Random Spin Models Research Articles

Griffiths-Type Theorems for Short-Range Spin Glass Models

We establish relations between different characterizations of order in spin glass models. We first prove that the broadening of the replica overlap distribution indicated by a nonzero standard deviation of the replica overlap R1,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R^{1,2}$$\\end{document} implies the non-differentiability of the two-replica free energy with respect to the replica coupling parameter λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda $$\\end{document}. In Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}_2$$\\end{document} invariant models such as the standard Edwards–Anderson model, the non-differentiability is equivalent to the spin glass order characterized by a nonzero Edwards–Anderson order parameter. This generalization of Griffiths’ theorem is proved for any short-range spin glass models with classical bounded spins. We also prove that the non-differentiability of the two-replica free energy mentioned above implies replica symmetry breaking in the literal sense, i.e., a spontaneous breakdown of the permutation symmetry in the model with three replicas. This is a general result that applies to a large class of random spin models, including long-range models such as the Sherrington-Kirkpatrick model and the random energy model. There is a 25-minute video that explains the main results of the present work:https://youtu.be/BF3hJiY1xvI

Signature of Scramblon Effective Field Theory in Random Spin Models.

Information scrambling refers to the propagation of information throughout a quantum system. Its study not only contributes to our understanding of thermalization but also has wide implications in quantum information and black hole physics. Recent studies suggest that information scrambling in large-N systems with all-to-all interactions is mediated by collective modes called scramblons. However, a criterion for the validity of scramblon theory in a specific model is still missing. In this work, we address this issue by investigating the signature of the scramblon effective theory in random spin models with all-to-all interactions. We demonstrate that, in scenarios where the scramblon description holds, the late-time operator size distribution can be predicted from its early-time value, requiring no free parameters. As an illustration, we examine whether Brownian circuits exhibit a scramblon description and obtain a positive confirmation both analytically and numerically. Our findings provide a concrete experimental framework for unraveling the scramblon field theory in random spin models using quantum simulators.

Concentration bounds for quantum states and limitations on the QAOA from polynomial approximations

We prove concentration bounds for the following classes of quantum states: (i) output states of shallow quantum circuits, answering an open question from \cite{DMRF22}; (ii) injective matrix product states; (iii) output states of dense Hamiltonian evolution, i.e. states of the form eιH(p)⋯eιH(1)|ψ0⟩ for any n-qubit product state |ψ0⟩, where each H(i) can be any local commuting Hamiltonian satisfying a norm constraint, including dense Hamiltonians with interactions between any qubits. Our proofs use polynomial approximations to show that these states are close to local operators. This implies that the distribution of the Hamming weight of a computational basis measurement (and of other related observables) concentrates.An example of (iii) are the states produced by the quantum approximate optimisation algorithm (QAOA). Using our concentration results for these states, we show that for a random spin model, the QAOA can only succeed with negligible probability even at super-constant level p=o(log⁡log⁡n), assuming a strengthened version of the so-called overlap gap property. This gives the first limitations on the QAOA on dense instances at super-constant level, improving upon the recent result [BGMZ22].

Engineering random spin models with atoms in a high-finesse cavity

All-to-all interacting, disordered quantum many-body models have a wide range of applications across disciplines, from spin glasses in condensed-matter physics over holographic duality in high-energy physics to annealing algorithms in quantum computing. Typically, these models are abstractions that do not find unambiguous physical realizations in nature. Here we realize an all-to-all interacting, disordered spin system by subjecting an atomic cloud in a cavity to a controllable light shift. Adjusting the detuning between atom resonance and cavity mode, we can tune between disordered versions of a central-mode model and a Lipkin–Meshkov–Glick model. By spectroscopically probing the low-energy excitations of the system, we explore the competition of interactions with disorder across a broad parameter range. We show how disorder in the central-mode model breaks the strong collective coupling, making the dark-state manifold cross over to a random distribution of weakly mixed light–matter, ‘grey’, states. In the Lipkin–Meshkov–Glick model, the ferromagnetic finite-sized ground state evolves towards a paramagnet as disorder is increased. In that regime, semi-localized eigenstates emerge, as we observe by extracting bounds on the participation ratio. These results present substantial steps towards freely programmable cavity-mediated interactions for the design of arbitrary spin Hamiltonians.

Parity Quantum Optimization: Benchmarks

We present benchmarks of the parity transformation for the Quantum Approximate Optimization Algorithm (QAOA). We analyse the gate resources required to implement a single QAOA cycle for real-world scenarios. In particular, we consider random spin models with higher order terms, as well as the problems of predicting financial crashes and finding the ground states of electronic structure Hamiltonians. For the spin models studied our findings imply a significant advantage of the parity mapping compared to the standard gate model. In combination with full parallelizability of gates this has the potential to boost the race for demonstrating quantum advantage.

Optimal Thresholds for Fracton Codes and Random Spin Models with Subsystem Symmetry.

