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arXiv:1106.2110 · New Journal of Physics

Geometric entanglement from matrix product state representations

Bing-Quan Hu, Xi-Jing Liu, Jin-Hua Liu, Huan-Qiang Zhou

Published Sep 30, 2011• 8 citations
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Abstract

An efficient scheme for computing the geometric entanglement (GE) per lattice site for quantum many-body systems on a periodic finite-size chain is proposed in the context of a tensor network algorithm based on matrix product state representations. It has been systematically tested for three prototypical critical quantum spin chains, which belong to the same Ising universality class. The simulation results lend strong support to the previous claim (Shi et al 2010 New J. Phys. 12 025008; Stéphan et al 2010 Phys. Rev. B 82 180406R) that the leading finite-size correction to the GE per lattice site is universal, with its remarkable connection to the celebrated Affleck–Ludwig boundary entropy corresponding to a conformally invariant boundary condition.

DOI
10.1088/1367-2630/13/9/093041

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An efficient scheme for computing the geometric entanglement (GE) per lattice site for quantum many-body systems on a periodic finite-size chain is proposed in the context of a tensor network algorithm based on matrix product state representations. It has been systematically tested for three prototypical critical quantum spin chains, which belong to the same Ising universality class. The simulation results lend strong support to the previous claim (Shi et al 2010 New J. Phys. 12 025008; Stéphan et al 2010 Phys. Rev. B 82 180406R) that the leading finite-size correction to the GE per lattice site is universal, with its remarkable connection to the celebrated Affleck–Ludwig boundary entropy corresponding to a conformally invariant boundary condition.
Recently, the finite-size corrections to the geometrical entanglement per lattice site in the spin-1/2 chain have been numerically shown to scale inversely with system size, and its prefactor b has been suggested to be possibly universal [Q-Q. Shi et al., New J. Phys. 12, 025008 (2010)]. As possible evidence of its universality, the numerical values of the prefactors have been confirmed analytically by using the Affleck-Ludwig boundary entropy with a Neumann boundary condition for a free compactified field [J-M. Stephan et al., Phys. Rev. B 82, 180406(R) (2010)]. However, the Affleck-Ludwig boundary entropy is not unique and does depend on conformally invariant boundary conditions. Here, we show that a unique Affleck-Ludwig boundary entropy corresponding to a finitesize correction to the geometrical entanglement per lattice site exists and show that the ratio of the prefactor b to the corresponding minimum groundstate degeneracy gmin for the Affleck- Ludwig boundary entropy is a constant for any critical region of the spin-1 XXZ system with the single-ion anisotropy, i.e., b/(2 log2 g min ) = −1. Previously studied spin-1/2 systems, including the quantum three-state Potts model, have verified the universal ratio. Hence, the inverse finite-size correction to the geometrical entanglement per lattice site and its prefactor b are universal for one-dimensional critical systems.
The global geometric entanglement (GE) is studied in the context of newly developed tensor network algorithms for finite systems. For one-dimensional quantum spin systems it is found that, at criticality, the leading finite-size correction to the global GE per site behaves as b / n , where n is the size of the system and b a given coefficient. Our conclusion is based on the computation of the GE per spin for the quantum Ising model in a transverse magnetic field and for the spin-1/2 XXZ model. We also discuss the possibility of coefficient b being universal.
We study the geometrical entanglement of the XXZXXZXXZ chain in its critical regime. Recent numerical simulations [Q.-Q. Shi, R. Or\'us, J. O. Fj\ae{}restad, and H.-Q. Zhou, New J. Phys. 12, 025008 (2010)] 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 XXXXXX 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.
The geometric entanglement per lattice site, as a holistic measure of the multipartite entanglement, serves as a universal marker to detect quantum phase transitions in quantum many-body systems. However, it is very difficult to compute the geometric entanglement due to the fact that it involves a complicated optimization over all the possible separable states. In this paper, we propose a systematic method to efficiently compute the geometric entanglement per lattice site for quantum many-body lattice systems in two spatial dimensions in the context of a newly-developed tensor network algorithm based on an infinite projected entangled pair state representation. It is tested for quantum Ising model in a transverse magnetic field and anisotropic spin 1/2 anti-ferromagnetic XYX model in an external magnetic field on an infinite-size square lattice. In addition, the geometric entanglement per lattice site is able to detect the so-called factorizing field. Our results are in a quantitative agreement with Quantum Monte Carlo simulations.
In this paper the geometric entanglement (GE) of systems in one spatial dimension (1D) and in the thermodynamic limit is analyzed focusing on two aspects. First, we reexamine the calculation of the GE for translation-invariant matrix product states (MPSs) in the limit of infinite system size. We obtain a lower bound to the GE which collapses to an equality under certain sufficient conditions that are fulfilled by many physical systems, such as those having unbroken space (P) or space-time (PT) inversion symmetry. Our analysis justifies the validity of several derivations carried out in previous works. Second, we derive scaling laws for the GE per site of infinite-size 1D systems with correlation length ξ≫1\xi \gg 1ξ≫1. In the case of MPSs, we combine this with the theory of finite-entanglement scaling, allowing to understand the scaling of the GE per site with the MPS bond dimension at conformally invariant quantum critical points.
We investigate the use of matrix product states (MPS) to approximate ground states of critical quantum spin chains with periodic boundary conditions (PBC). We identify two regimes in the (N,D)(N,D)(N,D) parameter plane, where NNN is the size of the spin chain and DDD is the dimension of the MPS matrices. In the first regime, MPS can be used to perform finite size scaling (FSS). In the complementary regime, the MPS simulations show instead the clear signature of finite entanglement scaling (FES). In the thermodynamic limit (or large NNN limit), only MPS in the FSS regime maintain a finite overlap with the exact ground state. This observation has implications on how to correctly perform FSS with MPS as well as on the performance of recent MPS algorithms for systems with PBC. It also gives clear evidence that critical models can actually be simulated very well with MPS by using the right scaling relations; in the appendixes, we give an alternative derivation of the result of Pollmann et al. [Phys. Rev. Lett. 102, 255701 (2009)] relating the bond dimension of the MPS to an effective correlation length.