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+{"text":"\\section{Introduction}\n\n\\noindent Emerging paradigms in machine learning (e.g., reinforcement learning) offer the potential for systems to learn complex behaviors through interaction with a learning environment. These advancements could help overcome the limitations of current autonomous systems, which largely rely on pre-determined rulesets and analytic control regimes to govern their behavior. Many open technical challenges must be overcome to realize this potential, including advancements in the safety and assurance of autonomous systems - especially goal-driven systems that learn for themselves. Recent works from the AI safety research community have identified a number of open challenges in this area. However, most existing AI testbeds and simulation environments do not explicitly address AI safety challenges. Those that do are relatively simplistic, such as gridworld environments, or narrowly-focused on isolating and illustrating a particular safety issue. \n\n\\begin{figure}[t]\n\\centering\n  \\includegraphics[width=1.0\\linewidth]{img\/tanksworld.png}\n  \\caption{The AI Safety TanksWorld\\xspace is a competitive multi-agent environment for exploring competing performance objectives, human-machine teaming, and multi-agent competition }\n  \\label{fig:tanksworld}\n\\end{figure} \n\n\nHence, there is a gap in environments that support AI safety research based on simulations that are sufficiently realistic to capture essential aspects of real-world AI applications while not requiring a prohibitive degree of domain expertise. To begin working towards addressing this gap, we introduce the AI Safety TanksWorld\\xspace. The AI Safety TanksWorld\\xspace is a multi-agent environment for exploring AI safety issues in applications that require competition and cooperation among multiple agents towards satisfying competing performance objectives. Specifically, the AI Safety TanksWorld\\xspace is a team-based tanks battle where one team aims to defeat the other while avoiding unintended consequences such as losing teammates or inflicting collateral damage on the environment. We highlight three essential components of the environment: dynamic and uncertain environments, safety concerns for human and machine teams, and complex tasks with competing objectives.\n\n\n\n\n\n\n\n\\emph{Dynamic and Uncertain Environments --}\nMany of the challenges surrounding AI safety arise from unknowns and uncertainty in the environment. For AI safety research, there is an important distinction between known-unknowns and unknown-unknowns. A known-unknown can often be characterized via a model that captures a degree of partial observability or structured uncertainty. Although challenging, an AI system can often be trained to mitigate the safety risks associated with known-unknowns. Real-world environments usually offer the even greater challenge of unknown-unknowns - novel situations not adequately represented during training experiences. The AI Safety TanksWorld\\xspace environment includes parameters that enable the complexity, observably, and novelty of the scenario to be systematically modified. For example, each tank in our simulation can only sense a local region of the environment, making communication among allies advantageous. Our explicit modeling of the range and quality of communication enables the ability to add complexity in unexpected ways. \n\n\n\n\\emph{AI Safety for Human-Machine Teams --}\nHuman-machine teaming can present unique challenges for learning agents. Even when people are highly skilled at performing a given task, they may take actions that do not optimize for near-term rewards either unintentionally or intentionally due to hidden objectives. This unpredictability creates the need for human-aware adaptation in goal-driven agents. This can be particularly challenging in safety-critical applications that require safe exploration of possible actions. Further complicating the human-machine teaming relationship is that humans also need to develop accurate models of the behavior of machine teammates even after training is complete. To support research in the area of human-machine teams, the AI Safety TanksWorld\\xspace includes several human surrogate policies built using behavior cloning \\cite{dart} from human demonstrations.\n\n\\emph{Complex Tasks with Competing Objectives --}\nLastly, due to the dynamic and competitive nature of the AI Safety TanksWorld\\xspace, AI agents are forced to tradeoff between the performance of a complex task (i.e., collaborate to defeat the opposing tanks) and safety objectives. This tension supports a broad exploration of methods that attempt to optimize over multiple, competing objectives. This can be particularly challenging when human developers and operators may desire to express specific preferences for certain outcomes over others. For example, a certain amount of collateral damage to property in an operating environment may be tolerable with less being preferable and more being unacceptable. Optimizing performance in these risk-sensitive regimes is an open research challenge. \n\n\n\n\n\n\n\n\n\\section{Related Work}\n\\noindent The AI community has a rich history of developing and adopting simulations and games to challenge the research community. Checkers, Chess~\\cite{deepblue}, and Go~\\cite{alphago,alphazero} have historically been grand challenge problems in AI. Variants of these classics have produced new AI challenges~\\cite{rbmc}. With a groundbreaking result from DeepMind~\\cite{ataridqn}, a suite of Atari games became benchmarks for exploring performance of algorithms across tasks. OpenAI introduced the Gym architecture along with a suite of environments for exploring the performance of reinforcement learning algorithms in classic control, robotics, and Atari environments~\\cite{openaigym}. DeepMind released their internally developed suite of benchmark tasks for evaluating the performance of reinforcement learning algorithms~\\cite{deepmindlab}. Unity released the obstacle tower challenge environment, which provides a range of difficulty with the increasing capabilities of reinforcement learning agents~\\cite{tower}. OpenAI released Neural MMO as an environment to test AI in highly multi-agent environments~\\cite{neural_mmo}.\n\n\nAll of these are very useful simulation environments, but they are not specifically focused on AI Safety issues. The AI Safety TanksWorld\\xspace, by contrast, is specifically focused on safety concepts related to competing performance objectives, human-machine teaming, and multi-agent competition.\n\nThe AI community has also developed several test suites for AI safety scenarios. For example, a team from DeepMind released AI Safety Gridworlds~\\cite{gridworlds} as a series of environments to illustrate concrete problems in AI safety~\\cite{concrete}. However, the Gridworlds scenarios present only minimalistic examples of each AI safety concept. While the minimalism of the Gridworlds is useful for illustration, it may be beneficial to have environments that reflect more realistic challenges for AI safety. The existing AI Safety Gridworlds are also not specifically designed to explore issues that arise around human-machine teams, since each Gridworld environment consists of only a single AI-controlled entity. Other platforms like the Safety Gym~\\cite{safetygym} focus specifically on safe exploration in single agent environments.\n\nAdditionally, most of the Gridworlds scenarios present only the most difficult variations of each safety problem. This is very useful for illustrative purposes, but we believe that it would be useful to have less difficult variations of the same problems which could then be scaled up to address the more difficult varieties. For example, the ``avoiding side effects\" Gridworlds scenario requires the agent to avoid pushing a box into a corner in pursuit of its goal, when the reward function it is given does not include anything at all about boxes. Essentially, the agent has to be able to successfully deal with an ``unknown unknown,\" which is an especially difficult problem. It may be useful for the community to first try to tackle easier ``known unknowns\" problems, with the hope of using some of the lessons to scale up to the more difficult ``unknown unknowns\" scenarios.\n\n\nThe rise of autonomous vehicles is a clear example of the AI safety hazards associated with human-machine teams. The autonomous vehicle research community has released simulation frameworks for training and evaluation of self-driving cars \\cite{car}. While self-driving cars are an excellent example of AI safety issues surrounding human-machine teams, these simulations tend to provide a limited view of AI safety through the lens of individual cars.\n \n\n\n\\section{The AI Safety TanksWorld\\xspace Environment}\n\n\n\n\n\nThe AI Safety TanksWorld\\xspace is a team-based N vs. N tanks battle (where N is a variable parameter) that motivates the design of safe multi-agent control policies that are effective, collaborative and cope with uncertainty. Figure \\ref{fig:tanksworld} illustrates a multi-agent scenario. The components of the environment are largely derived from assets distributed by Unity.\n\n\n\\subsection{State and Action Representation}\n Scenarios are composed of a closed arena, obstacles, buildings, trees, and tanks. The AI Safety TanksWorld\\xspace contains multiple views including a tank-centric first person view, a top down overview, and an isometric view. The state space is a 128x128 4-channel image where each channel conveys different information. The 4-channels include: position and orientation of allies, position and orientation of visible threats, position of neutral tanks, and position of obstacles. The state space given to each tank is unique in that the 4-channel state is rotated and translated to be relative to each tank's position and heading. The threats visible to each tank are dependent on its position relative to allies and threats as described in the Active Sensing and Teamwork section. Image \\ref{fig:tanksworld_state} illustrates the unique state provided to each agent.\n \n Three continuous actions are available to each agent: velocity (forward\/reverse), turning (left\/right), and shooting (yes\/no). All actions are in the range $-1$ to $1$. Shots are taken for actions greater than zero, and shot frequency is rate limited.\n \n \\begin{figure}[t]\n\\centering\n  \\includegraphics[width=\\linewidth]{img\/tanksworld_state.png}\n  \\caption{State is conveyed as a tank-relative 4-channel 128x128 image. The channels include position of allies, position of threats, position of neutral entities, and position of obstacles. The first three channels are shown as blue, green, and red respectively. (Top row) shows state for tanks on the red team. (Bottom row) shows states for tanks on the blue team.}\n  \\label{fig:tanksworld_state}\n\\end{figure} \n \n\\subsection{Parameterization}\n\nThe AI Safety TanksWorld\\xspace environment includes the following parameters, which may be selected to emphasize different research challenges:\n\\begin{itemize}\n    \\item Communication range\n    \\item Number of neutral tanks and obstacles\n    \\item Control policies of teammates\n\\end{itemize}\n\n\\emph{Communication range --} We parameterize communication range among teammates as a representation of environmental uncertainty. Since teammates may share information about the location of opposing tanks, this model controls the overall visibility of the threat environment. Opposing tanks can be seen if they are within the designated radius of a teammate and the teammate is within the radius of the current tank. This dynamic encourages teammates to stay close enough to each other to share information and far enough apart to maintain broad situational awareness of the environment. \n\n\\emph{Number of neutral tanks and obstacles --} The number of neutral tanks and density of obstacles can be varied to control the risk of collateral damage. The neutral tanks move around the scene at random creating hazards for both teams. The density of obstacles including trees, rocks, and buildings is also controlled by a parameter. The positions of obstacles are randomized on reset. \n\n\\emph{Control policies of teammates --} We parameterize the skill level of teammates to represent the variability that can arise in human-machine teams to the unpredictability of human decision-making. Human surrogate teammates are policies that may be obtained via behavior cloning~\\cite{dart} from human demonstrations or through explicit behavior modeling. \n\n\\subsection{Rewards}\nThe environment returns separate metrics for allied, neutral, and opponent kills. The flexibility in how a reward function or constraint can be defined based on these metrics will help researchers explore AI safety across a spectrum of performance-risk trade-offs. The specific numbers described here are used for illustration.\n\n\n\n\n\n\nOne possible reward scheme is: the penalty for death is $1$, the penalty for destroying neutral tanks is $1$, and the penalty for allied kills is $1$. With two neutral tanks in the scene, the minimum possible score is $-7$. Given a $1$ reward for each enemy kill and 5 enemy tanks, the maximum possible reward is $5$. The components of the reward (allied, opponent, and neutral kills) are returned separately for each tank.\n\n\n\\section{Discussion}\n\n\\subsection{Active Sensing and Teamwork}\nA relatively uncommon aspect of the AI Safety TanksWorld\\xspace that differentiates it from other environments is that each tank receives a unique and partial view of the world that is governed by its teamwork with allies. Within a parameterized distance, allies can communicate the threats that they observe. The communication between allies is transitive. The benefit of team coordination is better threat visibility for all allies.\n\n \\begin{figure}[th]\n\\centering\n  \\includegraphics[width=.5\\linewidth]{img\/tank_comms.png}\n  \\caption{Partial observability and communications. From the perspective of Blue1, Red2 is visible because Blue1 is within a parameterized radius of Blue2 and Blue2 is within a specified radius to Red2. Red1 is not visible to either Blue1 or Blue2 because neither one is within the specified radius to Red1 or Blue3. }\n  \\label{fig:ppo_results}\n\\end{figure} \n\nThe parameter controlling communication can alter the scenario from full observability to partial observability. Distributional shifts in this parameter force policies to cope with situations that resemble unreliable communications.\n\n\\subsection{Minimizing Collateral Damage}\nOne of the concrete~\\cite{concrete} problems in AI safety is avoiding unintended consequences. This is made more challenging in human-machine teaming because of the difficulty in reliably modeling human decision-making in dynamic scenarios.\n\nNeutral tanks in the AI Safety TanksWorld\\xspace environment present the risk of generating collateral damage. Distributional shifts in the number of neutral tanks in the environment present additional collateral damage risk.\n\n\\subsection{Human-machine Teaming}\nAI that have been trained to partner with other AI teammates may be ill equipped to partner with humans. To model human teammate behavior, we can substitute control policies for teammates with human surrogate policies. We learn these human surrogate policies from demonstration using behavior cloning \\cite{dart}. By substituting the policies of teammates with human surrogate policies, we can evaluate the challenges and safety issues that arise from an AI being partnered with human-like teammates of variable skill.\n\n\\subsection{Next Steps}\nOur ultimate goal is to host a competition for the AI research community focused on the AI safety aspects of the AI Safety TanksWorld\\xspace. In the coming months, we are interested in simultaneously exploring  parameterizations of the environment that elicit different AI safety challenges. We will establish expected performance using recent reinforcement learning algorithms. Once the competition rules are stable and baselines established, we will aim to host a competition track at an established AI workshop. By publishing this work in progress, we hope to attract both feedback and potential collaborators.\n\n\n\\section{Acknowledgements}\nThe authors would like acknowledge the APL CIRCUIT program and its organizers for the training and coordination of the interns on this project. \n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{introduction}\n\nTopological band theory underpins fertile topological states of matter~\\cite{HasanRevModPhys823045,QiRevModPhys8310572011}, ranging from conventional to higher-order topological insulators.\nAccording to the bulk-boundary correspondence (BBC)~\\cite{PhysRevB4811851,PhysRevLett713697}, topological invariants of bulk bands are intimately related to robust boundary states.\nTaking the Su-Schrieffer-Hegger (SSH) model~\\cite{SuWPPhysRevLett1979,PhysRevB101134423} as an example, the bulk topology characterized by a winding number $\\nu$ corresponds to $\\nu$ pairs of edge states under open boundary condition.\nAnother typical example is the Benalcazar-Bernevig-Hughes (BBH) model~\\cite{2017Science35761,2017PhysRevB96245115}, in which the nontrivial quantized quadrupole moment indicates four corner states in two dimensions.\nNot limited to fermionic particles, the above wisdom has been widely applied to study topological bosonic excitations such as magnons~\\cite{PhysRevLett.90.167204,PhysRevB.96.224414,Malki2020,ZXLiPhysRep2021,Bonbien2021,PhysRevX11021061}, phonons~\\cite{ZhangPhysRevLett105225901} and photons~\\cite{PhysRevLett100013904}.\n\t\nTopological magnons, a kind of collective excitations over trivial ground states of magnetic materials, have provided new insights into topological states and potential applications such as topological magnon laser~\\cite{GalHarariScience2018,MiguelScience2018}, magnon spintronics~\\cite{ChumakNatPhy2015,DanielNatC2019} and topological magnetic memory~\\cite{ARMellnikNature2014,PhysRevLett119077702}.\nParallel to the electronic counterpart, magnon Hall effect was theoretically predicted and experimentally observed~\\cite{ScienceYOnose2010}, and Dirac magnons~\\cite{PhysRevB.94.075401,PhysRevLett.119.247202,PhysRevX.8.011010,SBaoNatureComm2018,PhysRevX.10.011062}, Weyl magnons~\\cite{FYLiNatureComm2016,PhysRevLett.117.157204,PhysRevB.95.224403,PhysRevB.96.104437,PhysRevB.97.115162,PhysRevB.99.214413,PhysRevResearch.2.013063}, nodal line magnons~\\cite{PhysRevB.95.014418,PhysRevResearch.2.023282}, topological magnon polarons~\\cite{PhysRevLett117217205,PhysRevLett123167202,PhysRevLett123237207,PhysRevLett124147204}, higher-order topological magnons~\\cite{ASilJPCM2020,PhysRevB104024406} were theoretically predicted.\n\nBecause it is generally difficult to measure bulk topology of magnons, according to the BBC, edge states have been utilized as an experimental signature.\nMost of the topological magnon excitations previously focused on are the isotropic Heisenberg interaction, where the role of the longitudinal spin-spin interaction is weak enough to be neglected~\\cite{ScienceYOnose2010,2013PhysRevB87144101,2014NatCommun54815,2014PhysRevB90024412,2015PhysRevLett115147201,2016PhysRevLett117227201,2018PhysRevX8041028,2018PhysRevB97081106,2021PhysRevX11021061,PhysRevLett.117.157204,FYLiNatureComm2016,PhysRevX.8.011010,ASilJPCM2020,PhysRevB104024406}.\nThere have emerged many interesting topological boundary states with nontrivial magnon bands, such as  chiral magnon edge states~\\cite{ScienceYOnose2010,2013PhysRevB87144101,2014NatCommun54815,2014PhysRevB90024412,2015PhysRevLett115147201,2016PhysRevLett117227201,2018PhysRevX8041028,2018PhysRevB97081106,2021PhysRevX11021061}, magnon surface states of Dirac and Weyl magnons~\\cite{PhysRevLett.117.157204,FYLiNatureComm2016,PhysRevX.8.011010}, and topological magnon corner states~\\cite{ASilJPCM2020,PhysRevB104024406}.\nThe conventional BBC breaks down in various situations, including non-Hermitian topological systems with the skin effect~\\cite{2018PhysRevLett121086803,2018PhysRevLett121136802,2019PhysRevB99075130} and defect edge states~\\cite{2016PhysRevLett116133903,2020PhysRevB101121116,2020IntJModPhysB}, hole-edge states in spinless SSH model with interaction~\\cite{2017PhysRevB95115443}, two-dimensional topological insulators with the coexistence isolated corner states and gapless edge states~\\cite{2109arXiv}, Floquet topological systems with anomalous edge modes~\\cite{2013PhysRevX3031005}, and continuous models with ghost edge modes~\\cite{2020PhysRevResearch2013147}.\nAlthough the intrinsic longitudinal spin-spin interaction is ubiquitous,\nit is still unclear {\\em how the longitudinal spin-spin interactions affects topological magnon excitations in spin systems}.\n\t\n\nIn this article, we reveal the effect of longitudinal spin-spin interactions on magnon boundary states both in one- and two-dimensional topological spin systems as variants of SSH and BBH models, respectively.\nEven for single-magnon excitations, topological edge states (mainly distributed at the second or the next to last site) and defect edge states (mainly distributed at the first or the last site) may coexist in a dimerized spin chain.\nThis is because the strong longitudinal spin-spin interaction gives rise to defects at boundaries, and make inter-cell and intra-cell couplings swap their positions.\nBy analyzing the variation of energy spectrum, spin magnetization and adiabatical topological pumping, two types of edge states are well distinguished.\nThe staggered magnetization is proposed to identify the transitions from topological trivial (nontrivial) phase to nontrivial (trivial) one in the parameter space spanned by longitudinal spin-spin interaction and transverse dimerization.\nFurthermore, we find the topology of bound magnons sensitively depends on the number of magnon excitations, which is termed as even-odd effect.\nRemarkably, besides to defect corner states and topological sub-corner states, there have first-order topological edge states in two-dimensional BBH-type spin systems which is understood via a second-order tunneling process.\nSuch three types of boundary states following distinct adiabatic pumping provide alternative ways for designing spin devices.\n\nThis paper is organized as follows.\nIn Sec.~\\ref{onedimensional}, we examine the effect of longitudinal spin-spin interaction on a dimerized XXZ chain for topological single-magnon excitations in Sec.~\\ref{topologicalsinglemagnon} and  topological multi-magnon excitations in Sec.~\\ref{topologicalmultimagnon}.\nSpin dynamics is used to detect the magnon-excitation states in Sec.~\\ref{appendixDet}.\nIn Sec.~\\ref{appendixHOMI}, we analyze the longitudinal spin-spin interaction in a two-dimensional BBH-type spin systems.\nIn Sec.~\\ref{conclusiondicussion}, we give a conclusion and discussion.\n\n\n\\section{One-dimensional dimerized XXZ chains} \\label{onedimensional}\n\nWe first consider an open Heisenberg XXZ chain with transverse dimerization,\n\\begin{equation}\\label{DimerizedSpinChain}\n\t\\hat{H}=-\\sum_{l}^{}\\left\\{\\left[\\left(J+(-1)^{l} \\delta_0\\right)\\hat{S}^+_l\\hat{S}^-_{l+1}+\\mathrm{H.c.}\\right]+\\Delta \\hat{S}_{l}^{z} \\hat{S}_{l+1}^{z} \\right\\}\n\\end{equation}\nwith the lattice index $l$, the spin-$\\frac{1}{2}$ operators $\\hat{S}^{(x,y,z)}_l$, and the spin-raising (-lowing) operators $\\hat{S}^{\\pm}_l=\\hat{S}^x_l\\pm i\\hat{S}^y_l$.\n$J\\pm \\delta_0$ respectively denote the inter-cell and intra-cell spin-exchange strengths, and $\\Delta$ is the longitudinal spin-spin interaction.\nWithout loss of generality, we set $J=1$ and $\\Delta>0$.\nFor a large $\\Delta$, all spins downward $|\\downarrow \\downarrow \\cdots \\downarrow\\rangle$ is a ferromagnetic ground state, in which flipping a spin upward creates a magnon excitation.\nIn contrast to studies of topological ground states~\\cite{PhysRevB81014505,PhysRevLett1100753032013,PhysRevLett1102153012013,PhysRevLett1102604052013,PhysRevB93155133,PhysRevA990121222019,PhysRevLett1280434022022}, topological magnon excitations are associated with excited states.\nBecause $[\\hat{H}, \\sum_l\\hat{S}_{l}^{z}]=0$, the subspaces with different magnon numbers are decoupled.\nBelow we study a $2L$-site spin chain of $L$ two-site cells, except the adiabatic topological pumping of edge states (where the site number is odd).\n\n\nViewing magnons as quasiparticles, the spin exchange and longitudinal spin-spin interaction correspond to nearest-neighbor hopping and magnon-magnon interaction, respectively.\nAt the first glance, since the magnon-magnon interaction is absent in single-magnon excitations, the topology of single-magnon excitations was supposed to behave as the celebrated SSH model~\\cite{SuWPPhysRevLett1979,PhysRevB101134423}.\nIn such a SSH model, the bulk topology is characterized by the winding number $\\nu$, where $\\nu=1$ for $\\delta_0>0$ and $\\nu=0$ for $\\delta_0 <0$.\nThe conventional BBC indicates that a pair of edge states do (not) exist when the intra-cell spin exchange is weaker (stronger) than the inter-cell spin exchange.\nHowever, even for single-magnon excitations, we find that the conventional BBC becomes invalid in our system, in which the longitudinal spin-spin interaction still plays a crucial role.