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Physical review. B, Condensed matter

Solvable model of the quantum spin glass in a transverse field

Yadin Y. Goldschmidt

Published Mar 1, 1990• 110 citations
Publisher

Abstract

The spin-glass model with p-spin interactions in the presence of a transverse field is solved in the limit p\ensuremath{\rightarrow}\ensuremath{\infty}. The phase diagram is obtained and consists of three phases: A spin-glass phase and two paramagnetic phases. The paramagnetic phases are distinguished by transverse ordering. The spin-glass phase is similar to that of the random-energy model.

DOI
10.1103/physrevb.41.4858

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Physical review. B, Condensed matter•Mar 1, 1990•7 citations
The spin-glass model with p-spin interactions in the presence of a transverse field is solved in the limit p\ensuremath{\rightarrow}\ensuremath{\infty}. The phase diagram is obtained and consists of three phases: A spin-glass phase and two paramagnetic phases. The paramagnetic phases are distinguished by transverse ordering. The spin-glass phase is similar to that of the random-energy model.
The phase diagram of the ppp-spin-interacting spin glass model in a transverse field is investigated in the limit p→∞p \to \inftyp→∞ under the presence of ferromagnetic bias. Using the replica method and the static approximation, we show that the phase diagram consists of four phases: Quantum paramagnetic, classical paramagnetic, ferromagnetic, and spin-glass phases. We also show that the static approximation is valid in the ferromagnetic phase in the limit p→∞p \to \inftyp→∞ by using the large-ppp expansion. Since the same approximation is already known to be valid in other phases, we conclude that the obtained phase diagram is exact.
The phase diagram of the ppp-spin-interacting spin glass model in a transverse field is investigated in the limit p→∞p \to \inftyp→∞ under the presence of ferromagnetic bias. Using the replica method and the static approximation, we show that the phase diagram consists of four phases: Quantum paramagnetic, classical paramagnetic, ferromagnetic, and spin-glass phases. We also show that the static approximation is valid in the ferromagnetic phase in the limit p→∞p \to \inftyp→∞ by using the large-ppp expansion. Since the same approximation is already known to be valid in other phases, we conclude that the obtained phase diagram is exact.
The van Hemmen model with a transverse field is studied to describe the quantum Ising spin glass. The free energy and phase diagrams (T(T(T versus $\ensuremath{\Omega},$ and T versus Jo/J,{J}_{o}/J,Jo​/J, where $\ensuremath{\Omega}$ is the transverse field, Jo{J}_{o}Jo​ and J are the ferromagnetic and random exchange interactions, respectively) are calculated for the model with two-peaked and Gaussian exchange distribution. The system presents three ordered phases, namely, spin glass, mixed, and ferromagnetic phases, besides the paramagnetic disordered phase. The influence of the transverse field $\ensuremath{\Omega}$ is to destroy the ordered phases. In the $(T,\ensuremath{\Omega})$ plane our results are compared with those obtained by the replica-symmetry-breaking solution and the same qualitative behavior is observed for the spin-glass transition.
The classical transverse field Ising spin-glass model with short-range interactions is investigated beyond the mean-field approximation for a real d -dimensional lattice. We use an appropriate non-trivial modification of the Bethe-Peierls method recently formulated for the Ising spin-glass. The zero-temperature critical value of the transverse field and the linear susceptibility in the paramagnetic phase are obtained analytically as functions of dimensionality d . The phase diagram is also calculated numerically for different values of d . In the limit , known mean-field results are consistently reproduced.
We analyze the zero-temperature phase diagram of two kinds of one-dimensional quantum-spin models in a transverse magnetic field. The first kind has uniform nearest-neighbor ferromagnetic and next-nearest-neighbor antiferromagnetic couplings. A spin-wave analysis indicates the presence of several (possibly an infinite number of) commensurate modulated phases in the phase diagram. The three most prominent phases are paramagnetic, ferromagnetic, and a q=(1/4 phase of period four. The second kind of model is a spin-glass with only nearest-neighbor ferromagnetic interactions whose strengths are random and are characterized by a probability function. A classical large-S analysis shows the existence of a Griffiths phase lying in between a low-magnetic-field ferromagnetic phase and a high-magnetic-field paramagnetic phase. We present two order parameters that can together distinguish between the three phases.