In this manuscript I present an analysis on the performance of OpenAI O1-preview model in solving random K-SAT instances for K$in {2,3,4}$ as a function of $alpha=M/N$ where $M$ is the number of clauses and $N$ is the number of variables of the satisfiable problem. I show that the model can call an external SAT solver to solve the instances, rather than solving them directly. Despite using external solvers, the model reports incorrect assignments as output. Moreover, I propose and present an analysis to quantify whether the OpenAI O1-preview model demonstrates a spark of intelligence or merely makes random guesses when outputting an assignment for a Boolean satisfiability problem.
{"title":"Fast Analysis of the OpenAI O1-Preview Model in Solving Random K-SAT Problem: Does the LLM Solve the Problem Itself or Call an External SAT Solver?","authors":"Raffaele Marino","doi":"arxiv-2409.11232","DOIUrl":"https://doi.org/arxiv-2409.11232","url":null,"abstract":"In this manuscript I present an analysis on the performance of OpenAI\u0000O1-preview model in solving random K-SAT instances for K$in {2,3,4}$ as a\u0000function of $alpha=M/N$ where $M$ is the number of clauses and $N$ is the\u0000number of variables of the satisfiable problem. I show that the model can call\u0000an external SAT solver to solve the instances, rather than solving them\u0000directly. Despite using external solvers, the model reports incorrect\u0000assignments as output. Moreover, I propose and present an analysis to quantify\u0000whether the OpenAI O1-preview model demonstrates a spark of intelligence or\u0000merely makes random guesses when outputting an assignment for a Boolean\u0000satisfiability problem.","PeriodicalId":501066,"journal":{"name":"arXiv - PHYS - Disordered Systems and Neural Networks","volume":"187 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253820","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Quantum coherence, a fundamental resource in quantum computing and quantum information, often competes with localization effects that affects quantum states in disordered systems. In this work, we prove exact trade-off relations between quantum coherence and a measure of localization and many-body localization, namely, the inverse participation ratio (IPR). We prove that the l1-norm of quantum coherence and the relative entropy of coherence for a pure quantum state satisfy complementarity relations with IPR. For a mixed state, IPR and the l2-norm of quantum coherence as well as relative entropy of coherence satisfy trade-off inequalities. These relations suggest that quantum coherence, in disordered quantum systems is also an ideal characterization of the delocalisation to many-body localisation transition, much like IPR, which is a well-known diagnostic of MBL. These relations also provide insight into the unusual properties of bipartite entanglement entropy across the MBL transition. We believe that these trade-off relations can help in better understanding of how coherence can be preserved or lost in realistic many-body quantum systems, which is vital for developing robust quantum technologies and uncovering new phases of quantum matter.
{"title":"Trade-off relations between quantum coherence and measure of many-body localization","authors":"Arti Garg, Arun Kumar Pati","doi":"arxiv-2409.10449","DOIUrl":"https://doi.org/arxiv-2409.10449","url":null,"abstract":"Quantum coherence, a fundamental resource in quantum computing and quantum\u0000information, often competes with localization effects that affects quantum\u0000states in disordered systems. In this work, we prove exact trade-off relations\u0000between quantum coherence and a measure of localization and many-body\u0000localization, namely, the inverse participation ratio (IPR). We prove that the\u0000l1-norm of quantum coherence and the relative entropy of coherence for a pure\u0000quantum state satisfy complementarity relations with IPR. For a mixed state,\u0000IPR and the l2-norm of quantum coherence as well as relative entropy of\u0000coherence satisfy trade-off inequalities. These relations suggest that quantum\u0000coherence, in disordered quantum systems is also an ideal characterization of\u0000the delocalisation to many-body localisation transition, much like IPR, which\u0000is a well-known diagnostic of MBL. These relations also provide insight into\u0000the unusual properties of bipartite entanglement entropy across the MBL\u0000transition. We believe that these trade-off relations can help in better\u0000understanding of how coherence can be preserved or lost in realistic many-body\u0000quantum systems, which is vital for developing robust quantum technologies and\u0000uncovering new phases of quantum matter.","PeriodicalId":501066,"journal":{"name":"arXiv - PHYS - Disordered Systems and Neural Networks","volume":"201 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Silvio Franz, Cosimo Lupo, Flavio Nicoletti, Giorgio Parisi, Federico Ricci-Tersenghi
