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Physical Review Letters

Quantum Spin Glass on the Bethe Lattice

T. K. Kopeć, Klaus D. Usadel

Published Mar 10, 1997• 24 citations
Publisher

Abstract

We present an exact analytic solution of the short-range interaction disordered quantum spherical model with symmetric binary $\ifmmode\pm\else\textpm\fi{}J$ bond distribution on the Bethe lattice (with coordination number $z$). We find the zero temperature quantum spin glass phase transition for coordinations $z\ensuremath{\ge}3$ but not for the one-dimensional system $(z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2)$. Quantum dynamics is examined via various correlation functions on the infinite tree which are evaluated in a closed form. The relationship between the results for a fully connected lattice $(z\ensuremath{\rightarrow}\ensuremath{\infty})$ and the usual mean-field approach is established.

DOI
10.1103/physrevlett.78.1988

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We present an exact analytic solution of the short-range interaction disordered quantum spherical model with symmetric binary \ifmmode±\else\textpm\fiJ\ifmmode\pm\else\textpm\fi{}J\ifmmode±\else\textpm\fiJ bond distribution on the Bethe lattice (with coordination number zzz). We find the zero temperature quantum spin glass phase transition for coordinations z\ensuremath≥3z\ensuremath{\ge}3z\ensuremath≥3 but not for the one-dimensional system (z=2)(z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2)(z=2). Quantum dynamics is examined via various correlation functions on the infinite tree which are evaluated in a closed form. The relationship between the results for a fully connected lattice (z\ensuremath→\ensuremath∞)(z\ensuremath{\rightarrow}\ensuremath{\infty})(z\ensuremath→\ensuremath∞) and the usual mean-field approach is established.
We consider the short-range-interaction disordered quantum spherical model with a symmetrical binary -bond distribution on the Bethe lattice (with coordination number z ). The system exhibits a quantum phase transition separating the spin-glass and disordered phases, where the quantum effects are regulated by a parameter describing the kinetic energy. The model admits, in the spherical limit, an exact solution both in the spin-glass phase and in the disordered phase. The quantum dynamics is examined via various correlation functions on the infinite tree, which are evaluated in closed form. A closed-form solution for the averaged free energy has been obtained, and various thermodynamic functions are explicitly derived in the form of low-temperature expansions.
Abstract We consider the short‐range interaction disordered quantum Ising model with symmetric binary ±J bond distribution on the Bethe lattice (with coordination number z). The system exhibits quantum phase transition separating the spin glass and disordered phases where the quantum effect are regulated by a param‐ eter Δ describing the transverse field. By introducing a mapping of the quantum Hamiltonian of the model onto a soft‐spin action we consider it truncated version in a form of the solvable quantized spherical model. Quantum dynamics is examined via various correlation functions on the infinite tree which are evaluated in a closed form. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
Bethe Lattice Spin Glasses (BLSG) are models with finite connectivity which undergo a Replica Symmetry Breaking (RSB) phase transition in field, even at zero temperature. The order parameter describing this phase is a function related to the non-trivial organization of the states. We compute numerically the spin glass order parameter at zero temperature in BLSG with coordination number z=3z=3z=3 and couplings J=±1J=\pm1J=±1, near the transition. The method is based on a universal formula which relates the order parameter to the joint probability distribution of the energy gaps of the lowest lying states. Such distribution is derived numerically by introducing a convenient bulk perturbation to the Hamiltonian.
We consider quantum rotors or Ising spins in a transverse field on a d-dimensional lattice, with random, frustrating, short-range, exchange interactions. The quantum dynamics are associated with a finite moment of inertia for the rotors, and with the transverse field for the Ising spins. For a suitable distribution of exchange constants, these models display spin-glass and quantum paramagnet phases and a zero-temperature (T) quantum transition between them. An earlier exact solution for the critical properties of a model with infinite-range interactions cna be reproduced by minimization of a Landau effective-action functional for the model in finite d with short-range interactions. The functional is expressed in terms of a composite spin field which is bilocal in time. The mean-field phase diagram near the T=0 critical point is mapped out as a function of T, strength of the quantum coupling, and applied fields. The spin-glass phase has replica symmetry breaking; but, as in the classical Ising spin glass, the order parameter becomes replica symmetric as T\ensuremath{\rightarrow}0. Next we examine the consequences of fluctuations about the mean field for the critical properties. Above d=8, and with certain restrictions on the values of the Landau couplings, we find that the transition is controlled by a Gaussian fixed point with mean-field critical exponents. For couplings not attracted by the Gaussian fixed point above d=8, and for all physical couplings below d=8, we find runaway renormalization-group flows to strong coupling. General scaling relations that should be valid even at the strong-coupling fixed point are proposed and compared with Monte Carlo simulations.
