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Theoretical and Mathematical Physics

Weak First-Order Transition and Pseudoscaling Behavior in the Universality Class of the O(N) Ising Model

А. О. Сорокин

Published Aug 1, 2019• 5 citations
Publisher

Abstract

Using Monte Carlo and renormalization group methods, we investigate systems with critical behavior described by two order parameters: continuous (vector) and discrete (scalar). We consider two models of classical three-dimensional Heisenberg magnets with different numbers of spin components N = 1,…,4: the model on a cubic lattice with an additional competing antiferromagnetic exchange interaction in a layer and the model on a body-centered lattice with two competing antiferromagnetic interactions. In both models, we observe a first-order transition for all values of N. In the case where competing exchanges are approximately equal, the first order of a transition is close to the second order, and pseudoscaling behavior is observed with critical exponents differing from those of the O(N) model. In the case N = 2, the critical exponents are consistent with the well-known indices of the class of magnets with a noncollinear spin ordering. We also give a possible explanation of the observed pseudoscaling in the framework of the renormalization group analysis.

DOI
10.1134/s0040577919080117

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Using Monte Carlo and renormalization group methods, we investigate systems with critical behavior described by two order parameters: continuous (vector) and discrete (scalar). We consider two models of classical three-dimensional Heisenberg magnets with different numbers of spin components N = 1,…,4: the model on a cubic lattice with an additional competing antiferromagnetic exchange interaction in a layer and the model on a body-centered lattice with two competing antiferromagnetic interactions. In both models, we observe a first-order transition for all values of N. In the case where competing exchanges are approximately equal, the first order of a transition is close to the second order, and pseudoscaling behavior is observed with critical exponents differing from those of the O(N) model. In the case N = 2, the critical exponents are consistent with the well-known indices of the class of magnets with a noncollinear spin ordering. We also give a possible explanation of the observed pseudoscaling in the framework of the renormalization group analysis.
The Ising model with nearest and next nearest neighbor antiferromagnetic interactions on the triangular lattice displays, for Jnnn/Jnn = 0.1, three phase transitions in different universality classes as the magnetic field is increased. We have studied this model using Monte Carlo and renormalization group techniques. The transition from the paramagnetic to the 2 × 1 phase (universality class of the Heisenberg model with cubic anisotropy) is found to be first order; the transition from the paramagnetic phase to the [Formula: see text] phase (universality class of the three state Potts model) is continuous; and the transition from the paramagnetic to the 2 × 2 phase (universality class of the four state Potts model) is found to change from first order to continuous as the field is increased. We have mapped out the phase diagram and determined the critical exponents for the continuous transitions. A novel technique, using a Landau-like free energy functional determined from Monte Carlo calculations, to distinguish between first order and continuous transitions, is described.
We show that phase transitions in spin-1 Bose gases and stacked triangular Heisenberg antiferromagnets---an example of frustrated magnets with competing interactions---are described by the same Landau-Ginzburg-Wilson Hamiltonian with $\mathrm{O}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{O}(2)$ symmetry. In agreement with previous nonperturbative-renormalization-group studies of the three-dimensional $\mathrm{O}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{O}(2)$ model, we find that the transition from the normal phase to the superfluid ferromagnetic phase in a spin-1 Bose gas is weakly first order and shows pseudoscaling behavior. The (nonuniversal) pseudoscaling exponent $\ensuremath{\nu}$ is fully determined by the scattering lengths a0{a}_{0}a0​ and a2{a}_{2}a2​. We provide estimates of $\ensuremath{\nu}$ in 87Rb,41K^{87}\mathrm{Rb},^{41}\mathrm{K}87Rb,41K, and 7Li^{7}\mathrm{Li}7Li atom gases which can be tested experimentally. We argue that pseudoscaling comes from either a crossover phenomenon due to proximity of the O(6) Wilson-Fisher fixed point (87Rb(^{87}\mathrm{Rb}(87Rb and 41K)^{41}\mathrm{K})41K) or the existence of two unphysical fixed points (with complex coordinates) which slow down the RG flow (7Li)(^{7}\mathrm{Li})(7Li). These unphysical fixed points are a remnant of the chiral and antichiral fixed points that exist in the $\mathrm{O}(N)\ifmmode\times\else\texttimes\fi{}\mathrm{O}(2)$ model when NNN is larger than ${N}_{c}\ensuremath{\simeq}5.3$ (the transition being then second order and controlled by the chiral fixed point). Finally, we discuss a $\mathrm{O}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{O}(2)$ lattice model and show that our results, even though we find the transition to be first order, are compatible with Monte Carlo simulations yielding an apparent second-order transition.
