Entanglement negativity after a global quantum quench

We study the time evolution of the logarithmic negativity after a global quantum quench. In a 1+1 dimensional conformal invariant field theory, we consider the negativity between two intervals which can be either adjacent or disjoint. We show that the negativity follows the quasi-particle interpretation for the spreading of entanglement. We check and generalise our findings with a systematic analysis of the negativity after a quantum quench in the harmonic chain, highlighting two peculiar lattice effects: the late birth and the sudden death of entanglement.We study the time evolution of the logarithmic negativity after a global quantum quench. In a 1+1 dimensional conformal invariant field theory, we consider the negativity between two intervals which can be either adjacent or disjoint. We show that the negativity follows the quasi-particle interpretation for the spreading of entanglement. We check and generalise our findings with a systematic analysis of the negativity after a quantum quench in the harmonic chain, highlighting two peculiar lattice effects: the late birth and the sudden death of entanglement.

arXiv:1410.0900Journal of Statistical Mechanics Theory and ExperimentDec 16, 2014136 citations

Entanglement negativity after a local quantum quench in conformal field theories

Xueda Wen, Po-Yao Chang, Shinsei Ryu

We study the time evolution of the entanglement negativity after a local quantum quench in (1 + 1)-dimensional conformal field theories (CFTs), which we introduce by suddenly joining two initially decoupled CFTs at their end points. We calculate the negativity evolution for both adjacent intervals and disjoint intervals explicitly. For two adjacent intervals, the entanglement negativity grows logarithmically in time right after the quench. After developing a plateau-like feature, the entanglement negativity drops to the ground-state value. For the case of two spatially separated intervals, a light-cone behavior is observed in the negativity evolution; in addition, a long-range entanglement, which is independent of the distance between two intervals, can be created. Our results agree with the heuristic picture that quasiparticles, which carry entanglement, are emitted from the joining point and propagate freely through the system. Our analytical results are confirmed by numerical calculations based on a critical harmonic chain.We study the time evolution of the entanglement negativity after a local quantum quench in (1 + 1)-dimensional conformal field theories (CFTs), which we introduce by suddenly joining two initially decoupled CFTs at their end points. We calculate the negativity evolution for both adjacent intervals and disjoint intervals explicitly. For two adjacent intervals, the entanglement negativity grows logarithmically in time right after the quench. After developing a plateau-like feature, the entanglement negativity drops to the ground-state value. For the case of two spatially separated intervals, a light-cone behavior is observed in the negativity evolution; in addition, a long-range entanglement, which is independent of the distance between two intervals, can be created. Our results agree with the heuristic picture that quasiparticles, which carry entanglement, are emitted from the joining point and propagate freely through the system. Our analytical results are confirmed by numerical calculations based on a critical harmonic chain.

arXiv:1501.00568Physical Review BAug 6, 201592 citations

Time evolution of entanglement negativity from black hole interiors

Vinay Malvimat, Sayid Mondal, Gautam Sengupta

Abstract We investigate the time evolution of entanglement negativity following a global quench for mixed state configurations of two disjoint and adjacent intervals in a (1 + 1)-dimensional conformal field theory (CFT 1+1) dual to the eternal black hole sliced in half by an end of the world brane, through the AdS 3 /CFT 2 correspondence. To this end we obtain the time evolution of the holographic entanglement negativity for such mixed states from a dual bulk eternal black hole geometry and elucidate the relevant geodesic structures. The holographic entanglement negativity for such mixed states, following a global quench is described by half of the results for the eternal black hole. Significantly our results exactly match with the corresponding CFT 1+1 computations.Abstract We investigate the time evolution of entanglement negativity following a global quench for mixed state configurations of two disjoint and adjacent intervals in a (1 + 1)-dimensional conformal field theory (CFT 1+1) dual to the eternal black hole sliced in half by an end of the world brane, through the AdS 3 /CFT 2 correspondence. To this end we obtain the time evolution of the holographic entanglement negativity for such mixed states from a dual bulk eternal black hole geometry and elucidate the relevant geodesic structures. The holographic entanglement negativity for such mixed states, following a global quench is described by half of the results for the eternal black hole. Significantly our results exactly match with the corresponding CFT 1+1 computations.