We study numerically the Hessian of low-lying minima of vector spin glass models defined on random regular graphs. We consider the two-component (XY) and three-component (Heisenberg) spin glasses at zero temperature, subjected to the action of a randomly oriented external field. Varying the intensity of the external field, these models undergo a zero temperature phase transition from a paramagnet at high field to a spin glass at low field. We study how the spectral properties of the Hessian depend on the magnetic field. In particular, we study the shape of the spectrum at low frequency and the localization properties of low energy eigenvectors across the transition. We find that in both phases the edge of the spectral density behaves as $lambda^{3/2}$: such a behavior rules out the presence of a diverging spin-glass susceptibility $chi_{SG}=langle 1/lambda^2 rangle$. As to low energy eigenvectors, we find that the softest eigenmodes are always localized in both phases of the two models. However, by studying in detail the geometry of low energy eigenmodes across different energy scales close to the lower edge of the spectrum, we find a different behavior for the two models at the transition: in the XY case, low energy modes are typically localized; at variance, in the Heisenberg case low-energy eigenmodes with a multi-modal structure (sort of ``delocalization'') appear at an energy scale that vanishes in the infinite size limit. These geometrically non-trivial excitations, which we call Concentrated and Delocalised Low Energy Modes (CDLEM), coexist with trivially localised excitations: we interpret their existence as a sign of critical behavior related to the onset of the spin glass phase.
{"title":"Soft modes in vector spin glass models on sparse random graphs","authors":"Silvio Franz, Cosimo Lupo, Flavio Nicoletti, Giorgio Parisi, Federico Ricci-Tersenghi","doi":"arxiv-2409.10312","DOIUrl":"https://doi.org/arxiv-2409.10312","url":null,"abstract":"We study numerically the Hessian of low-lying minima of vector spin glass\u0000models defined on random regular graphs. We consider the two-component (XY) and\u0000three-component (Heisenberg) spin glasses at zero temperature, subjected to the\u0000action of a randomly oriented external field. Varying the intensity of the\u0000external field, these models undergo a zero temperature phase transition from a\u0000paramagnet at high field to a spin glass at low field. We study how the\u0000spectral properties of the Hessian depend on the magnetic field. In particular,\u0000we study the shape of the spectrum at low frequency and the localization\u0000properties of low energy eigenvectors across the transition. We find that in\u0000both phases the edge of the spectral density behaves as $lambda^{3/2}$: such a\u0000behavior rules out the presence of a diverging spin-glass susceptibility\u0000$chi_{SG}=langle 1/lambda^2 rangle$. As to low energy eigenvectors, we find\u0000that the softest eigenmodes are always localized in both phases of the two\u0000models. However, by studying in detail the geometry of low energy eigenmodes\u0000across different energy scales close to the lower edge of the spectrum, we find\u0000a different behavior for the two models at the transition: in the XY case, low\u0000energy modes are typically localized; at variance, in the Heisenberg case\u0000low-energy eigenmodes with a multi-modal structure (sort of ``delocalization'')\u0000appear at an energy scale that vanishes in the infinite size limit. These\u0000geometrically non-trivial excitations, which we call Concentrated and\u0000Delocalised Low Energy Modes (CDLEM), coexist with trivially localised\u0000excitations: we interpret their existence as a sign of critical behavior\u0000related to the onset of the spin glass phase.","PeriodicalId":501066,"journal":{"name":"arXiv - PHYS - Disordered Systems and Neural Networks","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253822","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Spin glasses have played a fundamental role in statistical mechanics field. Purpose of this work is to analyze a variation on theme of the mean field case of them, when the Ising spins are replaced to Boolean ones, i.e. {0,1} possible values. This may be useful to continue building a solid bridge between statical mechanics of spin glasses and Machine Learning techniques. We have drawn a detailed framework of this model: we have applied Guerra and Toninelli's approach to prove the existence of the thermodynamic quenched statistical pressure for this model recovering its expression using Guerra's interpolation. Specifically, we have supposed Replica Symmetric assumption and first step of Replica Symmetry Breaking approximation for the probability distribution of the order parameter of the model. Then, we analyze the stability of the resolution in both assumptions via de Almeida-Thouless line, proving that the Replica Symmetric one well describes the model apart for small values of temperature, when the Replica Symmetry Breaking is better. All the theoretical parts are supported by numerical techniques that demonstrate perfect consistency with the analytical results.
