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arXiv:1507.01471 · Physical review. E

Fine structure of the entanglement entropy in the O(2) model

Li‐Ping Yang, Yuzhi Liu A, Haiyuan Zou, Z. Y. Xie, Yannick Meurice

Published Jan 22, 2016• 35 citations

Abstract

We compare two calculations of the particle density in the superfluid phase of the O(2) model with a chemical potential μ in 1+1 dimensions. The first relies on exact blocking formulas from the Tensor Renormalization Group (TRG) formulation of the transfer matrix. The second is a worm algorithm. We show that the particle number distributions obtained with the two methods agree well. We use the TRG method to calculate the thermal entropy and the entanglement entropy. We describe the particle density, the two entropies and the topology of the world lines as we increase μ to go across the superfluid phase between the first two Mott insulating phases. For a sufficiently large temporal size, this process reveals an interesting fine structure: the average particle number and the winding number of most of the world lines in the Euclidean time direction increase by one unit at a time. At each step, the thermal entropy develops a peak and the entanglement entropy increases until we reach half-filling and then decreases in a way that approximately mirrors the ascent. This suggests an approximate fermionic picture.

DOI: 10.1103/physreve.93.012138

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A bstract We propose a method to compute the entanglement entropy (EE) using the tensor renormalization group (TRG) method. The reduced density matrix of a d -dimensional quantum system is represented as a ( d + 1)-dimensional tensor network. We develop an explicit algorithm for d = 1 that enables the calculation of EE for single-interval subsystems of arbitrary size. We test our method in two-dimensional tensor network of the Ising model. The central charge is obtained as c = 0 . 49997(8) for D = 96, which agrees with the theoretical prediction within an error, demonstrating the accuracy and reliability of our proposed method.

We discuss the reformulation of the O(2) model with a chemical potential and the Abelian Higgs model on a 1+1 dimensional space-time lattice using the Tensor Renormalization Group (TRG) method. The TRG allows exact blocking and connects smoothly the classical Lagrangian approach to the quantum Hamiltonian approach. We calculate the entanglement entropy in the superfluid phase of the O(2) model and show that it approximately obeys the logarithmic Calabrese-Cardy scaling obtained from Conformal Field Theory (CFT). We calculate the Polyakov loop in the Abelian Higgs model and discuss the possibility of a deconfinement transition at finite volume. We propose Bose-Hubbard Hamiltonians implementable on optical lattices as quantum simulators for CFT models.

We report on tensor renormalization group calculations of entanglement entropy in one-dimensional quantum systems. The reduced density matrix of a Gibbs state can be represented as a

1 + 1

-dimensional tensor network, which is analogous to the tensor network representation of the partition function. The HOTRG method is used to approximate the reduced density matrix for arbitrary subsystem sizes, from which we obtain the entanglement entropy. We test our method in the quantum Ising model and obtain the entanglement entropy of the ground state by taking the size of time direction to infinity. The central charge

c

is obtained as

c = 0.49997(8)

for a bond dimension

D_\text{cut}=96

, which agrees with the theoretical value

c=1/2

within the error.

We study (1+1)-dimensional O(3) nonlinear sigma model using the tensor renormalization group method with the infinite limit of the bond dimension

D_{\rm cut}\rightarrow \infty

. At the vanishing chemical potential

μ=0

, we investigate the von Neumann and Rényi types of entanglement entropies. The central charge is determined to be

c=1.97(9)

by using the asymptotic scaling properties of the entropies. We also examine the consistency between two entropies. In the finite density region with

μ\ne 0

, where this model suffers from the sign problem in the standard Monte Carlo approach, we investigate the properties of the quantum phase transition. We determine the transition point

μ_{\rm c}

and the critical exponent of the correlation length

ν

from the

μ

dependence of the number density in the thermodynamic limit. The dynamical critical exponent

z

is also extracted from the scaling behavior of the temporal correlation length as a function of

μ

. This is the first successful calculation of the dynamical critical exponent with the TRG method.

We study (1+1)-dimensional O(3) nonlinear sigma model using the tensor renormalization group method with the infinite limit of the bond dimension

D_{\rm cut}\rightarrow \infty

. At the vanishing chemical potential

\mu=0

, we investigate the von Neumann and R\'enyi types of entanglement entropies. The central charge is determined to be

c=1.97(9)

by using the asymptotic scaling properties of the entropies. We also examine the consistency between two entropies. In the finite density region with

\mu\ne 0

, where this model suffers from the sign problem in the standard Monte Carlo approach, we investigate the properties of the quantum phase transition. We determine the transition point

\mu_{\rm c}

and the critical exponent of the correlation length

\nu

from the

\mu

dependence of the number density in the thermodynamic limit. The dynamical critical exponent

z

is also extracted from the scaling behavior of the temporal correlation length as a function of

\mu

. This is the first successful calculation of the dynamical critical exponent with the TRG method.