Absence of Replica Symmetry Breaking in the Transverse and Longitudinal Random Field Ising Model

It is proved that replica symmetry is not broken in the transverse and longitudinal random field Ising model. In this model, the variance of spin overlap of any component vanishes in any dimension almost everywhere in the coupling constant space in the infinite volume limit. The weak Fortuin–Kasteleyn–Ginibre property in this model and the Ghirlanda–Guerra identities in artificial models in a path integral representation based on the Lie–Trotter–Suzuki formula enable us to extend Chatterjee’s proof for the random field Ising model to the quantum model.     

Broken Replica Symmetry Bounds in the Mean Field Spin Glass Model

 By using a simple interpolation argument, in previous work we have proven the existence of the thermodynamic limit, for mean field disordered models, including the Sherrington-Kirkpatrick model, and the Derrida p-spin model. Here we extend this argument in order to compare the limiting free energy with the expression given by the Parisi Ansatz, and including full spontaneous replica symmetry breaking. Our main result is that the quenched average of the free energy is bounded from below by the value given in the Parisi Ansatz, uniformly in the size of the system. Moreover, the difference between the two expressions is given in the form of a sum rule, extending our previous work on the comparison between the true free energy and its replica symmetric Sherrington-Kirkpatrick approximation. We give also a variational bound for the infinite volume limit of the ground state energy per site. Received: 6 May 2002 / Accepted: 6 September 2002 Published online: 13 January 2003     

Replica Symmetry Breaking in Short-Range Spin Glasses: Theoretical Foundations and Numerical Evidences

We discuss replica symmetry breaking (RSB) in spin glasses. We update work in this area, from both the analytical and numerical points of view. We give particular attention to the difficulties stressed by Newman and Stein concerning the problem of constructing pure states in spin glass systems. We mainly discuss what happens in finite-dimensional, realistic spin glasses. Together with a detailed review of some of the most important features, facts, data, and phenomena, we present some new theoretical ideas and numerical results. We discuss among others the basic idea of the RSB theory, correlation functions, interfaces, overlaps, pure states, random field, and the dynamical approach. We present new numerical results for the behaviors of coupled replicas and about the numerical verification of sum rules, and we review some of the available numerical results that we consider of larger importance (for example, the determination of the phase transition point, the correlation functions, the window overlaps, and the dynamical behavior of the system).     

Replica Symmetry Breaking Condition Exposed by Random Matrix Calculation of Landscape Complexity

We start with a rather detailed, general discussion of recent results of the replica approach to statistical mechanics of a single classical particle placed in a random N(?1)-dimensional Gaussian landscape and confined by a spherically symmetric potential suitably growing at infinity. Then we employ random matrix methods to calculate the density of stationary points, as well as minima, of the associated energy surface. This is used to show that for a generic smooth, concave confining potentials the condition of the zero-temperature replica symmetry breaking coincides with one signaling that both mean total number of stationary points in the energy landscape, and the mean number of minima are exponential in N. For such systems the (annealed) complexity of minima vanishes cubically when approaching the critical confinement, whereas the cumulative annealed complexity vanishes quadratically. Different behaviour reported in our earlier short communication (Fyodorov et al. in JETP Lett. 85:261, 2007) was due to non-analyticity of the hard-wall confinement potential. Finally, for the simplest case of parabolic confinement we investigate how the complexity depends on the index of stationary points. In particular, we show that in the vicinity of critical confinement the saddle-points with a positive annealed complexity must be close to minima, as they must have a vanishing fraction of negative eigenvalues in the Hessian.     

Replica Symmetry Breaking in FRET‐Assisted Random Laser Based on Electrospun Polymer Fiber

Spin‐glass theory has been widely introduced to describe the statistical behaviors in complex physical systems. By analogy between disorder photonics and other complex systems, the glassy behavior, especially the replica symmetry breaking (RSB) phenomenon, has been observed in random lasers. However, previous studies only analyzed the statistical properties of the random laser systems with single gain material. Here, the first experimental evidence of the glassy behavior in a random laser with complex energy level structure is reported. This novel random laser is demonstrated based on the electrospun polymer fibers with the assistance of Förster resonance energy transfer (FRET). The electrospinning technology employed in the experiment herein promises high‐volume production of random laser devices with multiple energy levels, enabling the comprehensive investigation of lasing properties in multi‐energy level random laser system. Clear paramagnetic phase and spin‐glass phase are observed in the FRET‐assisted random laser under different pump energies. The RSB phase transition is verified to occur at the laser threshold, which is robust among the random lasers with different donor–acceptor ratio. The finding of RSB in FRET‐assisted random laser provides a new statistical analysis method toward the laser system with complex energy level, for example, quantum cascade laser.     

