Replica-symmetry Solution Research Articles

Teacher-Student Learning for a Binary Perceptron with Quantum Fluctuations

We analysed the generalisation performance of a binary perceptron with quantum fluctuations using the replica method. An exponential number of local minima dominate the energy landscape of the binary perceptron. Local search algorithms often fail to identify the ground state of a binary perceptron. In this study, we considered the teacher-student learning method and computed the generalisation error of a binary perceptron with quantum fluctuations. Due to the quantum fluctuations, we can efficiently find robust solutions that have better generalisation performance than the classical model. We validated our theoretical results through quantum Monte Carlo simulations. We adopted the replica symmetry (RS) ansatz assumption and static approximation. The RS solutions are consistent with our simulation results, except for the relatively low strength of the transverse field and high pattern ratio. These deviations are caused by the violation of ergodicity and static approximation. After accounting for the deviation between the RS solutions and numerical results, the enhancement of generalisation performance with quantum fluctuations holds.

Inverse freezing in the Hopfield fermionic Ising spin glass with a transverse magnetic field

The Hopfield fermionic Ising spin glass (HFISG) model in the presence of a magnetic transverse field Γ is used to study the inverse freezing transition. The mean field solution of this model allows introducing a parameter a that controls the frustration level. Particularly, in the present fermionic formalism, the chemical potential μ and the Γ provide a magnetic dilution and quantum spin flip mechanism, respectively. Within the one step replica symmetry solution and the static approximation, the results show that the reentrant transition between the spin glass and the paramagnetic phases, which is related to the inverse freezing for a certain range of μ, is gradually suppressed when the level of frustration a is decreased. Nevertheless, the quantum fluctuations caused by Γ can destroy this inverse freezing for any value of a.

Effect of asymmetry in a binary state on the collective behavior of a system with spatially modulated interaction and quenched randomness

The properties of solid states, biophysical materials, neuronal circuits, and equilibrium states of a many-body system can be studied by using techniques in statistical physics. It has been common practice to represent a system composed of binary state units by using an Ising spin network where each unit has symmetric {-1,1} states. However, the asymmetry or symmetry of the binary states of the units can affect the property and ergodicity of the system, but better understanding of the quantitative difference is still needed. We compare systems of binary units with symmetric or asymmetric states. The network has spatially modulated interaction with quenched randomness. We can bridge the Ising spin network and McCulloch-Pitts neuron network and analyze the stability of the system via replica method by introducing an interpolating parameter. The effects of the asymmetry states affect the multistability of the system and the stability of replica-symmetry solutions.

Some large linear systems in communications: estimation, information and statistical physics

Linear systems are frequently used to model physical channels in communication problems. In such systems, the goal is to recover the information bearing input signal to the linear transformation based on noisy observations of the output. This paper investigates the performance of linear systems randomly selected from some ensembles in the large-dimension limit. Previous analysis of such systems using statistical physics techniques is strengthened, where it is shown that the posterior of each individual input symbol is asymptotically identical to the posterior of a Gaussian channel with the same input. Interestingly, the replica symmetry solution takes an identical form as the rigorous result obtained for an ensemble of sparse linear systems. Some challenges to the information theory and statistical physics communities are highlighted.

One-step replica symmetry breaking solution for a highly asymmetric two-sublattice fermionic Ising spin-glass model in a transverse field

The one-step replica symmetry breaking (RSB) is used to study a two-sublattice fermionic infinite-range Ising spin glass (SG) model in a transverse field $\Gamma$. The problem is formulated in a Grassmann path integral formalism within the static approximation. In this model, a parallel magnetic field $H$ breaks the symmetry of the sublattices. It destroys the antiferromagnetic (AF) order, but it can favor the nonergodic mixed phase (SG+AF) characterizing an asymmetric RSB region. In this region, intra-sublattice disordered interactions $V$ increase the difference between the RSB solutions of each sublattice. The freezing temperature shows a higher increase with $H$ when $V$ enhances. A discontinue phase transition from the replica symmetry (RS) solution to the RSB solution can appear with the presence of an intra-sublattice ferromagnetic average coupling. The $\Gamma$ field introduces a quantum spin flip mechanism that suppresses the magnetic orders leading them to quantum critical points. Results suggest that the quantum effects are not able to restore the RS solution. However, in the asymmetric RSB region, $\Gamma$ can produce a stable RS solution at any finite temperature for a particular sublattice while the other sublattice still presents RSB solution for the special case in which only the intra-sublattice spins couple with disordered interactions.

