\nwhere the wavefunction $\\ket{\\Psi_{\\rm G}}$ is defined as\n\\begin{align}\\label{Eq_Gutzwav_def}\n{\\Psi_{\\rm G}} = \\Gpg{} \\ket{\\Psi_{0}} = \\prod_{{\\bm R}} \\Gpl{{\\bm R}} \\ket{\\Psi_{0}}\\,.\n\\end{align}\nIn the above equation, $\\ket{\\Psi_{0}}$ is still a Slater determinant, and the elements of novelty are \nthe operators $\\Gpl{{\\bm R}}$, which are linear transformations acting on the configurational space of \na chosen set of local orbitals at lattice site ${\\bm R}$. As in \\ac{ldau}, this set of orbitals $\\phi_{m,{\\bm R}}$ \nretain well defined atomic angular momentum $l$, $m$ being its projection on a given quantization axis. \nThe operator $\\Gpl{{\\bm R}}$ can be generally written as\n\\begin{align}\\label{Eq_multiband_projector}\n\\Gpl{{\\bm R}} = \\sum_{\\Gamma \\Gamma'} \\Lambda_{\\Gamma\\Gamma',{\\bm R}}\\,\n\\ket{\\Gamma,{\\bm R}}\\bra{\\Gamma',{\\bm R}}\\,,\n\\end{align}\nwhere $\\ket{\\Gamma,{\\bm R}}$ denote many-body configurations of electrons occupying the \norbitals $\\phi_{m,{\\bm R}}$. Differently from \\ac{ldau}, the expectation value of the kinetic plus atomic \ninteraction operators will not depend solely on the Slater determinant $\\ket{\\Psi_{0}}$, but also on the variational \nparameters $\\Lambda_{\\Gamma\\Gamma',{\\bm R}}$ that define $\\Gpl{{\\bm R}}$. \n\nComputing exact expectation values on the Gutzwiller wavefunction for lattices of finite coordination \nis a task that can be accomplished only numerically, e.g. through Variational Quantum Monte Carlo.\\cite{Sorella_SR,Sorella-VMC} \nFor infinite-coordination lattices, an exact expression can be instead computed analytically. There is in fact \na close connection between the Gutzwiller variational approach in the limit of infinite lattice coordination \nand dynamical mean field theory.\\cite{DMFT} In that limit, the single particle self-energy matrix \n$\\Sigma(\\epsilon,{\\bm k}) = \\Sigma(\\epsilon)$ becomes purely local, hence momentum independent. \nDMFT allows to evaluate exactly $\\Sigma(\\epsilon)$ by solving an auxiliary Anderson impurity model \nconstructed in such a way as to have the same self-energy. The Gutzwiller variational approach is instead \na consistent approximation to the exact solution, which assumes a Fermi-liquid expression \n$\\Sigma(\\epsilon) \\simeq \\Sigma(0) + \\left(1-Z^{-1}\\right)\\epsilon$, where $Z$ is commonly refereed \nto as the quasiparticle weight. Because of this assumption, the Gutzwiller wavefunction can describe only states \nwhose elementary excitations are quasiparticles,\nsuch as \nLandau-Fermi liquids and insulators that can be \nrepresented through a Slater determinant. However, the additional freedom brought by the parameter $Z$, \nwhose value is strictly \n$Z = 1$ within Hartree-Fock \nand in \\ac{ldau}, opens the possibility to access \nstrongly correlated metals, $Z\\ll 1$, \nand thus the approach to a \nMott transition, where $Z\\to 0$. \nAlthough DMFT is exact only in the limit of infinite coordination, it is currently used as an approximation \nin realistic finite-coordination lattices, under the hypothesis that (strong) correlation effects beyond Hartree-Fock (HF) are well represented by \n$\\Sigma(\\epsilon,{\\bm k}) \\simeq \\Sigma_{\\text{HF}}({\\bm k}) + \\Sigma(\\epsilon)$, where \n$\\Sigma_{\\text{HF}}({\\bm k}) $ is the HF self-energy, eventually including frequency-dependent \nrandom-phase-like contributions,\\cite{LDA+cRPA+DMFT} and the correction $\\Sigma(\\epsilon)$ is momentum independent and can be obtained by DMFT. Under the same assumptions, one can keep using the \nformal results of the Gutzwiller variational approach, that are strictly valid only in infinite-coordination lattices, also in finite-coordination ones, an approximation refereed to as the {\\sl Gutzwiller approximation} (\\acs{ga}). \nIn other words, the GA should be better regarded as an approximation to DMFT, when either of them are used in finite-coordination lattices, rather than an approximation to the exact evaluation of average values on the Gutzwiller wavefunction, \n\\eqn{Eq_Gutzwav_def}. This viewpoint, which we underwrite, is our motivation for adopting the Gutzwiller \napproximation in combination with \\ac{ldau} as an alternative to \\ac{ldadmft}, \nat the cost of less rigor, but as we shall show with gain in simplicity and flexibility. \n\\subsubsection{Expectation values in the Gutzwiller Approximation}\\label{Sect:Gutz_expect}\nIn order to determine \nthe functional $T_{\\rm G}[n({\\bm r})]$, one should be able to compute expectation values \nof both many-body on-site operators such as those contained in $\\hat{H}_{\\rm at}$, and off-site single-particle \noperators, which are present in the definition of the kinetic operator $\\hat{T}$. In all what follows, \nwe shall use the formalism presented in Ref.~\\onlinecite{BaroneLanata}. \n\nFirst of all, the Slater determinant $\\mid\\Psi_0\\rangle$ defines the uncorrelated one-body local \ndensity-matrix $\\hat{n}^{(0)}_{\\bm R}$ (the same matrix that enters the \\ac{ldau} energy \ncorrection term \\eqn{Eq:LDAU_total_ham}), with elements \n\\begin{equation}\nn^{(0)}_{{\\bm R} m\\sigma,{\\bm R} m'\\sigma'} = \\langle\\Psi_0\\mid \nc^\\dagger_{{\\bm R},m\\sigma} c^{\\phantom{\\dagger}}_{{\\bm R},m'\\sigma'}\\mid\\Psi_0\\rangle, \\label{II.B-n0}\n\\end{equation}\nwhere \n$c^\\dagger_{{\\bm R},m\\sigma}$ \ncreates a spin-$\\sigma$ electron in orbital $\\phi_{m,{\\bm R}}$. \n$\\hat{n}^{(0)}_{\\bm R}$ is diagonalized by a unitary transformation that turns the \noriginal basis of operators \n$c^\\dagger_{{\\bm R}, m\\sigma}$ \ninto the natural basis of operators \n$c^\\dagger_{{\\bm R}, \\gamma\\sigma}$\n, assuming \ninvariance with respect to spin rotations around the $z$-axis. In the natural basis, the one-body \ndensity matrix is therefore diagonal, with eigenvalues $n^{(0)}_{{\\bm R},\\gamma\\sigma}$. \nIn the natural-orbital Fock basis, with states \n\\[\n\\mid \\{n_{{\\bm R},\\gamma\\sigma}\\}\\rangle \n\\equiv \\prod_{\\gamma\\sigma}\\,\\left(c^\\dagger_{{\\bm R},\\gamma\\sigma}\\right)^{n_{{\\bm R},\\gamma\\sigma}}\\,\n\\mid 0\\rangle,\n\\] \nit follows that the probability matrix \n\\begin{eqnarray}\n&&P^{({\\bm R})}_{0,\\{n_{{\\bm R},\\gamma\\sigma}\\}\\{m_{{\\bm R},\\gamma\\sigma}\\}} \\equiv \n\\langle \\Psi_0\\mid \\, \n\\mid \\{m_{{\\bm R},\\gamma\\sigma}\\}\\rangle\\langle \\{n_{{\\bm R},\\gamma\\sigma}\\}\\mid\\, \n\\mid \\Psi_0\\rangle \\nonumber\\\\\n&&~~~~~~~~~=\nP^{({\\bm R})}_{0,\\{n_{{\\bm R},\\gamma\\sigma}\\}}\\,\\delta_{\\{n_{{\\bm R},\\gamma\\sigma}\\}\n\\{m_{{\\bm R},\\gamma\\sigma}\\}} \\nonumber\\\\\n&& ~~~~~~~~=\\prod_{\\gamma\\sigma} \\left(n^{(0)}_{{\\bm R},\\gamma\\sigma}\\right)^{n_{{\\bm R},\\gamma\\sigma}}\\,\n\\left(1-n^{(0)}_{{\\bm R},\\gamma\\sigma}\\right)^{1-n_{{\\bm R},\\gamma\\sigma}}\\,, \\label{Eq_pizero_explicit}\n\\end{eqnarray}\nis diagonal, too. It is actually convenient\\cite{BaroneLanata} to rewrite the operator \\eqn{Eq_multiband_projector} in a mixed basis representation\nas \n\\begin{align}\\label{IIB.Eq_multiband_projector}\n\\Gpl{{\\bm R}} = \\sum_{\\Gamma \\{n_{{\\bm R},\\gamma\\sigma}\\}} \\left(\n\\fract{\\Phi_{\\Gamma \\{n_{{\\bm R},\\gamma\\sigma}\\},{\\bm R}} }{P^{({\\bm R})}_{0,\\{n_{{\\bm R},\\gamma\\sigma}\\}}}\\right)\n\\ket{\\Gamma,{\\bm R}}\\bra{\\{n_{{\\bm R},\\gamma\\sigma}\\}}\\,,\n\\end{align}\nwhere $\\mid \\Gamma,{\\bm R}\\rangle$ is a state, e.g. a Fock state, \nin the original basis, whereas $\\mid \\{n_{{\\bm R},\\gamma\\sigma}\\}\\rangle$ is a Fock state in the natural basis. This mixed representation \nsimplifies considerably the calculations. \nIn order to use the Gutzwiller approximation, we \nneed to impose the two following \nconstraints on the matrix $\\hat{\\Phi}_{\\bm R}$ with elements $\\Phi_{\\Gamma \\{n_{{\\bm R},\\gamma\\sigma}\\},{\\bm R}}$:\\cite{BaroneLanata}\n\\begin{align}\n\\Tr{\\hat{\\Phi}^\\dagger_{{\\bm R}}\\hat{\\Phi}^{\\phantom{\\dagger}}_{{\\bm R}}} &= 1\\,, \\label{Eq_Gw2_constr1}\\\\\n\\Tr{\\hat{\\Phi}^\\dagger_{{\\bm R}}\\hat{\\Phi}^{\\phantom{\\dagger}}_{{\\bm R}}\\hat{c}^{\\dagger}_{{\\bm R}, \\gamma\\sigma}\\oc_{{\\bm R}, \\gamma'\\sigma'}} &= \nn^{(0)}_{{\\bm R},\\gamma\\sigma}\\delta_{\\gamma\\gamma'}\\,\\delta_{\\sigma\\sigma'}\\,,\\label{Eq_Gw2_constr2}\n\\end{align}\nwhere $\\hat{c}^{\\dagger}_{{\\bm R}, \\gamma\\sigma}$ is the matrix representation of the Fermi operator in its Fock basis. \nIf these constraints are fulfilled, then within the Gutzwiller approximation, which we recall is exact for infinite-coordination lattices,\nwe have\n\\begin{align}\\label{Eq:renorm_onsite}\n\\expect{{\\Psi_{\\rm G}}}{\\hat{O}_{{\\bm R}}}= \\Tr{\\hat{\\Phi}^\\dagger_{{\\bm R}}\\hat{O}_{{\\bm R}}\\hat{\\Phi}^{\\phantom{\\dagger}}_{{\\bm R}}}\\,,\n\\end{align}\nwhere $\\hat{O}_{{\\bm R}}$ is the matrix representation of any local operator. \nThe inter-site density matrix can be computed from\n\\begin{align}\\label{Eq:renorm_offsite}\n\\expect{{\\Psi_{\\rm G}}}{\\hat{c}^{\\dagger}_{{\\bm R},m\\sigma}\\oc_{{\\bm R},m'\\sigma}} = \n&\\sum_{\\gamma\\gamma'}\\, R_{\\gamma m;\\sigma,{\\bm R}}^\\dagger \\,R_{m'\\gamma';\\sigma,{\\bm R'}}^{\\phantom{\\dagger}}\\nonumber\\\\\n& \\expect{\\Psi_{0}}{c^\\dagger_{{\\bm R},\\gamma\\sigma'}c^{\\phantom{\\dagger}}_{{\\bm R'},\\gamma'\\sigma'}} \\,,\n\\end{align}\nwhere\n\\begin{align}\\label{Eq_Rparam2} \nR^{\\dagger}_{\\gamma m,\\sigma,{\\bm R}} = \\fract{\\Tr{\\hat{\\Phi}_{\\bm R}^\\dagger\\,\\hat{c}^{\\dagger}_{{\\bm R}, m \\sigma}\\,\n\\hat{\\Phi}_{\\bm R}^{\\phantom{\\dagger}}\\,\\oc_{{\\bm R}, \\gamma \\sigma}}}{\\sqrt{n^{(0)}_{{\\bm R},\\gamma\\sigma} (1-n^{(0)}_{{\\bm R},\\gamma\\sigma})}} \\,,\n\\end{align}\ncan be regarded as a wavefunction renormalization matrix. \nHere $\\hat{c}^{\\dagger}_{{\\bm R}, m \\sigma}$ is the matrix representation of the original operators in the basis of states \n$\\mid \\Gamma,{\\bm R}\\rangle$. When this is the Fock basis constructed by the same original operators, \ntheir matrix representation is actually independent of the basis of single-particle wavefunctions