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Journal of Physics Condensed Matter

The phase diagram and critical properties of the two-dimensional anisotropicXYmodel

Antonio S. T. Pires, L.S. Lima, M. E. Gouvêa

Published Nov 29, 2007• 39 citations
Publisher

Abstract

In this paper we present a study of the phase diagram and critical properties of the square-lattice quantum XY model, with a single-ion anisotropy and spin S = 1, as a function of the anisotropy parameter D . For D less than a critical value D C , which is the critical point for a quantum phase transition at T = 0, the model presents a Berezinskii–Kosterlitz–Thouless (BKT) transition. This region is adequately described by the self-consistent harmonic approximation. We show that, if we want to use a Schwinger boson theory in this region, we should include fluctuations around the mean-field approximation (which leads to a gauge field) to describe correctly the BKT transition. However both methods are inadequate to describe the large- D phase, and we show that this phase can be studied using the bond operator formalism which gives the correct behaviour for the correlation length as a function of the temperature in the critical point and above. Our results agree with scaling arguments.

DOI
10.1088/0953-8984/20/01/015208

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In this paper we present a study of the phase diagram and critical properties of the square-lattice quantum XY model, with a single-ion anisotropy and spin S = 1, as a function of the anisotropy parameter D . For D less than a critical value D C , which is the critical point for a quantum phase transition at T = 0, the model presents a Berezinskii–Kosterlitz–Thouless (BKT) transition. This region is adequately described by the self-consistent harmonic approximation. We show that, if we want to use a Schwinger boson theory in this region, we should include fluctuations around the mean-field approximation (which leads to a gauge field) to describe correctly the BKT transition. However both methods are inadequate to describe the large- D phase, and we show that this phase can be studied using the bond operator formalism which gives the correct behaviour for the correlation length as a function of the temperature in the critical point and above. Our results agree with scaling arguments.
Monte Carlo simulations of the two-dimensional XY model are performed in a square geometry with fixed boundary conditions. Using a conformal mapping, it is very easy to deduce the exponent η σ ( T ) of the order parameter correlation function at any temperature in the critical phase of the model. The temperature behaviour of η σ ( T ) is obtained numerically with a good accuracy up to the Kosterlitz-Thouless transition temperature. At very low temperatures, a good agreement is found with Berezinskii's harmonic approximation. Surprisingly, we show some evidence that there are no logarithmic corrections to the behaviour of the order parameter density profile (with symmetry-breaking surface fields) at the Kosterlitz-Thouless transition temperature.
Monte Carlo simulations of the two-dimensional XY model are performed in a square geometry with various boundary conditions (BC). Using conformal mappings we deduce the exponent η σ ( T ) of the order parameter correlation function and its surface analogue η ∥ ( T ) as a function of the temperature in the critical (low-temperature) phase of the model. The temperature dependence of both exponents is obtained numerically with a good accuracy up to the Kosterlitz–Thouless transition temperature. The bulk exponent follows from simulations of correlation functions with periodic boundary conditions or order parameter profiles with open boundary conditions and with symmetry breaking surface fields. At very low temperatures, η σ ( T ) is found in pretty good agreement with the linear temperature-dependence of Berezinskii's spin-wave approximation. We also show some evidence that there are no noticeable logarithmic corrections to the behaviour of the order parameter density profile at the Kosterlitz–Thouless (KT) transition temperature, while these corrections exist for the correlation function. At the KT transition the value η σ ( T KT ) = 1/4 is accurately recovered. The exponent associated with the surface correlations is similarly obtained after a slight modification of the boundary conditions: the correlation function is computed with free BC, and the profile with mixed fixed–free BC. It exhibits a monotonic behaviour with temperature, starting linearly according to the spin-wave approximation and increasing up to a value η ∥ ( T KT ) ≃ 1/2 at the Kosterlitz–Thouless transition temperature. The thermal exponent η ε ( T ) is also computed and we give some evidence that it keeps a constant value in agreement with the marginality condition of the temperature field below the KT transition.
Abstract In this paper, we thoroughly examined the Berezinskii-Kosterlitz-Thouless (BKT) phase transition in the two-dimensional XY model on the honeycomb lattice. To address its thermodynamical behavior, we combined standard numerical Monte Carlo simulations with the simulated annealing (SA) protocol and entropic simulations based on the Wang-Landau (WL) algorithm. The transition temperature was determined using the second (ϒ) and fourth-order (ϒ4) helicity modulus as the order parameter. Our best finite-size scaling analysis suggests T BKT = 0.575(8) from SA and T BKT = 0.576(3) from WL. These values deviate significantly from the expected theoretical value of 1 / 2 . We believe that this discrepancy suggests that the theoretical assumptions regarding the analytical calculation may need to be revisited. Additionally, we calculated the vortex density and the formation energy of the vortex-antivortex pairs, where the obtained vortex formation energy is 2μ = 5.80(12). Upon comparison with the square lattice, our results support the notion of instability of the honeycomb lattice to support the spin long-range order and give additional backing to the critical behavior we found.
The anisotropic quantum spin- 1 2 XY model on a linear chain was solved by Lieb, Schultz, and Mattis [] and shown to display a continuous quantum phase transition at the O(2) symmetric point separating two gapped phases with competing Ising long-range order. For the square lattice, the following is known. The two competing Ising ordered phases extend to finite temperatures, up to a boundary where a transition to the paramagnetic phase occurs, and meet at the O(2) symmetric critical line along the temperature axis that ends at a tricritical point at the Berezinskii-Kosterlitz-Thouless transition temperature where the two competing phases meet the paramagnetic phase. We show that the first-order zero-temperature (quantum) phase transition that separates the competing phases as a function of the anisotropy parameter is smoothed by thermal fluctuations into deconfined classical criticality.
We study the critical and thermodynamic properties of an S=1/2 two-dimensional quantum XXZ antiferromagnet with weak easy-plane anisotropy using the continuous-time quantum Monte Carlo method based on the loop-cluster algorithm. A detailed finite-size scaling analysis reveals that the model displays a finite temperature Berezinskii–Kosterlitz–Thouless (BKT) transition for anisotropy as small as 10−3, the dominant isotropic coupling. For larger anisotropy (10−2) we single out relevant features of anisotropic behavior that anticipate the BKT transition. Implications for experimental observation of a BKT phase in real magnets are discussed.
Abstract The Berezinskii–Kosterlitz–Thouless (BKT) transition of the two-dimensional XYXYXY model on the honeycomb lattice is investigated using both the techniques of Neural network (NN) and Monte Carlo simulations. It is demonstrated in the literature that, with certain plausible assumptions, the associated critical temperature TBKT,HT_{\text{BKT,H}}TBKT,H​ is found to be 1/2{1}/{\sqrt{2}}1/2​ exactly. Surprisingly, the value of TBKT,HT_{\text{BKT,H}}TBKT,H​ obtained from our NN calculations is 0.572(3), which deviates significantly from 1/2{1}/{\sqrt{2}}1/2​. In addition, based on the helicity modulus, the TBKT,HT_{\text{BKT,H}}TBKT,H​ determined is 0.576(4), agreeing well with that resulting from the NN estimation. It will be interesting to carry out a more detailed analytic calculation to obtain a theoretical value consistent with the numerical result reached here.