Abstract Nishimori’s gauge theory is extended to the quantum XYZ p-spin glass model in finite dimensions. This enables us to obtain useful correlation equalities, which show also that Duhamel correlation functions at an arbitrary temperature are bounded by those in the corresponding classical model on the Nishimori line. These bounds give that the spontaneous magnetization vanishes in any low temperature even if the model enters the Z 2 -symmetry broken spin glass phase. This theory explains well-known fact from experiments and numerical calculations that the magnetic susceptibility does not diverge in the spin glass transition. The new gauge theory together with the known phase diagram of the Edwards–Anderson model can specify the spin glass region in the coupling constant space of the quantum Heisenberg XYZ spin glass model.
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.
Numerical evidence that the number of pure states in the spin glass phase is infinite near the ferromagnetic phase boundary for the three dimensional (3D) ± J Ising model and 3D gauge glass models is presented. It is consistent with the mean field theory which suggests the multivalley structure of the spin glass phase. We will show that the onset of the ferromagnetic long-range order and that of the ferromagnetic spontaneous symmetry breaking do not coincide with each other, and this phenomenon characterizes the structure of the phase space.
We investigate the balanced
M=4,
p=4 spin-glass model for a one-dimensional long-range proxy for the finite dimensional short-range
p-spin glass model to examine the nature of the glass transition beyond mean-field theory. We perform large-scale Monte Carlo equilibrated simulations for both fully connected and power-law diluted versions of the model. The critical temperatures extracted from the finite-size scaling (FSS) analysis of spin-glass susceptibility are in good agreement with theoretical predictions for
σ=0,0.25, and 0.55. For these values of the long-range exponent
σ (which is the power of the decrease of the interactions between the spins with their separation), one might have expected that mean-field theory would provide a good description of the system. However, the spin-overlap distribution and the value of the
λ-parameter do not provide numerical evidence for a one-step replica symmetry breaking (1RSB) phase transition. Instead, our results indicate a direct transition from the paramagnetic state to a full replica symmetry broken phase, with a renormalized value of
λ≡ω2/ω1<1 suggesting a continuous FRSB transition, despite this ratio being equal to 2 at mean-field level. A value of
λ>1 is required for the discontinuous 1RSB transition. We argue that strong finite-size effects and closely spaced transition temperatures remove the expected 1RSB transition for the system sizes which we can study. For values of the exponent
σ=0.85, which roughly corresponds to a three dimensional system, we find that the renormalized value of
λ is again less than 1, with no signs of either the 1RSB transition or the continuous FRSB transition, suggesting that the Kauzmann temperature
TK in three dimensions might be zero and the complete absence of phase transitions in structural glasses.
We present a scaling theory for the low-temperature phase transition of the Ising spin glass in a transverse magnetic field (\ensuremath{\Gamma}). The theory provides relations and bounds on some critical exponents, which are supported by results from actual calculations. A renormalization-group analysis of the zero-temperature transition in the presence of a longitudinal symmetry-breaking field enables us to estimate the critical exponents \ensuremath{\beta} and \ensuremath{\gamma}, associated with the Edwards-Anderson order parameter and the nonlinear susceptibility, respectively. We have found no indications of the transition being of first order.
The infinite-range Ising spin-glass in the presence of a transverse field is studied by Monte Carlo simulations. The overlap probability distribution P ( q ) is defined and calculated for the present quantum case. The P ( q ) in the spin-glass phase has a long tail extended down to q = 0 qualitatively similar to the classical Sherrington-Kirkpatrick model. The phase boundary of the glass transition temperature vs. the transverse field is obtained. Our results for the phase diagram agree with our recent replica symmetry-breaking solution but disagree with the results of Yokata and of Ray et al.