We report on the ensemble inequivalence in a many-body spin-glass model with integer spin. The spin-glass phase transition is of first order for certain values of the crystal field strength and is dependent whether it was derived in the microcanonical or the canonical ensemble. In the limit of infinitely many-body interactions, the model is the integer-spin equivalent of the random-energy model, and is solved exactly. We also derive the integer-spin equivalent of the de Almeida-Thouless line.
We investigate the ensemble inequivalence of the spherical spin glass model with nonlinear interactions of polynomial order $p$. This model is solved exactly for arbitrary $p$ and is shown to have first-order phase transitions between the paramagnetic and spin glass or ferromagnetic phases for $p \geq 5$. In the parameter region around the first-order transitions, the solutions give different results depending on the ensemble used for the analysis. In particular, we observe that the microcanonical specific heat can be negative and the phase may not be uniquely determined by the temperature.
We apply the method of gauge transformation to spin glasses under the microcanonical ensemble to study the possibility of ensemble inequivalence in systems with long-range interactions and quenched disorder. It is proved that all the results derived under the canonical ensemble on the Nishimori line (NL) can be reproduced by the microcanonical ensemble irrespective of the range of interactions. This establishes that ensemble inequivalence should take place away from the NL if it happens in spin glasses. It is also proved on the NL that the microcanonical configurational average of the energy a…
We study the effect that randomness has on long-range interacting systems by using the ferromagnetic Ising model with $p$-body interactions in random fields. The case with p=2 yields a phase diagram similar to that of previously studied models and shows known features that inequivalence of the canonical and microcanonical ensembles brings with it, for example negative specific heat in a narrow region of the phase diagram. When p>2, however, the canonical phase diagram is completely different from the microcanonical one. The temperature does not necessarily determine the microcanonical phases u…
We investigate the p-spin model with Gaussian-distributed random interactions in the microcanonical ensemble using the replica theory. For p=2, there are only second-order phase transitions and we recover the results of Sherrington and Kirkpatrick obtained in the canonical ensemble. For p > 2, the transition between the ferromagnetic and paramagnetic phases is of first order, and the microcanonical and canonical ensembles give different results. We also discuss the ensemble inequivalence of the random energy model, corresponding to the limit p => infinity. This is the first systematic treatmen…
Logarithmic Aging Diffusion from a Multiplicative Event Clock: Rare Event Statistics, Ultraslow Transport, and Ensemble-Time Inequivalence
Center for Advanced Quantum Studies, School of Physics and Astronomy, Beijing Normal University（北京师范大学物理与天文学部先进量子研究中心）；Key Laboratory of Multiscale Spin Physics, Ministry of Education（教育部多尺度自旋物理重点实验室）；School of Science, Tianjin University of Technology and Education（天津职业技术师范大学理学院）；Interdisciplinary Center for Theoretical Physics and Information Sciences, Fudan University（复旦大学理论物理与信息科学交叉中心）；International Center for Quantum Materials, School of Physics, Peking University（北京大学物理学院量子材料国际中心）；Hefei National Laboratory（合肥国家实验室）
For the quantum Ising chain, the self-dual block renormalization procedure of Fernandez-Pacheco [Phys. Rev. D 19, 3173 (1979)] is known to reproduce exactly the location of the zero-temperature critical point and the correlation length exponent $ν=1$. Recently, Miyazaki and Nishimori [Phys. Rev. E 87, 032154 (2013)] have proposed to study the disordered quantum Ising model in dimensions $d>1$ by applying the Fernandez-Pacheco procedure successively in each direction. To avoid the inequivalence of directions of their approach, we propose here an alternative procedure where the $d$ directions ar…
We generalize the de Almeida-Thouless line for the many-body Ising spin glass to the microcanonical ensemble and show that it coincides with the canonical one. This enables us to draw a complete microcanonical phase diagram of this model.
The recent description of the highly viscous flow in terms of irreversible structural Eshelby rearrangements is extended to calculate the heat capacity of a glass former at a constant cooling rate through the glass transition. The result is compared to measured data from the literature, showing that the explanation works both for polymers and other glass formers.