Replicon Modes in the Renormalization Group with Replica Symmetry Breaking

Abstract The renormalization group with replica symmetry (RSB) breaking is studied for the disordered ferromagnets. The ε-expansion (ε= 4-D, D is the spatial dimension) is used. For fixed points with step-like RSB structufe replicon eigqnvalpes are defined to describe the stability of the fixed points with respect to the continuous lESB perturbation.It is shown that this metliod can be generalized to the cases in which the number of the steps in the step-like RSB structure ahd the order of expansion are arbitrary.     

The State(s) of Replica Symmetry Breaking: Mean Field Theories vs. Short-Ranged Spin Glasses

We prove the impossibility of recent attempts to decouple the Replica Symmetry Breaking (RSB) picture for finite-dimensional spin glasses from the existence of many thermodynamic (i.e., infinite-volume) pure states while preserving another signature RSB feature—space filling relative domain walls between different finite-volume states. Thus revisions of the notion of pure states cannot shield the RSB picture from the internal contradictions that rule out its physical correctness in finite dimensions at low temperature in large finite volume.     

Continuous Replica Symmetry Breaking in Critical Phenomena in Disordered Ferromagnets

The replica symmetry breaking (RSB) in critical phenomena in disordered ferromagnets is studied. We modified the assumption of Dotsenko et al. about the local minimum solutions of the Hamiltonian with random temperature. It is shown explicitly that the continuous RSB is generated in the renormalization group (RG). The physical regime of the coupling constants are investigated through the RG equations.     

Replica Symmetry Breaking in Critical Phenomena in Long-Range Disordered Ferromagnets

The replica symmetry breaking (RSB) is investigated in m-component ferromagnetic spin systems with a long-range disorder, with a correlation function obeying a power law ~s^-a. The Landau-Ginzburg Hamiltonjan with an RSB quartic interaction term is studied, with the renormalization-group expansion in ε = 4-D and δ = 4-a, where D is the spatial dimension. It is shown that the long-range disorder fixed point found previously is unstable under the perturbation of RSB for m<4 and stable for m≥4. The RSB fixed points are calculated and there is no physical stable fixed point for m<4.     

Universal Nature of Replica Symmetry Breaking in Quantum Systems with Gaussian Disorder

We study quantum spin systems with quenched Gaussian disorder. We prove that the variance of all physical quantities in a certain class vanishes in the infinite volume limit. We study also replica symmetry breaking phenomena, where the variance of an overlap operator in the other class does not vanish in the replica symmetric Gibbs state. On the other hand, it vanishes in a spontaneous replica symmetry breaking Gibbs state defined by applying an infinitesimal replica symmetry breaking field. We prove also that the finite variance of the overlap operator in the replica symmetric Gibbs state implies the existence of a spontaneous replica symmetry breaking.     

Critical Behavior of the Annealed Ising Model on Random Regular Graphs

In Giardinà et al. (ALEA Lat Am J Probab Math Stat 13(1):121–161, 2016), the authors have defined an annealed Ising model on random graphs and proved limit theorems for the magnetization of this model on some random graphs including random 2-regular graphs. Then in Can (Annealed limit theorems for the Ising model on random regular graphs, arXiv:1701.08639, 2017), we generalized their results to the class of all random regular graphs. In this paper, we study the critical behavior of this model. In particular, we determine the critical exponents and prove a non standard limit theorem stating that the magnetization scaled by \(n^{3/4}\) converges to a specific random variable, with n the number of vertices of random regular graphs.     

