Clifford algebras and new singular Riemannian foliations in spheres

Using representations of Clifford algebras we construct indecomposable singular Riemannian foliations on round spheres, most of which are non-homogeneous. This generalizes the construction of non-homogeneous isoparametric hypersurfaces due to by Ferus, Karcher and Munzner.Comment: 21 pages. Construction of foliations in the Cayley plane added. Proofs simplified and presentation improved, according to referee's suggestions. To appear in Geom. Funct. Ana

Gauge Theory for Quantum Spin Glasses

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.Comment: 15 pages, 2 figure

A New Method to Calculate the Spin-Glass Order Parameter of the Two-Dimensional +/-J Ising Model

A new method to numerically calculate the $n$th moment of the spin overlap of the two-dimensional $\pm J$ Ising model is developed using the identity derived by one of the authors (HK) several years ago. By using the method, the $n$th moment of the spin overlap can be calculated as a simple average of the $n$th moment of the total spins with a modified bond probability distribution. The values of the Binder parameter etc have been extensively calculated with the linear size, $L$, up to L=23. The accuracy of the calculations in the present method is similar to that in the conventional transfer matrix method with about $10^{5}$ bond samples. The simple scaling plots of the Binder parameter and the spin-glass susceptibility indicate the existence of a finite-temperature spin-glass phase transition. We find, however, that the estimation of $T_{\rm c}$ is strongly affected by the corrections to scaling within the present data ($L\leq 23$). Thus, there still remains the possibility that $T_{\rm c}=0$, contrary to the recent results which suggest the existence of a finite-temperature spin-glass phase transition.Comment: 10 pages,8 figures: final version to appear in J. Phys.

Criticality in the two-dimensional random-bond Ising model

The two-dimensional (2D) random-bond Ising model has a novel multicritical point on the ferromagnetic to paramagnetic phase boundary. This random phase transition is one of the simplest examples of a 2D critical point occurring at both finite temperatures and disorder strength. We study the associated critical properties, by mapping the random 2D Ising model onto a network model. The model closely resembles network models of quantum Hall plateau transitions, but has different symmetries. Numerical transfer matrix calculations enable us to obtain estimates for the critical exponents at the random Ising phase transition. The values are consistent with recent estimates obtained from high-temperature series.Comment: minor changes, 7 pages LaTex, 8 postscript figures included using epsf; to be published Phys. Rev. B 55 (1997

Non-equilibrium Relations for Spin Glasses with Gauge Symmetry

We study the applications of non-equilibrium relations such as the Jarzynski equality and fluctuation theorem to spin glasses with gauge symmetry. It is shown that the exponentiated free-energy difference appearing in the Jarzynski equality reduces to a simple analytic function written explicitly in terms of the initial and final temperatures if the temperature satisfies a certain condition related to gauge symmetry. This result is used to derive a lower bound on the work done during the non-equilibrium process of temperature change. We also prove identities relating equilibrium and non-equilibrium quantities. These identities suggest a method to evaluate equilibrium quantities from non-equilibrium computations, which may be useful to avoid the problem of slow relaxation in spin glasses.Comment: 8 pages, 2 figures, submitted to JPS

Ground-State Phase Diagram of the Two-Dimensional Quantum Heisenberg Mattis Model

The two-dimensional $S=1/2$ asymmetric Heisenberg Mattis model is investigated with the exact diagonalization of finite clusters. The N\'eel 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 N\'eel 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, N\'eel order, spin glass orderComment: 10 pages, LaTeX, 6 compressed/uuencoded postscript figures, J. Phys. Soc. Jpn. 65 (1996) No. 2 in pres

Finite Size Scaling Analysis of Exact Ground States for +/-J Spin Glass Models in Two Dimensions

With the help of EXACT ground states obtained by a polynomial algorithm we compute the domain wall energy at zero-temperature for the bond-random and the site-random Ising spin glass model in two dimensions. We find that in both models the stability of the ferromagnetic AND the spin glass order ceases to exist at a UNIQUE concentration p_c for the ferromagnetic bonds. In the vicinity of this critical point, the size and concentration dependency of the first AND second moment of the domain wall energy are, for both models, described by a COMMON finite size scaling form. Moreover, below this concentration the stiffness exponent turns out to be slightly negative \theta_S = -0.056(6) indicating the absence of any intermediate spin glass phase at non-zero temperature.Comment: 7 pages Latex, 5 postscript-figures include

Nishimori point in random-bond Ising and Potts models in 2D

We study the universality class of the fixed points of the 2D random bond q-state Potts model by means of numerical transfer matrix methods. In particular, we determine the critical exponents associated with the fixed point on the Nishimori line. Precise measurements show that the universality class of this fixed point is inconsistent with percolation on Potts clusters for q=2, corresponding to the Ising model, and q=3Comment: 11 pages, 3 figures. Contribution to the proceedings of the NATO Advanced Research Workshop on Statistical Field Theories, Como 18-23 June 200

Nishimori point in the 2D +/- J random-bond Ising model

We study the universality class of the Nishimori point in the 2D +/- J random-bond Ising model by means of the numerical transfer-matrix method. Using the domain-wall free-energy, we locate the position of the fixed point along the Nishimori line at the critical concentration value p_c = 0.1094 +/- 0.0002 and estimate nu = 1.33 +/- 0.03. Then, we obtain the exponents for the moments of the spin-spin correlation functions as well as the value for the central charge c = 0.464 +/- 0.004. The main qualitative result is the fact that percolation is now excluded as a candidate for describing the universality class of this fixed point.Comment: 4 pages REVTeX, 3 PostScript figures; final version to appear in Phys. Rev. Lett.; several small changes and extended explanation