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:1805.06935 · New Journal of Physics

Sharp entanglement thresholds in the logarithmic negativity of disjoint blocks in the transverse-field Ising chain

Younes Javanmard, Daniele Trapin, Soumya Bera, Jens Hjorleifur Bardarson, Markus Heyl

Published Aug 13, 2018• 28 citations
arXivHTMLPDF

Abstract

Entanglement has developed into an essential concept for the characterization of phases and phase transitions in ground states of quantum many-body systems. In this work we use the logarithmic negativity to study the spatial entanglement structure in the transverse-field Ising chain both in the ground state and at nonzero temperatures. Specifically, we investigate the entanglement between two disjoint blocks as a function of their separation, which can be viewed as the entanglement analog of a spatial correlation function. We find sharp entanglement thresholds at a critical distance beyond which the logarithmic negativity vanishes exactly and thus the two blocks become unentangled, which holds even in the presence of long-ranged quantum correlations, i.e., at the system's quantum critical point. Using time-evolving block decimation, we explore this feature as a function of temperature and size of the two blocks and present a simple model to describe our numerical observations.

DOI
10.1088/1367-2630/aad9ba

Related papers

Scale Invariant Entanglement Negativity at the Many-Body Localization Transition

Johnnie Gray, Abolfazl Bayat, Arijeet Pal +1 more

The exact nature of the many-body localization transition remains an open question. An aspect which has been posited in various studies is the emergence of scale invariance around this point, however the direct observation of this phenomenon is still absent. Here we achieve this by studying the logarithmic negativity and mutual information between disjoint blocks of varying size across the many-body localization transition. The two length scales, block sizes and the distance between them, provide a clear quantitative probe of scale invariance across different length scales. We find that at the transition point, the logarithmic negativity obeys a scale invariant exponential decay with respect to the ratio of block separation to size, whereas the mutual information obeys a polynomial decay. The observed scale invariance of the quantum correlations in a microscopic model opens the direction to probe the fractal structure in critical eigenstates using tensor network techniques and provide constraints on the theory of the many-body localization transition.

arXiv:1908.02761•Aug 7, 2019•7 citations

Critical and noncritical long-range entanglement in Klein-Gordon fields

S. Marcovitch, Alex Retzker, Martin Bodo Plenio +1 more

We investigate the entanglement between two spatially separated intervals in the vacuum state of a free one-dimensional Klein-Gordon field by means of explicit computations in the continuum limit of the linear harmonic chain. We demonstrate that the entanglement, which we quantify by the logarithmic negativity, is finite with no further need for renormalization. We find that in the critical regime, the quantum correlations are scale invariant as they depend only on the ratio of distance to length. They decay much faster than the classical correlations as in the critical limit long-range entanglement decays exponentially for separations larger than the size of the blocks, while classical correlations follow a power-law decay. With decreasing distance of the blocks, the entanglement diverges as a power law in the distance. The noncritical regime manifests richer behavior, as the entanglement depends both on the size of the blocks and on their separation. In correspondence with the von Neumann entropy also long-range entanglement distinguishes critical from noncritical systems.

arXiv:0811.1288•Physical Review A•Jul 21, 2009•115 citations

Nonzero-temperature entanglement negativity of quantum spin models: Area law, linked cluster expansions, and sudden death

Nicholas E. Sherman, Trithep Devakul, Matthew B. Hastings +1 more

We show that the bipartite logarithmic entanglement negativity (EN) of quantum spin models obeys an area law at all nonzero temperatures. We develop numerical linked cluster (NLC) expansions for the "area-law" logarithmic entanglement negativity as a function of temperature and other parameters. For one-dimensional models the results of NLC are compared with exact diagonalization on finite systems and are found to agree very well. The NLC results are also obtained for two dimensional XXZ and transverse field Ising models. In all cases, we find a sudden onset (or sudden death) of negativity at a finite temperature above which the negativity is zero. We use perturbation theory to develop a physical picture for this sudden onset (or sudden death). The onset of EN or its magnitude are insensitive to classical finite-temperature phase transitions, supporting the argument for absence of any role of quantum mechanics at such transitions. On approach to a quantum critical point at T=0, negativity shows critical scaling in size and temperature.

arXiv:1510.08005•Physical review. E•Feb 18, 2016•71 citations

Entanglement of two blocks of spins in the critical Ising model

Paolo Facchi A, Giuseppe Florio, Carmen Invernizzi +1 more

We compute the entropy of entanglement of two blocks of $L$ spins at a distance $d$ in the ground state of an Ising chain in an external transverse magnetic field. We numerically study the von Neumann entropy for different values of the transverse field. At the critical point we obtain analytical results for blocks of size $L=1$ and 2. In the general case, the critical entropy is shown to be additive when $d\ensuremath{\rightarrow}\ensuremath{\infty}$. Finally, based on simple arguments, we derive an expression for the entropy at the critical point as a function of both $L$ and $d$. This formula is in excellent agreement with numerical results.