自旋玻璃在统计力学领域发挥着基础性作用。这项工作的目的是分析其均值场情况的主题变体，即当伊辛自旋被替换为布尔自旋时，即{0,1}可能值。这可能有助于继续在自旋玻璃的静力学和机器学习技术之间架起一座坚实的桥梁。我们绘制了该模型的详细框架：我们应用 Guerra 和 Toninelli 的方法证明了该模型热力学淬火统计压力的存在，并利用 Guerra 的插值法恢复了其表达式。具体而言，我们假定了复制对称假设，并对模型阶次参数的概率分布进行了第一步复制对称破坏近似。然后，我们通过 de Almeida-Thouless 线分析了这两种假设中分辨率的稳定性，证明除了温度值较小时，复制对称近似能更好地描述模型。所有理论部分都得到了数值技术的支持，证明与分析结果完全一致。
{"title":"Boolean mean field spin glass model: rigorous results","authors":"Linda Albanese, Andrea Alessandrelli","doi":"arxiv-2409.08693","DOIUrl":"https://doi.org/arxiv-2409.08693","url":null,"abstract":"Spin glasses have played a fundamental role in statistical mechanics field.\u0000Purpose of this work is to analyze a variation on theme of the mean field case\u0000of them, when the Ising spins are replaced to Boolean ones, i.e. {0,1} possible\u0000values. This may be useful to continue building a solid bridge between statical\u0000mechanics of spin glasses and Machine Learning techniques. We have drawn a\u0000detailed framework of this model: we have applied Guerra and Toninelli's\u0000approach to prove the existence of the thermodynamic quenched statistical\u0000pressure for this model recovering its expression using Guerra's interpolation.\u0000Specifically, we have supposed Replica Symmetric assumption and first step of\u0000Replica Symmetry Breaking approximation for the probability distribution of the\u0000order parameter of the model. Then, we analyze the stability of the resolution\u0000in both assumptions via de Almeida-Thouless line, proving that the Replica\u0000Symmetric one well describes the model apart for small values of temperature,\u0000when the Replica Symmetry Breaking is better. All the theoretical parts are\u0000supported by numerical techniques that demonstrate perfect consistency with the\u0000analytical results.","PeriodicalId":501066,"journal":{"name":"arXiv - PHYS - Disordered Systems and Neural Networks","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142253823","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Elena Agliari, Andrea Alessandrelli, Adriano Barra, Martino Salomone Centonze, Federico Ricci-Tersenghi
While auto-associative neural networks (e.g., the Hopfield model implementing the standard Hebbian prescription for learning) play as the reference setting for pattern recognition and associative memory in statistical mechanics, hetero-associative extensions (despite much less investigated) display richer emergent computational skills. Here we study the simplest generalization of the Kosko's Bidirectional Associative Memory (BAM), namely a Three-directional Associative Memory (TAM), that is a tripartite neural network equipped with generalized Hebbian weights. We study its information processing capabilities analytically (via statistical mechanics and signal-to-noise techniques) and computationally (via Monte Carlo simulations). Confined to the replica symmetric description, we provide phase diagrams for this network in the space of the control parameters, highlighting the existence of a region where the machine can successful perform recognition as well as other tasks. For instance, it can perform pattern disentanglement, namely when inputted with a mixture of patterns, the network is able to return the original patterns, namely to disentangle the signal's components. Further, they can also perform retrieval of (Markovian) sequences of patterns and they can also disentangle mixtures of periodic patterns: should these mixtures be sequences that combine patterns alternating at different frequencies, these hetero-associative networks can perform generalized frequency modulation by using the slowly variable sequence of patterns as the base-band signal and the fast one as the information carrier.