Beyond Storage Capacity in a Single Model Neuron: Continuous Replica Symmetry Breaking

A single McCulloch–Pitts neuron, that is, the simple perceptron is studied, with focus on the region beyond storage capacity. It is shown that Parisi's hierarchical ansatz for the overlap matrix of the synaptic couplings with so called continuous replica symmetry breaking is a solution, and as we propose it is the exact one, to the equilibrium problem. We describe some of the most salient features of the theory and give results about the low temperature region. In particular, the basics of the Parisi technique and the way to calculate thermodynamical expectation values is explained. We have numerically extremized the replica free energy functional for some parameter settings, and thus obtained the order parameter function, i.e., the probability distribution of overlaps. That enabled us to evaluate the probability density of the local stability parameter. We also performed a simulation and found a local stability density closer to the theoretical curve than previous numerical results were.     

Finite-Size Scaling in the Energy-Entropy Plane for the 2D ± Ising Spin Glass

For L × L square lattices with L ≤ 20 the 2D Ising spin glass with +1 and −1 bonds is found to have a strong correlation between the energy and the entropy of its ground states. A fit to the data gives the result that each additional broken bond in the ground state of a particular sample of random bonds increases the ground state degeneracy by approximately a factor of 10/3. For x=0.5 (where x is the fraction of negative bonds), over this range of L, the characteristic entropy defined by the energy-entropy correlation scales with size as L ^1.78(2). Anomalous scaling is not found for the characteristic energy, which essentially scales as L ². When x=0.25, a crossover to L ² scaling of the entropy is seen near L=12. The results found here suggest a natural mechanism for the unusual behavior of the low temperature specific heat of this model, and illustrate the dangers of extrapolating from small L. PACS numbers: 75.10.Nr, 75.40.Mg, 75.50.Lk     

Parity Symmetry and Parity Breaking in the Quantum Rabi Model with Addition of Ising Interaction

We explore the possibility to generate new parity symmetry in the quantum Rabi model after a bias is introduced. In contrast to a mathematical treatment in a previous publication [J. Phys. A 46(2013) 265302], we consider a physically realistic method by involving an additional spin into the quantum Rabi model to couple with the original spin by an Ising interaction, and then the parity symmetry is broken as well as the scaling behavior of the ground state by introducing a bias. The rule can be found that the parity symmetry is broken by introducing a bias and then restored by adding new degrees of freedom. Experimental feasibility of realizing the models under discussion is investigated.     

Translational Symmetry Breaking and Soliton Sectors for Massive Quantum Spin Models in 1 + 1 Dimensions

We consider the classification of pure infinite volume ground states and that of soliton sectors for 1+1 dimensional massive quantum spin models. We obtain a proof that non-translationally invariant ground state cannot exist for a class of translationally invariant Hamiltonians including the spin 1 AKLT (Affleck Kennedy Lieb Tasaki) antiferromagnetic spin model. We also obtain a complete classification of soliton sectors (up to unitary equivalence) for certain massive models (e.g. ferromagnetic XXZ models). Received: 13 January 1997 / Accepted: 11 March 1997     

Finite-Size Scaling of the Domain Wall Entropy Distributions for the 2D ± J Ising Spin Glass

The statistics of domain walls for ground states of the 2D Ising spin glass with +1 and −1 bonds are studied for L × L square lattices with L ≤ 48, and p = 0.5, where p is the fraction of negative bonds, using periodic and/or antiperiodic boundary conditions. When L is even, almost all domain walls have energy E _dw = 0 or 4. When L is odd, most domain walls have E _dw = 2. The probability distribution of the entropy, S _dw , is found to depend strongly on E _dw . When E _dw = 0, the probability distribution of |S _dw | is approximately exponential. The variance of this distribution is proportional to L, in agreement with the results of Saul and Kardar. For E _dw = k > 0 the distribution of S _dw is not symmetric about zero. In these cases the variance still appears to be linear in L, but the average of S _dw grows faster than L. This suggests a one-parameter scaling form for the L-dependence of the distributions of S _dw for k> 0. PACS: 75.10.Nr, 75.40.Mg, 75.60.Ch, 05.50.+q     

Path Integral Approach of Quantum Heisenberg Planar Spin Glass Model in the Presencesof Random Field

A quantum version of the site-random Heisenberg planar XY model in a random field is presented in the boson representation. Like classical spherical model in the spin space, the model can be solved exactly within the coherent state path integral-representation. The phase diagram is obtained, and the effects of the randomness and quantum fluctuations on the onset of a spin glass phase are discussed.