Scaling Theories of Kosterlitz-Thouless Phase Transitions

Abstract

We propose scaling theories for Kosterlitz-Thouless (KT) phase transitions on the basis of the hallmark exponential growth of their correlation length. Finite-size scaling, finite-entanglement scaling, short-time critical dynamics, and finite-time scaling, as well as some of their combinations are studied. Relaxation times of both a usual power-law and an unusual power-law with a logarithmic factor are considered. Finite-size and finite-entanglement scaling forms somehow similar to a frequently employed ansatz are presented. The Kibble-Zurek scaling of topological defect density for a linear driving across the KT transition point is investigated in detail. An implicit equation for a rate exponent in the theory is derived and the exponent varies with the distance from the critical point and the driving rate consistent with relevant experiments. To verify the theories, we utilize the KT phase transition of a one-dimensional Bose-Hubbard model. The infinite time-evolving-block-decimation algorithm is employed to solve numerically the model for finite bond dimensions. Both a correlation length and an entanglement entropy in imaginary time and only the entanglement entropy in real-time driving are computed. Both the short-time critical dynamics in imaginary time and the finite-time scaling in real-time driving, both including the finite bond dimension, for the measured quantities are found to describe the numerical results quite well via surface collapses. The critical point is also estimated and confirmed to be $0.302(1)$ at the infinite bond dimension on the basis of the scaling form.

16 pages, 9 figures. A mistake in a finite-size scaling form corrected

References in corpus (15)

• The density-matrix renormalization group in the age of matrix product states
• Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems
• Classical simulation of infinite-size quantum lattice systems in one spatial dimension
• Scaling of entanglement support for Matrix Product States
• Observation of scale invariance and universality in two-dimensional Bose gases
• Emergence of coherence and the dynamics of quantum phase transitions
• Time evolution algorithms for Matrix Product States and DMRG
• Critical Point of an Interacting Two-Dimensional Atomic Bose Gas
• The trapped two-dimensional Bose gas: from Bose-Einstein condensation to Berezinskii-Kosterlitz-Thouless physics
• Scaling of the gap, fidelity susceptibility, and Bloch oscillations across the superfluid to Mott insulator transition in the one-dimensional Bose-Hubbard model
• Large-scale Monte Carlo simulation of two-dimensional classical XY model using multiple GPUs
• Dynamics of a quantum phase transition in the 1D Bose-Hubbard model: excitations and correlations induced by a quench
• Gap scaling at Berezinskii-Kosterlitz-Thouless quantum critical points in one-dimensional Hubbard and Heisenberg models
• Renormalization-group flow and asymptotic behaviors at the Berezinskii-Kosterlitz-Thouless transitions
• High-temperature expansion of the magnetic susceptibility and higher moments of the correlation function for the two-dimensional XY model

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• Nonequilibrium Dynamics of Dirac Quantum Criticality in Imaginary Time
• Theory of Critical Phenomena with Long-Range Temporal Interaction
• Theory of Critical Phenomena with Memory
• Quantum phases of the biased two-chain-coupled Bose-Hubbard Ladder
• Relaxation Critical Dynamics in Measurement-induced Phase Transitions
• Berezinskii-Kosterlitz-Thouless Renormalization Group Flow at a Quantum Phase Transition
• Short-time critical dynamics in the classical cubic dimer model