arXiv:1812.04424Journal of High Energy PhysicsMay 1, 201923 citations

Entanglement Negativity in Quantum Field Theory

Pasquale Calabrese, John Cardy, Erik Tonni C

We develop a systematic method to extract the negativity in the ground state of a $1+1$ dimensional relativistic quantum field theory, using a path integral formalism to construct the partial transpose ${\ensuremath{\rho}}_{A}^{{T}_{2}}$ of the reduced density matrix of a subsystem $A={A}_{1}\ensuremath{\cup}{A}_{2}$, and introducing a replica approach to obtain its trace norm which gives the logarithmic negativity $\mathcal{E}=\mathrm{ln}\ensuremath{\Vert}{\ensuremath{\rho}}_{A}^{{T}_{2}}\ensuremath{\Vert}$. This is shown to reproduce standard results for a pure state. We then apply this method to conformal field theories, deriving the result $\mathcal{E}\ensuremath{\sim}(c/4)\mathrm{ln}[{\ensuremath{\ell}}_{1}{\ensuremath{\ell}}_{2}/({\ensuremath{\ell}}_{1}+{\ensuremath{\ell}}_{2})]$ for the case of two adjacent intervals of lengths ${\ensuremath{\ell}}_{1}$, ${\ensuremath{\ell}}_{2}$ in an infinite system, where $c$ is the central charge. For two disjoint intervals it depends only on the harmonic ratio of the four end points and so is manifestly scale invariant. We check our findings against exact numerical results in the harmonic chain.We develop a systematic method to extract the negativity in the ground state of a $1+1$ dimensional relativistic quantum field theory, using a path integral formalism to construct the partial transpose ${\ensuremath{\rho}}_{A}^{{T}_{2}}$ of the reduced density matrix of a subsystem $A={A}_{1}\ensuremath{\cup}{A}_{2}$, and introducing a replica approach to obtain its trace norm which gives the logarithmic negativity $\mathcal{E}=\mathrm{ln}\ensuremath{\Vert}{\ensuremath{\rho}}_{A}^{{T}_{2}}\ensuremath{\Vert}$. This is shown to reproduce standard results for a pure state. We then apply this method to conformal field theories, deriving the result $\mathcal{E}\ensuremath{\sim}(c/4)\mathrm{ln}[{\ensuremath{\ell}}_{1}{\ensuremath{\ell}}_{2}/({\ensuremath{\ell}}_{1}+{\ensuremath{\ell}}_{2})]$ for the case of two adjacent intervals of lengths ${\ensuremath{\ell}}_{1}$, ${\ensuremath{\ell}}_{2}$ in an infinite system, where $c$ is the central charge. For two disjoint intervals it depends only on the harmonic ratio of the four end points and so is manifestly scale invariant. We check our findings against exact numerical results in the harmonic chain.

arXiv:1206.3092Physical Review LettersSep 28, 2012463 citations

Entanglement negativity in extended systems: a field theoretical approach

Pasquale Calabrese, John Cardy, Erik Tonni C

We report on a systematic approach for the calculation of the negativity in the ground state of a one-dimensional quantum field theory. The partial transpose of the reduced density matrix of a subsystem A = A 1 ∪ A 2 is explicitly constructed as an imaginary-time path integral and from this the replicated traces are obtained. The logarithmic negativity is then the continuation to n → 1 of the traces of the even powers. For pure states, this procedure reproduces the known results. We then apply this method to conformally invariant field theories (CFTs) in several different physical situations for infinite and finite systems and without or with boundaries. In particular, in the case of two adjacent intervals of lengths ℓ 1 ,ℓ 2 in an infinite system, we derive the result ∼ ( c /4)ln(ℓ 1 ℓ 2 /(ℓ 1 + ℓ 2 )), where c is the central charge. For the more complicated case of two disjoint intervals, we show that the negativity depends only on the harmonic ratio of the four end points and so is manifestly scale invariant. We explicitly calculate the scale invariant functions for the replicated traces in the case of the CFT for the free compactified boson, but we have not so far been able to obtain the n → 1 continuation for the negativity even in the limit of large compactification radius. We have checked all our findings against exact numerical results for the harmonic chain which is described by a non-compactified free boson.We report on a systematic approach for the calculation of the negativity in the ground state of a one-dimensional quantum field theory. The partial transpose of the reduced density matrix of a subsystem A = A 1 ∪ A 2 is explicitly constructed as an imaginary-time path integral and from this the replicated traces are obtained. The logarithmic negativity is then the continuation to n → 1 of the traces of the even powers. For pure states, this procedure reproduces the known results. We then apply this method to conformally invariant field theories (CFTs) in several different physical situations for infinite and finite systems and without or with boundaries. In particular, in the case of two adjacent intervals of lengths ℓ 1 ,ℓ 2 in an infinite system, we derive the result ∼ ( c /4)ln(ℓ 1 ℓ 2 /(ℓ 1 + ℓ 2 )), where c is the central charge. For the more complicated case of two disjoint intervals, we show that the negativity depends only on the harmonic ratio of the four end points and so is manifestly scale invariant. We explicitly calculate the scale invariant functions for the replicated traces in the case of the CFT for the free compactified boson, but we have not so far been able to obtain the n → 1 continuation for the negativity even in the limit of large compactification radius. We have checked all our findings against exact numerical results for the harmonic chain which is described by a non-compactified free boson.

