Bethe Peierls Research Articles

Quantum phase transitions in non-Hermitian [formula omitted]-symmetric transverse-field Ising spin chains

We present a theoretical study of quantum phases and quantum phase transitions occurring in non-Hermitian PT-symmetric superconducting qubits chains described by a transverse-field Ising spin model. A non-Hermitian part of the Hamiltonian is implemented via imaginary staggered longitudinal magnetic field, which corresponds to a local staggered gain and loss terms. By making use of a direct numerical diagonalization of the Hamiltonian for spin chains of a finite size N, we explore the dependencies of the energy spectrum, including the energy difference between the first excited and the ground states, the spatial correlation function of local polarization (z-component of local magnetization) on the adjacent spins interaction strength J and the local gain (loss) parameter γ. A scaling procedure for the correlation length ξ allows us to establish a complete quantum phase diagram of the system. We obtain two quantum phases for J<0, namely, PT-symmetry broken antiferromagnetic state and PT-symmetry preserved paramagnetic state, and the quantum phase transition line between them is the line of exception points. For J>0 the PT-symmetry of the ground state is retained in a whole region of parameter space of J and γ, and a system shows two intriguing quantum phase transitions between ferromagnetic and paramagnetic states for a fixed parameter γ>1. We also provide the qualitative quantum phase diagram γ−J derived in the framework of the Bethe–Peierls approximation that is in a good accord with numerically obtained results.

A simple approximation for Ising spin models

A simple method is introduced for Ising spin systems based on the correlated molecular-field theory. Here, we consider an interacting triple in the mean field of its surrounding lattice points taking into account the product of three spins in the cluster. The present results for Curie temperature are compared to the mean-field theory(MFT), Bethe–Peierls–Weiss(BPW) approach, and self-consistent correlated field(SCCF) approximation, respectively. A procedure of triangular clusters is investigated that yields very distinguishable improvement of the critical temperature, as well.

A mean field approach to the Ising chain in a transverse magnetic field

We evaluate a mean field method to describe the properties of the ground state of the Ising chain in a transverse magnetic field. Specifically, a method of the Bethe–Peierls type is used by solving spin blocks with a self-consistency condition at the borders. The computations include the critical point for the phase transition, exponent of magnetisation and energy density. All results are obtained using basic quantum mechanics at an undergraduate level. The advantages and the limitations of the approach are emphasised.

Statistical mechanics of clonal expansion in lymphocyte networks modelled with slow and fast variables

We use statistical mechanical techniques to model the adaptive immune system, represented by lymphocyte networks in which B cells interact with T cells and antigen. We assume that B- and T-clones evolve in different thermal noise environments and on different timescales, and derive stationary distributions and study expansion of B clones for the case where these timescales are adiabatically separated. We compute characteristics of B-clone sizes, such as average concentrations, in parameter regimes where T-clone sizes are modelled as binary variables. This analysis is independent of the network topology, and its results are qualitatively consistent with experimental observations. To obtain the full distributions of B-clone sizes we assume further that the network topologies are random and locally equivalent to trees. This allows us to compete these distributions via the Bethe–Peierls approach. As an example we calculate B-clone distributions for immune models defined on random regular networks.

Stability of the (2+2)-fermionic system with zero-range interaction

We introduce a 3D model, and we study its stability, consisting of two distinct pairs of identical fermions coupled with a two-body interaction between fermions of different species, whose effective range is essentially zero (a so called (2+2)-fermionic system with zero-range interaction). The interaction is modelled by implementing the celebrated (and ubiquitous in the literature of this field) Bethe–Peierls contact condition with given two-body scattering length within the Krein–Višik–Birman theory of extensions of semi-bounded symmetric operators, in order to make the Hamiltonian a well-defined (self-adjoint) physical observable. After deriving the expression for the associated energy quadratic form, we show analytically and numerically that the energy of the model is bounded below, thus describing a stable system.