arXiv:0808.0600•Physical Review A•Nov 3, 2008•44 citations

A contour for the entanglement negativity of bosonic Gaussian states

Gioele Zambotti, Erik Tonni

We construct a contour function for the logarithmic negativity and the logarithm of the moments of the partial transpose of the reduced density matrix for multimode bosonic Gaussian states of a free lattice model. In one spatial dimension, numerical results are obtained for harmonic chains either in the ground state or at finite temperature, by considering, respectively, either a subsystem made by two adjacent or disjoint blocks on the line or a bipartition of the circle. The contour function of the logarithmic negativity diverges only at the entangling points, while the contour function for the logarithm of the moments of the partial transpose is divergent also at the boundary of the bipartite subsystem, as functions of the position. In a two-dimensional conformal field theory, analytic expressions that describe these divergencies are discussed. In one spatial dimension, we explore the partial derivative of the logarithmic negativity of two adjacent intervals with respect to the logarithm of the harmonic ratio of their lengths while their ratio and the other parameters are kept fixed. Considering the ground state of the harmonic chain on the line and in the massive regime, we report numerical results showing that this quantity displays a monotonically decreasing behaviour.

cond-mat.stat-mechCross-listedarXiv:2602.18099•SciPost Physics•Feb 23, 2026•0 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
Entanglement has developed into an essential concept for the characterization of phases and phase transitions in ground states of quantum many-body systems. In this work we use the logarithmic negativity to study the spatial entanglement structure in the transverse-field Ising chain both in the ground state and at nonzero temperatures. Specifically, we investigate the entanglement between two disjoint blocks as a function of their separation, which can be viewed as the entanglement analog of a spatial correlation function. We find sharp entanglement thresholds at a critical distance beyond which the logarithmic negativity vanishes exactly and thus the two blocks become unentangled, which holds even in the presence of long-ranged quantum correlations, i.e., at the system's quantum critical point. Using time-evolving block decimation, we explore this feature as a function of temperature and size of the two blocks and present a simple model to describe our numerical observations.
The exact nature of the many-body localization transition remains an open question. An aspect which has been posited in various studies is the emergence of scale invariance around this point, however the direct observation of this phenomenon is still absent. Here we achieve this by studying the logarithmic negativity and mutual information between disjoint blocks of varying size across the many-body localization transition. The two length scales, block sizes and the distance between them, provide a clear quantitative probe of scale invariance across different length scales. We find that at the transition point, the logarithmic negativity obeys a scale invariant exponential decay with respect to the ratio of block separation to size, whereas the mutual information obeys a polynomial decay. The observed scale invariance of the quantum correlations in a microscopic model opens the direction to probe the fractal structure in critical eigenstates using tensor network techniques and provide constraints on the theory of the many-body localization transition.
We investigate the entanglement between two spatially separated intervals in the vacuum state of a free one-dimensional Klein-Gordon field by means of explicit computations in the continuum limit of the linear harmonic chain. We demonstrate that the entanglement, which we quantify by the logarithmic negativity, is finite with no further need for renormalization. We find that in the critical regime, the quantum correlations are scale invariant as they depend only on the ratio of distance to length. They decay much faster than the classical correlations as in the critical limit long-range entanglement decays exponentially for separations larger than the size of the blocks, while classical correlations follow a power-law decay. With decreasing distance of the blocks, the entanglement diverges as a power law in the distance. The noncritical regime manifests richer behavior, as the entanglement depends both on the size of the blocks and on their separation. In correspondence with the von Neumann entropy also long-range entanglement distinguishes critical from noncritical systems.
We show that the bipartite logarithmic entanglement negativity (EN) of quantum spin models obeys an area law at all nonzero temperatures. We develop numerical linked cluster (NLC) expansions for the "area-law" logarithmic entanglement negativity as a function of temperature and other parameters. For one-dimensional models the results of NLC are compared with exact diagonalization on finite systems and are found to agree very well. The NLC results are also obtained for two dimensional XXZ and transverse field Ising models. In all cases, we find a sudden onset (or sudden death) of negativity at a finite temperature above which the negativity is zero. We use perturbation theory to develop a physical picture for this sudden onset (or sudden death). The onset of EN or its magnitude are insensitive to classical finite-temperature phase transitions, supporting the argument for absence of any role of quantum mechanics at such transitions. On approach to a quantum critical point at T=0, negativity shows critical scaling in size and temperature.
We compute the entropy of entanglement of two blocks of LLL spins at a distance ddd in the ground state of an Ising chain in an external transverse magnetic field. We numerically study the von Neumann entropy for different values of the transverse field. At the critical point we obtain analytical results for blocks of size L=1L=1L=1 and 2. In the general case, the critical entropy is shown to be additive when $d\ensuremath{\rightarrow}\ensuremath{\infty}$. Finally, based on simple arguments, we derive an expression for the entropy at the critical point as a function of both LLL and ddd. This formula is in excellent agreement with numerical results.
We construct a contour function for the logarithmic negativity and the logarithm of the moments of the partial transpose of the reduced density matrix for multimode bosonic Gaussian states of a free lattice model. In one spatial dimension, numerical results are obtained for harmonic chains either in the ground state or at finite temperature, by considering, respectively, either a subsystem made by two adjacent or disjoint blocks on the line or a bipartition of the circle. The contour function of the logarithmic negativity diverges only at the entangling points, while the contour function for the logarithm of the moments of the partial transpose is divergent also at the boundary of the bipartite subsystem, as functions of the position. In a two-dimensional conformal field theory, analytic expressions that describe these divergencies are discussed. In one spatial dimension, we explore the partial derivative of the logarithmic negativity of two adjacent intervals with respect to the logarithm of the harmonic ratio of their lengths while their ratio and the other parameters are kept fixed. Considering the ground state of the harmonic chain on the line and in the massive regime, we report numerical results showing that this quantity displays a monotonically decreasing behaviour.
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).