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Physical review. B, Condensed matter

Infinite-range-interactionM-component quantum spin glasses: Statics and dynamics in the large-Mlimit

T. K. Kopeć

Published Oct 1, 1994• 14 citations
Publisher

Abstract

The infinite-range M-component models of quantum rotors and anharmonic oscillators coupled by random exchange interactions with Gaussian distributions are considered. Both models exhibit a quantum phase transition separating the spin glass and disordered phases where the quantum effects are regulated by a parameter \ensuremath{\Delta} (inverse momentum of inertia for rotor model and inverse mass for oscillator model) parametrizing the kinetic energy. The models allow, in the limit M\ensuremath{\rightarrow}\ensuremath{\infty}, for an exact solution in both the spin glass and disordered phases. Employing the quantum finite-temperature real-time description, we examine the static and dynamic properties of the systems without making use of the n-replica method. We find that the dynamical solution is marginally stable in the spin-glass phase with a constant static susceptibility below the spin-glass freezing temperature. A closed-form solution for the averaged free energy has been obtained and various thermodynamic functions are derived. In particular, the entropy is always positive in the spin-glass plane and goes to zero with a ${\mathit{T}}^{3}$ dependency in the low-temperature region. This is in contrast to the classical M-component Heisenberg model in the M\ensuremath{\rightarrow}\ensuremath{\infty} limit.

