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ひねり境界条件を使った多重臨界点付近の相転移線の計算手法

守屋, 俊志 MORIYA, Shunji モリヤ, シュンジ 九州大学

2022.03.23

概要

A point where several critical lines intersect is called a multicritical point. Near such a point, multiple critical phenomena interfere and a finite-size correction becomes large. Therefore, conventional methods cannot be applied near the multicritical point. We propose a new method to numerically calculate a transition point for a quantum spin chain near the multicritical point.

We treat a bond-alternating (BA) XXZ model for S = 1/2, 1, 3/2. In a ground-state phase diagram of this model, a 2D Gaussian universality transition line bifurcates into two 2D Ising universality transition lines, which make a multicritical point. At this point, Berezinskii-Kosterlitz-Thouless transition occurs, where the correlation length diverges singularly.

To deal with a 2D Ising universality, we review a transverse field 1D quantum Ising (TFI) model, corresponding to the classical 2D Ising model with transfer matrix. A relation between a disorder phase and an order phase is known as the Kramers-Wannier duality transformation. We find a proper duality transformation that is exact in the finite-size TFI model with a periodic and anti-periodic boundary condition. This shows that the energies for the two boundary conditions are crossing at the transition point. A free fermion field theory is derived from the critical Ising model by taking a continuum limit. From a conformal field theory, we verify that the energy-crossing is realized in the 2D Ising universality class, not only in the TFI model.

We apply our method to the BA XXZ model. The energy-crossing happens in boundary conditions that are twisted around a z-axis and y-axis. In an anisotropic limit, two energies in a finite-size are crossing at the transition point since the BA XXZ model is identical to the TFI model. on the other hand, at the multicritical point, the finite-size correction vanishes by an isotropy and a twist translation symmetry. Therefore, near the multicritical point, our method has a smaller finite-size correction.

In this paper, we also review the 2D Gaussian universality. By a bosonization, the BA XXZ model can be transformed to a phase Hamiltonian composed of boson operators. The degenerate energies for a twisted boundary condition are split by a perturbation around a Gaussian fixed point.

By our method, we numerically calculate a transition point of the BA XXZ model for S = 1/2, 1, 3/2. As expected, the finite-size correction of numerical results becomes very small near the multicritical point.

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