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arXiv:2502.02030 · International Symposium on Lattice Field Theory (LATTICE)

Entanglement entropy by tensor renormalization group approach

Gota Tanaka, Takahiro Hayazaki, Daisuke Kadoh, Shinji Takeda

Published Feb 5, 2025• 0 citations
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Abstract

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.

DOI
10.22323/1.466.0360

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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+11 + 11+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 ccc is obtained as c=0.49997(8)c = 0.49997(8)c=0.49997(8) for a bond dimension Dcut=96D_\text{cut}=96Dcut​=96, which agrees with the theoretical value c=1/2c=1/2c=1/2 within the error.
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 have discussed the tensor-network representation of classical statistical or interacting quantum lattice models, and given a comprehensive introduction to the numerical methods we recently proposed for studying the tensor-network states/models in two dimensions. A second renormalization scheme is introduced to take into account the environment contribution in the calculation of the partition function of classical tensor-network models or the expectation values of quantum tensor-network states. It improves significantly the accuracy of the coarse-grained tensor renormalization-group method. In the study of the quantum tensor-network states, we point out that the renormalization effect of the environment can be efficiently and accurately described by the bond vector. This, combined with the imaginary-time evolution of the wave function, provides an accurate projection method to determine the tensor-network wave function. It reduces significantly the truncation error and enables a tensor-network state with a large bond dimension, which is difficult to be accessed by other methods, to be accurately determined.
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.
Density matrix renormalization group (DMRG) and its extensions in the form of matrix product states are arguably the choice for the study of one-dimensional quantum systems in the last three decades. However, due to the limited entanglement encoded in the wave-function ansatz, to maintain the accuracy of DMRG with the increase of the system size in the study of two-dimensional systems, exponentially increased resources are required, which limits the applicability of DMRG to only narrow systems. We introduce a new ansatz in which DMRG is augmented with disentanglers to encode area-law-like entanglement entropy (entanglement entropy supported in the new ansatz scales as l for an l × l system). In the new method, the O(D 3) low computational cost of DMRG is kept (with an overhead of O(d 4) and d the dimension of the physical degrees of freedom). We perform benchmark calculations with this approach on the two-dimensional transverse Ising and Heisenberg models. This new ansatz extends the power of DMRG in the study of two-dimensional quantum systems.
We carry out a systematic study of entanglement entropy in relativistic quantum field theory. This is defined as the von Neumann entropy S A = −Tr ρ A logρ A corresponding to the reduced density matrix ρ A of a subsystem A . For the case of a 1+1-dimensional critical system, whose continuum limit is a conformal field theory with central charge c , we re-derive the result of Holzhey et al when A is a finite interval of length in an infinite system, and extend it to many other cases: finite systems, finite temperatures, and when A consists of an arbitrary number of disjoint intervals. For such a system away from its critical point, when the correlation length ξ is large but finite, we show that , where is the number of boundary points of A . These results are verified for a free massive field theory, which is also used to confirm a scaling ansatz for the case of finite size off-critical systems, and for integrable lattice models, such as the Ising and XXZ models, which are solvable by corner transfer matrix methods. Finally the free field results are extended to higher dimensions, and used to motivate a scaling form for the singular part of the entanglement entropy near a quantum phase transition.