which \nthey refer to, hence it is the same as for the $\\hat{c}^{\\dagger}_{{\\bm R}, \\gamma \\sigma}$ operators of the natural basis. \nIn reality, in most cases that are relevant for real materials the natural basis that diagonalizes the local density \nmatrix is determined fully by the lattice symmetry, hence it is possible and convenient to write the Hamiltonian \ndirectly in that basis. In the above formulas, this corresponds to identifying the set of labels \n$\\{m\\}$ with $\\{\\gamma\\}$. Since the natural basis is such both for the uncorrelated on-site density matrix \n\\begin{align}\nn^{(0)}_{{\\bm R} m\\sigma,m'\\sigma'} &= \\langle\\Psi_0\\mid \n\\hat{c}^{\\dagger}_{{\\bm R},m\\sigma} \\oc_{{\\bm R},m'\\sigma'}\\mid\\Psi_0\\rangle \\nonumber\\\\\n&= \\Tr{\\hat{\\Phi}_{\\bm R}^\\dagger\\, \\hat{\\Phi}_{\\bm R}^{\\phantom{\\dagger}}\\,\\hat{c}^{\\dagger}_{{\\bm R}, m \\sigma}\\oc_{{\\bm R}, m' \\sigma}\n} \\nonumber\\\\\n&= \\delta_{mm'}\\, n^{(0)}_{{\\bm R}, m\\sigma},\\label{II.B-density-matrix-var}\n\\end{align}\nand for the correlated one\n\\begin{align}\nn_{{\\bm R} m\\sigma,m'\\sigma'} &= \\langle\\Psi_{\\text{G}}\\mid \n\\hat{c}^{\\dagger}_{{\\bm R},m\\sigma} \\oc_{{\\bm R},m'\\sigma'}\\mid{\\Psi_{\\rm G}}\\rangle \\nonumber \\\\\n&= \\Tr{\\hat{\\Phi}_{\\bm R}^\\dagger\\, \\hat{c}^{\\dagger}_{{\\bm R}, m \\sigma}\\oc_{{\\bm R}, m' \\sigma}\\, \n\\hat{\\Phi}_{\\bm R}^{\\phantom{\\dagger}}} \\nonumber\\\\\n&=\n\\delta_{mm'}\\, n_{{\\bm R}, m\\sigma},\\label{II.B-density-matrix-true}\n\\end{align}\ngenerally with different eigenvalues, it is not difficult to realize that the wavefunction renormalization \nmatrix \\eqn{Eq_Rparam2} becomes diagonal, i.e. \n\\begin{align}\\label{Eq_Rparam2-1} \nR^{\\dagger}_{m' m,\\sigma,{\\bm R}} = \\fract{\\Tr{\\hat{\\Phi}_{\\bm R}^\\dagger\\,\\hat{c}^{\\dagger}_{{\\bm R}, m \\sigma}\\,\n\\hat{\\Phi}_{\\bm R}^{\\phantom{\\dagger}}\\,\\oc_{{\\bm R}, m' \\sigma}}}{\\sqrt{n^{(0)}_{{\\bm R},m'\\sigma} (1-n^{(0)}_{{\\bm R},m'\\sigma})}} \n= \\delta_{mm'}\\,R_{m\\sigma,{\\bm R}}^\\dagger\\,.\n\\end{align} \n\nThe Eqs.~(\\ref{Eq:renorm_onsite})--(\\ref{Eq_Rparam2-1}) are the basic formulas that allow to \nevaluate the average value of the Hamiltonian as a functional of the Slater determinant and of the matrices \n$\\hat{\\Phi}_{\\bm R}$, hence to solve the variational problem.\n\\section{The Gutzwiller functional in practice}\\label{Sec:Gutz_practice}\nIn this section we show how to perform a density-self-consistent \\ac{ldag} calculation on a realistic system, \nnamely\n$bcc$ Fe which,\nas mentioned in the Introduction, although a\nbasic and supposedly simple system, still exhibits controversial aspects. \n\nWe first have to select the {\\sl correlated} orbitals to be treated by the Gutzwiller operator. In the present \ncase the choice is simple: the $3d$ orbitals of Fe.\nThis case \nis one of those mentioned earlier \nin which the natural basis is determined by symmetry and corresponds to the cubic crystal field split \n$d$ orbitals, namely the $e_g$ doublet and the $t_{2g}$ triplet. In this representation the \nformulas Eqs.~(\\ref{II.B-density-matrix-var})--(\\ref{Eq_Rparam2-1}) hold, which is a great simplification. \nFurthermore, since $bcc$ is a Bravais lattice, the positions ${\\bm R}$ of Fe atoms also label unit cells, hence \nby translational symmetry we can safely assume that the variational matrix parameters $\\hat{\\Phi}_{\\bm R} \n= \\hat{\\Phi}$ are independent of ${\\bm R}$. So are therefore the eigenvalues of the local density matrices, \n$n^{(0)}_{{\\bm R},m\\sigma} = n^{(0)}_{m\\sigma}$ and $n_{{\\bm R},m\\sigma}=n_{m\\sigma}$, as well as the wavefunction \nrenormalization $R_{m\\sigma,{\\bm R}}=R_{m\\sigma}$. To lighten notations, in what follows the orbital labels \n$m$ will refer both to the correlated set and to the uncorrelated ones, unaffected by the action of the Gutzwiller operator. In the last paragraph of this section we shall come back to this point. \n\nWe define the Gutzwiller density functional as\n\\begin{align}\\label{Eq:F_functional-1}\n&{\\cal F}[n({\\bm r})] = \\min_{{\\Psi_{\\rm G}} \\rightarrow n({\\bm r})} {\\cal E}[{\\Psi_{\\rm G}},n({\\bm r})]\\,.\n\\end{align}\nwhere the quantity ${\\cal E}[{\\Psi_{\\rm G}},n({\\bm r})]$ undergoing constrained minimization is\n\\begin{align}\\label{Eq:E_functional}\n&{\\cal E}[{\\Psi_{\\rm G}},n({\\bm r})] = \\expect{{\\Psi_{\\rm G}}}{\\hat{T}+\\hat{H}_{\\rm int}}+\\nonumber\\\\&+\\int V_{\\rm ext}({\\bm r}) n({\\bm r}) \\de{{\\bm r}}+\\tilde{E}_{\\rm H}[n({\\bm r})] + \\tilde{E}_{\\rm xc}[n({\\bm r})] - {E_{\\rm dc}}[n({\\bm r})]\\,.\n\\end{align}\nFor our purposes, it is convenient to rewrite \\eqn{Eq:F_functional-1} as a minimization constrained \nwith respect to the ``uncorrelated'' density $n^{(0)}({\\bm r})$,\n\\begin{align}\\label{Eq:F_functional}\n&{\\cal F}[n^{(0)}({\\bm r})] = \\min_{\\Gpl{},\\Psi_{0} \\rightarrow n^{(0)}({\\bm r})} {\\cal E}[\\Psi_{0},\\Gpl{},n^{(0)}({\\bm r})]\\,,\n\\end{align}\nwhere ${\\cal E}[\\Psi_{0},\\Gpl{},n^{(0)}({\\bm r})] = {\\cal E}[{\\Psi_{\\rm G}}(\\Psi_{0},\\Gpl{}),n(\\Psi_{0},\\Gpl{})]$.\nThe dependence of the ``correlated'' density $n({\\bm r})$ upon the ``uncorrelated'' density $n^{(0)}({\\bm r})$ can \nbe made explicit once one writes them in terms of the one-body ``correlated'' density-matrix of the periodic system\n\\begin{align}\\label{Eq:1b_corr_densmat}\n{D}_{mm',\\sigma,{\\bm R}} = \\langle \\Psi_{\\text{G}}\\mid \\hat{c}^{\\dagger}_{{\\bm R},m\\sigma}\n\\oc_{\\bm 0,m'\\sigma}\\mid\\Psi_{\\text{G}}\\rangle,\n\\end{align}\nand of the ``uncorrelated'' density-matrix \n\\begin{align}\\label{Eq:1b_uncorr_densmat}\nD^{(0)}_{mm',\\sigma,{\\bm R}}= \\langle \\Psi_{0}\\mid \\hat{c}^{\\dagger}_{{\\bm R},m\\sigma}\n\\oc_{\\bm 0,m'\\sigma}\\mid\\Psi_{0}\\rangle,\n\\end{align}\nnamely\n\\begin{align}\nn^{(0)}({\\bm r}) &= \\sum_\\sigma \\, n^{(0)}_\\sigma({\\bm r}) \\nonumber\\\\\n&= \\sum_{m,m',\\sigma, {\\bm R}} D^{(0)}_{mm',\\sigma,{\\bm R}}\\, \n\\phi^\\ast_{m,{\\bm R}}({\\bm r}) \\,\n\\phi_{m',\\bm 0}(\\bm{r})\\, , \\label{Eq_dens_uncorr}\\\\\nn(\\bm{r}) &= \\sum_\\sigma \\, n_\\sigma({\\bm r}) \\nonumber\\\\\n&= \n\\sum_{m,m',\\sigma, \\bm R} {D}_{mm',\\sigma,{\\bm R}} \\, \\phi^\\ast_{m,{\\bm R}}({\\bm r}) \\,\n\\phi_{m',\\bm 0}(\\bm{r})\\, .\n\\label{Eq_dens_corr}\n\\end{align}\nIndeed, ${D}_{mm',\\sigma,{\\bm R}}$ can be obtained by $D^{(0)}_{mm',\\sigma,{\\bm R}}$ using the recipe of the Gutzwiller Approximation:\n\\begin{align}\\label{Eq_rendensmat}\n{D}_{mm',\\sigma,{\\bm R}} = \\begin{cases}\nR_{m\\sigma}^\\dagger\\, \nD^{(0)}_{mm',\\sigma,{\\bm R}} \\,R_{m'\\sigma}\\,, & \\hspace{-0.3cm}{\\bm R}\\neq \\bm 0\\,,\\vspace{0.3cm}\\\\\n\\Tr{\\hat{\\Phi}^\\dagger \\,\\hat{n}_{mm',\\sigma}\\,\\hat{\\Phi}^{\\phantom{\\dagger}}\\hspace{-0.2cm}}= \n\\delta_{mm'} n_{m\\sigma}\\,,\n& \\hspace{-0.3cm}{\\bm R} = {\\bm 0}\\,,\n\\end{cases}\n\\end{align}\nwhere $\\hat{n}_{mm',\\sigma}$ is the matrix representation on the local Fock space at site ${\\bm R}$ of \n$\\hat{c}^{\\dagger}_{{\\bm R},m\\sigma}\\hat{c}^{\\dagger}_{{\\bm R},m'\\sigma}$, which is independent of ${\\bm R}$ for a periodic system, \nand where $n_{m\\sigma}$ is equal to $n_{{\\bm R}=0,m\\sigma}$ defined in \\eqn{II.B-density-matrix-true}.\n\nIn order to write ${\\cal E}[\\Psi_{0},\\Gpl{},n^{(0)}({\\bm r})]$ explicitly in terms of the new variables, we start from \nthe first and second terms of \\eqn{Eq:E_functional}.\nWe can now treat the kinetic and the external potential terms on the same footing through \n\\begin{align}\\label{Eq_kinextgutz_trace}\n& \\expect{{\\Psi_{\\rm G}}}{\\hat{T}} + \\int n({\\bm r}) V_{\\rm ext}({\\bm r}) d{\\bm r} = \\nonumber\\\\\n& \\sum_{m,m',\\sigma,{\\bm R}} \\Big(T_{mm',{\\bm R}} + V^{(\\rm ext)}_{mm',{\\bm R}}\\Big)\n\\,{D}_{mm',\\sigma,{\\bm R}}\\,,\n\\end{align}\nwhere values of $T_{mm',{\\bm R}}$ and $V^{(\\rm ext)}_{mm',{\\bm R}}$ are the spin-independent matrix elements of \nthe kinetic and external potential operators computed between our basis orbitals at sites ${\\bm R}$ and $\\bm 0$, \ni.e. \n\\begin{align}\nV^{(\\rm ext)}_{mm',{\\bm R}} = \\int \\phi^\\ast_{m,{\\bm R}}({\\bm r}) V_{\\rm ext}({\\bm r}) \n\\phi_{m',\\bm 0}({\\bm r})d {\\bm r}\\,,\\\\\nT_{mm',{\\bm R}} = -\\frac{\\hbar^2}{2m} \\int \\phi^\\ast_{m,{\\bm R}}({\\bm r}) \n\\Big[\\nabla^2 \\phi_{m',\\bm 0}({\\bm r})\\Big] d {\\bm r}\\,\n\\end{align}\nand compute the value of the atomic interaction energy $\\expect{{\\Psi_{\\rm G}}}{\\hat{H}_{\\rm at}}$ using the Gutzwiller Approximation recipe\n\\begin{align}\\label{Eq:eat}\nE_{\\rm at}[\\Psi_{0},\\Gpl{}] = \\expect{{\\Psi_{\\rm G}}}{\\hat{H}_{\\rm at}} = \\Tr{\\hat{\\Phi}^\\dagger \\hat{H}_{\\rm at} \\hat{\\Phi}^{\\phantom{\\dagger}}}\n\\end{align}\nIn order to simplify the density self-consistent \\ac{ldag} minimization we decided to use the\nHartree $\\tilde{E}_{\\text{H}}[n({\\bm r})]$ and exchange-correlation $\\tilde{E}_{\\text{xc}}[n({\\bm r})]$ \nfunctionals as the \\ac{lda} functionals linearized around the uncorrelated density $n^{(0)}({\\bm r})$. \nWe checked {\\sl a posteriori} the accuracy of such a linearization. \nThe modified Hartree functional then reads\n\\begin{align}\\label{Eq:Hartree1}\n\\tilde{E}_{\\rm H}\\left[n^{(0)}({\\bm r}),n({\\bm r})\\right] &\\simeq \\frac{e^2}{2}\\int d{\\bm r} d{\\bm r'}\\; \\fract{n^{(0)}({\\bm r}) n^{(0)}({\\bm r'})}{|{\\bm r} - {\\bm r'}|}\n\\nonumber \\\\\n&~~~+ \\int d{\\bm r} \\,\\delta n({\\bm r})\\,v_{\\rm H}[n^{(0)}({\\bm r})]\\,,\n\\end{align}\nwhere $\\delta n({\\bm r}) = \\sum_\\sigma \\delta n_\\sigma({\\bm r})=\\sum_\\sigma n_\\sigma({\\bm r})-n^{(0)}_\\sigma({\\bm r})$ \nand $v_{\\rm H}[n^{(0)}({\\bm r})]$ is the conventional Hartree potential, whereas the exchange-correlation functional is \n\\begin{align}\\label{Eq:exc1}\n\\tilde{E}_{\\rm xc}\\left[n^{(0)}({\\bm r}),n({\\bm r})\\right] &= \\sum_\\sigma\\, \\int \\de{{\\bm r}} n^{(0)}_\\sigma({\\bm r})\\, \n\\epsilon_{\\text{xc},\\sigma}[n^{(0)}({\\bm r})] \n\\nonumber\\\\\n&~~+ \\int d{\\bm r} \\, v_{\\text{xc},\\sigma}[n^{(0)}({\\bm r})] \\, \\delta n_\\sigma({\\bm r})\\,,\n\\end{align}\n$v_{\\rm xc}[n^{(0)}({\\bm r})]$ being the \\ac{lda} exchange-correlation potential.