Fracton models provide examples of novel gapped quantum phases of matter that host intrinsically immobile excitations and therefore lie beyond the conventional notion of topological order. Here, we calculate optimal error thresholds for quantum error correcting codes based on fracton models. By mapping the error-correction process for bit-flip and phase-flip noises into novel statistical models with Ising variables and random multibody couplings, we obtain models that exhibit an unconventional subsystem symmetry instead of a more usual global symmetry. We perform large-scale parallel tempering MonteCarlo simulations to obtain disorder-temperature phase diagrams, which are then used to predict optimal error thresholds for the corresponding fracton code. Remarkably, we found that the X-cube fracton code displays a minimum error threshold (7.5%) that is much higher than 3D topological codes such as the toric code (3.3%), or the color code (1.9%). This result, together with the predicted absence of glass order at the Nishimori line, shows great potential for fracton phases to be used as quantum memory platforms.

Disconnecting a traversable wormhole: Universal quench dynamics in random spin models

Understanding strongly interacting quantum matter and quantum gravity are both important open issues in theoretical physics, and the holographic duality between quantum field theory and gravity theory nicely brings these two topics together. Nevertheless, direct connections between gravity physics and experimental observations in quantum matter are still rare. Here we utilize the gravity physics picture to understand quench dynamics experimentally observed in a class of random spin models realized in several different quantum systems, where the dynamics of magnetization are measured after the external polarization field is suddenly turned off. Two universal features of the magnetization dynamics, namely, a slow decay described by a stretched exponential function and an oscillatory behavior, are respectively found in different parameter regimes across different systems. This work addresses the issues of generic conditions under which these two universal features can occur, and we find that a natural answer to this question emerges in the gravity picture. By the holographic duality bridged by a model proposed by Maldacena and Qi, the quench dynamics after suddenly turning off the external polarization field is mapped to disconnecting an eternal traversable wormhole. Our studies show that insight from gravity physics can help unifying different experiments in quantum systems.

Generic Local Hamiltonians are Gapless

We prove that generic quantum local Hamiltonians are gapless. In fact, we prove that there is a continuous density of states above the ground state. The Hamiltonian can be on a lattice in any spatial dimension or on a graph with a bounded maximum vertex degree. The type of interactions allowed for include translational invariance in a disorder (i.e., probabilistic) sense with some assumptions on the local distributions. Examples include many-body localization and random spin models. We calculate the scaling of the gap with the system's size when the local terms are distributed according to a Gaussian β orthogonal random matrix ensemble. As a corollary, there exist finite size partitions with respect to which the ground state is arbitrarily close to a product state. When the local eigenvalue distribution is discrete, in addition to the lack of an energy gap in the limit, we prove that the ground state has finite size degeneracies. The proofs are simple and constructive. This work excludes the important class of truly translationally invariant Hamiltonians where the local terms are all equal.

Thermoelectric phenomena in disordered open quantum systems

Using a stochastic quantum approach, we study thermoelectric transport phenomena at low temperatures in disordered electrical systems connected to external baths. We discuss three different models of one-dimensional disordered electrons, namely, the Anderson model of random on-site energies, the random-dimer model, and the random-hopping model---also relevant for random-spin models. We find that although the asymptotic behavior of transport in open systems is closely related to that in closed systems for these noninteracting models, the magnitude of thermoelectric transport in finite systems strongly depends on the boundary conditions and the baths spectral properties. This shows the importance of employing theories of open quantum systems in the study of energy transport.

A simple one dimensional glassy Kac model

We define a new family of random spin models with one-dimensional structure, finite-rangemultispin interactions, and bounded average degree (number of interactions in which eachspin participates). Unfrustrated ground states can be described as solutions of asparse, band diagonal linear system, thus allowing for efficient numerical analysis.In the limit of infinite interaction range, we recover the so-called XORSAT (dilutedp-spin) model that is known to undergo a random first-order phase transition as the averagedegree is increased. Here we investigate the most important consequences of a large butfinite interaction range: (i) fluctuation-induced corrections to thermodynamic quantities;(ii) the need of an inhomogeneous (position-dependent) order parameter; (iii) the emergenceof a finite mosaic length scale. In particular, we study the correlation length divergence atthe (mean field) glass transition.

Statistical-mechanical approach to image processing

The basic frameworks and techniques of the Bayesian approach to image restoration are reviewed from the statistical-mechanical point of view. First, a few basic notions in digital image processing are explained to convince the reader that statistical mechanics has a close formal similarity to this problem. Second, the basic formulation of the statistical estimation from the observed degraded image by using the Bayes formula is demonstrated. The relationship between Bayesian statistics and statistical mechanics is also listed. Particularly, it is explained that some correlation inequalities on the Nishimori line of the random spin model also play an important role in Bayesian image restoration. Third, the framework of Bayesian image restoration for binary images by means of the Ising model is reviewed. Some practical algorithms for binary image restoration are given by employing the mean-field and the Bethe approximations. Finally, Bayesian image restoration for a grey-level image using the Gaussian model is reviewed, and the Gaussian model is extended to a more practical probabilistic model by introducing the line state to treat the effects of edges. The line state is also extended to quantized values.