arXiv:1210.5359Journal of Statistical Mechanics Theory and ExperimentFeb 1, 2013311 citations

Entanglement Hamiltonians in 1D free lattice models after a global quantum quench

Giuseppe Di Giulio, Raúl Arias, Erik Tonni C

Abstract We study the temporal evolution of the entanglement Hamiltonian of an interval after a global quantum quench in free lattice models in one spatial dimension. In a harmonic chain we explore a quench of the frequency parameter. In a chain of free fermions at half filling we consider the evolution of the ground state of a fully dimerised chain through the homogeneous Hamiltonian. We focus on critical evolution Hamiltonians. The temporal evolutions of the gaps in the entanglement spectrum are analysed. The entanglement Hamiltonians in these models are characterised by matrices that provide also contours for the entanglement entropies. The temporal evolution of these contours for the entanglement entropy is studied, also by employing existing conformal field theory results for the semi-infinite line and the quasi-particle picture for the global quench.Abstract We study the temporal evolution of the entanglement Hamiltonian of an interval after a global quantum quench in free lattice models in one spatial dimension. In a harmonic chain we explore a quench of the frequency parameter. In a chain of free fermions at half filling we consider the evolution of the ground state of a fully dimerised chain through the homogeneous Hamiltonian. We focus on critical evolution Hamiltonians. The temporal evolutions of the gaps in the entanglement spectrum are analysed. The entanglement Hamiltonians in these models are characterised by matrices that provide also contours for the entanglement entropies. The temporal evolution of these contours for the entanglement entropy is studied, also by employing existing conformal field theory results for the semi-infinite line and the quasi-particle picture for the global quench.

arXiv:1905.01144Journal of Statistical Mechanics Theory and ExperimentDec 1, 201957 citations

We develop a systematic method to extract the negativity in the ground state of a

1+1

dimensional relativistic quantum field theory, using a path integral formalism to construct the partial transpose ${\ensuremath{\rho}}_{A}^{{T}_{2}}$ of the reduced density matrix of a subsystem $A={A}_{1}\ensuremath{\cup}{A}_{2}$, and introducing a replica approach to obtain its trace norm which gives the logarithmic negativity $\mathcal{E}=\mathrm{ln}\ensuremath{\Vert}{\ensuremath{\rho}}_{A}^{{T}_{2}}\ensuremath{\Vert}$. This is shown to reproduce standard results for a pure state. We then apply this method to conformal field theories, deriving the result $\mathcal{E}\ensuremath{\sim}(c/4)\mathrm{ln}[{\ensuremath{\ell}}_{1}{\ensuremath{\ell}}_{2}/({\ensuremath{\ell}}_{1}+{\ensuremath{\ell}}_{2})]$ for the case of two adjacent intervals of lengths ${\ensuremath{\ell}}_{1}$, ${\ensuremath{\ell}}_{2}$ in an infinite system, where

c

is the central charge. For two disjoint intervals it depends only on the harmonic ratio of the four end points and so is manifestly scale invariant. We check our findings against exact numerical results in the harmonic chain.