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Journal of Applied Physics

Correlation decay in low-dimensional spin glasses

ARMEN N. KOCHARIAN, A. Sogomonian

Published Nov 15, 1994• 1 citation
Publisher

Abstract

A new method to investigate the two-point correlation function decay in a finite size random spin system and at finite temperatures is proposed. It is applicable to systems with cyclic boundary conditions and in any local or uniform magnetic field. The efficiency of this method is presented in the case of an isotropic XY spin glass model with Gaussian bond distribution in external field. Using unitary U(1) gauge transformation and operator inequalities we have obtained explicit upper bounds for generalized spin-glass susceptibility κ(T,R) that describes off-diagonal transverse magnetization in two dimensional random XY spin lattices at finite temperatures. It is found that the slowest possible decay for susceptibility in random XY spin glass is of the Kosterlitz–Thouless-type with power-law decay in 2D, and the exponential type in 1D, at low temperatures. These upper bounds rule out the possibility of the corresponding magnetic ordering in 1D and 2D isotropic XY spin glasses at finite temperatures.

DOI
10.1063/1.358537

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A new method to investigate the two-point correlation function decay in a finite size random spin system and at finite temperatures is proposed. It is applicable to systems with cyclic boundary conditions and in any local or uniform magnetic field. The efficiency of this method is presented in the case of an isotropic XY spin glass model with Gaussian bond distribution in external field. Using unitary U(1) gauge transformation and operator inequalities we have obtained explicit upper bounds for generalized spin-glass susceptibility κ(T,R) that describes off-diagonal transverse magnetization in two dimensional random XY spin lattices at finite temperatures. It is found that the slowest possible decay for susceptibility in random XY spin glass is of the Kosterlitz–Thouless-type with power-law decay in 2D, and the exponential type in 1D, at low temperatures. These upper bounds rule out the possibility of the corresponding magnetic ordering in 1D and 2D isotropic XY spin glasses at finite temperatures.
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We calculate the two-dimensional probability for the magnetization in a two-dimensional XY model of finite size. We show that, for arbitrarily large N, there is a topological difference between the distributions in the low-temperature spin wave regime and in the high-temperature paramagnetic regime. In the low-temperature phase is a well-defined ring function and we calculate an upper bound for . Even so, this is consistent with the susceptibility per spin, , being divergent. We show further that the distribution function Q(M) for the scalar magnetization has a universal form, scaling with the single variable , where L is the system size and J is the coupling constant. We show that has considerable structure, with two terms of order N, one due to the spin waves and the other due to vortices. This leads to a peak in at the Kosterlitz - Thouless - Berezinskii transition.
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In a general class of one- and two-dimensional Hubbard models, we prove upper bounds for the two-point correlation functions at finite temperatures for electrons, electron pairs, and spin. The upper bounds decay exponentially in one dimension, and with power laws in two dimensions. The bounds rule out the possibility of the corresponding condensation of superconducting electron pairs, and of the corresponding magnetic ordering. Our method is general enough to cover other models such as the t-J model.
We study the finite-size behavior of two-dimensional spin-glass models. We consider the $\ifmmode\pm\else\textpm\fi{}J$ model for two different values of the probability of the antiferromagnetic bonds and the model with Gaussian distributed couplings. The analysis of renormalization-group invariant quantities, the overlap susceptibility, and the two-point correlation function confirms that they belong to the same universality class. We analyze in detail the standard finite-size scaling limit in terms of $T{L}^{1/\ensuremath{\nu}}$ in the $\ifmmode\pm\else\textpm\fi{}J$ model. We find that it holds asymptotically. This result is consistent with the low-temperature crossover scenario in which the crossover temperature, which separates the universal high-temperature region from the discrete low-temperature regime, scales as ${T}_{c}(L)\ensuremath{\sim}{L}^{\ensuremath{-}{\ensuremath{\theta}}_{S}}$ with ${\ensuremath{\theta}}_{S}\ensuremath{\approx}0.5$.
The spin-spin correlation function of the two-dimensional (2D) XYXYXY model decays as a power law at all temperatures below the Berezinskii-Kosterlitz-Thouless transition point with a temperature-dependent exponent $\ensuremath{\eta}=\ensuremath{\eta}(T/J)$ (JJJ is the ferromagnetic coupling strength). It is known from computer experiments that in the 2D XYXYXY model with site or bond dilution this exponent depends on the concentration ppp of removed sites or bonds as well. Knowing the slope $\ensuremath{\partial}\ensuremath{\eta}/\ensuremath{\partial}p$ at point p=0p=0p=0, one can predict the value of the exponent for small dilution concentrations: ${\ensuremath{\eta}(p)\ensuremath{\simeq}\ensuremath{\eta}(0)+p(\ensuremath{\partial}\ensuremath{\eta}/\ensuremath{\partial}p)|}_{p=0}$. As shown in this paper, the spin-wave Hamiltonian allows one to obtain exact results for this slope: ${(\ensuremath{\partial}\ensuremath{\eta}/\ensuremath{\partial}p)|}_{p=0}=T/(2J)+O({(T/J)}^{2})$ and $T/(\ensuremath{\pi}J)+O({(T/J)}^{2})$ for site and for bond dilution, respectively.