The nature of the phase transition of antiferromagnetic vector spin models on the stacked triangular lattice (STA) has been an issue of debate. In order to account for the coplanar non-collinear order in the ground state one needs two vector fields in a Ginzburg–Landau–Wilson Hamiltonian. The field theoretical renormalization group analysis shows that there is a threshold value Nc for the number of spin component N, and for N > Nc the transition is of second order belonging to a new ‘‘chiral’’ universality class, while for N < Nc the chiral fixed points ceases to exist in the real parameter space and the transition is of first order. According to a recent estimate based on a Monte Carlo renormalization group study, 3 < Nc < 8. 2) Therefore a first order transition is expected for the physically relevant XY and Heisenberg spin systems. The phase transition of the XY or Heisenberg STA is of very weak first order and is said to be of ‘‘almost second order’’ with large (but finite) correlation lengths due to a slow velocity region in the renormalization group flow. Monte Carlo simulations on systems with moderate size appear to show second order transitions with pseudo-critical behavior. Strong first order transitions were found for variants of the model on which a constraint of local rigidity was imposed which still belong to the same universality class as the original STA. Recently the first order nature of the phase transition in XY antiferromagnet on the stacked triangular lattice (XY-STA model) was demonstrated directly by canonical Monte Carlo simulations on large systems, in which the energy probability distribution shows a double peak structure near the transition temperature. The purpose of the present paper is to show the first order transition of the XY-STA model by applying a novel technique: a microcanonical method. To distinguish a weak first order transition from a second order one can be difficult, especially if the energy jump, the characteristic of a first order transition, is tiny as is the case for frustrated spin systems. Limited to accessible system sizes one may analyze the different power law behavior of the maximum of specific heat or susceptibility, with an exponent equal to three for a first order transition and less than three for a second order transition. The obvious solution to increase the size of the system runs into difficulties for the canonical simulations (at constant temperature) because of the growing autocorrelation times. Also the histogram method is then restricted to smaller intervals of temperature shrinking as the inverse power of the system size. In this short note we will show that the microcanonical simulations (at constant energy) can help to solve the problems encountered in canonical simulations. The method has been tested for strong first order transitions and also for second order ones. Before presenting the method and the results we have to describe the model we study. It is given by the usual Hamiltonian
The critical properties of the antiferromagnetic Heisenberg model on the three-dimensional stacked-triangular lattice are studied by means of a large-scale Monte Carlo simulation in order to get insight into the controversial issue of the criticality of the noncollinear magnets with the $O(3)\ifmmode\times\else\texttimes\fi{}O(2)$ symmetry. The maximum size studied is 3843{384}^{3}3843, considerably larger than the sizes studied by the previous numerical works on the model. Availability of such large-size data enables us to examine the detailed critical properties including the effect of corrections to the leading scaling. Strong numerical evidence of the continuous nature of the transition is obtained. Our data indicate the existence of significant corrections to the leading scaling. Careful analysis by taking account of the possible corrections yields critical exponents estimates, $\ensuremath{\alpha}=0.44(3), \ensuremath{\beta}=0.26(2), \ensuremath{\gamma}=1.03(5), \ensuremath{\nu}=0.52(1), \ensuremath{\eta}=0.02(5)$, and the chirality exponents, ${\ensuremath{\beta}}_{\ensuremath{\kappa}}=0.40(3)$ and ${\ensuremath{\gamma}}_{\ensuremath{\kappa}}=0.77(6)$, supporting the existence of the O(3)O(3)O(3) chiral [or $O(3)\ifmmode\times\else\texttimes\fi{}O(2)$] universality class governed by a ``chiral'' fixed point. We also obtain an indication that the underlying fixed point is of the focus type, characterized by the complex-valued correction-to-scaling exponent, $\ensuremath{\omega}=0.{1}_{\ensuremath{-}0.05}^{+0.4}+i\phantom{\rule{4pt}{0ex}}0.{7}_{\ensuremath{-}0.4}^{+0.1}$. The focus-like nature of the chiral fixed point accompanied by the spiral-like renormalization-group (RG) flow is likely to be the origin of the apparently complicated critical behavior. The results are compared and discussed in conjunction with the results of other numerical simulations, several distinct types of RG calculations including the higher-order perturbative massive and massless RG calculations and the nonperturbative functional RG calculation, and the conformal-bootstrap program.
The disordering transition of half a monolayer of chemisorbed oxygen on Ru(001) can be classified in the universality class of the two-dimensional Heisenberg model with face-centered anisotropy. Theoretical studies of this class suggest fluctuation-induced first-order-transition behavior. Experimental study of the O/Ru(001) system, on the other hand, reveals a continuous phase transition with exponents near those of the three-state Potts model. We have investigated this discrepancy via Monte Carlo simulations of the Heisenberg model using the histogram method to determine transition order, focusing on two possible resolutions of the discrepancy: (1) finite-size rounding might be expected to render the experimental transition apparently continuous even if it is actually first order; or (2) transition order might depend on the details of the interaction energies. Despite its seeming inconsistency with the standard understanding of universality, we have strong evidence for the latter explanation; there appears to be a parameter regime of second-order behavior within standard ferromagnetic Heisenberg symmetry. We delineate this breakdown of universality and speculate as to its mechanism.
Abstract Phase transitions and critical properties of the antiferromagnetic Heisenberg model on a body-centered cubic lattice are investigated by the Monte Carlo method, based on the replica algorithm with allowance of the interactions between the first and second nearest neighbors. Analysis is performed for intensity ratios r of exchange interaction between the first and second nearest neighbors in the interval 0.0 ≤ r ≤ 1.0. The phase diagram of the dependence of the critical temperature on the intensity of interaction of the second nearest neighbors is constructed. On this diagram, a region in which the transition from the antiferromagnetic to the paramagnetic phase is of the first order is detected. The entire set of the main static critical indices is calculated. It is shown that the universality class of the critical behavior is preserved in the interval 0.0 ≤ r ≤ 0.6. It is found that the variation of the second nearest neighbor interaction intensity in the range 0.8 ≤ r ≤ 1.0 leads to nonuniversal critical behavior.