\n\t\n\n\\subsection{Topological single-magnon excitations} \\label{topologicalsinglemagnon}\n\n\\begin{figure}[htp]\n\\center\n\\includegraphics[width=0.5\\textwidth]{Fig1.eps}\n\\caption{(Color online) Single-magnon excitations. (a) and (b): single-magnon energy spectra respectively with $\\Delta=0.01$ and $\\Delta=100$.\n\t%\nThe eigenstate index is ordered for increasing values of the energy.\n\t%\nThe other parameters are chosen as $J=1$, $\\delta_{0}=-0.5$ and $L=50$.\n\t%\n(c) and (d): spin magnetization distributions $S_{l}^{z}=\\langle\\Psi|\\hat{S}_{l}^{z}| \\Psi\\rangle$ for the isolated states far from the bottom band and lying in the energy gap in (b).\n\t%\nThe insets respectively correspond to the sketches of edge states.\n\t%\n\\label{fig:onemagnonspectrum}}\n\\end{figure}\n\nBelow we discuss how longitudinal spin-spin interaction affects topological edge states in single-magnon systems, which is ignored in the observation of topological magnon insulator states~\\cite{CaiPhysRevLett1230805012019}.\n\t%\nWe show single-magnon energy spectra for weak and strong longitudinal spin-spin interactions in Figs.~\\ref{fig:onemagnonspectrum}(a) and~\\ref{fig:onemagnonspectrum}(b), respectively.\n\t%\nFor $\\Delta=0.01$, there are only two separated energy bands, similar to the trivial SSH model.\n\t%\nHowever, for $\\Delta=100$, four additional isolated edge states appear:\nthe two below the bottom band (the first two states) correspond to a single magnon strongly confined at the leftmost or rightmost site,\nand the other two in the band gap (the ${(L+1)}$-th and ${(L+2)}$-th states) correspond to a single magnon strongly localized at the second or the next to last site;\nsee the spin magnetization $S_{l}^{z}=\\langle\\Psi|\\hat{S}_{l}^{z}| \\Psi\\rangle$ in Figs.~\\ref{fig:onemagnonspectrum}(c) and (d), respectively.\n\t%\nThe insets in Figs.~\\ref{fig:onemagnonspectrum}(c) and (d) schematically display the two types of edge states at left, while their degenerate counterparts are symmetrically localized at right.\n\t%\nSurprisingly, the edge states appear in the system of strong longitudinal spin-spin interaction even when the bulk topology was supposed to be trivial with $\\delta_0<0$.\n   %\nA question naturally arises: {\\em what is the emergence mechanism of these edge states?}\n\t\n\nTo understand the origin of these edge states, we analyze their behaviors across the topological transition point $(J-\\delta_{0})\/ (J+\\delta_{0})=1$ of the conventional SSH model.\nWhen the longitudinal spin-spin interaction is weak enough, similar to a conventional SSH model, edge states appear in the band gap when $\\delta_{0}>0$, which corresponds to nontrivial topology of the bulk band.\nHowever, for strong enough $\\Delta$, the edge states in the band gap appear when $\\delta_0<0$; see Fig.~\\ref{fig:topogicalinvariant}(a) with $\\Delta=100$.\nThese edge states in the band gap indicate that the conventional BBC is broken by the strong longitudinal spin-spin interaction.\nBesides, the edge states below the bottom band always exist regardless of the dimerization strength $\\delta_0$.\nThis means that the edge states below the bottom band are not related to topology but are solely induced by the longitudinal spin-spin interaction.\n\n\\begin{figure}[htp]\n\\center\n\\includegraphics[width=0.5\\textwidth]{Fig2.eps}\n\\caption{(Color online) Appearance of topological edge states.\n(a) Single-magnon excitation energy versus the transverse spin-exchange ratio $(J-\\delta_{0})\/(J+\\delta_{0})$ for longitudinal spin-spin interaction $\\Delta=100$ with $L=50$.\nThe energy spectrum (b) and adiabatic topological pumping of topological (c) and defect (d) edge states for a 19-site system with $\\Delta=100$.\nThe other parameter is $J=1$.\n\\label{fig:topogicalinvariant}}\n\\end{figure}\n\n\nTo uncover these mysterious edge states, we revisit the mapping from longitudinal spin-spin interaction to magnon-magnon interaction.\nFrom the perspective of magnon excitations under open boundaries, we discover that intrinsic longitudinal spin-spin interaction has two main effects: one is defect potential at the end points, and the other is magnon-magnon interaction.\nTaking the open three-spin chain as an example, single-magnon non-edge states $(\\left|\\downarrow \\uparrow \\downarrow\\right\\rangle)$ and the edge states $(\\left|\\uparrow \\downarrow \\downarrow\\right\\rangle, \\left|\\downarrow \\downarrow \\uparrow\\right\\rangle)$ have onsite energies $\\Delta\/4 \\pm \\Delta\/4$, respectively.\nThe longitudinal interaction contributes an energy offset $\\Delta\/2$ between the magnon at the edge sites and the other bulk sites~\\cite{LiuPhysRevA2021}.\nThis means that the cooperation between the open boundary condition and the longitudinal spin-spin interaction induces effective on-site defects at the edge sites.\nA large $\\Delta\/2$ will trap a magnon at an end point to form the non-topological edge state, which can be dubbed as defect edge state.\nIn the limit of $\\Delta\\rightarrow \\infty$, the first and the last sites are decoupled from other bulk sites and can then be understood as a new interface.\nThus the second and the next to last sites act as a new ``boundary'' of a topological SSH lattice, in which the intra- and inter-cell spin exchanges are swapped.\nConsequently, the topology of the renormalized SSH model is opposite to the original trivial one, and topological edge states appear at the second and the next to last sites.\nHence, these sub-edge states in the band gap belong to topological edge states.\n\n\nTo further distinguish two types of edge states, we adiabatically sweep $(J-\\delta_0)\/(J+\\delta_0)$ across the phase transition point in a 19-site system with $\\Delta=100$.\nIts instantaneous energy spectrum is shown in Fig.~\\ref{fig:topogicalinvariant}(b).\nWith the two types of edge states for $(J-\\delta_0)\/(J+\\delta_0)=0$ as initial states, after tracking the change of spin magnetization we find that the topological edge state can be adiabatically transferred from one end to the other end [see Fig.~\\ref{fig:topogicalinvariant}(c)], while the defect edge state remains unchanged [see Fig.~\\ref{fig:topogicalinvariant}(d)].\nThis means that observation of edge states is not enough to support bulk-boundary correspondence and that adiabatic topological pumping of edge states is a more rigorous method to demonstrate the bulk topology.\n\n\\begin{figure}[htp]\n\\center\n\\includegraphics[width=0.5\\textwidth]{Chosendelta0.eps}\n\\caption{(Color online) The band-edge gap $\\Delta E$ as a function of the longitudinal spin-spin interaction $\\Delta$: (a) topological edge state and (b) non-topological edge state.\nThe staggered magnetization $m_{\\mathrm{st}}^{z}$ as a function of the longitudinal spin-spin interaction $\\Delta$ for (c) the ($L+1$)-th state and (d) the first state.\nThe inset of (b) zooms in its region around critical points.\nOur calculations are performed with $\\delta_{0}=-0.5$ (red solid lines) and $\\delta_{0}=0.5$ (blue dotted lines) and $L=5000$.\n\\label{fig:chosendelta0}}\n\\end{figure}\n\n\n\nOwing to the longitudinal spin-spin interactions, there exist transitions between topologically trivial and nontrivial phases as $\\Delta$ increases.\nWe investigate the band-edge gap $\\Delta E$ to determine the critical point $\\Delta_c$, both analytically and numerically.\nFor topological edge state, in the thermodynamic limit $\\Delta E$ naturally tends to zero in the topologically trivial phase, and takes a finite value due to the appearance of topological edge states in the topologically nontrivial phase.\nWe calculate its band-edge gap as a function of $\\Delta$ for $(J-\\delta_{0})\/ (J+\\delta_{0})=3$ with $\\delta_0=-0.5$ (red solid line) and $(J-\\delta_{0})\/ (J+\\delta_{0})=1\/3$ with $\\delta_0=0.5$ (blue dotted line) for a lattice size $L=5000$; see Fig.~\\ref{fig:chosendelta0}(a).\nThe band-edge gap can successfully identify the critical points $\\Delta_c$, no matter for the increase from zero to finite value or vice versa.\nThe corresponding $m_{\\mathrm{st}}^{z}=\\sum_{l=1}^{2L}(-1)^{l}\\langle\\Psi|\\hat{S}_{l}^{z}|\\Psi\\rangle$ of the ($L+1$)-th state is exhibited in Fig.~\\ref{fig:chosendelta0}(c).\nThere are two degenerate topological edge states in topologically nontrivial phases where the ($L+1$)-th state is used to represent the left topological edge state.\n$m_{\\mathrm{st}}^{z}\\approx-1$ means the corresponding left topological edge state almost distributes at odd sites where a single-magnon excitation mainly locates at the first site and decays at other odd sites.\nHowever, differently, if left topological edge state mainly distributes at even sites, it has $m_{\\mathrm{st}}^{z}\\approx1$.\n\nFor non-topological edge state, $\\Delta E=E_3-E_2$ represents the energy gap between the second and third eigenstates, which remains vanishingly small when $\\Delta<\\Delta_c$.\nUnlike the topological edge states finally lying in the middle of two bulk bands, the non-topological edge states linearly increase with $\\Delta$ after separating from the bottom bands, as shown in Fig.~\\ref{fig:chosendelta0}(b).\nThe staggered magnetization of the first state in Fig.~\\ref{fig:chosendelta0}(d) reflects, after crossing the critical point $\\Delta_c$, the left non-topological edge state considerably distributes at odd sites as $\\Delta$ increases.\nTwo degenerate non-topological edge states always appear as a pair where the first state is used to represent the left non-topological edge state.\nThe transition point induced by longitudinal spin-spin interaction depends on the appearance or disappearance of non-topological states.\nImportantly, using the transfer matrix method, the condition for the appearance of the defect edge states can be analytically given by $\\Delta>\\Delta_c$ with $\\Delta_c=2(J+\\delta_0)$ (see Appendix~\\ref{appendixCritical} for details).\nThe analytical critical points $\\Delta_c=1$ and $\\Delta_c=3$ for two cases are added in Fig.~\\ref{fig:chosendelta0} with gray dashed lines which are well consistent with the numerical ones.\n\n\n\t\n\\begin{figure}[htp]\n\\center\n\\includegraphics[width=0.38\\textwidth]{Fig3.eps}\n\\caption{(Color online) The staggered magnetization $m_{\\mathrm{st}}^{z}$ of ($L+1$)-th state as a function of the longitudinal spin-spin interaction $\\Delta$ and the transverse dimerization $(J-\\delta_{0})\/ (J+\\delta_{0})$ for a $2L$-site system with $L=1000$.\nGray shading is a region where the staggered magnetization cannot be integer owing to the absence of chiral symmetry.\nThe critical lines $\\Delta_c=2(J+\\delta_0)$ and $(J+\\delta_0)\/(J-\\delta_0)=1$ are marked by the red and white dashed lines, respectively.\nThe other parameter is $J=1$.\n\\label{fig:windingnumber}}\n\\end{figure}\n\nBecause the longitudinal spin-spin interaction breaks the chiral symmetry and invalidates topological invariant, we calculate the staggered magnetization to witness the transition from topologically trivial to nontrivial phases.\nFig.~\\ref{fig:windingnumber} manifests the staggered magnetization $m_{\\mathrm{st}}^{z}$ of ($L+1$)-th state in the parameter space spanned by $\\Delta$ and $(J-\\delta_{0})\/ (J+\\delta_{0})$.\nAlthough $m_{\\mathrm{st}}^{z}$ vanishes in topologically trivial phases, it takes a finite negative (positive) value in the left-bottom (right-top) region.\nFor a weak longitudinal spin-spin interaction, the topological characterization behaves like the conventional SSH model: topologically nontrivial phase for $(J-\\delta_{0})\/ (J+\\delta_{0})<1$ while for $(J-\\delta_{0})\/ (J+\\delta_{0})>1$ it turns into the trivial one.\nTuning the longitudinal spin-spin interaction strong enough accompanied with the appearance of the non-topological edge states, topological edge states exist for $(J-\\delta_{0})\/ (J+\\delta_{0})>1$ while disappear for $(J-\\delta_{0})\/ (J+\\delta_{0})<1$.\nThe analytical critical lines $\\Delta_c=2(J+\\delta_0)$ and $(J-\\delta_{0})\/ (J+\\delta_{0})=1$ are respectively added in Fig.~\\ref{fig:windingnumber} with red and white dashed lines,\nindicating that the analytical results agree well with the numerical ones.\nSimilar to the nonlinear Thouless pumping~\\cite{2021Nature59663}, near the critical line $\\Delta_c$ there exists a narrow region (the gray region) in which $0<|m_{\\mathrm{st}}^{z}|<1$.\nFrom the perspective of distribution properties of topological edge states, such a staggered magnetization can serve as an effective topological indicator to understand the topological phase transitions.\n\n\n\n\t\n\\subsection{Topological multi-magnon excitations} \\label{topologicalmultimagnon}\n\n\nThe longitudinal spin-spin interaction provides not only boundary defects but also nearest-neighbor magnon-magnon interaction in multi-magnon excitations.\nThe magnon-magnon interaction can bind magnons together as bound states~\\cite{2012PhysRevLett108077206,2013Nature50276,2014PhysRevLett112257204,2017PhysRevB96195134}.\n\t%\nWhen the longitudinal spin-spin interaction is sufficiently strong,\none can treat the transverse term as a perturbation to the longitudinal one to analytically derive an effective model for well explaining the bound states.\nHere, we just present results for multi-magnon bound states (see Appendix~\\ref{appendixA} and Appendix~\\ref{appendixB} for unbound magnons).\n\t%\nThe bound states can be treated as a quasiparticle that is governed by an effective Hamiltonian.\n\t%\nTaking two and three magnons as examples, the effective hopping strengths of bound states in the bulk are given by $J_{\\mathrm{Eff}}^{(2)}={\\left(J+\\delta_{0}\\right)\\left(J-\\delta_{0}\\right)}\/{\\Delta}$ and $J_{\\mathrm{Eff}}^{(3)}={\\left(J-\\delta_{0}\\right)\\left(J+\\delta_{0}\\right)\\left[J+\\delta_{0}(-1)^l\\right]}\/{\\Delta^2}$\n(see Appendix~\\ref{appendixA} and Appendix~\\ref{appendixB} for details).\nBecause the effective hopping strengths of two- and three-magnon bound states are respectively uniform and site-dependent,\nwe know that two-magnon bound states are topologically trivial and three-magnon bound states may inherit topology from the SSH model.\nDue to no more than one magnon at one site, the effective hopping originates from the $n$-th order process for $n$-magnon excitations.\nFrom the effective $n$-th order hopping process, we find that the corresponding effective hopping strength is proportional to $[J+\\delta_0(-1)^l]^{mod(n,2)}(J+\\delta_0)^{\\lfloor n\/2 \\rfloor}(J-\\delta_0)^{\\lfloor n\/2 \\rfloor}$, where $\\lfloor{x\\rfloor}$ takes the closest integer value less than or equal to $x$.\n    %\nWe generalize this result to an even-odd effect, that is, even-magnon bound states are trivial and odd-magnon bound states may have nontrivial topology.\n\n\\begin{figure}[htp]\n\\center\n\\includegraphics[width=0.45\\textwidth]{Fig9.eps}\n\\caption{(Color online) Two-magnon bound-state subspace for a strong longitudinal spin-spin interaction ($\\Delta=100$).\n(a) Bound-state energy spectrum in ascending order for the values of the energies.\nThe black dots denote the energies regarding the Hamiltonian~\\eqref{DimerizedSpinChain},\nand the red circles are the energies obtained by the effective model $\\hat{H}_{\\mathrm{Eff}}^{(2)}$~\\eqref{Heff2}.\nThe parameters are chosen as $J=1$, $\\delta_{0}=-0.5$ and $L=24$.\n(b)-(d) are respectively two-magnon correlation distributions of chosen eigenstates in (a).\n\\label{fig:TwoBoundMagnons}}\n\\end{figure}\n\n\t\nWe further analyze topological states in the two- and three-magnon bound-state subspaces.\nFor a strong longitudinal interaction, we obtain the bound-state subspace consisting of two-magnon bound states in Fig.~\\ref{fig:TwoBoundMagnons}(a).\nThe eigenstate index on the horizontal axis is ordered with increasing values of the energy.\nThe parameters are chosen as $J=1$, $\\delta_{0}=-0.5$, $\\Delta=100$ and $L=24$.\nThe energy spectrum of the effective model $\\hat{H}_{\\mathrm{Eff}}^{(2)}$ is added in Fig.~\\ref{fig:TwoBoundMagnons}(a) with red circles.\nIt can be observed that the numerical results (black dots) from the Hamiltonian~\\eqref{DimerizedSpinChain} perfectly agree with the analytical ones (red circles) given by the effective model $\\hat{H}_{\\mathrm{Eff}}^{(2)}$~\\eqref{Heff2} whose validity has been analyzed in Fig.~\\ref{fig:ValidityTwoMagnons}.\nThe bound-state subspace contains four isolated bands which are ordered for increasing values of the energy.\nWe extract one state of the first band to calculate the two-magnon correlation function $C_{i j}=\\langle\\Psi|\\hat{S}_i^{+} \\hat{S}_j^{+} \\hat{S}_j^{-} \\hat{S}_i^{-}| \\Psi\\rangle$ in Fig.~\\ref{fig:TwoBoundMagnons}(b), which indicates eigenstates with two magnons bound at one end point as a two-magnon bound edge state.\nThe correlation properties of the second-band states show that the bound magnons are mostly distributed at one sub-edge to form a two-magnon bound sub-edge state [Fig.~\\ref{fig:TwoBoundMagnons}(c)].\nThe two remaining continuum bands include eigenstates with two magnons as bound pairs [Fig.~\\ref{fig:TwoBoundMagnons}(d)].\nThe emergence of bound edge states is due to the fact that the longitudinal spin-spin interaction creates potentials trapping one magnon at one end point and binding the other together.\nThe presence of two-magnon bound sub-edge states results from the effective potentials at the sub-edges (the second and the next to last sites) in the effective model $\\hat{H}_{\\mathrm{Eff}}^{(2)}$~\\eqref{Heff2}.\n\n\n\\begin{figure}[htp]\n\\center\n\\includegraphics[width=0.45\\textwidth]{Fig4.eps}\n\\caption{(Color online) Three-magnon bound-state subspace for a strong longitudinal spin-spin interaction ($\\Delta=100$). (a) Bound-state energy spectrum as a function of eigenstate index in the ascending order of energy.\n\tThe black dots denote the eigen-energies of the Hamiltonian~\\eqref{DimerizedSpinChain}, and the red circles are the eigen-energies of the effective Hamiltonian $\\hat{H}_{\\mathrm{Eff}}^{(3)}$~\\eqref{Heff3}.\n\t(b)-(d): spin magnetization distributions $S_{l}^{z}$ and three-magnon correlations $C_{l,l+1,l+2}$ for three types of three-magnon bound edge states in (a).\n\tHere, $\\sum_{l=1}^{2L-2} C_{l, l+1, l+2} \\geqslant 0.9997$.\n\tThe parameters are chosen as $J=1$, $\\delta_{0}=0.5$ and $L=24$.\n\\label{fig:Threemagnons}}\n\\end{figure}\n\n\t%\nFig.~\\ref{fig:Threemagnons}(a) shows the energy spectrum for three-magnon bound states.\nThe energy spectrum of the effective model $\\hat{H}_{\\mathrm{Eff}}^{(3)}$~\\eqref{Heff3} is added in Fig.~\\ref{fig:Threemagnons}(a) with red circles.\nIt can be observed that the numerical results (black dots) from the Hamiltonian~\\eqref{DimerizedSpinChain} agree well with the analytical ones (red circles) given by the effective model $\\hat{H}_{\\mathrm{Eff}}^{(3)}$~\\eqref{Heff3} whose validity has been analyzed in Fig.~\\ref{fig:ValidityThreeMagnons}.\n\t%\nThere are three types of three-magnon bound edge states: (i) bounded to the first site, (ii) bounded to the second site, and (iii)  bounded to the third site with energy in the band gap, whose spin magnetizations and three-magnon correlations are shown in Fig.~\\ref{fig:Threemagnons}(b, c, d), respectively.\nHere, the three-magnon correlation is defined as $C_{l,l+1,l+2}=\\langle\\Psi|\\hat{S}_{l}^{+} \\hat{S}_{l+1}^{+} \\hat{S}_{l+2}^{+}\\hat{S}_{l+2}^{-}\\hat{S}_{l+1}^{-} \\hat{S}_{l}^{-}| \\Psi\\rangle$.\nThe type-(i,ii) three-magnon bound states are attributed to the emergent defects at the first and second sites, while the type-(iii) three-magnon bound states are due to nontrivial topology of the bulk band (see the effective Hamiltonian $\\hat{H}_{\\mathrm{Eff}}^{(3)}$~\\eqref{Heff3} in Appendix~\\ref{appendixB} for details).\n\t\n\\subsection{Detecting magnon-excitation states via spin dynamics} \\label{appendixDet}\n\n\\begin{figure*}[htp]\n\\center\n\\includegraphics[width=0.9\\textwidth]{Fig13.eps}\n\\caption{(Color online) The time evolution of the spin magnetization distribution $S_{l}^{z}(t)=\\langle\\Psi(t)|\\hat{S}_{l}^{z}|\\Psi(t)\\rangle$ obtained using TEBD algorithm.\nThe top, middle, and bottom rows correspond to single-magnon excitations $N_m=1$ with $\\delta_0=-0.5$, two-magnon excitations $N_m=2$ with $\\delta_0=-0.5$, and\nthree-magnon excitations $N_m=3$ with $\\delta_0=0.5$, respectively.\nFor all three rows, the chosen initial states are shown from left to right, respectively.\nThe evolved time is set as $t=2000$.\nThe other parameters are chosen as $J=1$ and $\\Delta=100$. The system size is $2L=48$.\n\\label{fig:spindynamics}}\n\\end{figure*}\n\nSpin dynamics provides new insights into the detection of magnon-excitation states.\nReferring to the above analysis, a rich variety of magnon-excitation states appear as tuning intrinsic system parameters and magnon-excitation numbers, especially for topological magnon edge states.\nTopologically protected edge states have been employed to implement quantum state transfer~\\cite{2012PhysRevLett109106402,2013NatCommun41585,2017QuantumSciTechnol2015001,2018PhysRevA98012331,2019PhysRevLett123034301}, disorder-immune photonic and phononic transport~\\cite{2014NatPhotonics8821,2016NatPhys12621,2019RevModPhys91015006}, topological quantum computation~\\cite{2015RevModPhys87137}, and topological laser~\\cite{GalHarariScience2018,MiguelScience2018}.\nWe calculate the time evolution of the spin magnetization distribution $S_{l}^{z}(t)=\\langle\\Psi(t)|\\hat{S}_{l}^{z}|\\Psi(t)\\rangle$ by using the time-evolving block decimation\n(TEBD) algorithm~\\cite{VidalPhysRevLett911479022003,VidalPhysRevLett930405022004}.\nInitially, magnon excitations are located at different lattice sites, as shown in Fig.~\\ref{fig:spindynamics} from $N_m=1$ to 3 magnons.\nWhen initial magnon excitations are prepared with a high enough fidelity with the corresponding edge states, dynamical localization, where magnon excitations almost stay at the initial positions as time evolves, appears as a consequence of edge states.\nThe existence of non-topological edge states and topological edge state offers promising applications for manipulating the spin transports and designing the magnon spintronic devices.\n\n\\section{Two-dimensional BBH-type spin systems} \\label{appendixHOMI}\n\n\\begin{figure*}[htp]\n\\center\n\\includegraphics[width=1\\textwidth]{Threeedgestates.eps}\n\\caption{(Color online) Single-magnon excitations in a two-dimensional BBH-type spin system.\nThree parts of single-magnon excitation spectra: (a) defect corner states, (b) topological corner states as well as their neighboring eigenstates and (c) first-order topological edge states as well as their neighboring eigenstates.\nThe eigenstate index is ordered for increasing values of the energy.\nSpin magnetization ${S}_{i,j}^{z}$ for defect corner states (d), topological corner states (e), lower (f) and upper four topological edge states (g).\nThe other parameters are chosen as $J=1$, $\\delta_{0}=-0.5$ and $L=16$.\n\\label{fig:threeedgestates}}\n\\end{figure*}\n\n\n\nAfter revealing the effects of intrinsic spin-spin interaction on magnon edge states in a dimerized spin chain,\nit is reasonable to predict that the longitudinal spin-spin interaction plays a crucial role in topological magnon corner states by generating non-topological corner states.\nBased on the quantized quadrupole moment in a Benalcazar-Bernevig-Hughes (BBH) model~\\cite{2017Science35761,2017PhysRevB96245115}, our two-dimensional BBH-type spin system with longitudinal spin-spin interaction described by\n\\begin{eqnarray}\\label{TDimerizedSpinChain}\n\\hat{H}&=&-\\sum_{j=1}^{2 L} \\sum_{i=1}^{2 L-1}\\left[J+(-1)^{i} \\delta_{0}\\right] \\hat{S}_{i, j}^{+} \\hat{S}_{i+1, j}^{-}+\\text {H.c. } \\nonumber \\\\\n&-& \\sum_{i=1}^{2 L} \\sum_{j=1}^{2 L-1}(-1)^i\\left[J+(-1)^{j} \\delta_{0}\\right] \\hat{S}_{i, j}^{+} \\hat{S}_{i, j+1}^{-} +\\text {H.c. }  \\nonumber \\\\\n&-&\\Delta \\hat{S}_{i, j}^{z} \\hat{S}_{i+1, j}^{z}-\\Delta \\hat{S}_{i, j}^{z} \\hat{S}_{i, j+1}^{z}.\n\\end{eqnarray}\nWe restrict to discuss the subspace of single-magnon excitations spanned by the basis $\\left\\{|i, j\\rangle=\\hat{S}_{i, j}^{+}|\\downarrow \\downarrow \\downarrow \\ldots \\downarrow\\rangle\\right\\}$ with $1 \\leq i,j \\leq 2 L$ and $|\\downarrow \\downarrow \\ldots \\downarrow\\rangle$ being the ferromagnetic ground state.\nWhen $\\Delta=0$, the Hamiltonian~\\eqref{TDimerizedSpinChain} for single-magnon excitations is equal to a BBH model~\\cite{2017Science35761,2017PhysRevB96245115}, where two phases are distinct:\none is the topological phase that supports localized corner states almost distributing at four outmost corners when $\\delta_0>0$, and the other is the trivial phase lacking the corner states when $\\delta_0<0$.\n\nAs the longitudinal spin-spin interaction increases, the topological boundaries are gradually shifted one lattice site inward to support topological sub-corner states.\nBy analyzing the spin magnetization $S_{i, j}^z$ at position ($i$,$j$) for $\\Delta=100$, four defect corner states occupy four outmost corners [see Fig.~\\ref{fig:threeedgestates}(d)] away from the bulk bands [see Fig.~\\ref{fig:threeedgestates}(a)].\nIn contrast to the quadrupolar topological insulator, four topological sub-corner states [see Fig.~\\ref{fig:threeedgestates}(e)] are in the midgap of bulk bands [see Fig.