Multipartite entanglement measures and quantum criticality from matrix and tensor product states

Huang, Ching-Yu; Lin, Feng Li

doi:10.1103/physreva.81.032304

Cited by 42 publications

(30 citation statements)

References 47 publications

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“…We consider that this log 2 L dependence on n kink is closely related to the ln L dependence on S in Eq. (16). It should be noted that only few states are concerned with the long-range spin correlation of the critical behavior in our expression.…”

Section: Eigenvalue Distribution At T Cmentioning

confidence: 99%

“…Furthermore, the eigenvalue distribution has strong T dependence, particularly near T c as shown later. In the following sections, we examine the physical origins of the scaling equation (16) at and above T c , separately.…”

Section: A Scaling Relation For the Entropy At T T Cmentioning

confidence: 99%

“…Numerical tests of this prediction have been performed in models with different central charges, respectively: free fermions (c = 1), the transverse-field Ising chain (c = 1/2), the S = 1 XXZ chain with uniaxial anisotropy (c = 1), and the S = 1 Heisenberg chain with biquadratic term (c = 3/2, 2) [13][14][15][16]. For c = 1/2, S χ ∼ (1/6) ln χ .…”

Section: Introductionmentioning

confidence: 99%

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Holographic entanglement entropy in Suzuki-Trotter decomposition of spin systems

Matsueda

2012

Phys. Rev. E

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In quantum spin chains at criticality, two types of scaling for the entanglement entropy exist: one comes from conformal field theory (CFT), and the other is for entanglement support of matrix product state (MPS) approximation. On the other hand, the quantum spin-chain models can be mapped onto two-dimensional (2D) classical ones by the Suzuki-Trotter decomposition. Motivated by the scaling and the mapping, we introduce information entropy for 2D classical spin configurations as well as a spectrum, and examine their basic properties in the Ising and the three-state Potts models on the square lattice. They are defined by the singular values of the reduced density matrix for a Monte Carlo snapshot. We find scaling relations of the entropy compatible with the CFT and the MPS results. Thus, we propose that the entropy is a kind of "holographic" entanglement entropy. At T(c), the spin configuration is fractal, and various sizes of ordered clusters coexist. Then, the singular values automatically decompose the original snapshot into a set of images with different length scales, respectively. This is the origin of the scaling. In contrast to the MPS scaling, long-range spin correlation can be described by only few singular values. Furthermore, the spectrum, which is a set of logarithms of the singular values, also seems to be a holographic entanglement spectrum. We find multiple gaps in the spectrum, and in contrast to the topological phases, the low-lying levels below the gap represent spontaneous symmetry breaking. These contrasts are strong evidence of the dual nature of the holography. Based on these observations, we discuss the amount of information contained in one snapshot.

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Section: Eigenvalue Distribution At T Cmentioning

confidence: 99%

Section: A Scaling Relation For the Entropy At T T Cmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Holographic entanglement entropy in Suzuki-Trotter decomposition of spin systems

Matsueda

2012

Phys. Rev. E

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“…7 The GE has also recently gained interest in many-body and condensed-matter physics. [8][9][10][11] Since it generally scales linearly with the volume ͑N͒ of the system one also defines the GE per site…”

Section: Introductionmentioning

confidence: 99%

“…[9][10][11] In this Rapid Communication we focus on the case of the ground state ͉⌿͘ = ͉G͘ of the Hamiltonian of a periodic spin-1/2 XXZ chain…”

Section: Introductionmentioning

confidence: 99%

Geometric entanglement and Affleck-Ludwig boundary entropies in criticalXXZand Ising chains

2010

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We study the geometrical entanglement of the XXZ chain in its critical regime. Recent numerical simulations ͓Q.-Q. Shi, R. Orús, J. O. Fjaerestad, and H.-Q. Zhou, New J. Phys. 12, 025008 ͑2010͔͒ indicate that it scales linearly with system size, and that the first subleading correction is constant, which was argued to be possibly universal. In this work, we confirm the universality of this number, by relating it to the Affleck-Ludwig boundary entropy corresponding to a Neumann boundary condition for a free compactified field. We find that the subleading constant is a simple function of the compactification radius, in agreement with the numerics. As a further check, we compute it exactly on the lattice at the XX point. We also discuss the case of the Ising chain in transverse field and show that the geometrical entanglement is related to the Affleck-Ludwig boundary entropy associated to a ferromagnetic boundary condition.

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Tensor Network Contractions

Ran

Tirrito

Peng

et al. 2020

Lecture Notes in Physics

119

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PrefaceTensor network (TN), a young mathematical tool of high vitality and great potential, has been undergoing extremely rapid developments in the last two decades, gaining tremendous success in condensed matter physics, atomic physics, quantum information science, statistical physics, and so on. In this lecture notes, we focus on the contraction algorithms of TN as well as some of the applications to the simulations of quantum many-body systems. Starting from basic concepts and definitions, we first explain the relations between TN and physical problems, including the TN representations of classical partition functions, quantum many-body states (by matrix product state, tree TN, and projected entangled pair state), time evolution simulations, etc. These problems, which are challenging to solve, can be transformed to TN contraction problems. We present then several paradigm algorithms based on the ideas of the numerical renormalization group and/or boundary states, including density matrix renormalization group, time-evolving block decimation, coarsegraining/corner tensor renormalization group, and several distinguished variational algorithms. Finally, we revisit the TN approaches from the perspective of multilinear algebra (also known as tensor algebra or tensor decompositions) and quantum simulation. Despite the apparent differences in the ideas and strategies of different TN algorithms, we aim at revealing the underlying relations and resemblances in order to present a systematic picture to understand the TN contraction approaches. v AcknowledgementsWe are indebted to

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Multipartite entanglement measures and quantum criticality from matrix and tensor product states

Cited by 42 publications

References 47 publications

Holographic entanglement entropy in Suzuki-Trotter decomposition of spin systems

Holographic entanglement entropy in Suzuki-Trotter decomposition of spin systems

Geometric entanglement and Affleck-Ludwig boundary entropies in criticalXXZand Ising chains

Tensor Network Contractions

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