\nNote that the choice of $\\tilde{E}_{\\rm H}$ involves neglecting a term\n\\begin{align}\n\\DeltaE_{\\rm H}\\left[n^{(0)}({\\bm r}),n({\\bm r})\\right] &= \\tilde{E}_{\\rm H}\\left[n^{(0)}({\\bm r}),n({\\bm r})\\right]-E_{\\rm H}[n({\\bm r})] =\\nonumber\\\\\n&=\\frac{e^2}{2}\\int d{\\bm r} d{\\bm r'}\\, \\fract{\\delta n({\\bm r}) \\delta n({\\bm r'})}{|{\\bm r} - {\\bm r'}|}\n\\end{align}\nwhich can be interpreted as the energy of correlation-induced charge fluctuations. This term, together with \nthe corresponding one neglected for the exchange-correlation functional, $\\DeltaE_{\\rm xc}\\left[n^{(0)}({\\bm r}),n({\\bm r})\\right]$, \ncan be computed at the end of the \\ac{ldag} calculation in order to provide an estimate of the error due to \napproximations~\\eqref{Eq:Hartree1} and~\\eqref{Eq:exc1} (see \\rtab{Tab:delta_energies}).\nIt is worth mentioning that the linearization~\\eqref{Eq:exc1} of exchange-correlation energy around \nthe ``uncorrelated'' density does not spoil the sum rule for the \\ac{lda} exchange-correlation hole.\nAs for the double-counting term, similarly to what is done within \\ac{ldau}, it is chosen as a function \nof the local ``uncorrelated'' density-matrix $n^{(0)}$ only, ${E_{\\rm dc}}[n({\\bm r})] = {E_{\\rm dc}}[n^{(0)}]$. In \\sect{Sec:results} we take\nas its explicit form the one of \\eqn{double-counting-Hund}, having chosen our atomic interaction Hamiltonian $\\hat{H}_{\\rm at}$\nto be the expression of \\eqn{Eq:hHat_approssimata}.\n\n\\subsection{Three-step minimization of the LDA+Gutzwiller functional}\\label{Sec:threestep_mini}\n\nThe two densities $n({\\bm r})$ and $n^{(0)}({\\bm r})$ must be such that Gutzwiller constraints are fulfilled. \nIn our case where original and natural basis coincide, the constraints on the density matrix can be written as\n\\begin{align}\nD^{(0)}_{mm', \\sigma,{\\bm R}= \\bm 0} &= n^{(0)}_{m\\sigma}\\,\\delta_{mm'}\\,,\\label{Eq_df2_gwconst1}\\\\\n\\Tr{\\hat{\\Phi}^\\dagger\\hat{\\Phi}^{\\phantom{\\dagger}} \\hat{n}_{mm',\\sigma}} &= n^{(0)}_{m\\sigma}\\,\\delta_{mm'}\\,,\\label{Eq_df2_gwconst2}\n\\end{align}\nwhere we regard $n^{(0)}_{m\\sigma}$ as an additional independent variational parameter of the density \nfunctional. These constraints can be enforced with Lagrange multipliers, together with the first Gutzwiller constraint \n\\begin{align}\n\\Tr{\\hat{\\Phi}^\\dagger\\hat{\\Phi}^{\\phantom{\\dagger}}} &= 1\\,.\\label{Eq_df_gwconst}\n\\end{align}\n\nSumming up all contributions and adding the electrostatic ion-ion interaction $E_{\\rm ion}$, \nwe find that the overall functional we need to minimize has the form\n\\begin{widetext} \n\\begin{align}\\label{Eq_Gwdensfunc_full}\n{\\cal F}\\Big[n({\\bm r}),n^{(0)}({\\bm r}),n^{(0)}_{m\\sigma}\\Big] &= \\max_{\\lambda\\lambda'\\lambda_0} \n\\Bigg[{\\cal K}[n({\\bm r})] +E_{\\rm at}[n({\\bm r})] -{E_{\\rm dc}}[n^{(0)}_{m\\sigma}]+E^{(0)}_{\\rm H}[n^{(0)}({\\bm r})] + E^{(0)}_{\\rm xc}[n^{(0)}({\\bm r})]\n- \\lambda_0 \\left(\\Tr{\\hat{\\Phi}^\\dagger\\hat{\\Phi}^{\\phantom{\\dagger}}} - 1\\right)\\nonumber \\\\\n& -\\sum_{mm'\\sigma}\\, \n\\lambda'_{mm',\\sigma}\\left(D^{(0)}_{mm',\\sigma, {\\bm R}=\\bm 0}-n^{(0)}_{m\\sigma}\\delta_{mm'}\\right) \n- \\lambda_{mm',\\sigma} \\left(\\Tr{\\hat{\\Phi}^\\dagger\\hat{\\Phi}^{\\phantom{\\dagger}} \\hat{n}_{mm',\\sigma}}\n-n^{(0)}_{m\\sigma}\\delta_{mm'}\\right)\n\\Bigg] +E_{\\rm ion},\n\\end{align}\n\\end{widetext}\nwhere the functional ${\\cal K}[n({\\bm r})]$ contains all terms which depend on $n({\\bm r})$ linearly through \nthe renormalized density matrix ${D}$, namely\n\\begin{align}\n{\\cal K}({D}) &= \\sum_{mm',\\sigma,{\\bm R}} \\bigg[T_{mm',{\\bm R}}+V^{(\\rm H)}_{mm',{\\bm R}}\n+\nV^{(\\rm xc)}_{mm',\\sigma,{\\bm R}}\\nonumber\\\\\n&~~~~~~~~~~~~~~~~~~~~~~~~~ +V^{(\\rm ext)}_{mm',{\\bm R}}\\bigg]\\, {D}_{mm',\\sigma,{\\bm R}}\\nonumber\\\\\n&\\equiv \\sum_{mm',\\sigma,{\\bm R}} \\, {\\cal K}_{mm',\\sigma,{\\bm R}}\\, {D}_{mm',\\sigma,{\\bm R}}\\,,\n\\end{align}\nwhere $V^{(\\rm H)}_{mm',{\\bm R}}$ and $V^{(\\rm xc)}_{mm',\\sigma,{\\bm R}}$ are the matrix elements of $v_{\\rm H}$ and $v_{\\rm xc}$ \nbetween the basis orbitals. \nFor every fixed value of $n^{(0)}_{m\\sigma}$, we can optimize ${\\cal F}$ with respect to the two \ndensities $n^{(0)}({\\bm r})$ and $n({\\bm r})$. In practice, by inspection of equations~\\eqref{Eq_dens_uncorr}, \n\\eqref{Eq_dens_corr} and \\eqref{Eq_rendensmat} one can see that this is equivalent to a minimization \nwith respect to the Slater determinant $\\ket{\\Psi_{0}}$ and the Gutzwiller parameters\\ contained in the operator $\\hat{\\Phi}^{\\phantom{\\dagger}}$.\nThis minimization can be carried out in two separate steps:\n\\begin{enumerate}\n\\item first carry out a Siesta self-consistent calculation to find the Slater determinant $\\Psi_{0}$ \nthat optimizes ${\\cal F}[n({\\bm r}),n^{(0)}({\\bm r}), n^{(0)}_{m\\sigma}]$ with respect to $n^{(0)}({\\bm r})$, enforcing the \nconstraint~\\eqref{Eq_df2_gwconst1} through an Augmented Lagrangian Method~\\cite{Fletcher}.\nThe Gutzwiller parameters, and therefore the hopping renormalization parameters $R_{m\\sigma}$, are kept fixed throughout \nthis optimization. The atomic energy $E_{\\rm at}[n({\\bm r})]$ does not change, nor does the double-counting energy \n${E_{\\rm dc}}[n^{(0)}({\\bm r})]$, which is a function of $n^{(0)}({\\bm r})$ only through $n^{(0)}_{m\\sigma}$.\nThe self-consistent single-particle Kohn-Sham equations allowing the minimization with respect to $\\ket{\\Psi_{0}}$ are\n\\begin{align}\n\\sum_{m'{\\bm R}}\\,\\mathcal{H}_{mm',\\sigma,{\\bm R}}\\,\\psi_{m'\\sigma,{\\bm R}} &= \\varepsilon\\, \\psi_{m,\\sigma,\\bm0}\\,,\n\\end{align}\nwhere\n\\[\n\\mathcal{H}_{mm',\\sigma,{\\bm R}} = {\\cal K}_{mm',\\sigma,{\\bm R}}+V^{(\\rm 0)}_{mm',\\sigma,{\\bm R}} - \\lambda'_{mm',\\sigma}\n\\delta_{{\\bm R}\\bm 0},\n\\]\nand \n\\begin{align}\nV^{(\\rm 0)}_{mm',\\sigma,{\\bm R}} &= \\int \\de{{\\bm r}} \\phi^\\ast_{m,{\\bm R}}({\\bm r})\\Big\\{v_{\\rm H}[n^{(0)}({\\bm r})] \\nonumber\\\\\n&~~~~~~~~~+ v_{\\rm xc}[n^{(0)}({\\bm r})]\\Big\\}\\phi_{m',\\bm 0}({\\bm r})\\,.\n\\end{align}\n\\item next, optimize ${\\cal F}$ with respect to Gutzwiller parameters\\ by a Lanczos-improved \\ac{lm} algorithm (see ","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\subsubsection*{Introduction}\n\nVERITAS, located at the Fred Lawrence Whipple Observatory in southern\nArizona, is one of three major imaging atmospheric Cherenkov telescope\nfacilities in operation worldwide. It consists of an array of four,\n12-meter diameter telescopes, providing a $<1\\%$ Crab-flux sensitivity in\nthe energy range between $100\\U{GeV}$ and $50\\U{TeV}$. The array\nhas been operating since 2007 and has detected 42 objects from\n$\\sim10$ different source classes, including many new discoveries\n\\cite{veritas_status}.\n\nIn 2012, the VERITAS collaboration established a long-term plan\ndescribing a scientific strategy for operations. Many of the goals\noutlined address problems relevant to the charge of the CF6\nsubgroup. An additional white paper describing the VERITAS indirect\ndark matter detection program has been submitted to subgroup CF2\n\\cite{andyWP}. A prerequisite for achieving these goals was the\nsuccess of a major upgrade to the array which was completed, with no\ndisruption to the array operations, in the summer of 2012. The upgrade\ninvolved the installation of a new trigger system, and the replacement\nof all of the photodetectors with super-bialkali photomultiplier\ntubes. This has resulted in an increase of at least 35\\% in photon\ncollection efficiency. VERITAS will remain the premier VHE facility in\nthe Northern Hemisphere for some time, while the next-generation\nCherenkov Telescope Array (CTA) project is under development. The\ncontemporaneous overlap of VERITAS operations with Fermi-LAT, HAWC,\nIceCube and Auger will be of critical importance to many of the\nscience goals described here.\n\n\\subsubsection*{Cosmic Particle Acceleration as Signal and Background}\n\nVERITAS has made significant contributions to the study of Galactic\nparticle accelerators, including pulsars and their nebulae, gamma-ray\nbinary systems and supernova remnants. Highlight results include the\ndetection of $>100\\U{GeV}$ emission from the Crab pulsar\n\\cite{crabpulsar}, and the first TeV detection of Tycho's SNR\n\\cite{tycho}. The Crab pulsar result can only be easily explained by a\nnew emission mechanism, or an additional component at high energies,\nThe Tycho detection, combined with results from the Fermi-LAT,\nprovides compelling evidence for hadronic particle acceleration in\nSNR. Complementary evidence for a link between cosmic ray production\nand star formation activity was provided by the discovery of\ngamma-ray emission from the starburst galaxy M82 \\cite{m82}. \n\nHAWC \\cite{HAWC} will soon begin observations, and will provide a complete\nTeV map of the northern sky. Follow-up observations with high\nsensitivity and better angular and energy resolution will be performed\nby VERITAS. In particular, this is key to determining the nature of\nunidentified 'dark accelerators', as already demonstrated by VERITAS\nobservations of Milagro sources. The resolution of gamma-ray emission\nin the region of MGRO~J2019+37 and the Cygnus OB1 association into at\nleast two distinct sources, one clearly associated with the pulsar\nwind nebula CTB~87, demonstrates the importance of the excellent\nangular resolution provided by the imaging technique. Contemporaneous\noperation of VERITAS and HAWC will also allow a rapid response to\ntransient events, such as blazar flares and gamma-ray bursts.