In this paper, we introduce a dynamical Monte Carlo algorithm for spin models in which the number of the spins fluctuates from zero to a given number by addition and deletion of spins with a probabilistic rule. Such simulations are realized with a variable-system-size ensemble, a mixture of canonical ensembles each of which corresponds to a system with different size. The weight of each component of the mixture is controlled by a penalty term and systematically tuned in a preliminary run in a way similar to the multicanonical algorithm. In a measurement run, the system grows and shrinks withou…
We introduce a new method, based on the recently developed random tensor theory, to study the p-spin glass model with non-Gaussian, correlated disorder. Using a suitable generalization of Gurau's theorem on the universality of the large N limit of the p-unitary ensemble of random tensors, we exhibit an infinite family of such non-Gaussian distributions which leads to same low temperature phase as the Gaussian distribution. While this result is easy to show (and well known) for uncorrelated disorder, its robustness with respect to strong quenched correlations is surprising. We show in detail ho…
We propose and study a renormalization group transformation that can be used also for models with strong quenched disorder, like spin glasses. The method is based on a mapping between disorder distributions, chosen such as to keep some physical properties (e.g., the ratio of correlations averaged over the ensemble) invariant under the transformation. We validate this ensemble renormalization group by applying it to the hierarchical model (both the diluted ferromagnetic version and the spin glass version), finding results in agreement with Monte Carlo simulations.
We study the bimodal Edwards-Anderson spin glass comparing established methods, namely the multicanonical method, the $1/k$-ensemble and parallel tempering, to an approach where the ensemble is modified by simulating power-law-shaped histograms in energy instead of flat histograms as in the standard multicanonical case. We show that by this modification a significant speed-up in terms of mean round-trip times can be achieved for all lattice sizes taken into consideration.
The crossover from back-and-forth jumps between structural minima to the no-return jumps of the viscous flow is modeled in terms of an ensemble of double-well potentials with a finite decay probability. The ensemble is characterized by the Kohlrausch-exponent $β$ of the time dependence $t^β$ of the response at short times. The model is applied to shear and dielectric data from the literature.
In the paper [Angelini M C, Parisi G, and Ricci-Tersenghi F, Ensemble renormalization group for disordered systems, Phys. Rev. B 87 134201 (2013)] we introduced a real-space renormalization group called Ensemble Renormalization Group (ERG) and we applied it to the Edwards-Anderson model, obtaining estimates for the critical exponents in good agreement with those from Monte Carlo simulations. Recently the paper [Castellana M, Real-space renormalization-group methods for hierarchical spin glasses, J. Phys. A: Math. Theor. 52 445002 (2019)] re-examined the ERG method from a different perspective,…
We consider the dynamics of a disordered p-spin model with p=2, analyzing the dynamics within Ruelle's thermodynamic formalism, We use an indicator of the dynamical activity to construct the relevant dynamical Gibbs ensemble. We prove that the dynamics in the low-temperature (spin glass) phase of the model take place at a second-order phase transition between dynamically active and inactive trajectories. We also show that the same behaviour is found in a related model of a three-dimensional ferromagnet.
