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+{"text":"\\section{Introduction}\n\nQuantum spin liquids represent nowadays one of the paradigms of strongly-correlated systems, hosting elementary excitations with fractional\nquantum numbers (e.g., $S=1\/2$ spinons), emerging gauge fields (e.g., visons or magnetic monopoles), and topological degeneracy of the\nground-state manifold (in case of a gapped spectrum)~\\cite{savary2017,zhou2017}. Their unconventional properties are ultimately triggered\nby a large entanglement between spins, which remain highly correlated down to zero temperature. In this respect, standard mean-field approaches \ncompletely fail to give a correct description of spin liquids. In order to overcome this difficulty, elegant {\\it escamotages} have been \nintroduced, based upon what is now generically known as {\\it parton construction}~\\cite{baskaran1988,affleck1988,arovas1988,read1989,wen1991}.\nHere, the spin degrees of freedom are written in terms of elementary objects (spinons), enlarging the original Hilbert space and introducing \ngauge fields (related to the redundancy of the spinon representation). Mean-field approximations of the spinon Hamiltonian can be performed, \nfixing gauge fields to a given spatial configuration; within fermionic or bosonic representations of spinons, the mean-field approximation \ngives rise to non-interacting models that can be easily handled~\\cite{baskaran1988,affleck1988,arovas1988,read1989,wen1991}. Remarkably, in \nsome very fortunate cases, the mean-field approximation gives the exact solution of the problem, most notably for the Kitaev compass model \non the honeycomb lattice~\\cite{kitaev2006}. However, for generic spin models, the accuracy of parton-based mean-field methods is questionable. \nParticularly delicate cases arise for gapless spin liquids, which are defined by mean-field theories with a gapless spinon spectrum: here, \nsmall perturbations (e.g., gauge-field fluctuations) may lead to instabilities towards some spontaneous symmetry breaking, the most celebrated \none being valence-bond order~\\cite{read1989b}. A stable spin liquid (i.e., a state that is not limited to isolated points in the phase diagram) \ncan be obtained when the gauge fields have a discrete symmetry and gapped excitations, as in the Kitaev model~\\cite{kitaev2006}. Instead, the \ncase where the gauge fields have a continuous symmetry and sustain gapless excitations is more problematic. Here, the validity of the \nmean-field approximation has been discussed only in some limiting cases~\\cite{hermele2004}, while general implications of low-energy gauge \nfluctuations on the fermionic properties are still unknown. \n\nOne technical aspect that strongly limits the use of mean-field wave functions is the fact that they are defined in the artifically enlarged\nHilbert space. For example, in the fermionic approach for $S=1\/2$ models~\\cite{wen2002}, the enlarged Hilbert space contains four states per \nsite (with zero, one, and two fermions) in contrast to the original spin Hilbert space that is limited to two states per site (up and down \nspins). Therefore, in general, these mean-field wave functions do not give reliable insights into the exact ground-state properties. In order \nto go beyond this approximation, quantum Monte Carlo methods have been very insightful. Here, starting from an auxiliary Hamiltonian for free \nfermions (not necessarily the best mean-field solution), along the Monte Carlo procedure, it is possible to restrict the calculations to the \noriginal Hilbert space of the spin model by including the Gutzwiller projector~\\cite{becca2011}. Then, genuine variational wave functions \ncan be defined, thus allowing quantitative predictions for the original spin model. \n\nThe variational approach is particularly suited within the fermionic representation, for which there are polynomial algorithms that allow to \nperform large-size calculations. This approach has been demonstrated insightful to describe $S=1\/2$ Heisenberg models on frustrated lattices \nin two spatial dimensions, such as the $J_1-J_2$ models on square~\\cite{hu2013,ferrari2020}, triangular~\\cite{iqbal2016,ferrari2019}, and \nkagome~\\cite{iqbal2013,iqbal2015} lattices. In all these cases, Gutzwiller-projected wave functions have suggested the possibility that a \n{\\it gapless} spin liquid may be stabilized in highly-frustrated regimes, the free-fermion spectrum displaying Dirac points. On the honeycomb\nlattice the gapless spin liquid does not represent the optimal state, having a slightly higher energy than a state with valence-bond \norder~\\cite{ferrari2017}. It is worth mentioning that recent density-matrix renormalization group (DMRG) calculations on the kagome~\\cite{he2017} \nand triangular~\\cite{hu2019} lattices have given further support to this conclusion. Still, obtaining a gapless spin liquid within DMRG is \nnot easy; first of all, because gapped low-entangled states are favored, thus disfavoring highly-entangled gapless phases. In addition, not \nless impactfully, DMRG simulations are usually done on cylindrical geometries, i.e., on $N$-leg ladders with a relatively large number of legs. \nWithin this choice of the clusters, the nature of the ground state can be altered with respect to the truly two-dimensional case. Indeed, only \nby using modified boundary conditions along the rungs (namely introducing a fictitious magnetic field piercing the cylinder), it was possible \nto detect gapless points in the kagome and triangular lattices~\\cite{he2017,hu2019,zhu2018}. A fictitious magnetic field has been also employed \nin the calculation of the dynamical structure factor for the Heisenberg model on the kagome lattice~\\cite{zhu2019}.\n\nGapless spin liquids represent particularly fragile states, which are expected to be stable only to a limited number of perturbations. In this \nrespect, a tiny modification to the Hamiltonian may immediately open a gap in the spectrum or lead to a spinon condensation, giving rise to \ntopological order or some sort of spontaneous symmetry breaking. For example, the gapless spin liquid of the Kitaev model, is locally stable \nfor small variations of the super-exchange couplings around the isotropic point $J_x=J_y=J_z$, but a magnetic field immediately opens a gap, \ndriving the ground state into a topological phase with anyon excitations~\\cite{kitaev2006}. In general, considering a ladder geometry, which \nexplicitly breaks point-group symmetries, represents also a relevant perturbation that may open a gap.\n\nIn this work, we study the fate of the gapless spin-liquid phase obtained on frustrated square and kagome lattice by Gutzwiller-projected \nfermionic wave functions, when cylindrical geometries are considered. We restrict to the case of {\\it even} number of legs, in order not to \nintroduce further frustration in the system. On cylindrical clusters, either gapless or gapped states can be constructed by playing with the \nboundary conditions of the fermionic operators (still keeping periodic boundary conditions on the physical spins). For the models under \ninvestigation, the best variational {\\it Ansatz} on cylindrical geometries is achieved by states with a gapped excitation spectrum in the \nthermodynamic limit, in striking constrast with the results for isotropic two-dimensional clusters, where the spin liquid states have a gapless \nspectrum and the effect of the fermionic boundary conditions is irrelevant in the thermodynamic limit (the same energy per site is achieved by \nany choice). In addition, the gapped ground state found on cylinders is unique, thus featuring neither spontaneous symmetry breaking nor \ntopological degeneracy. It is worth mentioning that this is not forbidden by the Lieb-Schultz-Mattis theorem and its generalizations to \ncylindrical geometries~\\cite{lieb1961,affleck1988b}. Our results are relevant for recent studies on both square~\\cite{mambrini2006,richter2010,jiang2012b,mezzacapo2012,hu2013,gong2014,doretto2014,morita2015,haghshenas2018,wang2018,liu2018,liao2019,choo2019,ferrari2020,nomura2020,liu2020,hasik2021}\nand kagome lattices~\\cite{yan2011,depenbrock2012,jiang2012,iqbal2013,clark2013,he2017,liao2017,hering2019}, highlighting the fact that gapless \nspin liquids, as obtained in the truly two-dimensional limit, may turn into trivial gapped phases when constrained into cylindrical geometries.\n\nThe paper is organized as follows: in section~\\ref{sec:methods}, we outline the parton construction for the Gutzwiller-projected wave function;\nin section~\\ref{sec:results}, we discuss the results for the square and kagome geometries; finally, in section~\\ref{sec:concl}, we draw our\nconclusions.\n\n\\section{Models and methods}\\label{sec:methods}\n\nIn the following, we will consider frustrated Heisenberg models\n\\begin{equation}\\label{eq:hamilt}\n{\\cal H} = \\sum_{R,R^\\prime} J_{R,R^\\prime} {\\bf S}_R \\cdot {\\bf S}_{R^\\prime},\n\\end{equation}\ndefined on either the square lattice, with primitive vectors ${\\bf a}_1=(1,0)$ and ${\\bf a}_2=(0,1)$, or the kagome lattice, with ${\\bf a}_1=(1,0)$ \nand ${\\bf a}_2=(1\/2,\\sqrt{3}\/2)$. For the square lattice, both nearest- ($J_1$) and next-nearest-neighbor ($J_2$) super-exchange couplings are \nconsidered (with $J_2\/J_1=0.5$). For the kagome lattice, only nearest-neighbor ($J$) terms are included. Clusters are identified by two vectors \n${\\bf T}_1=L_1 {\\bf a}_1$ and ${\\bf T}_2=L_2 {\\bf a}_2$, with $L_1 \\gg L_2$. The finite width $L_2$, which is taken to be even, defines the \ncylindrical geometry; periodic boundary conditions along both directions will be considered, in order not to explicitly break translational \nsymmetries.\n \nWe consider the fermionic approach, in which the $S=1\/2$ spin operators are represented by the so-called Abrikosov fermions~\\cite{wen2002}:\n\\begin{equation}\\label{eq:Sabrikosov}\n{\\bf S}_R = \\frac{1}{2} \\sum_{\\alpha,\\beta} c_{R,\\alpha}^\\dagger \n\\boldsymbol{\\sigma}_{\\alpha,\\beta} c_{R,\\beta}^{\\phantom{\\dagger}}.\n\\end{equation}\nHere $c_{R,\\alpha}^{\\phantom{\\dagger}}$ ($c_{R,\\alpha}^\\dagger$) destroys (creates) a fermion with spin $\\alpha=\\uparrow,\\downarrow$ on site $R$, and the \nvector $\\boldsymbol{\\sigma}=(\\sigma_x,\\sigma_y,\\sigma_z)$ is the set of Pauli matrices. The fermionic representation enlarges the local Hilbert\nspace, including unphysical configurations with zero and two fermions per site. \n\nThe construction of spin-liquid wave functions goes as follows. First of all, an auxiliary (fermionic) Hamiltonian is defined:\n\\begin{equation}\\label{eq:generic_mf}\n{\\cal H}_{0} = \\sum_{R,R^\\prime} \\sum_{\\alpha} t_{R,R^\\prime} c_{R,\\alpha}^\\dagger c_{R^\\prime,\\alpha}^{\\phantom{\\dagger}} +\n\\sum_{R,R^\\prime}  \\Delta_{R,R^\\prime} c_{R,\\downarrow}^{\\phantom{\\dagger}} c_{R,\\uparrow}^{\\phantom{\\dagger}} + h.c.,\n\\end{equation}\nwhere $t_{R,R^\\prime}^{\\phantom{\\dagger}}=t_{R^\\prime,R}^*$ and $\\Delta_{R,R^\\prime}=\\Delta_{R^\\prime,R}$ are singlet hopping and pairing terms. Then, the \nunprojected state is obtained by the ground state $|\\Phi_0\\rangle$ of ${\\cal H}_0$. Finally, the variational {\\it Ansatz} for the spin model \nis obtained by enforcing the constraint of one fermion per site, thus going back to the original Hilbert space of $S=1\/2$ spins. \nThis can be achieved by applying the Gutzwiller projector ${\\cal P}_G = \\prod_R (n_{R,\\uparrow} - n_{R,\\downarrow})^2$ to $|\\Phi_0\\rangle$ \n($n_{R,\\alpha}=c_{R,\\alpha}^\\dagger c_{R,\\alpha}^{\\phantom{\\dagger}}$ being the fermion density on the site $i$):\n\\begin{equation}\\label{eq:finalwf}\n|\\Psi_0\\rangle = {\\cal P}_G |\\Phi_0\\rangle.\n\\end{equation}\nThe physical properties of the projected wave function can be assessed by using standard quantum Monte Carlo techniques~\\cite{becca2017}.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig1a.pdf}\n\\includegraphics[width=0.83\\columnwidth]{fig1b.pdf}\n\\caption{\\label{fig:square} \nTop panel: schematic illustration of the square lattice cylindrical geometry with $L_2=4$ legs. The site numbering indicates how periodic \nboundary conditions are chosen along the ${\\bf T}_2$ direction, namely how the cylinder is wrapped up. Black (red) lines represent positive \nand negative hoppings in the fermionic Hamiltonian~(\\ref{eq:generic_mf}) for the $\\pi$-flux state~\\cite{affleck1988c}. Bottom panels: cuts \nof momenta of the fermionic partons which are allowed by the cylindrical geometry in the top panel. Depending on the choice of the fermionic \nboundary conditions along ${\\bf T}_2$, the cuts can hit or avoid the Dirac points in the spinon spectra [${\\bf k}=(\\pm \\pi\/2, \\pm \\pi\/2)$, \nindicated by the crosses]. The black square represents the original Brillouin zone, while the dashed lines delimit the reduced Brillouin \nzone of the $\\pi$-flux state.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig2a.pdf}\n\\includegraphics[width=0.85\\columnwidth]{fig2b.pdf}\n\\caption{\\label{fig:kagome}\nThe same as in Fig.~\\ref{fig:square} but for the kagome lattice and the corresponding $\\pi$-flux state~\\cite{ran2007}.}\n\\end{figure}\n\nIn the following, we will focus on specific spin-liquid {\\it Ans\\\"atze} defined by auxiliary fermionic Hamiltonians which possess a gapless\nspectrum with Dirac points in the two-dimensional limit. In particular, we will consider either simple cases with only nearest-neighbor hopping \nparameters, defining non-trivial magnetic fluxes, or cases with both hopping and pairing terms. In the former case, no variational parameters \nare present in the wave function (the hopping parameter defines the energy scale of the auxiliary Hamiltonian); in the latter case, a few \nvariational parameters are present and optimized by using standard Monte Carlo schemes~\\cite{becca2017}.\n\n\\begin{figure*}\n\\includegraphics[width=\\columnwidth]{fig3a.pdf}\n\\includegraphics[width=\\columnwidth]{fig3b.pdf}\n\\includegraphics[width=\\columnwidth]{fig3c.pdf}\n\\includegraphics[width=\\columnwidth]{fig3d.pdf}\n\\caption{\\label{fig:square_sqw} \nDynamical structure factor for the $J_1-J_2$ Heisenberg model on the square lattice with $J_2\/J_1=0.5$. The cylindrical geometry with $L_1=84$ \nand $L_2=6$ is taken. The results for the gapped state with PBC-PBC (upper panels) and the gapless one with ABC-ABC (lower panels) are reported for \ntwo values of the transverse momentum, $q_y=0$ and $q_y=\\pi$. A Gaussian smearing of the delta-functions of Eq.~(\\ref{eq:dsf}) is applied, with \n$\\sigma=0.03J_1$. In the lower-right panel, a negative excitation is observed at ${\\bf q}=(\\pi,\\pi)$. This is an artifact of the numerical method \ncaused by the fact that the gapless state is not the optimal variational {\\it Ansatz} for the $J_1-J_2$ model on the cylinder (see also the \ndiscussion in the main text).}\n\\end{figure*}\n\nDifferent choices for the boundary conditions on fermionic operators are possible, affecting the free-fermion Hamiltonian~(\\ref{eq:generic_mf})\nand, consequently, the variational wave function, but not the spin operators~(\\ref{eq:Sabrikosov}). In the following, we will restrict ourselves \nto real wave functions, which implies that either periodic boundary conditions (PBC) or anti-periodic boundary conditions (ABC) are possible:\n\\begin{eqnarray}\nc_{R+T_1,\\alpha}^{\\phantom{\\dagger}} &=& e^{i \\theta_1} c_{R,\\alpha}^{\\phantom{\\dagger}}, \\\\\nc_{R+T_2,\\alpha}^{\\phantom{\\dagger}} &=& e^{i \\theta_2} c_{R,\\alpha}^{\\phantom{\\dagger}}, \n\\end{eqnarray}\nwhere $\\theta_1, \\theta_2=0(\\pi)$ for PBC (ABC).\n\nThe important point is that the finiteness of the cylinder width $L_2$ implies a coarse discretization of the momenta along the direction of \nthe reciprocal lattice vector ${\\bf b}_2$. The discretization is present also in the limit of a infinitely long cylinder, $L_1 \\to \\infty$, \nwhere the set of allowed momenta form parallel ``cuts'' running along the ${\\bf b}_1$ direction. As a consequence, gapless points in the \nspinon spectrum of the auxiliary Hamiltonian~(\\ref{eq:generic_mf}) may be absent on cylinders, if the cuts of momenta do not run through them. \nTherefore, the unprojected spectrum will be gapped also for $L_1 \\to \\infty$, only because of the finite number of legs $L_2$. As an example, \nlet us consider the $\\pi$-flux phase on the square lattice, which has Dirac points at ${\\bf k}=(\\pm \\pi\/2,\\pm \\pi\/2)$~\\cite{affleck1988c}; \nfor $L_2=4n$ ($L_2=4n+2$) and ABC (PBC) along ${\\bf T}_2$ these k-points are not allowed and the spinon spectrum is gapped, for any value of the \ncylinder length $L_1$. By contrast, for $L_2=4n$ ($L_2=4n+2$) and PBC (ABC) along ${\\bf T}_2$, zero-energy modes are present or absent depending \non the boundary conditions along ${\\bf T}_1$: with one choice the gapless points are present for any value of $L_1$, while for the opposite \nchoice the spectrum has a finite-size gap, which vanishes in the limit of an infinitely long cylinder $L_1\\to \\infty$. The case for $L_2=4$ is \nshown in Fig.~\\ref{fig:square}. In order to have a unique and well defined variational state, zero-energy modes in Eq.~(\\ref{eq:generic_mf}) \nmust be avoided on any finite size, thus leaving to only three possibilities of boundary conditions for each choice of $L_1$ and $L_2$: two of \nthem will correspond to fully-gapped states, while the third one will correspond to a state that will become gapless for $L_1 \\to \\infty$ and \nwill be dubbed ``gapless'' for simplicity. A similar situation happens in the kagome lattice, for the $\\pi$-flux phase defined in \nRef.~\\cite{ran2007}. The case with $L_2=4$ is shown in Fig.~\\ref{fig:kagome}. Notice that the unprojected (spinon) spectrum is gauge invariant, \neven though the position of the Dirac points in $k$-space is gauge dependent. Therefore, a given choice of the boundary conditions will give \nrise to a gapped or gapless spectrum, independently on the gauge choice. In both $\\pi$-flux phases on square and kagome lattices, the fermionic \nHamiltonian~(\\ref{eq:generic_mf}) can accomodate the magnetic fluxes by breaking the translational symmetry (e.g., the unit cell is doubled \nalong ${\\bf a}_2$). Nevertheless, after Gutzwiller projection, the wave function~(\\ref{eq:finalwf}) is fully translationally \ninvariant~\\cite{wen2002}.\n\nThe properties of these variational states can be assessed by computing equal-time spin-spin and dimer-dimer correlations as a function of \ndistance along the long side of the cylinder. The former ones are defined as:\n\\begin{equation}\n{\\cal S}({\\bf R}) = \\langle \\mathbf{S}_0 \\cdot \\mathbf{S}_R \\rangle,\n\\end{equation}\nwhere $\\langle \\cdots \\rangle$ denotes the expectation value over the variational state. Instead, dimer-dimer correlations are given by:\n\\begin{equation}\n{\\cal D}({\\bf R}) = \\langle D_0 D_R \\rangle - \\langle D_0 \\rangle \\langle D_R \\rangle,\n\\end{equation}\nwhere $D_R=\\mathbf{S}_R \\cdot \\mathbf{S}_{R+a_1}$. \n\nUp to now, in the definition of gapped and gapless states, we have made reference to the unprojected spectrum of the fermionic Hamiltonian.\nThe physical excitations of the spin model may be assessed within the variational method by constructing a set of Gutzwiller-projected states \nfor each momentum ${\\bf q}$~\\cite{li2010,ferrari2018}. For the case of a Bravais lattice (e.g., the square lattice), we define a set of triplet \nstates:\n\\begin{equation}\\label{eq:qRstate}\n|q,R\\rangle =  \\mathcal{P}_G \n\\frac{1}{\\sqrt{N}} \\sum_{R^\\prime}\\sum_{\\alpha} e^{i {\\bf q} \\cdot {\\bf R^\\prime}} s_{\\alpha} \nc^\\dagger_{R^\\prime+R,\\alpha}c^{\\phantom{\\dagger}}_{R^\\prime,\\alpha} |\\Phi_0\\rangle.\n\\end{equation}\nwhere $N=L_1L_2$ and $s_{\\alpha}=1$ ($s_{\\alpha}=-1$) for $\\alpha=\\uparrow$ ($\\alpha=\\downarrow$); these states are labelled by $R$, \nwhich runs over all lattice vectors. Then, the adopted PBC or ABC are imposed to $c^\\dagger_{R^\\prime+R,\\alpha}$, whenever $R^\\prime+R$ falls \noutside of the original cluster. Low-energy excited states are given by suitable linear combinations:\n\\begin{equation}\\label{eq:psinq}\n|\\Psi_n^q\\rangle=\\sum_R A^{n,q}_R |q,R\\rangle.\n\\end{equation}\nThe parameters $\\{ A^{n,q}_R \\}$ are obtained once the spin Hamiltonian of Eq.~(\\ref{eq:hamilt}) is fully specified. In fact, for any given \nmomentum ${\\bf q}$, we can consider the Schr{\\\"o}dinger equation, restricting the form of its eigenvectors to the one of Eq.~(\\ref{eq:psinq}),\ni.e., ${ {\\cal H}|\\Psi_n^q\\rangle = E_n^q |\\Psi_n^q\\rangle }$. Then, expanding everything in terms of $\\{|q,R\\rangle\\}_R$, we get to the \nfollowing generalized eigenvalue problem\n\\begin{equation}\\label{eq:general_eig_prob}\n\\sum_{R^\\prime} \\langle q,R|{\\cal H}|q,R^\\prime \\rangle  A^{n,q}_{R^\\prime} = E_n^q \\sum_{R^\\prime} \\langle q,R|q,R^\\prime \\rangle \nA^{n,q}_{R^\\prime},\n\\end{equation}\nwhich is solved to find the expansion coefficients $A^{n,q}_R$ and the energies $E_n^q$ of the excitations. All the matrix elements,\n$\\langle q,R|{\\cal H}|q,R^\\prime \\rangle$ and $\\langle q,R|q,R^\\prime \\rangle$, are evaluated within the Monte Carlo procedure, by \nsampling according to the variational ground-state wave function~\\cite{ferrari2018}. Finally the dynamical structure factor is computed by:\n\\begin{equation}\\label{eq:dsf}\nS^{z}({\\bf q},\\omega) = \\sum_n |\\langle \\Psi_{n}^q | S^{z}_q | \\Psi_0 \\rangle|^2 \\delta(\\omega-E_{n}^q+E_0^{\\rm var}),\n\\end{equation}\nwhere $E_0^{\\rm var}$ is the variational energy of $|\\Psi_0 \\rangle$ and \n\\begin{equation}\nS^{z}_q=\\frac{1}{\\sqrt{N}} \\sum_R e^{i{\\bf q} {\\bf R}} S^{z}_R.\n\\end{equation}\nIn the case of a lattice with a basis of $n_b$ sites (e.g., the kagome lattice with $n_b=3$), the generalization is straighforward, \nwith the only modification that particle-hole excitations~(\\ref{eq:qRstate}) acquire basis indices, attached to creation and annihilation \noperators. The dynamical structure factor~(\\ref{eq:dsf}) becomes a $n_b \\times n_b$ matrix, constructed from spin operators on the elements\nof the basis (with a phase factor that depends upon the actual spatial position of each sites). Then, the physical (experimentally measurable) \nquantity is the sum of all its elements.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig4a.pdf}\n\\includegraphics[width=\\columnwidth]{fig4b.pdf}\n\\caption{\\label{fig:piflux_spinspin} \nComparison of the absolute value of the spin-spin ($|{\\cal S}({\\bf R})|$, upper panel) and dimer-dimer ($|{\\cal D}({\\bf R})|$, lower panel) \ncorrelations for the $\\pi$-flux and the optimal spin-liquid wave functions on the square lattice for $J_2\/J_1=0.5$. Calculations are done \non the cylindrical cluster with $84 \\times 6$ sites. Empty symbols refer to the gapped state with PBC-PBC, while full symbols refer to the \ngapless state with ABC-ABC.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig5.pdf}\n\\caption{\\label{fig:piflux_4xL} \nAbsolute value of the spin-spin correlations ($|{\\cal S}({\\bf R})|$) of the gapless $\\pi$-flux state (with ABC-PBC) on $L_1 \\times 4$ cylinders \n(square lattice). Inset: size scaling of the spin-spin correlations at the maximum distance along ${\\bf a}_1$.}\n\\end{figure}\n\n\\section{Results}\\label{sec:results}\n\n\\subsection{Preliminary considerations on the Kitaev model}\nA useful insight on how a gapless spin liquid reveals itself in a finite-size cluster comes from the investigation of the Kitaev compass \nmodel on the honeycomb lattice~\\cite{kitaev2006}. The model can be mapped to free Majorana fermions in a static magnetic field, where fluxes \npiercing the hexagons are either $0$ or $\\pi$. This auxiliary model is defined in an extended Hilbert space and the physical states are \nobtained by a projection. Most importantly, all the physical energies of the spin model also belong to the spectrum of the auxiliary Hamiltonian. \nThen, the study of the flux states of free Majorana fermions on the honeycomb lattice provides the ground state energy and the excitation \nspectrum of the Kitaev model. In the thermodynamic limit, the isotropic model ($J_x=J_y=J_z$) displays a gapless spinon branch, due to the \nexcitations of Majorana fermions, and a gapped vison spectrum, corresponding to changes in the flux pattern. On the cylindrical clusters \ndefined by ${\\bf T}_1$ and ${\\bf T}_2$ (with $L_1=4n$ and $L_2=2m$), the Lieb theorem~\\cite{lieb1994} holds, implying zero flux through all \nhexagonal plaquettes. This choice of the flux pattern leaves four possible orthogonal candidates for the ground state, corresponding to the \nfour possible boundary conditions. Zero-energy modes are present only when both $L_1$ and $L_2$ are multiple of $3$. In this case, there are \ntwo gapped states and one gapless state (for $L_1 \\to \\infty$), while the fourth one has zero-energy modes. Remarkably, the lowest energy is \nnever obtained by the gapless wave function and, therefore, on such cylindrical geometries, the Kitaev model with $J_x=J_y=J_z$ is fully gapped.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig6a.pdf}\n\\includegraphics[width=\\columnwidth]{fig6b.pdf}\n\\caption{\\label{fig:largeL_spinspin} \nAbsolute value of the spin-spin ($|{\\cal S}({\\bf R})|$, upper panel) and dimer-dimer ($|{\\cal D}({\\bf R})|$, lower panel) correlations on \n$20L \\times L$ cylinders for the $\\pi$-flux state on the square lattice. Empty symbols refer to gapped states (with ABC-ABC for $L=4n$ and \nABC-PBC for $L=4n+2$), while full symbols refer to gapless states (with ABC-PBC for $L=4n$ and ABC-ABC for $L=4n+2$).