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arXiv:cond-mat/0605465 · Physical Review B

The m -component spin glass on a Bethe lattice

A. Braun, Timo Aspelmeier

Published Oct 26, 2006• 5 citations
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Abstract

We study the $m$-component vector spin glass in the limit $m\ensuremath{\rightarrow}\ensuremath{\infty}$ on a Bethe lattice. The cavity method allows for a solution of the model in a self-consistent field approximation and for a perturbative solution of the full problem near the phase transition. The low-temperature phase of the model is analyzed numerically and a generalized Bose-Einstein condensation is found, as in the fully connected model. Scaling relations between four distinct zero-temperature exponents are found.

DOI
10.1103/physrevb.74.144205

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We study the mmm-component vector spin glass in the limit m\ensuremath→\ensuremath∞m\ensuremath{\rightarrow}\ensuremath{\infty}m\ensuremath→\ensuremath∞ on a Bethe lattice. The cavity method allows for a solution of the model in a self-consistent field approximation and for a perturbative solution of the full problem near the phase transition. The low-temperature phase of the model is analyzed numerically and a generalized Bose-Einstein condensation is found, as in the fully connected model. Scaling relations between four distinct zero-temperature exponents are found.
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.
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 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 zzz). We find the zero temperature quantum spin glass phase transition for coordinations $z\ensuremath{\ge}3$ 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 $(z\ensuremath{\rightarrow}\ensuremath{\infty})$ and the usual mean-field approach is established.
We propose a generalization of the cavity method to quantum spin glasses on fixed connectivity lattices. Our work is motivated by the recent refinements of the classical technique and its potential application to quantum computational problems. We numerically solve for the phase structure of a connectivity q=3q=3q=3 transverse field Ising model on a Bethe lattice with $\ifmmode\pm\else\textpm\fi{}J$ couplings and investigate the distribution of various classical and quantum observables.
We investigate zero and finite-temperature properties of the one-dimensional spin-glass model for vector spins in the limit of an infinite number mmm of spin components where the interactions decay with a power, $\ensuremath{\sigma}$, of the distance. A diluted version of this model is also studied, but found to deviate significantly from the fully connected model. At zero temperature, defect energies are determined from the difference in ground-state energies between systems with periodic and antiperiodic boundary conditions to determine the dependence of the defect-energy exponent $\ensuremath{\theta}$ on $\ensuremath{\sigma}$. A good fit to this dependence is $\ensuremath{\theta}=3/4\ensuremath{-}\ensuremath{\sigma}$. This implies that the upper critical value of $\ensuremath{\sigma}$ is 3/43/43/4, corresponding to the lower critical dimension in the ddd-dimensional short-range version of the model. For finite temperatures, the large mmm saddle-point equations are solved self-consistently, which gives access to the correlation function, the order parameter, and the spin-glass susceptibility. Special attention is paid to the different forms of finite-size scaling effects below and above the lower critical value, $\ensuremath{\sigma}=5/8$, which corresponds to the upper critical dimension 8 of the hypercubic short-range model.
We describe results of Monte Carlo simulation studies on the Ising spin glass in four dimensions on a hypercubic lattice with nearest neighbor bonds. Studies of the equilibrium static properties show that the system undergoes a genuine phase transition to an ordered spin glass phase. Critical dynamical behavior is analyzed to obtain the dynamic exponent. Finally, we describe results on the spin glass phase, in particular the finite size scaling of the order parameter distribution function, and compare it with existing models of the spin glass phase, namely the droplet model and the Parisi solution for the low temperature phase of the infinite range spin glass.