The combination of the random interaction fields’ method and the Bethe–Peierls method for studying two-sublattice magnets

In this paper, the concentration magnetic phase transitions are described using the random interaction fields’ method in combination with the Bethe–Peierls method. This approach allows us to take into account the correlation between magnetic moments and obtain a more accurate solution for the calculation of the Curie point in comparison with the use of the Kikuchi method, which is considered as the most accurate method in the molecular field theory. As an example, we study the dependence of the Curie point on the concentration of titanium in titanomagnetite. This material is one of the typical natural ferrimagnetics.

Spin systems on hypercubic Bethe lattices: a Bethe–Peierls approach

We study spin systems on Bethe lattices constructed from d-dimensional hypercubes. Although these lattices are not tree-like, and therefore closer to real cubic lattices than Bethe lattices or regular random graphs, one can still use the Bethe–Peierls method to derive exact equations for the magnetization and other thermodynamic quantities. We compute phase diagrams for ferromagnetic Ising models on hypercubic Bethe lattices with dimension d = 2, 3, and 4. Our results are in good agreement with the results of the same models on d-dimensional cubic lattices, for low and high temperatures, and offer an improvement over the conventional Bethe lattice with connectivity .

Six–vertex model with domain wall boundary conditions in the Bethe–Peierls approximation

We use the Bethe–Peierls method combined with the belief propagation algorithm to study the arctic curves in the six vertex model on a square lattice with domain-wall boundary conditions, and the six vertex model on a rectangular lattice with partial domain-wall boundary conditions. We show that this rather simple approximation yields results that are remarkably close to the exact ones when these are known, and allows one to estimate the location of the phase boundaries with relative little effort in cases in which exact results are not available.

Lowering the error floor of Gallager codes: a statistical-mechanical view

The error correction for Gallager's low-density parity-check codes is a problem famously equivalent to that of computing marginal Boltzmann probabilities for an Ising-like model with multispin interactions in a non-uniform magnetic field. Since the graph of interactions is locally a tree, the solution is very well approximated by a generalized mean-field (Bethe–Peierls) approximation. Belief propagation (BP) and similar iterative algorithms are an efficient method to perform the calculation, but they sometimes fail to converge, or converge to non-codewords, giving rise to a non-negligible residual error probability (error floor). On the other hand, provably-convergent algorithms are far too complex to be implemented in a real decoder. In this work we consider the application of the probability-damping technique, which can be regarded either as a variant of BP, from which it retains the property of low complexity, or as an approximation of a provably-convergent algorithm, from which it is expected to inherit better convergence properties. We investigate the algorithm behaviour on a real instance of Gallager code, and compare the results with state-of-the-art algorithms.

Short-range correlations in binary alloys: Spin model approach to AgcAu1−c and AgcPd1−c

Short-range correlations in Ag–Au and Ag–Pd alloys are investigated by analyzing the ab initio total energy of face centered cubic (fcc) based random AgcAu1−c and AgcPd1−c. Since the information on the atomic interactions is incorporated in the energetics of alloys it is possible with a suitable model, Bethe–Peierls–Weiss model is used in the present work, to invert the problem, i.e. to obtain information on the short-range correlation from the total energy of a random system. As an example we demonstrate how site correlations can be extracted from random alloy data. Bethe–Peierls–Weiss model predicts negative (positive) first neighbor correlator for substitutional fcc Ag–Au and (Ag–Pd) alloys at low temperature which can be related to the optimal structures of Ag0.5Au0.5 (and Ag0.5Pd0.5).

Static properties of 2D spin-ice as a sixteen-vertex model

We present a thorough study of the static properties of 2D models of spin-ice type on the square lattice or, in other words, the sixteen-vertex model. We use extensive Monte Carlo simulations to determine the phase diagram and critical properties of the finite-dimensional system. We put forward a suitable mean-field approximation, by defining the model on carefully chosen trees. We employ the cavity (Bethe–Peierls) method to derive self-consistent equations, the fixed points of which yield the equilibrium properties of the model on the tree-like graph. We compare mean-field and finite-dimensional results. We discuss our findings in the context of experiments in artificial two-dimensional spin-ice.