}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig7.pdf}\n\\caption{\\label{fig:squakag}\nAbsolute value of the spin-spin correlations ($|{\\cal S}({\\bf R})|$) on the isotropic clusters for the square ($48 \\times 48$) and kagome \n($28 \\times 28$) lattices. In both cases the simple $\\pi$-flux states are considered, with ABC-ABC. Dashed lines are linear fits, compatible \nwith a power-law behavior with an exponent $\\beta \\approx 2$.}\n\\end{figure}\n\n\\subsection{Square-lattice geometry}\nHere, we present the results for the square-lattice geometry, see Fig.~\\ref{fig:square}, and the spin Hamiltonian~\\eqref{eq:hamilt} with \n${J_2\/J_1=0.5}$. In the two-dimensional limit, the Gutzwiller-projected wave function suggested the existence of a gapless spin-liquid \nphase~\\cite{hu2013,ferrari2020}. The best variational {\\it Ansatz} within this class of states has, in addition to uniform nearest-neighbor \nhopping, pairing terms with $d_{x^2-y^2}$ symmetry (at first neighbors, connected by ${\\bf a}_1$ and ${\\bf a}_2$) and $d_{xy}$ symmetry (at \nfifth neighbors, connected by $2{\\bf a}_1 \\pm 2{\\bf a}_2$). While the point-group symmetry of the hopping and pairing couplings is determined \nby the projective symmetry group of the spin liquid {\\it Ansatz}~\\cite{wen2002}, the actual values of the parameters are fully optimized by \nthe numerical minimization of the variational energy. We note that the simpler case with only hopping and nearest-neighbor pairing corresponds \nto a $U(1)$ spin liquid (gauge equivalent to the the staggered flux state~\\cite{affleck1988c}), while the addition of $d_{xy}$ pairing terms \nleads to a $Z_2$ spin liquid~\\cite{wen2002}. In both cases, the Hamiltonian~(\\ref{eq:generic_mf}) possesses Dirac points at \n${\\bf k}=(\\pm \\pi\/2, \\pm \\pi\/2)$. Variational Monte Carlo calculations of the dynamical structure factors suggested that Dirac points survive \nthe Gutzwiller projection and are a genuine feature of the low-energy spectrum~\\cite{ferrari2018b}. It is worth noting that the simple $\\pi$-flux \nstate discussed previously, despite having a slightly higher energy than the $Z_2$ spin liquid, gives a quite accurate approximation of the best \nvariational {\\it Ansatz}.\n\nLet us start by considering a cylinder with $L_1=84$ and $L_2=6$; the large aspect ratio ensures that results are indistinguishable from the \nlimit $L_1 \\to \\infty$. For this cluster, the fermionic Hamiltonian~(\\ref{eq:generic_mf}) is gapped by taking PBC along ${\\bf T}_2$ and either \nPBC or ABC along ${\\bf T}_1$. Instead, the gapless wave function corresponds to ABC along ${\\bf T}_1$ and ${\\bf T}_2$. The remaining option,\nwith ABC along ${\\bf T}_2$ and PBC along ${\\bf T}_1$, is not considered because it implies zero-energy modes in the unprojected spectrum. \n\n\\begin{figure*}\n\\includegraphics[width=\\columnwidth]{fig8a.pdf}\n\\includegraphics[width=\\columnwidth]{fig8b.pdf}\n\\caption{\\label{fig:kagome_sqw} \nDynamical structure factor for the Heisenberg model on the kagome lattice. The cylindrical geometry with $L_1=24$ and $L_2=4$ is taken. \nThe results for the gapped state with ABC-ABC (left panel) and the gapless one with ABC-PBC (right panel) are reported along the path in \nthe extended Brillouin zone indicated in the inset. A Gaussian smearing of the delta-functions of Eq.~(\\ref{eq:dsf}) is applied, with \n$\\sigma=0.03J$.}\n\\end{figure*}\n\nThe best variational energy, ${E\/J_1=-0.49837(1)}$, is obtained by the two gapped states (with PBC-PBC and ABC-PBC), which yield equivalent \nresults within the statistical errors. Indeed, these two gapped {\\it Ans\\\"atze} correspond to the same physical state, as demonstrated by the \nfact that the overlap between them (directly computed within Monte Carlo sampling) gives $1$ for the $84 \\times 6$ cylinder. On the other hand, \nthe gapless {\\it Ansatz} has a considerably higher energy, i.e., ${E\/J_1=-0.48881(2)}$. Size effects, due to the finiteness of $L_1$, are \nnegligible. This represents the first important result of this work, showing that a gapless two-dimensional spin liquid may turn into gapped \nwhen confined in a cylindrical geometry. The fact that the best variational state actually describes a gapped phase is proven by the dynamical \nstructure factor of Eq.~(\\ref{eq:dsf}), see Fig.~\\ref{fig:square_sqw}. The gapped state shows an intense and dispersive branch, which has a \nminimum at ${\\bf q}=(\\pi,\\pi)$. A weak continuum is also visible at higher energies. By contrast, the gapless state sustains low-energy \nexcitations and even a few states with negative energies are present close to ${\\bf q}=(\\pi,\\pi)$. Within this approach, negative excitation \nenergies are possible when the reference state $|\\Phi_0\\rangle$ is not the optimal variational minimum.\nThen, it may happen that some of the states defined in Eq.~(\\ref{eq:psinq}) end up having a lower energy than the one of $|\\Psi_0\\rangle$. \nA similar situation was detected in the unfrustrated Heisenberg model by using a spin-liquid state as variational \n{\\it Ansatz}~\\cite{dallapiazza2015}. However, it is worth stressing that all the wave functions~(\\ref{eq:psinq}) constructed from the gapless \n{\\it Ansatz} have much higher energies than the one of the gapped state, which remains the best variational approximation for the ground state.\n\nGapped and gapless states show also remarkably different spin-spin and dimer-dimer correlation functions, see Fig.~\\ref{fig:piflux_spinspin}. \nThe gapped state possesses rapidly decaying spin-spin and dimer-dimer correlations. The vanishingly small values of the long-range dimer-dimer \ncorrelations suggests that no valence-bond order is present. Combining this observation with the fact that the two gapped {\\it Ans\\\"atze} \n(with PBC-PBC and ABC-PBC) correspond to the same physical state, we conclude that the ground state on cylindrical geometries is consistent \nwith a trivial quantum paramagnet, with neither valence-bond nor topological order.\n\nIn order to perform a more detailed analysis of the size-dependence of the correlation functions, we adopt a simpler variational {\\it Ansatz}, \nnamely the $\\pi$-flux state of Fig.~\\ref{fig:square}, which contains only hopping terms. The results obtained by the $\\pi$-flux state are very \nsimilar to the ones given by the optimal variational wave function. For example, the energies of gapped and gapless states on the $84 \\times 6$ \ncylinder (with $J_2\/J_1=0.5$) are $E\/J_1=-0.49661(1)$ and $E\/J_1=-0.48769(1)$, respectively. Moreover, also static correlation fluctions are \nvery similar to those of the optimized $Z_2$ spin liquid, see Fig.~\\ref{fig:piflux_spinspin}. Indeed, the overlap between the optimal wave \nfunction and the simple $\\pi$-flux state is approximately $0.8$ ($0.9$) for gapped (gapless) cases. First of all, we prove that the gapless \n{\\it Ansatz} has power-law spin-spin correlations, which might be difficult to detect from the results shown in Fig.~\\ref{fig:piflux_spinspin}. \nIn order to perform a convincing size scaling, we take $L_2=4$ and perform calculations for different values of $L_1$, see Fig.~\\ref{fig:piflux_4xL}. \nFrom these results, we obtain that the spin-spin correlations decay to zero as $1\/L_1$ that is fully compatible with a power-law behavior \n$|{\\cal S}({\\bf R})| \\approx 1\/|\\mathbf{R}|^\\beta$ with an exponent $\\beta \\approx 1$.\n\nFinally, we compute the spin-spin and dimer-dimer correlations of the $\\pi$-flux state for different cylindrical geometries with $20L \\times L$ \nsites, as shown in Fig.~\\ref{fig:largeL_spinspin}. These calculations are easily affordable because the wave function, which does not correspond \nto the optimal state, does not contain any free variational parameter. Thus, no numerical optimization are required. We observe that the \nshort-distance correlations of the gapped state strengthen when increasing $L$, rapidly approaching the ones of the gapless state. Still, even \nfor cylinders with a relative large width (i.e., $L=10$), the results obtained from gapless and gapped states are quite different from each other, \nindicating a particularly slow convergenece to the truly two-dimensional limit. Interestingly, the spin-spin correlations of the gapless wave \nfunction show a crossover between a power-law behavior with $\\beta \\approx 2$ for $|\\mathbf{R}| \\lesssim L$ and $\\beta \\approx 1$ for \n$|\\mathbf{R}| \\gg L$. The former one is typical of the two-dimensional limit, see Fig.~\\ref{fig:squakag}, while the latter one is characteristic \nof quasi-one-dimensional systems, as also obtained in Fig.~\\ref{fig:piflux_4xL}.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig9.pdf}\n\\caption{\\label{fig:kagome_4xL}\nAbsolute value of the spin-spin correlations ($|{\\cal S}({\\bf R})|$) on $L_1 \\times 4$ cylinders for the $\\pi$-flux spin-liquid on the kagome lattice. \nEmpty symbols refer to the gapped state with ABC-ABC, while full symbols refer to the gapless state with ABC-PBC.}\n\\end{figure}\n\n\\subsection{Kagome-lattice geometry}\nLet us now turn to discuss the results for the kagome-lattice geometry, see Fig.~\\ref{fig:kagome}, with nearest-neighbor super-exchange $J$.\nAlso for this highly-frustrated problem, in the two-dimensional limit, the Gutzwiller-projected fermionic state suggested that the ground\nstate corresponds to a gapless spin liquid~\\cite{ran2007,iqbal2013}. In this case, a very accurate {\\it Ansatz} has only nearest-neighbor\nhopping, such that fermions experience a $\\pi$-flux piercing the elementary unit cell (i.e., $\\pi$-flux through hexagons and $0$-flux \nthough triangles). As for the $\\pi$-flux on the square lattice, the unprojected fermionic spectrum possesses Dirac points~\\cite{ran2007}.\nThis {\\it Ansatz} corresponds to a $U(1)$ spin liquid. \n\nWe start by taking a cylinder with $L_1=24$ and $L_2=4$. Here, the fermionic Hamiltonian~(\\ref{eq:generic_mf}) is gapped by taking ABC along \n${\\bf T}_2$ and either PBC or ABC along ${\\bf T}_1$; the gapless wave function has ABC along ${\\bf T}_1$ and PBC along ${\\bf T}_2$. The fourth\npossibility with PBC along both ${\\bf T}_1$ and ${\\bf T}_2$ leads to zero-energy modes in the spectrum. As for the square lattice, the two\noptions for the gapped states correspond to the same physical wave function, after Gutzwiller projection (as verified by computing the overlap\nbetween them). \n\nThe gapped state has the best variational energy per site $E\/J=-0.43021(3)$, while the gapless wave function has a much higher energy, i.e., \n$E\/J=-0.42672(3)$, confirming that also in this case the gapped {\\it Ansatz} is favored on the cylindrical geometry, even though the \ntwo-dimensional case is gapless. In Fig.~\\ref{fig:kagome_sqw}, we show the dynamical structure factor for both gapped and gapless states. \nNotice that, in contrast to the square-lattice geometry, here no negative-energy states are obtained for the gapless {\\it Ansatz}. In addition, \nthe results for this latter case is qualitatively similar to the spectrum recently obtained in the truly two-dimensional case~\\cite{zhang2020}. \nA broad continuum of excitations is observed for both the gapped and the gapless state. Notably, in both cases, the maximum of the spectral \nintensity is concentrated around the $\\Gamma^\\prime$ point (as observed also in Ref.~\\cite{zhu2019}), within a weakly dispersive branch of \nexcitations which is not separated from the continuum at higher energy.\n\nAnother remarkable feature of this {\\it Ansatz} on the kagome lattice is the rapid decay of the spin-spin correlation functions, even when \ncomputed for the gapless wave function. In Fig.~\\ref{fig:kagome_4xL}, we report the correlations between spins on the same sublattice (a small \ndifference for the three possible sublattices is detected, not changing the qualitative behavior). Here, gapped and gapless {\\it Ans\\\"atze} \ngive almost indistinguishable correlation functions for distances larger than $4$ lattice spacings. This aspect is very different from what\nwe obtained in the square lattice, where gapped and gapless states show rather distinct correlation functions, see Fig~\\ref{fig:piflux_spinspin}.\nThe behavior is not much modified by increasing the number of legs, as reported for $10L \\times L$ cylinders in Fig.~\\ref{fig:kagome_10LxL}.\nAlso in this case, there is no apreciable difference between gapped and gapless states, at variance with the square lattice case, where the\ngapless state displays sizable correlations, substantially different from the ones of the gapped case. Quite surprisingly, also in the truly \ntwo-dimensional clusters, spin-spin correlations decay very rapidly with the distance, still being compatible with a power-law decay with an \nexponent $\\beta \\approx 2$ (similarly to the one obtained in the square lattice), see Fig.~\\ref{fig:squakag}. However, in the kagome lattice, \na small prefactor strongly renormalizes the whole behavior of correlations.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig10.pdf}\n\\caption{\\label{fig:kagome_10LxL}\nAbsolute value of the spin-spin correlations ($|{\\cal S}({\\bf R})|$) on $10L \\times L$ cylinders for the $\\pi$-flux spin-liquid on the kagome \nlattice. Empty symbols refer to the gapped state (with ABC-ABC for $L=4n$ and ABC-PBC for $L=4n+2$), while full symbols refer to the gapless \nstate (with ABC-PBC for $L=4n$ and ABC-ABC for $L=4n+2$).}\n\\end{figure}\n\n\\section{Conclusions}\\label{sec:concl}\n\nIn conclusions, we performed a systematic study of gapless spin liquids, defined within a fermionic parton construction, on cylindrical geometries. \nWe focused on cases that have been widely investigated in the past and have been demonstrated to accurately represent the exact ground-state\nproperties of two-dimensional square and kagome lattice models~\\cite{hu2013,iqbal2013}. In both cases, Dirac points are present in the unprojected \nspectrum of the fermionic spinons. In this work, we considered $L_1 \\times L_2$ cylindrical clusters with $L_1 \\gg L_2$, in order to include a \nfinite-size gap due to the finiteness of the cylinder width $L_2$. By playing with the boundary conditions of fermionic operators, both gapless \nand gapped wave functions can be constructed. Strikingly, the gapped states have a lower variational energy, better approximating the true ground \nstate on cylinders. In addition, the gapped states are trivial, i.e., they do not possess local or non-local (topological) order, which is not in \ncontrast to the Lieb-Schultz-Mattis theorem ($L_2$ is taken to be even).\n\nOur results are important for two reasons. First of all, they show that two-dimensional gapless spin liquids are quite delicate states of matter\nand can be destabilized on constrained geometries, i.e., on cylinders. Here, the same construction that has been used in two dimensions leads\nto both gapless and gapped states, just by changing boundary conditions in the underlying parton picture. Gapped states give rise to much lower\nvariational energies, implying their stabilization on cylinders with an even number of legs. Second, they give a transparent argument to explain \nwhy previous DMRG calculations~\\cite{yan2011,depenbrock2012,jiang2012,jiang2012b,gong2014} failed to detect gapless phases in frustrated models. \nEven though our approach does not give an exact solution of the problem, it strongly suggests that the actual ground state on cylinders is gapped,\nthus giving a neat interpretation of DMRG results for the models considered here. In this regard, an important step forward has been the inclusion \nof fictitious magnetic fluxes (in the original spin model), which allowed DMRG to detect the existence of Dirac points in kagome and triangular \nlattices~\\cite{he2017,hu2019}.\n\n\\section*{Acknowledgements}\nWe thank D. Poilblanc for discussions in the early stage of the project. F.F. acknowledges support from the Alexander von Humboldt Foundation \nthrough a postdoctoral Humboldt fellowship.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe statistical features of coupled systems have attracted considerable\ninterest in many branches of physics. Random matrix theory (RMT) has\nbeen successfully used in many of those investigations. RMT was\nfounded by Wigner~\\cite{Wigner}. It is a schematic model~\\cite{Mehta}\nin which the Hamiltonian, or, more generally, the wave operator of the\nsystem, is replaced by a random matrix. The necessary prerequisite is\nthat the system be sufficiently ``complex,'' implying that the matrix\nelements of the Hamiltonian, or wave operator, calculated in an\narbitrary basis, behave like random numbers. It has been shown that\nthe spectral fluctuations in numerous different systems, if measured\non the scale of the local mean-level spacing, are very well modeled by\nRMT; see the reviews in Refs.~\\cite{GMGW,Stockmann,Haake}. Due to the\nconnection with chaos, one frequently refers to those systems as\nquantum chaotic which show correlations of RMT type.  Similarly,\nsystems are often referred to as regular if they lack spectral\ncorrelations.\n\nWe consider two coupled systems. We assume that either the two systems\nare chaotic before they are coupled or that the coupling itself\nintroduces chaoticity if the separate systems are regular. This\nscenario is equivalent to the breaking of symmetries, if only two\nvalues of the quantum number belonging to that symmetry are taken into\naccount.  The statistical features crucially depend on the strength of\nthe coupling measured on the scale of the local mean-level spacing.\nMany studies have been devoted to this issue of chaotically coupled\nsystems or, equivalently, to symmetry breaking. We mention isospin\nbreaking in nuclear physics~\\cite{values,GW}, symmetry breaking in\nmolecular physics~\\cite{Leitner}, symmetry breaking in resonating\nquartz crystals~\\cite{cristais2} and coupled microwave\nbilliards~\\cite{nos}. While these studies addressed the spectral\ncorrelation, several investigations in nuclear\nphysics~\\cite{26Al,22Na,BGH,HNSA} focused on the statistics of the\nwave functions and related observables in the presence of symmetry\nbreaking or similar effects.  In all these cases, RMT approaches in\nthe spirit of the Rosenzweig-Porter model~\\cite{Porter} were\nsuccessful.\n\nSometimes observables in the time domain such as spectral form factors\nare more appropriate than the eigenenergy correlations\nfunctions~\\cite{Mehta,GMGW,Stockmann,Haake}. This is so, for example,\nin the case of the presently much discussed fidelity; see\nRefs.~\\cite{CT,BCV,PSZ,VH} and references therein. Another example is\nthe study of the energy spread in chaotic\nsystems~\\cite{cohen00,kottos01}.  In the context of coupled systems,\nthe time evolution of wave packets was investigated in\nRef.~\\cite{BTU}.\n\nIn the present contribution, we study energy localization in two\ncoupled systems in the time domain. This problem was addressed in a\nrecent work by Weaver and Lobkis~\\cite{WL} who measured the time\ndependence of the wave intensity distribution in two coupled\nreverberation rooms. To this end, these authors recorded the time\nresponse to an elastic excitation of two coupled aluminum cubes.\nMoreover, they investigated the same problem theoretically and they\nnumerically calculated the response in coupled two-dimensional\nmembranes. In our study, we se tup and analyze an RMT model, based on\nthe approaches in Refs.~\\cite{Porter,GW,BGH}. Its general character\nmakes our model useful for similar problems in different physics\ncontexts. In particular, we expect that our RMT approach also applies\nto coupled quantum dots.\n\nThe article is organized as follows. In Sec.~\\ref{sec:exp_and_model}\nwe sketch the work of Weaver and Lobkis~\\cite{WL}. In\nSec.~\\ref{sec:RMT} we set up the RMT model and analyze it analytically\nand numerically.  We compare our results to those of Weaver and Lobkis\nin Sec.~\\ref{sec:RMT_result}. Discussion and conclusions are given in\nSec.~\\ref{sec:dic_concl}.\n\n\n\\section{Experiment and numerical calculations}\n\\label{sec:exp_and_model}\n\nAs we aim at a comparison with their findings, we present the work of\nWeaver and Lobkis~\\cite{WL} in some detail. Thereby, we also introduce\nthe notation and conventions. The system studied experimentally consists\nof two aluminum cubes coupled by a solid connection, manufactured out\nof a solid aluminum block.  The corners of the cubes were removed to\ndesymmetrize the structure. This was done to ensure ``chaotic''\nmotion. Elastomechanical wave modes were excited in one room, and the\nresponse was measured in the other room. In this way, 16\ndifferent curves of energy intensity versus time were recorded, each\nin a small region around a different frequency. The results show that\nthe energy does not always spread equally over the two rooms. If the\ncoupling is weak, then the wave intensity is higher in the room where\nthe initial excitation was performed than in the other room,\nregardless of how long one waits. Hence, the energy ratio never\napproaches unity. This deviation from the equipartition of the energy\nin the two rooms is referred to as energy localization. The resulting\ndata are shown in Fig.~\\ref{fig:WL_exp}, and as expected there is\nlocalization in the bins of larger mean-level spacing, but not in the\nbins of small mean-level spacing. We will discuss these results\nfurther in Sec.~\\ref{sec:RMT_result}.\n\n\\begin{figure}\n  \\centering \n  \\subfigure[Low frequency\n  bins.]{\\includegraphics[height=4cm]{tccs_fig_1a.eps} \n  \\label{fig:exp_loc}} \\quad \n  \\subfigure[High frequency bins.]{\\includegraphics[height=4cm]{tccs_fig_1b.eps}\\label{fig:exp_nonlocalised}} \n  \\caption{Results from the experiment by Weaver and Lobkis. The\n  energy ratio between the different rooms is plotted versus time (in\n  ms). We note the different\n  scales on the $y$ axes. Reprinted from Ref.~\\cite{WL}.}   \n\\label{fig:WL_exp}\n\\end{figure}\n\nTwo numerical studies were performed on membranes with rough\nboundaries~\\cite{WL}. The dynamics of the system was in both cases\ngoverned by discretized wave equations in each of the rooms, and the\ncoupling was realized in different ways in the two numerical\ncalculations, to which we refer as N1 and N2 in the sequel. In the\nfirst one, N1, the connection had the form of a window between the two\nmembranes similar to the situation in the experimental setup. In the\nsecond numerical calculation, N2, the rooms were separated, but\nsprings were attached to a few different sites in rooms 1 and \n2, thereby coupling those sites. In N1 and N2, a nonvanishing\ninitial condition was given to one site in one of the rooms, and the\nresponse was calculated at different sites in the other room. The\nresulting time series were cosine-bell time windowed to focus on a\nspecific instant in time. Then, the time series were Fourier\ntransformed and integrated over a small region in frequency to\naccumulate data around a certain frequency. As in the experiment,\n16 different curves of intensity versus time were obtained around\ndifferent frequencies. At different frequencies, the systems have\ndifferent effective couplings. Therefore, one expects~\\cite{WL} the\ntime behavior of the different curves to differ in the degree of\nlocalization, as well as in the way in which this asymptotic\nsaturation value is reached. Due to the differences in the coupling\nmechanism, there will also be differences between the results of N1\nand N2.\n\nMoreover, Weaver and Lobkis performed an analytical model study.  The\nelastic wave equation for the state of the system $U(t)$, say, is of\nsecond order in time. To focus on the response in a narrow interval\naround a certain frequency $\\Omega$, the ansatz $U(t) =\nu(t)\\exp(-i\\Omega t)$ is made with the assumption that $u(t)$ varies\nslowly with time. This leads to a first-order differential equation in\ntime for $u(t)$ which has the form\n\\begin{equation}\n-i\\frac{\\partial}{\\partial t} u(t) = \\frac{C+\\Omega^2}{2 \\Omega} u(t) \\ , \n\\label{eq:H_v_def}\n\\end{equation}   \nwhere $C$ is the wave operator of the original second-order equation.\nThe energies $E_1(t)$ and $E_2(t)$ in rooms 1 and 2 at time $t$\nare defined as the total probability density of finding the system\nstate $u(t)$ in one of the states $\\psi_{ik}, \\ i=1,2$, which are good\neigenstates in room $i$:\n\\begin{equation}\nE_i(t) = \\sum_k \\left| u(t) \\cdot \\psi_{ik} \\right| ^2 \\ .\n\\end{equation}\nStrictly speaking, $E_i(t)$ is no energy. Nevertheless, we find this\nterminology introduced in Ref.~\\cite{WL} appropriate and use it as\nwell, because $E_i(t)$ measures the degree of motion in room $i$.  If\nno energy dissipates into the surrounding environment, the total\nenergy $E=E_1(t)+E_2(t)$ is conserved -- i.e., independent of time. In\nthe sequel, it is always assumed that the system is excited in room\n1, and the energy is measured in room 2.  Analytical solutions for\ntwo coupled states are presented in Ref.~\\cite{WL} by employing different\nstatistical assumptions.\n\n\n\\section{Random matrix model}\n\\label{sec:RMT}\n\nWe set up the model in Sec.~\\ref{sec:RMT_model}. The connection to the\ntwo-level form factor is established in Sec.~\\ref{RMTff}, and a\n$2\\times 2$ version of the model is evaluated in Sec.~\\ref{M22}.  We\ndiscuss numerical simulations of the RMT model in\nSec.~\\ref{sec:RMT_num}. Finally, we comment on chaotically coupled\nregular systems in Sec.~\\ref{sec:RMT_pos}.\n\n\n\\subsection{Setup of the model}\n\\label{sec:RMT_model}\n\nSpectral correlations in elastomechanics have been shown to be well\ndescribed by RMT \\cite{Weaver}. This is also true in the case of\nsymmetry breaking~\\cite{cristais2}, which is of direct relevance for\nthe present study. Thus, RMT is also likely to be capable of modeling\nthe time behavior of elastomechanical systems.  