~\\ref{fig:threeedgestates}(b)] when $\\delta_0<0$.\nUnlike the absence of first-order topological edge states due to the vanishing dipole moment in BBH models, one can find that eight first-order topological edge states, lying inside the energy gap, are grouped into four degenerate pairs with energy splitting; see Fig.~\\ref{fig:threeedgestates}(c).\nFigs.~\\ref{fig:threeedgestates}(f) and (g) respectively manifest the spin magnetization for the lower and upper four topological edge states in Fig.~\\ref{fig:threeedgestates}(c).\nThe eigenstate index on the horizontal axis is ordered with increasing values of the energy.\nThe other parameters are chosen as $J=1$, $\\delta_{0}=-0.5$ and $L=16$.\nThe emergence of first-order topological edge states is due to the fact that\nthe longitudinal spin-spin interaction creates potentials, making four boundary lines decouple to the bulk sites.\nEach boundary line is equal to a one-dimensional dimerized spin chain that supports the coexistence of defect edge states and topological edge states.\nTwo defect edge states meeting at a corner create a defect corner state.\nEight first-order topological edge states originate from the four boundary lines.\nHowever, the spin magnetization of first-order topological edge states in Figs.~\\ref{fig:threeedgestates}(f) and (g)\nis still unclear.\n\n\\begin{figure*}[htp]\n\t\\center\n\t\\includegraphics[width=1\\textwidth]{CornerTransport.eps}\n\t\\caption{(Color online) Instantaneous energy spectra of adiabatic pumping for defect corner states (a1), topological corner state as well as its neighboring states (b1) and first-order topological edge states as well as their neighboring states (c1).\n\t\t%\n\t\t(a2), (b2), (c2) Average positions $\\bar{x}$ with red lines ($\\bar{y}$ with blue dashed lines) in $x$ ($y$) direction and IPRs with yellow dashed-dotted lines of one of defect corner state, topological corner state and the lowest first-order topological edge state, respectively.\n\t\t%\n\t\t(a3), (b3), (c3) The spin magnetizations at moments $(J-\\delta_0)\/(J+\\delta_0)=0$,1,4 from left to right corresponding to (a2), (b2) and (c2), respectively.\n\t\t%\n\t\tThe other parameters are chosen as $J=1$, $\\delta_{0}=-0.5$ and $L=5$.\n\t\t%\n\t\t\\label{fig:cornertransport}}\n\\end{figure*}\n\n\nWithout loss of generality, we focus on four positions [($1$,$1$), ($1$,$2$), ($2$,$1$), ($2$,$2$)] to explore such first-order topological edge states.\nAn energy difference $\\Delta\/2$ between the magnon at the edge sites and the other bulk sites comes from the longitudinal spin-spin interaction, which becomes $\\Delta$ between the magnon at the corners and the other bulk sites.\nOnce ignoring an energy constant, the matrix of the above space spanned by the basis ($|1,1\\rangle$, $|1,2\\rangle$, $|2,1\\rangle$, $|2,2\\rangle$) is written as\n\\begin{equation}\n\\hat{H}_S=\\left(\\begin{array}{cccc}\n\\Delta & J-\\delta_0 & -\\left(J-\\delta_0\\right) & 0 \\\\\nJ-\\delta_0 & \\Delta \/ 2 & 0 & J-\\delta_0 \\\\\n-\\left(J-\\delta_0\\right) & 0 & \\Delta \/ 2 & J-\\delta_0 \\\\\n0 & J-\\delta_0 & J-\\delta_0 & 0\n\\end{array}\\right).\n\\end{equation}\nWhen the longitudinal spin-spin interaction is strong enough, the population in states $|1,1\\rangle$ and $|2,2\\rangle$ can be adiabatically eliminated.\nThen the effective $2\\times2$ model in basis ($|1,2\\rangle$, $|2,1\\rangle$) can be obtained as\n\\begin{equation}\n\t\\left(\\begin{array}{cc}0 & 4\\left(J-\\delta_0\\right)^2 \/ \\Delta \\\\ 4\\left(J-\\delta_0\\right)^2 \/ \\Delta & 0\\end{array}\\right).\n\\end{equation}\nBy solving the eigenequation, its eigenstates are $(|1,2\\rangle \\pm|2,1\\rangle) \/ \\sqrt{2}$ with eigenvalues $\\pm 4\\left(J-\\delta_0\\right)^2 \/ \\Delta$, respectively.\nThe remaining six first-order topological edge states can be understood in the same way.\nThe topological edge states of four boundary lines hybridize through a second-order tunneling process to form the hybrid first-order topological edge states in Figs.~\\ref{fig:threeedgestates}(f) and (g).\n\nAfter removing two arrays from the $x$ and $y$ directions yielding a $(2L-1)$$\\times$$(2L-1)$ square lattice, the adiabatic evolution is shown in Fig.~\\ref{fig:cornertransport} with $L=5$.\nThe other parameters are chosen as $J=1$ and $\\delta_{0}=-0.5$.\nBy adiabatically sweeping $(J-\\delta_0)\/(J+\\delta_0)$, there are still four defect corner states with energy far away from the other states; see the instantaneous energy spectrum in Fig.~\\ref{fig:cornertransport}(a1).\nWe choose one of four defect corner states as initial state, and the evolved state $|\\psi(t)\\rangle=\\sum_{i,j} p_{i,j}(t) |i,j\\rangle$ is governed by the\nSchr\\\"{o}dinger equation $i \\hbar \\frac{\\partial}{\\partial t}|\\psi(t)\\rangle=\\hat{H}|\\psi(t)\\rangle$.\nThe average position $\\bar{x}=\\sum_i i\\left|p_{i,j}\\right|^2$ ($\\bar{y}=\\sum_j j\\left|p_{i,j}\\right|^2$) of single-magnon excitations in the $x$ ($y$) direction is shown in Fig.~\\ref{fig:cornertransport}(a2) with the red line and the blue dashed line, respectively.\nIt means the magnon excitation is well localized at the initial corner whose corresponding inverse participation ratios $\\operatorname{IPR}=\\sum_{i,j}\\left|p_{i,j}\\right|^4$ are shown in Fig.~\\ref{fig:cornertransport}(a2) with the yellow dashed-dotted line for a localized state with a high value.\nThe spin magnetizations at moments $(J-\\delta_0)\/(J+\\delta_0)=0$, $1$ and $4$ are also analyzed in Fig.~\\ref{fig:cornertransport}(a3) from left to right, which clearly exhibits the unchanged corner state.\nThe other three defect corner states all behave like Figs.~\\ref{fig:cornertransport}(a1), (a2) and (a3).\n\nFig.~\\ref{fig:cornertransport}(b1) shows the instantaneous energy spectrum of topological corner state as well as its neighboring states.\nStarting from the initial topological corner state, its average position and IPR are revealed in Fig.~\\ref{fig:cornertransport}(b2).\nIt can be seen from Fig.~\\ref{fig:cornertransport}(b3) from left to right that the initial state is set to be topological sub-corner state at the right-top sub-corner, then gradually spreads over the bulk sites around moment $(J-\\delta_0)\/(J+\\delta_0)=1$, and finally transfers to be the topological sub-corner state at the left-bottom sub-corner for $(J-\\delta_0)\/(J+\\delta_0)=4$.\nIn Fig.~\\ref{fig:cornertransport}(c1), we plot the instantaneous energy spectrum of first-order topological edge states as well as their neighboring states.\nIt is found that four first-order topological edge states appear in the middle of two bands.\nThe energy splitting resulting from the second-order tunneling process allows the lowest and highest topological edge states to be separated from the middle two.\nSpecifically, we show the adiabatic transfer of the lowest first-order topological edge state in Fig.~\\ref{fig:cornertransport}(c2).\nThe state transfer between two hybrid first-order topological edge states at diagonal lines is established through the extended edge states around the moment $(J-\\delta_0)\/(J+\\delta_0)=1$; see Fig.~\\ref{fig:cornertransport}(c3).\n\n\n\\section{Conclusion and discussion} \\label{conclusiondicussion}\n\nWe elaborate the role of the longitudinal spin-spin interaction on topological magnon excitaitons ranging from\none-dimensional dimerized XXZ chains to two-dimensional BBH-type spin systems.\nIn a dimerized XXZ chain, we explore the appearance of topological edge states at the sub-edges, which can be distinguished by spin magnetization, variation of energy spectrum, and adiabatic topological pumping.\nThe topological phase transition induced by longitudinal spin-spin interaction in our system is accompanied by a bulk-edge gap closing of topological edge states rather than a bulk-band gap closing.\nIts phase boundary is analytically derived via the transfer matrix method, and different phases are faithfully identified via the staggered magnetization.\nWe analytically obtain an effective model for revealing the even-odd effect of magnon-excitation number on topological magnons based upon the many-body degenerate perturbation theory.\nRemarkably, the odd-even effect we find here is the first example in which the particle number serves as a degree of freedom to tune the topological properties.\nOur results demonstrate that the magnon-magnon correlation of multi-magnon excitations plays a crucial role in topological multi-magnon states.\nThe interplay among longitudinal spin-spin interaction, transverse dimerization and magnon-magnon correlations leads to a rich variety of magnon excitations which are able to be flexibly detected via spin dynamics.\nFor a two-dimensional BBH-type spin system, the longitudinal spin-spin interaction is responsible for the coexistence of defect corner states, second-order topological corner states and first-order topological edge states.\nBeyond the absence of first-order topological edge states in a typical BBH model, the hybrid first-order topological edge states are well explained by a second-order tunneling process.\nThese three types of magnon-excitation states with distinct adiabatic pumping offer alternative ways for state transfer.\n\n\nWe have demonstrated that longitudinal spin-spin interactions provide a versatile tool to engineer topological magnon states, which have been overlooked for a long time.\nIt deserves further study to generalize our method to other interacting systems such as long-range spin systems and extended Hubbard systems.\n\t\n\n\\begin{acknowledgments}\nWe acknowledge useful discussions with Linhu Li, Ling Lin and Li Zhang.\nThis work is supported by the National Natural Science Foundation of China (Grants No. 12025509, No. 11874434), the Key-Area Research and Development Program of GuangDong Province (Grants No. 2019B030330001), and the Science and Technology Program of Guangzhou (China) (Grants No. 201904020024).\nW.L. is partially supported by the National Natural Science Foundation of China (Grant No. 12147108) and the\nFundamental Research Funds for the Central Universities, Sun Yat-Sen University (Grant No. 22qntd3101).\nY.K. is partially supported by the National Natural Science Foundation of China (Grant No. 11904419; No. 12275365).\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection{CDV Methodology}\\label{sc:cdvmethod}\nIn CDV, an iterative process of test generation, execution, coverage collection and analysis is used to achieve coverage closure over several cycles. In practice, engineering input is required to interpret the data and to guide test generation towards closing coverage holes. This is either achieved simply by allowing further pseudorandom tests to be generated, by adding constraints to bias test generation, by employing model-based test generation or, as a last resort, by directed testing. If model-based test generation has already been applied, modifications to the formal model may yield new tests.  \n\nIt is important to note that further test generation is not always the only appropriate response to a coverage hole or a requirement violation.  The following options should also be considered: 1) the SUT has a bug, to be referred to the design team; 2) modifications to one or more of the requirements models (e.g.\\ assertions or formal properties) are needed to more accurately reflect the actual requirements and\/or design of the SUT; and\/or 3) modifications to one or more of the testbench components are needed. This third decision may be reached if the tests and requirements models are deemed appropriate but the testbench does not allow the SUT's full range of functions to be exercised and observed.\n\n\n\\section{Coverage-Driven Verification} \\label{sc:CDV}\n\n\\subsection{Structure of a CDV Testbench}\nIn CDV, a verification plan must be constructed before the testing process begins~\\cite{Piziali2004}. This plan includes the aspects of the SUT that need to be verified, e.g.\\ a requirements list or a functional description of the SUT, and a coverage strategy. The coverage strategy indicates how to achieve effective coverage, i.e.\\ the exploration of the SUT and advancement of the verification progress, through the design of the testbench components, especially the Test Generator, the Checker and the Coverage Collector. The coverage strategy also specifies how to measure the coverage, e.g.\\ a requirements model or a functional model to traverse.\n\nIn Testing, the SUT is placed into a test environment, a {\\em testbench}. The testbench represents (a model of) the universe, or of its target environment. The process of testing is realised using the following four core components in a testbench, as shown in Fig.~\\ref{testbench}:\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{cdvtestbench.pdf}\n\\caption{Structure of a basic CDV testbench}\n\\label{testbench}\n\\end{figure} \n\n\\begin{itemize}\n\\item the {\\bf Test Generator} is the component that generates stimulus for\n  the SUT;\n\\item the {\\bf Driver} is the component that takes a test, potentially at a\n  high level of abstraction, translates it into the level of abstraction used\n  in the simulation, and drives it to stimulate the SUT;\n\\item the {\\bf Checker} is the component that checks the response of the SUT to\n  the stimulus and detects failures; \n\\item the {\\bf Coverage Collector} is the component that records the quality\n  of the generated tests with respect to a set of complementing coverage models.\n\\end{itemize}\n\nA key objective in the design of a CDV testbench is to achieve a fully autonomous environment, so that verification engineers can\nconcentrate on areas requiring intelligent input, namely efficient and effective test generation, bug detection, reliable tracking of progress and timely completion.\n\nIn the following sections we describe each testbench component in more detail, explaining how they can be used for verification in robotics. \n\n\n\n\n\\subsection{Checker}\\label{sc:checker}\nThe automation of test generation prompts the need for automatic and test independent checkers, i.e.\\ self-checking testbenches. Assertion-based verification~\\cite{abd} allows locating checkers, in the form of assertion monitors, close to the code that is being observed.\n\nRequirements to verify can be expressed as Temporal Logic properties. Assertion monitors can be derived automatically from these\nproperties~\\cite{Havelund2002}, in an automata-based form. Since the simulations are time bound, some safety properties defined over infinite traces (e.g., using an \\verb+always+ Temporal Logic operator) are bound over the duration of a simulation run. Relevant work in~\\cite{Armoni2006} mentions the advantages of the automatic generation of monitors as automata, including the reduction of errors caused by manual translation.\n\nFor requirements about the low-level continuous behaviour of the SUT (e.g., trajectories computed by the motion planning), the monitoring can be performed in a quasi-continuous manner, considering computational limitations. Otherwise, over-approximations or interpolation can be performed to predict events at instants of time between computations, such as the overlapping of regions in the 3D space for collision avoidance.\n\n\n\\section{Conclusions}\\label{sc:conclusions}\n\nWe advocated the use of CDV for robot code in the context of HRI. By promoting automation, CDV can provide a faster route to coverage closure, compared with manually directed testing. CDV is typically used in Software-in-the-Loop simulations, but it can also be used in conjunction with Hardware-in-the-Loop simulation, Human-in-the-Loop simulation or with emulation.\nThe flexibility of CDV with regard to the levels of abstraction used in both the requirements models and the SUT makes it particularly well suited to verification of HRI.\n\nThe principal drawback of CDV, compared with directed testing, is the overhead effort associated with building an automated testbench. Directed testing produces early results, but CDV significantly accelerates the approach towards coverage closure once the testbench is in place. Hence CDV is an appropriate choice for systems in which complete coverage is difficult to achieve due to a broad and varied state space that includes rare but important events, as is typically the case for HRI.\n\n\nWe proposed implementations of four automatic testbench components, the Test Generator, the Driver, the Checker and the Coverage Collector, that suit the HRI domain. Different test generation strategies were considered: pseudorandom, constrained pseudorandom and model-based to complement each other in the quest for meaningful tests and exploration of unexpected behaviours. Assertions were proposed for the Checker, accommodating requirements at different levels of abstraction, an important feature for HRI. Different coverage models were proposed for the Coverage Collector: requirements (assertion), code statements, and cross-product. \n\nThe potential for CDG (Coverage-Driven test Generation), through the implementation of automated feedback loops, has been considered. Nevertheless, we believe a great part of the feedback work needs to be performed by the verification engineer, since CDG is difficult to implement in practice. \n\nA handover example demonstrated the feasibility of implementing the CDV testbench as a ROS-Gazebo based simulator. The results show the relative merits of our proposed testbench components, and indicate how feedback loops in the testbench can be explored to seek coverage closure. Several key observations can be noted from these results. Pseudorandom test generation allows a degree of unpredictability in the environment, so that unexpected behaviours of the SUT may be exposed. Model-based test generation usefully complements this technique by systematically directing tests according to the requirements of the SUT. This requires the development of a formal model of the system, which additionally enables exhaustive verification through formal methods, as explored by previous authors for HRI~\\cite{BFS09:HRIshort,Cowley2011,Kouskoulas2013,Mohammed2010,Muradore2011,webster14formalshort}.\n\nIf the requirements are translated into Temporal Logic properties for model checking, assertion monitors can be derived automatically. In future work, we will be exploring generation of monitors for different levels of abstraction in the simulation (e.g., events-based, or checked at every clock cycle) in a more formal manner. We will further explore the use of bisimulation analysis to ensure equivalence between a robot's high-level control code and any associated formal models. We intend to incorporate probabilistic models of the human, the environment and other elements in the simulator, to enable more varied stimulation of an SUT. We also intend to verify a more comprehensive set of requirements for the handover task, e.g., according to the safety standard ISO~15066 (currently under development) for collaborative industrial robots.\n\n \nOur approach is scalable, as more complex systems can be verified using the same CDV approach, for the actual system's code. We are confident CDV can be used for the verification and validation of autonomous systems in general. Open source platforms and established tools can serve to create simulators and models at different abstraction levels for the same SUT. \n\\subsection{Coverage Collector}\\label{sc:coverage}\nAutomatic test generation necessitates monitoring the quality of the tests to gain an understanding of the verification progress. To achieve this, statistics can be collected on the tests, the driven stimulus (external events), the SUT's response, and the SUT's internal state, including assertion monitors. In general, we distinguish between {\\em code} coverage models and {\\em functional} coverage models. A comprehensive account on coverage can be found in~\\cite{Piziali2004}. \n\nThe collected coverage data provides information on unexplored (coverage ``holes'') or lightly covered areas. {\\em Coverage closure} is the process of identifying coverage holes and creating new tests that address these holes. This introduces a feedback loop from coverage collection\/analysis to test generation, termed Coverage Directed test Generation (CDG)~\\cite{Piziali2004}. Attempts have been made to automate CDG using machine learning techniques~\\cite{ioannides}. However, CDG remains a difficult challenge in practice.\n\nIn principle, coverage collection and analysis techniques can be transferred directly into the domain of robotics verification. In fact, it is interesting to note that functional coverage in the form of ``cross-product'' coverage~\\cite{wile}, as widely used in hardware design verification, has recently been proposed (independently) for the verification of autonomous robots in~\\cite{Alexander2015}, where it is termed {\\em situation} coverage and includes combinations of external events only.\n\n\\subsection{Driver}\\label{sc:driver}\n\nThe Driver is a fully automated component that translates a (potentially high-level) description of a test into signal-level stimulus that can be applied to the interfaces of the SUT in order to expose the SUT to the situation prescribed by the test. The Driver may comprise an interacting network of modules corresponding to the distinct interfaces of the SUT. The SUT reacts to the stimuli provided on its interfaces. The Driver runs in parallel with the SUT and responds to it, if necessary; i.e., the Driver can be reactive. The automation of the Driver makes it feasible to execute batches of abstract tests, to accelerate testing.\n\nIn HRI, the Driver comprises a model of the human, a physics model, and communication channels to represent any interactions that do not require detailed physical simulation. For example, if the human element in the simulator is driving the robot's code, the Driver would execute the corresponding high-level action sequence, one item at a time, by translating it into the respective sequence of input signals, potentially passing through the physics model before exposing the signals to the robot's input channels.\n\n\\section{CDV Implementation}\\label{sc:implementation}\nA case study from a collaborative manufacture scenario is presented. We demonstrate the transferability of CDV into the HRI domain by constructing a CDV testbench for this case study using a combination of established open-source tools and custom components. Our implementation showcases the potential of CDV to verify robotic code used in HRI.\n\n\n\\subsection{Case Study: Robot to Human Object Handover Task}\\label{Example}\nOur case study is an object handover, a critical subtask in a broader scenario where a robot (BERT2~\\cite{lenz2010bert2}) and a person work together to assemble a table. The handover starts with an activation signal from the person to the robot. The robot then picks up an object, and holds it near the person. The robot indicates it is ready for the person to receive the object. Then, the person is expected to hold the object simultaneously, moving closer if necessary, and to look at it --- indicating readiness of the person. The robot collects data through two different sensing systems: ``pressure'', sensors that determine whether just the robot, or simultaneously the robot and the person, are holding the object; and ``location'' and ``gaze'' sensors, an `EgoSphere' system that tracks whether the human hand is close to the object and whether the human head is directed towards the object~\\cite{lenz2010bert2}. Based on the sensors, the robot determines whether the release condition is satisfied, and decides on a course of action: the robot will release the object and allow the person to take it, if the human was ready; if not, the robot will not release the object. The robot or human may disengage from the task (look or move away). The sensors are considered perfect. \n\nAccording to the handover task's interaction protocol, a robot ROS `node' was developed in Python, comprising 209 code statements. This node was structured as a state machine, using the SMACH modules~\\cite{SMACH}, to facilitate modularity. \nThe states, with their transitions, can be enumerated as shown below.  Each state transitions to the next in sequence, except where indicated otherwise. The code is also depicted as a flow chart in Figure~\\ref{codecoverage}. \n\n\\begin{enumerate}\n\\item \\verb|reset| - The robot moves to its starting position, with gripper open.\n\\item \\verb|receive_signal| - Read signals.  If `startRobot' is received, transition to \\verb|move|; elseif timeout, transition to \\verb|done|; else, loop back to present state.\n\\item \\verb|move| - Plan trajectory of hand to piece. Move arm. Close gripper. Plan trajectory of hand to human. Move arm.\n\\item \\verb|send_signal| - Send signal to inform human of handover start.\n\\item \\verb|receive_signal| - Read signals.  If `humanIsReady' is received, transition to \\verb|sense|; elseif timeout, transition to \\verb|done|; else loop back to present state.\n\\item \\verb|sense| - Read sensors. If timeout, transition to \\verb|done|; elseif not all signals available, loop back to present state; else, transition to \\verb|decide|.\n\\item \\verb|decide| - If all sensors are satisfied, transition to \\verb|release|; else, transition to \\verb|done| (without releasing).