Modified Bethe-Peierls boundary condition for ultracold atoms with spin-orbit coupling

We show that the Bethe-Peierls (BP) boundary condition should be modified for ultracold atoms with spin-orbit (SO) coupling. Moreover, we derive a general form of the modified BP boundary condition, which is applicable to a system with arbitrary kind of SO coupling. In the modified BP condition, an anisotropic term appears and the inter-atomic scattering length is normally SO-coupling dependent. For the special system in the current experiments, however, it can be proved that the scattering length is SO-coupling independent, and takes the same value as in the case without SO coupling. Our result is helpful for the study of both few-body and many-body physics in SO-coupled ultracold gases.

CORRELATED REDUCED TRANSFER MATRIX APPROACH FOR ISING MODEL

In this paper we present a simple approximate transfer matrix method for 2D and 3D hyper cubic nearest neighbor Ising models with various coordination number z to calculate the corresponding critical coupling strengths Kc. The critical coupling strengths of the Ising ferromagnets are obtained quite accurately by simple improvements over the self-consistent correlated field (SCCF) approximation. The important physical effect included in this work is some of the fluctuation effects of the systems by the help of a reduced transfer matrix method. When used in combination with the accuracy of the average magnetization obtained from the SCCF approximation, this reduced transfer matrix method leads to estimate of Kc more accurate than those obtained from the Bethe–Peierls–Weiss approximation and also SCCF approximation. Therefore, we believe that the approach we refer to as the correlated reduced transfer matrix method is potentially very useful scheme for obtaining approximate values of the critical coupling strengths of the Ising models with a mathematically easy meaner.

Bethe–Peierls approximation and the inverse Ising problem

We apply the Bethe–Peierls approximation to the inverse Ising problem and show how thelinear response relation leads to a simple method for reconstructing couplings and fields ofthe Ising model. This reconstruction is exact on tree graphs, yet its computational expenseis comparable to those of other mean-field methods. We compare the performance of thismethod to the independent-pair, naive mean-field, and Thouless–Anderson–Palmerapproximations, the Sessak–Monasson expansion, and susceptibility propagation on theCayley tree, SK model and random graph with fixed connectivity. At low temperatures,Bethe reconstruction outperforms all of these methods, while at high temperatures it iscomparable to the best method available so far (the Sessak–Monasson method).The relationship between Bethe reconstruction and other mean-field methods isdiscussed.

Partition function loop series for a general graphical model: free-energy corrections and message-passing equations

A loop series expansion for the partition function of a general statistical model on a graph is carried out. If the auxiliary probability distributions of the expansion are chosen to be a fixed point of the belief-propagation equation, the first term of the loop series gives the Bethe–Peierls (BP) free-energy functional at the replica-symmetric level of the mean-field spin-glass theory, and corrections are contributed only by subgraphs that are free of dangling edges. This result generalizes the early work of Chertkov and Chernyak on binary statistical models. If the belief-propagation equation has multiple fixed points, a loop series expansion is performed for the grand partition function. The first term of this series gives the BP free-energy functional at the first-step replica-symmetry-breaking (RSB) level of the mean-field spin-glass theory, and corrections again come only from subgraphs that are free of dangling edges, provided that the auxiliary probability distributions of the expansion are chosen to be a fixed point of the survey-propagation equation. The same loop series expansion can be performed for higher-level partition functions, obtaining the higher-level RSB BP free-energy functionals (and the correction terms) and message-passing equations without using the BP approximation.

A COMPOSITE FERMION APPROACH TO THE ULTRACOLD DILUTE FERMI GAS

It is argued that the recently observed Fermi liquids in strongly interacting ultracold Fermi gases are adiabatically connected to a projected Fermi gas. This conclusion is reached by constructing a set of Jastrow wavefunctions, following Tan's observations on the structure of the physical Hilbert space [Annals of Physics 323, 2952 (2008)]. The Jastrow projection merely implements the Bethe–Peierls condition on the BCS and Fermi gas wavefunctions. This procedure provides a simple picture of the emergence of Fermi polarons as composite fermions in the normal state of the highly polarized gas. It is also shown that the projected BCS wavefunction can be written as a condensate of pairs of composite fermions (or Fermi polarons). A Hamiltonian for the composite fermions is derived. Within a mean-field theory, it is shown that the ground state and excitations of this Hamiltonian are those of a non-interacting Fermi gas although they are described by Jastrow–Slater wavefunctions.