As the first-order\nequation~\\eqref{eq:H_v_def} has proved to be a good approximation to\nthe experimental situation, we also base our model on this\nSchr\\\"odinger type of equation. Thus, it is more natural to\nreplace $(C+\\Omega^2)\/2\\Omega$ by the random matrix $H$ than to\nreplace $C$ itself. It turns out that this is indeed the best choice.\n\nThe appropriate RMT model is an extension of the one employed in\nRefs.~\\cite{GW,BGH}. The random matrix $H$ modeling the operator\n$(C+\\Omega^2)\/2\\Omega$ reads\n\\begin{equation}\nH = \\left[ \\begin{array}{cc} H_1 & 0 \\\\ \n                               0 & H_2 \\end{array} \\right] + \n    \\alpha \\left[ \\begin{array}{cc}  0 & V \\\\ \n                                     V^\\dagger & 0 \\end{array} \\right] \\ .\n\\label{eq:RMT_model_def}\n\\end{equation}  \nThe two matrices $H_i, \\ i=1,2$, model the uncoupled rooms 1 and\n2. They are real symmetric and have random entries. We draw them\nfrom (two independent) Gaussian orthogonal ensembles (GOE's). As the\nrooms in the experiments and the numerical calculations N1 and N2 were\nof the same size, the level densities were also the same. This can be\nadjusted in the RMT model by giving the matrices $H_i$ the same\nstatistical weights and the same dimension $N$, such that $2N$ is the\ntotal dimension of $H$. The matrix $V$ also has random entries. The\nstrength of the coupling is measured by the dimensionless parameter\n$\\alpha$. It is sufficient to always assume $\\alpha \\ge 0$. The\nstatistical weight of $V$ is chosen such that the total $H$ is in the\nGOE of $2N\\times 2N$ matrices for $\\alpha = 1$.\n\nWe write the eigenvalue equation for the total Hamiltonian $H$ in the form\n\\begin{eqnarray}\nH\\Psi_{n} = \\omega_{n}\\Psi_{n} \\ , \\qquad n=1,\\ldots,2N \\ .\n\\label{evh}\n\\end{eqnarray}\nThe eigenvalues $\\omega_{n}$ and eigenvectors $\\Psi_{n}$ are functions\nof the coupling parameter $\\alpha$. It is convenient to introduce the\nnotation\n\\begin{eqnarray}\n\\Psi_{n} =  \\left[ \\begin{array}{c} \\Psi_{1n} \\\\ \n                                    \\Psi_{2n} \\end{array} \\right] \\ ,\n\\label{psi12}\n\\end{eqnarray}\nwhere $\\Psi_{in}, \\ i=1,2$, is the projection of $\\Psi_n$ onto the\nsubspace $i$. We emphasize that $\\Psi_{in}, \\ i=1,2$, are\nfunctions of the coupling parameter $\\alpha$. The eigenvalue equations\nfor the Hamiltonians $H_i$ are written as\n\\begin{eqnarray}\nH_1\\psi_{1n} &=& \\omega_{1n}\\psi_{1n} \\ , \\qquad n=1,\\ldots,N \\ , \\nonumber\\\\\nH_2\\psi_{2n} &=& \\omega_{2n}\\psi_{2n} \\ , \\qquad n=1,\\ldots,N \\ ,\n\\label{evhi}\n\\end{eqnarray}\nwhere the eigenvalues $\\omega_{in}$ and eigenvectors $\\psi_{in}$ are\nnot functions of $\\alpha$. This difference between $\\Psi_{in}$ and\n $\\psi_{in}$ is an immediate consequence of the fact that $H$\ndepends on $\\alpha$, while $H_1$ and $H_2$ do not. We will also use\nthe notation $\\widehat{\\psi}_{2n}=(0,\\psi_{2n})^\\dagger$ for the\ncorresponding $2N$-dimensional vector with zeros in the first $N$\ncomponents.\n\nAt time $t=0$, the system is excited in room one such that the state\nof the total system can be written as\n\\begin{eqnarray}\nS =  \\left[ \\begin{array}{c} s \\\\ \n                             0 \\end{array} \\right] \\ ,\n\\label{source}\n\\end{eqnarray}\nwhere $s$ in room one is not specified in detail. We refer to $S$ as\nto the source. The time evolution of the source is then simply\n\\begin{eqnarray}\nu(t) = T(t)S \\ , \\qquad {\\rm where} \\qquad\nT(t) = \\exp\\left(iHt\\right)\n\\label{teo}\n\\end{eqnarray}\nis the time evolution operator.  Using the eigenvectors $\\widehat{\\psi}_{2n}$,\nthe energy in room 2 is thus given by\n\\begin{eqnarray}\nE_2(t,\\alpha) &=& \\sum_{n=1}^{N} \\left| u(t) \\cdot \\widehat{\\psi}_{2n} \\right|^2 \n        = \\sum_{n=1}^{N} \\left( T(t)S \\right) ^{\\dagger}\\widehat{\\psi}_{2n}\n             \\widehat{\\psi}_{2n}^\\dagger \\left( T(t)S \\right) \\nonumber \\\\\n       &=& S^{\\dagger} T^{\\dagger}(t) \n             \\left[\\begin{array}{cc} 0 & 0\\\\\n                                     0 & 1_N \\end{array} \\right]\n             T(t) S \\ .\n\\label{eq:RMT_E2}\n\\end{eqnarray}\nThe block matrix only contains the unit matrix $1_N$ for room 2.  We\nwrite $E_2(t,\\alpha)$ instead of $E_2(t)$ to emphasize the dependence\non the coupling parameter $\\alpha$ in the RMT model. We will study\naverages $\\overline{E_2}(t,\\alpha)$ over the ensemble of matrices\nintroduced in Eq.~\\eqref{eq:RMT_model_def}.  Occasionally, we will\nalso consider averages over the direction of the source. This average\nis denoted by angular brackets $\\langle \\cdots \\rangle$.\n\nThe total energy $E$ is trivially conserved in the framework of our\nmodel. We have\n\\begin{eqnarray}\nE = E_1(t,\\alpha)+E_2(t,\\alpha) = S \\cdot S = s \\cdot s \\ .\n\\label{etot}\n\\end{eqnarray}\nHence, we can always construct $E_1(t,\\alpha)$, once $E_2(t,\\alpha)$\nhas been calculated.\n\nAs all correlations have to be\nmeasured~\\cite{Mehta,GMGW,Stockmann,Haake} on the local scale of the\nmean-level spacing $D$, we introduce an unfolded time $\\tau=Dt$. This\nis also the scale on which $\\alpha$ acts; we introduce the unfolded\ncoupling parameter $\\lambda=\\alpha\/D$. The energy on the unfolded\nscale is then given by\n\\begin{eqnarray}\n\\varepsilon_2(\\tau,\\lambda) =\n            \\lim_{N\\to\\infty} \\overline{E_2}(\\tau\/D,D\\lambda).\n\\label{eun}\n\\end{eqnarray}\nSmall values of $\\alpha$ have a large effect on the correlation\nfunctions if the mean-level spacing $D$ is small as well. \n\n\n\\subsection{Relation to the two-level form factor}\n\\label{RMTff}\n\nWe now derive an estimate for the time evolution of $E_2(t,\\alpha)$\nwhich should apply to strong coupling strength-- i.e., to a parameter\n$\\alpha$ which is large on the unfolded scale of the mean level\nspacing. If one assumes that the source $S$ comprises excitations into\nall states $\\psi_{1n}$, one may average over the direction of the\nsource. We find, from Eq.~\\eqref{eq:RMT_E2},\n\\begin{eqnarray}\n\\langle E_2\\rangle (t,\\alpha) = B {\\rm tr}^{(11)}\\,T^{\\dagger}(t) \n             \\left[\\begin{array}{cc} 0 & 0\\\\\n                                     0 & 1_N \\end{array} \\right] T(t) \\ ,\n\\label{e2est1}\n\\end{eqnarray}\nwhere ${\\rm tr}^{(11)}$ is the trace over the $(11)$ sector of the\nwhole matrix -- i.e., over the upper left block. The constant $B$ results\nfrom the average. Expanding the time evolution operator in terms of\nthe eigenvectors,\n\\begin{eqnarray}\nT(t) = \\sum_{n=1}^{2N} \\Psi_n \\exp\\left(i\\omega_nt\\right) \\Psi_n^\\dagger \\ ,\n\\label{teoexp}\n\\end{eqnarray}\nwe arrive after a short calculation at\n\\begin{eqnarray}\n\\langle E_2\\rangle (t,\\alpha) = B \\sum_{n,m} \n                        \\left(\\Psi_{1m}\\cdot\\Psi_{1n}\\right)\n                        \\left(\\Psi_{2n}\\cdot\\Psi_{2m}\\right)\n                        \\exp\\left[i(\\omega_n-\\omega_m)t\\right] \\ .\n\\label{e2est2}\n\\end{eqnarray}\nThe vectors $\\Psi_{1n}$ and $\\Psi_{2n}$ are, according to\nEq.~\\eqref{psi12}, the projections of the full eigenvector $\\Psi_{n}$\nonto the subspaces corresponding to the two rooms. They depend on the\ncoupling parameter $\\alpha$. These vectors coincide with the\neigenvectors for the matrices $H_1$ and $H_2$ only for $\\alpha=0$.\nHence, they are only orthogonal in this special case. For all values\n$\\alpha>0$, the scalar products $\\Psi_{1m}\\cdot\\Psi_{1n}$ and\n$\\Psi_{2n}\\cdot\\Psi_{2m}$ are neither zero nor given by $\\delta_{nm}$.\nThe ensemble average over the eigenenergies $\\omega_n$ and over the\neigenfunctions $\\Psi_{n}$ decouples only for the parameter values\n$\\alpha=0$ and $\\alpha=1$-- i.e., if the system either falls into two\nGOE's or is represented by one GOE. We now estimate the ensemble average\nof the energy $\\langle E_2\\rangle (t,\\alpha)$ by making the\napproximation that the averages over eigenenergies and eigenfunctions\ndecouple. For strong coupling this should yield a reasonable result,\nand we will use this approximation in that limit only.  We average\nover the eigenfunctions. For each term in the double sum in\nEq.~\\eqref{e2est2} the product of scalar products gives the same\ncontribution which we can take out of the sums. Thus, we find\n\\begin{eqnarray}\n\\overline{\\langle E_2\\rangle} (t,\\alpha) \n = B \\overline{\\sum_{n,m}\\exp\\left[i(\\omega_n-\\omega_m)t\\right]} \\ ,\n\\label{e2est3}\n\\end{eqnarray}\nwhere we absorbed the contributions from the average over the scalar\nproducts into the constant $B$. The average over the\neigenvalues is yet to be performed.  Luckily, the average in\nEq.~\\eqref{e2est3} is recognized as the Fourier transform of the\ntwo--level correlation function~\\cite{Mehta,GMGW}. This yields, on the\nunfolded scale,\n\\begin{eqnarray}\n\\varepsilon_2(\\tau,\\lambda) = B \\left[\\delta(\\tau\/2\\pi) \n                            + K_2(\\tau\/2\\pi,\\lambda)\\right] \\ ,\n\\label{e2est4}\n\\end{eqnarray}\nwhere the function $K_2(\\tau,\\lambda)$ is referred to as the two-level\nform factor. The term $\\delta(\\tau)$ is due to the diagonal\ncontributions in the double sum of Eq.~\\eqref{e2est3}. We notice that\nthe conventions used here require a rescaling of time with a factor of\n$2\\pi$.  The expression~\\eqref{e2est4} is an estimate for the energy\n$\\varepsilon_2(\\tau,\\lambda)$, exclusively in terms of the \nFourier-transformed two-level spectral correlation function for the\ntransition from two GOE's to one GOE. Unfortunately, the two-level form\nfactor is not known analytically for all values of $\\lambda$.  The\nresult for one GOE, corresponding to $\\lambda\\to\\infty$,\nreads~\\cite{Mehta}\n\\begin{equation}\nK_2(\\tau,\\infty) = \\left\\{ \n            \\begin{array}{ll} 2\\tau-\\tau \\ln(2\\tau+1) & \n                                   {\\rm for} \\quad 0<\\tau\\le 1 , \\\\ \n                              2-\\tau \\ln\\frac{{\\textstyle 2\\tau+1}}\n                                             {{\\textstyle 2\\tau-1}} &\n                                   {\\rm for} \\quad \\tau\\ge 1 .\n            \\end{array} \\right. \n\\label{eq:formfactor_result}\n\\end{equation} \nWe notice that the function $\\varepsilon_2(t,\\infty)$ has the\nlimit properties\n\\begin{eqnarray}\n\\varepsilon_2(0,\\infty) &=& 0 , \\nonumber \\\\\n\\lim _{\\tau \\rightarrow \\infty} \\varepsilon_2(\\tau,\\infty) &=& B \\ .\n\\label{eq:ff_props}\n\\end{eqnarray}\nThe second property will be used to fix the scale in comparison with\nthe results of Ref.~\\cite{WL}.  The estimate~\\eqref{e2est4} should\napply to large coupling -- i.e., to parameter values $\\lambda\\gg 1$.\n\n\n\\subsection{Two-by-two model}\n\\label{M22}\n\nQuite often, one obtains surprisingly good information about an RMT\nmodel by restricting it to the smallest possible matrix dimension such\nthat the nontrivial specific characteristics of the model are still\npresent. This is successful in the case of the nearest-neighbor\nspacing distribution for the Gaussian orthogonal, unitary and\nsymplectic ensembles (GGOE, GUE and GSE, respectively) ~\\cite{GMGW,Stockmann,Haake}, but also for crossover\ntransitions~\\cite{lenz}. Here, we proceed analogously.  We obtain a\n$2\\times 2$ RMT model by setting $N=1$ in\nEq.~\\eqref{eq:RMT_model_def}.  It turns out convenient to absorb the\nparameter $\\alpha$ into the matrix element $V$ such that\n\\begin{equation}\n  H = \\left[ \\begin{array}{cc} H_1 & V \\\\ V & H_2 \\end{array} \\right]\n\\label{m1}\n\\end{equation} \nand to readjust the probability density function $P(H)$ for the\nensemble average accordingly:\n\\begin{equation}\n  P(H) = \\frac{\\sqrt{2}}{\\pi\\sqrt{\\pi\\alpha^2}} \n               \\exp\\left(-H_1^2-H_2^2-2\\frac{V^2}{\\alpha^2}\\right) \\ .\n\\label{m2}\n\\end{equation} \nAs before, the GOE is recovered for $\\alpha=1$. We introduce\neigenvalue and angle coordinates\n\\begin{equation}\n  \\left[ \\begin{array}{cc} H_1 & V \\\\ \n                             V & H_2 \\end{array} \\right] = \n  \\left[ \\begin{array}{cc} \\cos\\varphi & -\\sin\\varphi \\\\ \n                           \\sin\\varphi &  \\cos\\varphi \\end{array} \\right]\n  \\left[ \\begin{array}{cc} \\omega_1 & 0 \\\\ \n                                  0 & \\omega_2 \\end{array} \\right] \n  \\left[ \\begin{array}{cc} \\cos\\varphi & \\sin\\varphi \\\\ \n                          -\\sin\\varphi & \\cos\\varphi \\end{array} \\right] \\ ,\n\\label{m3}\n\\end{equation} \nwhich implies that the time evolution can be written in the same form\nwith eigenvalues $\\exp(i\\omega_1t)$ and $\\exp(i\\omega_2t)$. For the\nintegration measure one has\n\\begin{equation}\ndH_1 dH_2 dV = \\frac{1}{4} |\\omega_1-\\omega_2| d\\omega_1 d\\omega_2 d\\varphi \\ .\n\\label{m4}\n\\end{equation} \nThe source in this $2\\times 2$ model is simply given by $S=(1,0)$.\nThus, collecting everything, we find, with Eq.~\\eqref{eq:RMT_E2},\n\\begin{eqnarray}\n\\overline{E_2}(t,\\alpha) &=& \\frac{\\sqrt{2}}{4\\pi\\sqrt{\\pi\\alpha^2}}\n             \\int\\limits_{-\\infty}^{+\\infty}d\\omega_1\n             \\int\\limits_{-\\infty}^{+\\infty}d\\omega_2|\\omega_1-\\omega_2|   \n             \\int\\limits_0^{2 \\pi} d\\varphi \\nonumber\\\\\n    & & \\qquad \\exp\\left[-\\frac{(\\omega_1+\\omega_2)^2}{2}-\n        \\frac{(\\omega_1-\\omega_2)^2}{2} \\left(\\cos ^2 2\\varphi + \\frac{\\sin ^2\n             2\\varphi}{\\alpha^2} \\right) \\right] \\nonumber \\\\ \n    & & \\qquad \\sin^22\\varphi \\, \\sin^2\\frac{\\omega_1-\\omega_2}{2}t \\ .\n\\label{m5}\n\\end{eqnarray}\nWe introduce $x=(\\omega_1+\\omega_2)\/\\sqrt{2}$ and\n$y=(\\omega_1-\\omega_2)\/\\sqrt{2}$ as new integration variables, perform\nthe $x$ and $\\varphi$ integrals, and arrive at\n\\begin{eqnarray}\n\\overline{E_2}(t,\\alpha) &=& \\frac{\\alpha}{2 (\\alpha + 1)} - \n \\nonumber\\\\\n & & \\qquad \\alpha \\int\\limits_0^\\infty dy\\, y \\, \n                   \\exp\\left[-y^2(1+\\alpha^2)\\right] \n \\nonumber\\\\\n & & \\qquad \\qquad \n\\left[{\\rm I}_0 (y^2 (1-\\alpha^2)) - {\\rm I}_1(y^2 (1-\\alpha^2))\\right] \n \\cos 2\\alpha yt \\, .\n\\label{eq:2d_result}\n\\end{eqnarray}\nHere, ${\\rm I}_0$ and $\\rm{I}_1$ denote the modified Bessel functions\nof zeroth and first order, respectively.  \n\nFormula~\\eqref{eq:2d_result} is the final result in the framework of\nour $2\\times 2$ model. In Ref.~\\cite{WL}, solutions for two-state\nmodels based on Eq.~\\eqref{eq:H_v_def} with different statistical\nassumptions were given. Although the general behavior is similar to\nour $2\\times 2$ RMT result~\\eqref{eq:2d_result}, the analytical forms\nof these solutions in Ref.~\\cite{WL} are different.\n\nFor obvious reasons, the unfolding of formula~\\eqref{eq:2d_result} is\nmeaningless.  Nevertheless, experience with $2\\times 2$ model for the\nspacing distribution tells that meaningful statements can be achieved\nif the transition parameter -- i.e., the coupling strength $\\alpha$ in\nour case -- is interpreted properly.  Here, we can do that in the\nfollowing manner. For large times $t$, the function\n$\\overline{E_2}(t,\\alpha)$ becomes constant, because it reaches its\nsaturation limit. The latter will depend on $\\alpha$. Thus, comparing\nthe saturation limit for the $2\\times 2$ model with that of the\n$2N\\times 2N$ RMT simulation or with those of the experiment and the\nnumerical calculations N1 and N2 allows one, in principle, to\ninterpret $\\alpha$ on the unfolded scale.  The saturation limit is\neasily obtained from Eq.~\\eqref{eq:2d_result}.  Due to the\nRiemann-Lebesgue lemma~\\cite{GR}, we have\n\\begin{eqnarray}\n\\lim_{t\\to\\infty}\\overline{E_2}(t,\\alpha) = \\frac{\\alpha}{2 (\\alpha + 1)} \\ ,\n\\label{m7} \n\\end{eqnarray}\nbecause the cosine function in the integrand oscillates so rapidly\nfor large $t$, that the integral gives zero in the limit $t\n\\rightarrow \\infty$.\n\nWe notice that there is no equipartition of the energies for\n$t\\to\\infty$ at $\\alpha = 1$. This may seem a bit unexpected, because\n$H$ is a $2\\times 2$ GOE matrix for that parameter value. It simply\nreflects the need to properly interpret the parameters, as just\ndiscussed. We study a two-state system and compare the probability\nfor a transition between the two states with the probability of\nstaying in one state. There is no reason why these should be\nequal. In the $N \\rightarrow \\infty$ limit, however, we expect\nequipartition for the full GOE. As the coupling parameter has to be\nmeasured on the scale of the local mean level spacing, the limit $N\n\\rightarrow \\infty$ corresponds to the limit $\\alpha \\rightarrow \\infty$ in\nthe $2\\times 2$ version, and in that limit the right-hand side of\nEq.~\\eqref{m7} tends to $1\/2$ and therefore to equipartition.\n\n\n\\begin{figure}[htb]\n  \\centering\n  \\includegraphics[height=4cm]{tccs_fig_2.eps}\n  \\caption{Energy ratio $E_2 \/ E_1$ versus (dimensionless) time. Data\n  from N1 of Ref.~\\cite{WL}, bin 16, plotted\n  as dashed line. The result from \n    Eq.~\\eqref{e2est4} for $\\lambda \\rightarrow \\infty$, $B$ set by property 2\n    of Eq.~\\eqref{eq:ff_props} and a rescaling of time to make the two curves\n    fit for early times, is plotted as a solid line.}\n  \\label{fig:WL_num1_vs_form_factor}\n\\end{figure} \n\n\\begin{figure}[htb]\n  \\centering \\subfigure[$2 \\times 2$ RMT\n  model, fitted to N2 bins 2,3,4 from top to bottom.]{\\includegraphics[height=4cm]{tccs_fig_3a.eps}\n\\label{fig:2d_result}} \\quad \\subfigure[Data from N2, bins\n  2,3,4 from top to\n  bottom.]{\\includegraphics[height=4cm]{tccs_fig_3b.eps}\n\\label{fig:num2_for_2d}}\n  \\caption{Comparison between the $2\\times 2$ RMT model and the low\n  frequency bins of N2 of Ref.~\\cite{WL}. The energy ratio $E_2\/E_1$ is plotted versus\n  (dimensionless) time.} \n\\label{fig:2d_and_num2}\n\\end{figure} \n\n\n\\subsection{Numerical simulations}\n\\label{sec:RMT_num}\n\nThe numerical simulations of the RMT model are performed as in\nRefs.~\\cite{GW,BGH}. The unfolding of the results, however, is not\ndone in the standard way. To compare with the numerical calculations\nof Weaver and Lobkis, it is more convenient to first unfold the level\ndensities and then ``refold'' the spectra of our model with the\nlevel densities of Ref.~\\cite{WL}. Thereby we ensure that the \nmean-level densities of the RMT model acquire a form given by the Weyl\nformula for the billiardlike system of Ref.~\\cite{WL}. The\nappropriate Weyl formula for the modal density in a square membrane\nwith Dirichlet boundary conditions reads~\\cite{WL}\n\\begin{equation}\n  \\rho^{({\\rm smooth})}(\\Omega) = \\frac{\\Omega A}{2\\pi c^2} -\n                                  \\frac{L}{4c} \\ ,\n\\label{Weyl}\n\\end{equation} \ndepending on the frequency $\\Omega$. Here, $A$ is the area of the\nmembrane and $L$ is the side length of one room. Moreover, $c$ is\nthe wave velocity and is chosen as unity here. Using twice the area of a\nroom ($A = 2 \\times 198^2$) and a value of $L$ which takes the roughness\nof the boundaries into account ($L = 2 \\times 3 \\times 198$), we aquire an\nexpression for the level density of the entire system.  A comparison\nwith the Weyl formula of Ref.~\\cite{berry_sinai} shows that the terms\nfrom the extra edges introduced to model the disorder cancel. Hence,\nwe use the formula for a square membrane, which is precisely\nEq.~\\eqref{Weyl}.  This Weyl formula is employed to refold our RMT\nmodel for both numerical calculations N1 and N2. In the latter case,\nthe system is not really a billiard, due to the springs used for the\ncoupling. Nevertheless, the Weyl formula should be a good\napproximation to the real level density.\n\nFor every simulation, we generate random matrices of dimension $2N\n\\times 2N=100 \\times 100$, unfold, refold, calculate $E_2(t,\\alpha)$,\nand average over the results of 800 such simulations. As the functions\n$E_2(t,\\alpha)$ are not measured on the unfolded scale, but on that of\nthe Weyl formula, we do not introduce the notation\n$\\varepsilon_2(\\tau,\\lambda)$ in the present context.  Moreover, we\nnotice that the numerical calculations N1 and N2 depend on the mean\nfrequency $\\Omega$ in the bin under consideration. Accordingly, we\narrive at a two-parameter family of time-dependent curves\n$\\overline{E_2}(t,\\alpha,\\Omega)$. We now associate each bin in the\nnumerical calculation performed by Weaver and Lobkis with the\ncorresponding frequency $\\Omega$. This leaves us with a one-parameter\nfamily of time-dependent curves $\\overline{E_{2\\mu}}(t,\\alpha), \\\n\\mu=1,\\ldots,16$, for each bin labeled by $\\mu$.\n\n\n\\subsection{Chaotically coupled regular systems}\n\\label{sec:RMT_pos}\n\nIn the RMT model defined by Eq.~\\eqref{eq:RMT_model_def} and in its\nsubsequent analysis, we always drew the matrices $H_i, \\ i=1,2$, from\n(two independent) GOE's, implying that we model the two rooms\nindividually as chaotic systems --- before the coupling is considered.\nOne can also assume that the two rooms are regular systems before they\nare coupled. In that case, the matrices $H_i, \\ i=1,2$, would be drawn\nfrom Poisson ensembles~\\cite{GMGW}.  We also did such numerical\nsimulations. The results are qualitatively the same if the coupling --\ni.e., the matrix $V$ -- introduces enough chaos. The main effect is an\nadjustment of the scales. What matters for the qualitative behavior is\nthe chaoticity of the total system.\n\n\n\\section{Comparison with experiment and numerical calculations}\n\\label{sec:RMT_result}\n\n\\begin{figure}\n\\centering\n\\subfigure[Bin 2.]{\n\\includegraphics[height=4cm]{tccs_fig_4a.eps}\n\\label{fig:RMT_N1_bin2}}\n\\quad\n\\subfigure[Bin 5.]{\n\\includegraphics[height=4cm]{tccs_fig_4b.eps}}\n\\\\\n\\subfigure[Bin 9.]{\n\\includegraphics[height=4cm]{tccs_fig_4c.eps}}\n\\quad\n\\subfigure[Bin 13.]{\n\\includegraphics[height=4cm]{tccs_fig_4d.eps}\n}\n\\\\\n\\subfigure[Bin 16.]{\n\\includegraphics[height=4cm]{tccs_fig_4e.eps}\n\\label{fig:RMT_N1_bin16}}\n\\quad\n\\caption{Energy ratio $E_2\/E_1$ versus (dimensionless) time. Data from\n  N1 of Ref.~\\cite{WL} plotted as dashed lines and from refolded\n  numerical simulations using the RMT model as solid lines. We notice the\n  different $E_2\/E_1$ scale in plot (a).}\n\\label{fig:RMT_result_N1}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\subfigure[Bin 2.]{\n\\includegraphics[height=4cm]{tccs_fig_5a.eps}\n\\label{fig:RMT_N2_bin2}}\n\\quad\n\\subfigure[Bin 5.]{\n\\includegraphics[height=4cm]{tccs_fig_5b.eps}}\n\\\\\n\\subfigure[Bin 9.]{\n\\includegraphics[height=4cm]{tccs_fig_5c.eps}}\n\\quad\n\\subfigure[Bin 13.]{\n\\includegraphics[height=4cm]{tccs_fig_5d.eps}\n}\n\\\\\n\\subfigure[Bin 16.]{\n\\includegraphics[height=4cm]{tccs_fig_5e.eps}\n\\label{fig:RMT_N2_bin16}}\n\\quad\n\\caption{Energy ratio $E_2\/E_1$ versus (dimensionless) time. Data from\n  N2 of Ref.~\\cite{WL} plotted as dashed lines and from refolded\n  numerical simulations using the RMT model as solid lines. We notice the\n  different $E_2\/E_1$ scale in plot (a).}\n\\label{fig:RMT_result_N2}\n\\end{figure}\n\nWe now compare the results of our RMT model with those of\nRef.~\\cite{WL}. The quantity studied in Ref.~\\cite{WL} is mostly the\nfraction of energy in room 2, or $E_2(t)\/E$. In our figures, we plot\n$E_2(t)\/E_1(t)$ instead. We first consider the estimate,\nEq.~\\eqref{eq:formfactor_result}, involving the GOE form factor and the\nresult~\\eqref{e2est4} of the $2\\times 2$ RMT model and compare with\nthe numerical calculations N1 and N2.  As formula~\\eqref{e2est4} was\nderived under the assumption of strong coupling on the scale of the\nlocal mean-level spacing, it should apply to the high-frequency bins\nof N1. Anticipating the later extraction of the coupling parameter, we\nalready now mention that indeed $\\lambda>1$ in that bin. In\nFig.~\\ref{fig:WL_num1_vs_form_factor} we compare Eq.~\\eqref{e2est4}\nwith data from bin 16 (the highest frequency bin) of N1.  A good\ndescription is obtained, although no equipartition is reached.  This\nimplies that the effective coupling $\\lambda$ is large, but not very\nlarge. We recall that formally $\\lambda\\to\\infty$ corresponds to the\nstrongest coupling $\\alpha=1$ -- i.e., to one single system.  Expansion\nof the form factor for short times $\\tau$ reveals a linear short time\ndependence of the ensemble-averaged energy\n$\\varepsilon_2(\\tau,\\lambda)$. This is in agreement with the\nanalytical discussion of Ref.~\\cite{WL}.\n\n\\begin{table}\n\\caption{Coupling parameters $\\alpha$ for the $2 \\times 2$ model used in\nFig.~\\ref{fig:2d_result}. The asymptotic saturation $E_2\/E$ values for\nN2 in bins 2, 3, and 4 are taken from Ref.~\\cite{WL}. The corresponding\ncoupling parameters are calculated from Eq.~\\eqref{m7}.}\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\nbin & $E_2\/E$ (asymptotic) & $\\alpha$ \\\\\n\\hline\n2 & 0.35 &  2.33 \\\\\n3 & 0.2  &  0.67 \\\\\n4 & 0.14 &  0.39 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:2d_cplngs}\n\\end{table}\n\nWe turn to the $2\\times 2$ RMT model and compare to bins 2, 3, and 4 of\nN2. The values of the coupling parameter $\\alpha$ are determined from\nEq.~\\eqref{m7} and given in Table ~\\ref{tab:2d_cplngs}, together with\nthe $E_2\/E$ saturation values of N2. The $2\\times 2$ RMT model curves\nand data from N2 are shown in Figs.~\\ref{fig:2d_result}\nand~\\ref{fig:num2_for_2d}, respectively. As expected, smaller values\nof $\\alpha$ correspond to a higher degree of localization -- i.e., to a\nstronger deviation from equipartition.  