\n\\item \\verb|release| - Open the gripper.  Wait for 2 seconds.\n\\item \\verb|done| - End of sequence.\n\\end{enumerate}\n\n\n\n\\subsection{Requirements}\\label{ssc:requirements}\nRequirements were derived from ISO~13482:2014 and desired functionality of the robot in the interaction~\\cite{Grigore2011}: \n\n\\begin{enumerate}\n\\item If the gaze, pressure and location are sensed as correct, then the object shall be released. \n\\item If the gaze, pressure or location are sensed as incorrect, then the object shall not be released. \n\\item The robot shall make a decision before a threshold of time. \n\\item The robot shall always either time out, decide to release the object, or decide not to release the object.\n\\item The robot shall not close the gripper when the human is too close.\n\\end{enumerate}\n\nRequirements 1 to 4 refer to sequences of high-level events over time, whereas Requirement 5 refers to a lower-level safety requirement of the continuous state space of the robot in the HRI. Thus, the former can be both targeted with model-based techniques and implemented as assertion monitors, whereas the latter is only suitable for implementation as an assertion monitor. \n\n\\subsection{CDV Testbench Implementation}\nROS is a widely used open-source platform for the design of code for robots in C++ and\/or Python. ROS allows interfacing directly with robots. Gazebo is a robot simulation tool designed for compatibility with ROS, that is able to emulate the physics of our world. Thus, the combination ROS-Gazebo provides a means of developing a robotic simulator, as shown in Figure~\\ref{Simulatorphoto}. \n\n\\begin{figure}[h]\n\\centering\n      \\includegraphics[width=0.4\\textwidth]{bert.png}\n      \\includegraphics[width=0.4\\textwidth, trim= 70mm 40mm 170mm 30mm, clip]{sim3.jpg}\n\\caption{BERT2 robot and a human, and the simulator in ROS-Gazebo}\n\\label{Simulatorphoto}\n\\end{figure}\n\nFigure~\\ref{simstructure} shows the structure of our CDV testbench implementation, incorporating the robot's high-level control code. The Driver incorporates the Gazebo physics simulator and the MoveIt!\\footnote{http:\/\/moveit.ros.org\/} packages for path-planning and inverse kinematics of the robot's motion. The human is embodied as a floating head and hand for simplicity; in future, this representation can be replaced by one that is anatomically accurate. The implementation in ROS ensures that assertion monitors  and coverage collection can access parameters internal to the robot code as well as the external physics model and other interfaces, such as signals. Observability of the external behaviour allows validating the robot's actions. In real life experiments, this is equivalent to observing the robot's physical behaviour to see if its responses are as expected.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{simulator_testbench.pdf}\n\\caption{Testbench and simulator elements in ROS-Gazebo}\n\\label{simstructure}\n\\end{figure}\n\n\n\n\\subsection{Test Generator and Driver}\nTests were generated pseudorandomly, by concatenating randomly selected elements from the set of high-level actions belonging to the handover workflow, forming random action sequences and instantiating relevant parameters. These randomized sequences represent environmental settings that do not necessarily comply with the interaction protocol. Thus, pseudorandom action sequence generation produces stimulus that correspond to unexpected behaviours that were not previously considered in the requirements. Posteriorly, constraints were introduced to bias the pseudorandom generation to obtain tests that do comply with the interaction protocol (e.g., enforcing particular sequences of actions). \n\nThe handover interaction protocol was formalized as a set of six automata, in particular Probabilistic-Timed Automata (PTA) ~\\cite{Hartmanns2009}, comprising the robot, the workflow, the gaze, the location, the pressure, and the sensors. Behaviours of the different elements (e.g., protocol compliant actions to activate the robot through signals) were abstracted in terms of state transitions and variable assignments. The structure of the robot's code guided the abstraction process, and the abstraction was verified via bisimulation analysis~\\cite{CCSbook}. \n\n\nModel-based test templates were obtained from witness traces (examples or counterexamples) produced by model checking the product automaton~\\cite{Nielsen2003}. These witnesses contain combined sequences of states from the different automata. Requirements 1 to 4 (Section~\\ref{ssc:requirements}) were used to derive model-based test templates that would trigger corresponding assertion monitors. We employed UPPAAL\\footnote{http:\/\/www.uppaal.org\/}, a model checker for PTA that produces witnesses automatically. Projections over these traces with respect to the workflow, gaze, location, pressure and sensors automata remove the elements that correspond to the robot's activities, to form a test template. Based on these test templates, tests were generated pseudorandomly. \n\nA test template for our simulator consists of a sequence of high-level actions (workflow) to activate the robot expressed as a state machine. A test comprises, besides the high-level actions, the pseudorandom instantiation of parameters, from well defined sets (e.g., ranges of values for gaze correct or gaze incorrect). An example is shown in Figure~\\ref{exampletest}. The Driver produces responses in the physical model in Gazebo, signals to be communicated to the robot, and sensor readings. \n \n\\begin{figure}[h]\n\\scriptsize\n\\centering\n\\begin{tabular}{r|lllll|lll}\n\\cline{2-6}\n1&&\\verb+sendsignal+&&\\verb+activateRobot+ &&& &\\\\\n2&&\\verb+setparam+ && \\verb+time = 40+& && This \\verb+time+ instantiation produces &\\\\\n3&&\\verb+receivesignal+ && \\verb+informHumanOfHandoverStart+ &&&a waiting time of $40 \\times 0.05$ seconds.&\\\\\n4&&\\verb+sendsignal+ && \\verb+humanIsReady+ &&&&\\\\\n5&&\\verb+setparam+ && \\verb+time = 10+ &&&&\\\\\n6&&\\verb+setparam+ && \\verb+honTask = true+ && &&\\\\\n7&&\\verb+setparam+ && \\verb+hgazeOk = true+ && &Gaze instantiation for \\verb+true+: choosing\toffset,&\\\\\n8&&\\verb+setparam+ && \\verb+hpressureOk = true+ && & distance and angle, from ranges $ \\{[0.1,0.2],$&\\\\\n9&&\\verb+setparam+ && \\verb+hlocationOk = true+ && & $[0.5,0.6],[15,40)\\}$, e.g., $(0.1,0.5,30)$&\\\\ \n\\cline{2-6}\n\\end{tabular}\n\\caption{Example test from a test template, comprising high-level actions and some parameter instantiations (time and gaze)}\n\\label{exampletest}\n\\end{figure}\n\nAn example of a constraint for constrained pseudorandom generation is the enforcement of the sequence of actions in lines 1 to 4 of Figure~\\ref{exampletest}, followed by any other action sequence. This constraint ensures the immediate activation of the robot, when a simulation starts. \n\nAn added benefit from the development of a formal model for test generation is that this allows formal verification through model checking~\\cite{ClarkeMC}. Formal verification can thus complement CDV. However, properties that hold for abstract models must still be verified at the code level. Model checkers for code (e.g., CBMC\\footnote{http:\/\/www.cprover.org\/cbmc\/}, Java PathFinder\\footnote{http:\/\/javapathfinder.sourceforge.net\/}) target runtime bugs in general, such as arrays out of bounds or unbounded loop executions. These are, however, at a different level than the complex functional behaviours we aim to verify. In~\\cite{webster14formalshort}, the runtime detail is abstracted, giving way to high-level behaviour models where functional requirements can be verified with respect to the model only. \n\n\\subsection{Checker}\n\nAssertion monitors were implemented for all the requirements in Section~\\ref{ssc:requirements}. Requirements 1 to 4 were translated first into CTL properties, and then automata-based assertion monitors were generated manually. This process will be automated in the future. For example, Requirement 1 corresponds to the property:\n$$E <> sgazeOk \\wedge spressureOk \\wedge slocationOk \\wedge releasedTrue.$$\n\nThe resulting monitor is triggered when reading a sensors signal indicating the gaze, pressure and location are correct. Then, the automaton transitions when receiving a signal of the object's release. If the latter signal event happens within a time threshold (3 seconds), a \\verb+True+ result is reported. Finally, the automaton returns to the initial state.\n\nRequirement 2 corresponds to the CTL property:\n$$E<> (sgazeNotOk \\vee spressureNotOk \\vee slocationNotOk) \\wedge releasedFalse.$$\n\nThis monitor is triggered when any of the gaze, pressure or location are incorrect in a sensing signal. Then, the automaton transitions to either a \\verb+False+ result when receiving a signal of the object's release, or a \\verb+True+ result if some time has elapsed (2 seconds) and no release signal has been received. Finally, the automaton returns to the initial state. \n\nRequirement 5 refers to physical space details abstracted from our PTA model, and it cannot be expressed as a Temporal Logic property. Hence, it was directly implemented as an automaton-based assertion monitor. When the robot grabs the object, it needs to make sure the human's hand (or any other body part) is at a distance. The monitor is triggered every time the code invokes the \\verb+hand(close)+ function, which causes the motion of the robot's hand joints. The location of the human hand is then read from the Gazebo model (the head is ignored, since the model is abstracted to a head and a hand). If this location is close to the robot's hand (within a 0.05\\,m distance of both mass centres), the monitor registers a \\verb+False+ result, or otherwise \\verb+True+. \n\nThe monitors automatically generate report files, indicating their activation time, and the result of the checks if completed. \n\n\\subsection{Coverage Collector}\nWe implemented two coverage models: code (statement) coverage and functional coverage in the form of requirements (assertion) coverage. The statement coverage was implemented through the `coverage'~\\footnote{http:\/\/nedbatchelder.com\/code\/coverage\/} module for Python. For each test run, statistics on the number of executed code statements are gathered. The assertion coverage is obtained by recording which assertion monitors are triggered by each test. If all the assertions are triggered at the end of the test runs, the testbench has achieved 100\\% requirements coverage. \n\n\n\n\n\\section{Introduction}\\label{sc:introduction}\nHuman-Robot Interaction (HRI) is a rapidly advancing sector within the field of robotics. Robotic assistants that engage in collaborative physical tasks with humans are increasingly being developed for use in industrial and domestic settings. However, for these technologies to translate into commercially viable products, they must be demonstrably safe and functionally sound, and they must be deemed trustworthy by their intended users~\\cite{ROMAN14}. In existing industrial robotics, safety is achieved predominantly by physical separation or through limiting the robot's physical capabilities (e.g., speed, force) to thresholds, according to predefined interaction zones. To fully realize the potential of collaborative robots, the correctness of the software with respect to safety and functional (liveness) requirements needs to be verified. \n\nHRI systems present a substantial challenge for software verification --- the process used to gain confidence in the correctness of an implementation, i.e.\\ the robot's code, with respect to the requirements. {\\em The robot responds to an environment that is multifaceted and highly unpredictable.} This is especially true for robots involved in direct interaction with humans, whether this is in an unstructured home environment or in the more structured setting of collaborative manufacturing. We require a verification methodology that is sufficiently realistic (models the system with sufficient detail) while thoroughly exploring the range of possible outcomes, without exceeding resource constraints. \ns. \nPrior work~\\cite{BFS09:HRIshort,Cowley2011,Kouskoulas2013,Mohammed2010,Muradore2011,webster14formalshort} has explored the use of formal methods to verify HRI. Formal methods can achieve full coverage of a highly abstracted model of the interactions, but are limited in the level of detail that can practically be modelled. {\\em Sensors, motion and actuation in a continuous world present a challenge for models and requirement formulation in formal verification.} Physical experiments or simulation-based testing may be used to achieve greater realism, and to allow a larger set of requirements to be verified over the real robot's code.  However, neither of these can be performed exhaustively in practice. \n\n{\\em Robotic code is typically characterised by a high level of concurrency between the communicating modules} (e.g., nodes and topics used in the Robot Operating System, ROS\\footnote{http:\/\/www.ros.org\/}) that control and monitor the robot's sensors and actuators, and its decision making. Parallels can be drawn here to the design of microelectronics hardware, which consists of many interacting functional blocks, all active at the same time. Hence it is natural to ask: `Can techniques from the microelectronics field be employed to achieve comprehensive verification of HRI systems?' \n\nIn this paper, we present the use of Coverage-Driven Verification (CDV) for the high-level control code of robotic assistants, in simulation-based testing. CDV is widely used in functional verification of hardware designs, and its adoption in the HRI domain is an innovative response to the challenge of verifying code for robotic assistants. CDV is a systematic approach that promotes achieving coverage closure efficiently, i.e.\\ generation of effective tests to explore a System Under Test (SUT), efficient coverage data collection, and consequently efficient verification of the SUT with respect to the requirements. The resulting efficiency is critical in our application, given the challenge of achieving comprehensive verification with limited resources. \n\n{\\em The extension of CDV to HRI requires the development of practical tools that are compatible with established robotics tools and methods.} The microelectronics industry benefits from the availability of hardware description languages, which streamline the application of systematic V\\&V techniques. No practical verification tool exists for Python or C++, common languages for robotics code~\\cite{Trojanek2014}. A novel contribution of this paper is the development of a CDV testbench specifically for HRI; this implementation makes use of established open-source tools where possible, while custom tools have been created as necessary to complete and connect the testbench components (Test Generator, Driver, Checker and Coverage Collector). Additionally, we outline the relevant background to ensure robust implementation of CDV.\n\nTo demonstrate the feasibility and potential benefits of the method, we applied CDV to an object-handover task, a critical component of a cooperative manufacture scenario, implemented as a ROS and Gazebo\\footnote{http:\/\/gazebosim.org\/} based simulator. Our automated testbench conveniently allows the actual robot code to be used in the simulation. Model-based and constrained pseudorandom test generation strategies form the Test Generator. A Driver applies the tests to the simulation components. The Checker comprises assertion monitors, collecting requirement coverage. The Coverage Collector, besides requirement, includes code coverage. \n\nWe verified selected safety and liveness (functional) requirements of the handover task to showcase the potential of CDV in the HRI domain.\n\nThe paper proceeds with an overview of the CDV testbench components and verification methodology in Section~\\ref{sc:CDV}. The handover scenario is introduced in Section~\\ref{sc:implementation}, where we then present the CDV testbench we used to verify the code that implements the robot's part of the handover task. Section~\\ref{sc:results} discusses the verification and coverage results for this example. Conclusions and future work are given in Section~\\ref{sc:conclusions}.\n\n\n\\section{Experiments and Verification Results}\\label{sc:results} \nThe CDV testbench described in Section~\\ref{sc:implementation} was used \n\\textit{(a)} to demonstrate the benefits of CDV in the context of HRI; \n\\textit{(b)} to obtain an insight into the verification results, including unexpected behaviours or requirement violations; and \n\\textit{(c)} to explore options to achieve coverage closure (from Section~\\ref{sc:cdvmethod}).\n\nThe requirements mentioned in Section~\\ref{Example} were verified using a CDV testbench in ROS (version Indigo) and Gazebo (2.2.5), and through model checking in UPPAAL (version 4.0.14), using the model we developed for model-based test generation. We used a PC with Intel i5-3230M 2.60\\,GHz CPU, 8\\,GB of RAM, running Ubuntu 14.04.\n\nTable~\\ref{results1} presents the assertion coverage for the handover, and the verification results from model checking. In model checking, the requirements were verified as true (T) or false (F). Through model checking, we were only able to cover Requirements 1 to 4. From each of the model checking witnesses (test templates) of Requirements 1 to 4, we generated a test (model-based generation). We also generated 100 pseudorandom (unconstrained) tests, and 100 constrained pseudorandom tests that enforced the activation of the robot as explained in Section~\\ref{sc:implementation}. We verified Requirements 1 to 5 in simulation, and recorded the results of the assertion monitors: Pass (P), Fail (F), Not Triggered (NT), or Inconclusive (U) when the monitor was triggered but the check was not completed within the simulation run. The same setup was used to compute both assertion and statement coverage, allowing the comparison of the test generation strategies in terms of coverage efficiency. \n\n\\begin{table}\n\\caption{Requirements (assertion) coverage and model checking results}\n\\centering\n\\scriptsize\n\\renewcommand{\\arraystretch}{1.2}\n\\begin{tabular}{|c|c|cccc|cccc|cccc|}\n\\hline \nReq. & Model\t& \\multicolumn{12}{|c|}{Simulation-Based Testing} \\\\ \\cline{3-14}\n\t& Checking&\\multicolumn{4}{|c|}{Pseudorandom} & \\multicolumn{4}{|c|}{Constrained-Pseudorandom} &\\multicolumn{4}{|c|}{Model-Based}  \\\\\n \\cline{3-14}\n\t\t\t&  \t\t\t\t& P & F & NT & I\t\t & P & F & NT & I \t\t\t   & P & F & NT & I\\\\\n\\hline\n1 \t\t& T \t& 0\/100  &0\/100 & 100\/100 & 0\/100\t\t& 0\/100  & 0\/100 & 100\/100 & 0\/100 \t&  3\/4 & 0\/4 & 1\/4 & 0\/4\\\\\n2 \t\t& T \t& 33\/100 & 0\/100 & 67\/100  & 0\/100\t\t& 87\/100 & 0\/100 & 13\/100 & 0\/100\t&  1\/4 & 0\/4 & 3\/4 & 0\/4\\\\\n3 \t\t& T \t& 33\/100 & 0\t\/100 & 67\/100 & 0\/100\t\t& 87\/100 & 0\/100 & 13\/100 & 0\/100\t&  4\/4 & 0\/4 & 0\/4 & 0\/4 \\\\\n4 \t\t& T \t& 98\/100 & 0\/100 & 0\/100  & 2\/100\t\t& 98\/100 & 0\/100 & 0\/100 & 2\/100\t\t&  4\/4 & 0\/4 & 0\/4 & 0\/4\\\\ \\hline\n5\t\t& - \t& 46\/100 & 0\/100 & 54\/100 & 0\/100 \t\t& 93\/100 & 0\/100 & 7\/100 & 0\/100 \t&  4\/4 & 0\/4 & 0\/4 & 0\/4\\\\\n\\hline\n\\end{tabular} \\label{results1}\n\\end{table}\n\nThe results in Table~\\ref{results1} confirm our expectations for the different test generation strategies. For assertion-based functional coverage, pseudorandom and constrained-pseudorandom test generation are less efficient than model-based test generation, which triggered all five assertions with just four tests. Requirement 1 was not covered by either the pseudorandom or the constrained pseudorandom strategy. If either of these strategies was used alone, the coverage hole could potentially be closed by adding further constraints or by using a more sophisticated test generation strategy such as model-based test generation.\n\nThe assertion monitor checks for Requirement 4 were inconclusive for some of the pseudorandom and constrained-pseudorandom generated tests. This occurs because in these tests the robot is activated long after the start of the handover task (when the robot is reset and proceeds to wait for a signal). These tests do not comply with the protocol which requires to activate the robot at the start and within a given time threshold. \n\nThis coverage result could trigger different actions, e.g.\\ the assertion monitor could be modified to choose either pass or fail at the end of the simulation; the Driver could be modified such that the simulation duration is extended; or, the inconclusive checks could be dismissed as trivial, in which case the efficiency of any further tests could be improved by directing them away from such cases. As noted in Section~\\ref{sc:cdvmethod}, further test generation is not always the sole appropriate response to a coverage hole. It is worth noting that this scenario was exposed only by pseudorandom and constrained-pseudorandom test generation, demonstrating the unique benefit of these approaches; by exploring the SUT's behaviour beyond the assumptions of the verification engineer, they provide a useful complement to the more directed approach of model-based test generation. \n\nFigure~\\ref{codecoverage} illustrates the code coverage (statements) achieved with each test generation strategy over 206 statements (the actual percentages may vary $\\pm 2$\\% due to decision branches with 1 or 2 lines of code each). The lines of code are grouped using the state machine structure in the Python module, to facilitate visualization. The block of code corresponding to the ``release'' state is not covered by the pseudorandom and constrained pseudorandom generated tests, hence it is shown in white. This coverage hole could be closed by applying the test template produced by model-based test generation for Requirement 1. \n\nBecause our code is structured as a finite state machine (FSM), it would be appropriate to also incorporate structural coverage models in the future.  A comprehensive test suite would include tests that visit all states, trigger all possible state transitions, and traverse all paths.\n\n\n\\begin{figure}[h]\n \\subfloat[\\label{subfig-3:a}]{\n      \\includegraphics[width=0.27\\textwidth]{coverage3.pdf}\n    }\n    \\hfill\n    \\subfloat[\\label{subfig-3:b}]{\n      \\includegraphics[width=0.36\\textwidth]{coverage2.pdf}\n    } \n    \\hfill\n    \\subfloat[\\label{subfig-3:c}]{\n      \\includegraphics[width=0.27\\textwidth]{coverage1.pdf}\n    }\n\\caption{Code coverage (percent values) obtained in simulation with (a) model-based (4 tests), (b) pseudorandom (100 tests), and (c) constrained-pseudorandom test generation (100 tests)\n}\n\\label{codecoverage}\n\\end{figure}\n\n\nThe generation of effective tests, that target both the exploration of the SUT and the verification progress, is fundamental to maximising the efficiency of a CDV testbench reaching for coverage closure. From the overall results, it can be seen that the three test generation approaches applied have complementary strengths that overcome their respective weaknesses in terms of coverage. While model-based test generation ensures that the requirements are covered in an efficient manner, pseudorandom test generation can construct scenarios that the verification engineer has not foreseen. Such cases are useful for exposing flawed or missing assumptions in the design of the testbench or the requirements.\n\n\n\n\n\n\\subsection{Test Generator}\\label{sc:testgen}\n\nThe test generator aims to exercise the SUT for verification (activation of faults), while working towards full coverage. Test generators in CDV make use of pseudorandom generation techniques. Using pseudorandom as opposed to random generation allows repeatability of the tests. The generated tests must be valid (realistic, like a sensor input that reflects a valid scene). An effective set of tests includes a good variety that explores unexpected conditions and addresses the scenarios of interest as per the requirements list. An efficient set of tests maximises the coverage and verification progress, whilst minimizing the number of tests needed. To achieve the former while allowing for the latter, pseudorandom test generation can be biased using constraints.  These constraints can be derived from the SUT's functional requirements or from the verification plan~\\cite{Piziali2004}. However, supplying effective constraints requires significant engineering skill and application knowledge. It is particularly difficult to generate meaningful sequences of actions, whether these are transactions on the interface of a system-on-chip, or interactions between humans and robots.\n \nConstrained pseudorandom test generation can be complemented with model-based techniques~\\cite{Haedicke2012,Lakhotia2009} to generate sequences that address specific use cases, such as interaction protocols between human and robot in a collaborative manufacturing environment. In model-based test generation, a model is explored or traversed to obtain abstract tests, i.e.\\ tests at the same level of abstraction as the model. These abstract tests can serve as test templates, or constraints, for tests that target specific behaviours~\\cite{Lackner2012,Nielsen2003}. For this, a model needs to be implemented, e.g.