We investigate the low temperature phase of a vector spin glass in the limit of large component numbers mmm of the spins, where this model is expected to be replica symmetric. On a Bethe lattice, a random graph with fixed and finite connectivity, we employ a self-consistent effective-field approach to the cavity method to describe equilibrium states of the system in terms of the configuration of cavity fields, which are reduced mean fields onto each spin. We make use of an iterative algorithm to find such equilibrium states for finite system sizes NNN and observe a phase transition, which is indicated by an order parameter describing the preferred direction each spin points into in terms of the configuration of cavity, resp. mean local fields. On single realizations of Bethe lattices the transition is sharp and accompanied by the Generalized-Bose-Einstein condensation, in which the cavity, resp. local fields of the spins condense into a subspace of the available mmm-dimensional space for large mmm. The dimension of this subspace, meff(T)m_{\rm eff}(T)meff​(T) depends on the temperature. Is either one or two right at the critical temperature and increases below. Furthermore, for fixed temperature it is expected to scale with the system size NNN with an exponent μm\mu_mμm​ which depends on the connectivity. Averaging over many realizations of disorder we find meff‾(Tc)=2\overline{m_{\rm eff}}(T_c)=2meff​​(Tc​)=2 and meff‾(T)∼Nμm\overline{m_{\rm eff}}(T)\sim N^{\mu_m}meff​​(T)∼Nμm​. Even more, the cavity fields are subject to rotations in the low temperature phase, i.e. there is no freezing in time as generally expected. In case of meff(Tc)=2m_{\rm eff}(T_c)=2meff​(Tc​)=2 the whole configuration rotates at the critical temperature about the same angle and the same axis of rotation corresponding to a single droplet. In addition, we find replica symmetry breaking below the large-mmm limit, i.e. for meff(T)>mm_{\rm eff}(T) > mmeff​(T)>m. These results are supported by results of parallel tempering Monte Carlo simulations. Replica symmetry breaking is further investigated by an analytical method in the large connectivity limit, where we compare the free energy in terms of spin-spin-correlations of different replicas calculated with a replica symmetric and a one-step broken replica symmetry ansatz. In the large-mmm limit, we find that both solutions coexist for finite connectivities and that replica symmetry holds for full connectivity up to corrections in second order around the saddle point for m→∞m\to\inftym→∞. In the last chapter we calculate the sample-to-sample fluctuations of the free energy of the SK model of the large-mmm limit. We derive an exact connection between these fluctuations and bond chaos. Making use of a replica calculation for the {\it large deviations} and a scaling ansatz for the {\it small deviations}, we quantify bond chaos, i.e. calculate the probability of finding a given link overlap for two systems with possibly different sets of bonds. With this, we find the sample-to-sample fluctuations to scale with the number of spins, NNN, with an exponent μ\muμ in between 1/51/51/5 and 3/103/103/10.
We investigate the balanced M=4M=4M=4, p=4p=4p=4 spin-glass model for a one-dimensional long-range proxy for the finite dimensional short-range ppp-spin glass model to examine the nature of the glass transition beyond mean-field theory. We perform large-scale Monte Carlo equilibrated simulations for both fully connected and power-law diluted versions of the model. The critical temperatures extracted from the finite-size scaling (FSS) analysis of spin-glass susceptibility are in good agreement with theoretical predictions for σ=0,0.25σ= 0, 0.25σ=0,0.25, and 0.55. For these values of the long-range exponent σσσ (which is the power of the decrease of the interactions between the spins with their separation), one might have expected that mean-field theory would provide a good description of the system. However, the spin-overlap distribution and the value of the λλλ-parameter do not provide numerical evidence for a one-step replica symmetry breaking (1RSB) phase transition. Instead, our results indicate a direct transition from the paramagnetic state to a full replica symmetry broken phase, with a renormalized value of λ≡ω2/ω1<1λ\equiv ω_2/ω_1 < 1λ≡ω2​/ω1​<1 suggesting a continuous FRSB transition, despite this ratio being equal to 2 at mean-field level. A value of λ>1λ> 1λ>1 is required for the discontinuous 1RSB transition. We argue that strong finite-size effects and closely spaced transition temperatures remove the expected 1RSB transition for the system sizes which we can study. For values of the exponent σ=0.85σ= 0.85σ=0.85, which roughly corresponds to a three dimensional system, we find that the renormalized value of λλλ is again less than 1, with no signs of either the 1RSB transition or the continuous FRSB transition, suggesting that the Kauzmann temperature TKT_KTK​ in three dimensions might be zero and the complete absence of phase transitions in structural glasses.