\n\nThe study of VHE gamma-ray emission from particle accelerators is, of\ncourse, interesting from a purely astrophysical perspective. It is\nalso critical to understand the nature and properties of astrophysical\nbackgrounds in searches for new physical effects (this issue is\naddressed in detail in a separate CF6 white paper\n\\cite{amandaWP}). The interpretation of indirect dark matter searches,\nLorentz Invariance Violation (LIV) tests, studies of gamma-ray and\nantimatter backgrounds and searches for axion-like particles all rely\non an accurate knowledge of the potential astrophysical backgrounds\nand their spectral, morphological and temporal properties. A classic\nexample of this is the case of the Galactic Center, which has the\nhighest local concentration of dark matter, but also hosts multiple\nknown and potential astrophysical TeV sources, both point-like and\nextended \\cite{andyWP}. VERITAS observations of the Galactic Center\nare ongoing as part of our long-term observing plan. For this southern\nsource, the observations take place at low elevation angles, resulting\nin a high energy threshold but providing an increase in the effective\ncollection area at high energies. This allows us to probe the end\npoint of the spectrum of the Galactic Center gamma-ray emission, which\nmay hold the key to resolving the nature of the source.\n\n\n\\subsubsection*{Probing Fundamental Physics}\n\nVERITAS is a mature experiment, and has moved beyond the initial\nsource discovery phase. Fundamental physics topics now play an\nincreasingly important role in the observing plan. The success of\nthese studies, which often require long and technically challenging\nexposures, relies on stable operation and a thorough knowledge of\nthe detector performance, calibration and associated Monte Carlo\nsimulations. After five years of operations, all of these aspects of\nVERITAS are very well understood. Indirect dark matter searches are\ndescribed elsewhere \\cite{andyWP}. Other topics which we plan to\ninvestigate with VERITAS in the coming years include:\n\n\\begin{itemize}\n\\item{\\bf Antimatter studies:} The rising positron fraction identified\n by PAMELA \\cite{PAMELA} and confirmed by Fermi \\cite{fermipositron,\n fermipositron2} up to a few hundred GeV is an intriguing result. It may be explained by a contribution from local astrophysical\n sources, or possibly by annihilating dark matter. First results from\n AMS-02 confirm that the positron fraction continues to rise up to at\n least $250\\U{GeV}$, at which point it appears to flatten \\cite{AMS}. A\n measurement of the positron fraction at higher energies would\n provide a key discriminant between the competing\n explanations. VERITAS is attempting to make such a measurement by\n observing the shadow of the Moon in both electrons and positrons, as\n proposed by Colin \\cite{colin}. This is technically challenging, due\n to the optical sky brightness close to the Moon, and the limited\n amount of observing time available at high elevations. We have\n developed short-wavelength optical filter plates for the telescope cameras\n to enable us to observe close to the Moon, and the results of\n preliminary test observations are encouraging. Observations over the\n next few years should allow us to build up the necessary exposure\n required for this unique measurement.\n\n\\item{\\bf Primordial Black Holes:} In addition, or as an alternative\n to, particle dark matter, primordial black holes (PBHs) formed\n during the early universe can serve as a viable candidate for\n cosmological dark matter (see \\cite{PBH}). PBHs can evaporate\n through Hawking radiation, where the evaporation rate is directly\n coupled to their mass. Consequently, during the final seconds of\n their lifetime, PBHs can release a large flux of gamma rays within\n the sensitivity range of VERITAS. Dedicated searches for these PBH\n signals have already commenced with VERITAS \\cite{gordana}, and an\n evaporation rate limit of\n $\\rho_{PBH}<1.29\\times10^5\\UU{pc}{-3}\\UU{yr}{-1}$ has been placed\n using only 700 hours of VERITAS observations. This limit is already\n an order of magnitude below previous limits. VERITAS accrues\n approximately 800 hours of Moonless observations each year, so a\n significant refinement of the result can be expected.\n\n\\item{\\bf Cosmological measurements using the EBL and IGMF:} The\n gamma-ray spectra of blazars are modified by interactions with\n intergalactic radiation fields through pair-production and\n subsequent cascade processes. As a result, these spectra contain an\n imprint of the extragalactic background light (EBL) and the\n intergalactic magnetic field (IGMF). The EBL comprises the combined\n flux of all extragalactic sources integrated over the history of the\n Universe, and carries unique information regarding the epoch of\n galaxy formation and the history of galaxy evolution. This topic is\n discussed in detail in a related white paper \\cite{frankWP}. The\n IGMF strength is only weakly constrained, and impossible to measure\n directly. VERITAS observations of the spectra, angular distribution\n and arrival times of gamma-rays from distant blazars will provide\n constraints to, or a measurement of, the IGMF strength which is not\n accesible to other techniques. A positive measurement would be\n important, possibly implying the existence of a primordial\n field produced in the early Universe. Both EBL and IGMF measurements\n require deep, multi-year exposures of numerous blazars over a range\n of redshifts out to $z\\sim0.5$, as envisaged in our long-term\n observing plan.\n\n\\item{\\bf Tests of Lorentz Invariance Violation (LIV):} Blazar observations\n provide the most stringent tests of LIV for VERITAS, thanks to their\n large distance and rapid timescale of variability. Four bright,\n high-energy peaked BL Lac objects have been identified for deep\n monitoring exposures of $\\geq100\\U{hours}$ in our long-term plan. An\n additional target-of-opportunity program allows us to respond\n rapidly to alerts of enhanced emission from instruments at other\n wavelengths. The detection of VHE emission from the Crab pulsar also\n raises the possibility of using pulsar time profiles to constrain LIV\n \\cite{crabLIV}, and we plan to substantially augment our already\n extensive Crab pulsar dataset over the coming years, as well as to\n search for pulsed emission from other candidate sources. The energy\n threshold reduction provided by the 2012 upgrade will be\n particularly important in this regard.\n\n\\end{itemize}\n\n\\subsubsection*{UHECRs and Neutrino Astrophysics}\n\nVERITAS observations impact the related fields of ultra-high energy\ncosmic rays (UHECRs) and neutrino astrophysics. The UHECRs are most\nlikely extragalactic in origin, with active galactic nuclei (AGN)\namong the best candidates for the accelerators. Gamma-ray observations\nin the GeV-TeV range are essential to constrain models of particle\nacceleration and gamma-ray\/ neutrino emission in these sources (see\n\\cite{dummWP} for more details). Our long-term plan calls for regular\nmonitoring of most of the northern hemisphere VHE blazar population\nover the next five years, allowing us to accumulate deep exposures of\nthe sources in various emission states, and maximizing our chances of\ndetecting bright VHE flares. Observations of the nearby radio galaxy\nM87 will also continue, and will be complemented by high resolution\nX-ray and radio observations in the event of a flare. A clear\ncorrelation between morphological changes in the jet structure and the\nVHE emission state could help to finally pin down the particle\nacceleration and photon emission region in AGN jets.\n\nVERITAS can also act as a flare alert system for the UHECR and\nneutrino observatories, and provide rapid, high sensitivity follow-up\nobservations. In response to the early Auger reports of a correlation\nbetween ultra-high energy cosmic rays and AGN, VERITAS was the first\ninstrument to provide follow-up TeV gamma-ray observations\n\\cite{UHECRs}. No gamma-ray emission was seen, and the evidence for a\ncorrelation has diminished over time, but the observations demonstrate\nthe substantial overlap between the two instruments, despite their\nlocations in different hemispheres. IceCube, conversely, can easily\nview the northern sky, and VERITAS and IceCube are very well-matched\nin energy range (IceCube has a minimum neutrino energy threshold of\n50-100GeV, and an optimal response above $1\\U{TeV}$\n\\cite{IceCube}). Numerous predictions of measurable neutrino fluxes\nassociated with astrophysical particle accelerators exist in the\nliterature, including both Galactic (SNRs, binary systems,\nunidentified TeV sources and pulsar wind nebulae \\cite{neutrinos_gal1,\n neutrinos_gal2}) and extragalactic (GRBs, active and starburst\ngalaxies \\cite{neutrinos_xgal}) objects. VERITAS is the best instrument\nto search for and characterize the electromagnetic signatures of\nparticle interactions in these objects, which will be necessary to\nassess the relative contributions of leptonic and hadronic particle\npopulations. We will perform follow-up gamma-ray observations of any\nreported neutrino sources, and have established a memorandum of\nunderstanding between VERITAS and IceCube which allows us to rapidly\ntrigger observations of any transient neutrino excess. IceCube and\nAuger will be at their most productive over the coming five years, and\nVERITAS observations will both complement and augment their results.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Motivation}\n\\section{Motivation}\\label{s:motivation}\n\nEvidence is accumulating that star formation follows rapidly upon molecular\ncloud formation (e.g. \\citealp{2001ApJ...562..852H} and \\citealp{2007RMxAA..43..123B}\nfor the solar neighborhood; \\citealp{2003ApJS..149..343E} for M33;\n\\citealp{2007ApJ...668.1064E} in the context of M51). This rapid onset\nsuggests that the clouds need to acquire high, non-linear density enhancements\nduring their formation, since massive, finite clouds are highly susceptible\nto global gravitational collapse which could overwhelm small-scale fragmentation\nnecessary for (local) star formation \\citep{2004ApJ...616..288B}. \nThus to understand the initial conditions for star formation, we need to \nunderstand the formation of the parental clouds.