The general behavior of the\nresults from the $2 \\times 2$ model stays the same for large values of\n$\\alpha$. We notice that the time scale is different, because the\ndata from N2 were not unfolded.  The similarity shown in\nFig.~\\ref{fig:2d_and_num2} between the $2\\times 2$ RMT model and the\nWeaver-Lobkis results~\\cite{WL} is remarkable. This parallels the\nsuccess of $2\\times 2$ RMT models for the spacing distributions.\n\nThe RMT simulations yield a one-parameter family of curves for each\nbin, to be compared with the numerical calculations N1 and N2. We use\nvisual inspection to determine, for each bin, which curve fits best to\nthe numerical calculations N1 and N2. Typical results for some of the\nbins are shown in Figs.~\\ref{fig:RMT_result_N1}\nand~\\ref{fig:RMT_result_N2}. In the low-frequency bins of N2 there is\na discrepancy; see Fig.~\\ref{fig:RMT_N2_bin2}. The RMT simulation does\nnot overshoot its saturation value as clearly as the data of\nRef.~\\cite{WL}. This very large overshoot within a short-time interval\nis, however, borne out in the $2\\times 2$ RMT model; see\nFig.~\\ref{fig:2d_and_num2}.  By the visual fit we determine the\ncoupling parameter $\\alpha$.  We measure it on the scale of the local\nmean-level spacing. In Table ~\\ref{tab:num_RMT}, we list the in this\nmanner obtained coupling parameter $\\lambda$ for N1 and N2 for the\nbins under consideration. The $\\lambda$ values extracted for N1 go up\nwith higher bins, because the saturation value comes closer to\nequipartition. This effect is also visible for N2, but not so\npronounced, because the saturation value does not change much for\nhigher bins.\n\n\\begin{table}\n\\caption{Coupling constants $\\lambda$ on the unfolded scale resulting \nfrom the RMT simulation, determined for some bins of the numerical\ncalculations N1 and N2 with mean frequencies $\\Omega$: see\nFigs.~(\\ref{fig:RMT_result_N1}) and (\\ref{fig:RMT_result_N2}).}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\nbin & $\\Omega$ & $\\lambda$ for N1 & $\\lambda$ for N2 \\\\\n\\hline\n2 & 0.1562 & 0.4750  & 3.2316 \\\\\n5 & 0.4686 & 2.0222  & 0.7754 \\\\\n9 & 0.8851 & 3.0586  & 0.7502 \\\\\n13 & 1.3016 & 3.9093  & 0.8441 \\\\\n16 & 1.6140 & 4.2427  & 0.8899 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:num_RMT}\n\\end{table}\n\nWe find good qualitative agreement between the RMT simulation and N1\nand N2. Importantly, fluctuations between different random matrices\nare quite large, and the RMT curves presented here are averages of\n$E_2$ over 800 random matrices. Among those some show very much closer\nsimilarity to the curves of Weaver and Lobkis~\\cite{WL}, and we\nbelieve that most of the discrepancy in the low-frequency bins of N1\nare due to fluctuations resulting from the specific choice of the\nboundary and perhaps other parameters in the numerical calculations of\nRef.~\\cite{WL}. The reason for this interpretation of the discrepancy\nis the peculiar form of the fluctuations in bin 2 of N1; see\nFig.~\\ref{fig:RMT_N1_bin2}. Since these lower bins have a lower level\ndensity, they should also be more sensitive to this kind of\nfluctuations. In the intermediate-frequency bins, we find that the\nnumerical calculations N1 and N2 are well reproduced by the RMT model.\nIn the figures for the high-frequency bin, Figs.\n\\ref{fig:RMT_N1_bin16} and \\ref{fig:RMT_N2_bin16}, the RMT results are\nseen to deviate  \ndownwards for large times. It is conceivable that the short-time\nbehavior strongly depends on the specific realization of the coupling\nwhile the long-time behavior is universal. \n\nThe result from the experiment is shown in Fig.~\\ref{fig:WL_exp}. The\nqualitative features are well described by the RMT model.  It can be\nseen that the system localizes in the low-frequency bins, but does not\ndo so in the high-frequency bins. In other words, the system\napproaches equipartition of the energy. The localization behavior\ndepends on the strength of the coupling measured on the scale of the\nlocal mean-level spacing, and the mean-level density is significantly\nhigher in the high-frequency bins. It seems that the saturation value\nhas not yet been reached on the time scale visible in\nFig.~\\ref{fig:WL_exp}. A closer inspection shows an initial behavior\nsimilar to the one in the results from our numerical RMT calculations\nand a slight upwards trend of the energy ratio towards the end of the\ntime window studied. According to Ref.~\\cite{WL} this means that,\nfirst, the expected asymptotic value of $E_2\/E_1$ is reached in the\nmiddle of the time window studied and that, second, there is another\nunknown effect acting on longer time scales adding to the energy spread\nwhich is likely to be due to the coupling of the setup to the\nenvironment.  Thus, we do not attempt to extract a quantitative\nestimate of the coupling parameter $\\alpha$ for the experiment.\n\n\n\\section{Conclusion}\n\\label{sec:dic_concl}\n\nWe have set up and analyzed an RMT model to describe the time behavior\nof coupled reverberation rooms. This system shows localization effects\nunder certain conditions. Within our RMT model, we gave an estimate of\nthe strong coupling behavior which involved the two-level GOE\nform factor. Moreover, we studied the $2\\times 2$ version of our RMT\nmodel analytically for arbitrary coupling strength and performed numerical simulations for the\n$2N\\times 2N$ version. \n\n{}From the comparison with the work of Weaver and Lobkis, we conclude\nthat the RMT model yields a good qualitative description.  Moreover,\nwe find an interpretation of the localization effect by relating it to\nthe universal features of RMT models for crossover transitions.\n\nFormally similar RMT models have been studied in connection with\nsymmetry breaking. Then, the parameter $\\alpha$ measures the degree of\nsymmetry breaking. This is so for isospin breaking in nuclear\nphysics~\\cite{values,GW,26Al,BGH}, symmetry breaking in molecular\nphysics~\\cite{Leitner}, and symmetry breaking in resonating quartz\ncrystals~\\cite{cristais2}. The experiment that comes closest to the\npresent situation is the study of spectral correlations in coupled\nmicrowave billiards~\\cite{nos}. In this work and in the experiment of\nWeaver and Lobkis~\\cite{WL}, an excitation initially in system one\nwould stay there for all times if the coupling is zero. This is\nformally analogous to a conserved quantum number.\n\nIn the RMT model the parameter measuring the size of the connection\nhas a most natural counterpart: the root-mean-square matrix element\ndue to the coupling measured on the scale of the local mean-level\nspacing. Thus, there are two ways of making the effective,\ndimensionless coupling parameter $\\lambda$ small and to thereby\nintroduce localization: either the original coupling parameter\n$\\alpha$ which always has a dimension is small or the mean-level\nspacing $D$ is made large.\n\nIt might be surprising that no equipartition of the energy is seen for\nall coupling strengths even if one waits very long. This would be the\nexpectation if one compares the system with two water basins coupled\nby a channel. Suppose that initially the two basins are empty and the\nchannel is closed by a gate. Now one of the basins is filled with\nwater and the gate is opened at $t=0$. Obviously, the water levels in\nthe two basins will be equal after sufficiently long time. The speed\nwith which this equipartition is reached simply depends on the cross\nsection of the channel. \n\nIn the present case of the two coupled acoustic rooms the situation is\ndifferent, because the wave character of the excitations has to be\ntaken into account. Thus, the crucial parameter entering is the size\nof the coupling connection -- i.e., its geometrical width -- compared to a\ntypical wavelength. The first waves after the excitation which come\nfrom system 1 into system 2 enter a silent territory and cause the\nfirst excitations there. The next waves coming from system 1,\nhowever, encounter these first excitations in system 2 and interfere\nwith them constructively or destructively. This process continues and,\nof course, after being excited in system 2, waves also travel back\ninto system 1. For smaller times, this complicated dynamics\ncertainly depends strongly on the realization of the coupling.  This\nis clearly so in the numerical calculation of\nRef.~\\cite{WL}. Nevertheless, the loneg-time behavior, in particular\nthe saturation limit, shows universal characteristics, consistent with\ngeneral features of quantum chaotic systems; see Ref.~\\cite{GMGW}. In\nthe energy domain, this is borne out in the fact that the correlations\non smaller energy scales are described by universal RMT features,\nwhile system-specific properties show up on larger scales, leading to\ndeviations from the RMT prediction. To avoid confusion, we emphasize\nthat these universal RMT features include those in the presence of\nsymmetry breaking.  By system-specific properties we mean, most\nimportantly, the scales set by the shortest periodic orbits.\n\nIt is worthwhile to realize that the chaoticity of the individual\nsubsystems before the coupling is not crucial, if the coupling itself\nintroduces enough chaos. We tested numerically that the behavior of\nthe corresponding model for two regular subsystems coupled chaotically\nshows the same qualitative behavior. The saturation value is reached\nslightly faster, which simply means that time is rescaled.  Finally,\nwe mention that our model is not restricted to elastomechanics. It\nwould also apply to coupled quantum dots and other coupled systems.\n\n\n\\section*{Acknowledgment}\n\nThis project was prompted by a conversation between R.L.~Weaver and\none of the present authors (T.G.) during a workshop at the Centro\nInternacional de Ciencias (CIC\/UNAM), Cuernavaca, Mexico. We thank\nR.L.~Weaver for providing us with the data for the numerical\ncalculations N1 and N2. We acknowledge financial support from Det\nSvenska Vetenskapsr\\aa det.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n{\\let\\thefootnote\\relax\\footnotetext{$^*$ Equal contribution}}\n{\\let\\thefootnote\\relax\\footnotetext{$^\\dagger$ Corresponding author email: liuzy@tsinghua.edu.cn}}\n\n\n\nLarge-scale Transformer-based pretrained language models (\\plm s)~\\cite[][\\textit{inter alia}]{BERT,ALBERT,ELECTRA,CharBERT,DeBERTa} have achieved great success in recent years and attracted wide research interest, in which the tokenization plays a fundamental role.\n\nUnfortunately, current tokenization methods are mostly developed primarily for English~\\cite{BPEsuboptimal}. Almost all the current \\plm s adopt the sub-word tokenization method originating from machine translation, such as the Byte-Pair Encoding~\\cite{BPE}, WordPiece~\\cite{WordPiece,BERT} and SentencePiece based on the unigram language model~\\cite{sentencepiece}. While the idea of sub-word tokenization is intuitive and effective for morphological-rich synthetic languages, it is \\textbf{not} the case for Chinese.\n\nWe believe that it is crucial to develop tailored techniques for the languages beyond English because there can be huge differences between different languages~\\cite[\\#BenderRule]{BenderRule}. Towards this end, we devote this work to analysing three unique linguistic characteristics of Chinese (writing system) compared to English: 1) The Chinese writing system is morphemic~\\cite{hill2016typology}, which means the Chinese characters poorly reflect the pronunciation, resulting in the conventional character-based tokenization misses much more phonological information. 2) Modern Chinese words basically do not undergo morphological alternations~\\cite{morphology_chinese}, thus rendering sub-word tokenization inapplicable. However, Chinese characters are mainly logograms, which means their glyphs, the composition of stokes and radicals, also contain rich semantic information~\\cite{glyce}. 3) In Chinese writing, there is no natural word boundary like the space in English writing. Although it is possible to inject word boundaries via Chinese word Segmentation (\\cws), there is no study on how this works for Chinese \\plm s.\n\n Targeting the three factors, we then explore three corresponding tokenization strategies: 1) A pronunciation-based tokenizer family called \\camelabr{Shuo}{Wen}, which first romanizes the Chinese characters based on their pronunciations, and then constructs the vocabulary with the romanized scripts using the unigram language model~\\cite{sentencepiece}. 2) A glyph-based tokenizer family called \\camelabr{Jie}{Zi}, which decomposes characters into combinations of Chinese strokes or radicals, and then constructs the vocabulary with the stroke or radical sequences using the unigram language model. 3) A word segmented tokenizer family, which first uses a Chinese word segmenter to segment Chinese texts into words, and then constructs the vocabulary with the segmented word sequences using the unigram language model.\n\nWe pretrain BERT-style \\plm s using the proposed tokenizers from scratch and evaluate the resultant models on various downstream tasks. Through comprehensive evaluation on ten Chinese NLU tasks, we find that our pronunciation-based (\\camelabr{Shuo}{Wen}) and glyph-based (\\camelabr{Jie}{Zi}) tokenizers outperform conventional single-character tokenizers in most tasks. Furthermore, as they have the unique advantage to learn the meanings of complex characters through the composition of simpler sub-characters, they are naturally more robust on handling noisy input. \n %\nSurprisingly, we find that Chinese Word Segmentation (\\cws) has no benefit for Chinese language model pretraining. \n\nOur work suggests that linguistically informed techniques based on the characteristics of different languages need more attention. We will release the code, pretrained models, and the \\camelabr{Shuo}{Wen}-\\camelabr{Jie}{Zi}~tokenizers to serve as a better foundation for future research on Chinese \\plm .\n\n\n\n\\section{Related Work}\n\\textbf{Chinese \\plm}. Several previous works have explored techniques to improve Chinese language model pretraining. \\citet{MVP-BERT} and~\\citet{AMBERT} expanded BERT vocabulary with Chinese words apart from the single characters and incorporated them in the pretraining objectives. \\citet{ERNIE-GRAM} and~\\citet{WWM} considered coarse-grained information through masking n-gram and whole words during the masked language modeling pretraining. \\citet{ZEN} incorporated word-level information via superimposing the character and word embeddings.~\\citet{Lattice-BERT} incorporated Chinese word lattice structures in the pretraining. \n  \n\\noindent \\textbf{Linguistically Informed Techniques for Chinese}. \\cws{} is a common preprocessing step for Chinese NLP tasks~\\cite{THULAC}. \\citet{CWSnecessary} empirically analysed whether \\cws{} is helpful for downstream Chinese NLP tasks before the \\plm{} era and found that in many cases the answer is negative. We examine the impact of \\cws{} for \\plm{} instead.~\\citet{glyce} incorporated glyph information of Chinese characters though adding extra encoders to encode the images of Chinese characters and then combine them with the character embeddings. We do not intend to fuse in additional information from sources like images, but instead, \nall of our proposed tokenzation methods are drop-in replacements to the existing single-character tokenizers, without adding any extra layers or parameters. ~\\citet{wubiNMT} explore to Chinese text into Wubi sequences that represent character glyph information for the task of machine translation.\n \n\n\n\n\n\\section{Method}\n\nIn this section, we introduce our proposed tokenization methods.\n\n\n\\subsection{\\camelabr{Shuo}{Wen}: Pronunciation-based Tokenizers}\n\nThe Chinese writing system is morphemic~\\cite{hill2016typology} and barely convey phonological information. However, the pronunciation of Chinese characters also reveals semantic patterns~\\cite{phonology} and has long been widely used as input methods in China (\\textit{e.g., pinyin}).\nIn order to capture such information, we propose a pronunciation-based tokenizer named \\camelabr{Shuo}{Wen}{}.\n\nOn raw Chinese input texts (\\textit{e.g., \u9b51\u9b45\u9b4d\u9b49}), \\camelabr{Shuo}{Wen}{} performs the following steps:\n\n\\begin{enumerate}\n  \\item Romanize the text using Chinese transliteration systems. In this work, we explore two different transliteration methods: \\textit{pinyin} and \\textit{zhuyin (i.e., bopomofo)}. \\textit{Pinyin} uses the Latin alphabet and four~\\footnote{The light tone is sometimes considered as the fifth tone but we omit it for simplicity.} different tones (\\={}, \\'{}, \\v{}, \\`{}) to romanize pronunciations of characters, \\textit{e.g., \u9b51\u9b45\u9b4d\u9b49} $\\rightarrow$ \\textit{Chi\\={} Mei\\`{} Wang\\v{} Liang\\v{}}. On the other hand, \\textit{zhuyin} uses a set of self-invented characters and the same four tones to romanize the characters, \\textit{e.g., \u9b51\u9b45\u9b4d\u9b49} $\\rightarrow$ \\textit{\u3114 \u3107\u311f\\`{} \u3128\u3124\\v{} \u310c\u3127\u3124\\v{}}. Note that in \\textit{zhuyin}, the first tone mark (\\={}) is usually omitted.\n  \n  \\item Insert special separation symbols (\\textit{+}) after each character's romanized sequence, \\textit{e.g., Chi\\={}+Mei\\`{}+Wang\\v{}+Liang\\v{}+, \u3114+\u3107\u311f\\`{}+\u3128\u3124\\v{}+\u310c\u3127\u3124\\v{}+}. This prevents cases where romanized sequences of different characters are mixed together, especially when there are no tone markers to split them in \\textit{zhuyin}.\n  \n  \\item Different Chinese characters often have the same pronunciation. For disambiguation, we append different indices after the romanized sequences for the homophonic characters, so that allowing a biunique mapping between each Chinese character and its romanized sequence, \\textit{e.g., Chi\\={}33+Mei\\`{}24+Wang\\v{}25+Liang\\v{}13+, \u311410+\u3107\u311f\\`{}3+\u3128\u3124\\v{}6+\u310c\u3127\u3124\\v{}1+}. \n  \n  \\item Apply a unigram language model (\\ulm{}) as in \\citet{sentencepiece} on the romanized sequences to build the final vocabulary. \n  \n\\end{enumerate}\n\nWe do not set any constraint on the vocabulary other than the vocabulary size. The resultant vocabulary contains tokens corresponding to flexible combinations of characters and sub-characters.\n\n\n\n \n\\subsection{\\camelabr{Jie}{Zi}: Glyph-based Tokenizers}\n\nThe word shapes of Chinese characters contain rich semantic information and can help NLP models~\\cite{cw2vec}. For example, most Chinese characters can be broken down into semantically meaningful radicals. Characters that share common radicals often share related semantic information, such as the four characters \\textit{`\u9b51\u9b45\u9b4d\u9b49'} all share the same radical \\textit{`\u9b3c'} (meaning ghost), and their meanings are indeed related to ghosts and monsters.~\\footnote{Interestingly, the word \\textit{`\u9b51\u9b45\u9b4d\u9b49'} is in fact a Chinese idiom, which is now often used to refer to bad people who are like monsters.} \n\n\nHowever, the prevailing tokenization method for Chinese treats each Chinese character as a separate token and hence preventing the model to learn the shared semantics of characters with common radicals. In order to solve this problem, we propose the glyph-based tokenizer \\camelabr{Jie}{Zi}{}, which performs the following steps on raw Chinese input (\\textit{e.g., \u9b51\u9b45\u9b4d\u9b49}):\n\n\\begin{enumerate}\n \\item Convert each character into a stroke or radical sequence. To convert into stroke sequences, we use Latin alphabet to represent the basic strokes and convert the characters based on the standard stroke orders\\footnote{\\url{https:\/\/en.wikipedia.org\/wiki\/Stroke_order}}, \\textit{e.g., \u9b51} $\\rightarrow$ \\textit{\\underline{pszhshpzzn}nhpnzsszshn}; \\textit{\u9b45} $\\rightarrow$ \\textit{\\underline{pszhshpzzn}hhspn}. To convert into radical sequences, we adopt three existing glyph-based Chinese input methods: \\textit{Wubi, Zhengma, Cangjie}. These methods group strokes together in different ways to form radicals or stroke combinations, and then represent characters with them. We use Latin alphabet to represent these radicals or stroke combinations, \\textit{e.g., \u9b51\u9b45\u9b4d\u9b49} $\\rightarrow$ \\textit{Wubi: \\underline{rqc}c \\underline{rqc}i \\underline{rqc}n \\underline{rqc}w}; \\textit{Zhengma: \\underline{nj}lz \\underline{nj}bk \\underline{nj}ld \\underline{nj}oo}; \\textit{Cangjie: \\underline{hi}yub \\underline{hi}jd \\underline{hi}btv \\underline{hi}mob}.\n \n \\item Similar to the pronunciation-based tokenizers, we add the same separation symbol after each character, and also add the disambiguation indices for characters whose stroke sequences are identical (\\textit{e.g., \u4eba} (people) and \\textit{\u516b} (eight)). \n\\end{enumerate} \n\nIn the converted sequences, we can see how common radicals naturally appear (the \\underline{underlined} parts). Please refer to the Appendix~\\ref{app:inputmethod} for a detailed explanation on the differences between these different input methods involved.\n\n\n\\begin{table*}[t]\n    \\centering\n    \\small {\n    \\begin{tabular}{ lcccc|cccc }\n        \\toprule Dataset & Task & MaxLen & Batch & Epoch & \\#Train & \\#Dev & \\#Test & Domain \\\\\n        \\midrule\n            TNEWS    & DC  & 256 & 32 & 6 & 53.4K   & 10K   & 10K   & News \\\\\n            IFLYTEK  & DC  & 256 & 32 & 6 & 12.1K  & 2.6K  & 2.6K   & App Description \\\\\n            BQ       & SPM  & 256 & 32 & 6 & 100K   & 10K   & 10K    & Bank Service \\\\\n            THUCNEWS & DC  & 256 & 32 & 6 & 669K & 83.6K & 83.6K  & News \\\\\n            CLUEWSC  & WSC & 256 & 32 & 24 & 1.2K & 0.3K   & 0.3K   & Literature \\\\\n            AFQMC    & SPM  & 256 & 32 & 6 & 34.3K  & 4.3K  & 3.9K   & Financial  \\\\\n            CSL      & SPM & 256 & 32 & 6 & 20K    & 3K    & 3K     & Academic Papers \\\\\n            OCNLI    & SPM & 256 & 32 & 6 & 45.4K  & 5K    & 3K    & Mixed  \\\\\n            CHID     & MRC & 96  & 24 & 6 & 519K & 57.8K & 23K   & Mixed  \\\\\n            C3       & MRC & 512 & 24 & 6 & 12K  & 3.8K  & 3.9K  & Mixed  \\\\\n        \\bottomrule\n    \\end{tabular}}\n    \\caption{Hyper-parameters and statistics of different datasets. DC: document classification. SPM: sentence pair matching (including natural language inference). WSC: Winograd Schema Challenge. MRC: machine reading comprehension.} \n    \\label{tab:hyper_parameters}\n\\end{table*}\n\n\n\\subsection{Word Segmented Tokenizer}\n\nChinese Word Segmentation (\\cws{}) is a common technique to split Chinese chunks into a sequence of Chinese words. The resultant segmented words sometimes provide better granularity for downstream tasks~\\cite[\\textit{e.g.,}][]{CWS-NMT}. However, the impact of \\cws{} is unclear in the context of pretraining, especially how it interplays with statistical approaches like BPE and unigram LM. Hence, we study word segmented tokenizers that performs the following process on raw Chinese input, e.g., ``\u8fd9\u7bc7\u8bba\u6587\u6709\u610f\u601d\u3002(this paper is interesting)'' :\n\n\\begin{enumerate}\n  \\item We use a state-of-the-art segmenter THULAC~\\cite{THULAC} to segment the sentence into a sequence of words joined by spaces, \\textit{e.g., `\u8fd9\u7bc7\\textbackslash \u8bba\u6587\\textbackslash \u6709\u610f\u601d\\textbackslash  \u3002'} (We use \\textbackslash to indicate a blank space for easier reading.)\n\n  \\item We directly apply \\ulm{}~on these space-joined sequences to construct the vocabulary.\n  \n\\end{enumerate}\n\nIn other words, \\cws{} is used as a preprocessing step on training corpora when we build the vocabulary using above-mentioned methods. When we perform the actual pretraining and finetuning, we also perform \\cws{} as preprocessing before tokenization using the word segmented tokenizer. \n\n\n\n\n\\begin{table*}[t]\n\\centering\n\\small\n\\begin{adjustbox}{width=\\linewidth,center}\n\\setlength{\\tabcolsep}{3.5pt}{\n\\begin{tabular}{ lccccccccccl }\n\\toprule \n & TNEWS\n & IFLY\n & THUC\n & BQ \n & WSC\n & AFQMC\n & CSL\n & OCNLI\n & CHID\n & C3\n & AVG\n \\\\\n \\hline  \n\\multicolumn{11}{c}{\\textit{6-layer}} \\\\\n\\hline \nBERT-Chinese & 64.10 & 57.77 & 96.97 & 81.98 & 62.39 & 68.95 & 82.60 & 68.46 & 72.33 & 53.51 & 70.91 \\\\\n\\sentp{}-\\ulm{} & 64.26 & 55.44 & \\textbf{97.09} & 81.52 & 62.06 & 69.88 & 83.16 & 68.98 & 72.77 & 51.73 & 70.69 (-0.22) \\\\\n\\camelabr{Jie}{Zi}-CangJie & 63.86 & 59.51 & 97.04 & 81.59 & 63.27 & \\textbf{70.47} & 82.91 & 69.03 & 72.73 & 52.67 & \\textbf{71.31 (+0.40)} \\\\\n\\camelabr{Jie}{Zi}-Stroke & 63.81 & 58.74 & 96.87 & 81.55 & 62.94 & 69.66 & 82.44 & 68.02 & 72.21 & 53.35 & 70.96 (+0.05) \\\\\n\\camelabr{Jie}{Zi}-Zhengma & 63.96 & 58.74 & 96.99 & \\textbf{82.27} & 61.95 & 69.86 & \\textbf{83.46} & 68.56 & 72.12 & \\textbf{54.91} & 71.28 (+0.37) \\\\\n\\camelabr{Jie}{Zi}-Wubi & \\textbf{64.91} & 59.39 & 97.03 & 81.41 & 62.72 & 69.14 & 82.60 & 69.12 & 72.02 & 53.99 & 71.16 (+0.25) \\\\\n\\camelabr{Shuo}{Wen}-Pinyin & 63.58 & \\textbf{59.55} & 97.04 & 81.65 & 63.60 & 68.60 & 82.66 & 67.93 & \\textbf{72.81} & 53.02 & 71.04 (+0.13) \\\\\n\\camelabr{Shuo}{Wen}-Zhuyin & 64.11 & 59.16 & 97.01 & 81.64 & \\textbf{63.93} & 68.53 & 82.86 & \\textbf{69.39} & 71.48 & 54.59 & 71.27 (+0.36) \\\\\n\\hline \n\\multicolumn{11}{c}{\\textit{12-layer}} \\\\\n\\hline \nBERT-Chinese & 65.07 & 58.01 & 97.05 & 82.33 & 73.14 & 71.04 & 83.90 & 70.19 & 76.61 & 55.90 & 73.32 \\\\\n\\sentp{}-\\ulm{} & 65.01 & 58.98 & \\textbf{97.20} & 82.99 & \\textbf{73.36} & 70.93 & 83.45 & 70.46 & 77.28 & 57.70 & 73.74 (+0.42) \\\\\n\\camelabr{Jie}{Zi}-CangJie & 64.26 & 60.29 & 97.15 & \\textbf{83.48} & 71.16 & 71.48 & 83.68 & 71.50 & 76.82 & 57.99 & 73.78 (+0.46) \\\\\n\\camelabr{Jie}{Zi}-Stroke & \\textbf{65.11} & 59.75 & 97.09 & 82.88 & 70.72 & 71.64 & 83.63 & 70.03 & \\textbf{77.45} & \\textbf{59.68} & 73.53 (+0.21) \\\\\n\\camelabr{Jie}{Zi}-Zhengma & 64.51 & 60.78 & 97.14 & 83.15 & 72.15 & 70.76 & 83.68 & 71.22 & 76.72 & 57.49 & 73.76 (+0.44) \\\\\n\\camelabr{Jie}{Zi}-Wubi & 64.47 & 60.05 & 97.16 & 82.76 & 72.70 & \\textbf{72.00} & 83.62 & 70.77 & 76.34 & 58.31 & 73.82 (+0.50) \\\\\n\\camelabr{Shuo}{Wen}-Pinyin & 64.50 & \\textbf{60.40} & 97.17 & 83.13 & 70.18 & 71.37 & \\textbf{84.12} & \\textbf{71.97} & 76.11 & 58.05 & 73.70 (+0.38) \\\\\n\\camelabr{Shuo}{Wen}-Zhuyin & 64.50 & 59.98 & 97.09 & 82.99 & 73.03 & 71.83 & 83.82 & 71.74 & 76.74 & 57.23 & \\textbf{73.90 (+0.58)} \\\\\n\\bottomrule\n\\end{tabular}}\n\\end{adjustbox}\n\\caption{Results for standard evaluation. Best result on each dataset of each model size is \\textbf{boldfaced}. The numbers in brackets in the last column indicate the average difference compared to the BERT-Chinese baseline.