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Geometric Entanglement in Valance-Bond-Solid state

H. T. Cui, C.M. Wang, S. Z. Yuan

Published Aug 17, 2016• 2 citations
OpenAlex

Abstract

Multipartite entanglement, measured by the geometric entanglement(GE), is discussed for integer spin Valance-Bond-Solid (VBS) state respectively with periodic boundary condition(PBC) and open boundary condition(OBC) in this paper. The optimization in the definition of geometric entanglement can be reduced greatly by exploring the symmetry of VBS state, and then the fully separable state can be determined explicitly. Numerical evaluation for GE by the random simulation is also implemented in order to demonstrate the validity of the reductions. Our calculations show that GE is saturated by a finite value with the increment of particle number, that means that the total entanglement for VBS state would be divergent under the thermodynamic limit. Moreover it is found that the scaling behavior of GE with spin number $s$ is fitted as $\alpha\log (s+\tfrac{\beta}{s}+\gamma)+\delta$, in which the values of the parameters $\alpha, \beta, \gamma, \sigma$ are only dependent on the parity of spin $s$. A comparison with entanglement entropy of VBS state is also made, in order to demonstrate the essential differences between multipartite and bipartite entanglement in this model.

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Multipartite entanglement, measured by the geometric entanglement(GE), is discussed for integer spin Valance-Bond-Solid (VBS) state respectively with periodic boundary condition(PBC) and open boundary condition(OBC) in this paper. The optimization in the definition of geometric entanglement can be reduced greatly by exploring the symmetry of VBS state, and then the fully separable state can be determined explicitly. Numerical evaluation for GE by the random simulation is also implemented in order to demonstrate the validity of the reductions. Our calculations show that GE is saturated by a finite value with the increment of particle number, that means that the total entanglement for VBS state would be divergent under the thermodynamic limit. Moreover it is found that the scaling behavior of GE with spin number sss is fitted as αlog⁡(s+βs+γ)+δ\alpha\log (s+\tfrac{\beta}{s}+\gamma)+\deltaαlog(s+sβ​+γ)+δ, in which the values of the parameters α,β,γ,σ\alpha, \beta, \gamma, \sigmaα,β,γ,σ are only dependent on the parity of spin sss. A comparison with entanglement entropy of VBS state is also made, in order to demonstrate the essential differences between multipartite and bipartite entanglement in this model.
This article reviews the quantum entanglement in Valence-Bond-Solid (VBS) states defined on a lattice or a graph. The subject is presented in a self-contained and pedagogical way. The VBS state was first introduced in the celebrated paper by I. Affleck, T. Kennedy, E. H. Lieb and H. Tasaki (abbreviation AKLT is widely used). It became essential in condensed matter physics and quantum information (measurement-based quantum computation). Many publications have been devoted to the subject. Recently entanglement was studied in the VBS state. In this review, we start with the definition of a general AKLT spin chain and the construction of VBS ground state. In order to study entanglement, a block subsystem is introduced and described by the density matrix. Density matrices of 1-dimensional models are diagonalized and the entanglement entropies (the von Neumann entropy and Rényi entropy) are calculated. In the large block limit, the entropies also approach finite limits. Study of the spectrum of the density matrix led to the discovery that the density matrix is proportional to a projector.
We investigate the entropy of a subsystem for the valence-bond solid (VBS) state with general open boundary conditions. This entropy quantifies the entanglement of a bipartite state, and the effects of the boundary on the entropy are studied. We show that the effect of the boundary on the entropy decays exponentially fast in the distance between the subsytem considered and the boundary sites. Further, we show that as the size of the subsystem increases, its entropy exponentially approaches the summation of the entropies of the two ends, the exponent being related to the size. We also study the entanglement between two spins. Curiously, while the boundary operators decrease the entropy of L spins, they increase the entanglement between two spins.
Starting with the valence-bond solid (VBS) ground state of the 1D AKLT Hamiltonian, we make a partition of the system in two subsystems A and B , where A is a block of L consecutive spins and B is its complement. In that setting we compute the partial transpose density matrix with respect to A , . We obtain the spectrum of the transposed density matrix of the VBS pure system. Subsequently, we define two disjoint blocks, A and B , containing L A and L B spins, respectively, separated by L sites. Tracing away the spins which do not belong to A ∪ B , we find an expression for the reduced density matrix of the A and B blocks, ρ( A , B ). With this expression (in the thermodynamic limit), we compute the entanglement spectrum and other several entanglement measures, such as the purity P = tr(ρ( A , B ) 2 ), the negativity and the mutual entropy.
Geometric entanglement (GE), as a measure of multipartite entanglement, has been investigated as a universal tool to detect phase transitions in quantum many-body lattice models. In this paper we outline a systematic method to compute GE for two-dimensional (2D) quantum many-body lattice models based on the translational invariant structure of infinite projected entangled pair state (iPEPS) representations. By employing this method, the qqq-state quantum Potts model on the square lattice with $q\ensuremath{\in}{2,3,4,5}$ is investigated as a prototypical example. Further, we have explored three 2D Heisenberg models: the antiferromagnetic spin-1/2 XXXXXXXXX and anisotropic XYXXYXXYX models in an external magnetic field, and the antiferromagnetic spin-1 XXZXXZXXZ model. We find that continuous GE does not guarantee a continuous phase transition across a phase transition point. We observe and thus classify three different types of continuous GE across a phase transition point: (i) GE is continuous with maximum value at the transition point and the phase transition is continuous, (ii) GE is continuous with maximum value at the transition point but the phase transition is discontinuous, and (iii) GE is continuous with nonmaximum value at the transition point and the phase transition is continuous. For the models under consideration, we find that the second and the third types are related to a point of dual symmetry and a fully polarized phase, respectively.
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.