\n \n\\citet{1999ApJ...527..285B} and \\citet{2001ApJ...562..852H} proposed that the build-up\nof clouds in large-scale, converging flows of diffuse atomic gas\ncould explain the crossing time problem, i.e. the observation that the stellar age \nspreads in a large number of local star forming regions are substantially smaller \nthan the lateral crossing time \\citep{2001ApJ...562..852H,2007RMxAA..43..123B}. \nIn this picture, there need not be a causal connection between star formation events \nin the plane perpendicular to the large-scale flows (see also \\citealp{2000ApJ...530..277E}). \nRapid star formation is a necessary requirement for this scenario to work.\n\nNumerical models of flow-driven cloud formation (we give only an early and the most\nrecent numerical work of each group, namely \n\\citealp{1999A&A...351..309H} and \\citealp{2008A&A...486L..43H};\n\\citealp{2000ApJ...532..980K} and \\citealp{2008ApJ...687..303I};\n\\citealp{2005ApJ...633L.113H} and \\citealp{2008ApJ...674..316H}; \n\\citealp{2006ApJ...643..245V} and \\citealp{2007ApJ...657..870V})\nhave identified the thermal and dynamical instabilities that are responsible for the \nrapid fragmentation of the nascent cloud (see \\citealp{2008ApJ...683..786H} \nfor an assessment of the roles of the physical processes). Despite these promising successes, \nmany questions about the physics at play remain unanswered, among one of the most pressing is \nthe role of magnetic fields during the cloud formation process. \n\nThe role of magnetic fields in the flow-driven cloud formation scenario has been largely\nenvisaged as one of ``guiding the flows'' to assemble the clouds, whether in\nform of the Parker instability along galactic spiral arms \n\\citep{1966ApJ...145..811P,1967ApJ...149..517P,1974A&A....33...73M},\nin a generally turbulent interstellar medium (ISM) \n\\citep{1995ApJ...455..536P,2001ApJ...562..852H}, \nor during the sweep-up of gas in spiral shocks \\citep{2006ApJ...646..213K,2008MNRAS.383..497D}.\nBased on the models of Passot et al., Hartmann et al. suggested that the field orientation\nwith respect to the flows selects the locations of cloud formation, namely that\nclouds will only form if the fields are aligned with the flows. A perpendicular\nfield will reduce the compression of the post-shock gas and thus will limit the strong cooling and \nthe thermal instability (TI, \\citealp{1965ApJ...142..531F}) necessary for the rapid \nflow fragmentation and the build-up of high-density contrasts \n\\citep{2004ApJ...616..288B,2008ApJ...674..316H,2008ApJ...683..786H}. \n\nGiven sufficiently high strengths,\nfields aligned with the inflows can suppress the dynamical instabilities responsible\nfor the generation of turbulence, namely the non-linear thin shell instability (NTSI, \n\\citealp{1994ApJ...428..186V}; for a magnetic version see \\citealp{2007ApJ...665..445H}) \nand the Kelvin-Helmholtz instability (KHI, e.g. \\citealp{1961hhs..book.....C}, \nmore recently \\citealp{2008MNRAS.385.1494K}, and for numerical studies \\citealp{2008ApJ...678..234P}).\nYet magnetic fields are intrinsically three-dimensional, and already two-dimensional\nmodels by \\citet{2008ApJ...687..303I} show that even for fields perpendicular to the inflow,\ncold (albeit diffuse) clouds can form. \n\nThus, three-dimensional models of flow-driven cloud formation including magnetic fields \nare needed. \\citet{2008A&A...486L..43H} present a first approach to the problem, \nmodeling the formation of a cloud in converging, perturbed flows, including fields and\nself-gravity. Here, we focus on the \neffects of magnetic field strength and orientation on the early stages\nof flow-driven cloud formation. We work in the ideal MHD limit (i.e. we do not\nexplicitly consider ambipolar drift or resistivity), and we do not include gravity in the models.\n\nAll our models start out with field strengths below equipartition with the kinetic energy\nof the inflows. At a factor of $4.3$ below equipartition -- corresponding to an\nabsolute field strength of $5\\mu$G at flow densities and velocities of $1$~cm$^{-3}$ \nand $16$~km~s$^{-1}$ -- , the fields already suppress the dynamical instabilities \n(and thus the generation of turbulence) leading to slab-like molecular clouds, while weaker \nfields -- at $2.5\\mu$G, corresponding to a factor of $17$ below equipartition -- lead to clouds \nclosely resembling the hydrodynamical case, albeit with more coherent filaments. \nFields at $0.5\\mu$G perpendicular to the inflows suppress the build-up of massive clouds in the collision\nplane, while they lead to the formation of diffuse, cold filaments perpendicular\nto the mean field, reminiscent of the cold HI clouds discussed by \\citet{2003ApJ...586.1067H}.\nA tangled field allows the assembly of substantial column densities\nin regions where the lateral field component is small or vanishing.\nOur results are consistent with the notion \nthat magnetic fields select the location of cloud formation \\citep{2001ApJ...562..852H}.\n\n\\section{Technical Details \\& Parameters}\n\n\\subsection{Athena}\nCalculations were performed with Athena \\citep{2005JCoPh.205..509G,2008JCoPh.227.4123G}, \nan unsplit, second-order accurate Godunov scheme, using the corner transport upwind method\n\\citep{2008JCoPh.227.4123G} and a linearized Roe solver \\citep{1981JCoPh..43..357R}. \nThe divergence of the magnetic field is kept zero by using constrained transport\n\\citep{1988ApJ...332..659E}. Dissipative terms (viscosity, heat conduction and resistivity) \nare not explicitly included.\nFor a detailed description and test results, the reader is referred to \n\\citet{2005JCoPh.205..509G,2008JCoPh.227.4123G} and \\citet{2008ApJS..178..137S}.\n\nWe implemented heating and cooling as an additional energy source term at 2nd order in time.\nA tabulated cooling function provides the energy change rate as function of density and temperature\nat each grid cell. We decided to keep the iterative approach we had used in our earlier studies\nof cloud formation \\citep{2005ApJ...633L.113H,2006ApJ...648.1052H,2008ApJ...674..316H}, \nwith a slight modification. Instead of advancing the fluid evolution at the usual time step\ngiven by the Courant-Friedrichs-Levy (CFL) condition and subcycling on the energy equation in case\nthe cooling timescale is shorter than the CFL timestep, we lower the CFL timestep according to \n\\begin{equation}\n \\Delta t = \\Delta t_{CFL} \\min(1,(\\tau_c\/\\Delta t_{CFL})^p),\n\\end{equation}\nwith $0\\leq p\\leq 1$. For increasing $p$, small cooling timesteps will control the overall CFL\ntimestep. The earlier version (see references above) would be equivalent\nto $p\\equiv 0$. Yet this choice can lead to inconsistencies in the hydrodynamical evolution \nonce the cooling timesteps get substantially shorter than the fluid timesteps, leading to a \nnumerical overemphasis of the acoustic mode of the TI, since regions can cool substantially \nwithout accounting for the resulting pressure drop in the dynamics. While these inconsistencies \nmay not affect the overall results, they turn out to affect the stability of the solution. \nFor the models presented here, $0.51$ should be expected. \n\nThere are four classes of models: hydrodynamical (series H), field aligned with flow (series X),\nfield perpendicular to the flow (series Y), and (series XR) a uniform field component aligned with \nthe flow plus a random field component of similar size, consistent with (although a little smaller than) \nobserved magnetic field \nstrength estimates (e.g. \\citealp{1996ASPC...97..457H}; \\citealp{2004Ap&SS.289..293B};\n\\citealp{2006ChJAS...6b.211H}). \nIn the latter series, we do not perturb the \ncollision interface but rely on the tangled field component to trigger the dynamical instabilities. \nTable~\\ref{t:param} summarizes the model parameters. Self-gravity is not included in the models.\n\nTo initialize the random field component, we set the amplitudes and phases of e.g. the $x$-component\nof the (edge-centered) vector potential to \n\\begin{equation}\n A_x(x,y,z) = \\sum_{i,j,k=1}^{max} |k|^{-p}\\sin(k_xx+k_yy+k_zz+\\phi_{i,j,k}^x),\\label{e:vecpot}\n\\end{equation}\nwhere $|k|\\equiv k_x^2+k_y^2+k_z^2$ and e.g. $k_x\\equiv 2\\pi i\/L_x$ with the box length $L_x$. \nWe set $p\\equiv 4$, mimicking a (steep) turbulent energy spectrum as observed in detailed\nnumerical simulations of magneto-hydrodynamic turbulence (e.g. \\citealp{2003MNRAS.345..325C}).\nThe wavenumbers $k_{x,y,z}$ are chosen such that $1\\leq |k| \\leq 4$, i.e. all\ncombinations of $(i,j,k)$ in the sum over $k$-space are used that satisfy the constraint on $|k|$. \nThe phases $\\phi_{i,j,k}^x$ in $k$-space are chosen from a uniform random distribution. Each vector\npotential component $A_{x,y,z}$ requires a separate phase array $\\phi^{x,y,z}$. \n\nThis formulation in real space instead of in Fourier space (see e.g. \n\\citealp{1998PhRvL..80.2754M}; \\citealp{1998ApJ...508L..99S}; \\citealp{2008ApJ...682L..97L} for\nvelocity fields) allows us to easily regenerate \nthe vector potential (and the field) at the inflow boundaries by\n\\begin{equation}\n A_x(\\pm L_x\/2,y,z,t) = A_x(\\pm (L_x\/2+v_0t),y,z),\\label{e:bc}\n\\end{equation}\nwhere the negative value refers to the lower $x$-boundary, and the positive to the upper one. \nThe face-centered fields are then computed from the vector potential by \n$\\mathbf{B}=\\nabla\\times\\mathbf{A}$. \n\nThe choice of the wave-number range $1 \\leq |k| \\leq 4$ does not constitute a restriction\nin terms of generality of our simulations, since the energy distribution over spatial scales\nis determined by the (steep) power law index $p$. This is fortunate in a sense, since the \ngeneration of the boundary conditions (eq.~[\\ref{e:bc}]) would consume substantially more time\nif we had to sum over all available $|k|$ in equation~(\\ref{e:vecpot}).