}\n\\label{tab:main_results}\n\\end{table*}\n\n\n\\begin{table*}[t]\n\\centering\n\\small\n\\begin{adjustbox}{width=\\linewidth,center}\n\\addtolength{\\tabcolsep}{-2pt}\n\\setlength{\\tabcolsep}{2mm}{\n\\begin{tabular}{ lcccccccl }\n\\toprule \n & TNEWS\n & IFLYTEK\n & CLUEWSC\n & AFQMC\n & CSL\n & OCNLI\n & C3\n & AVG\n \\\\\n \\hline  \n\\sentp{}-\\ulm{} & 64.26 & 55.44 & 62.06 & 69.88 & 83.16 & 68.98 & 51.73 & 65.07 \\\\\n\\sentp{}-\\ulm{} + \\cws{} & 64.26 & 54.15 & 63.05 & 69.62 & 82.87 & 68.64 &  51.77 & 64.91 \\\\\n\\camelabr{Jie}{Zi}{}-Wubi & 64.91 & 59.39 & 62.72 & 69.14 & 82.60 & 69.12 & 53.99 & 65.98 \\\\\n\\camelabr{Jie}{Zi}{}-Wubi + \\cws{} & 63.66 & 59.22 & 63.16 & 68.65 & 82.21 & 68.81 & 52.76 & 65.50 \\\\\n\\camelabr{Shuo}{Wen}{}-Zhuyin & 64.11 & 59.16 & 63.93 & 68.53  & 82.86 & 69.39 & 54.59 & 66.08 \\\\\n\\camelabr{Shuo}{Wen}{}-Zhuyin + \\cws{} & 63.37 & 57.24 & 62.83 & 68.94 & 82.12 & 68.69 & 51.48 & 64.95 \\\\\n\\bottomrule\n\\end{tabular}}\n\\end{adjustbox}\n\\caption{Results of models trained with Word Segmented Tokenization. All models are 6-layer. }\n\\label{tab:cws_results}\n\\end{table*}\n\n\n\n\n\\section{Experiment}\n\n\\subsection{Baselines}\n\nWe use two strong baseline tokenizers in this work. The first one is the conventional single-character tokenizer as used in BERT-Chinese and many other follow-up Chinese \\plm{}~\\cite[\\textit{e.g.,}][]{WWM,MacBERT}. We name this tokenizer BERT-Chinese as it originated from the Chinese version of BERT.\n\nFor the second baseline, we directly apply SentencePiece with unigram LM on the raw Chinese corpus to generate the vocabulary. \nAs a result, the vocabulary contains both single characters and words (\\textit{i.e.,} character combinations). \nThis approach resembles the vocabulary of some recent multi-granularity Chinese \\plm{} variants such as AMBERT~\\cite{AMBERT} and Lattice-BERT~\\cite{Lattice-BERT}. Unlike them, we do not add any new model designs or pretraining objectives, but instead use the original BERT architecture and masked LM objective. We name this baseline \\sentp{}-\\ulm{}.\n\nTo ensure a fair comparison, we set the same vocabulary size of 22675 for all tokenizers. \nWe use the same training corpus to train all the tokenizers. We use the SentencePiece library's unigram LM implementation to train the tokenizers.\n\nIn order to evaluate the effectiveness of the tokenizers, we pretrain a BERT model using each tokenizer and compare their performance on downstream tasks. \nWhen pretraining the BERT models using each tokenizer, we use the same pretraining corpus and the same set of hyper-parameters for all models being compared. \nNotably, we also re-pretrain the BERT model using the BERT-Chinese tokenizer on our pretraining corpus instead of just loading from existing checkpoints to ensure that all baselines and proposed methods are directly comparable.\nSince our proposed tokenizers are direct drop-in replacements for the baseline tokenizers, they do not incur any extra parameters. As a result, all the models being compared have the same number of parameters, allowing for a truly apple-to-apple comparison.\n\n\\subsection{Datasets}\n\n\nWe evaluate the trained models with different tokenization methods on a total of ten different downstream datasets, including single-sentence tasks, sentence-pair tasks, as well as reading comprehension tasks. We briefly introduce each dataset below and present the dataset statistics in Table~\\ref{tab:hyper_parameters}.\n\n\\noindent \\textbf{TNEWS}~\\cite{TNEWS} is a news title  classification dataset containing 15 classes. We use the split as released in \\citet{CLUEBenchmark}.\n\n\\noindent \\textbf{IFLYTEK}~\\cite{IFLYTEK} is a long text classification dataset containing 119 classes. The task is to classify mobile applications into corresponding categories given their description.\n\n\\noindent \\textbf{BQ}~\\cite{BQ} is a sentence-pair question matching dataset extracted from an online bank customer service log. The goal is to evaluate whether two questions are semantically identical or can be answered by the same answer. \n\n\\noindent \\textbf{THUCNEWS}~\\cite{THUCNEWS} is a document classification dataset with 14 classes. The task is to classify news into the corresponding categories given their title and content.\n\n\\noindent \\textbf{CLUEWSC}~\\cite{CLUEBenchmark} is a coreference resolution dataset in the format of Winograd Schema Challenge. The task is to determine whether the given noun and pronoun in the sentence co-refer.\n\n\\noindent \\textbf{AFQMC} is the Ant Financial Question Matching Corpus for the question matching task that aims to predict whether two sentences are semantically similar\n\n\\noindent \\textbf{CSL} is the Chinese Scientific Literature dataset extracted from academic papers. Given an abstract and some keywords, the goal is to determine whether they belong to the same paper. It is formatted as a sentence-pair matching task.\n\n\\noindent \\textbf{OCNLI}~\\cite{OCNLI} is a natural language inference dataset. The task is to determine whether the relationship between the hypothesis and premise is entailment, neural, or contradiction.\n\n\\noindent \\textbf{CHID}~\\cite{CHID} is a  cloze-style reading comprehension dataset where . Given contexts where some idioms are masked, the task is to select the appropriate idiom from a list of candidates.\n\n\\noindent \\textbf{C3}~\\cite{C3} is a multiple choice machine reading comprehension dataset. The goal is to choose the correct answer for some questions given a context. \n\n\n\n\n\\begin{table*}[t]\n\\centering\n\\small\n\\addtolength{\\tabcolsep}{-2pt}\n\\setlength{\\tabcolsep}{3mm}{\n\\begin{tabular}{ lccccc }\n\\toprule \n & clean\n & 15\\%\n & 30\\%\n & 45\\% \n & 60\\%\n \\\\\n \\hline  \n\\multicolumn{6}{c}{TNEWS} \\\\\n\\hline \nBERT-Chinese & 63.99 & 62.18 & 60.49 & 57.64 & 53.97 \\\\\n\\sentp{}-\\ulm{} & 63.99 & 62.88 & 60.79 & 59.20 & 55.42 \\\\\n\\camelabr{Jie}{Zi}-Wubi & 64.05 & 62.80 & 62.56 & 62.71 & 62.81 \\\\\n\\hline \n\\multicolumn{6}{c}{OCNLI} \\\\\n\\hline \nBERT-Chinese & 68.07 & 62.60 & 56.83 & 51.37 & 46.00 \\\\\n\\sentp{}-\\ulm{} & 69.10 & 64.00 & 56.57 & 52.43 & 46.73 \\\\\n\\camelabr{Jie}{Zi}-Wubi & 68.43 & 67.10 & 65.47 & 65.40 & 64.80 \\\\\n\\bottomrule\n\\end{tabular}}\n\\caption{Results for noisy evaluation with glyph noises.}\n\\label{tab:noisy_results_glyph}\n\\end{table*}\n\n\n\\begin{table*}[t]\n\\centering\n\\small\n\\addtolength{\\tabcolsep}{-2pt}\n\\setlength{\\tabcolsep}{3mm}{\n\\begin{tabular}{ lccccc }\n\\toprule \n & clean\n & 15\\%\n & 30\\%\n & 45\\% \n & 60\\%\n \\\\\n \\hline  \n\\multicolumn{6}{c}{TNEWS} \\\\\n\\hline \nBERT-Chinese & 63.99 & 60.87 & 57.70 & 52.21 & 45.51 \\\\\n\\sentp{}-\\ulm{} & 63.99 & 61.52 & 58.60 & 53.30 & 46.81 \\\\\n\\camelabr{Shuo}{Wen}-Pinyin & 63.35 & 60.60 & 57.45 & 53.61 & 49.29 \\\\\n\\camelabr{Shuo}{Wen}-Zhuyin & 63.79 & 60.99 & 57.69 & 53.15 & 48.98 \\\\\n\\hline \n\\multicolumn{6}{c}{OCNLI} \\\\\n\\hline \nBERT-Chinese & 68.07 & 61.73 & 54.50 & 49.97 & 44.80 \\\\\n\\sentp{}-\\ulm{} & 69.10 & 62.23 & 54.77 & 50.33 & 45.70 \\\\\n\\camelabr{Shuo}{Wen}-Pinyin & 67.83 & 61.00 & 54.33 & 49.87 & 45.10 \\\\\n\\camelabr{Shuo}{Wen}-Zhuyin & 69.63 & 60.77 & 54.77 & 50.67 & 47.70 \\\\\n\\bottomrule\n\\end{tabular}}\n\\caption{Results for noisy evaluation with phonology noises.}\n\\label{tab:noisy_results_phonology}\n\\end{table*}\n\n\n\n\\subsection{Experiment Setup: Standard Evaluation}\n\n\nFor each tokenizer, we pretrain a 6-layer and 12-layer BERT style model using the Baidu Baike corpus~\\cite{CPM} which has 2.2G of raw text before processing. \nThe model configuration is exactly the same for all models: 6 or 12 layers, 12 attention heads, intermediate size 3072, hidden size 768.\nFor pretraining, we follow the original BERT paper's two-stage pretraining procedure where we first train with sequence length 128 for 8k steps and then train with sequence length 512 for 4k steps.  We only keep the masked language modeling objective in pretraining and discard the next sentence prediction objective as suggested in RoBERTa~\\cite{RoBERTa}.\nDuring fine-tuning, we use the set of hyper-parameters as shown in Table~\\ref{tab:hyper_parameters}.\nFor all experiments in this paper, we report results of the average run of three different random seeds.\n\n\\subsection{Experiment Setup: Noisy Evaluation}\n\nApart from evaluating on the standard benchmarks, we also evaluate in a noisy setting to illustrate the advantage of our proposed tokenization methods to handle noisy inputs.\nSpecifically, we inject two types of synthetic noises into both the training and test data in order to test whether the models can learn from noisy training data and also perform robustly on noisy test data. \nWe vary the ratio of noise in the data to examine the impact. The two types of noises we inject are:\n\n\\begin{itemize}\n  \\item Glyph-based noise: we replace original characters with other characters that have similar glyph but have different semantic meanings (\\textit{e.g., \u58c1} (wall) and \\textit{\u74a7} (jade)). Specifically, we obtain a substitution candidate list for each character, where the candidates are selected so that they share at least one common radical with the original character. Then, we randomly sample a certain ratio $r\\%$ of the original characters, for each of them, we randomly sample a substitution character from its candidate list for substitution.\nThis simulates common noises when people use glyph-based input methods where these similar characters could be chosen since their input encoding are similar. \n\n  \\item Pronunciation-based noise: we replace original characters with other characters that have the same pronunciation but different semantic meanings (\\textit{e.g., \u771f} (real) and \\textit{\u9488} (needle)).\n  Specifically, we obtain a substitution candidate list for each character, where all the candidates have the same pronunciation as the original character. Then, similarly, we randomly sample a certain ratio $r\\%$ of the original characters, for each of them, we randomly sample a substitution character from its candidate list for substitution.\n  This simulates the common noise when users use pronunciation-based input methods where the input encoding of these characters and their substitutions are the same.\n\\end{itemize}\n\nFor our experiment, we vary the noise ratio $r\\%$ within the range of \\{$0$, $15\\%$, $30\\%$, $45\\%$ $60\\%$\\}. For the sampled characters to be replaced with noises, we randomly sample a substitution character from their candidate list for every appearance of the character, instead of substituting with the same candidate. This induces more noise variations. Note that some substitutions are both glyph-based noises and pronunciation-based noises (\\textit{e.g., \u5feb} and \\textit{\u5757} share a same radical and also have the same pronunciation), we keep them in both types of noises.\n\nIntuitively, our \\camelabr{Shuo}{Wen}{} tokenizer could be robust to pronunciation-based noises and \\camelabr{Jie}{Zi}{} tokenizer could be robust to glyph-based noises because the substitution characters share similar pronunciation or glyph components with the original characters, which may be captured by our tokenizers. \n\nThis noisy setup is reflective of real-life use cases where user queries often contain such noises. Since most Chinese people use either glyph-based input methods (\\textit{e.g., wubi}) or pronunciation-based input methods (\\textit{e.g., pinyin, zhuyin}), such mis-typed characters can be very common. This highlights the potential impact of our work.\n \n\n\\section{Results}\n\n\\subsection{Standard Evaluation}\nWe compare the results of the baseline tokenizers (BERT-Chinese, \\sentp{}-\\ulm{}) with our proposed \\camelabr{Jie}{Zi}{} (including four variants: \\camelabr{Jie}{Zi}{}-Cangjie, \\camelabr{Jie}{Zi}{}-Stroke, \\camelabr{Jie}{Zi}{}-Zhengma, \\camelabr{Jie}{Zi}{}-Wubi) and \\camelabr{Shuo}{Wen}{} (including two variants: \\camelabr{Shuo}{Wen}{}-Pinyin and \\camelabr{Shuo}{Wen}{}-Zhuyin) tokenizers in Table~\\ref{tab:main_results}. \n\nDespite some variations across different datasets, we observe that in terms of the average score over ten datasets, all of our proposed tokenizers outperform the BERT-Chinese baseline. Notably, for the 6-layer model size, our \\camelabr{Jie}{Zi}{}-Cangjie tokenizer obtains an average of 0.40 points over the BERT-Chinese tokenizer, for the 12-layer model size, our \\camelabr{Shuo}{Wen}{}-Zhuyin tokenizer achieves an average of 0.58 points of improvement over the BERT-Chinese tokenizer.\n\nThese results indicate that on standard benchmarks, our proposed tokenizers can match or outperform the existing tokenizers for Chinese. \n\nOn the other hand, we examine the impact of \\cws{} by comparing three tokenizers with their word segmented counterparts in Table~\\ref{tab:cws_results}. We can see that adding \\cws{} as a preprocessing actually slightly decreased the average performance on downstream tasks. \n\n\\subsection{Noisy Evaluation}\n\nWe perform noisy evaluation on two datasets: TNEWS and OCNLI. For glyph-based noises, we compare baselines BERT-Chinese and \\sentp{}-\\ulm{} with our \\camelabr{Jie}{Zi}{}-Wubi. The results are presented in Table~\\ref{tab:noisy_results_glyph}. We observe that when the noise ratio increases, the advantage of \\camelabr{Jie}{Zi}{} is particularly large. For example, when $60\\%$ characters are substituted, \\camelabr{Jie}{Zi}{}-Wubi still performs close to the original performance, while other baselines suffer large drops in performance. On OCNLI, the gap can be as large as 18 points in accuracy. \n\nFor pronunciation-based noises, we compare baselines BERT-Chinese and \\sentp{}-\\ulm{} with our \\camelabr{Shuo}{Wen}{}-Pinyin and \\camelabr{Shuo}{Wen}-Zhuyin. The results are shown in Table~\\ref{tab:noisy_results_phonology}. Unlike the cases on glyph-based noises, we observe that the advantage of our \\camelabr{Shuo}{Wen}{} tokenizers are not so significant compared to the baselines. One potential reason is that there many Chinese characters with the same pronunciation. Unlike how the semantic meanings of radicals can be rather consistent across different characters, phoneme combinations can have vastly different meanings across different characters (\\textit{i.e.,} characters may pronounce the same but have totally different semantic meanings), which makes it difficult to learn these different semantic meanings in the pronunciation-based token embedding. \n\n\\section{Conclusion}\n\nIn this paper, we have explored three linguistically informed tokenization methods motivated by the unique linguistic characteristics of the Chinese writing system. Specifically, we find that pronunciation-based and glyph-based tokenizers can match or outperform the conventional Chinese tokenizers and Chinese word segmentation is not a useful addition for the tokenizer.\nMoreover, we find that our glyph-based tokenizers achieve large gains on noisy input as compared to the baselines, while our pronunciation-based tokenizers obtain limited success. This highlights the potential advantage of our proposed methods in real-life scenarios with noisy data.\nWe believe that our work sets an important example of exploiting the unique linguistic property of a language beyond English to develop more tailored techniques, which should be an important direction for the global NLP community. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Methodological approach}\n\nGiven two probability distributions, $1$ and $2$,\nwith $1$ nested within $2$,\nthe likelihood ratio test evaluates $\\hat L_{2}\/\\hat L_{1}$,\nwhere $\\hat L_{2}$ is the likelihood (at maximum)\nof the ``bigger'' or ``full'' model (either Tap or TrG)\nand $\\hat L_{1}$ corresponds to the nested or null model\n(power law in our case). Taking logarithms we get\nthe log-likelihood-ratio\n\\[\n\\mathcal{R}=\\ln \\frac{\\hat L_2}{\\hat L_1}= l_{2}-l_{1},\n\\]\nwith $ l_{j}=\\ln \\hat L_j=\\sum_{i=1}^{N} \\ln f_{j}(M_i;\\hat{\\Theta}_j) $,\nwhere $f_{i}$ denotes the probability density function of the distribution $j$\nfor every $j=1,2$, and the MLE corresponds to $\\hat \\Theta_1=\\hat\n\\beta_1$ and $\\hat\\Theta_2=\\{\\hat\\beta_2,\\hat\\theta_2\\}$. In order\nto compare the fit provided by the two distributions, it is\nnecessary to characterize the distribution of $\\mathcal{R}$.\n\nLet $n_1$ and $n_2$ be the number of free parameters in the models\n$1$ and $2$, respectively.\nIn general, if the models are nested,\nand under the null\nhypothesis that the data comes from the simpler model, the\nprobability distribution of the statistic $2 \\mathcal{R}$ in the limit\n$N\\rightarrow \\infty$ is\na chi-squared distribution\nwith degrees of freedom equal to $n_2-n_1 > 0$.\nSo,\n\\begin{equation}\n2 \\mathcal{R}>3.84\n\\end{equation}\nwith a level of risk equal to 0.05.\n{Note that the chi-squared distribution\nprovides a penalty for\n``model complexity''\nas the {``width''} of the distribution is given directly by\nthe number of the degrees of freedom.}\nThis likelihood ratio test\nconstitutes the best option to choose among models 1 and 2, in the\nsense that it has a convergence to its asymptotic distribution\nfaster\nthan any other test \\cite{mccullagh1986}.\nThe null and\nalternative hypotheses correspond to accept model $1$ or $2$,\nrespectively, although the acceptance of model $1$ does not imply the rejection\nof $2$, it is simply that\nthe ``full'' model $2$\ndoes not bring any significant improvement with respect the simpler\nmodel $1$.\n\n{However, when the nesting of distribution $1$ within $2$ takes place\nin such a way that the space of parameters of the former one\nlies within a boundary of the space of parameters of distribution $2$,\nthe approach just explained for\nthe asymptotic distribution of $2 \\mathcal{R}$\nis not valid \\cite{Self1987,Geyer1994}.\nThis is the case when testing both the Tap or the TrG distributions\nin front of the power-law distribution,\nas the $\\theta \\rightarrow \\infty$ limit of the latter\ncorresponds to the boundary of the parameter space of the two other distributions.\nBut the asymptotic theory of Refs. \\cite{Self1987,Geyer1994}\nis also invalid, as the power-law distribution lacks\nthe necessary regularity conditions,\ndue to the divergence of their moments.\nThis illustrates part of the difficulties\nof performing proper model selection when\nfractal-like distributions are involved\n\\cite{Kagan_calcutta}.\nIn order to obtain the distribution of $2\\mathcal{R}$ and from there\nthe $p-$values of the LR tests,\nwe are left to the simulation of the null hypothesis.\nWe advance that the results seem to indicate that the distribution of $2\\mathcal{R}$,\nfor high percentiles, is chi-square with one degree of freedom,\nbut we lack a theoretical support for this fact.}\n\nLet us proceed, using this {method},\nby comparing the performance of the power-law and Tap fits\nwhen applied to the global shallow seismic activity,\nfor time windows starting always in 1977 and ending in the successive times\nindexed by the abscissa in Fig. \\ref{graf:densities}(a)\n(as in Ref. \\cite{Bell_Naylor_Main}).\nThe log-likelihood-ratio of these fits (times 2),\nis shown in the figure\ntogether with the critical region of the test. In agreement with\nBell {\\it et al.} \\cite{Bell_Naylor_Main}, we find that: (i) the\npower-law fit can be safely rejected in front of the Tap\ndistribution for any time window ending between 1980 and before\n2004; and (ii) the results change drastically after the occurrence\nof the great 2004 Sumatra earthquake,\n{for which the power law cannot be rejected at the 0.05 level.\nSo, for parsimony reasons, the power law becomes preferable in front of the\nTap distribution for time windows\nending later than 2004.\nThe fact that the Tap distribution cannot be distinguished from the power law\nis also in agreement with previous results showing that the contour lines\nin the likelihood maps of the Tap distribution are highly non-symmetric\nand may be unbounded for smaller levels of risk \\cite{Kagan_Schoenberg,Kagan_gji02,Geist_Parsons_NH}.\n}\n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\centerline{\\includegraphics[width=9cm]{explore_earthquake_figura2_resumlikeli_10000.pdf}}\n\\caption{ Results of likelihood ratio tests for nested and\nnon-nested models. The points denote the value of the statistic $2\n\\mathcal{R}$ or $\\mathcal{R}$ (depending on the test) and the dashed\nlines show the critical value of the corresponding test {(at level\n0.05)}. {For the nested case, boxplots show the distribution of\n$2\\mathcal{R}$ for 10000 simulations of the power-law null\nhypothesis, from which the critical value is computed.} The abscissa\ncorresponds to the ending point of a time window starting always in\n1 Jan 1977. Note that the year is considered a continuous variable\n(not a categorical variable), so, the time window ending on 31 Dec\n2004 takes value $2004.99\\dots \\simeq 2005$. (a) Tap distribution\nversus power law. (b) Truncated gamma versus power law. (c)\nTruncated gamma versus Tap {(non-nested case)}.\n\\label{graf:densities}}\n\\end{center}\n\\end{figure}\n\nWhen we compare the power-law fit with the truncated gamma, using\nthe same test, for the same data, the results are more\nsignificant, see Fig. \\ref{graf:densities}(b). The situation\nprevious to 2004 is the same, with an extremely poor performance of\nthe power law; but after 2004, despite a big jump again in the value\nof the likelihood ratio, the power law continues as being\nnon-acceptable, at the 0.05 level.\nIt is only after the great Tohoku earthquake of 2011\nthat the $p-$value of the test {enters into} the acceptance region,\nbut keeping values not far from the 0.05 limit.\nFrom here we\nconclude that, in order to find an alternative to the power-law\ndistribution, the truncated gamma distribution is a better\noption than the Tap distribution, as it is more clearly distinguishable\nfrom the power law (for this particular data).\nNevertheless, a comparison between\nthese two distributions (Tap and TrG) seems pertinent.\n\n\n\n\n\nWhen the models are not nested, as it happens if we want a direct\ncomparison between the Tap and the TrG distributions,\nthe procedure we use\nis the likelihood ratio test of Vuong for non-nested models \\cite{Vuong,Clauset}.\nIn this case the critical values\ndepend on the sample size, $N$,\nturning out to be that, when $N$ is large, $\\mathcal{R}$ is normally\ndistributed with standard deviation $s \\sqrt N$,\nwhere $s$ denotes the standard deviation of the set\n$$\n\\{\\ln f_{trg}(M_i;\\hat{\\beta}_{trg},\\hat{\\theta}_{trg})-\\ln\nf_{tap}(M_i;\\hat{\\beta}_{tap},\\hat{\\theta}_{tap})\\}\n$$\nfor $i=1,\\dots, N$ and $\\mathcal{R}= l_{trg}-l_{tap}$.\nThen, we accept that there exists a\nsignificant difference between the models if\n\\begin{equation}\\label{reg}\n|\\mathcal{R}|>\\:1.96 s \\sqrt{N}\n\\end{equation}\nat a level of risk equal to 0.05,\nwith the model with larger log-likelihood being the preferred one.\nThe critical value of the test arises because\nthe null hypothesis is that the mean value of $\\mathcal{R}$ is zero (i.e.,\nboth models are equally close to the true distribution).\nNote that the alternative hypothesis corresponds to accept that the\ndifference between the fit provided by the models is significant. As\nthe number of parameters is the same for the Tap and TrG models,\ntheir log-likelihood-ratio coincides with the difference in BIC or\nAIC, {but,} as mentioned above, the LR test incorporates\na statistical test which specifies the distribution\nof the statistic under consideration.\n\n\nFigure \\ref{graf:densities}(c) shows the evolution of the\nlog-likelihood-ratio between the two models, for different time windows (starting always in\n1977), together with the critical region of the test given by the\nEq. (\\ref{reg}). One can see how the fits provided by the Tap and TrG\ndistributions do not exhibit significant difference, although the\nTrG provides, in general, slightly higher likelihoods. After the\nmega-event in 2004 the performance of the TrG fit improves,\napproaching the limit of significance.\n{This reinforces our conclusion that the TrG distribution is preferred\nin front of the Tap and power-law distributions.}\n\nIn order to gain further insight, we simulate\nrandom samples following the truncated gamma distribution, with the\nparameters $\\hat \\beta_{trg}$ and $\\hat \\theta_{trg}$ obtained from\nML estimation of the complete dataset (Table \\ref{taula}), with the\nsame truncation parameter $a$ and number of points ($N=6150$) also.