We present an effcient Monte-Carlo method for lattice glass models which are characterized by hard constraint conditions. The basic idea of the method is similar to that of the $N$-fold way method. By using a list of sites into which we can insert a particle, we avoid trying a useless transition which is forbidden by the constraint conditions. We applied the present method to a lattice glass model proposed by Biroli and M{é}zard. We first evaluated the efficiency of the method through measurements of the autocorrelation function of particle configurations. As a result, we found that the effici…
We introduce a Sherrington-Kirkpatrick spin-glass model with the addition of elastic degrees of freedom. The problem is formulated in terms of an effective four-spin Hamiltonian in the pressure ensemble, which can be treated by the replica method. In the replica-symmetric approximation, we analyze the pressure-temperature phase diagram, and obtain expressions for the critical boundaries between the disordered and the ordered (spin-glass and ferromagnetic) phases. The second-order para-ferromagnetic border ends at a tricritical point, beyond which the transition becomes discontinuous. We use th…
We consider a microcanonical local algorithm to be applied on the $\pm J$ spin glass model. We have compared the results coming from a microcanonical Monte Carlo simulation with those from a canonical one: Thermalization times, spin glass susceptibilities and Binder parameters. For a fixed lattice size we found different results between the two thermodynamic ensembles, which tend to vanish at bigger volumes. Moreover, microcanonical thermalization times are longer than the canonical ones. Finally we have checked that one of the Guerra relations is satisfied with good precision for the two larg…
We define an ensemble of random Clifford quantum circuits whose output state undergoes an entanglement phase transition between two volume-law phases as a function of measurement rate. Our setup maps exactly the output state to the ground space of a spin glass model. We identify the entanglement phases using an order parameter that is accessible on a quantum chip. We locate the transition point and evaluate a critical exponent, revealing spin glass criticality. Our work establishes an exact statistical mechanics theory of an entanglement phase transition.
Using the discrete $\pm J$ bond distribution for the Sherrington-Kirkpatrick spin glass, all ground states for the entire ensemble of the bond disorder are enumerated. Although the combinatorial complexity of the enumeration severely restricts attainable system sizes, here $N\leq9$, some remarkably intricate patterns found in previous studies already emerge. The analysis of the exact ground state frequencies suggests a direct construction of their probability density function. Against expectations, the result suggests that its highly skewed appearance for finite $N$ evolves logarithmically slo…
We propose a new class of phenomenological models for dynamic glass transitions. The system consists of an ensemble of mesoscopic regions to which local energies are allocated. At each time step, a region is randomly chosen and a new local energy is drawn from a distribution that self-consistently depends on the global energy of the system. Then, the transition is accepted or not according to the Metropolis rule. Within this scheme, we model an energy threshold leading to a mode-coupling glass transition as in the p-spin model. The glassy dynamics is characterized by a two-step relaxation of t…
We show that the phonon-saddle transition in the ensemble of generalized inherent structures (minima and saddles) happens at the same point as the dynamical phase transition in glasses, that has been studied in the framework of the mode coupling approximation. The Boson peak observed in glasses at low temperature is a remanent of this transition.
Answering the question of existence of efficient quantum algorithms for NP-hard problems require deep theoretical understanding of the properties of the low-energy eigenstates and long-time coherent dynamics in quantum spin glasses. We discovered and described analytically the property of asymptotic orthogonality resulting in a new type of structure in quantum spin glass. Its eigen-spectrum is split into the alternating sequence of bands formed by quantum states of two distinct types ($x$ and $z$). Those of $z$-type are non-ergodic extended eigenstates (NEE) in the basis of $\{σ_z\}$ operators…
We analyse biased ensembles of trajectories for the random-field Ising model on a fully-connected lattice, which is described exactly by mean-field theory. By coupling the activity of the system to a dynamical biasing field, we find a range of dynamical phase transitions, including spontaneous symmetry breaking into ordered states. For weak bias, the phase behaviour is controlled by extrema of the free energy, which may be local minima or saddle points. For large bias, the system tends to states of extremal activity, which may differ strongly from free energy minima. We discuss connections of …
We provide here a brief perspective on the glass transition field. It is an assessment, written from the point of view of theory, of where the field is and where it seems to be heading. We first give an overview of the main phenomenological characteristics, or "stylised facts", of the glass transition problem, i.e. the central observations that a theory of the physics of glass formation should aim to explain in a unified manner. We describe recent developments, with a particular focus on real space properties, including dynamical heterogeneity and facilitation, the search for underlying spatia…
The K-core of a graph is the maximal subgraph within which each vertex is connected to at least K other vertices. It is a fundamental network concept for understanding threshold cascading processes with a discontinuous percolation transition. A minimum attack set contains the smallest number of vertices whose removal induces complete collapse of the K-core. Here we tackle this prototypical optimal initial-condition problem from the spin-glass perspective of cycle-tree maximum packing and propose a cycle-tree guided attack (CTGA) message-passing algorithm. The good performance and time efficien…