\n\n\\begin{deluxetable}{lcccccc}\n\\tablewidth{0pt}\n\\tablecaption{Model Parameters\\label{t:param}}\n\\tablehead{\\colhead{Name}&\\colhead{$B_{x0}$ [$\\mu$G]}\n &\\colhead{$B_{y0}$ [$\\mu$G]}\n &\\colhead{$B_{rms}$ [$\\mu$G]}\n &\\colhead{$\\beta_{th}$}\n &\\colhead{$\\beta_{ram}$}}\n\\startdata\nH & $0.0$ & $0.0$ & $0.0$ & $\\infty$ & $\\infty$ \\\\\nX25 & $2.5$ & $0.0$ & $0.0$ & $4.5$ & $17$ \\\\\nX50 & $5.0$ & $0.0$ & $0.0$ & $1.1$ & $4.3$ \\\\\nY05 & $0.0$ & $0.5$ & $0.0$ & $110$ & $430$ \\\\\nXR25 & $2.5$ & $0.0$ & $2.5$ & $2.2$ & $8.5$ \n\\enddata\n\\tablecomments{1st column: model name, \n2nd: magnetic field strength $B_x$; 3rd: $B_y$, 4th: random field $B_{rms}$, \n5th: thermal plasma $\\beta$, 6th: ram plasma $\\beta$.}\n\\end{deluxetable}\n\n\\subsection{Physical Interpretation of the Initial Conditions}\n\nObviously, our initial conditions are somewhat idealized, e.g. \ngenerally, the flows would be expected to have substructure, \nthe flows might not be expected to collide always head-on, and the\nmagnetic fields will have parallel and perpendicular components with respect\nto the inflows. Yet the initial conditions can be seen as idealized versions \nof different physical environments.\n\nThe case of uniform fields aligned with the inflows (models X25, X50) \ncould be identified with the sweep-up of material by an expanding supernova \nshell along an ordered background field, or with the collision of two expanding \nshells in such a field. The initial field strength of $5\\mu$G (model X50) \nis close to the local median (total) field strength in the CNM (e.g. \nHT05; \\citealp{2005ASPC..343...64T}). \nUsing Nakano \\& Nakamura's (\\citeyear{1978PASJ...30..671N})\nexpression for the critical surface density $N_c\\equiv B\/\\sqrt{4\\pi^2 G}$ \nabove which gravitational collapse is possible under flux-freezing conditions, \nthe swept-up clouds would reach approximately $0.5N_c$ after $12$~Myr, while model\nX25 ($B_{x0}=2.5\\mu$G) would be marginally critical at the same time. We defer\nthe discussion of the mass-to-flux ratio in the clouds to a subsequent paper\nincluding gravity. \n\nAn ordered field aligned with the flows plus a large-scale random\ncomponent of similar amplitude (model XR25) introduces a large-scale shear and \nmight be considered a general situation for sweep-up of gas in spiral shocks, \nwhile the perpendicular field case (Y05) would address the (probably common) \nsituation of an oblique field whose lateral component is amplified by flow compressions.\n\nWe emphasize that while we attempt to address the extreme situations\nof field orientations, the finite size of our simulation domains cannot fully capture the \neffects of the magnetic field's boundary conditions. These will be set on larger \nscales than our local simulations can cover. In that sense, our results should be \nviewed as providing insight into magnetized cloud formation under idealized conditions \nrather than under physically realistic ones.\n\n\\subsection{A Comment on Resolution}\n\nWe decided to keep the resolution of our models constant, foregoing a resolution\nstudy in favor of a parameter study. Resolution effects have been discussed \nby \\citet{2007A&A...465..431H}. In addition, we have performed a systematic\nresolution study for two-dimensional cloud formation models (unpublished -- the\nmodels are similar to the ones discussed by \n\\citet{2005ApJ...633L.113H,2006ApJ...648.1052H}), covering a factor of $32$ in\nspatial resolution (from $256^2$ to $8192^2$ cells). As has been pointed out,\nthe critical length scale to resolve is the cooling \nlength of the thermal instability. If not resolved, the thermal instability will \nbe partially suppressed. At parameters of the WNM, the cooling\nlength is on the order of a parsec, while for the cold neutral medium (CNM), \nit drops to a fraction of a parsec. Thus, while more substructures should form with\nincreasing resolution, we expect our models to follow the general evolution\nof the thermal and dynamical instabilities sufficiently accurately for our purposes.\n\n\\section{Results}\n\n\\subsection{Morphologies}\n\nFigure~\\ref{f:polmap} summarizes the morphological effects of magnetic fields during\nthe build-up of a cloud. From top to bottom, it shows logarithmic column density maps of \nthe hydrodynamical model H, and the four MHD models X25 through XR25. The three columns stand \nfor projections along each coordinate axis, namely along the inflow ($x$-axis, {\\em left}), \nand perpendicular to the inflow (along $y$ and $z$-axes, {\\em center} and {\\em right}). The \nmaps of the MHD-models show polarization vectors which have been determined by integrating \nthe density-weighted Stokes $Q$ and $U$ parameters along the respective line-of-sight \n(see \\citealp{1996ASPC...97..486Z}; \\citealp{2001ApJ...561..800H}). \n\n\\begin{figure*}\n \\begin{center}\n \\includegraphics[width=0.7\\textwidth]{f1.eps}\n \\end{center}\n \\caption{\\label{f:polmap}Logarithmic column density projections (in cm$^{-2}$) \n along the three \n grid axes ({\\em left}: along inflow, {\\em center} and {\\em right}:\n perpendicular to the inflow) for models H, X25, X50, Y05 and XR25 as indicated,\n at $t=9.5$~Myr after flow collision. The mean field direction is denoted by \n the symbols in the panel labels ($\\rightarrow$,$\\uparrow$,$\\times$).}\n\\end{figure*}\n\n\\subsubsection{Field parallel to inflow}\n\nModel H shows the strong fragmentation due to thermal and dynamical instabilities\nsimilar to the models discussed by \\citet{2008ApJ...674..316H}. \nSpecifically, the large-scale initial perturbation\ntriggers the NTSI, due to whose rapid growth some of the dense material has already reached\nthe inflow boundaries. Viewed along the inflow (top left panel in Fig.~\\ref{f:polmap}), the\ncold dense fragments appear clumpy rather than filamentary.\n\nIntroducing a magnetic field {\\em aligned} with the flow (model X25, second row)\nsuppresses fragmentation compared to model H.\nThe face-on view ({\\em left}) shows several large-scale coherent filaments with\ndenser cores. The edge-on views ({\\em center} and {\\em right}) \ndemonstrate that the magnetic field is not dynamically\ndominant. The polarization vectors are aligned with local structures.\n\nIncreasing the magnetic field (model X50, third row from top) \nseems to suppress much of the fragmentation. Specifically, the NTSI is only very weakly \n(if at all) present, since the magnetic field is\nstrong enough to suppress the lateral momentum transport necessary for triggering the NTSI\n\\citep{2007ApJ...665..445H}. Nonetheless, the flows still fragment, albeit into a tight \nnetwork of filaments\n(X50, left) instead of a few large, more clumpy and fuzzy structures \n(models H, X25). The suppression of the NTSI leads to the formation of a more or less coherent \nfilament in the lateral projection (center and right column for model X50). Local shear modes \nlead to strong distortions of the field from its initial alignment with the inflow, as \nindicated by the polarization vectors which mostly trace out the mean background field.\n\n\\subsubsection{Field perpendicular to inflow}\n\nThe introduction of a field {\\em perpendicular} to the inflow changes the\nmorphology completely (4th row of Fig.~\\ref{f:polmap}, model Y05), despite the by a \nfactor of $10$ \nweaker field (see Table~\\ref{t:param}). The perpendicular field breaks the symmetry in the plane of\nthe flow collision, leading to filaments perpendicular to the mean field direction (note that\nthe mean field in the left panel of the 4th row of Fig.~\\ref{f:polmap} is oriented horizontally). \nThese filaments form due to motions along the field lines, but perpendicular to the incoming flows\n(see also \\citealp{2007ApJ...665..445H} and \\citealp{2008ApJ...687..303I} \nfor two-dimensional models). The magnetic field suppresses \none degree of freedom in the gas motions, also leading to lower column density contrasts\nthan in models H and X50. The two lateral views of model Y05 exhibit another effect of \nthe perpendicular field. Seen along the mean field direction, a large scale NTSI-driven mode \nis discernible, while\nthe projection perpendicular to the inflow and to the mean field (Y05 right) just shows a \nslab (albeit with substructure). In the former, the field lines are just shuffled around and\ncontribute to the dynamics only via the pressure term in the Lorentz force, thus lowering\nthe column densities and broadening the slab (compare to center panel of model X50).\nIn the latter, the tension term of the Lorentz force prevents the growth of the NTSI.\nThis is evidence for the presence of interchange modes in the NTSI, similar to e.g. \nthe Rayleigh-Taylor instability \\citep{2007ApJ...671.1726S}.\nStill, material is free to move along the field lines (and thus perpendicular to the inflows), \nleading to the formation of the filaments {\\em parallel} to the inflows. \n\nThe magnetic field perpendicular to the inflow resists compression, leading\nto a suppression of the thermal instability, which is also mirrored in the total mass budget\nof all models (Fig.~\\ref{f:masses}). Model X50 has the highest fraction of cold gas, due to \nthe strong guide field leading to a strong compression of the gas, while model Y05 shows the\nsmallest cold mass fraction, because the lateral field resists compression by the flows, and thus\nreduces the cooling rates. \n\n\\begin{figure}\n \\includegraphics[width=\\columnwidth]{f2.eps}\n \\caption{\\label{f:masses}Total masses against time, below and above $T=100$~K. \n The perpendicular field\n reduces the compression and thus lowers the cold gas mass, while the \n field aligned with the flow leads to higher compressibility.}\n\\end{figure}\n\n\\subsubsection{Tangled field}\n\nThe bottom row of Figure~\\ref{f:polmap} shows the maps for model XR25, which starts out\nwith a uniform field aligned with the flow at $2.5\\mu$G and a random field component of \nequal magnitude. Although the (varying) lateral field components contain $10$ times\nas much energy as the perpendicular field in model Y05, the fields do not suppress\nthe formation of clouds with column densities in excess of $10^{22}$~cm$^{-2}$; a tangled\nfield is substantially less efficient in preventing compression. Since there are regions\nwhere the field will be aligned with the flow, it leads \nto a selection effect in the sense that the clouds form at positions where\nthe lateral random components of the fields are weakest over time and\/or where bends\nin the fields determine the position of cloud formation (see \\citealp{2001ApJ...562..852H}).