\nTo avoid that the conclusions depend on the time correlation of magnitudes,\nwe reshuffle the simulated data in such a way that the occurrence of\nthe order statistics is the same as for the empirical data; in other\nwords, the largest simulated event is assigned to take place at the\ntime of the 2011 Tohoku earthquake (the largest of the CMT catalog\n\\cite{Bell_Naylor_Main}), the second largest at the time of the 2004\nSumatra event, and so on. In this way, we model earthquake seismic\nmoments as\narising from a gamma distribution with fixed parameters and with\noccurrence times given by the empirical times and with the same\nseismic moment correlations\nas the empirical data, approximately.\n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\centerline{\\includegraphics[trim = 0mm 130mm 0mm\n10mm,clip=true,width=9cm]{explore_earthquake_figura3_resumlikeli_1000.pdf}}\n\\caption{Comparison of the empirical log-likelihood-ratios between\nthe TrG and power law with those of 1000 simulations of the TrG\ndistribution, using the final parameters of Table \\ref{taula}\n{(i.e., $\\beta=0.681$ and $m_c=9.15$)}. Simulated seismic moments\nare reshuffled as explained in the text {to make the comparison\npossible}. Simulation results are displayed using boxplots,\nrepresenting the three quartiles of the distribution of\n$2\\mathcal{R}$.\n\\label{l_boxplots}}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\centerline{\\includegraphics[trim = 0mm 0mm 0mm\n60mm,clip,width=9cm]{explore_earthquake_final_3.pdf}}\n\\caption{Comparison of the values of the estimated parameters of the\nTrG distribution, $\\hat \\beta_{trg}$ and $\\hat m_{c\\,trg}$, for the\nempirical data and for 1000 simulations of the TrG distribution,\nusing the final parameters of Table \\ref{taula}. Simulated seismic\nmoments are reshuffled as explained in the text. The different\nstability of both parameters is apparent. \\label{gr4}}\n\\end{center}\n\\end{figure}\n\n\n\n\n\nWe simulate 1000 datasets with $N=6150$ each. The results, displayed\nin Fig. \\ref{l_boxplots}, show that the behavior of the empirical\ndata is not atypical in comparison with this gamma modeling. In\nnearly all time windows the empirical data lies in between the first\nand third quartile of the simulated data, although before 2004 the\nempirical values are close to the third quartile whereas after 2004\nthey lay just below the median. This leads us to compute the\nstatistics of the jump in the log-likelihood-ratio between 2004 and\n2005. The estimated probability of having a jump larger than the\nempirical value is around 4.5 \\%, which is not far from what one\ncould {accept} from the gamma modeling explained above.\n{Thus, a TrG distribution, with fixed parameters, is able\nto reproduce the empirical findings, if the peculiar time ordering of magnitude\nof the real events is taken into account.\nNotice also that, although the simulated data come from a TrG distribution,\nthey are not distinguishable from a power law for\nabout half of the simulations of the last time windows,\nas the critical region is close to the median\nindicated by the boxplots.}\n\n\n\nWe can also compare the evolution of the estimated parameters\nfor the empiral dataset and for the reshuffled TrG simulations, with a good agreement\nagain, see Fig. \\ref{gr4}. There, it is clear that although the exponent $\\beta$\nreaches very stable values relatively soon,\nthe scale parameter $\\theta$ (equivalent to $m_c$) is largely unstable,\nand the occurrence of the biggest events makes its value increase.\n\n\n\n{As a complementary control\nwe invert the situation, simulating\n1000 syntetic power-law datasets\nwith $\\beta=0.685$ (Table \\ref{taula}),\n$a= 5.3 \\cdot 10^{17}$ N$\\cdot$m,\nand $N=6150$, for which the same time reshuffling is perfomed,\nin such a way that the order of the order statistics is the same.\nIn this case, the results of the simulations lead to much smaller values\nof the log-ratio, for which the power-law distribution cannot be\nrejected (as expected), in contrast with the empirical data,\nsee Fig. \\ref{figlanueva}.}\n\n\\begin{figure}[t!]\n\\begin{center}\n\\centerline{\\includegraphics[trim = 0mm 70mm 0mm\n70mm,clip,width=9cm]{explore_earthquake_figura3_resumlikeli_1000.pdf}\n} \\caption{ As Fig. \\ref{l_boxplots}, but simulating a power law\nwith parameter $\\beta=0.685$ (Table \\ref{taula}) instead of a TrG\ndistribution. The reshuffling is also as in Fig. \\ref{l_boxplots},\nas explained in the text.\nThe\nsimulations cannot explain the large empirical values of the log-likelihood-ratio.\n\\label{figlanueva}}\n\\end{center}\n\\end{figure}\n\n\n\n\n\nIn summary, the truncated gamma distribution represents the best\nalternative to model global shallow earthquake seismic moments, in\ncomparison with the tapered GR distribution and the power law. The\npreponderance of the gamma model is maintained after the occurrence\nof the mega-earthquakes taking place from 2004\n{and it is only after the 2011 Tohoku earthquake\nthat it is difficult to decide between power law and TrG.\nWe have verified that these results are qualitatively similar if\nwe restrict our study to subduction zones, as defined by\nthe Flinn-Engdahl's regionalization \\cite{Kagan_pageoph99}, \nwith the main difference that\nthe values of $l_{trg}-l_{pl}$ become somewhat smaller\nand therefore the power-law hypothesis cannot be rejected\nafter the Tohoku earthquake.}\n\n\n{In order to reproduce the\ntime evolution of the statistical results\nit suffices that independent gamma seismic moments,\nwith fixed parameters, are reshuffled\nso that the peculiar empirical time correlations of magnitudes are maintained.}\nSo, although the scale parameter\n$\\theta$ is not stabilized, and the occurrence or not of more\nmega-earthquakes could significantly change its value\n\\cite{Zoller_grl},\n{the current value is enough to explain the\navailable data}.\nIt would be very interesting to investigate if\nthe high values of the likelihood ratio attained before the 2004\nSumatra event could be employed to detect the end of periods of low\nglobal seismic activity. Certainly, more data would be necessary {for that purpose}\n\\cite{Zoller_grl}.\n\nAs an extra argument in favor of the truncated gamma\ndistribution in front of the tapered GR,\nwe can bring not a statistical evidence but\nphysical plausibility; indeed,\nthe former distribution can be justified as coming from a\nbranching process that is slightly below its critical point\n\\cite{Christensen_Moloney,Corral_FontClos}.\n{Further reasons that may support\nthe truncated gamma\nare that\nthis arises\n(i) as the maximum entropy outcome\nunder the constrains of fixed (arithmetic) mean and fixed geometric mean of the seismic moment\n\\cite{Main_information};\n(ii) as the closest to the power law, in terms of the Kullback-Leibler ``distance'',\nwhen the mean seismic moment is fixed \\cite{Sornette_Sornette_bssa}; and\n(iii) as a stable distribution under a fragmentation process with a power-law\ntransition rate \\cite{Sornette_Sornette_bssa}.}\n\n\n\n\n\n\n\n\n\n\nWe are grateful to J. del Castillo, Y. Y. Kagan, I. G. Main, and M. Naylor for their feedback.\nResearch expenses were founded by projects\nFIS2012-31324 and MTM2012-31118 from Spanish MINECO,\n2014SGR-1307 from AGAUR, and the Collaborative Mathematics\nProject from La Caixa Foundation.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThemis is one of the most statistically reliable families in\nthe asteroid belt. First discovered by Hirayama (1918), it has been identified as a\nfamily in all subsequent works, and it has 550 members according to Zappal\\'{a} et al. (1995), and more than 5000 members as determined by Nesvorny (2012). \nThe Themis family is characterized by asteroids with 3.05 $ \\leq a \\leq $ 3.22 AU,  0.12 $\\leq e \\leq$ 0.19, in low inclination orbit $0.7^{o} \\leq i \\leq 2.22^{o}$ (Zappal\\'{a} et al. 1990). It is spectrally dominated by primitive C- and B-type asteroids, as reported by spectroscopic investigation of several members (Moth\\'e-Diniz et al. 2005; Florczak et al. 1999; de Leon et al. 2012; Kaluna et al., 2016). The family is old with an estimated age of $\\sim$ 2.5$\\pm$1.0 Gyr (Bro\\v{z} et al. 2013), and it was formed from a large-scale catastrophic disruption of a $\\sim$ 270 km (Bro\\v{z} et al. 2013) or a $\\sim$ 400 km sized parent body (Marzari et al. 1995; Durda et al. 2007). Nesvorny et al. (2008) confirmed that this family is quite old and formed more than 1 Gyr ago.\\\\\nInterestingly, Rivkin \\& Emery (2010) and Campins et al. (2010) found spectroscopic evidence of the presence of water ice and organics on (24) Themis. Rivkin and Emery (2010) concluded that its surface contains very fine water frost, probably in the form of surface grain coatings, and that the infrared spectral signatures can be fully explained by a mixture of spectrally neutral material, water ice, and organics. At the same time, Campins et al. (2010) suggested that water ice is evenly distributed over the entire Themis surface using spectra obtained at four different rotational phases. \nNevertheless the nature of the 3.1 $\\mu$m feature on (24) Themis is still a matter of debate, and Beck et al. (2011) proposed the hydrated iron oxide goethite as alternative interpretation of this feature, even if Jewitt \\& Guilbert-Lepoutre (2012) stress that goethite, when found in meteorites, is a product of aqueous alteration in the terrestrial environment and that extraterrestrial goethite in freshly fallen meteorites has not been detected. \\\\\nAbsorption bands in the visible region related to hydrated silicates have been detected on the surface of several Themis family members (Florczak et al. 1999). These materials are produced by the aqueous alteration process, that is a low temperature ($<$ 320 K) chemical alteration of materials by liquid water (Vilas \\& Sykes, 1996). The presence of hydrated minerals implies that liquid water was present on these asteroids in the past, and suggests that post-formation heating took place.\nThe  main belt comets 133P\/Elst-Pizarro, 176P\/LINEAR, 288P\/(300163) 2006 VW139, and possibly 238P\/Read may be related to the Themis family (Nesvorny et al. 2008; Hsieh et al. 2006, 2009, 2012; Haghighipour 2009) because of orbital proximity and spectral properties analogies. According to Licandro et al. (2011, 2013) works, the visible spectra of the main belt comets 133P, 176P, and 288P  are compatible with those of Themis and Beagle families members.\nIn particular Nesvorny et al. (2008) propose that 133P could potentially be one member of the Beagle sub-family within the Themis group. This sub-family is a young cluster with an estimates age $< 10$ Myr (Nesvorny et al. 2008).\n\nThe detection of water ice on asteroid (24) Themis, of hydrated silicates on several Themis members and the spectral and dynamical link between the Themis family and some main belt comets all indicate that Themis family is an important  reservoir of water in the outer main belt. These discoveries support the suggestion that at least some of Earth's current supply of water was delivered by asteroids some time following the collision that produced the Moon (Morbidelli et al. 2000, Lunine 2006). \\\\\nWith this paper we investigate both regular Themis family members (22) and younger ones (8), belonging to the Beagle sub-family, aiming to constrain their surface composition, to establish spectral links with meteorites, and to investigate potential space weathering effects and spectral variability between the two families members.\n\n\n\\section{Observations and data reduction} \n\nSpectroscopic observations in the visible and near infrared range of Themis and Beagle families members were made at the 3.56 m Italian Telescopio Nazionale Galileo (TNG) of the European Northern Observatory (ENO) in la Palma, Spain, in two runs, on 16-20 February and 16-18 December 2012.\nFor visible spectroscopy we used the Dolores (Device Optimized for the \nLOw RESolution) instrument equipped with the low resolution red (LRR) and blue (LRB) grisms.\nThe LRR grism covers the 0.52--0.95 $\\mu$m range with a spectral dispersion \nof 2.9 \\AA\/px, while the LRB grism covers the 0.38-0.80 $\\mu$m range with a spectral dispersion \nof 2.8 \\AA\/px. \nThe Dolores detector is a 2048 $\\times$ 2048 pixels Loral thinned and \nback-illuminated CCD, having a pixel size of \n15 $\\mu$m and a pixel scale of 0.275 arcsec\/px. \nThe red and blue spectra in the visible range were separately reduced and finally combined together to obtain spectral coverage from 0.38 to 0.95 $\\mu$m. Like most of the Loral CCDs, the Dolores chip is affected by moderate-to-strong fringing\nat red wavelengths. Despite taking as much care as possible in\nthe data reduction process, some asteroid spectra acquired with the LRR grism show residual\nfringing that impedes identification of absorption bands and an accurate analysis in the 0.81-0.95 $\\mu$m range when NICS observations are not available. \\\\\nFor the infrared spectroscopic investigation  we used the\nnear infrared camera and spectrometer (NICS) equipped with an Amici prism disperser. \nThis equipment covers the 0.78--2.40 $\\mu$m range during a single \nexposure with a spectral resolution of about 35 (Baffa et al. 2001). \nThe detector is a 1024 $\\times$ 1024 pixel Rockwell HgCdTe Hawaii array. \nThe spectra were acquired nodding the object\nalong the spatial direction of the slit, in order to obtain alternating pairs \n(named A and B) of near--simultaneous images for the  background removal. \nFor both the visible and near infrared observations we utilized a 2 arcsec \nwide slit, oriented along the parallactic angle to minimize \nthe effect of atmospheric differential refraction.\n\n\nVisible and near-infrared spectra were reduced using ordinary procedures \nof data reduction with the software packages Midas as described in\nFornasier et al. (2004, 2008). \nFor the visible spectra, the procedure includes the subtraction of the bias \nfrom the raw data, flat--field correction, cosmic ray removal, sky subtraction,\ncollapsing the two--dimensional spectra to one dimension, wavelength calibration,\nand atmospheric extinction correction, using La Palma atmospheric extinction coefficients.\nThe spectra were normalized at 5500 \\AA. \nThe reflectivity of each asteroid was \nobtained by dividing its spectrum by that of the solar analog star closest in\ntime and airmass to\nthe object. Spectra were finally smoothed with a median filter \ntechnique, using a box of 19 pixels in the spectral direction for each point of\nthe \nspectrum. The threshold was set to 0.1, meaning that the original value was\nreplaced by \nthe median value if the median value differs by more than 10\\% from the original\none.\n\nFor observations in the infrared range, spectra were first corrected for flat\nfielding, \nthen bias and sky subtraction\nwas performed by producing A-B and B-A frames. The positive spectrum of the B-A\nframe was \nshifted and added to the positive spectrum of \nthe A-B frame. The final spectrum is the result of the mean of all pairs of \nframes previously combined. The spectrum was extracted and wavelength \ncalibrated. Due to the very low resolution of the Amici prism, \nthe lines of Ar\/Xe lamps are blended and cannot be easily used for standard reduction procedures. \nFor this reason, the wavelength calibration was obtained using a look-up table\nwhich is  based on the theoretical dispersion predicted by ray-tracing and adjusted \nto best fit the observed spectra of calibration sources.  Finally, the extinction correction and solar removal was obtained by \ndivision of each asteroid spectrum by that of the solar analog star closest in\ntime and airmass to the object. The stellar and asteroid spectra\nwere cross-correlated and, if necessary, sub-pixel shifts were\nmade before the asteroid-ratio star division was done. \\\\\nThe infrared and visible spectral ranges of each asteroid were finally\ncombined by overlapping the spectra, merging the two wavelength\nregions at the common wavelengths and utilizing the zone\nof good atmospheric transmission to find the normalization factor\nbetween the two spectral parts. The\noverlapping region goes from about 0.78-0.92 $\\mu$m. For most of the data  we took the\naverage value in reflectance of the visible spectrum in the 0.89--0.91 $\\mu$m region \nto normalize the reflectance of the infrared spectrum. For the spectra where LRR grism data were not available or present but with strong fringing problems, we joint the visible and NIR data using the mean reflectance value of the visible spectrum in the 0.79-0.81 $\\mu$m region. \\\\\nAsteroids having both visible and NIR data or only visible spectra were {\\bf finally all normalized in relative reflectance at 1 at the wavelength 0.55 $\\mu$m}, while those observed only in the NIR region were normalized  at 1.25 $\\mu$m. \nDetails on the observing conditions are reported in Table~\\ref{tab1}, and the spectra of the observed asteroids are shown in Figs.~\\ref{f1}-\\ref{f4}.\n\n\n[Here Figs 1, 2, 3, 4]\n\n[Here Table 1]\n\n                          \\section{Results}\n\n                          [Here Table 2 and Figure 5]\n\nWe present new spectra of 8 Beagle and 22 Themis families members.\nMost of the asteroids were observed in the V+NIR range, except for some targets for which\nwe did not get the visible spectra (objects 621, 954, 1778, 2009, and 2203; for asteroids 621 and 954 we complete our data with the visible spectra from the\n$S^3S0^2$ survey (Lazzaro et al. 2004)), or the NIR ones (62, 383, 268, 526, 1247, and 2228; for objects 62 and 383 we got the IR spectra from Clark et al. (2010)). \nThe observed asteroids and their proper elements are reported in Table~\\ref{slope}. \\\\\nThe members of each family were derived from the family lists from analytic proper elements\n(Nesvorny et al. 2012).  A hierarchical clustering method has been applied to the asteroids proper elements to identify the dynamical families (Nesvorny et al., 2008). Themis is an old and huge family with 5425 members identified at a cutoff velocity of 60 m\/s (Nesvorny et al. 2012). Beagle is a young smaller cluster within Themis, having 149 members at a cutoff velocity of 20 m\/s, and an estimated age $<$ 10Myr (Nesvorny et al. 2008, 2012).  The cutoff velocity is related to the relative velocity of the fragments as they were ejected from the sphere of influence of the parent body, and lower values implies a higher statistical significance of the family. This means that the Beagle family is very robust.\n        \nAll the objects investigated belong to the C or B spectral types, following the Tholen (1984) classification scheme. The majority of the spectra are featureless, and in particular none of the investigated spectra show water ice absorption bands at 1.5 and 2 micron. A few Themis members ((24) Themis, (90) Antiope, (461) Saskia, and (846) Lipperta) show weak spectral absorption features in the visible range associated with hydrated minerals (Table~\\ref{hydra}, and Fig.~\\ref{f5bis}), indicating that they have experienced the aqueous alteration process, as already found for other Themis family members (Florczak et al. 1999).\nWe considered only the absorption features deeper than the peak-to-peak  scatter (that is depth $>$ 0.8\\%) in the spectrum, and we characterized them following the method described in Fornasier et al. (1999, 2014). This method includes first a smoothing of the spectra using a box of 6 pixels in the spectral direction, then the evaluation of the linear continuum at the edges of the band identified, the division of the original spectrum by the continuum, and the fitting  of the absorption band with a polynomial of order 2--4.  The band center was then calculated as the position where the minimum of the spectral reflectance curve occurs (on the polynomial fit), and the band depth as the minimum of the polynomial fit. The main absorption band identified is the one centered in the 0.68-0.73 $\\mu$m region which is attributed to $Fe^{2+}\\rightarrow Fe^{3+}$ charge transfer absorptions in phyllosilicate minerals (Vilas \\& Gaffey 1989; Vilas et al. 1993). This band is often associated with an evident UV absorption below $\\sim$ 0.5 $\\mu$m, due to a strong ferric oxide intervalence charge transfer transition (IVCT) in oxidized iron, and is often coupled with\nother visible absorption features related to the presence of aqueous alteration products (e.g. phyllosilicates, oxides, etc) (Vilas et al. 1994). We see the UV drop-off in reflectivity in most of the asteroids observed with the LRB grism, that is on Themis members (24) Themis, (268) Adorea, (461) Saskia, (492) Gismonda, (526) Jena, (1623) Vivian, (222) Lermontov, (2228) Soyuz-Apollo, (2270) Yazhi, and on Beagle members (656) Beagle, (1687) Glarona, (2519) Annagerman, (3174) Alcock, and (4903) Ichikawa,  but not on (90) Antiope and (846) Lipperta, that have a faint absorption centered at $\\sim$ 0.7 $\\mu$m. These IVCTs comprise multiple absorptions that are not uniquely indicative of phyllosilicates, but are present in the spectrum of any object\ncontaining Fe$^{2+}$ and Fe$^{3+}$ in its surface material (Vilas 1994). Thus, even if most of these spectra are featureless in the VIS-NIR range, their UV drop-off in reflectivity may indicate the presence of phillosilicates (Vilas 1994).  \\\\ \nSaskia also shows two absorption features in the NIR region centered at about 1.32 $\\mu$m and 1.82 $\\mu$m, as determined by a 4$^{th}$ order polynomial fit to the absorption bands. However, the center of these bands fall in the region of very low atmosphere transmission so their center position cannot be precise and their identification needs to be confirmed by other observations. If real, these absorptions look similar to those found on some hydrous Fe-bearing hydrated silicates (Bishop \\& Pieters 1995), thus potentially associated to the presence of hydrated minerals.  The continuum removed spectra of asteroids showing faint absorption features are shown in Fig.~\\ref{f5bis}. \\\\\nWe identified features associated to hydrated silicates only on Themis members. Thus only 18\\% of the 22 Themis members investigated show evidence of aqueous alteration in the visible and near infrared ranges. Recently Kaluna et al. (2016) investigated a sample of 22 Themis and 23 Beagle members. The asteroids they observed are all different from those of our survey, and they are all smaller than 15 km. Interestingly their analysis also indicates a lack of the 0.7 $\\mu$m features in the observed asteroids, with only one Beagle member having that absorption. It must be noted that the absence of the 0.7 $\\mu$m feature does not necessarily imply a lack of aqueous alteration, considering that this absorption is much fainter than the main absorption feature associated to hydrated minerals and centered at 3 $\\mu$m (Lebofsky 1980; Jones et al. 1990), and that not all the aqueous altered  CM\/CI meteorites show this feature (McAdam et al. 2015). Considering the results of other surveys,  Florczak et al. (1999) found  absorption bands associated to hydrated silicates in 15 out of 36 Themis members observed in the visible range, while Fornasier et al. (2014, Table 4), observed the $\\sim$ 0.7 $\\mu$m band only on 8 out of 47 Themis members spectra.\\\\\nTwo out of the three Themis members observed by Licandro et al. (2011) are in common with our observations: (62) Erato and (383) Janina. For (62) Erato Licandro et al. found a slope value (-2$\\pm$1 \\%\/(1000\\AA) similar to the one we found (Table~\\ref{slope}), while the Janina spectrum is quite different, having a positive visible slope in our data and a negative one in both Licandro et al. (2011) and  Bus \\& Binzel (2003) observations. Additional data covering different rotational phases are needed to cast light on potential surface heterogeneities of this asteroid.    \n\n[Here Table 3]\n\n\\subsection{(24) Themis: surface heterogeneities}\n\n[Here FIGURE 6]\n\n(24) Themis is a big asteroid with a diameter of  218$\\pm$1 km and a low albedo value of  6$\\pm$1\\% (Hargrove et al. 2015).  \nWe observed (24) Themis three times in the visible range and once in the NIR range during the December 2012 run, finding some spectral variability (Fig.~\\ref{f5}). Unfortunately  all the LRR spectra of (24) Themis  suffered of strong fringing problems, and they have a very bad S\/N ratio in the 0.8-0.95 $\\mu$m. We thus complete the {\\bf 18 and 19 Decem} LRB observations with the available NICS spectrum, that has a better S\/N ratio in 0.8-0.95 $\\mu$m range than the LRR data. The spectra acquired on 18 Dec. have a negative slope in the visible range and no hints of absorption bands, while the one taken on 19 Dec. shows an absorption band centered at $\\sim$ 7333$\\pm$48\\AA, having a depth of 2.6\\%, attributed to hydrated silicates. \nThe data also indicate a clear UV-drop-off of the reflectance for wavelength $<$  0.5 $\\mu$m for the observations acquired on 18 Dec. UT 22 and 19 Dec. Assuming a rotational period of 8.37677$\\pm$0.00002 h (Higley, 2008) and taking the LRB spectrum of 18 Dec. UT 02:16 as reference for the rotational phase, then the second and third LRB spectra of (24) Themis fall at rotational phases of 0.3020 and 0.7278, respectively. We thus covered different rotational phases and the observed spectral differences may be related to surface heterogeneities on (24) Themis. This conclusion is supported also by the comparison with (24) Themis spectra available in the literature (Fig.~\\ref{f5}), taken from Fornasier et al. (1999), Bus\\& Binzel (2002), and Lazzaro et al. (2004). In particular Fornasier et al. (1999) reported an absorption feature attributed to hydrated silicates, similar to the one found in this paper but centered at a shorter wavelength (6722$\\pm$39 \\AA), with a depth of 3.5\\% compared to the continuum, while the other literature spectra are featureless except for the potential presence of the UV-drop-ff in reflectivity.