\\ one that captures the intended behaviours of the robot when interacting with humans and\/or its environment. In robotics, the degree of abstraction between such a model and the simulation often differs significantly compared to that observed in microelectronics~\\cite{Nielsen2014}. Many low-level implementation details such as motion control, sensing models or path planning are abstracted from (e.g., as in~\\cite{webster14formalshort}) to keep these models within manageable size.  For model-based testing to be credible and effective, the correctness of the behavioural model with respect to the robot's code needs to be established. However, this is beyond the scope of this paper.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:intro}Introduction \nNano- to meso-scale simulations have seen abundant applications, especially in recent years, for porous media flows such as shale gas transport, groundwater flows, fuel cells, and metal-air batteries.~\\cite{Wang2016a} These applications typically involve multi-phase, multi-species, and reactive transport with the reactions commonly occurring only at the fluid\/solid-boundary interface. While in macroscopic flows, the solutions obtained from the classical Navier-Stokes equations are accurate, there is a degree of departure from their predictions in microscopic flows. This departure is characterized by the Knudsen number ($\\text{Kn} = \\lambda \/ L $, where $\\lambda$ is the mean free path of the fluid and $L$ is the characteristic dimension), which categorizes a flow as: (a) Continuum flow -- where Kn is assumed to be vanishingly small, typically $\\text{Kn} \\lesssim 10^{-3}$, (b) Slip flow ($10^{-3} \\lesssim \\text{Kn} \\lesssim 0.1$), (c) Transition flow ($0.1 \\lesssim \\text{Kn} \\lesssim 10$), and (d) Ballistic or free-molecular flow ($\\text{Kn} \\gtrsim 10$).\n In the slip flow regime, the departure from the continuum flow occurs only at the boundaries where the no-slip boundary condition is applied. Hence, this can be corrected using a slip boundary condition which allows for a corresponding velocity to be applied based on a theory such as the Maxwell-, Navier-, or higher-order slip  models.~\\cite{Wu2008sl,Fukui1990,Mitsuya1993,Hsia1983} However, this correction is not applicable for high Kn flows (i.e., in the transition-flow regime and beyond) as the non-linearity is observed in the bulk-flow and is not isolated to the boundary. This limits the use of conventional CFD, even with corrected boundary conditions in the applications mentioned above.\n\nAs a solution to this problem, the Lattice Boltzmann Method (LBM) has gained popularity because of its roots in the Boltzmann Transport Equation (BTE), although it was historically derived from the (failed) Lattice Gas Automata (LGA).~\\cite{succi2001lattice} While there was some initial argument that LBM is limited only to continuum flows, and that any noticeable departure is an artifact of discretization, it has been proven that the LBM matches the analytical solutions of the BTE, molecular dynamics simulations, and Direct Simulation Monte-Carlo methods.~\\cite{Ansumali2007,Kim2008} The LBM framework constitutes the splitting of the Boltzmann equation into two steps -- collision (which is local to each node), and streaming (which involves moving the value corresponding to a node to its neighbor along the direction of propagation). The simplicity of the framework makes the implementation of this method straightforward while keeping the computation costs low. Traditional LBM has a few disadvantages as the grid is constrained to follow the velocity set used to discretize the momentum-phase of the Boltzmann equation, and the grid spacing is tied to the time step as the streaming step involves a direct translation of values from one node to its neighbors. This induces uniform grids which may not conform to the local geometry. In addition, the framework requires a CFL number of unity, by construction. Furthermore, the coupling between the momentum and position space, as imposed by the LBM discretization, is not mandatory in capturing the correct flow-field dynamics.~\\cite{abe1997d,He1997a,Cao1997f}\n\nTo obtain a grid-independent variant of the LBM, several methods have been proposed that involve applying the Eulerian framework to the Discrete Boltzmann Equation (DBE). These approaches decouple the momentum phase (the discrete velocities) and the position space (spatial grid) using Finite Difference (FD), Finite Volume (FV), and Finite Element (FE) methods for spatial discretization. \n\nThe FD discretization of the DBE acts as a generalization of LBM but maintains the decoupling of the position and momentum space, allowing the scheme to utilize a non-unity CFL number.\\cite{Chen1998} As with most FD schemes, the accurate treatment of curved boundaries is complicated in FD-LBM schemes, and boundary conditions may introduce errors in the evaluation of slip-velocity and mass conservation. High-order FD schemes have been developed for Cartesian and curvilinear grids, such as the 4th-order scheme developed by Hejranfar et al.~\\cite{Hejranfar2014a,Hejranfar2014b} However, these schemes typically involve the use of filters to ensure numerical stability, leading to the convergence depending on the filtering coefficients.\n\nThe FV formulation of the DBE was first proposed by Amati et al.,~\\cite{Amati1997} where the volume-averaged values of the PDFs are obtained from a piecewise linear interpolation on a nonuniform coarse grid. Further developments of FV-LBM have been presented that improve the approximation of conversation laws and allow different element shapes to enable geometric flexibility at the boundaries.~\\cite{peng1998lattice,peng1999finite,xi1999finite} Recently, Chen et al. have developed implementations of FV-LBM with a cell-centered approach that improves the accuracy of obtained flow features in complex flows and along curved boundaries.~\\cite{Chen2018,Chen2019,Chen2015}\nIn some cases, stability of high-order FV-solvers requires the use of Essentially Non Oscillatory ENO~\\cite{Harten1986some} and Weighted Essentially Non Oscillatory WENO~\\cite{balsara2000monotonicity} schemes. However, the high-order FV formulations require large stencils. These stencils link elements from disparate parts of the mesh, hindering the geometric flexibility of the schemes near the boundary due to the necessity of flux reconstruction. This may also lead to a reduction in the order of accuracy of the formulation near boundaries. \n\nFE schemes can be considered as a promising alternative to high-order FV schemes, as they possess a more compact stencil. A variety of FE schemes have been proposed to obtain solutions to the DBE. Krivovichev~\\cite{Krivovichev2014} and Jo et al.~\\cite{jo2009finite} demonstrated the use of Continuous Galerkin FEM (CG-FEM) with equilibrium boundary conditions for viscous flows. However, due to the non-self-adjoint nature of the BTE, these schemes are often susceptible to spurious oscillations.~\\cite{jiang1998least} To resolve this problem, Lee et al.~\\cite{Lee2001} proposed a characteristic Galerkin approach involving a second-order accurate predictor-corrector step, whereas Li et al.~\\cite{li2004least} suggested the use of a least squares scheme with fourth-order accuracy in space and second-order accuracy in time. Discontinuous Galerkin (DG) solvers are generally preferred for discretizing the DBE as they allow for easier parallelization (due to their element-by-element computation) and are better-suited for advection-dominated equations such as the DBE. Shi et al.~\\cite{Shi2003} and D\\\"uster et al.~\\cite{Duster2006} have shown that the DG formulation is an efficient solver in obtaining high-order numerical solutions to the DBE. While only a first-order, forward Euler time stepping method was utilized by these researchers, the scheme allows coupling to higher-order time integrators. To further improve the computational efficiency, Min et al.~\\cite{Min2011} proposed the decoupling of the DBE into collision and streaming steps, similar to the LBM. Here, the DBE is integrated first using trapezoidal rule and following a transformation of the distribution function, the solution is obtained in two stages with a local collision step and a streaming step, the latter which is treated as an advection-only equation solved using DG-FEM. The Eulerian treatment of the streaming step enables the scheme to be grid-independent. This enables a trivially-diagonalizable mass matrix facilitating efficient computation even with low relaxation times at high CFL numbers. This approach has been used frequently in studying flows through\/past cylinders and porous media.~\\cite{Wardle2013,Wu2018A,Zadehgol2014} Although high-order temporal integrators are typically used for solving the streaming step, the scheme retains the native second-order accuracy in time due to the use of the trapezoidal rule, or equivalently, a second-order time accurate expansion using Strang splitting. Therefore, for higher temporal accuracy, the space-time coupled DBE is required to be solved with the corresponding higher-order time-integrator.~\\cite{Shao2018} Recently, other modifications to DG-DBE schemes have been proposed in order to improve the numerical efficiency.~\\cite{Coulette2018,Karakus2019} \n\nThe aforementioned FD, FV, and FE schemes share the ability to operate with non-unity CFL numbers, and (for the FV and FE schemes) to operate on unstructured grids. However, these schemes have only been applied to continuum flows in conjunction with small sets of discrete velocity directions. To capture non-continuum effects, Jaiswal et al.~have developed DG-based solvers for the BTE with Fourier-transform-based discretizations of velocity space capable of handling the full Boltzmann collision operator.~\\cite{jaiswal2019discontinuous} Of course, this comes at a significant computational cost due to the complexity of the associated integrals. To decrease the cost of these schemes, Guo et al.~\\cite{Guo2013} developed a FV Discrete Unified Gas Kinetic Scheme, using the BGK-collision operator and velocity discretizations similar to the LBM. Theoretically speaking, this scheme is capable of generating accurate solutions at all Knudsen numbers. However, with higher Knudsen numbers and non-linearity in the flow field, a larger velocity set is still required, increasing the computational cost. In addition, although there is no decoupling of the collision and streaming steps, the use of trapezoidal rule in time integration limits the scheme to second-order accuracy in time. \n\nIn this manuscript, we present a fully implicit DG-DBE method implemented for high-Kn flows with high-order accuracy in both time and space. The paper is structured as follows. In section \\ref{sec:methods}, we present the foundation of the DBE and the corresponding discretization using DG-FEM, along with the associated velocity sets and boundary conditions. In section \\ref{sec:results}, we show the high-order accuracy of the scheme for high-Kn Couette flow, and we examine its performance in conjunction with various velocity sets. Thereafter, we apply the scheme to a highly non-linear high-Kn lid-driven micro-cavity flow, to showcase the flexibility of the method for various flow regimes. Finally, some concluding remarks are provided in section~\\ref{sec:conclude}.\n\n\n\\section{\\label{sec:methods}Methods }\n\\subsection{The Discrete Boltzmann Equation}\n\\subsubsection{The BGK-Boltzmann equation}\nLet us begin with the isothermal body-force-free BTE %\n\\begin{equation}\n\\label{eqn:BTE}\n    \\frac{\\partial f}{\\partial t} + \\boldsymbol{\\xi}\\cdot\\nabla f = \\boldsymbol{\\Omega} \\equiv -\\frac{1}{\\tau}\\left(f - f^{eq} \\right),\n\\end{equation}\nwhich is a 6+1-dimensional equation with three dimensions each in space and velocity, and one dimension in time, forming the full phase space for the system. Here, \\(f = f(\\boldsymbol{x},\\boldsymbol{\\xi},t)\\) is the particle distribution function (also called the density distribution function) for particles traveling with a velocity \\(\\boldsymbol{\\xi}\\) at time \\(t\\) and position \\(\\boldsymbol{x}\\). \\(\\boldsymbol{\\Omega}\\) is the particle collision operator, typically truncated to two-body collisions. The formulation given for $\\boldsymbol{\\Omega}$ in eqn.~\\ref{eqn:BTE} is the simplification proposed by Bhatnagar-Gross-Krook (the so-called BGK collision operator) to make the BTE solvable for near-equilibrium non-trivial flows. In this operator, $\\tau$ is the relaxation time and $f^{eq}$ is the Maxwellian equilibrium distribution function \n\\begin{equation*}\n   \n    f^{eq} = \\frac{\\rho}{(2 \\pi RT)^{d\/2}} \\exp\\left( -\\frac{|\\boldsymbol{\\xi} - \\boldsymbol{u}|^2}{2RT}\\right),\n\\end{equation*}\nwith density $\\rho$, gas constant $R$, temperature $T$, number of spatial dimensions $d$, and bulk fluid velocity $\\boldsymbol{u}$. The macroscopic variables for isothermal flows are then defined by \n\\begin{align}\n    \\nonumber \\rho &= \\int f d\\boldsymbol{\\xi},\\\\\n    \\rho \\boldsymbol{u} &= \\int f \\boldsymbol{\\xi} d\\boldsymbol{\\xi}.  \\label{eqn:IntMacro}\n\\end{align}\n\\subsubsection{Discretization of the velocity space}\nThe discretization of the velocity space, i.e. $\\boldsymbol{\\xi} \\rightarrow \\cup_{i=0}^{N} \\boldsymbol{e}_i$, dictates the accuracy of the DBE as a reduced order model of the 7-dimensional BTE. For this purpose, typically Gauss-Hermite (GH) quadrature is used in conjunction with the moment expansion method proposed by Shan et al. \\cite{Shan1997} and He et al. \\cite{He1997APriori}  The procedure is briefly described below.\n\nThe distribution function is first expanded using the orthonormal Hermite polynomials in $\\boldsymbol{\\xi}$ associated with the rank-\\emph{m} tensor $\\mathcal{H}^{(m)}$ and weight function $\\omega$ as follows\n\\begin{align}\n    \\label{eqn:Herm1}\n    f(\\boldsymbol{x},\\boldsymbol{\\xi},t) = \\omega(\\boldsymbol{\\xi})\\mathlarger{\\mathlarger{ \\sum}_{m=0}^\\infty}\\frac{1}{m!}\\boldsymbol{a}_{\\boldsymbol{i}}^{(m)}(\\boldsymbol{x},t)\\mathcal{H}_{\\boldsymbol{i}}^{(m)}(\\boldsymbol{\\xi}),\n\\end{align}\nwhere\n\\begin{align*}\n   \n    \\boldsymbol{a}_{\\boldsymbol{i}}^{(m)}(\\boldsymbol{x},t) = \\int f(\\boldsymbol{x},\\boldsymbol{\\xi},t) \\mathcal{H}_{\\boldsymbol{i}}^{(m)}(\\boldsymbol{\\xi}) \\, d\\boldsymbol{\\xi}.\n\\end{align*}\nHere, the index $\\boldsymbol{i}$ refers to the $m$-fold indices $i_1i_2...i_m$. The summation in eqn.~\\ref{eqn:Herm1} can be truncated to $M$th order while retaining the first $M$ moments due to the orthonormality of the Hermite polynomials such that\n\\begin{align}\n    \\label{eqn:Herm3a}\n     f(\\boldsymbol{x},\\boldsymbol{\\xi},t) \\approx f^{M}(\\boldsymbol{x},\\boldsymbol{\\xi},t) = \\omega(\\boldsymbol{\\xi})\\mathlarger{\\mathlarger{ \\sum}_{m=0}^{M}}\\frac{1}{m!}\\boldsymbol{a}_{\\boldsymbol{i}}^{(m)}(\\boldsymbol{x},t)\\mathcal{H}_{\\boldsymbol{i}}^{(m)}(\\boldsymbol{\\xi}).\n\\end{align}\nThe first few Hermite polynomials are \n\\begin{align*}\n   \n    \\mathcal{H}^{(0)}(\\boldsymbol{\\xi}) &= 1  \\\\\n    \\mathcal{H}^{(1)}_{i_1}(\\boldsymbol{\\xi}) &= \\xi_{i_1}\\\\\n    \\mathcal{H}^{(2)}_{i_1i_2}(\\boldsymbol{\\xi}) &= \\xi_{i_1} \\xi_{i_2} -\\delta_{i_1i_2},\n\\end{align*}\nyielding\n\\begin{align*}\n    \\boldsymbol{a}^{(0)} &= \\rho,\\\\\n    \\boldsymbol{a}^{(1)} &= \\rho \\boldsymbol{u},\\\\\n    \\vdots\n\\end{align*}\nWe can discretize the velocity and expand $f^{eq}$ up to 2nd order as follows  \n\\begin{align}\n    \\label{eqn:feq}\n    f^{eq}_i &= \\omega_i \\rho \\left[1+\\frac{\\boldsymbol{e}_i\\cdot\\boldsymbol{u}}{RT} + \\frac{\\left(\\boldsymbol{e}_i\\cdot\\boldsymbol{u}\\right)^2}{2(RT)^2} - \\frac{\\boldsymbol{u}^2}{2RT} \\right],\n\\end{align}\nwhere $\\boldsymbol{e}_i$ is the discretization of velocity $\\boldsymbol{\\xi}$ in the $i^{th}$ direction and  $\\omega_i$ is the corresponding Gauss-Hermite weight.\n\nThis helps simplify the BTE and reduce it to 3+1 dimensions, allowing us to obtain the DBE with BGK collision operator\n\\begin{equation}\n    \\label{eqn:DBE}\n    \\frac{\\partial f_i}{\\partial t} + \\boldsymbol{e}_i\\cdot\\nabla f_i =  -\\frac{1}{\\tau}\\left(f_i - f^{eq}_i \\right).\n\\end{equation}\nThe above equation is valid only for low Mach number isothermal flows due to the functional form of $f^{eq}$ in eqn.~\\ref{eqn:feq}, with errors of the order $\\mathcal{O}(Ma^2)$. While this functional form is sufficient for the flows considered in this article, higher-order expansions including the dependency on temperature can be incorporated to improve its accuracy.\\cite{Shan2006}\n\nThe macroscopic flow variables in eqn.~\\ref{eqn:IntMacro} can be obtained from the discrete distribution functions for $N$-point discretization of $\\boldsymbol{\\xi}$, as follows\n\\begin{align*}\n    \\rho &= \\Sigma_i f_i,\\\\\n    \\rho\\boldsymbol{u} &= \\Sigma_i \\boldsymbol{e}_i f_i.\n\\end{align*}\n\n\n\\subsubsection{Choice of discrete velocities.}\nIn two-dimensional flows, the D2Q9 velocity set is obtained from the tensor product of 1D 3-point GH quadrature rules of order 5. Here, the discrete velocities $\\boldsymbol{e}_i$ are given by\n\\begin{align*}\n   \n    e_x &= \\sqrt{3}\\{0,1,0,-1,0,1,-1,-1,1\\},\\\\\n    e_y &= \\sqrt{3}\\{0,0,1,0,-1,1,1,-1,-1\\},\n\\end{align*}\nwith corresponding weights \n\\begin{align*}\n    w &= (1\/36)\\{16,4,4,4,4,1,1,1,1\\},\n\\end{align*}\nand with lattice speed of sound $c_s=1$. This quadrature set allows the DBE formulation to capture hydrodynamics up to the same fidelity as the Navier-Stokes equations. Higher-order quadrature can be used to obtain hydrodynamics beyond the Navier-Stokes limit. Table \\ref{tab:quadratures} shows a few higher-order GH quadratures in one dimension.\\cite{Kim2008} The corresponding 2D quadratures can be obtained as tensor products of the 1D quadratures.  \n\\begin{table*}[t]\n\\caption{One-dimensional quadratures of various orders\\label{tab:quadratures}}\n\\centering\n\\begin{tabular}{|p{3cm}|p{1.4cm}||p{5.6cm}|p{5.6cm}|}\n\n\n \\hline\n Quadrature & Order & velocities & weights\\\\\n \\hline\\hline\n\n D1Q4   &   7    & $\\pm \\sqrt{3-\\sqrt{6}} $& $(3+\\sqrt{6})\/12$ \\\\\n \\ & \\ & $\\pm \\sqrt{3+\\sqrt{6}}$ & $(3-\\sqrt{6})\/12$ \\\\\n D1Q5   &   9    &  $0$ &  $8\/15$ \\\\\n \\  & \\  &  $\\pm \\sqrt{5-\\sqrt{10}}$ & $(7+2\\sqrt{10})\/60$ \\\\\n \\  & \\  &  $\\pm \\sqrt{5+\\sqrt{10}}$ & $(7-2\\sqrt{10})\/60$ \\\\\n D1Q6  &   11   &  $\\pm 0.616706590193136$ & $4.088284695558080 \\times 10^{-1}$ \\\\\n \\ & \\ & $\\pm 1.88917587775414$ & $8.861574604199542 \\times 10^{-2}$ \\\\\n \\ & \\ & $\\pm 3.32425743355142$ & $2.555784402056898 \\times 10^{-3}$ \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\nWhile high-order quadratures are usually beneficial in capturing the non-linearity in the solution distribution in high Knudsen number flows, GH quadrature provides depreciating benefits due to the quadrature points far from the centroid having very small weights.~\\cite{Guo2013} To alleviate this issue, Newton-Cotes (NC) quadrature can be used to obtain the discrete velocities. \n\\subsubsection{Relaxation time and influence of Knudsen number} \nThe dependency on Kn is introduced in the above formulation via the relaxation time $\\tau$. Upon defining the mean free path $\\lambda = \\sqrt{3}\\tau c_s$ and recalling that $\\text{Kn} = \\lambda\/L$, we get\n\\begin{align}\n    \\label{eqn:KnTau}\n    \\tau = \\text{Kn} L\/\\sqrt{3}c_s.\n\\end{align}\nThis is consistent with the standard definition of $\\text{Kn} = \\sqrt{3\/2}\\alpha^{-1}$ with $\\alpha = \\frac{L}{\\tau}\\sqrt{2k_BT}$, where $k_B$ is the Boltzmann constant. Note: this is the usual definition of Kn for Direct Simulation Monte Carlo (DSMC) methods.\\cite{Ansumali2007} In addition, eqn.~\\ref{eqn:KnTau} can be reparameterized depending on the expression chosen for $\\lambda$. Kim and Pitsch \\cite{Kim2008} use $\\lambda$ derived from a first principles understanding of Kinetic Theory and arrive at an additional factor of $\\sqrt{\\pi\/6}$ in the expression for Kn. Further corrections can be introduced by scaling $\\tau$ with a Kn-dependent function such as $\\psi(\\text{Kn}) = \\frac{2}{\\pi}\\arctan(\\sqrt{2} \\text{Kn}^{-3\/4})$.\\cite{Zhang2014} \n\nTo enable comparisons to the work presented by Ansumali et al,\\cite{Ansumali2007} we shall use the definition in eqn.~\\ref{eqn:KnTau} in the remainder of this article. \n\n\\subsubsection{Boundary conditions for wall-fluid interactions}\nThe most common wall boundary condition in the LBM, the so-called bounce-back scheme and its variants, is designed to enforce the no-slip boundary condition.\\cite{ladd1994numerical,sukop2006dt,Krueger2003} The absence of specular reflection (i.e., bounce-forward) prevents the capturing of relative motion between the fluid and the wall leading to a slip velocity.\\cite{Krueger2003} Conversely, the bounce-forward-only scheme reproduces elastic collisions with an ideal wall by enforcing angle of reflection to equal angle of incidence of the incoming distribution function at a boundary node. This scheme enables uninhibited slip at the wall. Based on these ideas, a variety of methods have been proposed to accurately capture the wall-fluid interaction in the slip-flow and transition regimes. \\cite{Wang2016a}\n\n\nScattering-type boundary conditions enable a convenient middle-ground between uninhibited-slip and no-slip boundary conditions. To account for the non-elasticity of collisions and a rough wall, the diffuse-scattering (also known as the Maxwellian diffuse-reflection) boundary condition is typically used. Here, it is assumed that the fluid particles undergoing collision with the wall scatter back following the Maxwell distribution function, losing the memory associated with their movement prior to the collision. While in LBM, this can be implemented in different ways \\cite{Sofonea2005}, we shall follow the method proposed in references \\cite{Ansumali2007,Kim2008,Shi2011} as stated below. \n\nFor the distribution function $f_i$ satisfying the criteria $(\\boldsymbol{e}_i - \\boldsymbol{u_b}) \\cdot \\boldsymbol{n} < 0$, where $\\boldsymbol{u_b}$ is the true wall velocity and $\\boldsymbol{n}$ is the \\emph{outward} pointing normal, the diffuse-scattering kernel yields \\\\\n\\begin{align}\n        \\label{eqn:FullBC}\n        f_i(\\boldsymbol{x_b},t) = \\frac{\\Sigma_{(\\boldsymbol{e}_j-\\boldsymbol{u_b})>0} \\lvert(\\boldsymbol{e}_j-\\boldsymbol{u_b}) \\cdot \\boldsymbol{n} \\rvert  f_j}{\\Sigma_{(\\boldsymbol{e}_k-\\boldsymbol{u_b})<0} \\lvert(\\boldsymbol{e}_k-\\boldsymbol{u_b}) \\cdot \\boldsymbol{n} \\rvert f_k^{eq}} f_i^{eq}(\\rho_b,\\boldsymbol{u}_b),\n\\end{align}\nwhich reduces to\n\\begin{align}\n        \\label{eqn:UsedBC}\n        f_i(\\boldsymbol{x_b},t) = f_i^{eq}(\\rho_b,\\boldsymbol{u_b}),\n\\end{align}\nfor steady unidirectional flows.\\cite{Kim2008} \n\nDepending on the physics required to be described in the problem, further improvements to the diffuse-scattering kernel can be incorporated. The Maxwell-type second-order slip model estimates the slip velocity as a function of Kn, first- and second-derivatives of fluid velocity normal to the wall, and parameterized slip coefficients.\\cite{Wang2016a} The values of the coefficients are obtained by other models, such as a micro-scale molecular dynamics simulation.\\cite{Chibbaro2008} The Langmuir slip boundary condition resolves this issue by estimating the slip velocity as a function of the wall velocity, fluid velocity adjacent to the wall, and physical coefficients of the gas particles for a given interaction potential.\\cite{eu1987nonlinear,abe1997d} Nevertheless, in this article we shall only consider the simplified diffuse-scattering boundary condition described in eqn.~\\ref{eqn:UsedBC}, as it has been shown to be in good agreement with the DSMC solutions of the BTE for the flows considered here.\\cite{Ansumali2007,Guo2013}\n\n\n\\subsection{Discontinuous Galerkin Finite Element Method}\n\n\\subsubsection{Weak formulation of DG-FEM}\nConsider the spatial domain D tessellated by non-overlapping, \\emph{d}-dimensional, elements $\\text{D}^k$ of characteristic size \\emph{h}, forming the mesh $\\mathcal{T}_h$. While the size \\emph{h} can vary among the elements, it is required that the faces of the elements along the perimeter of the mesh conform exactly to the domain. Unlike continuous Galerkin FEM, we duplicate values of variables at nodal points $\\boldsymbol{x}^k$ on the vertices, edges, and faces of the elements in order to ensure locality of the scheme within each element. We denote the boundary of each element  with $\\partial \\text{D}^k$ and associate with it an outward-pointing normal $\\boldsymbol{n}$. The global solution $f_i$ can be approximated by a piecewise $p$-th order polynomial over $\\mathcal{T}_h$ as follows.\n\nOn each element, the local solution is approximated by a polynomial basis $\\eta(\\boldsymbol{x})$ as\n\\begin{align*}\n    \\boldsymbol{x}\\in\\text{D}^k : f_{i,h}^k(\\boldsymbol{x},t) = \\sum_{j=1}^{N_p} f_{i,h}^k(\\boldsymbol{x}_j^k,t)\\eta_j^k(\\boldsymbol{x}).