\nThe resulting clouds \nare more isolated, with larger voids between them (bottom left panel of Fig.~\\ref{f:polmap}).\nThe side view (bottom center and right) exhibits a diffuse halo of thermally unstable gas, \nmaterial which is caught in the tangled field between the bounding shocks and the dense cold\ngas. \n\n\\subsection{Dynamics}\n\nThe cold mass fractions of models H and X25 (Fig.~\\ref{f:masses}) are slightly lower than that\nof X50, indicating that the developing turbulence due to the flow fragmentation is also broadening\nthe slab. While this notion is already suggested by Figure~\\ref{f:polmap}, it is confirmed by comparing\nthe $rms$ velocity dispersion in the cold ($T<300$~K) gas (Fig.~\\ref{f:vrms}), and it can also be \ngleaned from a more detailed look at the laterally averaged pressure profiles (Fig.~\\ref{f:prssprof}). \n\n\\begin{figure}\n \\includegraphics[width=\\columnwidth]{f3.eps}\n \\caption{\\label{f:vrms}Density-weighted $rms$ velocity dispersion against time for\n all models. Fields parallel to the inflows seem to suppress turbulence in the \n cold gas.}\n\\end{figure}\n\n\\begin{figure*}\n \\includegraphics[width=\\textwidth]{f4.eps}\n \\caption{\\label{f:prssprof}Pressure profiles along the inflow direction $(x)$, averaged over\n the perpendicular directions $(y,z)$, for three times as indicated, and for all models.\n Shown are total (solid line), kinetic (dashed),\n internal (dash-dot) and -- if applicable -- magnetic (dash-3-dot) pressures. Note \n that we show the pressures, not the logarithm of the pressures.}\n\\end{figure*}\n\nShown is a time sequence of the pressure profiles along the $x$-axis (i.e. along the inflows) for \nall models. In the absence of gravity, the slabs are all overpressured by the ram pressure of\nthe colliding flows (solid lines). At early times, all five models show a drop in kinetic pressure\nand an increase in thermal pressure in the collision region. The flows have not fully fragmented\nyet, and the cooling is not in full strength yet because of the still low densities. With evolving\ntime, the thermal pressure peak for model H drops due to increasing cooling. \n\nThis is markedly different for model X50, where the thermal pressure continues to be enhanced\nby a factor of more than 2 above that of the inflow. Also, the kinetic pressure\ndrops, reaching an approximate equipartition with the thermal pressure. This is due\nto the strong magnetic guide field, which ``splits'' the cloud into a network of dense filaments \nwith low-density, high-temperature voids in between (see top panel of 2nd column of \nFig.~\\ref{f:polmap}, model X50). Figure~\\ref{f:prssdens} offers a different view of the same\nphenomenon, showing the pressure-density distributions for all four models, at $t=9.5$~Myr. \n\n\\begin{figure*}\n \\includegraphics[width=\\textwidth]{f5.eps}\n \\caption{\\label{f:prssdens}Greyscale-coded mass fraction of thermal pressure against volume density \n for all models as indicated in the plots, at $t=9.5$~Myr. The solid line indicates the\n thermal equilibrium curve, while the vertical and horizontal dashed lines denote the \n initial conditions in density and pressure. Diagonal dashed lines stand for isotherms\n at $T=10^4$, $300$ and $10$~K as indicated.}\n\\end{figure*}\n\nThe high-temperature voids of model X50 show up at $\\log n \\approx 0.5$ and $\\log P \\approx 4.3$, \nwhile the high-density filaments sit all on the stable low-temperature branch of the thermal\nequilibrium curve at $T\\approx 40$~K. Note that a substantial amount of the gas mass is actually\nthermally over-pressured, in contrast to model H. \n\nReducing the field aligned with the flow (model X25) leads to pressure profiles \n(Fig.~\\ref{f:prssprof}) and thermal states (Fig.~\\ref{f:prssdens}) similar to the hydrodynamical \nmodel H. In other words, while the field in model X25 is non-negligible in the sense that its \npresence still makes a morphological difference (see Fig.~\\ref{f:polmap}), it does not noticeably \naffect the overall dynamics of the cloud.\n\nThe field is obviously dynamically important in model Y05. Because of the strong flow \ncompression perpendicular to the field lines, the magnetic pressure takes over the role of the\nthermal pressure, which leads to a substantial amount of thermally underpressured gas \nin model Y05 (Fig.~\\ref{f:prssprof} and \\ref{f:prssdens}). There is only a small amount of \nmaterial at high ($\\log n > 2$) densities.\n\nIntroducing the tangled field component on top of a field aligned with the inflows leads to an\nover-pressurization of the slab by a factor of more than $2$ (Fig.~\\ref{f:prssprof} right, model XR25).\nThis is mainly due to a combined increase in magnetic and kinetic pressure, i.e. the tangled field\nleads to more turbulence than all other field geometries. The increase in kinetic pressure cannot\nbe solely due to enhanced densities -- model X50 should show a similar increase then. \nAlthough the tangled field in the diffuse gas phase is not force-free, it does not contribute\nperceptibly to turbulent motions in the inflows, as can be seen by comparing the kinetic pressure\nlevels in the inflows between models XR25 and e.g. X50. Also, the kinetic pressure of model\nXR25 does not increase when moving closer towards the midplane, until one enters the post-shock region.\nThe thermal pressure peaks in the diffuse\nenvelopes due to warm gas being unable to cool down (see bottom right panel of Fig.~\\ref{f:prssdens}),\nbut it drops back to the ambient {\\em thermal} pressure at the cloud midplane ($x=0$). Obviously,\nthe averaged thermal pressure alone is not a very accurate indicator of the cloud's physical state. \n\n\\begin{figure*}\n \\includegraphics[width=\\textwidth]{f6.eps}\n \\caption{\\label{f:prsstime}Pressures against time for all models, sorted according to temperature\n regimes (rows; total, $T>3000$K, $300300$~K) against the velocity\n dispersion in the CNM (Fig.~\\ref{f:vrms}), averaged between\n $7.5$ and $9.5$~Myr. The error bars show errors on the mean.}\n\\end{figure}\n\n\\section{Discussion}\n\n\\subsection{The Role of Magnetic Fields for Cloud Formation}\n\nThe field strength, and the orientation of the mean magnetic field with respect to the \nflows sweeping up the gas play a crucial role for the flow-driven formation of molecular clouds\n(Fig.~\\ref{f:polmap}). If the fields are dynamically dominant, the only chance to \nbuild up substantial clouds is by channeling the flows along the fields. \nThis is the situation shown in model X50, and it is also \nborne out by simulations of molecular cloud formation in galactic spiral arms\n\\citep{2006ApJ...646..213K}, where the clouds tend to be oriented perpendicularly to \nthe large-scale field (also possibly visible in the models by \n\\citealp{2008MNRAS.383..497D}), until sufficient material has been accumulated that they\ndecouple dynamically from the large-scale field. Similarly, for the sweep-up\nof material by e.g. H{\\small{II}}-regions or supernova shells, one would expect the densest clouds \nto appear at the locations where the field is perpendicular to the shell\n(see Fig.~2 of \\citealp{2006ApJ...641..905H}, although the effect might\nbe less clear in a highly turbulent environment, see Fig.~10 of \n\\citealp{2004ApJ...617..339B}). \n\n\\citet{2001ApJ...562..852H} point out -- based on simulations by\n\\citet{1995ApJ...455..536P} -- that dynamically weak (but not necessarily\nordered) fields would lead to a general selection effect for the formation of molecular\nclouds. Since $\\beta_{ram}>1$, the flows stretch out the\nfieldlines, leading to a natural alignment. In this picture, clouds form in the bends\nof large-scale fields (see Figs. 4 \\& 5 of \\citealp{2001ApJ...562..852H}).\nSuch a scenario is to some extent\naddressed by model XR25, where varying field orientations entail \na local selection effect, picking out the formation sites of molecular clouds \nover e.g. a broad shock front. Note that while the field is dynamically weak \n($\\beta_{ram}=8.5$, $\\beta_{th}=4.3$) in the initial conditions (and in the inflows) of model XR25, \n$\\beta_{th} < 1$ within the cloud (Fig.~\\ref{f:prssprof}).\n\nThis selection effect comes about because already a small oblique component can\nbe amplified sufficiently to withstand the compression, preventing the high densities\nneeded for cloud formation \\citep{2001ApJ...562..852H}. Such a situation is addressed in\nthe extreme by model Y05. \nA field perpendicular to the sweeping-up flow can suppress the formation of massive clouds,\nalthough the three-dimensional situation is much less clear-cut than its one-dimensional\ncounterpart (see e.g. \\citealp{1980ARA&A..18..219M}; \\citealp{2004ApJ...612..921B}). \nIn one dimension, a density increase by a factor of $100$ from e.g. $n=1$~cm$^{-3}$ to\n$100$~cm$^{-3}$ would entail the same factor for the magnetic field strength since $B\\propto n$. \nFigure~\\ref{f:babsdens} shows this is not the case in three dimensions.\nThe weak perpendicular field (model Y05) has been amplified by a peak factor of $\\approx 30$, \nwhile the density has increased by up to a factor of $300$. Generally, our models \nshow a weak correlation of field strength with density over the whole thermal range, from \nthe WNM to the CNM, consistent with observations of the field-density relation \nin the WNM and CNM (\\citealp{1986ApJ...301..339T}; HT05), and with\nnumerical results (e.g. \\citealp{2005A&A...436..585D,2008A&A...486L..43H}).\nFor models X50 and X25 a weak correlation between field and density is not overly surprising.