Complex replica zeros of ±J Ising spin glass at zero temperature

Zeros of the nth moment of the partition function [Zn] are investigated in a vanishing temperature limit β → ∞, n → 0 keeping y = βn ∼ O(1). In this limit, the moment parameterized by y characterizes the distribution of the ground-state energy. We numerically investigate the zeros for ±J Ising spin-glass models with tree and several other systems, which can be carried out with a feasible computational cost by a symbolic operation based on the Bethe–Peierls method. For several tree systems we find that the zeros tend to approach the real axis of y in the thermodynamic limit implying that the moment cannot be described by a single analytic function of y as the system size tends to infinity, which may be associated with breaking of the replica symmetry. However, examination of the analytical properties of the moment function and assessment of the spin-glass susceptibility indicate that the breaking of analyticity is relevant to neither one-step nor full replica symmetry breaking.

Fermions and loops on graphs: I. Loop calculus for determinants

This paper is the first in a series devoted to evaluation of the partition function instatistical models on graphs with loops in terms of the Berezin/fermion integrals. Thepaper focuses on a representation of the determinant of a square matrix in terms of a finiteseries, where each term corresponds to a loop on the graph. The representation is based ona fermion version of the loop calculus, previously introduced by the authors for graphicalmodels with finite alphabets. Our construction contains two levels. First, we represent thedeterminant in terms of an integral over anti-commuting Grassmann variables, withsome reparametrization/gauge freedom hidden in the formulation. Second, weshow that a special choice of the gauge, called the BP (Bethe–Peierls or beliefpropagation) gauge, yields the desired loop representation. The set of gauge fixing BPconditions is equivalent to the Gaussian BP equations, discussed in the past asefficient (linear scaling) heuristics for estimating the covariance of a sparse positivematrix.

Friendship and physicsLeeSabineThe Bethe–Peierls Correspondence. World Scientific, Singapore2007. pp. xi +506. £39.00 (hardback).13 978-981-277-135-3, 10 981-277-135-2.

Restricted accessMoreSectionsView Full TextView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Elliott Roger 2008Friendship and physicsNotes Rec. R. Soc.62401–403http://doi.org/10.1098/rsnr.2008.0037SectionRestricted accessFriendship and physics Roger Elliott Roger Elliott [email protected] Google Scholar Find this author on PubMed Search for more papers by this author Roger Elliott Roger Elliott [email protected] Google Scholar Find this author on PubMed Search for more papers by this author Published:13 October 2008https://doi.org/10.1098/rsnr.2008.0037 Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetails This Issue20 December 2008Volume 62Issue 4 Article InformationDOI:https://doi.org/10.1098/rsnr.2008.0037Published by:Royal SocietyPrint ISSN:0035-9149Online ISSN:1743-0178History: Manuscript received05/08/2008Manuscript accepted05/08/2008Published online13/10/2008Published in print20/12/2008 License:© 2008 The Royal Society Citations and impact PDF Download Subjectsbiographical history

Three fermions in a box at the unitary limit: universality in a lattice model

We consider three fermions with two spin components interacting on a lattice model with an infinite scattering length. Low-lying eigenenergies in a cubic box with periodic boundary conditions, and for a zero total momentum, are calculated numerically for decreasing values of the lattice period. The results are compared to the predictions of the zero-range Bethe–Peierls model in continuous space, where the interaction is replaced by contact conditions. The numerical computation, combined with analytical arguments, shows the absence of negative energy solution, and a rapid convergence of the lattice model towards the Bethe–Peierls model for a vanishing lattice period. This establishes for this system the universality of the zero-interaction range limit.