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Register a new userMulticritical point of Ising spin glasses on triangular and honeycomb lattices
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Added by
Sergio L. A. de Queiroz
Publication date
2005
fields
Physics
and research's language is
English
Authors
S.L.A. de Queiroz
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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.
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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.
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.
On Archimedean lattices, the Ising model exhibits spontaneous ordering. Three examples of these lattices of the majority-vote model with noise are considered and studied through extensive Monte Carlo simulations. The order/disorder phase transition is observed in this system. The calculated values of the critical noise parameter are q_c=0.089(5), q_c=0.078(3), and q_c=0.114(2) for honeycomb, Kagome and triangular lattices, respectively. The critical exponents beta/nu, gamma/nu and 1/nu for this model are 0.15(5), 1.64(5), and 0.87(5); 0.14(3), 1.64(3), and 0.86(6); 0.12(4), 1.59(5), and 1.08(6) for honeycomb, Kagome and triangular lattices, respectively. These results differs from the usual Ising model results and the majority-vote model on so-far studied regular lattices or complex networks. The effective dimensionalities of the system D_{eff}= 1.96(5) (honeycomb), D_{eff} =1.92(4) (Kagome), and D_{eff}= 1.83(5) (triangular) for these networks are just compatible to the embedding dimension two.
Using Monte Carlo simulations and finite-size scaling analysis, the critical behavior of self-assembled rigid rods on triangular and honeycomb lattices at intermediate density has been studied. The system is composed of monomers with two attractive (sticky) poles that, by decreasing temperature or increasing density, polymerize reversibly into chains with three allowed directions and, at the same time, undergo a continuous isotropic-nematic (IN) transition. The determination of the critical exponents, along with the behavior of Binder cumulants, indicate that the IN transition belongs to the q=1 Potts universality class.
We review our earlier studies on the order parameter distribution of the quantum Sherrington-Kirkpatrick (SK) model. Through Monte Carlo technique, we investigate the behavior of the order parameter distribution at finite temperatures. The zero temperature study of the spin glass order parameter distribution is made by the exact diagonalization method. We find in low-temperature (high-transverse-field) spin glass region, the tail (extended up to zero value of order parameter) and width of the order parameter distribution become zero in thermodynamic limit. Such observations clearly suggest the existence of a low-temperature (high-transverse-field) ergodic region. We also find in high-temperature (low-transverse-field) spin glass phase the order parameter distribution has nonzero value for all values of the order parameter even in infinite system size limit, which essentially indicates the nonergodic behavior of the system. We study the annealing dynamics by the paths which pass through both ergodic and nonergodic spin glass regions. We find the average annealing time becomes system size independent for the paths which pass through the quantum-fluctuation-dominated ergodic spin glass region. In contrast to that, the annealing time becomes strongly system size dependent for annealing down through the classical-fluctuation-dominated nonergodic spin glass region. We investigate the behavior of the spin autocorrelation in the spin glass phase. We observe that the decay rate of autocorrelation towards its equilibrium value is much faster in the ergodic region with respect to the nonergodic region of the spin glass phase.
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