\n\\end{align*}\nHere, $f_{i,h}^k(\\boldsymbol{x}_j^k,t)$ is the nodal value of the approximate solution $f_{i,h}^k$ at one of the $N_p$ nodal points. $\\eta$ can be chosen to be the interpolating Lagrange polynomial defined in 1D as $l_m = \\prod_{m=1,m \\neq n}^{p+1} \\left( \\frac{x - x_n}{x_m - x_n}\\right)$. Thereafter, we can represent the global approximate solution as \n\\begin{align*}\n   \n    \\boldsymbol{x}\\in\\text{D}: f_i(\\boldsymbol{x},t) \\simeq f_{i,h}(\\boldsymbol{x},t) = \\sum_{k=1}^{K} f_{i,h}^k(\\boldsymbol{x},t),\n\\end{align*}\nwhere $K$ is the total number of elements. Consider setting $\\boldsymbol{F}_i = \\boldsymbol{e}_i f_i$, where one should note that the two-fold appearance of the index $i$ does \\emph{not} imply a summation over $i$. With this in mind, the local residual on element $\\text{D}^k$ can be written as \n\\begin{align*}\n   \n    \\mathcal{R}_{i,h}^{k} (\\boldsymbol{x},t) = \\frac{\\partial f_{i,h}^k}{\\partial t} + \\boldsymbol{\\nabla} \\cdot  \\boldsymbol{F}_{i,h}^{k} + \\frac{1}{\\tau}(f_{i,h}^{k} - f_{i,h}^{eq,k}), \n\\end{align*}\nin accordance with eqn.~\\ref{eqn:DBE}.\nUpon multiplying the above expression with a test function and integrating over the domain $\\text{D}^k$, we impose the condition\n\\begin{equation*}\n   \n    \\int_{\\text{D}^k} \\phi_j^k(\\boldsymbol{x}) \\mathcal{R}_{i,h}^{k}(\\boldsymbol{x,t}) \\, d\\boldsymbol{x} = 0. \n\\end{equation*}\nHere, $\\phi_j^{k} (\\boldsymbol{x})$ is an arbitrary test function, typically chosen to be a Lagrange polynomial. Upon integrating by parts and substituting the expression for residual $\\mathcal{R}_{i,h}^{k}$, we get\n\\begin{align}\n    &\\int_{\\text{D}^k} \\phi_{j}^{k} \\frac{\\partial f_{i,h}^k}{\\partial t} d\\boldsymbol{x} - \\int_{\\text{D}^k} \\boldsymbol{F}_{i,h}^k \\cdot \\boldsymbol{\\nabla}\\phi_{j}^{k} d\\boldsymbol{x}\n    +\\frac{1}{\\tau} \\int_{\\text{D}^k} (f_{i,h}^k - f_{i,h}^{eq,k})\\phi_{j}^{k} d\\boldsymbol{x} = - \\int_{\\partial {\\text{D}^k}} \\phi_{j}^{k} \\boldsymbol{F}_{i,h}^* \\cdot \\boldsymbol{n} \\,  ds. \\label{eqn:byparts} \n\\end{align}\nIn the expression above, $\\boldsymbol{F}^*$ is the numerical flux which allows information to pass between neighboring elements. Two elements sharing a (\\emph{d}-1)-dimensional face \\emph{F} are considered to be face-neighbors. We denote the normal vector pointing from the positive to the negative side of a shared face on element $\\text{D}^k$ with $\\boldsymbol{n}^+$; and correspondingly $\\boldsymbol{n}^-$ is the normal vector pointing in the opposite direction. At this point, we introduce the notation for the mean and jump operators as\n\\begin{align*}\n        \\{\\{&f_i\\}\\} = \\frac{f_i^- + f_i^{+}}{2}, \\\\\n        [[&f_i]] = \\boldsymbol{n}^-f_i^- +  \\boldsymbol{n}^+f_i^+.\n\\end{align*}\nThe numerical flux $\\boldsymbol{F}^*_i$ can be represented as\n\\begin{align*}\n    \\boldsymbol{F}^*_i = (\\boldsymbol{e}_if_i)^* = \\{\\{\\boldsymbol{e}_if_i\\}\\}+\\lvert \\boldsymbol{e}_i \\rvert \\beta[[f_i]],\n\\end{align*}\nwhere $\\beta$ is a parameter that corresponds to the numerical dissipation added to the scheme. $\\beta = 0.5$ leads to the central-flux, inducing no additional dissipation, and $\\beta = 1$ reproduces the upwind flux.\n\n\\subsubsection{Implementation of boundary conditions}\nThe boundary condition described in eqn.~\\ref{eqn:UsedBC} is a Dirichlet boundary condition that is imposed over the boundary in the DG method by setting the value exterior to the element equal to a prescribed value, e.g.~$f_i^+ = f_i(\\rho_b,\\boldsymbol{u_b})$. For the elements on the boundary, this yields \n\\begin{align*}\n        \\{\\{&f_i\\}\\} = \\frac{f_i^- + f_i^{eq}(\\rho_b,\\boldsymbol{u_b})}{2}, \\\\\n        [[&f_i]] = \\boldsymbol{n}^-f_i^- +  \\boldsymbol{n}^+f_i^{eq}(\\rho_b,\\boldsymbol{u_b}).\n\\end{align*}\nThe same approach is followed for the full form of the BC described in eqn.~\\ref{eqn:FullBC}.\n\n\\subsubsection{Time marching}\n\nTo maintain stability and accuracy, we used Singly Diagonally Implicit Runge-Kutta (SDIRK)~\\cite{alexander1977diagonally,burrage1982efficiently} methods to discretize eqn.~\\ref{eqn:byparts}. Unlike explicit RK schemes, these implicit schemes allow a time step that is not constrained by the CFL limit. \n\nA four-stage 4\\textsuperscript{th}-order scheme was chosen for the order of accuracy analysis on micro-Couette flow (see the next section), and a 1-stage first-order RK method, which reduces to the backward difference method, was chosen for the more demanding micro-cavity flow simulations. The overall implementation was performed within the Solution Adaptive Numerical Solver (SANS).~\\cite{galbraith2015verification,galbraith2018sans} Newton's method was used to linearize the non-linear system at each stage of the SDIRK methods, and each linear system was subsequently solved using the Generalized Minimal Residual Method (GMRES).\\cite{saad1986gmres} The iterative solution was obtained using the Portable, Extensible Toolkit for Scientific Computation (PETSc). \\cite{balay2017petsc,galbraith2018sans}\n\n\n\n\\section{\\label{sec:results}Results}\n\\subsection{Micro-Couette flow}\nTo evaluate the accuracy of the scheme, we first consider Couette flow for fluids with 0 < Kn $\\leq$ 1.5. Here, we consider two parallel plates separated by a distance \\emph{L} confining the fluid in the \\emph{y}-axis. The top and bottom walls are prescribed with opposing velocities of magnitude $u_{w,x} = 0.16 c_s$. The plates are placed at \\emph{y} = 1 and \\emph{y} = 0 respectively. The wall BC is imposed using eqn.~\\ref{eqn:UsedBC} while periodic boundary conditions are applied to the domain boundaries along the \\emph{x}-axis. Following the definitions set by Asumali et al.\\cite{Ansumali2007}, the analytical solution to \\emph{x}-velocity from the linearized DBE for the D2Q16 velocity set is given by\n\\begin{align}\n    \\label{eqn:exact}\n    u_x = \\frac{1}{Z_{16}}\\sinh \\left(\\frac{y+\\frac{1}{2}}{\\text{Kn}L}\\right)\\Delta U + \\frac{1}{\\Theta_{16}}\\left(\\frac{y+\\frac{1}{2}}{L}\\right)\\Delta U+U, \n\\end{align}\nwhere\n\\begin{align*}\n    \\Delta U &= U_{top} - U_{bottom}, \\\\\n    U &= (U_{top} + U_{bottom})\/2, \\\\\n    \\mu &= \\sqrt{3-\\sqrt{6}} + \\sqrt{3+\\sqrt{6}}, \\\\\n    \\Theta_{16} &= 1 + 2\\text{Kn}\\left[\\frac{2\\cosh\\left(\\frac{1}{2\\text{Kn}}\\right)+\\mu \\sinh\\left(\\frac{1}{2\\text{Kn}}\\right)}{\\mu \\cosh\\left(\\frac{1}{2\\text{Kn}}\\right)+2\\sqrt{3}\\sinh\\left(\\frac{1}{2\\text{Kn}}\\right)}\\right],\\\\\n    Z_{16} &= \\frac{\\mu}{4\\text{Kn}}\\Bigg[(4\\text{Kn}+\\mu)\\cosh\\left(\\frac{1}{2\\text{Kn}}\\right)\n    2(\\mu \\text{Kn}+\\sqrt{3})\\sinh\\left(\\frac{1}{2\\text{Kn}}\\right)\\Bigg].\n\\end{align*}\nTo evaluate the order of accuracy of our DG-FEM, we consider the D2Q16 velocity set  at Kn = 1 for a variety of uniform structured grids with 8, 32, 128, and 512 triangle elements using polynomial interpolation degrees $p = 0, 1, \\text{and}$ 2. Here, the structured grids were generated by splitting Cartesian grids into triangles, e.g. the 32 element grid was formed by splitting a $4\\times4$ Cartesian grid.\nThe $L_2$-error is calculated using\n\\begin{align*}\n    E^{L_2} = \\sqrt{ \\sum_{k=1}^K \\int_{\\text{D}^k} (u_x^e(\\boldsymbol{x}) - u_x^k(\\boldsymbol{x}))^2 d{\\boldsymbol{x}}},\n\\end{align*}\nwhere $u_x^e$ is the exact solution evaluated using eqn.~\\ref{eqn:exact} and $u_x^k$ is the solution obtained from the DG-FEM. The error is evaluated after $t = 40 $ characteristic time units to ensure steadiness in the solution. In each case, we confirmed that $\\lvert (\\rho,u_x,u_y)_{40} - (\\rho,u_x,u_y)_{30}\\rvert < 10^{-12}$. A time step of $dt = 0.05$ was chosen for time marching. \nFigure \\ref{fig:OOA} shows the $L_2$ error for different polynomial orders and mesh sizes. The slope of each curve provides the order of accuracy for a given polynomial order. We find that the calculated order of accuracy is very close to the theoretically expected value of $p+1$.\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[scale=0.35]{Images\/convergence.png}\n    \\caption{$L_2$ error in the streamwise velocity $u_x$ for Couette flow vs. mesh size $h$ for different polynomial degrees $p$.}\n    \\label{fig:OOA}\n\\end{figure}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.275]{Images\/Couette_profile.png}\n    \\caption{Velocity profiles for Couette flow with different velocity sets and Knudsen numbers. (Left) Normalized streamwise velocity profiles. (Right) Predicted Knudsen layer defined as a deviation from a straight line profile constrained to pass through \\{$\\tfrac{1}{2}$, 0\\}. }\n    \\label{fig:profile}\n\\end{figure*}\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.35]{Images\/Slip.png}\n    \\caption{Comparison of normalized slip velocity for Couette flow at different values of Kn. DSMC values replotted using data from Ansumali et al.\\cite{Ansumali2007}}\n    \\label{fig:slip}\n\\end{figure*}\n\nFollowing the order of accuracy study, we performed a set of numerical experiments with different Knudsen numbers and velocity sets, using a 400-element structured grid with $p=2$.\nThe left column in Figure~\\ref{fig:profile} shows the normalized streamwise velocity profile for different values of Kn. For all values of Kn, we find that the profiles corresponding to odd velocity sets D2Q9 and D2Q25 are close to each other, and the profiles corresponding to even velocity sets D2Q16 and D2Q36 are near to each other in a similar fashion. As expected, D2Q9 underpredicts the slope of the velocity the most, in addition to overestimating the slip effects at the walls. We note that D2Q16 and D2Q36 are closer to the expected results, whereas D2Q25 predicts the incorrect slope and slip velocity although it is obtained using a quadrature rule of higher order than D2Q16.~\\cite{Kim2008,Shi2011} This is reflected in the prediction of the so-called Knudsen layer, as shown in the right column of Figure~\\ref{fig:profile}. The Knudsen layer, a characteristic of a flow that violates the continuum assumption, is the deviation of the velocity profile from a straight line as predicted by the Navier-Stokes equations. Here, the Navier-Stokes velocity profile ($u_{x,fit}$) is obtained using a least-squares fit of a straight line constrained to pass through the point $\\{\\frac{1}{2},0\\}$ of the velocity profile ($u_x$). The limitations of D2Q9 are explicitly demonstrated by the complete absence of any deviation from the straight line for all Kn. While D2Q25 predicts a non-negligible extent of Knudsen layer formation, it underpredicts the effect substantially in comparison to D2Q16 and D2Q36.  At $\\text{Kn} = 0.1$, we find a good match between D2Q16 and D2Q36 results. However, at $\\text{Kn} \\geq 0.5$ there is a departure of the flow-field predicted by D2Q16 from D2Q36. This attribute is well known, as D2Q16 begins to deviate from the correct flow-field (from DSMC or the linearized Boltzmann equation) at $\\text{Kn} \\simeq 0.5$.\\cite{Meng2011} \n\nFigure \\ref{fig:slip} shows the normalized slip velocity evaluated at the wall for different velocity sets and values of Kn, in conjunction with DSMC results for reference purposes. As expected, D2Q9 and D2Q25 overpredict the slip velocity, whereas, D2Q16 and D2Q36 maintain a trend close to the DSMC results. While D2Q36 delays the onset of the deviation from the correct slip velocity profile in comparison to D2Q16, we find that beyond $\\text{Kn} \\simeq 0.8$, velocity sets with higher accuracy are required. It must be noted that even at $\\text{Kn} = 0.1$, we find a slight mismatch in the Knudsen layer formation predictions of the D2Q16 and D2Q36 models. This indicates the importance of using higher-order quadrature rules in the discretization of the velocity in the phase-space. However, as mentioned previously, D2Q25 underperforms in comparison to the lower-order D2Q16 quadrature. This is explained by Shi et al.~\\cite{Shi2011} as a boundary condition artifact due to the mis-alignment of the discrete velocities in odd-numbered velocity sets. The discrete particle velocities parallel to the boundaries, i.e., parallel to the x-axis in the system simulated, do not contribute to the half-space moments required to be described at the boundaries. \nHence, the velocity sets with particle velocities parallel to the wall boundaries introduce errors of larger magnitude, especially when they are associated with larger weights. Therefore, we would expect the flow-field using D2Q49 to be closer to D2Q25 than D2Q36,\nand higher-order even-numbered schemes such as D2Q64 are more likely to match the DSMC results up to $\\text{Kn} = 1$ (see Meng and Zhang~\\cite{Meng2011}). \n\n\n\\subsection{Micro-Cavity flow\\label{sec:microcavity}}\n\nWe also applied the DG-FEM to a lid-driven cavity flow with Knudsen numbers of 1, 2 and 8. Here, a domain of characteristic length $L$ = 1 was chosen with the Maxwellian diffuse-scattering boundary condition (eqn.~\\ref{eqn:FullBC}) applied on all four boundaries.\nThe lid at $y = L$ is prescribed a velocity corresponding to a Mach number of 0.16. The simulations were performed on a uniform structured mesh comprised of 200 triangle elements of order $p$ = 1. For comparison, simulations at $\\text{Kn}=1$ were also performed on a uniform structured mesh with 450 triangle elements and an unstructured mesh with 315 triangle elements. The unstructured mesh is shown in Figure~\\ref{fig:mesh}. The solution was deemed steady if the flow-fields $(u_x,u_y)$ evaluated 5 seconds apart varied by $\\mathcal{O}(10^{-5})$ or less. Velocity discretizations based on Gauss-Hermite and Newton-Cotes quadrature were used. In particular, D2Q16, D2Q64, D2Q144, D2Q289 and D2Q324 velocity sets were considered based on the tensor products of 1D Gauss-Hermite quadrature of order 7, 15, 23, 33 and 35 respectively. In addition, the tensor product of 1D Newton-Cotes quadrature with 32 intervals (33 nodes, order 32), spanning the domain [-4$c_s$, 4$c_s$] $\\times$ [-4$c_s$, 4$c_s$], was considered to obtain a NC D2Q1089 velocity set.\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.25]{Images\/mesh.png}\n    \\caption{Unstructured mesh comprised of 315 triangle elements used in the micro-cavity simulations.}\n    \\label{fig:mesh}\n\\end{figure*}\n\nFigure \\ref{fig:LDquad} shows the $u_x$ profile along the micro-cavity center \\{x = 0.5\\} at $\\text{Kn}=1$ and $\\text{Kn}=8$ for different velocity sets, in conjunction with DSMC results for reference purposes. The DG-FEM results were all obtained on the 200-element grid. We note that D2Q16 exhibits large discontinuities in the first-derivative of the profile for $\\text{Kn}=1$, although it was adequate for the Couette flow cases presented in the previous section. These discontinuities are due to the \\emph{ray-effect}, i.e.~the preferential alignment of the macroscopic velocities along the directions of the discrete velocity space. Higher-order quadratures -- D2Q64, D2Q144, D2Q289, D2Q324, and D2Q1089 show improvements in matching the DSMC results. One should note that D2Q289 overestimates the slip velocity at the top wall due to the nature of the BC definition, as was seen for odd quadrature rules in Couette flow. However, other high-order quadrature rules, NC D2Q1089 and GH D2Q324  match the DSMC results well. At $\\text{Kn} = 8$, we find some slight discontinuities in the velocity profile with GH D2Q324, but NC D2Q1089 maintains a relatively smooth profile. As expected, GH D2Q16--289 predict sharp discontinuities. Here, the fact that NC D2Q1089 has more velocity directions seems to give it an advantage over GH D2Q324, despite its basis on a quadrature rule of slightly lower strength. We will offer an explanation for this phenomenon in what follows.\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.3]{Images\/NCvsGH.png}\n    \\caption{Comparison of $u_x$ along the cavity center line for different GH and NC quadratures for (left) $\\text{Kn}=1$, (right) $\\text{Kn}=8$. The DSMC data is replotted using results from reference \\cite{Guo2013}.}\n    \\label{fig:LDquad}\n\\end{figure*}\n\nThe sharp gradients introduced by the walls cause highly irregular regions of the flow to form locally, and hence low-order quadrature schemes, such as D2Q16, are insufficient. To reduce the unphysical oscillations caused by the deviation of the solution from the local equilibrium, higher numbers of discrete velocities are required. However, as we mentioned previously, GH-quadrature rules provide depreciating benefits with increasing  quadrature strength because the weights of the points far away from the centroid have vanishingly small values. For example, the D2Q324 model obtained from a tensor product of the D1Q18 quadrature rule has 24 points with weights lower than machine zero ($2.2 \\times 10^{-16}$), with the lowest weight of order $10^{-23}$. Hence, these 24 points do not contribute to the evaluation of the solution. Now, the NC D2Q1089 velocity set also contains points that do not contribute to the solution. However, only 4 points have weights below machine-zero, unlike 24 in GH D2Q324. In addition, D2Q1089 has significantly more velocity directions overall. Broadly speaking, the accuracy of NC D2Q1089 (despite having an odd number of points), emphasizes the importance of requiring many, more equally weighted velocity directions relative to GH velocity sets. \n\nLastly, one may consider some additional results for the NC D2Q1089 velocity set. Figure~\\ref{fig:LDprofiles} shows $u_x$ and $u_y$ profiles at all values of Kn, obtained on the 200-element grid, in conjunction with DSMC results. Furthermore, Figure \\ref{fig:LDcontour} shows contour plots that illustrate the significant reduction in the ray-effect with NC D2Q1089 compared against GH D2Q16 at $\\text{Kn} =1$.  Even for the NC velocity set, we find some slight evidence of the ray-effect's presence along the corners next to the moving wall in Figure \\ref{fig:LDcontour}b for the 200-element uniform structured grid. Figures \\ref{fig:LDcontour}c-f show the contours for the 450-element uniform structured and 315-element unstructured meshes, highlighting that the ray-effect is (mostly) grid-independent. The slight evidence of the effect at the upper corners suggests that NC D2Q1089 may not completely capture the significant non-linearity, unlike at the cavity center where it matches the DSMC results well. Guo et al.~\\cite{Guo2013} suggest using the NC D2Q10000 velocity set to capture this non-linearity. Nonetheless, our results are reasonable, while requiring far fewer velocity directions. Yet, we acknowledge that the number of velocity directions will need to be significantly increased for $\\text{Kn} \\gg 1$.\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.3]{Images\/LDCF_profiles.png}\n    \\caption{Normalized $u_x$ (red) and $u_y$ (blue) along the cavity center for different Knudsen numbers. The DSMC data is replotted using results from reference \\cite{Guo2013}. }\n    \\label{fig:LDprofiles}\n\\end{figure*}\n\\begin{figure*}\n    \\centering\n    \\includegraphics[scale=0.1575]{Images\/LDCF_contour.png}\n    \\caption{Contours of normalized $u_x$ at Kn = 1 for (left) GH D2Q16 and (right) NC D2Q1089. (a-b) uniform structured mesh with 200 triangle elements, (c-d) uniform structured mesh with 450 triangle elements, and (e-f) unstructured mesh with 315 triangle elements.}\n    \\label{fig:LDcontour}\n\\end{figure*}\n\n\n\n\\section{Conclusion} \\label{sec:conclude}\nWe have introduced a DG-FEM for solving the discrete Boltzmann BGK equation, for the purpose of capturing non-continuum effects in isothermal fluid flows. We show that the scheme exhibits a spatial convergence of order $p+1$ with a D2Q16 velocity set, for a Couette flow at $\\text{Kn} = 1$. We note that high-order accuracy is achieved with a local stencil, unlike finite volume and finite difference methods. In addition, the temporal accuracy is not limited to second order as the full coupling between the streaming and collision parts of the DBE is retained. Therefore, different time integration schemes, such as backward Euler (first order) or Runge-Kutta (any order), can be used depending on the temporal accuracy needed for one's application. Although only implicit schemes were implemented in this work to ensure stability for large time-steps, the retention of the coupling allows the use of explicit time integration methods of any order as well.\n\nIn addition, we analyzed the slip velocity and the Knudsen layer predicted in Couette flow for different velocity sets (D2Q9, D2Q16, D2Q25, and D2Q36). For the Maxwellian diffuse-scattering boundary condition, we found that the even-numbered quadratures were more accurate as all the discrete velocities contributed to the boundary condition, unlike odd-numbered quadratures. However, larger velocity sets were required for higher Knudsen numbers, as we found the onset of departure from the DSMC results at $\\text{Kn} \\simeq 0.5$ for D2Q16 and $\\text{Kn} \\simeq 0.8$ for~D2Q36. \n\nThe deficiency of low-order quadrature was exacerbated in a more complex, lid-driven micro-cavity flow. For $\\text{Kn} = 1$, we compared GH velocity sets D2Q16, D2Q64, D2Q144, D2Q289 and D2Q324, with NC D2Q1089. We found that most GH velocity sets show significant first-derivative discontinuities in the flow-field due to the ray-effect. More specifically, while NC D2Q1089 (order 32) exhibited a smooth profile and matched well with DSMC data at all Knudsen numbers, GH D2Q324 (order 35) showed the characteristic discontinuities at $\\text{Kn} = 8$. This is because high-order GH quadrature has points clustered near the boundaries with weights close to machine-zero, diminishing the contributions of its velocity directions. Conversely, NC quadrature contains points uniformly distributed with most weights larger than machine-zero, enabling the scheme to better capture the non-linearity of the flow-field. However, we still found the slight presence of the ray-effect with NC D2Q1089 near the upper corners of the micro-cavity, indicating the need to use larger velocity sets to accurately capture the flow-field as the Knudsen number increases. \n\n\\section*{Data availability}\nThe data that support the findings of this study are available from the corresponding author upon reasonable request.\n\n\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:Intro}\nAn irreducible monic polynomial $\\phi(t)\\in \\mathord{\\mathbb Z}[t]$ of even degree $2d>0$\nis called a \\emph{Salem polynomial} if \n$\\phi(t)$ is reciprocal,\n$\\phi(t)=0$ has two positive real roots, and\nthe other $2d-2$ complex roots are located on $\\shortset{z\\in \\mathord{\\mathbb C}}{|z|=1}\\setminus \\{\\pm 1\\}$.\n\\par\nThe notion of Salem polynomials plays an important role in the study \nof dynamics of automorphisms of algebraic varieties.\nWe have the following fundamental theorem due to McMullen~\\cite{MR1896103}.\nSee also~\\cite{MR3129006} and~\\cite[Proposition 3.1]{1406.2761}. \n\\begin{theorem}[\\cite{MR1896103}]\\label{thm:cycsSalem}\nLet $g$ be an automorphism of an algebraic  $K3$ surface $X$\ndefined over an algebraically closed field.\nThen the characteristic polynomial of the action of $g$ on  \nthe N\\'eron--Severi lattice $S_X$ of $X$ is\na product of cyclotomic polynomials and at most one Salem polynomial\ncounting with multiplicities.\n\\end{theorem}\nA $K3$ surface $X$ defined over an algebraically closed field $k$ of characteristic $p>0$ is said to be \\emph{supersingular}\nif the rank of its N\\'eron--Severi lattice $S_X$ is $22$.\nWe say that an automorphism $g$ of a supersingular $K3$ surface $X$ is \\emph{of irreducible Salem type}\nif the characteristic polynomial of the action of $g$  on  $S_X$\nis a Salem polynomial of degree $22$.\n\\par\nThe purpose of this note is to report the following theorems,\nwhich are the results of  computer-aided experiments.\nBy a \\emph{double plane involution} of a $K3$ surface $X$ in characteristic not equal to $2$,\nwe mean an automorphism of $X$ of order $2$\ninduced by the Galois transformation  of a generically finite morphism $X\\to \\P^2$ of degree $2$.\n\\begin{theorem}\\label{thm:main}\nLet $p$ be an odd prime less than or equal to $7919$.\nThen every supersingular $K3$ surface $X$ in characteristic $p$ has \na sequence  of \ndouble plane involutions $\\tau_1, \\dots,  \\tau_l$ of length at most  $22$\nsuch that their product $\\tau_1\\cdots \\tau_l$ is \nan automorphism \nof irreducible Salem type.\n\\end{theorem}\nLet $X$ be a supersingular $K3$ surface in characteristic $p>0$, and \nlet $S_X\\sp{\\vee}$ denote  the \\emph{dual lattice} $\\mathord{\\mathrm {Hom}}(S_X, \\mathord{\\mathbb Z})$ of $S_X$,\ninto which $S_X$ is embedded as a submodule of finite index by the intersection form of $S_X$.\nArtin~\\cite{MR0371899} showed that the discriminant group $S_X\\sp{\\vee}\/S_X$ of $S_X$ is isomorphic to \n$(\\mathord{\\mathbb Z}\/p\\mathord{\\mathbb Z})^{2\\sigma}$,\nwhere $\\sigma$ is a positive integer less than or equal to $10$.\nThis integer $\\sigma$ is called the \\emph{Artin invariant} of $X$.\nBy the result of Ogus~\\cite{MR563467, MR717616},\nthe supersingular $K3$ surfaces of Artin invariant $\\le \\sigma$ \ndefined over an algebraically closed field $k$ constitute  a moduli of dimension $\\sigma-1$,\nand \na supersingular $K3$ surface $X(p)$ with Artin invariant $1$ is unique up to isomorphism.\n\\par\nFor supersingular $K3$ surfaces  with Artin invariant  $\\sigma=10$ in characteristic  $p$ with $11\\le p\\le 17389$, \nwe found a class of sequences  of \ndouble plane involutions whose product is \\emph{frequently} of irreducible Salem type.