Monte Carlo study of surface critical behavior in the XY model

Extraordinary-Log Surface Phase Transition in the Three-Dimensional XY Model

Universality is a pillar of modern critical phenomena. The standard scenario is that the two-point correlation algebraically decreases with the distance r as g(r)∼r^{2-d-η}, with d the spatial dimension and η the anomalous dimension. Very recently, a logarithmic universality was proposed to describe the extraordinary surface transition of the O(N) system. In this logarithmic universality, g(r) decays in a power of logarithmic distance as g(r)∼(lnr)^{-η[over ^]}, dramatically different from the standard scenario. We explore the three-dimensional XY model by Monte Carlo simulations, and provide strong evidence for the emergence of logarithmic universality. Moreover, we propose that the finite-size scaling of g(r,L) has a two-distance behavior: simultaneously containing a large-distance plateau whose height decays logarithmically with L as g(L)∼(lnL)^{-η[over ^]^{'}} as well as the r-dependent term g(r)∼(lnr)^{-η[over ^]}, with η[over ^]^{'}≈η[over ^]-1. The critical exponent η[over ^]^{'}, characterizing the height of the plateau, obeys the scaling relation η[over ^]^{'}=(N-1)/(2πα) with the RG parameter α of helicity modulus. Our picture can also explain the recent numerical results of a Heisenberg system. The advances on logarithmic universality significantly expand our understanding of critical universality.

Extending multiple histogram reweighting to a continuous lattice spin system exhibiting a first-order phase transition

We present extensive Monte Carlo simulations on a two-dimensional XY model with a modified form of interaction potential. Thermodynamic quantities other than energy, specific heat, etc. (such as magnetization, susceptibility, and fourth-order cumulant of magnetization) are obtained using multiple histogram reweighting of the data obtained from the simulations. We employ an approach which eliminates the need to construct two-dimensional histograms. This approach makes judicious use of computer memory as well as CPU time. Lee-Kosterlitz's method of finite size scaling for a first-order transition and analysis using Binder's cumulant method allow us to make an accurate determination of the transition temperature.

Bulk and surface phase transitions in the three-dimensional O4 spin model

We investigate the O(4) spin model on the simple-cubic lattice by means of the Wolff cluster algorithm. Using the toroidal boundary condition, we locate the bulk critical point at coupling K(c) = 0.935 856(2), and determine the bulk thermal magnetic renormalization exponents as y(t) = 1.337 5(15) and y(h) = 2.482 0(2), respectively. The universal ratio Q=m(2)(2)/m(4) is also determined as 0.9142(1). The precision of these estimates significantly improves over that of the existing results. Then, we simulate the critical O(4) model with two open surfaces on which the coupling strength K(1) can be varied. At the ordinary transitions, the surface magnetic exponent is determined as y((o))(h1) = 1.020 2(12). Further, we find a so-called special surface transition at (k) = K(1)/K-1 = 1.258(20). At this point, the surface thermal exponent y(s)(t1) is rather close to zero, and we cannot exclude that the corresponding surface transition is Kosterlitz-Thouless-like. The surface magnetic exponent is y((s))/h1 = 1.816(2).

Surface and bulk transitions in three-dimensional O(n) models

Using Monte Carlo methods and finite-size scaling, we investigate surface criticality in the O(n) models on the simple-cubic lattice with n=1 , 2, and 3, i.e., the Ising, XY , and Heisenberg models. For the critical couplings we find Kc (n=2) =0.454 1659 (10) and Kc (n=3) =0.693 003 (2). We simulate the three models with open surfaces and determine the surface magnetic exponents at the ordinary transition to be y (o)(h1) =0.7374 (15) , 0.781 (2), and 0.813 (2) for n=1 , 2, and 3, respectively. Then we vary the surface coupling K1 and locate the so-called special transition at kappa(c) (n=1) =0.502 14 (8) and kappa(c) (n=2) =0.6222 (3), where kappa= K1 /K-1 . The corresponding surface thermal and magnetic exponents are y (s)(t1) =0.715 (1) and y (s)(h1) =1.636 (1) for the Ising model, and y(s)(t1) =0.608 (4) and y(s)(h1) =1.675 (1) for the XY model. Finite-size corrections with an exponent close to -1/2 occur for both models. Also for the Heisenberg model we find substantial evidence for the existence of a special surface transition.

Surface critical phenomena in three-dimensional percolation

Using Monte Carlo methods and finite-size scaling, we investigate surface critical phenomena in the bond-percolation model on the simple-cubic lattice with two open surfaces in one direction. We decompose the whole lattice into percolation clusters and sample the surface and bulk dimensionless ratios Q1 and Qb, defined on the basis of the moments of the cluster-size distribution. These ratios are used to determine critical points. At the bulk percolation threshold pbc, we determine the surface bond-occupation probability at the special transition as p(s)1c = 0.418 17(2), and further obtain the corresponding surface thermal and magnetic exponents as y(s)t1 = 0.5387(2) and y(s)h1 = 1.8014(6), respectively. Next, from the pair correlation function on the surfaces, we determine y(o)h1 = 1.0246(4) and y(e)h1 = 1.25(6) for the ordinary and the extraordinary transition, respectively, of which the former is consistent with the existing result y(o)h1 = 1.024(4). We also numerically derive the line of surface phase transitions occurring at pb < pbc, and determine the pertinent asymptotic values of the universal ratios Q1 and Qb.