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arXiv:1808.08025 · Journal of High Energy Physics

Three-dimensional finite temperature Z2 gauge theory with tensor network scheme

Yoshinobu Kuramashi, Yusuke Yoshimura

Published Aug 1, 2019• 35 citations
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Abstract

Abstract We apply a tensor network scheme to finite temperature Z2 gauge theory in 2+1 dimensions. Finite size scaling analysis with the spatial extension up to N σ = 4096 at the temporal extension of N τ = 2, 3, 5 allows us to determine the transition temperature and the critical exponent ν at high level of precision, which shows the consistency with the Svetitsky-Yaffe conjecture.

DOI
10.1007/jhep08(2019)023

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Abstract We apply a tensor network scheme to finite temperature Z2 gauge theory in 2+1 dimensions. Finite size scaling analysis with the spatial extension up to N σ = 4096 at the temporal extension of N τ = 2, 3, 5 allows us to determine the transition temperature and the critical exponent ν at high level of precision, which shows the consistency with the Svetitsky-Yaffe conjecture.
A bstract Tensor networks offer a sign-problem-free approach to study lattice gauge theories, but extracting precise universal information associated with the deconfinement transition remains challenging. In this work, we study the deconfinement transition of (2+1)-dimensional ℤ N lattice gauge theories at finite temperature using a thermal tensor network approach, where the partition functions at finite temperature are formulated as three-dimensional tensor networks. These tensor networks are first contracted in the temporal direction, and the subsequent coarse-graining in the spatial directions yields a renormalized transfer matrix, the spectrum of which directly encodes the universal conformal field theory data. In particular, by numerically extracting the central charge and scaling dimensions, we verify that the universality class of the thermal deconfinement transition matches the prediction of the Svetitsky-Yaffe conjecture for N = 2, 3, 5. Moreover, we show that the ℤ 5 theory at finite temperature exhibits an intermediate phase with an emergent U(1) symmetry. Critical couplings are determined via Gu-Wen ratios and agree with existing Monte Carlo simulations. Finally, extrapolating these critical couplings at finite temperature enables us to determine the deconfinement transition points for N = 2, 3 at zero temperature.
We propose a scheme to perform tensor network based finite-size scaling analysis for two-dimensional classical models. In the tensor network representation of the partition function, we use higher-order tensor renormalization group (HOTRG) method to coarse grain the weight tensor. The renormalized tensor is then used to construct the approximated transfer matrix of an infinite strip of finite width. By diagonalizing the transfer matrix we obtain the correlation length, the magnetization, and the energy density, which are used in finite-size scaling analysis to determine the critical temperature and the critical exponents. As a benchmark we study the two-dimensional classical Ising model. We show that the critical temperature and the critical exponents can be accurately determined. With HOTRG bond dimension D=70D=70D=70, the absolute errors of the critical temperature Tc{T}_{c}Tc​ and the critical exponent $\ensuremath{\nu}$, $\ensuremath{\beta}$ are at the order of ${10}^{\ensuremath{-}7},{10}^{\ensuremath{-}5}$, ${10}^{\ensuremath{-}4}$, respectively. Furthermore, the results can be systematically improved by increasing the bond dimension of the HOTRG method. Finally, we study the length scale induced by the finite cutoff in bond dimension and elucidate its physical meaning in this context.
Abstract We benchmark recently proposed tensor network based finite-size scaling analysis in (Phys. Rev. B 2023 107 205123) against two-dimensional classical 3-state clock model. Due to the higher complexity of the model, more complicated crossover behavior is observed. We advocate that the crossover behavior can be understood from the perspective of finite bond dimension inducing relevant perturbation. This leads to a general strategy to best estimate the critical properties for a given set of control parameters. For the critical temperature T c , the relative error at the order of 10−7 can be reached with bond dimension D = 70. On the other hand, with bond dimension D = 60, the relative errors of the critical exponents ν , β , α are at the order of 10−2. Increasing the bond dimension to D = 90, these relative errors can be reduced at least to the order of 10−3. In all cases our results indicate that the errors can be systematically reduced by increasing the bond dimension and the stacking number.
Determination and characterization of criticality in two-dimensional (2D) quantum many-body systems is one of the most important challenges and problems in modern physics. Based on the scaling theory in one-dimensional (1D) systems and tensor networks, we propose an efficient scheme to access the criticality of 2D quantum states. We demonstrate that the boundary state of a critical infinite projected entangled pair state can be described by 1D conformal field theory, which provides a solid indicator that allows one to identify and characterize the criticality of 2D states. Our scheme is verified on the resonating valence bond (RVB) states on kagome and square lattices, where the boundary state of the honeycomb RVB is found to be described by a c=1c=1c=1 conformal field theory. We apply our scheme also to the ground state of the spin-1/2 XXZ model on honeycomb lattice, illustrating the difficulties of standard variational tensor network approaches to study the critical ground states. Our scheme is of high versatility and flexibility, and can be further applied to investigate the quantum criticality in many other phenomena, such as finite-temperature and topological phase transitions.
The variational tensor network renormalization approach to two-dimensional (2D) quantum systems at finite temperature is applied to a model suffering the notorious quantum Monte Carlo sign problem---the orbital eg{e}_{g}eg​ model with spatially highly anisotropic orbital interactions. Coarse graining of the tensor network along the inverse temperature $\ensuremath{\beta}$ yields a numerically tractable 2D tensor network representing the Gibbs state. Its bond dimension DDD---limiting the amount of entanglement---is a natural refinement parameter. Increasing DDD we obtain a converged order parameter and its linear susceptibility close to the critical point. They confirm the existence of finite order parameter below the critical temperature Tc{T}_{c}Tc​, provide a numerically exact estimate of Tc{T}_{c}Tc​, and give the critical exponents within 11%1 of the 2D Ising universality class.