Operator content of entanglement spectra in the transverse field Ising chain after global quenches (original) (raw)
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Operator content of entanglement spectra after global quenches in the transverse field Ising chain
2019
Jacopo Surace,1, 2 Luca Tagliacozzo,1, 3 and Erik Tonni4 1Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, United Kingdom 2ICFO-Institut de Cincies Fotniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain 3Departament de Fı́sica Quàntica i Astrofı́sica and Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona, Martı́ i Franquès 1, 08028 Barcelona, Catalonia, Spain 4SISSA and INFN Sezione di Trieste, via Bonomea 265, 34136 Trieste, Italy
Entanglement Entropy for the Long-Range Ising Chain in a Transverse Field
Physical Review Letters, 2012
We consider the Ising model in a transverse field with long-range antiferromagnetic interactions that decay as a power law with their distance. We study both the phase diagram and the entanglement properties as a function of the exponent of the interaction. The phase diagram can be used as a guide for future experiments with trapped ions. We find two gapped phases, one dominated by the transverse field, exhibiting quasi long range order, and one dominated by the long range interaction, with long range Néel ordered ground states. We determine the location of the quantum critical points separating those two phases. We determine their critical exponents and central-charges. In the phase with quasi long range order the ground states exhibit exotic corrections to the area law for the entanglement entropy coexisting with gapped entanglement spectra. arXiv:1207.3957v1 [cond-mat.str-el]
2007
Quantum Ising model in one dimension is an exactly solvable example of a quantum phase transition. We investigate its behavior during a quench caused by a gradual turning off of the transverse bias field. The system is then driven at a fixed rate characterized by the quench time τQ across the critical point from a paramagnetic to ferromagnetic phase. In agreement with Kibble-Zurek mechanism (which recognizes that evolution is approximately adiabatic far away, but becomes approximately impulse sufficiently near the critical point), quantum state of the system after the transition exhibits a characteristic correlation lengthξ proportional to the square root of the quench time τQ: ξ = √ τQ. The inverse of this correlation length is known to determine average density of defects (e.g. kinks) after the transition. In this paper, we show that this sameξ controls the entropy of entanglement, e.g. entropy of a block of L spins that are entangled with the rest of the system after the transition from the paramagnetic ground state induced by the quench. For large L, this entropy saturates at 1 6 log 2ξ , as might have been expected from the Kibble-Zurek mechanism. Close to the critical point, the entropy saturates when the block size L ≈ξ, but -in the subsequent evolution in the ferromagnetic phase -a somewhat larger length scale l = √ τQ ln τQ develops as a result of a dephasing process that can be regarded as a quantum analogue of phase ordering, and the entropy saturates when L ≈ l. We also study the spin-spin correlation using both analytic methods and real time simulations with the Vidal algorithm. We find that at an instant when quench is crossing the critical point, ferromagnetic correlations decay exponentially with the dynamical correlation lengtĥ ξ, but (as for entropy of entanglement) in the following evolution length scale l gradually develops. The correlation function becomes oscillatory at distances less than this scale. However, both the wavelength and the correlation length of these oscillations are still determined byξ. We also derive probability distribution for the number of kinks in a finite spin chain after the transition.
Entanglement dynamics after a quench in Ising field theory: a branch point twist field approach
Journal of High Energy Physics, 2019
We extend the branch point twist field approach for the calculation of entanglement entropies to time-dependent problems in 1+1-dimensional massive quantum field theories. We focus on the simplest example: a mass quench in the Ising field theory from initial mass m 0 to final mass m. The main analytical results are obtained from a perturbative expansion of the twist field one-point function in the post-quench quasi-particle basis. The expected linear growth of the Rényi entropies at large times mt 1 emerges from a perturbative calculation at second order. We also show that the Rényi and von Neumann entropies, in infinite volume, contain subleading oscillatory contributions of frequency 2m and amplitude proportional to (mt) −3/2. The oscillatory terms are correctly predicted by an alternative perturbation series, in the prequench quasi-particle basis, which we also discuss. A comparison to lattice numerical calculations carried out on an Ising chain in the scaling limit shows very good agreement with the quantum field theory predictions. We also find evidence of clustering of twist field correlators which implies that the entanglement entropies are proportional to the number of subsystem boundary points.
