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Journal of Physics A Mathematical and General

Non-frustrated random spin systems with gauge symmetry

Yukiyasu Ozeki

Published Sep 21, 1996• 3 citations
Publisher

Abstract

We introduce a class of random spin systems without frustration, which satisfies the conditions for gauge symmetry. The method of gauge transformation provides several properties on the phase diagram of gauge symmetric models; this method is almost rigorous, while the derivation for the absence of re-entrance contains unproved assumption. In the present random models, the absence of re-entrance can be shown exactly. Furthermore, the phase diagram and the critical properties are exactly related with those in the original pure system. The present random systems correspond to the Mattis model with asymmetric bond distribution in the Ising case, and a kind of the gauge glass model in the XY case. In the case of the clock model in two dimensions, we find a new thermodynamic phase which has long-range spin-glass correlation with critical (power-decaying) ferromagnetic correlation.

DOI
10.1088/0305-4470/29/18/012

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We introduce a class of random spin systems without frustration, which satisfies the conditions for gauge symmetry. The method of gauge transformation provides several properties on the phase diagram of gauge symmetric models; this method is almost rigorous, while the derivation for the absence of re-entrance contains unproved assumption. In the present random models, the absence of re-entrance can be shown exactly. Furthermore, the phase diagram and the critical properties are exactly related with those in the original pure system. The present random systems correspond to the Mattis model with asymmetric bond distribution in the Ising case, and a kind of the gauge glass model in the XY case. In the case of the clock model in two dimensions, we find a new thermodynamic phase which has long-range spin-glass correlation with critical (power-decaying) ferromagnetic correlation.
The authors introduce a general class of random spin systems which are symmetric under local gauge transformations. Their model is a generalization of the usual Ising spin glass and includes the Z q , XY, and SU (2) gauge glasses. For this general class of systems, the internal energy and an upper bound on the specific heat are calculated explicitly in any dimensions on a special line in the phase diagram. Although the line intersects a phase boundary at a multicritical point, the internal energy and the bound on the specific heat are found to be written in terms of a simple function. They also show that the boundary between the ferromagnetic and nonferromagnetic phases is parallel to the temperature axis in the low-temperature region of the phase diagram. This means the absence of re-entrant transitions. All these properties are derived by simple applications of gauge transformations of spin and randomness degrees of freedom.
The gauge theory for random spin systems is extended to quantum spin glasses to derive a number of exact and/or rigorous results. The transverse Ising model and the quantum gauge glass are shown to be gauge invariant. For these models, an identity is proved that the expectation value of the gauge invariant operator in the ferromagnetic limit is equal to the one in the classical equilibrium state on the Nishimori line. As a result, a set of inequalities for the correlation function are proved, which restrict the location of the ordered phase. It is also proved that there is no long-range order in the two-dimensional quantum gauge glass in the ground state. The phase diagram for the quantum XY Mattis model is determined.
The two-dimensional S=1/2S=1/2S=1/2 asymmetric Heisenberg Mattis model is investigated with the exact diagonalization of finite clusters. The Néel order parameter and the spin glass order parameter can be smoothly extrapolated to the thermodynamic limit in the antiferromagnetic region, as in the pure Heisenberg antiferromagnet. The critical concentration of the Néel phase is consistent with that of the two-dimensional Ising Mattis model, and the spin glass order parameter increases monotonously as the ferro-bond concentration increases. These facts suggest that quantum fluctuation does not play an essential role in two-dimensional non-frustrated random spin systems. KEYWORDS: quantum spin system, ground state, randomness, Mattis model, Néel order, spin glass order
The two-dimensional S = 1/2 asymmetric Heisenberg Mattis model is investigated with the exact diagonalization of finite clusters. The Néel order parameter and the spin glass order parameter can be smoothly extrapolated to the thermodynamic limit in the antiferromagnetic region, as in the pure Heisenberg antiferromagnet. The critical concentration of the Néel phase is consistent with that of the two-dimensional Ising Mattis model, and the spin glass order parameter increases monotonously as the ferro-bond concentration increases. These facts suggest that quantum fluctuation does not play an essential role in two-dimensional non-frustrated random spin systems.
Ground-state properties of the two-dimensional S=1/2S=1/2S=1/2 random Heisenberg models are investigated by the exact-diagonalization method. The phase diagram of the bond-random model (the ±J\pm J±J model) is the same as that of the corresponding Ising spin glass model, in spite of quantum fluctuation. In the site-random model, the spin glass phase exists at least in the ferro-rich case. In comparison with the corresponding Ising model, the Néel phase of this model becomes much narrower in the ferro-rich case, while it is comparable in the antiferro-rich case. KEYWORDS: quantum spin glass, Heisenberg model, ground-state phase transition, site-random model, exact diagonalization
We investigate the temperature-disorder (TTT-$\ensuremath{\sigma}$) phase diagram of a three-dimensional gauge glass model, which is a cubic-lattice nearest-neighbor XY\mathit{XY}XY model with quenched random phase shifts Axy{A}_{\mathit{xy}}Axy​ at the bonds. We consider the uncorrelated phase-shift distribution $P({A}_{\mathit{xy}})~\mathrm{exp}[(\mathrm{cos}{A}_{\mathit{xy}})/\ensuremath{\sigma}]$, which has the pure XY\mathit{XY}XY model, and the uniform distribution of random phase shifts as extreme cases at $\ensuremath{\sigma}=0$ and $\ensuremath{\sigma}\ensuremath{\rightarrow}\ensuremath{\infty}$, respectively, and which gives rise to equal magnetic and overlap correlation functions when $T=\ensuremath{\sigma}$. Our study is mostly based on numerical Monte Carlo simulations. While the high-temperature phase is always paramagnetic, at low temperatures there is a ferromagnetic phase for weak disorder (small $\ensuremath{\sigma}$) and a glassy phase at large disorder (large $\ensuremath{\sigma}$). These three phases are separated by transition lines with different magnetic and glassy critical behaviors. The disorder induced by the random phase shifts turns out to be irrelevant at the paramagnetic-ferromagnetic transition line, where the critical behavior belongs to the 3D XY\mathit{XY}XY universality class of pure systems; disorder gives rise only to very slowly decaying scaling corrections. The glassy critical behavior along the paramagnetic-glassy transition line belongs to the gauge-glass universality class, with a quite large exponent $\ensuremath{\nu}=3.2(4)$. These transition lines meet at a multicritical point M, located at ${T}_{M}={\ensuremath{\sigma}}_{M}=0.7840(2)$. The low-temperature ferromagnetic and glassy phases are separated by a third transition line, from M down to the T=0T=0T=0 axis, which is slightly reentrant.