DOI
10.1103/physrevb.50.9963

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The infinite-range M-component models of quantum rotors and anharmonic oscillators coupled by random exchange interactions with Gaussian distributions are considered. Both models exhibit a quantum phase transition separating the spin glass and disordered phases where the quantum effects are regulated by a parameter \ensuremath{\Delta} (inverse momentum of inertia for rotor model and inverse mass for oscillator model) parametrizing the kinetic energy. The models allow, in the limit M\ensuremath{\rightarrow}\ensuremath{\infty}, for an exact solution in both the spin glass and disordered phases. Employing the quantum finite-temperature real-time description, we examine the static and dynamic properties of the systems without making use of the n-replica method. We find that the dynamical solution is marginally stable in the spin-glass phase with a constant static susceptibility below the spin-glass freezing temperature. A closed-form solution for the averaged free energy has been obtained and various thermodynamic functions are derived. In particular, the entropy is always positive in the spin-glass plane and goes to zero with a T3{\mathit{T}}^{3}T3 dependency in the low-temperature region. This is in contrast to the classical M-component Heisenberg model in the M\ensuremath{\rightarrow}\ensuremath{\infty} limit.
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 examine a model of M-component quantum rotors coupled by Gaussian-distributed random, infinite-range exchange interactions. A complete solution is obtained at M=\ensuremath{\infty} in the spin-glass and quantum-disordered phases. The quantum phase transition separating them is found to possess logarithmic violations of scaling, with no further modifications to the leading critical behavior at any order in 1/M; this suggests that the critical properties of the transverse-field Ising model (believed to be identical to the M\ensuremath{\rightarrow}1 limit) are the same as those of the M=\ensuremath{\infty} quantum rotors.
We study a generalization of the quantum model of fully isotropic uniaxial p-polar glasses with multicomponent orientational degrees of freedom and Gaussian-distributed random, infinite-range exchange interactions. The quantum effects are associated with a finite moment of inertia (treated quantum mechanically) of a particle with M orientational degrees of freedom. The system exhibits a quantum phase transition separating the glassy and paraorientational phases regulated by a parameter \ensuremath{\Delta} (inverse of the moment of inertia of a particle). The model allows in the limit M\ensuremath{\rightarrow}\ensuremath{\infty} for an exact solution both in the glassy and disordered phases for an arbitrary value of the parameter p. We have analytically established the closed form of the self-consistency solution for the dynamic order parameter within the replica method and show that different spin glass transition scenarios emerge as a function of the parameter p. Whereas for p=1 (dipolar case) the solution of the quantum glass model is exact within replica symmetry, with a continuous phase transition along the critical line Tc{\mathit{T}}_{\mathit{c}}Tc​(\ensuremath{\Delta}) we showed that the model for p\ensuremath{\ge}2 (multipolar case) undergoes a first-order phase transition. In the latter case within the Parisi replica-symmetry-breaking scheme we have determined a functional Parisi-like order parameter q(x) and demonstrated that for p\ensuremath{\ge}2 a single step of the replica symmetry breaking is the exact solution within the glassy phase. For the physically most relevant case of p=2 (quantum quadrupolar glass) the \ensuremath{\Delta}-T phase diagram has been computed numerically. \textcopyright{} 1996 The American Physical Society.
We determine the low-energy spectrum and Parisi replica-symmetry-breaking function for the spin-glass phase of the quantum Ising model with infinite-range random exchange interactions and transverse and longitudinal ( h ) fields. We show that, for all h , the spin-glass state has full replica symmetry breaking and the local spin spectrum is gapless, with a spectral density that vanishes linearly with frequency. These results are obtained using an action functional—argued to yield exact results at low frequencies—that expands in powers of a spin-glass order parameter, which is bilocal in time, and a matrix in replica space. We also present the exact solution of the infinite-range spherical quantum p -rotor model at nonzero h : here, the spin-glass state has one-step replica symmetry breaking and gaplessness only appears after imposition of an additional marginal stability condition. Possible connections to experiments on random arrays of trapped Rydberg atoms are noted. Published by the American Physical Society 2024
A class of infinite-ranged random model Hamiltonians is defined as a limiting case in which the appropriate form of mean-field theory, order parameters and phase diagram to describe spin-glasses may be established. It is believed that these Hamiltonians may be exactly soluble, although a complete solution is not yet available. Thermodynamic properties of the model for Ising and XY\mathrm{XY}XY spins are evaluated using a many-replica procedure. Results of the replica theory reproduce properties at and above the ordering temperature which are also predicted by high-temperature expansions, but are in error at low temperatures. Extensive computer simulations of infinite-ranged Ising spin-glasses are presented. They confirm the general details of the predicted phase diagram. The errors in the replica solution are found to be small, and confined to low temperatures. For this model, the extended mean-field theory of Thouless, Anderson, and Palmer gives physically sensible low-temperature predictions. These are in quantitative agreement with the Monte Carlo statics. The dynamics of the infinite-ranged Ising spin-glass are studied in a linearized mean-field theory. Critical slowing down is predicted and found, with correlations decaying as ${e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}$ for TTT greater than Tc{T}_{c}Tc​, the spin-glass transition temperature. At and below Tc{T}_{c}Tc​, spin-spin correlations are observed to decay to their long-time limit as ${t}^{\ensuremath{-}\frac{1}{2}}$.
A class of infinite-ranged random model Hamiltonians is defined as a limiting case in which the appropriate form of mean-field theory, order parameters and phase diagram to describe spin-glasses may be established. It is believed that these Hamiltonians may be exactly soluble, although a complete solution is not yet available. Thermodynamic properties of the model for Ising and XY\mathrm{XY}XY spins are evaluated using a "many-replica" procedure. Results of the replica theory reproduce properties at and above the ordering temperature which are also predicted by high-temperature expansions, but are in error at low temperatures. Extensive computer simulations of infinite-ranged Ising spin-glasses are presented. They confirm the general details of the predicted phase diagram. The errors in the replica solution are found to be small, and confined to low temperatures. For this model, the extended mean-field theory of Thouless, Anderson, and Palmer gives physically sensible low-temperature predictions. These are in quantitative agreement with the Monte Carlo statics. The dynamics of the infinite-ranged Ising spin-glass are studied in a linearized mean-field theory. Critical slowing down is predicted and found, with correlations decaying as ${e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}$ for TTT greater than Tc{T}_{c}Tc​, the spin-glass transition temperature. At and below Tc{T}_{c}Tc​, spin-spin correlations are observed to decay to their long-time limit as ${t}^{\ensuremath{-}\frac{1}{2}}$.