\n(See Section~\\ref{sec:sigma10} for the detail.)\nUsing this class, we obtain the following theorem:\n\\begin{theorem}\\label{thm:main2}\nLet $p$ be an odd prime less than or equal to $17389$.\nThen every supersingular $K3$ surface $X$ in characteristic $p$ with Artin invariant $10$ has \na sequence  of \ndouble plane involutions  of length at most  $22$\nsuch that their product is \nan automorphism \nof irreducible Salem type.\n\\end{theorem}\n\\par\nThe interest of an automorphism of irreducible Salem type  \nstems from the following observation due to Esnault and Oguiso~\\cite{1406.2761, 1411.0769}:\n\\begin{theorem}[\\cite{1406.2761, 1411.0769}]\\label{thm:nonliftable}\nLet $g$ be an automorphism of a supersingular $K3$ surface $X$.\nIf the characteristic polynomial of the action of $g$ on  $S_X$ is irreducible,\nthen the pair $(X, g)$ can never be lifted to characteristic $0$.\n\\end{theorem}\nHence we obtain the following corollary.\n\\begin{corollary}\\label{cor:main}\nLet $X$ be a supersingular $K3$ surface in odd characteristic $p$ with Artin invariant $\\sigma$.\nSuppose that $p\\le 7919$ or {\\rm (}$\\sigma=10$ and $p\\le 17389${\\rm )}.\nThen $X$  has an automorphism $g$\nsuch that the pair $(X, g)$ can never be lifted to characteristic $0$.\n\\end{corollary}\nRecently, several authors have studied  the non-liftability of automorphisms of supersingular $K3$ surfaces\nby means of Salem polynomials.\nSee~\\cite{1307.0361, 1406.2761, 1411.0769, 1502.06923}.\nIn particular,\nthe existence of a non-liftable automorphism has been  established\nfor a supersingular $K3$ surface $X(p)$  in characteristic $p$\n\\emph{with Artin invariant $1$}, \nexcept for the cases  $p= 7$ and $13$.\n\\begin{remark}\nIn~\\cite{JangLocal}, the existence of a non-liftable  automorphism of $X(p)$ \nwas proved for $p$ large enough by another method.\n\\end{remark}\nOur main theorems  not only  fill the remaining cases $X(7)$ and $X(13)$ for  supersingular $K3$ surfaces with Artin invariant $1$,\nbut also  suggest that  this result can be extended  to  supersingular $K3$ surfaces with arbitrary Artin invariant,\nat least in odd characteristics.\nThere exists no theoretical significance in\nthe bounds $p\\le 7919$ in Theorem~\\ref{thm:main} and $p\\le 17389$ in Theorem~\\ref{thm:main2}.\nWe merely stopped our computations at the 1000th prime ($p=7919$)\nand the 2000th prime ($p=17389$).\n\\par\nThe main tool of the proof of Theorems~\\ref{thm:main} and~\\ref{thm:main2}\nis the structure theorem of the N\\'eron--Severi lattices of supersingular $K3$ surfaces $X$\ndue to Rudakov and Shafarevich~\\cite{MR633161},\nwhich states that \n the isomorphism class of the lattice $S_X$ is uniquely determined by \n$p$ and the Artin invariant $\\sigma$ of $X$.\n\\par\nLet $X$ be a supersingular $K3$ surface $X$ in odd characteristic.\nIn this paper,  \nwe present a method to generate many matrix representations on $S_X$  \nof double plane involutions of $X$.\nComposing  some of these involutions,\nwe obtain an  automorphism of irreducible Salem type.\nIn order to produce double plane involutions,\nwe have to find the nef cone in $S_X\\otimes\\mathord{\\mathbb R}$.\nFor this purpose,\nwe introduce  a notion of an \\emph{ample list of vectors}.\n(See Section~\\ref{sec:lattice}  for the definitions.)\n\\par\nThe results of the experiments are presented in the author's web page~\\cite{compdataIrredSalem}.\n\\par\nThanks are due to Professors Junmyeong Jang, Toshiyuki Katsura, Jonghae Keum,   Keiji Oguiso,  Matthias Sch\\\"utt\nand Hirokazu Yanagihara \nfor  stimulating discussions.\n\\section{Lattices}\\label{sec:lattice}\nA \\emph{lattice} is a free $\\mathord{\\mathbb Z}$-module $L$ of finite rank\nwith a nondegenerate symmetric bilinear form\n$\\intfL{\\phantom{i}, \\phantom{i}}: L\\times L\\to \\mathord{\\mathbb Z}$,\nwhich we call the \\emph{intersection form}.\nWe let the group $\\mathord{\\mathrm {O}}(L)$ of isometries of $L$ act on $L$ from the \\emph{right},\nand write the action of $g\\in \\mathord{\\mathrm {O}}(L)$ on $L$ by $x\\mapsto x^g$.\nA lattice $L$ is \\emph{even} if $\\intfL{v,v}$ is even for any vector $v\\in L$.\nA lattice $L$ is \\emph{hyperbolic} if its rank $n$ is larger than $1$  and \nthe real quadratic space $L\\otimes\\mathord{\\mathbb R}$ is of signature $(1, n-1)$.\n\\par\nLet $L$ be an even hyperbolic lattice.\nThe open subset $\\shortset{x\\in L\\otimes\\mathord{\\mathbb R}}{\\intfL{x,x}>0}$ of $L\\otimes \\mathord{\\mathbb R}$\nhas two connected components,\neach of which is called \na \\emph{positive cone}.\nWe choose a positive cone $\\mathord{\\mathcal P}_L$,\nand denote by  $\\mathord{\\mathrm {O}}^+(L)$ the stabilizer subgroup  of $\\mathord{\\mathcal P}_L$ in $\\mathord{\\mathrm {O}}(L)$.\nA vector $r\\in L$ is called a \\emph{$(-2)$-vector}\nif $\\intfL{r, r}=-2$.\nLet $r$ be a $(-2)$-vector.\nWe put\n$$\n(r)\\sp{\\perp} :=\\set{x\\in \\mathord{\\mathcal P}_L}{\\intfL{x, r}=0},\n$$\nand call it a \\emph{$(-2)$-hyperplane}.\nThe reflection\n$$\ns_r: x\\mapsto x+\\intfL{x, r}\\cdot r\n$$ \nin $(r)\\sp{\\perp}$ is an element of $\\mathord{\\mathrm {O}}^+(L)$.\nWe denote by  $\\Weyl{L}$  the subgroup of $\\mathord{\\mathrm {O}}^+(L)$\ngenerated by all the reflections $s_r$ in $(-2)$-hyperplanes,\nand call $\\Weyl{L}$ the \\emph{Weyl group} of $L$.\nA \\emph{standard fundamental domain of $\\Weyl{L}$}\nis the closure in $\\mathord{\\mathcal P}_L$ of a connected component of\n$$\n\\mathord{\\mathcal P}_L\\;\\setminus \\;\\bigcup_{r}\\; (r)\\sp{\\perp},\n$$\nwhere $r$ ranges through the set of $(-2)$-vectors.\nNote that $\\Weyl{L}$ acts on the set of standard fundamental domains\ntransitively.\n\\par\nSuppose that a basis of an even hyperbolic lattice $L$\nand the Gram matrix of the intersection form $\\intfL{\\phantom{i}, \\phantom{i}}$ with respect to this basis\nare given.\nWe have the following algorithms. See~\\cite[Section 3]{MR3166075} for the details.\n\\begin{algorithm}\\label{algo:affES}\n Let $v$ be a vector in  $\\mathord{\\mathcal P}_L \\cap L$.\nThen, for  an integer $a$ and  an even integer $d$, the finite set\n$\\shortset{x\\in L }{ \\intfL{x,v}=a, \\intfL{x,x}=d}$ can be calculated.\nIn particular, the sets\n$$\n\\mathord{\\mathcal R}(v):=\\set{r\\in L}{\\intfL{r, v}=0, \\;\\intfL{r, r}=-2}\n$$ \nand \n$$\n\\mathord{\\mathcal F}(v):=\\set{f\\in L}{\\intfL{f, v}=1,\\; \\intfL{f, f}=0}\n$$\ncan be calculated.\n\\end{algorithm}\n\\begin{algorithm}\\label{algo:separating}\nLet $u$ and $v$ be vectors in $\\mathord{\\mathcal P}_L\\cap L$.\nThen, for a negative even integer $d$, the finite set\n$\\shortset{x\\in L }{ \\intfL{x,u}>0, \\intfL{x,v}<0, \\intfL{x, x}=d}$ can be calculated.\nIn particular, the set \n$$\n\\mathord{\\mathcal S}(u, v):=\\set{r\\in L }{ \\intfL{r,u}>0, \\;\\intfL{r,v}<0, \\;\\intfL{r, r}=-2}\n$$\ncan be calculated.\n\\end{algorithm}\nWe call an ordered nonempty set \n$$\n\\mathord{\\textbf{\\itshape a}}:=[h_0, \\rho_1, \\dots, \\rho_K]\n$$\nof vectors of $L$ an \\emph{ample list of vectors}\nif $h_0\\in \\mathord{\\mathcal P}_L\\cap L$ and,\nfor any $r\\in \\mathord{\\mathcal R}(h_0)$, there exists a member $\\rho_i$ of $\\{  \\rho_1, \\dots, \\rho_K\\}$ such that $\\intfL{r, \\rho_i}\\ne 0$.\n\\begin{example}\n(1) If vectors $ \\rho_1, \\dots, \\rho_K$ of $L$  span the linear space $L\\otimes \\mathord{\\mathbb  Q}$ over $\\mathord{\\mathbb  Q}$, then \n$[h_0, \\rho_1, \\dots, \\rho_K]$ is an ample list of vectors\nfor any vector $h_0\\in \\mathord{\\mathcal P}_L\\cap L$.\n\\par\n(2)  If a vector $h_0\\in \\mathord{\\mathcal P}_L\\cap L$ satisfies  $\\mathord{\\mathcal R}(h_0)=\\emptyset$, then the list $[h_0]$ is an ample list of vectors.\n\\par\n(3) If $[h_0, \\rho_1, \\dots, \\rho_K]$ is an ample list of vectors,\nthen $[h_0, \\rho_1, \\dots, \\rho_K, \\rho_{K+1}]$ is an ample list of vectors for any $\\rho_{K+1}\\in L$.\n\n \\end{example}\n %\n Let $\\mathord{\\textbf{\\itshape a}}=[h_0, \\rho_1, \\dots, \\rho_K]$ be an ample list of vectors.\n\n\n We define $D(\\mathord{\\textbf{\\itshape a}})$ to be \n the  unique standard  fundamental domain  of $\\Weyl{L}$ such that \n$$\n\\va_{\\varepsilon}:=h_0+\\varepsilon \\rho_1+\\cdots+\\varepsilon^K \\rho_K\n$$\nis contained in the interior of $D(\\mathord{\\textbf{\\itshape a}})$,\nwhere $\\varepsilon$  is a  sufficiently small positive real number.\nFor $x\\in \\mathord{\\mathcal P}_L$, \nwe write \n$$\n\\intfL{\\mathord{\\textbf{\\itshape a}}, x}>0\n$$ \nif  the real vector\n$$\n(\\;\\intfL{h_0, x}, \\, \\intfL{\\rho_1, x}, \\,\\dots, \\, \\intfL{\\rho_K, x}\\;) \\in \\mathord{\\mathbb R}^{K+1}\n$$\nis nonzero and its leftmost nonzero entry\nis positive;\nthat is, $\\intfL{\\va_{\\varepsilon}, x}\\in \\mathord{\\mathbb R}$  is positive for a sufficiently small positive real number $\\varepsilon$.\nFor $x_1, x_2\\in \\mathord{\\mathcal P}_L$, we write \n$$\n\\intfL{\\mathord{\\textbf{\\itshape a}}, x_1}>\\intfL{\\mathord{\\textbf{\\itshape a}}, x_2}\n$$ \nif $\\intfL{\\mathord{\\textbf{\\itshape a}}, x_1-x_2}>0$.\nWe put\n$$\n\\mathord{\\mathcal R}^+ (\\mathord{\\textbf{\\itshape a}}):=\\set{r\\in \\mathord{\\mathcal R}(h_0)}{\\intfL{\\mathord{\\textbf{\\itshape a}}, r}>0}.\n$$\nNote that $\\mathord{\\mathcal R}(h_0)$ is the disjoint union of $\\mathord{\\mathcal R}^+(\\mathord{\\textbf{\\itshape a}})$ and $-\\mathord{\\mathcal R}^+(\\mathord{\\textbf{\\itshape a}})$.\nThen $D(\\mathord{\\textbf{\\itshape a}})$ is the unique standard fundamental domain of $\\Weyl{L}$ that contains $h_0$ and is contained in the region\n$$\n\\set{x\\in \\mathord{\\mathcal P}_L}{\\intfL{x, r}\\ge 0\\;\\;\\textrm{for any vector}\\;\\; r\\in \\mathord{\\mathcal R}^+ (\\mathord{\\textbf{\\itshape a}})}.\n$$\nThe following lemma is obvious.\n\\begin{lemma}\\label{lem:inDa}\nA vector $v\\in \\mathord{\\mathcal P}_L\\cap L$ is contained in $D(\\mathord{\\textbf{\\itshape a}})$ if and only if\n$\\mathord{\\mathcal S}(h_0, v)=\\emptyset$ and $\\intfL{v, r}\\ge 0$ for any vector $r\\in \\mathord{\\mathcal R}^+(\\mathord{\\textbf{\\itshape a}})$.\n\\end{lemma}\nLet $d$ be an even positive integer.\nSuppose that a vector $v\\in \\mathord{\\mathcal P}_L\\cap L$ satisfies $\\intfL{v, v}=d$.\nFrom $v$, we can find a vector  $h_v$ in $D(\\mathord{\\textbf{\\itshape a}})\\cap L$ satisfying  $\\intfL{h_v, h_v}=d$ by the following method.\nFirst we calculate the union \n$$\n\\mathord{\\mathcal S}(h_0, v) \\cup \\mathord{\\mathcal R}\\sp\\prime =\\{r_1, \\dots, r_M\\}, \n$$ \nwhere \n$$\n\\mathord{\\mathcal R}\\sp\\prime:=\\set{r\\in \\mathord{\\mathcal R}^+(\\mathord{\\textbf{\\itshape a}})}{\\intfL{v, r}<0}.\n$$\nNote that we have  $\\intfL{v, r_i}<0$  and $\\intfL{\\mathord{\\textbf{\\itshape a}}, r_i}>0$\nfor each $r_i\\in \\mathord{\\mathcal S}(h_0, v) \\cup \\mathord{\\mathcal R}\\sp\\prime$.\nNote also that, if a $(-2)$-vector $r$ satisfies $\\intfL{v, r}<0$  and $\\intfL{\\mathord{\\textbf{\\itshape a}}, r}>0$,\nthen $r$ belongs to $\\mathord{\\mathcal S}(h_0, v) \\cup \\mathord{\\mathcal R}\\sp\\prime$.\nWe put\n$$\n\\mathord{\\textbf{\\itshape t}}_i:=\\frac{-1}{\\intfL{v, r_i}} \\left( \\;\\intfL{h_0, r_i}, \\; \\intfL{\\rho_1, r_i}, \\; \\dots, \\; \\intfL{\\rho_{K}, r_i}\\; \\right) \\in \\mathord{\\mathbb R}^{K+1}.\n$$\nIf $\\mathord{\\textbf{\\itshape t}}_i=\\mathord{\\textbf{\\itshape t}}_j$ holds  for some distinct indices $i$ and $j$,\nthen we choose a random vector $\\rho_{K+1}\\in L$ and replace $\\mathord{\\textbf{\\itshape a}}$ by a new ample list of vectors \n$$\n[h_0, \\rho_1, \\dots, \\rho_{K}, \\rho_{K+1}].\n$$\n(Note that this replacement of $\\mathord{\\textbf{\\itshape a}}$ does not change $D(\\mathord{\\textbf{\\itshape a}})$.)\nRepeating this process,\nwe can assume that $\\mathord{\\textbf{\\itshape t}}_1, \\dots, \\mathord{\\textbf{\\itshape t}}_M$ are  distinct.\nWe sort the vectors  $r_1, \\dots, r_M$ of $\\mathord{\\mathcal S}(h_0, v) \\cup \\mathord{\\mathcal R}\\sp\\prime$ in such a way that, \nif $i>j$,  then the leftmost  nonzero entry of $\\mathord{\\textbf{\\itshape t}}_i-\\mathord{\\textbf{\\itshape t}}_j$ is positive.\nConsider the half-line $\\ell$ in $\\mathord{\\mathcal P}_L$\ngiven by\n$$\n\\va_{\\varepsilon} + tv \\;\\; (t\\in \\mathord{\\mathbb R}_{\\ge 0}),\n$$\nwhere $\\varepsilon$ is a sufficiently small positive real number.\nThen $\\ell$ is not contained in any $(-2)$-hyperplane,\nthe $(-2)$-hyperplanes $(r_1)\\sp{\\perp}, \\dots, (r_M)\\sp{\\perp}$\nintersect $\\ell$ at distinct points,\nand any $(-2)$-hyperplane intersecting $\\ell$ is one of $(r_1)\\sp{\\perp}, \\dots, (r_M)\\sp{\\perp}$.\nMoreover,  the values $t_i$ of the parameter $t$ of $\\ell$ at which $\\ell$ intersects $(r_i)\\sp{\\perp}$ satisfy \n$$\nt_1>\\dots >t_M>0, \n$$\nbecause,  if $\\mathord{\\textbf{\\itshape t}}_i=(t_{i, 0}, t_{i, 1}, \\dots, t_{i, K})\\in \\mathord{\\mathbb R}^{K+1}$, then we have\n$$\nt_i=t_{i, 0}+\\varepsilon\\, t_{i, 1}+ \\dots + \\varepsilon^K \\,t_{i, K}.\n$$\nTherefore, if we denote by $s_i\\in \\Weyl{L}$ the reflection in $(r_i)\\sp{\\perp}$,\nthen the vector\n\\begin{equation}\\label{eq:hvsss}\nh_v:=v^{s_1\\dots s_M}\n\\end{equation}\nbelongs to  $D(\\mathord{\\textbf{\\itshape a}})\\cap L$.\n\\section{Polarizations of degree $2$}\\label{sec:poldeg2}\nLet $X$ be a $K3$ surface defined over an algebraically closed field $k$ of  characteristic not equal to $2$,\nand let $S_X$ denote the N\\'eron--Severi lattice of $X$ \nwith the intersection form $\\intfS{\\phantom{i}, \\phantom{i}}$.\nSuppose that  $\\operatorname{\\mathrm {rank}}\\nolimits S_X$ is larger than $1$.\nThen $S_X$ is an even hyperbolic lattice.\nWe let the automorphism group $\\operatorname{\\mathrm {Aut}}\\nolimits(X)$ act on $X$ from the left and act on $S_X$ from the right by the pull-back.\nLet $\\mathord{\\mathcal P}(X)$ denote the positive cone of $S_X$ that contains  an ample class.\nWe put\n$$\nN(X):=\\set{x\\in \\mathord{\\mathcal P}(X)}{\\intfS{x, [C]}\\ge 0\\;\\;\\textrm{for any curve}\\;\\; C\\subset X},\n$$\nwhere $[C]\\in S_X$ is the class of a curve $C$ on $X$.\nIt is well known that $N(X)$ is a standard fundamental domain of the Weyl group $\\Weyl{S_X}$.\nA vector $h\\in S_X$ with $\\intfS{h, h}=2$ is called a \\emph{polarization of degree $2$}\nif the complete linear system $|\\mathord{\\mathcal L}_h|$ of a line bundle $\\mathord{\\mathcal L}_h\\to X$ whose class  is $h$\nis fixed-component free.\nBy~\\cite{MR1260944}, \nwe have the following criterion.\n\\begin{proposition}\\label{prop:Nikulindeg2pol}\nA vector $h\\in S_X$ with $\\intfS{h, h}=2$ is  a polarization of degree $2$\nif and only if $h\\in N(X)$ and $\\mathord{\\mathcal F}(h)=\\emptyset$.\n\\end{proposition}\nSuppose that $h\\in S_X$ is a polarization of degree $2$.\nThen, by~\\cite{MR0364263},  the complete linear system $|\\mathord{\\mathcal L}_h|$ is base-point free, and hence defines a generically finite morphism \n$\\Phi_h: X\\to \\P^2$ of degree $2$.\nLet \n$$\nX\\;\\maprightsp{\\psi_h}\\; Y_h \\;\\maprightsp{\\pi_h} \\;\\P^2\n$$\nbe the Stein factorization of $\\Phi_h$,\nand let $B_h\\subset \\P^2$ be the branch curve of the double covering $\\pi_h$.\nThen $\\psi_h: X\\to Y_h$ is a contraction of smooth rational curves, and $B_h$ is a curve of degree $6$ with only simple singularities.\nFor each singular point $P$ of $B_h$,\nthe  curves contracted to $P$ by $\\Phi_h$\nform an indecomposable $ADE$-configuration of smooth rational curves.\nWe put\n$$\n\\mathord{\\mathcal E}_P(h)\\;:=\\; \\set{[C]}{\\textrm{$C$ is a smooth rational curve on $X$ contracted to $P$ by $\\Phi_h$}},\n$$\nand label the elements of $\\mathord{\\mathcal E}_P(h)$ in such a  way that their dual graph is indicated  in Figure~\\ref{fig:ADE}.\n\\begin{figure} \n\\def40{40}\n\\def37{37}\n\\def25{25}\n\\def22{22}\n\\def10{10}\n\\def7{7}\n\\setlength{\\unitlength}{1.2mm}\n\\vskip .5cm\n\\centerline{\n{\\small\n\\begin{picture}(100, 37)(-20, 7)\n\\put(0, 40){$A\\sb l$}\n\\put(10, 40){\\circle{1}}\n\\put(9.5, 37){$a\\sb 1$}\n\\put(10.5, 40){\\line(5, 0){5}}\n\\put(16, 40){\\circle{1}}\n\\put(15.5, 37){$a\\sb 2$}\n\\put(16.5, 40){\\line(5, 0){5}}\n\\put(22, 40){\\circle{1}}\n\\put(21.5, 37){$a\\sb 3$}\n\\put(22.5, 40){\\line(5, 0){5}}\n\\put(30, 40){$\\dots\\dots\\dots$}\n\\put(45, 40){\\line(5, 0){5}}\n\\put(50.5, 40){\\circle{1}}\n\\put(50, 37){$a\\sb {l}$}\n\\put(0, 25){$D\\sb m$}\n\\put(10, 25){\\circle{1}}\n\\put(9.5, 22){$d\\sb 2$}\n\\put(10.5, 25){\\line(5, 0){5}}\n\\put(16, 31){\\circle{1}}\n\\put(17.5, 30.5){$d\\sb 1$}\n\\put(16, 25.5){\\line(0,1){5}}\n\\put(16, 25){\\circle{1}}\n\\put(15.5, 22){$d\\sb 3$}\n\\put(16.5, 25){\\line(5, 0){5}}\n\\put(22, 25){\\circle{1}}\n\\put(21.5, 22){$d\\sb 4$}\n\\put(22.5, 25){\\line(5, 0){5}}\n\\put(30, 25){$\\dots\\dots\\dots$}\n\\put(45, 25){\\line(5, 0){5}}\n\\put(50.5, 25){\\circle{1}}\n\\put(50, 22){$d\\sb {m}$}\n\\put(0, 10){$E\\sb n$}\n\\put(10, 10){\\circle{1}}\n\\put(9.5, 7){$e\\sb 2$}\n\\put(10.5, 10){\\line(5, 0){5}}\n\\put(16, 10){\\circle{1}}\n\\put(15.5, 7){$e\\sb 3$}\n\\put(22, 16){\\circle{1}}\n\\put(23.5, 15.5){$e\\sb 1$}\n\\put(22, 10.5){\\line(0,1){5}}\n\\put(16.5, 10){\\line(5, 0){5}}\n\\put(22, 10){\\circle{1}}\n\\put(21.5, 7){$e\\sb 4$}\n\\put(22.5, 10){\\line(5, 0){5}}\n\\put(30, 10){$\\dots\\dots\\dots$}\n\\put(45, 10){\\line(5, 0){5}}\n\\put(50.5, 10){\\circle{1}}\n\\put(50, 7){$e\\sb {n}$}\n\\end{picture}\n}\n}\n\\vskip 10pt\n\\caption{ Indecomposable $ADE$-configurations}\\label{fig:ADE}\n\\end{figure}\n\\par\nWe denote by $\\tau(h)\\in\\operatorname{\\mathrm {Aut}}\\nolimits(X)$ the involution of $X$\ninduced by  the Galois transformation of \nthe double covering $\\pi_h$,\nand call it a \\emph{double plane involution}.\nSuppose that a basis of $S_X$ and the Gram matrix of $\\intfS{\\phantom{i}, \\phantom{i}}$\nwith respect to this basis are given.\nSuppose also that \nwe have an ample list of vectors $\\mathord{\\textbf{\\itshape a}}$ such that\n$$\nD(\\mathord{\\textbf{\\itshape a}})=N(X)\n$$\nholds.\nThen we can calculate the matrix representation $M(h)$ \nof the action \nof $\\tau(h)$ on $S_X$   by the following method.\nIt is well known that there exists a successive blowing up $\\beta_h: F_h\\to \\P^2$ of $\\P^2$\nat  (possibly infinitely near) points of the singular locus of $B_h$\nsuch that\n$\\Phi_h$ factors as \n$$\nX\\maprightsp{q_h} F_h\\maprightsp{\\beta_h} \\P^2,\n$$\nwhere $q_h$ is the quotient morphism by $\\tau(h)$.\nLet $S_F$ denote the N\\'eron--Severi lattice of the smooth  rational surface  $F_h$.\nThen the pull-back $q_h^*$ by $q_h$ identifies $S_F\\otimes \\mathord{\\mathbb  Q}$ with the eigenspace of $\\tau(h)$ in $S_X\\otimes\\mathord{\\mathbb  Q}$\nwith eigenvalue $1$,\nand hence $\\tau(h)$ acts on the orthogonal complement of $q_h^*S_F\\otimes\\mathord{\\mathbb  Q}$ in  $S_X\\otimes\\mathord{\\mathbb  Q}$ as\nthe scalar multiplication by $-1$.\nOn the other hand, the subspace $q_h^*S_F\\otimes\\mathord{\\mathbb  Q}$ \nis generated by $h$  and\nthe vectors of the form $r+r^{\\tau(h)}$,\nwhere $r\\in \\mathord{\\mathcal E}_P(h)$ and $P\\in \\operatorname{\\mathrm {Sing}}\\nolimits (B_h)$.\nThe action of $\\tau(h)$  on $\\mathord{\\mathcal E}_P(h)$ is as follows:\n\\begin{itemize}\n\\item If $P$ is of type $A_{l}$, then $a_i^{\\tau(h)}=a_{l+1-i}$ for $i=1, \\dots, l$.\n\\item If $P$ is of type $D_{2k}$, then $\\tau(h)$ acts on $\\mathord{\\mathcal E}_P(h)$ as the identity.\n\\item If $P$ is of type $D_{2k+1}$, then $d_1^{\\tau(h)}=d_{2}$, $d_2^{\\tau(h)}=d_{1}$,\nand $d_i^{\\tau(h)}=d_{i}$  for $i=3, \\dots, 2k+1$.\n\\item If $P$ is of type $E_6$, then $e_1^{\\tau(h)}=e_{1}$, and $e_i^{\\tau(h)}=e_{8-i}$ for $i=2, \\dots, 6$.\n\\item If $P$ is of type $E_7$ or $E_8$,  then $\\tau(h)$ acts on $\\mathord{\\mathcal E}_P(h)$ as the identity.\n\\end{itemize}\nHence,  in order to calculate \nthe matrix representation $M(h)$ of $\\tau(h)$ on $S_X$, \nit is enough to calculate the sets $ \\mathord{\\mathcal E}_P(h)$.\n\\par\nWe put\n$$\n\\mathord{\\mathcal E}(h):=\\bigcup_{P\\in \\operatorname{\\mathrm {Sing}}\\nolimits(B_h)} \\mathord{\\mathcal E}_P(h).\n$$\nFirst we calculate the finite set \n$$\n\\mathord{\\mathcal R}^+(h):=\\set{r\\in \\mathord{\\mathcal R}(h)}{\\intfS {\\mathord{\\textbf{\\itshape a}}, r}>0}.\n$$\nNote that, since  $D(\\mathord{\\textbf{\\itshape a}})$ is equal to $N(X)$ and any $r\\in \\mathord{\\mathcal E}(h)$ is the class of a curve, \nwe have $\\intfS{\\mathord{\\textbf{\\itshape a}}, r}>0$ for any vector $r\\in \\mathord{\\mathcal E}(h)$.\nMoreover, any vector $r\\sp\\prime\\in \\mathord{\\mathcal R}^+(h)$ is the class of an effective divisor,\neach irreducible component of which is a smooth rational curve contracted by $\\Phi_h$.\nTherefore, we have $\\mathord{\\mathcal E}(h)\\subset \\mathord{\\mathcal R}^+(h)$.\nMoreover, \na vector $r\\sp\\prime\\in \\mathord{\\mathcal R}^+(h)$ is \na linear combination with nonnegative integer coefficients of vectors in $\\mathord{\\mathcal E}(h)$.\nConsequently,  a vector \n$r\\sp\\prime\\in \\mathord{\\mathcal R}^+(h)$ does  \\emph{not} belong to $\\mathord{\\mathcal E}(h)$\nif and only if\n$r\\sp\\prime$ can be written as a linear combination with nonnegative integer coefficients of vectors $r\\sp{\\prime\\prime}$ in $ \\mathord{\\mathcal R}^+(h)$\nsatisfying  $\\intfS{\\mathord{\\textbf{\\itshape a}}, r\\sp{\\prime\\prime}}< \\intfS{\\mathord{\\textbf{\\itshape a}}, r\\sp\\prime}$.\nThus, starting from the vector $r_0$ of $\\mathord{\\mathcal R}^+(h)$ with the smallest $\\intfS{\\mathord{\\textbf{\\itshape a}}, r_0}$,\nwe can successively detect the elements of $\\mathord{\\mathcal E}(h)$ in $\\mathord{\\mathcal R}^+(h)$.\nWe connect two distinct elements $r, r\\sp\\prime$ of $\\mathord{\\mathcal E}(h)$ by an edge if and only if\n$\\intfS{r, r\\sp\\prime}=1$.\nThen \nthe vertices of each  connected component of $\\mathord{\\mathcal E}(h)$ form the set  $\\mathord{\\mathcal E}_P(h)$.\n\\begin{remark}\nThis method of calculating the action of $\\tau(h)$ on $S_X$ was also used in finding  a finite set of generators\nof  $\\operatorname{\\mathrm {Aut}}\\nolimits(X)$ by Borcherds method \nin~\\cite{MR3190354} and~\\cite{1412.6904},\nand in the study of projective models of the supersingular $K3$ surface $X(5)$\nin characteristic $5$ with Artin invariant $1$ in~\\cite{MR3166075}.\n\\end{remark}\n\\section{N\\'eron--Severi lattices of supersingular $K3$ surfaces}\\label{sec:RS}\nRudakov and Shafarevich~\\cite{MR633161} \nproved the following theorems. \nFor the proof of Theorem~\\ref{thm:RS1}, \nsee also~\\cite[Chapter 15]{MR1662447}.\n\\begin{theorem}\\label{thm:RS1}\nLet $p$ be an odd prime, and let $\\sigma$ be  a positive integer less than or equal to $10$.\nThen there exists a lattice $\\Lambda_{p, \\sigma}^-$, unique up to  isomorphism,\nwith the following properties.\n{\\rm (i)} $\\Lambda_{p, \\sigma}^-$ is an even hyperbolic lattice of rank $22$.\n{\\rm (ii)} The discriminant group $(\\Lambda_{p, \\sigma}^-)\\sp{\\vee}\/\\Lambda_{p, \\sigma}^-$ of $\\Lambda_{p, \\sigma}^-$ is isomorphic to $(\\mathord{\\mathbb Z}\/p\\mathord{\\mathbb Z})^{2\\sigma}$.\n\\end{theorem}\n\\begin{theorem}\\label{thm:RS2}\nLet $X$ be a supersingular $K3$ surface in odd characteristic $p$ with Artin invariant $\\sigma$.\nThen its N\\'eron--Severi lattice $S_X$ is isomorphic to $\\Lambda_{p, \\sigma}^{-}$.\n\\end{theorem}\nAn explicit method of constructing  $\\Lambda_{p, \\sigma}^-$ is also given in~\\cite{MR633161} \n(see also~\\cite{MR2036331}).\nWe use the following construction,\nwhich is slightly different from the one given in~\\cite{MR633161}.\nThe ingredients of the construction are the following  lattices.\n\\par\n(i) Let $U$ and $U^{(p)}$ be the even hyperbolic lattices of rank $2$ with the Gram matrices\n\\begin{equation}\\label{eq:GramUUp}\n\\left[\\begin{array}{cc} 0 & 1 \\\\ 1 & 0\\end{array}\\right]\n\\quad\\textrm{and}\\quad \n\\left[\\begin{array}{cc} 0 & p \\\\ p & 0\\end{array}\\right],\n\\end{equation}\nrespectively.\n\\par\n(ii) Let $q$ be a prime satisfying\n$$\nq\\equiv 3 \\bmod 8 \\quad\\textrm{and}\\quad  \\left(\\frac{-q}{p}\\right)=-1, \n$$\nand let $\\gamma$ be an integer satisfying $\\gamma^2+p\\equiv 0 \\bmod q$.\nLet $H^{(-p)}$ be the even \\emph{negative} definite lattice of rank $4$\nwith the Gram matrix\n$$\n\\renewcommand{\\arraystretch}{1.4}\n(-1)\\left[\\begin{array}{cccc} \n2 & 1 & 0 & 0 \\\\\n1 & (q+1)\/2 & 0 & \\gamma\\\\\n0 & 0  & p(q+1)\/2 & p \\\\\n0 & \\gamma & p & 2(p+\\gamma^2)\/q\n\\end{array}\\right].\n$$\nThen the discriminant group of $H^{(-p)}$ is isomorphic to $(\\mathord{\\mathbb Z}\/p\\mathord{\\mathbb Z})^2$.\nSee~\\cite{MR0568309} and~\\cite{MR2036331}.\n\\par\n(iii) \nLet $E_8$ denote the root lattice of type $E_8$,\nwhich is an even unimodular positive definite lattice of rank $8$.\nThen $E_8$ has a \\emph{standard basis }$e_1, \\dots, e_8$,\nwhose dual graph is given in Figure~\\ref{fig:ADE}.\nLet $E_8^{(-1)}$ be the lattice obtained from $E_8$ by multiplying the intersection form by $-1$,\nand let $E_8^{(-p)}$ be the lattice obtained from $E_8^{(-1)}$ by multiplying the intersection form by $p$.\nThen the discriminant group of $E_8^{(-p)}$ is isomorphic to $(\\mathord{\\mathbb Z}\/p\\mathord{\\mathbb Z})^8$.