\n\n(24) Themis shows strong evidence of surface water ice (Campins et al. 2010; Rivkin et al. 2010), that seems to be widespread present on its surface, according to Campins et al. (2010) results. Its emissivity spectrum  exhibits a rounded 10 $\\mu$m emission feature, found also on other Themis family members (Licandro et al. 2012), attributed to small silicate grains embedded in a relatively transparent matrix, or from a very under-dense surface structure (Hargrove et al. 2015). Recently McAdam et al. (2015) interpret the emissivity behavior of (24) Themis as due to complex surface mineralogy with approximately 70 vol.\\% phyllosilicates and 25 vol.\\% anhydrous silicate. \\\\\nConsidering our data and those of the literature, (24) Themis most likely has an heterogeneous surface composition, presenting both water ice and hydrated silicates together with dark phases like those found on carbonaceous chondrites. According to Castillo-Rogez \\& Schmidt (2010) models, large Themis family objects may contain a large fraction of the parent body ice shell, and this ice may not be completely pure, but possibly contaminated with other materials such as organics, oxides and  hydrated minerals. Jewitt \\& Guilbert-Lepoutre (2012) report no detection of gas from sublimated ice on (24) Themis, and conclude that ice must cover a relatively small fraction (10\\%) of the asteroid surface, having potentially been exposed on the surfaces of (24) Themis by recent impacts. Surface heterogeneity seems to be common on the largest asteroids\/dwarf planet imaged so far, as seen for Vesta and Ceres by the Dawn mission. For instance several bright spots are observed on Ceres, and potentially attributed to water ice associated with impact craters (Reddy et al. 2015), and groundbased observations underline spectral variability on its surface (Perna et al. 2015).\\\\\nFuture observations  with high S\/N rotationally resolved spectra in the visible and near infrared region are needed to fully investigate the (24) Themis surface heterogeneity. \n\n\n\n\\subsection{Spectral slopes and physical parameters}\n\n[Here Figs. 7 and 8]\n\nTo analyze the data, we compute spectral slope values with linear\nfits to different wavelength regions: $S_{cont}$ is the spectral slope in the whole\nrange observed for each asteroid, $S_{VIS}$ is the slope in the 0.55-0.80 $\\mu$m\nrange, S$_{NIR1}$ is the slope in the 0.9-1.4 $\\mu$m range, S$_{NIR2}$ is\nthe slope in the 1.4-2.2 $\\mu$m range, and  S$_{NIR3}$ is\nthe slope in the 1.1-1.8 $\\mu$m range. Values are reported in\nTable~\\ref{slope}. The slope error bars take into account the\n$1\\sigma$ uncertainty of the linear fit plus 0.5\\%\/$10^3$\\AA\\ attributable to the spectral variation due to the use of different solar analog stars during the night. \\\\\n\n[Here Figs. 9 and 10]\n\nOur observations clearly show a range of spectral behaviors exhibited by Themis family members in the visible and near-infrared (Figs.~\\ref{f1}-~\\ref{f3}, and~\\ref{f6}), including asteroids with blue\/neutral and moderately red spectra (relative to the Sun). Our findings are consistent with, and complement previous spectroscopic studies that hinted at spectral diversity within the Themis family in the visible (Florczak et al. 1999; Kaluna et al., 2016), the near-infrared (Ziffer et al. 2011; de Le\\'on et al. 2012; Al\\'i-Lagoa et al. 2013), and the mid-infrared (Licandro et al. 2012). On the other hand, the robust and younger Beagle family members look different, with a smaller spectral slope variability. This family seems dominated by objects with a blue\/neutral spectrum in the visible range, and with a neutral to moderately red spectral behavior in the near infrared range.\n\nTo better understand the difference between the two families, we look for correlations between spectral slopes and physical parameters such as the albedo and diameter, which were derived from the WISE data (Masiero et al. 2011).\nFigure~\\ref{f7} shows the visible and NIR3 spectral slopes versus the WISE albedo ($p_v$) for the Themis and Beagle families members here investigated. It clearly appears that the young Beagle members are not only spectrally bluer but tend to have a higher albedo value (mean $p_v$ = 0.0941$\\pm$0.0055) than most of the Themis members investigated (mean $p_v$ = 0.0743$\\pm$0.0054). \\\\\nTo test if this trend is true, we extended the analysis on a larger sample including 79 Themis and Beagle families members spectra available in the literature from several surveys in the visible wavelength range (Xu et al. 1995; Bus \\& Binzel 2002; Lazzaro et al. 2004; Fornasier et al. 1999, 2014; Kaluna et al., 2016). For most of these data, the spectral slope has been evaluated in a similar way to what is done here, i. e. between 0.55-0.8 $\\mu$m. Only the spectral slope for asteroids observed in Kaluna et al. (2016) was evaluated in a different manner, between 0.49 and 0.91 $\\mu$m. However, beeing their spectra normalized at the same wavelength than us and featureless, the variation in the spectral slope values associated to the  different wavelength ranges used must be minimal. For 11 out of 23 Beagle members and 47 out of 56 Themis members of the literature data the albedo value is available from the WISE observations. Figure~\\ref{f8} shows the visible spectral slope versus albedo of this extended sample. The trend seen in our data is confirmed and it is evident that there are no Beagle members with a red spectral slope.  Conversely, Themis members show an important spectral variety, having both blue and moderately red slopes and low to moderate albedo values (4-15\\%). This spectral variety is not related to different sizes of the asteroids investigated because Themis members having diameter similar to the Beagle ones span different VIS spectral slopes (-2.5 $\\%\/(10^{3}$\\AA) to 3 $\\%\/(10^{3}$\\AA)) and albedo values (4-12.5\\%), as shown in Figure~\\ref{f9}. \\\\\nThe difference in the albedo values is less pronunced, Themis members having a slightly lower mean albedo  (0.0784$\\pm$0.0031) than Beagle ones  (0.0819$\\pm$0.0045). However, analyzing the average albedo from WISE data on a larger sample and comparing members of similar sizes,  Kaluna et al. (2016) found that the small Themis members ($D <$ 15km) have a lower albedo ($p_v$=0.068$\\pm$0.001) than the larger asteroids ($p_v$=0.075$\\pm$0.001) of the family, and that their albedo is  significantly lower than that of the Beagle population ($p_v$=0.079$\\pm$0.005). Their results clearly indicate that Beagle members have a sligthly higher albedo value than the Themis members of similar size.\n\n[Here Fig. 11]\n\nWe also analyzed the near infrared albedo (p$_{IR}$), that is the albedo at 3.4--4.6 $\\mu$m as defined in Mainzer et al. (2011a), and the geometric (p$_v$) albedo for the Themis members observed with WISE. Figure~\\ref{f10} shows the ratio of the p$_{IR}$\/p$_v$ albedo versus p$_v$ for a sample of 211 Themis and 5 Beagle members. The bright Themis members have lower p$_{IR}$\/p$_v$ ratio, indicating a blue spectrum in the NIR region and\/or the presence of absorption bands in the 3 $\\mu$m region, potentially attributed to hydrated silicates, organics and eventually water ice.\\\\\nA similar {\\it waning moon} shape of the  p$_{IR}$\/p$_v$ ratio versus p$_v$  was also noticed by Al\\'i-Lagoa et al. (2013) analyzing B-type asteroids. Some biases may partially affect the distribution we see, in particular the lack of points in the lower left corner may be attributed to faint fluxes in WISE bands W1 and W2, not allowing a proper evaluation of the reflected sunlight contribution (Mainzer et al. 2011b).  Al\\'i-Lagoa et al. (2013), in the analysis of B-type asteroids and p$_{IR}$ albedo, found that an absorption in the 3 $\\mu$m region may be common on these bodies.  \\\\\nFigure~\\ref{f10} confirms the huge spectral variability of Themis members, which have small to moderate albedo values  (3--16\\%) and a P$_{IR}$\/p$_v$ ratio varying from $\\sim$ 0.5 to 2.2.\n\n\n\n\\subsection{Meteorite analogues}\n\n[Here Figs. 12-15 and Table 4]\n\nTo constrain the possible mineralogies of the investigated asteroids, we conducted\na search for meteorite and\/or mineral spectral matches. We\nsought matches only for objects observed across the entire VIS--NIR\nwavelength range. We used the publicly available RELAB spectrum library (Pieters,\n1983). For each  spectrum in the library, a filter was applied to find relevant  \nwavelengths (0.4 to 2.45 $\\mu$m). Then, a second filter was applied to reject spectra with irrelevant albedo values (i.e. brightness at 0.55 $\\mu$m $>0.2$). A  Chi-squared value was calculated relative to the normalized input asteroid spectrum. We used the asteroid wavelength sampling to resample the laboratory spectra by linear interpolation. This allowed a least-squares calculation with the number of points equal to the wavelength sampling of the asteroid spectrum. The RELAB data files were sorted according to the corresponding Chi-squared values, and finally visually examined. A complete description of the search methodology\ncan be found in Fornasier et al. (2010, 2011).\n \nThe best matches between the observed asteroids and meteorites from the RELAB database are reported in Table~\\ref{met}, and shown in Figures~\\ref{f11} -~\\ref{f14}.\nMost of the Themis and Beagle families members are matched by CM2 carbonaceous chondrites, with few objects having CV3 or CI as best match. We do not see differences in the meteorite class matches between  the Beagle and Themis families members, except that there are no CV3 best match for Beagle members. Even if the aqueous alteration band at 0.7 $\\mu$m has been identified just on one Beagle member (Kaluna et al., 2016), the meteorite matches indicate that these asteroids may have experienced aqueous alteration in the past. As previously stated, not all the aqueous altered  CM\/CI meteorites show this feature (McAdam et al. 2015). \\\\\nThe asteroid (2519) Annagerman is spectrally perfectly matched by the unusual CI\/CM Y-86720,77 with grain size  $<$ 125 $\\mu$m, even if there are some differences in the albedo values, 11\\% for the asteroid and  6\\% for the meteorite. However it must be noted that the meteorite reflectance is usually evaluated at phase angle $>$ 7 $^o$ so it does not include the opposition surge effect. The composition and structure of the CM2 Y-86720 indicates that this meteorite was thoroughly aqueous altered (Lipschutz et al. 1999) and heated. Interestingly, some of the targets are best matched by heated or unusual meteorites. This was  already noticed by Clark et al. (2010), who found that Themis family asteroids tend to be best fitted by CM, CI, CR, CM\/CI unusual meteorites, and\/or heated CM\/CI samples, and that 50\\% of these asteroids are consistent with thermally metamorphosed material. \\\\\nOur spectral matches are also in agreement with the de L\\'eon et al. (2012) results from the  analysis of B-type asteroids, including 8 Themis members. These last fall in the G2-G5 groups as defined by de L\\'eon et al. (2012), and are spectrally best matched by meteorites who experienced different stages of the aqueous alteration,  from the less altered CV3 meteorites up to the extensively altered CM2 chondrites. \nThe spectral analysis together with the meteorite matches indicate that Themis members show a large spectral variety and different carbonaceous chondrite matches (from CV3 to CM2-CI), some of them being unusual or heated, indicating that these asteroids were thermally metamorphosed in the past. \n\n\n\n\\section{Discussion}\n\nSpectroscopic studies of asteroid families can provide information about the interior of their parent bodies (e.g., Cellino et al. 2002); this can constrain models on the thermal and collisional evolution of asteroids.  More specifically, simulations of catastrophic disruption processes suggest that the ejected material coming from coherent sections of a heterogeneous progenitor body should maintain a coherent compositional structure (Michel et al. 2004). Therefore, compositional gradients within planetesimals should expose themselves within asteroid families (e.g., Jacobson et al. 2014).  To date, no remnants of a completely disrupted differentiated body have been identified. Vesta is the only family generated from a differentiated body (e.g., Lazzaro et al. 2004; Binzel and Xu 1993), but it is not a result of a catastrophic disruption, so its members come from the surface of the parent body.\n\nThe case of the Themis family seems to be unique. For example, Jacobson et al. (2014) used estimates of the progenitor mass and the mass of the largest remnant to assess the exposed nature of asteroid families, and he suggested that the Themis family is possibly the most exposed one.   We have found clear evidence of spectral diversity within the Themis family, and between Themis and Beagle families members.  There are several possible interpretations of this spectral diversity, which include a compositional gradient with depth in the original parent body (and\/or with fragment size), space weathering effects, the survival of projectile fragments with a different composition, or a combination of these (e.g., Campins et al. 2012), that we discuss in the following. \\\\\n\n\\subsection{Scenario a): The Themis\/Beagle family parent body was heterogeneous in composition} \n\nIn this scenario, the diversity we see nowdays in Themis reflects different source regions in the parent body that was originally heterogeneous showing a compositional gradient. \nWe clarify that a compositional gradient does not necessarily mean a metal core with a silicate mantle and crust. In fact, for primitive asteroids like the Themis parent body it was more likely a differentiation of rock and ice where the core underwent mild temperatures and no high thermal metamorphism. (e.g., Castillo-Rogez \\& Schmidt 2010).\nConsistent with this mild temperature model are the visible spectroscopic results of 36 members of the Themis family presented by Florczak et al. (1999). They found that about 50\\% of their sample showed evidence of aqueous alteration, indicating that the parent body was sufficiently altered thermally to mobilize water. The results of  Florczak et al. (1999) also suggest that the percentage of asteroids showing aqueous alteration may decrease with the diameter of the objects, as expected by Vilas \\& Sykes (1996) and proven by Rivkin (2012) and Fornasier et al. (2014) on a large sample of primitive main belt asteroids. The study  presented in this paper and in Fornasier et al. (2014) indicates a much lower percentage of hydration in the Themis family ($<$ 20\\%), while just one out of the 45 Themis and Beagle small members studied by Kaluna et al. (2016) shows the 0.7 $\\mu$m absorption band. Moreover the meteorite spectral matches found by us and in the literature point towards a similarity of Themis-Beagle family members with the CV3-CM2 chondrites, the meteorites having experienced aqueous alteration processes (see section 3.3; Clark et al. 2010; de L\\'eon et al. 2012). \\\\\nThe above results support the view that at least part of the parent body of the Themis family was thermally altered and that the distribution of compositions could be attributed to fragments coming from different depths in the original parent body, which was catastrophically disrupted by a large projectile.\\\\\nIn this scenario, the Beagle family may have been originated by a fragment of a blue and bright piece of Themis parent body possibly coming from its interior.\\\\\n\n\\subsection{Scenario b): The Themis\/Beagle family parent body was homogeneous and the projectile that collided with it was different in composition}\n\n  Marzari et al. (1995) modeled the collisional evolution of the Themis family. Their best results yield parent body and projectile radii of 380 km and 190 km, respectively. They suggest that the two bodies were not formed in the same region and had different compositions. Since the projectile body was relatively large, its fragments should be a considerable part of the remnants. Interestingly, Florczak et al. (1999) argue that except for the hydration (indicated by an absorption centered near 0.7 $\\mu$m) their visible spectra were homogeneous and they did not favor a different composition between parent body and projectile or significant differentiation.  Our visible and near-infrared spectra (Figs.~\\ref{f7} and ~\\ref{f8}) and those of de Le\\'on et al. (2012) and Kaluna et al. (2016) show a wide and smooth range of behaviors suggestive of a range of compositions among the Themis family members. These results does not necessarily support different compositions between the large projectile and the Themis family parent body. Instead of a smooth spectral diversity, one might expect a bi-modal one from a different projectile and parent body composition.\n  \n\n\\subsection{Scenario c): The family parent body was homogeneous and the spectral variability is produced by space weathering effects} \n\n  Assuming that the parent body of the Themis\/Beagle families was quite homogeneous, the spectral variability seen on {\\it old} Themis members and the fact the {\\it young} Beagle members are bluer and with an higher albedo value than the Themis ones may be, at least in part, a result of space weathering effects. \n\nIn the past 20 years enormous progress has been reached in the study of space weathering (SW) effects on silicates and S-type asteroids. The so-called ordinary chondrite paradox, that is lack of asteroids similar to the ordinary chondrites, which represent 80\\% of meteorite falls, has been solved. These meteorites are now clearly related to S-type asteroids, as proved  by direct measurements of the NEAR and HAYABUSA missions on the near Earth asteroids Eros and Itokawa (Clark et al. 2001; Noguchi et al. 2011) and laboratory experiments on irradiation of silicates\/ordinary chondrites (Moroz et al. 1996; Sasaki et al. 2002; Brunetto \\& Strazzulla 2005; Noble et al. 2007). Spectral differences between S-type asteroids and ordinary chondrites are caused by space weathering effects, which produce a darkening in the albedo, a reddening of the spectra, and diminish the silicates absorption bands on the asteroids surfaces, exposed to cosmic radiation and solar wind. \\\\\nOn the other hand, our understanding of space weathering effects on primitive asteroids is still poor.\nOnly few laboratory experiments have been devoted to the investigation of SW effects on low albedo carbonaceous chondrites and on complex organic materials, and they indicate a no linear trend but much more complex effects. \nIrradiation of transparent organic materials produces  firstly  redder and lower albedo materials  that upon  further processing  turn  flatter-bluishing and darker (Kanuchova et  al. 2012; Moroz et al. 2004a). This result is consistent with a recent study that was looking for spectral differences between Ch\/Cgh-types and CM chondrites (Lantz et al. 2013). On the other hand, Lazzarin et al. (2006) expected a global reddening for the whole asteroidal population. Laboratory experiments on carbonaceous chondrites  give different results, that is reddening and\/or blueing. Some ion irradiations on CV Allende and CO Frontier Mountain 95002 (Lazzarin et al. 2006) led to reddening and darkening, and this trend  is confirmed for the Allende meteorite using both ion (Brunetto et al. 2014) and laser irradiation (Gillis-Davis et al. 2015).  \\\\\nThe laser irradiated CM Mighei resulted in spectral reddening (Moroz et al. 2004b) while laser irradiation on Tagish Lake showed a blueing effect (Hiroi et al. 2004).  Vernazza et al. (2013) performed ion irradiations on Tagish Lake and observed a strong blueing with argon, while irradiation with He produced only minor spectral changes and a slight reddening at 4 keV dose. Hiroi et al. (2013) used laser irradiations on a CO, a CK, a CM, a CI and Tagish Lake: the first two present the same behavior as ordinary chondrites while the others show a clear blueing (but reflectance decreases so that a darkening effect is seen). \\\\\nFor the CM Murchison the results after irradiation are contradictory:  Keller et al. (2015) found reddening and darkening of the sample after ion irradiation,  Matsuoka et al. (2015) found a blueing and darkening effect with laser irradiation,  Gillis-Davis et al. (2015), using again laser irradiation, found a darkening effect but without slope variations, while Lantz et al. (2015) found no clear albedo and slope variations after ion irradiation.  \n\nIt appears complicated to define a general space weathering trend for the primitive objects when considering the various results of laboratory simulations on carbonaceous chondrites. That would explain why the few studies on asteroids\nproposed different conclusions between reddening (Lazzarin et al. 2006) and blueing of the slope (Nesvorny et al. 2005; Lantz et al. 2013).\nHere we compare spectra of young and old members of the Themis group (respectively Beagle and Themis families). Starting from the hypothesis that the parent body was homogeneous, the Beagle members have been less exposed to the harsh space environment, and so less space weathered compared to Themis members. If this hypothesis is correct, then our observations seem to indicate that the surfaces of the Themis family members have the same spectral answer than the S-type asteroids: young Beagle members have globally higher albedo and bluer slopes, while most of the older Themis members are redder and have a lower albedo value. A similar conclusion is found by Kaluna et al. (2016) on their analysis of Themis and Beagle members of similar size, smaller than 15 km. From the mean slopes values of the two families, Kaluna et al. (2016) estimate a slope reddening of 0.08 $\\mu$m$^{-1}$ in 2.3 Gyr for C-complex asteroids. However, if space weathering effects are the only cause of the spectral diversity between the two families, we would expect a distinct spectral and albedo distribution between old and young asteroids, while our analysis clearly shows that several Themis members have albedo and spectral properties similar to the Beagle ones (Figure~\\ref{f8}). Some rejuvenating processes might have taken place (Shestopalov et al. 2013), but they cannot solely explain the large percentage of blue and bright Themis members observed. \\\\\nAnother important point is the timescale needed to space weathering processes to alter asteroid surfaces. Laboratory simulations indicate that solar wind may alter the surfaces in a relatively short timescale, of the order of 10$^3$--10$^5$ years, producing darkening and reddening effects on ordinary chondrites (Strazzulla et al. 2005), on S-type asteroids (Vernazza et al. 2009a, 2009b), and on the Allende carbonaceous chondrite (Brunetto et al. 2014). This has also been confirmed by the Itokawa grain analysis brought back to the Earth by the sample return mission Hayabusa (Noguchi et al. 2011). If these short timescales turn out to be correct, it is possible that Beagle members are old enought to have already experienced some space weathering processes. All this considered, we cannot firmly conclude that the observed spectral diversity is related to space weathering processes. Considering the presence of hydrated minerals on some members and the thermal evolution models which suggest a differentiation in the Themis\/Beagle parent body, we favor the compositional heterogeneity in the original parent body as explanation of the spectral diversity within Themis\/Beagle members, even if we cannot exclude that space weathering processes may be responsible of at least part of the spectral differences observed.\n\n\\section{Conclusions}\n\nIn this work we investigate the spectral properties of primitive asteroids belonging to the $\\sim$ 2 Gyr old Themis family and to the young ($<$ 10 Myr) Beagle subcluster within the Themis family.  We present new visible and near infrared spectra of 22 Themis and 8 Beagle families members. The Themis family members show different spectral behaviors including asteroids \nwith blue\/neutral and moderately red spectra, while the Beagle members have less spectral variability and they are all neutral\/blue.\nWe include in our analysis most of the data available in the literature on Themis\/Beagle families (a sample of 79 objects) for a complete analysis of their spectral properties versus physical parameters. The spectral slope versus albedo distribution clearly shows that Beagle members are less red and tend to have an higher albedo value than the Themis one. To explain the spectral differences between the two families we propose different interpretations, including heterogeneity in the parent body having a compositional gradient with depth, the survival of projectile fragments having a different composition than the parent body, space weathering effects, or a combination of these. \\\\\nThe spectral differences observed between Beagle and Themis members may be partially attributed to space weathering effects, which would produce on primitive asteroids reddening and darkening, as seen on S-type asteroids. However, space weathering effects cannot solely explain the spectral variety within the two families, first because we observe several Themis members having albedo and spectral behaviors similar to the younger Beagle members, and second because we need to assume that the parent body of the family was homogeneous. However, our and literature data show the presence of hydrated minerals on some of the Themis members, and a spectral analogy of the Themis\/Beagle members with carbonaceous chondrites having experienced different degrees of aqueous alteration. These results, together with those from thermal evolution models, indicate that the parent body of the Themis family experienced mild thermal metamorphism in the past, with a possible compositional gradient with depth. We thus favor the compositional heterogeneity in the original parent body as main source of the spectral diversity within Themis\/Beagle members. \n\n\n\n\\vspace{0.3truecm}\n{\\bf Acknowledgment} \\\\\nThis paper is based on observations made with the Italian Telescopio Nazionale Galileo (TNG) operated on the island of La Palma by the Centro Galileo Galilei of the INAF (Istituto Nazionale di\nAstrofisica), both located at the Observatorio del Roque de los Muchachos, La Palma, Spain, of the Instituto de Astrofisica de Canarias.