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Register a new userCritical and Griffiths-McCoy singularities in quantum Ising spin-glasses on d-dimensional hypercubic lattices: A series expansion study
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We study the $pm J$ transverse-field Ising spin glass model at zero temperature on d-dimensional hypercubic lattices and in the Sherrington-Kirkpatrick (SK) model, by series expansions around the strong field limit. In the SK model and in high-dimensions our calculated critical properties are in excellent agreement with the exact mean-field results, surprisingly even down to dimension $d = 6$ which is below the upper critical dimension of $d=8$. In contrast, in lower dimensions we find a rich singular behavior consisting of critical and Griffiths-McCoy singularities. The divergence of the equal-time structure factor allows us to locate the critical coupling where the correlation length diverges, implying the onset of a thermodynamic phase transition. We find that the spin-glass susceptibility as well as various power-moments of the local susceptibility become singular in the paramagnetic phase $textit{before}$ the critical point. Griffiths-McCoy singularities are very strong in two-dimensions but decrease rapidly as the dimension increases. We present evidence that high enough powers of the local susceptibility may become singular at the pure-system critical point.
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The behavior of two-dimensional Ising spin glasses at the multicritical point on triangular and honeycomb lattices is investigated, with the help of finite-size scaling and conformal-invariance concepts. We use transfer-matrix methods on long strips to calculate domain-wall energies, uniform susceptibilities, and spin-spin correlation functions. Accurate estimates are provided for the location of the multicritical point on both lattices, which lend strong support to a conjecture recently advanced by Takeda, Sasamoto, and Nishimori. Correlation functions are shown to obey rather strict conformal-invariance requirements, once suitable adaptations are made to account for geometric aspects of the transfer-matrix description of triangular and honeycomb lattices. The universality class of critical behavior upon crossing the ferro-paramagnetic phase boundary is probed, with the following estimates for the associated critical indices: $ u=1.49(2)$, $gamma=2.71(4)$, $eta_1= 0.183(3)$, distinctly different from the percolation values.
We consider the paramagnetic phase of the random transverse-field Ising spin chain and study the dynamical properties by numerical methods and scaling considerations. We extend our previous work [Phys. Rev. B 57, 11404 (1998)] to new quantities, such as the non-linear susceptibility, higher excitations and the energy-density autocorrelation function. We show that in the Griffiths phase all the above quantities exhibit power-law singularities and the corresponding critical exponents, which vary with the distance from the critical point, can be related to the dynamical exponent z, the latter being the positive root of [(J/h)^{1/z}]_av=1. Particularly, whereas the average spin autocorrelation function in imaginary time decays as [G]_av(t)~t^{-1/z}, the average energy-density autocorrelations decay with another exponent as [G^e]_av(t)~t^{-2-1/z}.
Diffusion on a diluted hypercube has been proposed as a model for glassy relaxation and is an example of the more general class of stochastic processes on graphs. In this article we determine numerically through large scale simulations the eigenvalue spectra for this stochastic process and calculate explicitly the time evolution for the autocorrelation function and for the return probability, all at criticality, with hypercube dimensions $N$ up to N=28. We show that at long times both relaxation functions can be described by stretched exponentials with exponent 1/3 and a characteristic relaxation time which grows exponentially with dimension $N$. The numerical eigenvalue spectra are consistent with analytic predictions for a generic sparse network model.
The site-diluted transverse field Ising model in two dimensions is studied with Quantum-Monte-Carlo simulations. Its phase diagram is determined in the transverse field (Gamma) and temperature (T) plane for various (fixed) concentrations (p). The nature of the quantum Griffiths phase at zero temperature is investigated by calculating the distribution of the local zero-frequency susceptibility. It is pointed out that the nature of the Griffiths phase is different for small and large Gamma.
The de Almeida-Thouless (AT) line in Ising spin glasses is the phase boundary in the temperature $T$ and magnetic field $h$ plane below which replica symmetry is broken. Using perturbative renormalization group (RG) methods, we show that when the dimension $d$ of space is just above $6$ there is a multicritical point (MCP) on the AT line, which separates a low-field regime, in which the critical exponents have mean-field values, from a high-field regime where the RG flows run away to infinite coupling strength; as $d$ approaches $6$ from above, the location of the MCP approaches the zero-field critical point exponentially in $1/(d-6)$. Thus on the AT line perturbation theory for the critical properties breaks down at sufficiently large magnetic field even above $6$ dimensions, as well as for all non-zero fields when $dleq 6$ as was known previously. We calculate the exponents at the MCP to first order in $varepsilon=d-6>0$. The fate of the MCP as $d$ increases from just above 6 to infinity is not known.
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