{"title":"Generalized hetero-associative neural networks","authors":"Elena Agliari, Andrea Alessandrelli, Adriano Barra, Martino Salomone Centonze, Federico Ricci-Tersenghi","doi":"arxiv-2409.08151","DOIUrl":"https://doi.org/arxiv-2409.08151","url":null,"abstract":"While auto-associative neural networks (e.g., the Hopfield model implementing\u0000the standard Hebbian prescription for learning) play as the reference setting\u0000for pattern recognition and associative memory in statistical mechanics,\u0000hetero-associative extensions (despite much less investigated) display richer\u0000emergent computational skills. Here we study the simplest generalization of the\u0000Kosko's Bidirectional Associative Memory (BAM), namely a Three-directional\u0000Associative Memory (TAM), that is a tripartite neural network equipped with\u0000generalized Hebbian weights. We study its information processing capabilities\u0000analytically (via statistical mechanics and signal-to-noise techniques) and\u0000computationally (via Monte Carlo simulations). Confined to the replica\u0000symmetric description, we provide phase diagrams for this network in the space\u0000of the control parameters, highlighting the existence of a region where the\u0000machine can successful perform recognition as well as other tasks. For\u0000instance, it can perform pattern disentanglement, namely when inputted with a\u0000mixture of patterns, the network is able to return the original patterns,\u0000namely to disentangle the signal's components. Further, they can also perform\u0000retrieval of (Markovian) sequences of patterns and they can also disentangle\u0000mixtures of periodic patterns: should these mixtures be sequences that combine\u0000patterns alternating at different frequencies, these hetero-associative\u0000networks can perform generalized frequency modulation by using the slowly\u0000variable sequence of patterns as the base-band signal and the fast one as the\u0000information carrier.","PeriodicalId":501066,"journal":{"name":"arXiv - PHYS - Disordered Systems and Neural Networks","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Long-time tails, or algebraic decay of time-correlation functions, have long been known to exist both in many-body systems and in models of non-interacting particles in the presence of quenched disorder that are often referred to as Lorentz models. In the latter, they have been studied extensively by a wide variety of methods, the best known example being what is known as weak-localization effects in disordered systems of non-interacting electrons. This paper provides a unifying, and very simple, approach to all of these effects. We show that simple modifications of the diffusion equation due to either a random diffusion coefficient, or a random scattering potential, accounts for both the decay exponents and the prefactors of the leading long-time tails in the velocity autocorrelation functions of both classical and quantum Lorentz models.
{"title":"Diffusion, Long-Time Tails, and Localization in Classical and Quantum Lorentz Models: A Unifying Hydrodynamic Approach","authors":"T. R. Kirkpatrick, D. Belitz","doi":"arxiv-2409.08123","DOIUrl":"https://doi.org/arxiv-2409.08123","url":null,"abstract":"Long-time tails, or algebraic decay of time-correlation functions, have long\u0000been known to exist both in many-body systems and in models of non-interacting\u0000particles in the presence of quenched disorder that are often referred to as\u0000Lorentz models. In the latter, they have been studied extensively by a wide\u0000variety of methods, the best known example being what is known as\u0000weak-localization effects in disordered systems of non-interacting electrons.\u0000This paper provides a unifying, and very simple, approach to all of these\u0000effects. We show that simple modifications of the diffusion equation due to\u0000either a random diffusion coefficient, or a random scattering potential,\u0000accounts for both the decay exponents and the prefactors of the leading\u0000long-time tails in the velocity autocorrelation functions of both classical and\u0000quantum Lorentz models.","PeriodicalId":501066,"journal":{"name":"arXiv - PHYS - Disordered Systems and Neural Networks","volume":"432 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220847","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Understanding the flow dynamics of yield stress fluids in porous media presents a substantial challenge. Both experiments and extensive numerical simulations frequently show a non-linear relationship between the flow rate and the pressure gradient, deviating from the traditional Darcy law. In this article, we consider a tree-like porous structure and utilize an exact mapping with the directed polymer (DP) with disordered bond energies on the Cayley tree. Specifically, we adapt an algorithm recently introduced by Brunet et al. [Europhys. Lett. 131, 40002 (2020)] to simulate exactly the tip region of branching random walks with the help of a spinal decomposition, to accurately compute the flow on extensive trees with several thousand generations. Our results confirm the asymptotic predictions proposed by Schimmenti et al. [Phys. Rev. E 108, L023102 (2023)], tested therein only for moderate trees of about 20 generations.
