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Physical review. B, Condensed matter

Quantum spin glasses with cubic anisotropy

Z. Doman´ski, T. K. Kopeć, F. Pa´zma´ndi

Published Feb 1, 1994• 2 citations
Publisher

Abstract

An infinite-range quantum Ising spin glass with cubic anisotropy (K) is studied using the imaginary-time representation with the n-replica approach and the thermofield dynamic method. Mean-field-theory phase diagrams in the temperature-anisotropy plane (T,K) for quantum spins S ranging from 2 to 9/2 are presented. For integer-spin values and large cubic anisotropy (positive or negative, depending on S) a condensation into a nonmagnetic spin state occurs, accompanied by the destruction of the spin-glass order as indicated by the finite critical value ${\mathit{K}}_{\mathit{c}}$(T=0). For half-integer S and sufficiently low temperature the spin-glass phase persists for arbitrary K.

DOI
10.1103/physrevb.49.3378

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Physical review. B, Condensed matter•Jun 1, 1978•1,007 citations
An infinite-range quantum Ising spin glass with cubic anisotropy (K) is studied using the imaginary-time representation with the n-replica approach and the thermofield dynamic method. Mean-field-theory phase diagrams in the temperature-anisotropy plane (T,K) for quantum spins S ranging from 2 to 9/2 are presented. For integer-spin values and large cubic anisotropy (positive or negative, depending on S) a condensation into a nonmagnetic spin state occurs, accompanied by the destruction of the spin-glass order as indicated by the finite critical value Kc{\mathit{K}}_{\mathit{c}}Kc​(T=0). For half-integer S and sufficiently low temperature the spin-glass phase persists for arbitrary K.
The infinite-range quantum spin glasses with cubic anisotropy (K) are studied using a combination of the imaginary-time representation with the n-replica approach and the thermofield dynamic method. Mean-field theory phase diagrams in the temperature-anisotropy plane (T,K) are presented for quantum spin numbers S ranging from 2 to 9/2. At T=0, the ground state is nonmagnetic and the spin-glass order is absent for integer spin and large cubic anisotropy, the sign of which depends on S. For half-integer S and sufficiently low temperatures the spin-glass phase persists for arbitrary K.
An infinite-range quantum Heisenberg spin glass with uniaxial (D) and cubic (K) anisotropies is studied using the thermo-field-dynamic method. Mean-field theory phase diagrams in the temperature-anisotropy plane (T,D) for quantum spins S=2 and S=5/2 are presented for different values of K. Generally, for integer spin values and large anisotropy D a condensation into a nonmagnetic spin state occurs, accompanied by the destruction of the spin-glass order as indicated by the finite critical value Dc{\mathrm{D}}_{\mathrm{c}}Dc​(T=0). For half-integer S and sufficiently low temperature, the spin-glass phase persists for arbitrary D and K. In both cases the presence of a cubic term induces complicated crossovers from longitudinal to transverse freezing of spins.
The infinite-range quantum Heisenberg spin glass with uniaxial anisotropy (D) and coaxial external magnetic field (h) is studied by means of the thermo-field-dynamics as a substitute for the n-replica trick. Within the mean-field theory a multiplicity of spin-glass phases has been found for the spin values S=1 and S=3/2, respectively, including: longitudinal, transverse, and mixed phases. The stability of the mean-field-type solution against the action of fluctuations has been investigated and the existence of a crossover from the longitudinal-to-transverse freezing is demonstrated explicitly in the temperature-field (T-h) phase diagram for various values of the constant D.
We study a long-range quantum spin-glass model with uniaxial anisotropy. A calculation of the phase diagram as well as the susceptibilities above the freezing temperature is presented. For easy-axis systems Ising-like behavior is found, while for easy-plane systems with increasing anisotropy a change from XY\mathrm{XY}XY behavior to a condensation in a nonmagnetic singlet ground state takes place for S=1S=1S=1.
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 ${t}^{\ensuremath{-}\frac{1}{2}}$.