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arXiv:2608.03195

Fluctuations of topological charges in two-dimensional classical Heisenberg model through high-temperature and low-temperature expansions

Shan-Chang Tang

Published Aug 4, 2026• 0 citations
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Abstract

It is well known that KT transition in 2d XY model is driven by the binding and unbinding of topological defects, which can be characterized by the fluctuation of topological charges inside a region. We extend the idea into the 2d Heisenberg model and calculate the fluctuation through high-temperature and low-temperature expansion respectively. It is found that the fluctuation of topological charges is proportional to the area of the region at high temperatures while obeys the perimeter law at low temperatures.

DOI
10.48550/arxiv.2608.03195

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It is well known that KT transition in 2d XY model is driven by the binding and unbinding of topological defects, which can be characterized by the fluctuation of topological charges inside a region. We extend the idea into the 2d Heisenberg model and calculate the fluctuation through high-temperature and low-temperature expansion respectively. It is found that the fluctuation of topological charges is proportional to the area of the region at high temperatures while obeys the perimeter law at low temperatures.
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A numerical method to study defect-mediated phase transitions is proposed, which combines a standard Monte Carlo technique with a particular type of boundary conditions. The method is applied to the two-dimensional XY (plane rotator) model as well as to the two-dimensional SO(3) Heisenberg model which models a triangular-lattice Heisenberg antiferromagnet. Our results give a direct numerical support for the occurrence of a topological transition in these systems driven by the vortex binding-unbinding.
The dissipative XYXYXY model in two spatial dimensions belongs to a new universality class of quantum-critical phenomena with the remarkable property of the decoupling of the critical fluctuations in space and time. We have shown earlier that the quantum-critical point is driven by proliferation in time of topological configurations that we termed warps. We show here that a warp may be regarded as a configuration of a monopoles surrounded symmetrically by antimonopoles so that the total charge of the configuration is zero. Therefore the interaction with other warps is local in space. They, however, interact with other warps at the same spatial point logarithmically in time. As a function of dissipation warps unbind leading to a quantum phase transition. The critical fluctuations are momentum independent but have power-law correlations in time.
We prove the existence of a finite temperature Z 2 phase transition for the topological charge ordering within the fully frustrated XY model. Our method enables a proof of the topological charge confinement within the conventional XY models from a rather general vista. One of the complications that we face is the non-exact equivalence of the continuous (angular) XY model and its discrete topological charge dual. In reality, the energy spectra of the various topological sectors are highly nested, much unlike that suggested by the discrete dual models. We surmount these difficulties by exploiting the reflection positivity symmetry that this periodic flux phase model possesses. The techniques introduced here may prove binding of topological charges in numerous models and might be applied to examine transitions associated with various topological defects, e.g., the confinement of disclinations in the isotropic to nematic transition.
The critical behavior of 2D magnetic easy-plane systems has mainly been studied by the classical XY model. However, the z components of the spins have to be considered in order to describe real systems, and their fluctuations cannot positively be neglected when quantum effects are to be included, quantum spins being intrinsically three-component objects. Therefore, Monte Carlo simulations are performed for the Heisenberg ferromagnet with easy-plane anisotropy (XXZ model) on a two-dimensional square lattice with a twofold aim: first, to obtain accurate quantitative results about the critical behavior of the classical model, showing the relevant role played by the out-of-plane fluctuations; second, to open the way for approaching the quantum thermodynamics by means of the effective Hamiltonian method that reduces the quantum thermodynamics of the XXZ ferromagnet to the investigation of an effective classical model with temperature-dependent renormalized interaction parameters. Specific heat, magnetic susceptibility, and correlation length are calculated in the critical region for lattice sizes up to 128×128. These quantities preserve the Kosterlitz–Thouless behavior of the XY model. For the transition temperature of the classical XX0 model we obtain the estimate kBTc/(JS̃2)=0.69±0.01.
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.