\nFor model Y05, the decorrelation\\footnote{The seemingly strongly correlated B(n) for $n<1$~cm$^{-3}$\nin model Y05 does not affect the argument. These are a few regions (low mass fraction) at the edges\nof the expanding slab, subjected to numerical reconnection.} is a consequence of the fact that material is still free to \nmove along the field lines perpendicular to the original inflow \\citep{2007ApJ...665..445H}, \nthus leading to the build-up of \nhigher-density filaments perpendicular to the field (but aligned with the inflow), see\nFigure~\\ref{f:polmap}. Also, other effects, such as the acceleration of magnetic field\ntransport by turbulence (\\citealp{2002ApJ...567..962Z,2002ApJ...570..210F,2004ApJ...603..165H}\nfor ion-neutral drift, \\citealp{1999ApJ...517..700L} for reconnection),\nor a decorrelation due to MHD waves \\citep{2003A&A...398..845P}\ncould explain the observed weak correlation.\n\nMagnetic fields will rarely be completely uniform. Model XR25 tests the more\ngeneral case of a uniform field at $2.5\\mu$G and a random component of equal size, consistent\nwith (although slightly lower than) observational estimates for magnetic field strengths in the diffuse\ngas \\citep{1996ASPC...97..457H,2004Ap&SS.289..293B,2006ChJAS...6b.211H}. \n\\citet{2007ApJ...663L..41G} \nshowed in a two-dimensional numerical experiment that pre-existing perturbations in the \ninflows can lead to substantial magnetic field amplification due to a rippling of the \nshockfront and subsequent fieldline stretching. We observe a similar effect in model XR25, \nalthough our Mach numbers are substantially lower (their study addressed the propagation of a \nsupernova shock front). \\citet{2008A&A...486L..43H} perturb the velocities of the inflows and \nfind only a modest increase of the field strength. Clearly, the initially tangled field leads \nto rather different dynamics in the forming cloud (Figs~\\ref{f:polmap}, \\ref{f:prsstime}). \n\nModels X25 and X50 demonstrate that not only\nthe field orientation will play a role during cloud formation (see model Y05), but also\nthe field strength, since all the instabilities involved have threshold limits for the\nfield strength -- at least in two dimensions. It might well be that the stronger field\nin model X50 suppressing the formation of filaments could be offset by higher inflow speeds\nor substructures in the flows. This remains to be studied. \n\n\\begin{figure*}\n \\includegraphics[width=\\textwidth]{f9.eps}\n \\caption{\\label{f:babsdens}Magnetic field strength against volume density for models\n as indicated in the plots. The colors denote temperatures, and the intensity\n the mass fraction. Generally, there is no clear correlation between field strength\n and density. The steeper of the two dashed lines denotes $B\\propto n$, the \n flatter one $B\\propto n^{1\/2}$. Dotted lines denote the initial conditions.}\n\\end{figure*}\n\n\\subsection{Turbulence and Thermal States}\\label{ss:turbtherm}\nFields aligned with the inflows tend to suppress the NTSI, and thus lead to an approximate \nequipartition between the spatial components of the kinetic energy in the cold and\nin the thermally unstable gas \n(Fig.~\\ref{f:prssprof}, bottom couple of rows). For comparison, the hydrodynamical model \nH has the bulk of the kinetic energy in the (flow-aligned) $x$-component. Magnetic fields may play\nan important role to isotropize highly directional flows. Thus, searching for observational\nsignatures of flow-driven cloud formation should focus on the warm, diffuse gas phase, \nsince the inflow signature will be erased in the cold dense gas.\n\nFractions of thermally unstable gas (Fig.~\\ref{f:vrmsunm}) for flow-aligned\nfields (models X50, X25) are lower than values observed for diffuse CNM \nclouds \\citep{2003ApJ...586.1067H}. A lateral field component results in\na thermally unstable gas fraction of $\\approx 50$\\%, consistent with observations.\nBased on these findings, one could feel tempted to extend the above argument\nabout the selection effect introduced by magnetic fields: not only could\nmagnetic fields control the locations of molecular cloud formation, but they\nalso could lead to ``failed'' molecular clouds, i.e. diffuse atomic hydrogen\nclouds, if there is a non-negligible field component perpendicular to the\nsweeping-up flow (see also \\citet{2008ApJ...687..303I}\nfor a similar argument based on two-dimensional simulations). \n\n\\subsection{Ordered vs. Random Component}\\label{ss:components}\n\nAnother observational constraint is given by the ratio of the ordered \nover the unordered (or turbulent) field component. The observational\nevidence points to the components being of similar magnitude \n(e.g. \\citealp{1996ASPC...97..457H}; \\citealp{2004Ap&SS.289..293B};\n\\citealp{2004ASSL..315..277B}; \\citealp{2006ChJAS...6b.211H}; see also\ndiscussion in HT05). A direct comparison to\nour models is hampered by the fact that in order to see the varying \ncomponent, sufficiently large scales need to be addressed, which is why\nHT05 argue that their observed median field strength of $6\\mu$G should\nbe identified with the {\\em total} magnetic field strength. Likewise,\nit is not obvious that the components should be of equal magnitude locally \neverywhere. Bearing this limitation in mind, it is clear from \nFigure~\\ref{f:fieldtime} that only for models X25 and Y05 the components are \ncomparable. \n\n\\subsection{Gravity}\n\nWe deliberately left out self-gravity in our simulations, in order to get a clearer \nview of the role of magnetic fields during the early cloud formation phase. Thus, our\nclouds are only confined by the ram pressure of the inflows, and at later stages, \nthe dynamics of the clouds are probably underestimated since gravity as a source\nof turbulence is missing (e.g. \\citealp{2008MNRAS.385..181F}). \nAs a result of the restricted physics, a comparison of our models with observations\nis only meaningful for models where gravity is not expected to play a role, i.e. \nfor model Y05 addressing the formation of diffuse HI clouds. For all other models, \nwe expect gravity to be relevant during the cloud formation process \\citep{2008ApJ...689..290H}.\n\n\\section{Summary}\n\nExtending our previous work and complementing a model by \nHennebelle et al. (\\citeyear{2008A&A...486L..43H}; see also \\citealp{2008arXiv0808.0986B}),\nwe study the role of magnetic field strength and orientation on the process\nof flow-driven cloud formation. Our models include the usual heating and cooling effects,\nallowing rapid fragmentation of the flows, they use uniform inflows to study the most unfavorable\nconditions for structure formation, and they envisage the formation of clouds in two head-on\ncolliding flows, i.e. the extreme case for building up massive clouds. We do not include\nself-gravity, focusing on the early stages of cloud formation, during which gravity\nmight be less important. \n\nUnder these conditions, we find that the effects of magnetic fields on the morphology and\non the thermal state of the resulting clouds depend very strongly not only on the field\norientation with respect to the inflow, but also on the field strength. Initial field energies\nare below equipartition with the kinetic energies (by a factor of $4.3$, corresponding\nto a field strength of $5\\mu$G for our flow parameters) even for the strongest field\ncase in our study (model X50), yet they result in significantly different cloud properties\nthan those for a field weaker by a factor of $2$ (model X25, $2.5\\mu$G). \nMagnetic fields also lead to a redistribution of the inflow energy to the transverse spatial \ndirections (Fig.~\\ref{f:prsstime}). \nHence searching for signatures of colliding flows should focus on the diffuse\ngas phase, since the cold gas will have no memory of the original flow direction.\n\nNot surprisingly, weak magnetic fields ($0.5\\mu$G) perpendicular to the inflows can suppress the build-up\nof massive clouds (model Y05). Yet substructure still can arise in the post shock gas, in \nthe form of diffuse filaments perpendicular to the field, and of wave-like patterns \n(possibly magnetosonic waves). \nThe filaments are a consequence of lateral gas transport \n(see also \\citealp{2007ApJ...665..445H}; \\citealp{2008ApJ...687..303I}). \nThe straight-forward correlation $B\\propto n$ is not obeyed (Fig.~\\ref{f:babsdens}).\nMass fractions of thermally unstable gas for the model with a lateral field component (Y05)\nare consistent with observed values for diffuse HI clouds \\citep{2003ApJ...586.1067H}. For\nall other models, the fractions are lower (Fig.~\\ref{f:vrmsunm}). The ratio of ordered\nvs. random field component is consistent with observations only for the weak-field model\nX25, and for the diffuse HI cloud model Y05 (Fig.~\\ref{f:fieldtime}).\n\nA weak ($2.5\\mu$G) uniform field together with a random component of equal size \nleads to a strong over-pressurization of the cloud due to a combined \nincrease of magnetic and kinetic pressure (Fig.~\\ref{f:prssprof}), with the magnetic\npressure dominating the thermal pressure within the cloud.\nHigh column densities are assembled at locations\nwhere the perpendicular field component is weakest over time. Thus, a tangled field can lead\nto a selection effect for cloud formation while not preventing it globally. \n\nOur numerical models address the ideal MHD limit, i.e. we do not take into account ion-neutral\ndecoupling or resistive dissipation. It remains to be seen how non-ideal MHD processes affect\nthe structure formation during the build-up of the clouds (e.g. \\citealp{2008ApJ...687..303I}).\n\n\n\\acknowledgements\nWe thank the referee for a critical and very helpful report.\nComputations were\nperformed at the National Center for Supercomputing Applications\n(AST 060034) and on the local PC cluster Star, perfectly maintained and\nadministered by J.~Hallum \\& R. Bonser. FH is supported by the University\nof Michigan and NSF grant AST 0807305.\nThis work has made use of NASA's Astrophysics Data System.\n\n\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}