\" This project was supported by the French Planetology National Programme (INSU-PNP). The authors gratefully acknowledge the reviewers for their comments and suggestions who help us improving the paper.\n\n\\bigskip\n\n{\\bf References} \\\\\n\n\nAl\\'i-Lagoa, V., de Le\\'on, J., Licandro, J., Delb\\`o, M., Campins, H., Pinilla-Alonso, N., Kelley, M. S., 2013. Physical properties of B-type asteroids from WISE data. Astron. Astroph. 554, id. 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Spectra are shifted by 0.4 in reflectance for clarity.}\n\\label{f1}\n\\end{figure*}\n\n\\begin{figure*}\n\\centerline{\\psfig{file=I14484_fig2.ps,width=20truecm,angle=0}}\n\\caption{Reflectance spectra of the Themis family members investigated. Spectra are shifted by 0.4 in reflectance for clarity.}\n\\label{f2}\n\\end{figure*}\n\n\\begin{figure*}\n\\centerline{\\psfig{file=I14484_fig3.ps,width=20truecm,angle=0}}\n\\caption{Reflectance spectra of the Themis family members investigated. Spectra are shifted by 0.5 in reflectance for clarity.}\n\\label{f3}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\centerline{\\psfig{file=I14484_fig4.ps,width=20truecm,angle=0}}\n\\caption{Reflectance spectra of the Beagle family members investigated. Spectra are shifted by 0.4 in reflectance for clarity.}\n\\label{f4}\n\\end{figure*}\n\n\\begin{figure*}\n\\centerline{\\psfig{file=I14484_fig5.ps,width=15truecm,angle=0}}\n\\caption{Continuum removed spectra of the 4 asteroids showing absorption features. The visible spectrum of (24) Themis is the one acquired on 19 December.}\n\\label{f5bis}\n\\end{figure*}\n\n\\begin{figure*}\n\\centerline{\\psfig{file=I14484_fig6.ps,width=15truecm,angle=0}}\n\\caption{Relative reflectance spectra in the visible region of (24) Themis taken at different aspect and rotational phases from data presented here and from the literature. The asteroid shows surface heterogeneities. }\n\\label{f5}\n\\end{figure*}\n\n\n\n\\begin{figure*}\n\\centerline{\\psfig{file=I14484_fig7.ps,width=20truecm,angle=0}}\n\\caption{NIR3 spectral slope (1.1-1.8 $\\mu$m) versus VIS spectral slope (0.5-0.8 $\\mu$m) for the Themis (black circles) and Beagle (blue stars) members. The spectral slopes are in \\%\/(10$^{3}$\\AA). The size of the circles is proportional to the asteroids' diameters.}\n\\label{f6}\n\\end{figure*}\n\n\n\\begin{figure}\n\\centerline{\\psfig{file=I14484_fig8.ps,width=15truecm,angle=0}}\n\\caption{Spectral slopes in the visible (0.55-0.8 $\\mu$m) and in the NIR (1.1-1.8$\\mu$m) ranges versus albedo. The spectral slopes are in \\%\/(10$^{3}$\\AA). Beagle and Themis family members are represented with blue stars and black circles, respectively. The size of the circles is proportional to the asteroids' diameters.}\n\\label{f7}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\psfig{file=I14484_fig9.ps,width=15truecm,angle=0}}\n\\caption{Spectral slope in the visible wavelength range (0.55-0.8 $\\mu$m, units \\%\/(10$^{3}$\\AA)) versus the albedo for the Themis and Beagle members. Beagle family members are represented with stars, blue from our data and cyan from the literature data, while the Themis members are represented with circles, black from our data and red from the literature data. The size of the symbols is proportional to the asteroids' diameters.}\n\\label{f8}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\centerline{\\psfig{file=I14484_fig10.ps,width=15truecm,angle=0}}\n\\caption{Spectral slope in the visible wavelength range (0.55-0.8$\\mu$m, units \\%\/(10$^{3}$\\AA)) versus asteroids' diameter. Beagle family members are represented with star symbol, blue from our data and cyan from the literature data,  while the Themis members are represented as circles, black from our data and red from the literature data.}\n\\label{f9}\n\\end{figure}\n\n\n\n\\begin{figure*}\n\\centerline{\\psfig{file=I14484_fig11.ps,width=20truecm,angle=0}}\n\\caption{WISE albedos for the Themis family members: ratio of the infrared over the visible albedo (p$_v$) versus the visible albedo: Themis members having higher albedo value are also bluer in the IR range.\n  The size of the symbols is proportional to the asteroids' diameters. In the upper rigth part of the plot we report the mean error bars of these measurements.}\n\\label{f10}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\centerline{\\psfig{file=I14484_fig12.ps,width=20truecm,angle=0}}\n\\caption{Best spectral matches between the observed Beagle family members and meteorites from the RELAB database.}\n\\label{f11}\n\n\\end{figure*}\n\n\\begin{figure}\n\\centerline{\\psfig{file=I14484_fig13.ps,width=20truecm,angle=0}}\n\\caption{Best spectral matches between the observed Themis family members and meteorites from the RELAB database.}\n\\label{f12}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\psfig{file=I14484_fig14.ps,width=20truecm,angle=0}}\n\\caption{Best spectral matches between the observed Themis family members and meteorites from the RELAB database.}\n\\label{f13}\n\\end{figure}\n\n\\begin{figure}\n\\centerline{\\psfig{file=I14484_fig15.ps,width=20truecm,angle=0}}\n\\caption{Best spectral matches between the observed Themis family members and meteorites from the RELAB database.}\n\\label{f14}\n\\end{figure}\n\n\n\n\\newpage\n\n{\\bf Tables}\n\n{\\scriptsize\n       \\begin{center}\n     \\begin{longtable} {|l|l|c|c|c|c|c|c|l|}  \n\\caption[]{Observational circumstances. Solar analog stars named \"HIP\"\n  come from the Hipparcos catalogue and \"Lan\" from the\nLandolt photometric standard stars catalogue.}. \n        \\label{tab1} \\\\\n\\hline \\multicolumn{1}{|c|} {\\textbf{Object    }} & \\multicolumn{1}{c|}\n{\\textbf{Night}} & \\multicolumn{1}{c|} {\\textbf{UT$_{start}$}} &\n\\multicolumn{1}{c|} {\\textbf{T$_{exp}$ (s)}}  & \\multicolumn{1}{c|} {\\textbf{Instr.}} & \\multicolumn{1}{c|}\n{\\textbf{Grism}} & \\multicolumn{1}{c|} {\\textbf{airm.}} & \\multicolumn{1}{c|}\n{\\textbf{Solar Analog (airm.)}} \\\\  \\hline \n  \n\\endfirsthead\n\\multicolumn{8}{c}%\n{{\\bfseries \\tablename\\ \\thetable{} -- continued from previous page}} \\\\ \\hline \n\\endfoot\n\\hline \\multicolumn{1}{|c|} {\\textbf{Object    }} & \\multicolumn{1}{c|}\n{\\textbf{Night}} & \\multicolumn{1}{c|} {\\textbf{UT$_{start}$}} &\n\\multicolumn{1}{c|} {\\textbf{T$_{exp}$ (s)}}  & \\multicolumn{1}{c|} {\\textbf{Instr.}} & \\multicolumn{1}{c|}\n{\\textbf{Grism}} & \\multicolumn{1}{c|} {\\textbf{airm.}} & \\multicolumn{1}{c|}\n{\\textbf{Solar Analog (airm.)}} \\\\  \\hline \n\\endhead\n\\hline \\multicolumn{8}{r}{{Continued on next page}} \\\\ \n\\endfoot\n\\hline \\hline\n\\endlastfoot\n24 Themis & 18 Dec. 2012 & 00:38 & 20 &  NICS & Amici & 1.03 & HIP22536 (1.02) \\\\\n24 Themis & 18 Dec. 2012 & 02:16 & 10 &  Dolores & LRB & 1.22 & Lan98-978 (1.22) \\\\ \n24 Themis & 18 Dec. 2012 & 02:20 & 20 &  Dolores & LRR & 1.23 & Lan98-978 (1.22) \\\\ \n24 Themis & 18 Dec. 2012 & 21:33 & 10 &  Dolores & LRB & 1.14 & Lan115-271 (1.13) \\\\ \n24 Themis & 18 Dec. 2012 & 21:35 & 20 &  Dolores & LRR & 1.14 & Lan115-271 (1.13) \\\\ \n24 Themis & 19 Dec. 2012 & 01:07 & 60 &  Dolores & LRB & 1.07 & Lan115-271 (1.13) \\\\\n62 Erato & 19 Feb. 2012 &  05:38    & 900  &  Dolores & LRR & 1.24 & Lan98-978(1.18) \\\\\n90 Antiope & 18 Dec. 2012 & 01:48 & 30 &  Dolores & LRB & 1.20 & Lan102-1081 (1.15) \\\\ \n90 Antiope & 18 Dec. 2012 & 01:52 & 60 &  Dolores & LRR & 1.21 & Lan102-1081 (1.15) \\\\ \n90 Antiope & 17 Dec. 2012 & 23:29 & 60 &  NICS & Amici & 1.01 & HIP22536 (1.03) \\\\ \n268 Adorea & 20 Feb. 2012 &  06:08    & 300  &  Dolores & LRB & 1.33 & Hip59932 (1.32) \\\\\n268 Adorea & 20 Feb. 2012 &  06:15    & 300  &  Dolores & LRR & 1.33 & Hip59932 (1.32) \\\\\n383 Janina & 19 Feb. 2012 &  06:07    & 900  &  Dolores & LRR & 1.38 & Hip59932 (1.35) \\\\\n461 Saskia & 19 Feb. 2012 &  03:52    & 900  &  Dolores & LRB & 1.27 & Hip59932 (1.34) \\\\ \n461 Saskia & 19 Feb. 2012 &  04:09    & 900  &  Dolores & LRR & 1.33 & Hip59932 (1.35) \\\\ \n461 Saskia & 19 Feb. 2012 &  23:44    & 720  &  NICS    & Amici & 1.09 & HIP44103 (1.04) \\\\ \n468 Lina   & 19 Dec. 2012 & 06:08 & 300 &  Dolores & LRR & 1.12 & Lan115-271 (1.14) \\\\ \n492 Gismonda & 18 Dec. 2012 & 22:45 & 300 &  Dolores & LRB & 1.56 & HIP52192 (1.46) \\\\ \n492 Gismonda & 18 Dec. 2012 & 22:51 & 300 &  Dolores & LRR & 1.59 & HIP52192 (1.47) \\\\\n492 Gismonda & 16 Dec. 2012 & 20:47 & 480 &  NICS & Amici & 1.16 & Lan93-101 (1.14) \\\\\n526 Jena & 19 Dec. 2012 & 00:23 & 300 &  Dolores & LRB & 1.17 &  HIP44027 (1.17) \\\\\n526 Jena & 19 Dec. 2012 & 00:28 & 300 &  Dolores & LRR & 1.19 &  HIP44027 (1.17) \\\\ \n526 Jena & 17 Dec. 2012 & 22:54 & 240 &  NICS & Amici & 1.04 &  HIP22536 (1.03) \\\\ \n621 Werdandi  & 19 Feb. 2012 &  22:18    & 720 &  NICS    & Amici & 1.02 & HIP44103 (1.04) \\\\\n656 Beagle & 18 Dec. 2012 & 20:02 & 600 &  Dolores & LRB & 1.23 & Lan93-101 (1.23) \\\\ \n656 Beagle & 18 Dec. 2012 & 20:14 & 600 &  Dolores & LRR & 1.26 & Lan115-271 (1.14) \\\\ \n656 Beagle & 17 Dec. 2012 & 19:39 & 960 &  NICS & Amici & 1.21 & Lan115-271 (1.25) \\\\ \n846 Lipperta & 20 Feb. 2012 &  03:19    & 900  &  Dolores & LRB & 1.21 & Lan102-1081(1.31) \\\\\n846 Lipperta & 20 Feb. 2012 &  03:36    & 900  &  Dolores & LRR & 1.27 & Lan102-1081(1.31) \\\\\n846 Lipperta & 20 Feb. 2012 &  00:19    & 960  &  NICS    & Amici & 1.06 & HIP44103 (1.04) \\\\ \n954  Li      & 19 Feb. 2012 &  21:23    & 1440 &  NICS    & Amici & 1.23 & Hyades64 (1.22) \\\\ \n1027 Aesculapia & 18 Dec. 2012 & 20:42 & 1200 &  Dolores & LRR & 1.33 & Lan115-271 (1.14) \\\\ \n1027 Aesculapia & 17 Dec. 2012 & 20:07 & 1920 &  NICS & Amici & 1.29 & Lan115-271 (1.25) \\\\ \n1247 Memoria & 19 Dec. 2012 & 05:04 & 900 &  Dolores & LRR & 1.04 & Lan115-271 (1.14) \\\\ \n1623 Vivian & 20 Feb. 2012 &  05:06    & 900  &  Dolores & LRB & 1.32 & Lan102-1081(1.31) \\\\\n1623 Vivian & 20 Feb. 2012 &  05:24    & 900  &  Dolores & LRR & 1.39 & Lan102-1081(1.31) \\\\\n1623 Vivian & 20 Feb. 2012 &  01:26    & 1440 &  NICS    & Amici & 1.12 & HIP44103(1.04) \\\\ \n1687 Glarona & 19 Feb. 2012 &  03:01    & 600  &  Dolores & LRB & 1.24 & Lan98-978 (1.18) \\\\ \n1687 Glarona & 19 Feb. 2012 &  03:17    & 600  &  Dolores & LRR & 1.28 & Lan98-978 (1.18) \\\\ \n1687 Glarona & 19 Feb. 2012 &  01:26    & 720 &  NICS    & Amici & 1.04 & HIP59932 (1.09) \\\\ \n1778  Alfven & 17 Dec. 2012 & 06:04 & 1440 & NICS & Amici & 1.29 & HIP41815 (1.26) \\\\ \n1953 Rupertwildt & 18 Dec. 2012 & 04:19 & 300 &  Dolores & LRB & 1.17 & Lan102-1081 (1.15) \\\\ \n1953 Rupertwildt & 18 Dec. 2012 & 04:25 & 480 &  Dolores & LRR & 1.19 & Lan102-1081 (1.15) \\\\ \n1953 Rupertwildt & 17 Dec. 2012 & 03:48 & 960 & NICS & Amici & 1.11 & HIP22536 (1.13) \\\\ \n2009 Voloshina & 17 Dec. 2012 & 23:58 & 960 & NICS & Amici & 1.24 & Lan98-978 (1.28) \\\\ \n2203 Van Rhijn & 17 Dec. 2012 & 21:24 & 960 & NICS & Amici & 1.05 & HIP22536 (1.03) \\\\ \n2222 Lermontov  & 20 Feb. 2012 &  04:36    & 900  &  Dolores & LRB & 1.34 & Lan102-1081(1.31) \\\\\n2222 Lermontov  & 20 Feb. 2012 &  04:38    & 900  &  Dolores & LRR & 1.41 & Lan102-1081(1.31) \\\\\n2222 Lermontov  & 20 Feb. 2012 &  00:48    & 960 &  NICS    & Amici & 1.05 & Lan102-1081 (1.18) \\\\ \n2228  Soyuz-Apollo& 19 Dec. 2012 & 04:30 & 600 &  Dolores & LRB & 1.03 &  Lan115-271 (1.13) \\\\ \n2228  Soyuz-Apollo & 19 Dec. 2012 & 04:42 & 600 &  Dolores & LRR & 1.03 &  Lan115-271 (1.14) \\\\\n2264 Sabrina & 19 Dec. 2012 & 02:07 & 600 &  Dolores & LRB & 1.04 &  Lan115-271 (1.13) \\\\ \n2264 Sabrina & 18 Dec. 2012 & 06:21 & 600 &  Dolores & LRR & 1.45 &  Lan102-271 (1.15) \\\\ \n2264 Sabrina & 17 Dec. 2012 & 05:21 & 480 & NICS & Amici & 1.22 & Lan1021081 (1.16) \\\\ \n2270 Yazhi & 19 Dec. 2012 & 02:48 & 1100 &  Dolores & LRB & 1.01 &  Lan115-271 (1.13) \\\\ \n2270 Yazhi & 19 Dec. 2012 & 03:23 & 1100 &  Dolores & LRR & 1.02 &  Lan115-271 (1.13) \\\\ \n2270 Yazhi & 17 Dec. 2012 & 04:26 & 1920 &  NICS & Amici & 1.10 & HIP44027 (1.04) \\\\ \n2519    Annagerman & 19 Feb. 2012 &  04:46    & 900  &  Dolores & LRB & 1.16 & Land98-978 (1.17) \\\\ \n2519    Annagerman & 19 Feb. 2012 &  05:04    & 900  &  Dolores & LRR & 1.18 & Land98-978 (1.18) \\\\\n2519    Annagerman & 20 Feb. 2012 &  02:13    & 1920 &  NICS    & Amici & 1.18 & HIP59932 (1.09) \\\\ \n3174 Alcock & 19 Dec. 2012 & 01:22 & 600 &  Dolores & LRB & 1.26 &  Lan93-101 (1.23) \\\\ \n3174 Alcock & 19 Dec. 2012 & 01:34 & 600 &  Dolores & LRR & 1.30 &  HIP41815 (1.30) \\\\ \n3174 Alcock & 16 Dec. 2012 & 23:52 & 960 & NICS & Amici & 1.06 & HIP22536 (1.13) \\\\ \n3591 Vladimirskij & 18 Dec. 2012 & 21:53 & 2200 &  Dolores & LRR & 1.21 &  HIP44027 (1.18) \\\\ \n3591 Vladimirskij & 16 Dec. 2012 & 22:20 & 1920 & NICS & Amici & 1.36 & Lan93-101 (1.14) \\\\ \n3615 Safronov & 16 Feb. 2012 & 21:31 & 900 & Dolores & LRB & 1.21 & Lan98-978 (1.25) \\\\ \n3615 Safronov & 16 Feb. 2012 & 21:49 & 900 & Dolores & LRR & 1.21 & Lan98-978 (1.25) \\\\ \n3615 Safronov & 18 Dec. 2012 & 05:47 & 1200 & Dolores & LRR & 1.13 & Lan102-1081 (1.15) \\\\ \n3615 Safronov & 17 Dec. 2012 & 02:40 & 480 & NICS & Amici & 1.05 & HIP41815 (1.06) \\\\ \n4903 Ichikawa & 18 Dec. 2012 & 02:53 & 900 & Dolores & LRB & 1.12 & Lan102-1081 (1.15) \\\\ \n4903 Ichikawa & 18 Dec. 2012 & 03:25 & 900 & Dolores & LRR & 1.15 & Lan102-1081 (1.15) \\\\ \n4903 Ichikawa & 17 Dec. 2012 & 01:38 & 960 & NICS & Amici & 1.02 & HIP22536 (1.13) \\\\ \n\\hline\n\\hline\n\\end{longtable}\n\\end{center}\n}\n\n         \\begin{sidewaystable} \n       \\caption{Physical and orbital parameters of the Themis and Beagle families members investigated.  The albedo comes from WISE data (Masiero et al., 2011) when available, or from AKARI (indicated with the symbol $^*$, Usui et al., 2011). S$_{all}$ is the spectral slope value calculated \nin the whole {\\bf observed} wavelength range, S$_{VIS}$ for the  0.55-0.80 $\\mu$m range, S$_{NIR1}$ for the 0.9-1.4 $\\mu$m range, S$_{NIR2}$\nfor the 1.4-2.2 $\\mu$m range, and S$_{NIR3}$ for  1.1-1.8 $\\mu$m range). The family is indicated with $B$ for the Beagle, and $T$ for the Themis members. For (24) Themis we used the whole visible and NIR spectrum acquired on 18 Dec., 00:30-02:30 UT time.}\n        \\label{slope}\n        \\scriptsize{\n\\begin{tabular}{|l|c|c|c|c|c|c|c|c|c|c|c|c|} \\hline\nAsteroid &   S$_{all}$         & S$_{Vis}$          & S$_{NIR1}$       & S$_{NIR2}$ & S$_{NIR3}$               &  D    & p$_{v}$ & Fam. & a & e & $sin(i)$ & H$_v$ \\\\          \n         &  (\\%\/$10^{3}$\\AA)   &  (\\%\/$10^{3}$\\AA) & $(\\%\/10^{3}$\\AA) & $(\\%\/10^{3}$\\AA) & $(\\%\/10^{3}$\\AA)  &  (Km) &         &      & (UA) &  &  & \\\\ \\hline\n 656        Beagle  &  1.30$\\pm$0.50  &    -0.60$\\pm$0.50  & 1.51$\\pm$0.52 & 1.29$\\pm$0.51 & 1.79$\\pm$0.51& 47.58$\\pm$0.33\t& 0.0782$\\pm$0.0222&   B &  3.15598& 0.154640 & 0.0235182&  9.95   \\\\ \n 1027    Aesculapia  &  0.56$\\pm$0.50  &    -0.72$\\pm$0.51  & 0.94$\\pm$0.52 & 1.24$\\pm$0.51 & 1.37$\\pm$0.50& 35.38$\\pm$0.34\t& 0.0814$\\pm$0.0071&   B &  3.15903& 0.152254 & 0.0240919& 10.82  \\\\ \n1687       Glarona  &  1.05$\\pm$0.50  &    -1.36$\\pm$0.51  & 1.95$\\pm$0.53 & 1.69$\\pm$0.52 & 2.32$\\pm$0.51& 42.01$\\pm$0.51  & 0.0795$\\pm$0.0130&   B &  3.15870& 0.153173 & 0.0242712& 10.53  \\\\ \n2519    Annagerman  &  0.11$\\pm$0.50  &    -0.65$\\pm$0.51  &-0.56$\\pm$0.51 & 0.60$\\pm$0.54 & 0.62$\\pm$0.55& 20.55$\\pm$0.31  & 0.1152$\\pm$0.0289&   B &  3.14765& 0.151787 & 0.0225837& 11.5    \\\\ \n3174        Alcock  &  0.19$\\pm$0.50  &    -2.88$\\pm$0.50  & 1.37$\\pm$0.51 & 0.39$\\pm$0.52 & 1.73$\\pm$0.50& 18.66$\\pm$0.80\t& 0.1020$\\pm$0.0090&   B &  3.15479& 0.154074 & 0.0239525& 11.95  \\\\\n3591  Vladimirskij  & -0.57$\\pm$0.50  &    -3.24$\\pm$0.51  & 0.16$\\pm$0.52 & 0.88$\\pm$0.52 & 0.91$\\pm$0.51& 16.42$\\pm$0.25\t& 0.0947$\\pm$0.0248&   B &  3.15704& 0.151941 & 0.0244373& 12.07   \\\\ \n3615      Safronov  & -0.79$\\pm$0.51  &    -1.63$\\pm$0.50  &-3.06$\\pm$1.30 & 1.85$\\pm$1.16 &-0.17$\\pm$0.94& 26.24$\\pm$0.10\t& 0.0852$\\pm$0.0159&   B &  3.15974& 0.152726 & 0.0238029& 11.49   \\\\ \n4903      Ichikawa  &  0.40$\\pm$0.50  &    -1.56$\\pm$0.51  & 1.51$\\pm$0.55 & 0.77$\\pm$0.62 & 2.11$\\pm$0.52& 14.79$\\pm$0.39\t& 0.1167$\\pm$0.0087&   B &  3.15055& 0.15180  & 0.0240145& 12.54  \\\\ \\hline\n  24       Themis   &  0.97$\\pm$0.51  &    -0.61$\\pm$0.50  & 1.24$\\pm$0.51 & 1.95$\\pm$0.50 & 2.15$\\pm$0.50& 202.34$\\pm$6.05 & 0.0641$\\pm$0.0157&   T &  3.13450& 0.152779 & 0.0189278&  7.21  \\\\ \n  62         Erato  & -0.06$\\pm$0.51  &    -1.73$\\pm$0.50  &-0.23$\\pm$0.50 & 0.73$\\pm$0.50 & 0.32$\\pm$0.50& 95.40$\\pm$2.0   & 0.0610$\\pm$0.0030&   T &  3.12169& 0.149807 & 0.0225411&  8.62   \\\\ \n  90       antiope  &  0.88$\\pm$0.50  &    -1.19$\\pm$0.50  & 1.45$\\pm$0.51 & 2.08$\\pm$0.51 & 2.38$\\pm$0.51& 121.13$\\pm$2.47\t& 0.0593$\\pm$0.0096&   T &  3.14619& 0.153819 & 0.0231415&  7.84  \\\\  \n268         Adorea  &  2.55$\\pm$0.50  &     1.56$\\pm$0.50  & 1.29$\\pm$0.51 & 3.76$\\pm$0.50 & 3.42$\\pm$0.50&140.59$\\pm$3.18  & 0.0436$\\pm$0.0066&   T &  3.09684& 0.167397 & 0.0243772&  8.3      \\\\\n  383        Janina  &  0.61$\\pm$0.50  &     2.97$\\pm$0.51  & 0.59$\\pm$0.51 & 0.42$\\pm$0.50 & 0.48$\\pm$0.50& 44.64$\\pm$0.74  & 0.0964$\\pm$0.0127&   T &  3.13408& 0.151863 & 0.0246743&  9.74   \\\\ \n 461        Saskia  &  3.19$\\pm$0.50  &     2.79$\\pm$0.50  & 2.49$\\pm$0.51 & 3.04$\\pm$0.51 & 3.50$\\pm$0.51& 48.72$\\pm$0.24  & 0.0479$\\pm$0.0079&   T &  3.11426& 0.155859 & 0.0239453& 10.4     \\\\ \n468           Lina  &   -             &    -1.41$\\pm$0.53  &  -            &  -            &  -           & 64.59$\\pm$1.98  & 0.0495$\\pm$0.0094&   T &  3.14189& 0.158666 & 0.0215053&  9.69 \\\\\n 492      Gismonda  &  0.79$\\pm$0.50  &    -1.42$\\pm$0.50  & 0.84$\\pm$0.51 & 1.21$\\pm$0.51 & 1.22$\\pm$0.51& 59.92$\\pm$0.35  & 0.0592$\\pm$0.0047&   T &  3.11191& 0.151069 & 0.0220835&  9.85   \\\\ \n 526          Jena  & -0.26$\\pm$0.50  &    -0.95$\\pm$0.50  &-1.60$\\pm$0.51 &-0.63$\\pm$0.50 &-0.25$\\pm$0.51& 51.03$\\pm$0.74  & 0.0580$\\pm$0.0177&   T &  3.12313& 0.156463 & 0.0249680& 10.      \\\\ \n 621      Werdandi  &  1.05$\\pm$0.50  &    -0.51$\\pm$0.51  & 2.01$\\pm$0.51 & 1.98$\\pm$0.52 & 1.91$\\pm$0.51& 30.71$\\pm$0.50$^{*}$   & 0.1240$\\pm$0.0050$^{*}$ &   T &  3.1193 & 0.150097 & 0.0253958& 10.98    \\\\ \n 846      Lipperta  &  1.39$\\pm$0.50  &     0.50$\\pm$0.50  & 1.72$\\pm$0.51 & 2.11$\\pm$0.51 & 2.83$\\pm$0.51& 51.45$\\pm$0.76$^{*}$\t& 0.0530$\\pm$0.0020$^{*}$&   T &  3.12791& 0.150758 & 0.0268551& 10.25    \\\\ \n 954            Li  &  2.83$\\pm$0.50  &     1.26$\\pm$0.50  & 3.00$\\pm$0.53 & 3.06$\\pm$0.53 & 4.15$\\pm$0.53& 52.06$\\pm$0.81  & 0.0552$\\pm$0.0073&   T &  3.13701& 0.150656 & 0.0230923& 10.07    \\\\ \n1247       Memoria  &   -             &    -0.64$\\pm$0.51  &  -            &  -            &  -           & 42.07$\\pm$0.16  & 0.0618$\\pm$0.0060&   T &  3.13875& 0.160209 & 0.0297064& 10.58 \\\\\n 1623        Vivian  &  2.53$\\pm$0.50  &     1.62$\\pm$0.50  & 1.87$\\pm$0.51 & 2.62$\\pm$0.52 & 3.20$\\pm$0.52& 29.98$\\pm$1.74$^{*}$\t& 0.0780$\\pm$0.0100$^{*}$ &   T &  3.13466& 0.15294  & 0.0239821& 11.19     \\\\\n 1778        Alfven &   0.17$\\pm$0.50  &       --           & 0.14$\\pm$0.51 & -0.47$\\pm$0.53 & 0.29$\\pm$0.50 & 20.6$\\pm$0.24 & 0.0951$\\pm$0.0069 &  T   & 3.1441 &  0.155394 &  0.0224451 & 11.7 \\\\ \n 1953   Rupertwildt  &  0.55$\\pm$0.50  &    -1.60$\\pm$0.50  & 0.79$\\pm$0.51 & 0.89$\\pm$0.52 & 2.16$\\pm$0.51& 21.97$\\pm$0.27  & 0.0697$\\pm$0.0060&   T &  3.11586& 0.147538 & 0.0242019& 11.84   \\\\ \n2009     Voloshina  &  1.78$\\pm$0.50  &      -             & 1.43$\\pm$0.51 & 1.71$\\pm$0.51 & 2.35$\\pm$0.50& 26.56$\\pm$0.48  & 0.1199$\\pm$0.0242 &   T &  3.11914 & 0.150115 & 0.0282294 & 11.16 \\\\\n2203     vanRhijn   & 1.39$\\pm$0.50   &      --            &  1.41$\\pm$0.51&  1.00$\\pm$0.51 & 1.90$\\pm$0.51 & 22.28$\\pm$0.27 & 0.0898$\\pm$0.0058 &  T &  3.10989 & 0.149153 &  0.0209356 & 11.57 \\\\\n2222     Lermontov  &  2.39$\\pm$0.50  &     1.96$\\pm$0.50  & 2.05$\\pm$0.51 & 2.87$\\pm$0.52 & 2.85$\\pm$0.51& 32.22$\\pm$0.22  & 0.0469$\\pm$0.0061&   T &  3.11619& 0.159625 & 0.0228687& 11.29     \\\\\n2228  Soyuz-Apollo  &   -             &     1.65$\\pm$0.51  &  -            &  -            &  -           & 26.08$\\pm$0.29  & 0.1134$\\pm$0.0198&   T &  3.14342& 0.160273 & 0.0241862& 11.44  \\\\\n 2264       Sabrina  &  0.08$\\pm$0.50  &    -1.49$\\pm$0.50  & 0.44$\\pm$0.52 & 0.40$\\pm$0.51 & 1.56$\\pm$0.51& 37.34$\\pm$0.38  & 0.0799$\\pm$0.0138&   T &  3.13084& 0.154135 & 0.0234201& 10.92   \\\\ \n2270         Yazhi  &  1.82$\\pm$0.50 &    -0.07$\\pm$0.51  & 1.12$\\pm$0.55 & 0.06$\\pm$0.54 & 1.65$\\pm$0.52& 26.73$\\pm$0.30  & 0.1081$\\pm$0.0225&   T &  3.15117& 0.15233  & 0.0188940& 11.43    \\\\ \n \\hline\n\\end{tabular}\n}\n         \\end{sidewaystable}\n\n         \\thispagestyle{empty}    \n\n     \n \\begin{table}\n         \\begin{center} \n         \\caption{Identification of absorption bands attributed to hydrated silicates on the Themis family members.}\n        \\label{hydra}\n                \\begin{tabular}{|l|c|c|c|c|} \\hline\n         Asteroid     & W$_{in}$ (\\AA) & W$_{fin}$ (\\AA) & Band$_{center}$ (\\AA) & Depth (\\%) \\\\ \\hline\n         24 Themis       & 5715  &  8730  &   7333$\\pm$48 &   2.6$\\pm$0.1  \\\\\n         90 Antiope      & 5358  &  8375  &   6972$\\pm$43  & 1.4$\\pm$0.1  \\\\\n         461 Saskia      & 5648  &  8338  &   6881$\\pm$123  & 1.2$\\pm$0.1  \\\\\n     \t                 & 12397  &  15126  &  13175$\\pm$55  & 2.8$\\pm$0.2  \\\\\n                         & 15965  &  21378  &  18157$\\pm$31  & 2.7$\\pm$0.2  \\\\\n         846 Lipperta     & 5487  &  8480  &    6842$\\pm$65  & 1.02$\\pm$0.1 \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n \\newpage\n \n \\begin{table}\n         \\begin{center} \n         \\caption{RELAB matches. The first eight asteroids belong to the Beagle family.}\n         \\label{met}\n          \\scriptsize{\n        \\begin{tabular}{|l|c|c|c|c|c|} \\hline\n          Asteroid & Albedo & Best fit & Met refl. & Met Class \\& name &  Grain size \\\\ \\hline\n 656           Beagle  & 0.0782   &     NAMC02      & 0.033 &  CM2 LEW90500,45 &  $<$ 500 $\\mu$m           \\\\\n 1027        Aesculapia  & 0.0814 &     C1MP61      & 0.067 &  CM2 GRO95577,6 particulate & $<$ 125 $\\mu$m           \\\\\n 1687           Glarona  & 0.0795 &     camh52      & 0.068 &  CM2  Boriskino  & $<$ 45 $\\mu$m         \\\\\n2519       Annagerman  & 0.1152   &     c1mb20      & 0.063 &  CM2  Unusual Y-86720,77 &    $<$ 125 $\\mu$m            \\\\\n 3174            Alcock  & 0.102  &     MGP094      & 0.059 &  CM2 Murchison          & whole rock  \\\\ \n3591      Vladimirskij  & 0.0947  &     c1mb19      & 0.036 &  CI  unusual, Y-82162,79 & $<$ 150 $\\mu$m     \\\\\n3615          Safronov  & 0.0852  &     BKR1MA076   & 0.042 &  CM2\/CR Kaidun            & $> $  250 $\\mu$m        \\\\\n4903          Ichikawa  & 0.1167 &      CGP090      & 0.046 &  CM2 Cold Bokkevelt      & 150 -- 500 $\\mu$m    \\\\   \\hline\n  24   Themis (17 Dec)  & 0.0641 &      NAMC02      & 0.033 &  CM2 LEW90500,45         & $<$ 500 $\\mu$m           \\\\\n  62             Erato  & 0.061  &      ccmb19      & 0.050 &  CI unusual, Y-82162,79  &    \\\\\n  90           antiope  & 0.0593 &      NAMC02      & 0.033 &  CM2 LEW90500,45         & $<$ 500 $\\mu$m           \\\\\n  268           Adorea  & 0.0436 &      ccmb64      & 0.031 &  CM2 Murchison heated 600 C & $<$ 63 $\\mu$m \\\\\n 383            Janina  & 0.0964 &      c1mb18      & 0.06  &  CM2 unusual B-7904,108     &             \\\\ \n 461            Saskia  & 0.0479 &      s1rs42      & 0.037 &  CM2 Mighei                 &   \\\\\n 492          Gismonda  & 0.0592 &      C1RS46      & 0.036 &  CM2 Boriskino             & \\\\\n 526              Jena  & 0.0580 &      CGP130      & 0.047 &  CV3 Grosnaja             & 75--150$\\mu$m             \\\\\n 621          Werdandi  & 0.124  & BKR1MP022L0      & 0.046 &  CM2 MAC88100  laser irr.             & $<$ 125 $\\mu$m       \\\\    \n 846          Lipperta  &  0.053 &      ncmp22      & 0.033 &  CM2 MAC88100,30           & $<$ 125 $\\mu$m \\\\\n 954                Li  & 0.0552 &      s1rs42      & 0.035 &  CM2 Mighei               &  \\\\\n1623            Vivian  &  0.078 &      ccmb64      & 0.031 &  CM2 Murchison heated 600 C &  $<$ 500 $\\mu$m     \\\\\n 1953       Rupertwildt  & 0.0697 &     CGP142      & 0.046 &  CM2 ColdBokkevelt         & 75-500 $\\mu$m\\\\\n2222         Lermontov  & 0.0469 &      ccmb64      & 0.031 &  CM2 Murchison heated 600 C\t& $<$ 500 $\\mu$m  \\\\\n2264           Sabrina  & 0.0799 &      CGP094      & 0.057 &  CM2 Murchison            & whole rock\\\\\n2270             Yazhi  & 0.1081 &      CBMS01      & 0.035 &  CM2 Migei Separates      & 100 -- 200 $\\mu$m   \\\\ \\hline\n        \\end{tabular}\n        }\n\\end{center}\n \\end{table}\n\n\n\\end{document}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}