\n\\par\nThen $\\Lambda_{p, \\sigma}^-$ is isomorphic to the following lattices:\n\\begin{equation*}\\label{eq:UHEE}\n\\renewcommand{\\arraystretch}{1.4}\n\\begin{array}{ll}\nU\\oplus H^{(-p)} \\oplus E_8^{(-1)} \\oplus E_8^{(-1)} & \\textrm{if $\\sigma=1$, }\\\\\nU^{(p)}\\oplus H^{(-p)} \\oplus  E_8^{(-1)} \\oplus E_8^{(-1)} & \\textrm{if $\\sigma=2$, }\\\\\nU\\oplus H^{(-p)} \\oplus  H^{(-p)} \\oplus H^{(-p)}  \\oplus E_8^{(-1)} & \\textrm{if $\\sigma=3$, }\\\\\nU^{(p)}\\oplus H^{(-p)} \\oplus  H^{(-p)} \\oplus H^{(-p)}   \\oplus E_8^{(-1)} & \\textrm{if $\\sigma=4$, }\\\\\nU\\oplus H^{(-p)}  \\oplus E_8^{(-1)} \\oplus E_8^{(-p)} & \\textrm{if $\\sigma=5$, }\\\\\nU^{(p)}\\oplus H^{(-p)} \\oplus E_8^{(-1)} \\oplus E_8^{(-p)} & \\textrm{if $\\sigma=6$, }\\\\\nU\\oplus  H^{(-p)} \\oplus  H^{(-p)} \\oplus H^{(-p)}  \\oplus E_8^{(-p)} & \\textrm{if $\\sigma=7$, }\\\\\nU^{(p)}\\oplus H^{(-p)} \\oplus  H^{(-p)} \\oplus H^{(-p)}   \\oplus E_8^{(-p)} & \\textrm{if $\\sigma=8$, }\\\\\nU\\oplus H^{(-p)} \\oplus  E_8^{(-p)} \\oplus E_8^{(-p)} & \\textrm{if $\\sigma=9$, }\\\\\nU^{(p)}\\oplus H^{(-p)} \\oplus  E_8^{(-p)} \\oplus E_8^{(-p)} & \\textrm{if $\\sigma=10$.}\\\\\n\\end{array}\n\\end{equation*}\nLet $\\intfLL{\\phantom{i}, \\phantom{i}}$ denote the intersection form of $\\Lambda_{p, \\sigma}^-$.\nNote that $\\Lambda_{p, \\sigma}^-$ has the form of the orthogonal direct sum \n$$\nU\\sp\\prime\\oplus N,\n$$\nwhere $U\\sp\\prime$ is $U$ or $U^{(p)}$ according to the parity of $\\sigma$,\nand $N$ is an even negative definite lattice \nwith the intersection form $\\intfN{\\phantom{i}, \\phantom{i}}$.\nWe put\n$$\np\\sp\\prime:=\\begin{cases}\n1 & \\textrm{if $U\\sp\\prime$ is $U$}, \\\\\np & \\textrm{if $U\\sp\\prime$ is $U^{(p)}$}.\n\\end{cases}\n$$\nWe choose a vector $n\\in N$ randomly.\nIf $2-\\intfN{n, n}$ is divisible by $2\\,p\\sp\\prime$,\nthen we can find a vector $u\\in U\\sp\\prime$ such that $v:=u+n\\in \\Lambda_{p, \\sigma}^-$ satisfies \n$\\intfLL{v, v}=2$.\nBy this method, we can generate many vectors of $\\Lambda_{p, \\sigma}^-$ with square-norm $2$.\n\\section{Generating double plane involutions}\nWe fix an odd prime $p$ and a positive integer $\\sigma$ less than or equal to $10$.\nLet $X$ be a supersingular $K3$ surface in characteristic $p$\nwith Artin invariant $\\sigma$.\nWe make a set $\\mathord{\\mathcal M}$ of matrix representations \non $S_X$ of double plane involutions $\\tau(h)\\in \\operatorname{\\mathrm {Aut}}\\nolimits(X)$ associated with\n polarizations $h\\in S_X$ of degree $2$.\n %\n \\begin{itemize}\n \\item[(0)] We set $\\mathord{\\mathcal M}=\\{\\}$.\n \\item[(1)] We construct a Gram matrix of the lattice $\\Lambda_{p, \\sigma}^-$ by the result in Section~\\ref{sec:RS}.\n \\item[(2)] We find a vector $h_0\\in \\Lambda_{p, \\sigma}^-$ such that $\\intfLL{h_0, h_0}>0$.\nLet  $\\mathord{\\mathcal P}_{\\Lambda}$ be the positive cone of $\\Lambda_{p, \\sigma}^-$\n containing $h_0$.\n\\item[(3)]\n We calculate $\\mathord{\\mathcal R}(h_0)$,\n and choose an ample list of vectors\n $$\n \\mathord{\\textbf{\\itshape a}}:=[h_0, \\rho_1, \\dots, \\rho_K].\n $$\n\\item[(4)]\n By Theorem~\\ref{thm:RS2},\nthere exists an isomorphism $\\iota: \\Lambda_{p, \\sigma}^-\\mathbin{\\,\\raise -.6pt\\rlap{$\\to$}\\raise 3.5pt \\hbox{\\hskip .3pt$\\mathord{\\sim}$}\\,\\;} S_X$ of lattices.\n Multiplying $\\iota$ by $-1$ if necessary,\n we can assume that $\\iota$ maps $\\mathord{\\mathcal P}_{\\Lambda}$ to $\\mathord{\\mathcal P}(X)$.\n Composing $\\iota$ with an element of $\\Weyl{S_X}$  if necessary,\n we can further assume that $\\iota$ maps $D(\\mathord{\\textbf{\\itshape a}})$ to $N(X)$.\n From now on, we identify $\\Lambda_{p, \\sigma}^-$ with $S_X$,  and $D(\\mathord{\\textbf{\\itshape a}})$ with $N(X)$ by \n the isometry $\\iota$.\n \\item[(5)]\n We make a finite  set $\\mathord{\\mathcal V}$ of vectors $v\\in \\Lambda_{p, \\sigma}^-$ with $\\intfLL{v, v}=2$\n by the method described in Section~\\ref{sec:RS}.\n\\item[(6)] For each $v\\in \\mathord{\\mathcal V}$, we execute  the following calculations.\n\\begin{itemize}\n\\item[(6-1)] If $\\intfLL{v, h_0}<0$, then we replace $v$ with $-v$,\n so that we can assume that $v\\in \\mathord{\\mathcal P}_{\\Lambda}$.\n\\item[(6-2)]  We calculate $\\mathord{\\mathcal F}(v)$.\n If $\\mathord{\\mathcal F}(v)\\ne\\emptyset$, we proceed  to the next element of $\\mathord{\\mathcal V}$.\n If $\\mathord{\\mathcal F}(v)=\\emptyset$, we go to Step (6-3).\n\\item[(6-3)]\n From $v$, we construct the vector $h_v\\in \\Lambda_{p, \\sigma}^-$  with $\\intfLL{h_v, h_v}=2$ that belongs to $D(\\mathord{\\textbf{\\itshape a}})$\n by the method described in Section~\\ref{sec:lattice}.\n Since $h_v$ and $v$ are related by~\\eqref{eq:hvsss},\n we have  $\\mathord{\\mathcal F}(h_v)=\\emptyset$.\nBy the identification of $D(\\mathord{\\textbf{\\itshape a}})$ with $N(X)$,\n we see that $h_v$ is nef.\n Therefore, by Proposition~\\ref{prop:Nikulindeg2pol},\n we see that $h_v$ is a polarization of degree $2$.\n\\item[(6-4)]\n We then calculate the matrix representation $M(h_v)$ of \n the double plane involution $\\tau(h_v)\\in \\operatorname{\\mathrm {Aut}}\\nolimits(X)$ by the method described in Section~\\ref{sec:poldeg2},\n and append $M(h_v)$ to $\\mathord{\\mathcal M}$.\n %\n \\end{itemize}\n  \\end{itemize}\nOnce we make a sufficiently large set\n$$\n\\mathord{\\mathcal M}=\\{M(h_1), \\dots, M(h_N)\\}\n$$\nof $22\\times 22$ matrices representing the action of double plane involutions of $X$ on $S_X$,\nwe make a product\n$$\nM:=M(h_{i_1}) \\cdots M(h_{i_\\nu})\n$$\nof randomly chosen elements of $\\mathord{\\mathcal M}$,\nand calculate its characteristic polynomial $\\phi_{M}(t)$.\nBy Theorem~\\ref{thm:cycsSalem},\nif $\\phi_M(t)$ is irreducible in $\\mathord{\\mathbb Z}[t]$ and not equal to the cyclotomic polynomial $(t^{23}-1)\/(t-1)$,\nthen $\\phi_M(t)$ is a Salem polynomial.\n\\par\nBy this method,\nwe confirm that, if $p$ is an odd prime $\\le 7919$,\nthen $\\operatorname{\\mathrm {Aut}}\\nolimits(X)$ contains an automorphism of irreducible Salem type\nthat is a product of at most $22$ double plane involutions.\n\\begin{remark}\\label{rem:infample}\nLet $\\mathord{\\textbf{\\itshape e}}_1, \\dots, \\mathord{\\textbf{\\itshape e}}_{22}$ be a basis of $\\Lambda_{p, \\sigma}^-$,\nand let  $\\mathord{\\textbf{\\itshape e}}_1\\sp{\\vee}, \\dots, \\mathord{\\textbf{\\itshape e}}_{22}\\sp{\\vee}$ be the dual basis.\nNote that $p\\mathord{\\textbf{\\itshape e}}_i\\sp{\\vee} \\in \\Lambda_{p, \\sigma}^-$ holds for $i=1, \\dots, 22$.\nHence, in Step (3), \n we can choose \n$[h_0, p\\mathord{\\textbf{\\itshape e}}_1\\sp{\\vee}, \\dots, p\\mathord{\\textbf{\\itshape e}}_{22}\\sp{\\vee}]$\nas an ample list of vectors.\n\\end{remark}\n\\section{Supersingular $K3$ surfaces with Artin invariant $10$}\\label{sec:sigma10}\nWe consider a supersingular $K3$ surface $X$ in characteristic $p\\ge 11$ with Artin invariant $10$.\nWe have  \n$$\n\\Lambda_{p, 10}^{-}=U^{(p)}\\oplus H^{(-p)}\\oplus E_8^{(-p)} \\oplus E_8^{(-p)}.\n$$\nLet $u_1, u_2$ be the basis of $U^{(p)}$ with the Gram matrix ~\\eqref{eq:GramUUp},\nand let $e_1, \\dots, e_8$ (resp.~$e_1\\sp\\prime, \\dots, e_8\\sp\\prime$) be the standard basis of the first $E_8^{(-p)}$ (resp.  the second $E_8^{(-p)}$).\n In particular, each $e_{\\nu}$ or $e_{\\nu}\\sp\\prime$ is of square-norm $-2p$.\n For $v\\in H^{(-p)}$ and $a\\in \\mathord{\\mathbb Z}$,\n we denote by\n $$\n (a, 1, v)\\; \\in\\;  U^{(p)}\\oplusH^{(-p)}\n $$\n the vector $au_1+u_2+v$.\n Then the square-norm of $(a, 1, v)$ is $2pa+\\intfH{v, v}$,\n where  $\\intfH{\\phantom{a}, \\phantom{a}}$ is the intersection form of $H^{(-p)}$.\n Note that, if $(a, 1, v)\\in U^{(p)}\\oplusH^{(-p)}$ is of square-norm $2$, then  the vectors $(a+1, 1, v)+e_{\\nu}$ and $(a+1, 1, v)+e\\sp\\prime_{\\nu}$\n of $\\Lambda_{p, 10}^{-}$ are also of square-norm $2$\n for $\\nu=1, \\dots, 8$.\n\\par\nFor $p$ with $11\\le p\\le 17389$, we have found  six vectors $v_k \\in H^{(-p)}$ and six positive integers $a_k\\in \\mathord{\\mathbb Z}$\nwith the following properties (i)--(v).\n\\begin{itemize}\n\\item[(i)] The vector $h_k:=(a_k, 1, v_k)$ is of square-norm $2$ for $k=1, \\dots, 6$.\n\\end{itemize}\nWe put\n$$\nh_{6+\\nu} :=(a_k+1, 1, v_k)+e_{\\nu}, \\quad\nh_{14+\\nu} :=(a_k+1, 1, v_k)+e\\sp\\prime_{\\nu},\n$$\nfor $\\nu=1, \\dots, 8$.\nThen $h_7, \\dots, h_{22}$ are also of square-norm $2$.\n\\begin{itemize}\n\\item[(ii)] $\\intfLL{h_1, h_i}>0$ for $i=2, \\dots, 22$.\n\\item[(iii)] $\\mathord{\\mathcal S}(h_1, h_i)=\\emptyset$ for $i=2, \\dots, 22$.\n\\item[(iv)] $\\mathord{\\mathcal R}(h_i)=\\emptyset$ and $\\mathord{\\mathcal F}(h_i)=\\emptyset$ for $i=1, \\dots, 22$.\n\\end{itemize}\nSince $R(h_1)=\\emptyset$,\nthere exists a unique standard fundamental domain $D([h_1])$ of the Weyl group  $W(\\Lambda_{p, \\sigma}^-)$ \nthat contains $h_1$ in its interior.\nSince  $\\mathord{\\mathcal S}(h_1, h_i)=\\emptyset$ for $i=2, \\dots, 22$,\nwe see that $h_1, \\dots, h_{22}$ are also contained in $D([h_1])$.\nHence, under a suitable isometry $\\Lambda_{p, 10}^{-}\\mathbin{\\,\\raise -.6pt\\rlap{$\\to$}\\raise 3.5pt \\hbox{\\hskip .3pt$\\mathord{\\sim}$}\\,\\;} S_X$,\nwe can assume that each $h_i$ is a nef vector in $S_X$.\nSince $\\mathord{\\mathcal F}(h_i)=\\emptyset$ for $i=1, \\dots, 22$, \nwe see that each $h_i$ is \na polarization of degree $2$ on $X$.\nMoreover, since $\\mathord{\\mathcal R}(h_i)=\\emptyset$,\nthe branch curve $B_{h_i}\\subset \\P^2$ \nof the double plane involution $\\tau(h_i)$ is smooth.\nHence  $\\tau(h_i)$ acts on $h_i$ trivially,\nand on the orthogonal compliment of $h_i$ as the multiplication by $-1$.\n\\begin{itemize}\n\\item[(v)] The product $g:=\\tau(h_1)\\cdots \\tau (h_{22})$ is of irreducible Salem type.\n\\end{itemize}\nThis observation and a computer-aided calculation give the proof of Theorem~\\ref{thm:main2}.\n\\begin{example}\nConsider the case $p=17389$.\nThen $H^{(-p)}$ has a Gram matrix\n$$\n\\left[ \\begin {array}{cccc} -2&-1&0&0\\\\ \\noalign{\\medskip}-1&-30&0&-4\n\\\\ \\noalign{\\medskip}0&0&-521670&-17389\\\\ \\noalign{\\medskip}0&-4&-\n17389&-590\\end {array} \\right]\n$$\nunder a certain basis $\\eta_1, \\dots, \\eta_4$ of $H^{(-p)}$.\nThe vectors\n$$\n\\begin{array}{rccrrrrrrrc}\nh_1 &=& [&1, &1, &15, &31, &0, &-3&],\\\\\nh_2 &=&[&1, &1, &9, &18, &-1, &25&],\\\\\nh_3 &=&[&1, &1, &51, &4, &0, &-7&],\\\\\nh_4 &=&[&1, &1, &30, &29, &0, &3&],\\\\\nh_5 &=&[&1, &1, &55, &-4, &0, &7&],\\\\\nh_6 &=&[&2, &1, &19, &23, &-2, &56&],\n\\end{array}\n$$\nof $U^{(p)}\\oplus H^{(-p)}$ written with respect to the basis $u_1, u_2, \\eta_1, \\dots, \\eta_4$\nsatisfies the properties (i)--(v).\nThe characteristic polynomial on $S_X$  of the automorphism $g$\nobtained from these six vectors has a real root \n${4.2539 \\dots} \\times 10^{100}$.\n\\end{example}\n\\begin{remark}\nLet $g_p$ be the automorphism of a supersingular $K3$ surface $X$ with Artin invariant $10$\nin characteristic $p$\nobtained by the method described in this section,\nlet $\\rho_p$ be the real root $>1$ of the characteristic polynomial of $g_p$ on $S_X$, \nand let $\\lambda_p:=\\log \\rho_p$ be the \\emph{entropy} of $g_p$.\nThen, for $11\\le p\\le 17389$,\nwe 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9.181211){\\circle*{0.020}}\n\\put(9.718302, 9.158571){\\circle*{0.020}}\n\\put(9.719024, 9.223986){\\circle*{0.020}}\n\\put(9.719144, 9.291634){\\circle*{0.020}}\n\\put(9.720105, 9.157278){\\circle*{0.020}}\n\\put(9.720226, 9.251883){\\circle*{0.020}}\n\\put(9.720586, 9.349471){\\circle*{0.020}}\n\\put(9.720826, 9.244185){\\circle*{0.020}}\n\\put(9.721546, 9.293738){\\circle*{0.020}}\n\\put(9.722625, 9.213249){\\circle*{0.020}}\n\\put(9.722745, 9.198761){\\circle*{0.020}}\n\\put(9.723104, 9.334989){\\circle*{0.020}}\n\\put(9.723344, 9.263217){\\circle*{0.020}}\n\\put(9.724899, 9.101930){\\circle*{0.020}}\n\\put(9.725616, 9.252990){\\circle*{0.020}}\n\\put(9.725974, 9.188022){\\circle*{0.020}}\n\\put(9.726691, 9.333931){\\circle*{0.020}}\n\\put(9.726929, 9.077527){\\circle*{0.020}}\n\\put(9.728360, 9.109381){\\circle*{0.020}}\n\\put(9.729789, 9.164001){\\circle*{0.020}}\n\\put(9.730502, 9.262671){\\circle*{0.020}}\n\\put(9.730859, 9.066877){\\circle*{0.020}}\n\\put(9.730978, 9.433654){\\circle*{0.020}}\n\\put(9.731690, 9.317686){\\circle*{0.020}}\n\\put(9.733351, 9.298808){\\circle*{0.020}}\n\\put(9.733826, 9.421647){\\circle*{0.020}}\n\\put(9.734062, 9.239728){\\circle*{0.020}}\n\\put(9.734418, 9.113037){\\circle*{0.020}}\n\\put(9.735128, 9.220083){\\circle*{0.020}}\n\\put(9.735246, 9.324679){\\circle*{0.020}}\n\\put(9.736311, 9.357092){\\circle*{0.020}}\n\\put(9.736665, 9.332483){\\circle*{0.020}}\n\\put(9.736902, 9.212966){\\circle*{0.020}}\n\\put(9.737256, 9.272733){\\circle*{0.020}}\n\\put(9.737610, 9.254392){\\circle*{0.020}}\n\\put(9.738790, 9.369242){\\circle*{0.020}}\n\\put(9.739733, 9.250246){\\circle*{0.020}}\n\\put(9.739850, 9.261781){\\circle*{0.020}}\n\\put(9.740204, 9.370812){\\circle*{0.020}}\n\\put(9.740557, 9.420381){\\circle*{0.020}}\n\\put(9.741615, 9.320142){\\circle*{0.020}}\n\\put(9.742203, 9.215776){\\circle*{0.020}}\n\\put(9.742556, 9.271037){\\circle*{0.020}}\n\\put(9.742673, 9.344697){\\circle*{0.020}}\n\\put(9.742908, 9.335528){\\circle*{0.020}}\n\\put(9.743377, 9.251197){\\circle*{0.020}}\n\\put(9.743730, 9.160737){\\circle*{0.020}}\n\\put(9.744081, 9.355746){\\circle*{0.020}}\n\\put(9.745488, 9.395905){\\circle*{0.020}}\n\\put(9.746424, 9.175377){\\circle*{0.020}}\n\\put(9.746775, 9.167516){\\circle*{0.020}}\n\\put(9.747243, 9.421956){\\circle*{0.020}}\n\\put(9.747827, 9.194110){\\circle*{0.020}}\n\\put(9.748178, 9.205908){\\circle*{0.020}}\n\\put(9.748995, 9.114321){\\circle*{0.020}}\n\\put(9.750278, 9.329923){\\circle*{0.020}}\n\\put(9.750744, 9.250935){\\circle*{0.020}}\n\\put(9.751676, 9.092084){\\circle*{0.020}}\n\\put(9.752025, 9.151429){\\circle*{0.020}}\n\\put(9.752141, 9.198637){\\circle*{0.020}}\n\\put(9.752839, 9.337036){\\circle*{0.020}}\n\\put(9.753072, 9.226441){\\circle*{0.020}}\n\\put(9.753188, 9.265195){\\circle*{0.020}}\n\\put(9.754465, 9.055304){\\circle*{0.020}}\n\\put(9.754930, 9.263710){\\circle*{0.020}}\n\\put(9.755973, 9.497330){\\circle*{0.020}}\n\\put(9.757941, 9.116729){\\circle*{0.020}}\n\\put(9.758057, 9.219917){\\circle*{0.020}}\n\\put(9.758404, 9.243151){\\circle*{0.020}}\n\\put(9.759444, 9.388938){\\circle*{0.020}}\n\\put(9.759675, 9.154612){\\circle*{0.020}}\n\\put(9.760021, 9.174406){\\circle*{0.020}}\n\\put(9.760367, 9.212282){\\circle*{0.020}}\n\\put(9.760829, 9.385373){\\circle*{0.020}}\n\\put(9.761405, 9.216163){\\circle*{0.020}}\n\\put(9.761866, 9.337444){\\circle*{0.020}}\n\\put(9.762903, 9.362927){\\circle*{0.020}}\n\\put(9.763248, 9.250060){\\circle*{0.020}}\n\\put(9.763478, 9.284181){\\circle*{0.020}}\n\\put(9.763593, 9.268254){\\circle*{0.020}}\n\\end{picture}\n}\n\\caption{Growth of the entropy}\\label{fig:entropy}\n\\end{figure}\n\\section{An example with Artin invariant $1$}\\label{sec:example}\nWe denote by  $X(p)$ a supersingular $K3$ surface in characteristic $p$ with Artin invariant $1$,\nwhich is unique up to isomorphism by the result of Ogus~\\cite{MR563467, MR717616}. \nThe existence of an automorphism $g\\in \\operatorname{\\mathrm {Aut}}\\nolimits(X(p))$ of irreducible Salem type \nwas established by Blanc and Cantat~\\cite{1307.0361} for $p=2$,  by Esnault and Oguiso~\\cite{1406.2761} for $p=3$, and \nby Esnault, Oguiso, and Yu ~\\cite{1411.0769} for $p=11$ or $p \\ge 17$.\nOn the other hand, in~\\cite{1502.06923}, Sch\\\"utt showed that, \nif  $p$ is odd and  satisfies $p \\equiv 2 \\bmod 3$, then \nthere exists  a non-liftable automorphism  of  $X(p)$\nwhose characteristic polynomial  on  $S_{X(p)}$\nis divisible by  a Salem polynomial of degree $20$.\n\\par\nWe consider the supersingular $K3$ surface $X(7)$,\nwhich has not yet been  treated  by the previous works.\nThe lattice  $\\Lambda_{7, 1}^{-}=U\\oplus H^{(-7)} \\oplus E_8^{(-1)}\\oplus E_8^{(-1)}$ has a basis \n$\\mathord{\\textbf{\\itshape e}}_1, \\dots, \\mathord{\\textbf{\\itshape e}}_{22}$ such that \n$\\mathord{\\textbf{\\itshape e}}_1$ and $\\mathord{\\textbf{\\itshape e}}_2$ form a basis of $U$ with the Gram matrix~\\eqref{eq:GramUUp},\n$\\mathord{\\textbf{\\itshape e}}_3, \\dots, \\mathord{\\textbf{\\itshape e}}_6$  form a basis of $H^{(-7)}$ with the Gram matrix\n$$\n\\left[ \\begin {array}{cccc} -2&-1&0&0\\\\ \\noalign{\\medskip}-1&-6&0&-2\n\\\\ \\noalign{\\medskip}0&0&-42&-7\\\\ \\noalign{\\medskip}0&-2&-7&-2\n\\end {array} \\right] , \n$$\nand $\\mathord{\\textbf{\\itshape e}}_7, \\dots, \\mathord{\\textbf{\\itshape e}}_{14}$  (resp.,  $\\mathord{\\textbf{\\itshape e}}_{15}, \\dots, \\mathord{\\textbf{\\itshape e}}_{22}$) form the standard basis of the first $E_8^{(-1)}$ (resp., the second $E_8^{(-1)}$).\nWe put\n$$\nh_0 := [1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] \\in \\Lambda_{7, 1}^{-},\n$$\nwhich is of square-norm $2$.\nThe set $\\mathord{\\mathcal R}(h_0)$ consists of $486$ vectors.\nThe list\n$$\n\\mathord{\\textbf{\\itshape a}}:=[h_0, 7\\mathord{\\textbf{\\itshape e}}_1\\sp{\\vee} , \\dots, 7\\mathord{\\textbf{\\itshape e}}_{22}\\sp{\\vee}]\n$$\nis an ample list of vectors.\nWe identify $\\Lambda_{7, 1}^{-}$ with $S_{X(7)}$ by an isometry $\\Lambda_{7, 1}^{-}\\mathbin{\\,\\raise -.6pt\\rlap{$\\to$}\\raise 3.5pt \\hbox{\\hskip .3pt$\\mathord{\\sim}$}\\,\\;} S_{X(7)}$\nthat maps $D(\\mathord{\\textbf{\\itshape a}})$ to $N(X(7))$.\n(Since $\\mathord{\\mathcal F}(h_0)\\ne \\emptyset$,\nthe vector $h_0$ is \\emph{not} a polarization of degree $2$.)\n\\par\nWe consider the three vectors\n\\begin{eqnarray*}\nh_1 &:=&[5, 5, -2, 3, 2, -11, -12, -8, -16, -24, -20, -15, -10,\\\\\n&& \\qquad  -5, -8,  -5, -10, -15, -12, -9, -6, -3], \\\\\nh_2 &:=&[5, 5, -1, 0, 0, -2, -13, -9, -17, -25, -20, -15, -10, \\\\\n&& \\qquad  -5, -11,  -7, -14, -21, -17, -13, -9, -5],\\\\\nh_3 &:=&[3, 6, -2, 2, 2, -9, -5, -4, -7, -10, -8, -6, -4, -2, 0, 0, 0, 0,  0, 0, 0, 0],\n\\end{eqnarray*}\nof square-norm $2$.\n By means of Lemma~\\ref{lem:inDa},\n we can confirm that $h_1, h_2, h_3$ are located in $D(\\mathord{\\textbf{\\itshape a}})=N(X(7))$.\n Moreover we have $\\mathord{\\mathcal F}(h_1)=\\mathord{\\mathcal F}(h_2)=\\mathord{\\mathcal F}(h_3)=\\emptyset$.\n Hence these $h_i$ are polarizations of degree $2$,\n and induce double plane involutions $\\tau(h_i)$.\n The type of the singularities of the branch curve $B_{h_i}$ is\n %\n %\n\n %\n $$\nA_4+A_5+A_7, \\quad\n2A_1+A_7+A_9, \\quad\nA_2+D_7+E_8, \n $$\n respectively.\n %\n %\n\\begin{figure}\n{\\tiny \n\\setlength{\\arraycolsep}{1.8pt} \n$$\n\\left[ \\begin {array}{cccccccccccccccccccccc} 24&24&-10&15&10&-55&-57\n&-38&-76&-114&-95&-71&-48&-24&-40&-25&-50&-75&-60&-45&-30&-15\n\\\\ \\noalign{\\medskip}24&24&-10&15&10&-55&-57&-38&-76&-114&-95&-72&-48&\n-24&-40&-25&-50&-75&-60&-45&-30&-15\\\\ \\noalign{\\medskip}5&5&-3&3&2&-11\n&-12&-8&-16&-24&-20&-15&-10&-5&-8&-5&-10&-15&-12&-9&-6&-3\n\\\\ \\noalign{\\medskip}30&30&-12&17&12&-66&-72&-48&-96&-144&-120&-90&-60\n&-30&-48&-30&-60&-90&-72&-54&-36&-18\\\\ \\noalign{\\medskip}-35&-35&14&-\n21&-15&77&84&56&112&168&140&105&70&35&56&35&70&105&84&63&42&21\n\\\\ \\noalign{\\medskip}10&10&-4&6&4&-23&-24&-16&-32&-48&-40&-30&-20&-10&\n-16&-10&-20&-30&-24&-18&-12&-6\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&1&0&0\n&0&0&0&0&0&0&0&0&0&0&0&0\\\\ \\noalign{\\medskip}0&0&0&0&0&0&1&0&0&0&0&0&0\n&0&0&0&0&0&0&0&0&0\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0\n&0&0&0&0&0&0\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&0\n&0&0&0\\\\ \\noalign{\\medskip}4&5&-2&3&2&-11&-10&-7&-14&-20&-16&-12&-8&-4\n&-8&-5&-10&-15&-12&-9&-6&-3\\\\ \\noalign{\\medskip}1&-1&0&0&0&0&0&0&0&0&0\n&0&0&0&0&0&0&0&0&0&0&0\\\\ \\noalign{\\medskip}0&1&0&0&0&0&-3&-2&-4&-6&-5&\n-4&-3&-2&0&0&0&0&0&0&0&0\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0&0&0&0&0\n&1&0&0&0&0&0&0&0&0\\\\ \\noalign{\\medskip}5&5&-2&3&2&-11&-12&-8&-16&-24&-\n20&-15&-10&-5&-9&-6&-12&-18&-15&-12&-8&-4\\\\ \\noalign{\\medskip}0&0&0&0&0\n&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0\n&0&0&0&0&0&0&0&0&0&0&0&0&1&0\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0&0&0\n&0&0&0&0&0&0&0&0&1&0&0\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0&0&0&0&0&0\n&0&0&0&0&1&0&0&0\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\n&1&0&0&0&0\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0\n&0&0\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0\n\\end {array} \\right] \n$$\n}\n\\caption{$M(h_1)$}\\label{fig:Mh1}\n\\end{figure}\n\\begin{figure}\n{\\tiny\n\\setlength{\\arraycolsep}{2.4pt} \n$$\n\\left[ \\begin {array}{cccccccccccccccccccccc} 6&6&0&0&0&-3&-15&-10&-\n20&-29&-24&-18&-12&-6&-12&-8&-16&-24&-20&-16&-12&-6\n\\\\ \\noalign{\\medskip}6&6&0&0&0&-3&-15&-10&-20&-30&-24&-18&-12&-6&-12&-\n8&-16&-24&-20&-16&-12&-6\\\\ \\noalign{\\medskip}0&0&1&0&0&0&0&0&0&0&0&0&0\n&0&0&0&0&0&0&0&0&0\\\\ \\noalign{\\medskip}6&6&1&-1&0&-2&-16&-10&-20&-30&-\n24&-18&-12&-6&-12&-8&-16&-24&-20&-16&-12&-6\\\\ \\noalign{\\medskip}21&21&0\n&0&-1&-7&-56&-35&-70&-105&-84&-63&-42&-21&-42&-28&-56&-84&-70&-56&-42&\n-21\\\\ \\noalign{\\medskip}6&6&0&0&0&-3&-16&-10&-20&-30&-24&-18&-12&-6&-\n12&-8&-16&-24&-20&-16&-12&-6\\\\ \\noalign{\\medskip}0&1&0&0&0&-1&0&0&0&0&0\n&0&0&0&0&0&0&0&0&0&0&0\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0&0&0&0&0&0\n&0&0&0&0&0&0&0&1\\\\ \\noalign{\\medskip}0&1&0&0&0&0&0&0&0&0&0&0&0&0&-3&-2\n&-4&-6&-5&-4&-3&-2\\\\ \\noalign{\\medskip}1&-1&0&0&0&0&0&0&0&0&0&0&0&0&0&0\n&0&0&0&0&0&0\\\\ \\noalign{\\medskip}0&1&0&0&0&0&-3&-2&-4&-6&-5&-4&-3&-2&0\n&0&0&0&0&0&0&0\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0\n&0&0&0&0\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0\n&0\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&0\n\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0\n\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0\n\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0\n\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0\n\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0\n\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0\n\\\\ \\noalign{\\medskip}2&2&0&0&0&-1&-5&-4&-7&-10&-8&-6&-4&-2&-5&-4&-7&-\n10&-8&-6&-4&-2\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&0\n&0&0&0&0\\end {array} \\right] \n$$\n}\n\\caption{$M(h_2)$}\\label{fig:Mh2}\n\\end{figure}\n\\begin{figure}\n{\\tiny \n\\setlength{\\arraycolsep}{3.5pt} \n$$\n\\left[ \\begin {array}{cccccccccccccccccccccc} 14&27&-9&9&9&-42&-30&-\n24&-42&-60&-48&-36&-24&-12&0&0&0&0&0&0&0&0\\\\ \\noalign{\\medskip}8&14&-5\n&5&5&-23&-15&-12&-21&-30&-24&-18&-12&-6&0&0&0&0&0&0&0&0\n\\\\ \\noalign{\\medskip}5&9&-4&3&3&-14&-10&-8&-14&-20&-16&-12&-8&-4&0&0&0\n&0&0&0&0&0\\\\ \\noalign{\\medskip}21&39&-13&12&13&-60&-40&-32&-56&-80&-64\n&-48&-32&-16&0&0&0&0&0&0&0&0\\\\ \\noalign{\\medskip}-49&-84&28&-28&-29&\n133&105&84&147&210&168&126&84&42&0&0&0&0&0&0&0&0\\\\ \\noalign{\\medskip}1\n&3&-1&1&1&-5&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\\ \\noalign{\\medskip}0&0&0\n&0&0&0&0&0&1&0&0&0&0&0&0&0&0&0&0&0&0&0\\\\ \\noalign{\\medskip}3&6&-2&2&2&\n-9&-8&-5&-10&-15&-12&-9&-6&-3&0&0&0&0&0&0&0&0\\\\ \\noalign{\\medskip}0&0&0\n&0&0&0&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\\ \\noalign{\\medskip}0&0&0&0&0&0\n&0&0&0&1&0&0&0&0&0&0&0&0&0&0&0&0\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0\n&0&1&0&0&0&0&0&0&0&0&0&0&0\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0&0&0&1\n&0&0&0&0&0&0&0&0&0&0\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0&0&0&0&1&0&0\n&0&0&0&0&0&0&0\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0\n&0&0&0&0\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0\n&0\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0&0\n\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0&0\n\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0&0\n\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0\n\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0&0\n\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1&0\n\\\\ \\noalign{\\medskip}0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&1\n\\end {array} \\right] \n$$\n}\n\\caption{$M(h_3)$}\\label{fig:Mh3}\n\\end{figure}\nThe matrix representations $M(h_i)$ of $\\tau(h_i)$ on $S_{X(7)}$ are given in\nFigures~\\ref{fig:Mh1}--\\ref{fig:Mh3}. (Recall that $\\mathord{\\mathrm {O}}(S_X)$ acts on $S_X$ from the right.\nHence $M(h_i)$ satisfies $M(h_i) \\cdot G_{\\Lambda} \\cdot {}^t M(h_i)=G_{\\Lambda}$,\nwhere $G_{\\Lambda}$ is the Gram matrix of $\\Lambda_{7,1}^{-}$ with respect to $\\mathord{\\textbf{\\itshape e}}_1, \\dots, \\mathord{\\textbf{\\itshape e}}_{22}$.)\nThe characteristic polynomial\nof the product \n$$\nM:=M(h_1)M(h_2)M(h_3)\n$$\nis a Salem polynomial  \n\\begin{eqnarray*}\n&&t^{22}-993\\,t^{21}-1152\\,t^{20}-123\\,t^{19}+924\\,t^{18}+584\\,t^{17}-500\\,t^{16}-1022\\,t^{15}\\\\\n&&-661\\,t^{14}+105\\,t^{13}+476\\,t^{12}+878\\,t^{11}+476\\,t^{10}+105\\,t^9-661\\,t^8\\\\\n&&-1022\\,t^7-500\\,t^6+584\\,t^5+924\\,t^4-123\\,t^3-1152\\,t^2-993\\,t+1, \n\\end{eqnarray*}\nwhich has a positive real root\n$994.15889\\dots$.\n\\bibliographystyle{plain}\n\\\n\\def\\cftil#1{\\ifmmode\\setbox7\\hbox{$\\accent\"5E#1$}\\else\n  \\setbox7\\hbox{\\accent\"5E#1}\\penalty 10000\\relax\\fi\\raise 1\\ht7\n  \\hbox{\\lower1.15ex\\hbox to 1\\wd7{\\hss\\accent\"7E\\hss}}\\penalty 10000\n  \\hskip-1\\wd7\\penalty 10000\\box7} \\def$'$} \\def\\cprime{$'${$'$} \\def$'$} \\def\\cprime{$'${$'$}\n  \\def$'$} \\def\\cprime{$'${$'$} \\def$'$} \\def\\cprime{$'${$'$}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}