Linear CDMA Detection Algorithm Based on Statistical Neurodynamics and Belief Propagation and the Stability Conditions

Recently a modified algorithm of code-division multiple-access (CDMA) parallel interference canceler (PIC) has been proposed by Tanaka based on statistical neurodynamics. In this paper we apply the modified algorithm to the linear PIC (LPIC) and investigate its stability. We show that the stable (unstable) fixed points of the modified algorithm correspond to the stable (unstable) replica symmetry solutions with the Gaussian prior. We also show the modified algorithm is a special case of Kabashima’s belief-propagation algorithm with Gaussian prior.

Statistical Mechanical Analysis of CDMA Multiuser Detectors

A class of code-division multiple-access (CDMA) multiuser detectors in the large-system limit has been investigated by Tanaka by means of statistical mechanics, i.e. the replica method. He evaluated analytically the bit-error rate, the multiuser efficiency, etc. of Marginal-Posterior-Mode (MPM) detectors under the replica symmetry (RS) ansatz. In this paper, to investigate the breaking of the replica symmetry, we study the de Almeida-Thouless (AT) stability, the entropy, and the free energy of the RS solutions for MPM detectors including individually optimum (IO) ones.

Ghatak-Sherrington Model with SpinS

We investigate the Ghatak-Sherrington model with S ≥1, including S =∞, in a framework of the replica method, namely in terms of the replica symmetry (RS) solutions, the de Almeida and Thouless (AT) stability analysis and the one-step replica symmetry breaking (RSB) solutions. We find that there is a first order transition of van der Waals type between the paramagnetic phase and the spin glass phase and also a first order transition between the RS solution and the RSB solution in the model with integral S in some range of anisotropy parameter, while there is only a continuous transition in the model with half-integral S , including S =∞. We find that the model with integral S has a critical end point under the external magnetic field. We obtain phase diagrams as for the AT stability for the model under the external magnetic field.

The infinite-range two-real-Ising replicas spin glass

A spin-glass model, with two Ising coupled systems (real replicas), is investigated in the limit of infinite-range interactions, through the replica method. Besides the Ising representation, the model is also considered as a discrete two-component four-state clock spin glass. The phase diagram of the model is obtained, within the replica-symmetry approximation, exhibiting a collinear spin-glass phase, i.e., no transverse four-state clock ordering. The order parameter conjugated to the coupling field presents a discontinuity along the spin-glass phase, when the field is reversed. The validity limits of the replica-symmetry solution are analysed.

THREE-DIMENSIONAL QUANTUM SPIN GLASS THEORY(Ⅲ) ──REPLICA SYMMETRY BREAKING SOLUTION

Using generalized Parisi's replica symmetry breaking scheme, the quantum spin glass theory proposed by our previous work is extended to the first-step replica symmetry breaking. A set of new self-consistent equations of spin self-interact ions and spin glass order parameters are obtained. The corresponding local susceptibility is reduced. It is found that the replica symmetry solution for vector spin glass model at low temperature must be broken through calculating the free-energy of the replica symmetry breaking.

Ising spin glass in a transverse field: Replica-symmetry-breaking solution.

The replica-symmetry-breaking (RSB) solution of the infinite-range Ising spin glass in the presence of a transverse field is obtained. The quenched free energy and the phase boundary of the glass transition temperature versus the transverse field are calculated at first-step RSB without using the static approximation. We demonstrate that replica symmetry (RS) has to be broken in the spin-glass phase by comparing the free energies of the RSB and RS solutions. No evidence is found to support an intermediate spin-glass phase with replica symmetry