Diluted generalized random energy model

A layered random spin model, equivalent to the generalized random energy model (GREM), is introduced. In analogy with diluted spin systems, a diluted GREM (DGREM) is constructed. It can be applied to calculate approximately the thermodynamic properties of spin glass models in low dimensions. For the Edwards-Anderson model it gives the correct critical dimension and 5% accuracy for the ground state energy in two dimensions.

Random spin models and chemical kinetics

Lyapunov exponents constitute a class of parameters which describe the asymptotic behavior of a large class of dynamical processes in chemistry and physics. This paper gives a variational characterization of the largest Lyapunov exponent for a class of models of chemical kinetics described by products of random nonnegative matrices. We show that for this class of models the largest Lyapunov exponent satisfies an extremal principle formally identical to the minimization of the quenched free energy in random spin models. This extremal principle, which yields a computable expression for the Lyapunov exponent, implies that fluctuations in the Lyapunov exponent, due to a certain class of perturbations in the matrix elements, are determined by a macroscopic parameter which is the analog of the mean energy in random spin systems. These results characterize a class of random models in chemical kinetics that are thermodynamically stable in the sense that they possess an asymptotic limit in which analogs of the laws of equilibrium thermodynamics hold.

Random-temperature Ginzburg-Landau model: relation to conventional spin glass models

It is shown how an effective Ginzburg-Landau model having random-temperature quenched disorder can be related to more conventional random-spin models. Interaction terms beyond the usual (gradient)2 term are needed to describe spin glasses, but such terms arise naturally.

Critical behaviour of the random spin models with a coupling to a nonfluctuating parameter

physica status solidi (b)Volume 91, Issue 2 p. K143-K147 Short Note Critical behaviour of the random spin models with a coupling to a nonfluctuating parameter V. M. Laptev, V. M. Laptev Institute of Metal Physics, Academy of Sciences of the USSR, Ural Scientific Centre, SverdlovskSearch for more papers by this authorYu. N. Skryabin, Yu. N. Skryabin Institute of Metal Physics, Academy of Sciences of the USSR, Ural Scientific Centre, SverdlovskSearch for more papers by this author V. M. Laptev, V. M. Laptev Institute of Metal Physics, Academy of Sciences of the USSR, Ural Scientific Centre, SverdlovskSearch for more papers by this authorYu. N. Skryabin, Yu. N. Skryabin Institute of Metal Physics, Academy of Sciences of the USSR, Ural Scientific Centre, SverdlovskSearch for more papers by this author First published: 1 February 1979 https://doi.org/10.1002/pssb.2220910259Citations: 7AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume91, Issue21 February 1979Pages K143-K147 RelatedInformation

Critical properties of random-spin models from theεexpansion

In quenched random-spin systems, the renormalization group can be used to develop recursion relations for the probability distribution for random potentials. Alternatively, recursion relations for the average values of potentials and their higher cumulants can be obtained. In this paper, the above technique is used to study phase transitions in quenched random $n$-component classical spin systems using the $\ensuremath{\epsilon}$ expansion to second order in $\ensuremath{\epsilon}$. It there are long-range correlations in the random potentials (e.g., all potentials along a line are equal), there are no stable physical fixed points within the $\ensuremath{\epsilon}$ expansion. This is interpreted as a smeared transition. If there are no long-range correlations in the random potentials, there is a sharp transition with pure system exponents if the specific-heat exponent ${\ensuremath{\alpha}}^{H}$ of the pure system is negative. If ${\ensuremath{\alpha}}^{H}>0$ and $n>1$, there is a sharp transition with new exponents $\ensuremath{\eta}=[\frac{(5{n}^{2}\ensuremath{-}8n)}{256{(n\ensuremath{-}1)}^{2}}]{\ensuremath{\epsilon}}^{2}$ and $\ensuremath{\nu}=\frac{1}{2}+[\frac{3n}{32(n\ensuremath{-}1)}]\ensuremath{\epsilon}+[\frac{n(127{n}^{2}\ensuremath{-}572n\ensuremath{-}32)}{4096{(n\ensuremath{-}1)}^{3}}]{\ensuremath{\epsilon}}^{2}$. For $n=1$, there is no stable fixed point, which is again interpreted as a smeared transition.

Renormalization-Group Approach to the Critical Behavior of Random-Spin Models

A renormalization-group technique is used to study the critical behavior of spin models in which each interaction has a small independent random width about its average value. The cluster approximation of Niemeyer and Van Leeuwen indicates that the two-dimensional Ising model has the same critical behavior as the homogeneous system. The $\ensuremath{\epsilon}$ expansion for $n$-component continuous spins shows that this behavior holds to first order in $\ensuremath{\epsilon}$ for $n>4$. For $n<4$, there is a new stable fixed point with $2\ensuremath{\nu}=1+[\frac{3n}{16(n\ensuremath{-}1)}]\ensuremath{\epsilon}$.

Calorimetric study of a random Ising spin system: Co pZn 1− pCs 3Cl 5

Specific-heat and susceptibility measurements are presented on the magnetically diluted system Co p Zn 1− p Cs 3Cl 5. The results are in good agreement with theoretical predictions on the random-spin model for Ising interactions in three dimensions. High-resolution specific-heat measurements show that the singularity remains relatively sharp down to low concentrations of the magnetic ions.