Entanglement Entropy in Critical Quantum Spin Chains with Boundaries and Defects
Quantum science and technology, 2021
Entanglement entropy (EE) in critical quantum spin chains described by 1+1D conformal field theories contains signatures of the universal characteristics of the field theory. Boundaries and defects in the spin chain give rise to universal contributions in the EE. In this work, we analyze these universal contributions for the critical Ising and XXZ spin chains for different conformal boundary conditions and defects. For the spin chains with boundaries, we use the boundary states for the corresponding continuum theories to compute the subleading contribution to the EE analytically and provide supporting numerical computation for the spin chains. Subsequently, we analyze the behavior of EE in the presence of conformal defects for the two spin chains and describe the change in both the leading logarithmic and subleading terms in the EE.
Dynamics and steady state properties of entanglement in periodically driven Ising spin-chain
2017
We study the dynamics of microscopic quantum correlations, viz., bipartite entanglement and quantum discord, in Ising spin chain with periodically varying external magnetic field along the transverse direction. Depending upon system parameters, local quantum correlations in the evolved states of such systems may get saturated to non-zero values after sufficiently large number of driving cycles. Moreover, we investigate convergence of the local density matrices, from which the quantum correlations under study originate, towards the final steady-state density matrices as a function of driving cycles. We find that the geometric distance between the non-equilibrium and the steady-state reduced density matrices obey power-law scaling. The steady-state quantum correlations corresponding to various initial states in thermal equilibrium are studied as a function of drive time period of a square pulsed field. The steady-state quantum correlations are marked by presence of peaks in the freque...
Global geometric entanglement in transverse-field XY spin chains: finite and infinite systems
Quantum Information & Computation, 2011
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).
Entanglement negativity in the critical Ising chain
Journal of Statistical Mechanics: Theory and Experiment, 2013
We study the scaling of the traces of the integer powers of the partially transposed reduced density matrix Tr(ρ T2 A ) n and of the entanglement negativity for two spin blocks as function of their length and separation in the critical Ising chain. For two adjacent blocks, we show that tensor network calculations agree with universal conformal field theory (CFT) predictions. In the case of two disjoint blocks the CFT predictions are recovered only after taking into account the finite size corrections induced by the finite length of the blocks.
Dynamics of entanglement in a one-dimensional Ising chain
Physical Review A, 2008
The evolution of entanglement in a one-dimensional Ising chain is numerically studied under various initial conditions. We analyze two problems concerning the dynamics of the entanglement: (i) generation of the entanglement from the pseudopure separable state and (ii) transportation of the entanglement from one end of the chain to the other. The investigated model is a one-dimensional Ising spin-1/2 chain with nearest-neighbor interactions placed in an external magnetic field and irradiated by a weak resonant transverse field. The possibility of selective initialization of partially entangled states is considered. It was shown that, in spite of the use of a model with the direct interactions between the nearest neighbors, the entanglement between remote spins is generated.
Quench dynamics of the Ising field theory in a magnetic field
SciPost physics, 2018
We numerically simulate the time evolution of the Ising field theory after quenches starting from the E 8 integrable model using the Truncated Conformal Space Approach. The results are compared with two different analytic predictions based on form factor expansions in the pre-quench and post-quench basis, respectively. Our results clarify the domain of validity of these expansions and suggest directions for further improvement. We show for quenches in the E 8 model that the initial state is not of the integrable pair state form. We also construct quench overlap functions and show that their high-energy asymptotics are markedly different from those constructed before in the sinh/sine-Gordon theory, and argue that this is related to properties of the ultraviolet fixed point.