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arXiv:1111.3939 · Physical Review B

Spin-spin correlation function of the two-dimensional XY model with weak site or bond dilution

O. Kapikranian

Published Mar 5, 2012• 3 citations
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Abstract

The spin-spin correlation function of the two-dimensional (2D) $XY$ model decays as a power law at all temperatures below the Berezinskii-Kosterlitz-Thouless transition point with a temperature-dependent exponent $\ensuremath{\eta}=\ensuremath{\eta}(T/J)$ ($J$ is the ferromagnetic coupling strength). It is known from computer experiments that in the 2D $XY$ model with site or bond dilution this exponent depends on the concentration $p$ of removed sites or bonds as well. Knowing the slope $\ensuremath{\partial}\ensuremath{\eta}/\ensuremath{\partial}p$ at point $p=0$, 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.

DOI
10.1103/physrevb.85.094405

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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 \ensuremathη=\ensuremathη(T/J)\ensuremath{\eta}=\ensuremath{\eta}(T/J)\ensuremathη=\ensuremathη(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∂\ensuremathη/\ensuremath∂p\ensuremath{\partial}\ensuremath{\eta}/\ensuremath{\partial}p\ensuremath∂\ensuremathη/\ensuremath∂p at point p=0p=0p=0, one can predict the value of the exponent for small dilution concentrations: \ensuremathη(p)\ensuremath≃\ensuremathη(0)+p(\ensuremath∂\ensuremathη/\ensuremath∂p)∣p=0{\ensuremath{\eta}(p)\ensuremath{\simeq}\ensuremath{\eta}(0)+p(\ensuremath{\partial}\ensuremath{\eta}/\ensuremath{\partial}p)|}_{p=0}\ensuremathη(p)\ensuremath≃\ensuremathη(0)+p(\ensuremath∂\ensuremathη/\ensuremath∂p)∣p=0​. As shown in this paper, the spin-wave Hamiltonian allows one to obtain exact results for this slope: (\ensuremath∂\ensuremathη/\ensuremath∂p)∣p=0=T/(2J)+O((T/J)2){(\ensuremath{\partial}\ensuremath{\eta}/\ensuremath{\partial}p)|}_{p=0}=T/(2J)+O({(T/J)}^{2})(\ensuremath∂\ensuremathη/\ensuremath∂p)∣p=0​=T/(2J)+O((T/J)2) and T/(\ensuremathπJ)+O((T/J)2)T/(\ensuremath{\pi}J)+O({(T/J)}^{2})T/(\ensuremathπJ)+O((T/J)2) for site and for bond dilution, respectively.
The critical behavior of statistical models with long-range interactions exhibits distinct regimes as a function of $\ensuremath{\rho}$, the power of the interaction strength decay. For large enough $\ensuremath{\rho}, \ensuremath{\rho}>{\ensuremath{\rho}}_{\mathrm{sr}}$, the critical behavior is observed to coincide with that of the short-range model. However, there are controversial aspects regarding this picture, one of which is the value of the short-range threshold ${\ensuremath{\rho}}_{\mathrm{sr}}$ in the case of the long-range XY model in two dimensions. We study the 2D XY model on the diluted graph, a sparse graph obtained from the 2D lattice by rewiring links with probability decaying with the Euclidean distance of the lattice as ${|r|}^{\ensuremath{-}\ensuremath{\rho}}$, which is expected to feature the same critical behavior of the long-range model. Through Monte Carlo sampling and finite-size analysis of the spontaneous magnetization and of the Binder cumulant, we present numerical evidence that ${\ensuremath{\rho}}_{\mathrm{sr}}=4$. According to such a result, one expects the model to belong to the Berezinskii-Kosterlitz-Thouless universality class for $\ensuremath{\rho}\ensuremath{\ge}4$, and to present a second-order transition for $\ensuremath{\rho}<4$.
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.
We present detailed Monte Carlo results for the susceptibility \ensuremath{\chi}, correlation length \ensuremath{\xi}, and specific heat Cv{\mathit{C}}_{\mathit{v}}Cv​ for the XY model. The simulations are done on 642{64}^{2}642, 1282{128}^{2}1282, 2562{256}^{2}2562, and 5122{512}^{2}5122 lattices over the temperature range 0.98\ensuremath{\le}T\ensuremath{\le}1.43 corresponding to 270. Fits to \ensuremath{\chi} and \ensuremath{\xi} data favor a Kosterlitz-Thouless (KT) singularity over a second-order transition; however, unconstrained four-parameter KT fits do not confirm the predicted values \ensuremath{\nu}=0.5 and \ensuremath{\eta}=0.25. Our best estimate, Tc{\mathit{T}}_{\mathit{c}}Tc​=0.894(5), is obtained using KT fits with \ensuremath{\nu} fixed at 0.5. The exponent \ensuremath{\eta} is calculated as a function of temperature in the spin-wave phase using Monte Carlo renormalization-group and finite-size-scaling methods. Both methods give consistent results and we find \ensuremath{\eta}\ensuremath{\approxeq}0.235 at T=0.894. We also present results for the behavior of vortex density across the transition and exhibit how the dilute-gas approximation breaks down.
Generalizing the recent work of Collins, the wave-vector-dependent susceptibility and the static correlation function for a randomly diluted Heisenberg paramagnet on a Bravais lattice with quenched-site, or nearest-neighbor exchange-bond, dilution is studied by high-temperature series-expansion techniques. The first five coefficients for the spin correlation function $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}})$ and susceptibility $\ensuremath{\chi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}})$ are calculated for arbitrary $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$, the sign of the exchange and the spin magnitude and the magnetic site ccc and bond ppp concentrations. Numerical results for these functions are presented for various spins and for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ along the symmetry directions (1,1,1) and (1,0,0). Because of the generality of the present series, their length is rather limited, and as such they are not suitable for a careful analysis of the critical region. Nonetheless, to get a qualitative feel for the accuracy of the present results near the transition region, we have used Pad\'e approximants for determining the correlation length $\ensuremath{\xi}(T)$ as a function of the reduced temperature $\ensuremath{\epsilon}=\frac{[T\ensuremath{-}{T}_{C(N)}(\mathrm{random})]}{{T}_{C(N)}(\mathrm{random})}$. In this manner, we estimate that the numerical accuracy of our results should be reasonable as long as the system is moderately above the transition temperature, e.g., $\ensuremath{\epsilon}\ensuremath{\gtrsim}0.5$.
We study a percolation problem on a substrate formed by two-dimensional XY spin configurations using Monte Carlo methods. For a given spin configuration, we construct percolation clusters by randomly choosing a direction x in the spin vector space, and then placing a percolation bond between nearest-neighbor sites i and j with probability p(ij)=max(0,1-e(-2Ks(i)(x)s(j)(x))), where K>0 governs the percolation process. A line of percolation thresholds K(c)(J) is found in the low-temperature range J≥J(c), where J>0 is the XY coupling strength. Analysis of the correlation function g(p)(r), defined as the probability that two sites separated by a distance r belong to the same percolation cluster, yields algebraic decay for K≥K(c)(J), and the associated critical exponent depends on J and K. Along the threshold line K(c)(J), the scaling dimension for g(p) is, within numerical uncertainties, equal to 1/8. On this basis, we conjecture that the percolation transition along the K(c)(J) line is of the Berezinskii-Kosterlitz-Thouless type.