Abstract
Berezinskii-Kosterlitz-Thouless transition of the classical XY model is re-investigated, combining the Tensor Network Renormalization (TNR) and the Level Spectroscopy method based on the finite-size scaling of the Conformal Field Theory. By systematically analyzing the spectrum of the transfer matrix of the systems of various moderate sizes which can be accurately handled with a finite bond dimension, we determine the critical point removing the logarithmic corrections. This improves the accuracy by an order of magnitude over previous studies including those utilizing TNR. Our analysis also gives a visualization of the celebrated Kosterlitz Renormalization Group flow based on the numerical data.
Highlights
The Berezinskii-Kosterlitz-Thouless (BKT) transition was historically the first example of topological phase transitions, which is an essential concept in physics [1]
We demonstrate a successful implementation of Level Spectroscopy on the classical 2D XY model, based on the Tensor Network Renormalization (TNR) scheme [13,14,15,16,17,18]
The BKT transition is described in terms of two marginal couplings yK and yV, and the transition point can be identified with yK = yV where a hidden SU(2) symmetry emerges
Summary
The Berezinskii-Kosterlitz-Thouless (BKT) transition was historically the first example of topological phase transitions, which is an essential concept in physics [1]. The effective field theory in terms of boson field is equivalent to Eq (4), with the replacement 2φ → φ and θ → 2θ This implies that the fixed point Hamiltonian for the BKT transition point has the Luttinger parameter K = 2 instead of K = 1/2. It appears that the effective field theory in this case no longer has the SU(2) symmetry and the Level Spectroscopy may not apply. In order to apply the Level Spectroscopy to the 2D classical XY model, we need to calculate the spectrum of the transfer matrix (which corresponds to the energy levels of 1D quantum Hamiltonian), under the periodic and twisted boundary conditions
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