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.
We investigate the temperature-disorder (
T-$\ensuremath{\sigma}$) phase diagram of a three-dimensional gauge glass model, which is a cubic-lattice nearest-neighbor
XY model with quenched random phase shifts
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 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 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=0 axis, which is slightly reentrant.
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 phase diagrams and critical properties of a quenched random alloy of a ferromagnet and an antiferromagnet (or of two antiferromagnets with different periodicities) are studied in the mean-field approximation and by renormalization-group techniques in $d=6\ensuremath{-}\ensuremath{\epsilon}$ dimensions, using the $n\ensuremath{\rightarrow}0$ replica method. If only nearest-neighbor interactions are assumed, then one finds ferromagnetic (
F), antiferromagnetic (AF), and spin-glass (SG) phases. As the average strength of the next-nearest-neighbor interactions is increased (e.g., by an additional component in the alloy), the
F and the AF phases approach each other, and one may find regions in parameter space where a mixed
F-AF phase exists, as discussed in Paper II of this series. The phase diagram exhibits lines of multicritical points where the paramagnetic (
P),
F (or AF) and SG phases coexist (
P-SG-
F or
P-SG-AF), or where the
P,
F, and AF phases coexist ($P\ensuremath{-}F$-AF). These lines meet at a new multicritical point, where all four phases coexist (
P-SG-
F-AF). Recent renormalization-group analysis of
P-SG-
F, by Chen and Lubensky, yields complex exponents for the
XY and Heisenberg cases. We show that the phase diagram predicted by that analysis is difficult to understand even for the Ising case. The same method of analysis yields similar difficulties for
P-SG-
F-AF. A modified way to take the limit $n\ensuremath{\rightarrow}0$, which resolves these difficulties, is presented. This modified way still lacks rigorous justification.
An Ising spin glass of random bonds with the Gaussian and two-peaked distributions in two- and three-dimensions is investigated with the cumulant expansion and two-step decimation transformations. It is found that the spin glass state exists in two- and three-dimensions and that the ferromagnetic phase shrinks to that of the pure system in two-dimension for the two-peaked distribution. The specific heat exponent at the spin glass transition point is found to be -1.08 and -0.471 for the square and simple cubic lattices, respectively. Also the effect of bond correlations on the spin glass state is considered.
The spin-1/2 quantum Heisenberg spin-glass system is studied in all spatial dimensions d by renormalization-group theory. Strongly asymmetric phase diagrams in temperature and antiferromagnetic bond probability p are obtained in dimensions d>or=3. The asymmetry at high temperatures approaching the pure ferromagnetic and antiferromagnetic systems disappears as d is increased. However, the asymmetry at low but finite temperatures remains in all dimensions, with the antiferromagnetic phase receding from the ferromagnetic phase. A finite-temperature second-order phase boundary directly between the ferromagnetic and antiferromagnetic phases occurs in d>or=6, resulting in a new multicritical point. In d=3, 4, 5, a paramagnetic phase reaching zero temperature intervenes asymmetrically between the ferromagnetic and reentrant antiferromagnetic phases. There is no spin-glass phase in any dimension.