了解屈服应力流体在多孔介质中的流动动力学是一项巨大的挑战。实验和大量的数值模拟经常显示流速与压力梯度之间存在非线性关系，偏离了传统的达西定律。在本文中，我们考虑了树状多孔结构，并利用有向聚合物（DP）与 Cayleytree 上无序键能的精确映射。具体来说，我们调整了 Brunet 等人[Europhys. Lett. 131, 40002 (2020)]最近引入的算法，在脊柱分解的帮助下精确模拟分支随机漫步的顶端区域，从而精确计算数千代广泛树上的流动。我们的结果证实了 Schimmenti 等人[Phys.Rev. E 108, L023102 (2023)]提出的渐进预测，但他们只对约 20 代的中等树进行了测试。
{"title":"Numerical study of Darcy's law of yield stress fluids on a deep tree-like network","authors":"Stéphane Munier, Alberto Rosso","doi":"arxiv-2409.03480","DOIUrl":"https://doi.org/arxiv-2409.03480","url":null,"abstract":"Understanding the flow dynamics of yield stress fluids in porous media\u0000presents a substantial challenge. Both experiments and extensive numerical\u0000simulations frequently show a non-linear relationship between the flow rate and\u0000the pressure gradient, deviating from the traditional Darcy law. In this\u0000article, we consider a tree-like porous structure and utilize an exact mapping\u0000with the directed polymer (DP) with disordered bond energies on the Cayley\u0000tree. Specifically, we adapt an algorithm recently introduced by Brunet et al.\u0000[Europhys. Lett. 131, 40002 (2020)] to simulate exactly the tip region of\u0000branching random walks with the help of a spinal decomposition, to accurately\u0000compute the flow on extensive trees with several thousand generations. Our\u0000results confirm the asymptotic predictions proposed by Schimmenti et al. [Phys.\u0000Rev. E 108, L023102 (2023)], tested therein only for moderate trees of about 20\u0000generations.","PeriodicalId":501066,"journal":{"name":"arXiv - PHYS - Disordered Systems and Neural Networks","volume":"60 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220849","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Recent research has made significant progress in understanding localization transitions and mobility edges (MEs) that separate extended and localized states in non-Hermitian (NH) quasicrystals. Here we focus on studying critical states and anomalous MEs, which identify the boundaries between critical and localized states within two distinct NH quasiperiodic models. Specifically, the first model is a quasiperiodic mosaic lattice with both nonreciprocal hopping term and on-site potential. In contrast, the second model features an unbounded quasiperiodic on-site potential and nonreciprocal hopping. Using Avila's global theory, we analytically derive the Lyapunov exponent and exact anomalous MEs. To confirm the emergence of the robust critical states in both models, we conduct a numerical multifractal analysis of the wave functions and spectrum analysis of level spacing. Furthermore, we investigate the transition between real and complex spectra and the topological origins of the anomalous MEs. Our results may shed light on exploring the critical states and anomalous MEs in NH quasiperiodic systems.
最近的研究在理解非ermitian（NH）准晶体中分离扩展态和局部态的局部化转变和迁移率边缘（MEs）方面取得了重大进展。在这里，我们重点研究临界状态和反常移动边（ME），它们确定了两个不同的 NH 准周期模型中临界状态和局部状态之间的边界。具体来说，第一个模型是一个具有非互惠跳动项和现场势的准周期镶嵌晶格。与此相反，第二种模型具有无约束的准周期现场电势和非互惠跳变。为了证实这两个模型都出现了稳健临界态，我们对波函数进行了数值多分形分析，并对级距进行了谱分析。此外，我们还研究了真实光谱和复数光谱之间的过渡以及异常 ME 的拓扑起源。我们的研究结果可能有助于探索 NH 夸周期系统中的临界状态和反常 ME。
{"title":"Exact anomalous mobility edges in one-dimensional non-Hermitian quasicrystals","authors":"Xiang-Ping Jiang, Weilei Zeng, Yayun Hu, Lei Pan","doi":"arxiv-2409.03591","DOIUrl":"https://doi.org/arxiv-2409.03591","url":null,"abstract":"Recent research has made significant progress in understanding localization\u0000transitions and mobility edges (MEs) that separate extended and localized\u0000states in non-Hermitian (NH) quasicrystals. Here we focus on studying critical\u0000states and anomalous MEs, which identify the boundaries between critical and\u0000localized states within two distinct NH quasiperiodic models. Specifically, the\u0000first model is a quasiperiodic mosaic lattice with both nonreciprocal hopping\u0000term and on-site potential. In contrast, the second model features an unbounded\u0000quasiperiodic on-site potential and nonreciprocal hopping. Using Avila's global\u0000theory, we analytically derive the Lyapunov exponent and exact anomalous MEs.\u0000To confirm the emergence of the robust critical states in both models, we\u0000conduct a numerical multifractal analysis of the wave functions and spectrum\u0000analysis of level spacing. Furthermore, we investigate the transition between\u0000real and complex spectra and the topological origins of the anomalous MEs. Our\u0000results may shed light on exploring the critical states and anomalous MEs in NH\u0000quasiperiodic systems.","PeriodicalId":501066,"journal":{"name":"arXiv - PHYS - Disordered Systems and Neural Networks","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220848","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Djenabou Bayo, Burak Çivitcioğlu, Joseph J Webb, Andreas Honecker, Rudolf A. Römer
The detection of phase transitions is a fundamental challenge in condensed matter physics, traditionally addressed through analytical methods and direct numerical simulations. In recent years, machine learning techniques have emerged as powerful tools to complement these standard approaches, offering valuable insights into phase and structure determination. Additionally, they have been shown to enhance the application of traditional methods. In this work, we review recent advancements in this area, with a focus on our contributions to phase and structure determination using supervised and unsupervised learning methods in several systems: (a) 2D site percolation, (b) the 3D Anderson model of localization, (c) the 2D $J_1$-$J_2$ Ising model, and (d) the prediction of large-angle convergent beam electron diffraction patterns.
