944 research outputs found
Exact Ground-State Energy of the Ising Spin Glass on Strips
We propose a new method for exact analytical calculation of the ground-state
energy of the Ising spin glass on strips. An outstanding advantage of this
method over the numerical transfer matrix technique is that the energy is
obtained for complex values of the probability describing quenched randomness.
We study the and the site-random models using this method for strips of
various sizes up to . The ground-state energy of these models is
found to have singular points in the complex-probability plane, reminiscent of
Lee-Yang zeros in the complex-field plane for the Ising ferromagnet. The Ising model has a series of singularities which may approach a limiting
point around on the real axis in the limit of infinite width.Comment: 10 pages, 12 Postscript figures, LaTeX, uses subeqn.sty, minor
changes in tex-fil
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
Exact location of the multicritical point for finite-dimensional spin glasses: A conjecture
We present a conjecture on the exact location of the multicritical point in
the phase diagram of spin glass models in finite dimensions. By generalizing
our previous work, we combine duality and gauge symmetry for replicated random
systems to derive formulas which make it possible to understand all the
relevant available numerical results in a unified way. The method applies to
non-self-dual lattices as well as to self dual cases, in the former case of
which we derive a relation for a pair of values of multicritical points for
mutually dual lattices. The examples include the +-J and Gaussian Ising spin
glasses on the square, hexagonal and triangular lattices, the Potts and Z_q
models with chiral randomness on these lattices, and the three-dimensional +-J
Ising spin glass and the random plaquette gauge model.Comment: 27 pages, 3 figure
Aging Relation for Ising Spin Glasses
We derive a rigorous dynamical relation on aging phenomena -- the aging
relation -- for Ising spin glasses using the method of gauge transformation.
The waiting-time dependence of the auto-correlation function in the
zero-field-cooling process is equivalent with that in the field-quenching
process. There is no aging on the Nishimori line; this reveals arguments for
dynamical properties of the Griffiths phase and the mixed phase. The present
method can be applied to other gauge-symmetric models such as the XY gauge
glass.Comment: 9 pages, RevTeX, 2 postscript figure
Dynamical Probability Distribution Function of the SK Model at High Temperatures
The microscopic probability distribution function of the
Sherrington-Kirkpatrick (SK) model of spin glasses is calculated explicitly as
a function of time by a high-temperature expansion. The resulting formula to
the third order of the inverse temperature shows that an assumption made by
Coolen, Laughton and Sherrington in their recent theory of dynamics is
violated. Deviations of their theory from exact results are estimated
quantitatively. Our formula also yields explicit expressions of the time
dependence of various macroscopic physical quantities when the temperature is
suddenly changed within the high-temperature region.Comment: LaTeX, 6 pages, Figures upon request (here revised), To be published
in J. Phys. Soc. Jpn. 65 (1996) No.
High-Temperature Dynamics of Spin Glasses
We develop a systematic expansion method of physical quantities for the SK
model and the finite-dimensional model of spin glasses in
non-equilibrium states. The dynamical probability distribution function is
derived from the master equation using a high temperature expansion. We
calculate the expectation values of physical quantities from the dynamical
probability distribution function. The theoretical curves show satisfactory
agreement with Monte Carlo simulation results in the appropriate temperature
and time regions. A comparison is made with the results of a dynamics theory by
Coolen, Laughton and Sherrington.Comment: 24 pages, figures available on request, LaTeX, uses jpsj.sty, to be
published in J. Phys. Soc. Jpn. 66 No. 7 (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 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
Dynamical Gauge Theory for the XY Gauge Glass Model
Dynamical systems of the gauge glass are investigated by the method of the
gauge transformation.Both stochastic and deterministic dynamics are treated.
Several exact relations are derived among dynamical quantities such as
equilibrium and nonequilibrium auto-correlation functions, relaxation functions
of order parameter and internal energy. They provide physical properties in
terms of dynamics in the SG phase, a possible mixed phase and the Griffiths
phase, the multicritical dynamics and the aging phenomenon. We also have a
plausible argument for the absence of re-entrant transition in two or higher
dimensions.Comment: 3 figure
Closure of Macroscopic Laws in Disordered Spin Systems: A Toy Model
We use a linear system of Langevin spins with disordered interactions as an
exactly solvable toy model to investigate a procedure, recently proposed by
Coolen and Sherrington, for closing the hierarchy of macroscopic order
parameter equations in disordered spin systems. The closure procedure, based on
the removal of microscopic memory effects, is shown to reproduce the correct
equations for short times and in equilibrium. For intermediate time-scales the
procedure does not lead to the exact equations, yet for homogeneous initial
conditions succeeds at capturing the main characteristics of the flow in the
order parameter plane. The procedure fails in terms of the long-term temporal
dependence of the order parameters. For low energy inhomogeneous initial
conditions and near criticality (where zero modes appear) deviations in
temporal behaviour are most apparent. For homogeneous initial conditions the
impact of microscopic memory effects on the evolution of macroscopic order
parameters in disordered spin systems appears to be mainly an overall slowing
down.Comment: 14 pages, LateX, OUTP-94-24
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