Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0901.2863 · New Journal of Physics

Finite-size geometric entanglement from tensor network algorithms

Qian-Qian Shi, Román Orús, John Ove Fjærestad, Huan-Qiang Zhou

Published Feb 26, 2010• 38 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1367-2630/12/2/025008

Related papers

Geometric entanglement from matrix product state representations

Bing-Quan Hu, Xi-Jing Liu, Jin-Hua Liu +1 more

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.

arXiv:1106.2110•New Journal of Physics•Sep 30, 2011•8 citations

Universality of finite-size corrections to geometrical entanglement in one-dimensional quantum critical systems

Xijing Liu, Bing-Quan Hu, Sam Young Cho +2 more

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.

Journal of the Korean Physical Society•Oct 1, 2016•1 citation

Geometric entanglement of one-dimensional systems: bounds and scalings in the thermodynamic limit

Román Orús, Tzu-Chieh Wei

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 $\xi \gg 1$. 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.

arXiv:1006.5584•Quantum Information and Computation•Jul 1, 2011•2 citations

Global geometric entanglement in transverse-field XY spin chains: finite and infinite systems

Tzu-Chieh Wei, Smitha Vishveshwara, Paul M. Goldbart

The entanglement in quantum XY spin chains of arbitrary length is investigated via the geometric measure of entanglement. The emergence of entanglement is explained intuitively from the perspective of perturbations. The model is solved exactly and the energy spectrum is determined and analyzed in particular for the lowest two levels for both finite and infinite systems. The overlaps for these two levels are calculated analytically for arbitrary number of spins. The entanglement is hence obtained by maximizing over a single parameter. The corresponding ground-state entanglement surface is then determined over the entire phase diagram, and its behavior can be used to delineate the boundaries in the phase diagram. For example, the field-derivative of the entanglement becomes singular along the critical line. The form of the divergence is derived analytically and it turns out to be dictated by the universality class controlling the quantum phase transition. The behavior of the entanglement near criticality can be understood via a scaling hypothesis, analogous to that for free energies. The entanglement density vanishes along the so-called disorder line in the phase diagram, the ground space is doubly degenerate and spanned by two product states. The entanglement for the superposition of the lowest two states is also calculated. The exact value of the entanglement depends on the specific form of superposition. However, in the thermodynamic limit the entanglement density turns out to be independent of the superposition. This proves that the entanglement density is insensitive to whether the ground state is chosen to be the spontaneously $Z_2$ symmetry broken one or not. The finite-size scaling of entanglement at critical points is also investigated from two different view points. First, the maximum in the field-derivative of the entanglement density is computed and fitted to a logarithmic dependence of the system size, thereby deducing the correlation length exponent for the Ising class using only the behavior of entanglement. Second, the entanglement density itself is shown to possess a correction term inversely proportional to the system size, with the coefficient being universal (but with different values for the ground state and the first excited state, respectively).

arXiv:1012.4114•Quantum Information and Computation•Mar 1, 2011•10 citations

An efficient computation of geometric entanglement for two-dimensional quantum lattice systems

Honglei Wang, Qian‐Qian Shi, Sheng-Hao Li +1 more

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.

arXiv:1106.2129•Jun 10, 2011•0 citations

Susceptibility of entanglement entropy: a universal indicator of quantum criticality

Sarkar, Pritam

A measure of how sensitive the entanglement entropy is in a quantum system, has been proposed and its information geometric origin is discussed. It has been demonstrated for two exactly solvable spin systems, that thermodynamic criticality is directly \textit{indicated} by finite size scaling of the global maxima and turning points of the susceptibility of entanglement entropy through numerical analysis - obtaining power laws. Analytically we have proved those power laws for $| \ λ_c(N)-λ_c^{\infty}|$ as $N\to \infty$ in the cases of finite 1D transverse field ising model (TFIM) ($λ=h$) and XY chain ($λ=γ$). The integer power law appearing for XY model has been verified using perturbation theory in $\mathcal{O}(\frac{1}{N})$ and the fractional power law appearing in the case of TFIM, is verified by an exact approach involving Chebyshev polynomials, hypergeometric functions and complete elliptic integrals. Furthermore a set of potential applications of this quantity under quantum dynamics and also for non-integrable systems, are briefly discussed. The simplicity of this setup for understanding quantum criticality is emphasized as it takes in only the reduced density matrix of appropriate rank.

arXiv:2412.02236•Dec 3, 2024•0 citations
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.
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.
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.
The entanglement in quantum XY spin chains of arbitrary length is investigated via the geometric measure of entanglement. The emergence of entanglement is explained intuitively from the perspective of perturbations. The model is solved exactly and the energy spectrum is determined and analyzed in particular for the lowest two levels for both finite and infinite systems. The overlaps for these two levels are calculated analytically for arbitrary number of spins. The entanglement is hence obtained by maximizing over a single parameter. The corresponding ground-state entanglement surface is then determined over the entire phase diagram, and its behavior can be used to delineate the boundaries in the phase diagram. For example, the field-derivative of the entanglement becomes singular along the critical line. The form of the divergence is derived analytically and it turns out to be dictated by the universality class controlling the quantum phase transition. The behavior of the entanglement near criticality can be understood via a scaling hypothesis, analogous to that for free energies. The entanglement density vanishes along the so-called disorder line in the phase diagram, the ground space is doubly degenerate and spanned by two product states. The entanglement for the superposition of the lowest two states is also calculated. The exact value of the entanglement depends on the specific form of superposition. However, in the thermodynamic limit the entanglement density turns out to be independent of the superposition. This proves that the entanglement density is insensitive to whether the ground state is chosen to be the spontaneously Z2Z_2Z2​ symmetry broken one or not. The finite-size scaling of entanglement at critical points is also investigated from two different view points. First, the maximum in the field-derivative of the entanglement density is computed and fitted to a logarithmic dependence of the system size, thereby deducing the correlation length exponent for the Ising class using only the behavior of entanglement. Second, the entanglement density itself is shown to possess a correction term inversely proportional to the system size, with the coefficient being universal (but with different values for the ground state and the first excited state, respectively).
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.
A measure of how sensitive the entanglement entropy is in a quantum system, has been proposed and its information geometric origin is discussed. It has been demonstrated for two exactly solvable spin systems, that thermodynamic criticality is directly \textit{indicated} by finite size scaling of the global maxima and turning points of the susceptibility of entanglement entropy through numerical analysis - obtaining power laws. Analytically we have proved those power laws for ∣ λc(N)−λc∞∣| \ λ_c(N)-λ_c^{\infty}|∣ λc​(N)−λc∞​∣ as N→∞N\to \inftyN→∞ in the cases of finite 1D transverse field ising model (TFIM) (λ=hλ=hλ=h) and XY chain (λ=γλ=γλ=γ). The integer power law appearing for XY model has been verified using perturbation theory in O(1N)\mathcal{O}(\frac{1}{N})O(N1​) and the fractional power law appearing in the case of TFIM, is verified by an exact approach involving Chebyshev polynomials, hypergeometric functions and complete elliptic integrals. Furthermore a set of potential applications of this quantity under quantum dynamics and also for non-integrable systems, are briefly discussed. The simplicity of this setup for understanding quantum criticality is emphasized as it takes in only the reduced density matrix of appropriate rank.