{"title":"Machine learning of phases and structures for model systems in physics","authors":"Djenabou Bayo, Burak Çivitcioğlu, Joseph J Webb, Andreas Honecker, Rudolf A. Römer","doi":"arxiv-2409.03023","DOIUrl":"https://doi.org/arxiv-2409.03023","url":null,"abstract":"The detection of phase transitions is a fundamental challenge in condensed\u0000matter physics, traditionally addressed through analytical methods and direct\u0000numerical simulations. In recent years, machine learning techniques have\u0000emerged as powerful tools to complement these standard approaches, offering\u0000valuable insights into phase and structure determination. Additionally, they\u0000have been shown to enhance the application of traditional methods. In this\u0000work, we review recent advancements in this area, with a focus on our\u0000contributions to phase and structure determination using supervised and\u0000unsupervised learning methods in several systems: (a) 2D site percolation, (b)\u0000the 3D Anderson model of localization, (c) the 2D $J_1$-$J_2$ Ising model, and\u0000(d) the prediction of large-angle convergent beam electron diffraction\u0000patterns.","PeriodicalId":501066,"journal":{"name":"arXiv - PHYS - Disordered Systems and Neural Networks","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220850","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce a random matrix model for the stationary covariance of multivariate Ornstein-Uhlenbeck processes with heterogeneous temperatures, where the covariance is constrained by the Sylvester-Lyapunov equation. Using the replica method, we compute the spectral density of the equal-time covariance matrix characterizing the stationary states, demonstrating that this model undergoes a transition between stable and unstable states. In the stable regime, the spectral density has a finite and positive support, whereas negative eigenvalues emerge in the unstable regime. We determine the critical line separating these regimes and show that the spectral density exhibits a power-law tail at marginal stability, with an exponent independent of the temperature distribution. Additionally, we compute the spectral density of the lagged covariance matrix characterizing the stationary states of linear transformations of the original dynamical variables. Our random-matrix model is potentially interesting to understand the spectral properties of empirical correlation matrices appearing in the study of complex systems.
{"title":"Random matrix ensemble for the covariance matrix of Ornstein-Uhlenbeck processes with heterogeneous temperatures","authors":"Leonardo Ferreira, Fernando Metz, Paolo Barucca","doi":"arxiv-2409.01262","DOIUrl":"https://doi.org/arxiv-2409.01262","url":null,"abstract":"We introduce a random matrix model for the stationary covariance of\u0000multivariate Ornstein-Uhlenbeck processes with heterogeneous temperatures,\u0000where the covariance is constrained by the Sylvester-Lyapunov equation. Using\u0000the replica method, we compute the spectral density of the equal-time\u0000covariance matrix characterizing the stationary states, demonstrating that this\u0000model undergoes a transition between stable and unstable states. In the stable\u0000regime, the spectral density has a finite and positive support, whereas\u0000negative eigenvalues emerge in the unstable regime. We determine the critical\u0000line separating these regimes and show that the spectral density exhibits a\u0000power-law tail at marginal stability, with an exponent independent of the\u0000temperature distribution. Additionally, we compute the spectral density of the\u0000lagged covariance matrix characterizing the stationary states of linear\u0000transformations of the original dynamical variables. Our random-matrix model is\u0000potentially interesting to understand the spectral properties of empirical\u0000correlation matrices appearing in the study of complex systems.","PeriodicalId":501066,"journal":{"name":"arXiv - PHYS - Disordered Systems and Neural Networks","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142220851","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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