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Physical Review A

Scaling of entanglement entropy in a quantum phase transition in the transverse Ising model induced by a quench

Priyadarshi Majumdar, Pratul Bandyopadhyay

Published Jan 19, 2010• 8 citations
Publisher

Abstract

It is known that at the critical point of a zero-temperature quantum phase transition in a one-dimensional spin system the entanglement entropy of a block of $L$ spins with the rest of the system scales logarithmically with $L$ with a prefactor determined by the central charge of the relevant conformal field theory. When we introduce critical slowing down incorporating the Kibble-Zurek mechanism of defect formation induced by a quench, the implicit nonadiabatic transition disturbs the scaling behavior. We have shown that in this case the entanglement entropy also obeys a scaling law such that it increases logarithmically with $L$ but the prefactor depends on the quench time. This puts a constraint on the block size $L$ so that we cannot arbitrarily choose it. Thus, the entanglement entropy obeys the scaling law only in a restrictive sense due to the formation of defects.

DOI
10.1103/physreva.81.012311

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It is known that at the critical point of a zero-temperature quantum phase transition in a one-dimensional spin system the entanglement entropy of a block of LLL spins with the rest of the system scales logarithmically with LLL with a prefactor determined by the central charge of the relevant conformal field theory. When we introduce critical slowing down incorporating the Kibble-Zurek mechanism of defect formation induced by a quench, the implicit nonadiabatic transition disturbs the scaling behavior. We have shown that in this case the entanglement entropy also obeys a scaling law such that it increases logarithmically with LLL but the prefactor depends on the quench time. This puts a constraint on the block size LLL so that we cannot arbitrarily choose it. Thus, the entanglement entropy obeys the scaling law only in a restrictive sense due to the formation of defects.
We have studied quantum phase transition induced by a quench in different one-dimensional spin systems. Our analysis is based on the dynamical mechanism which envisages nonadiabaticity in the vicinity of the critical point. This causes spin fluctuation which leads to the random fluctuation of the Berry phase factor acquired by a spin state when the ground state of the system evolves in a closed path. The two-point correlation of this phase factor is associated with the probability of the formation of defects. In this framework, we have estimated the density of defects produced in several one-dimensional spin chains. At the critical region, the entanglement entropy of a block of LLL spins with the rest of the system is also estimated which is found to increase logarithmically with LLL. The dependence on the quench time puts a constraint on the block size LLL. It is also pointed out that the Lipkin-Meshkov-Glick model in point-splitting regularized form appears as a combination of the XXX\mathit{XXX}XXX model and Ising model with magnetic field in the negative zzz axis. This unveils the underlying conformal symmetry at criticality which is lost in the sharp point limit. Our analysis shows that the density of defects as well as the scaling behavior of the entanglement entropy follows a universal behavior in all these systems.
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 ${\ensuremath{\tau}}_{Q}$ 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 length $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$ proportional to the square root of the quench time ${\ensuremath{\tau}}_{Q}$: $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}=\sqrt{{\ensuremath{\tau}}_{Q}}$. 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 same $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$ controls the entropy of entanglement, e.g., entropy of a block of LLL spins that are entangled with the rest of the system after the transition from the paramagnetic ground state induced by the quench. For large LLL, this entropy saturates at $\frac{1}{6}\phantom{\rule{0.2em}{0ex}}{\mathrm{log}}_{2}\phantom{\rule{0.2em}{0ex}}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$, as might have been expected from the Kibble-Zurek mechanism. Close to the critical point, the entropy saturates when the block size $L\ensuremath{\approx}\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$, but---in the subsequent evolution in the ferromagnetic phase---a somewhat larger length scale $l=\sqrt{{\ensuremath{\tau}}_{Q}}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}{\ensuremath{\tau}}_{Q}$ develops as a result of a dephasing process that can be regarded as a quantum analog of phase ordering, and the entropy saturates when $L\ensuremath{\approx}l$. 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 length $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$, but (as for entropy of entanglement) in the following evolution length scale lll 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 by $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\xi}}$. We also derive probability distribution for the number of kinks in a finite spin chain after the transition.
We study the magnetic-field dependence of the entanglement entropy in quantum phase transition induced by a quench of the XXXXXX, XXX,XXX,XXX, and Lipkin-Meshkov-Glick (LMG) models. The entropy for a block of LLL spins with the rest follows a logarithmic scaling law where the block size LLL is restricted due to the dependence of the prefactor on the quench time. Within this restricted region the entropy undergoes a renormalization group (RG) flow. From the RG flow equation we have analytically determined the magnetic field dependence of the entropy. The anisotropy parameter dependence of the entropy for the XYXYXY and the LMG models has also been studied in this framework. The results are found to be in excellent agreement with that obtained by other authors from numerical studies without any quench.
We review quantum phase transitions of spin systems in transverse magnetic fields taking the examples of the spin-1/2 Ising and XY models in a transverse field. Beginning with an overview of quantum phase transitions, we introduce a number of model Hamiltonians. We provide exact solutions in one spatial dimension connecting them to conformal field theoretical studies. We also discuss Kitaev models and some other exactly solvable spin systems. Studies of quantum phase transitions in the presence of quenched randomness and with frustrating interactions are presented in detail. We discuss novel phenomena like Griffiths-McCoy singularities. We then turn to more recent topics like information theoretic measures of the quantum phase transitions in these models such as concurrence, entanglement entropy, quantum discord and quantum fidelity. We then focus on non-equilibrium dynamics of a variety of transverse field systems across quantum critical points and lines. After mentioning rapid quenching studies, we dwell on slow dynamics and discuss the Kibble-Zurek scaling for the defect density following a quench across critical points and its modifications for quenching across critical lines, gapless regions and multicritical points. Topics like the role of different quenching schemes, local quenching, quenching of models with random interactions and quenching of a spin chain coupled to a heat bath are touched upon. The connection between non-equilibrium dynamics and quantum information theoretic measures is presented at some length. We indicate the connection between Kibble-Zurek scaling and adiabatic evolution of a state as well as the application of adiabatic dynamics as a tool of a quantum optimization technique known as quantum annealing. The final section is dedicated to a detailed discussion on recent experimental studies of transverse Ising-like systems.
The quantum phase transition (QPT) of the one-dimensional (1D) quantum compass model in a transverse magnetic field is studied in this paper. An exact solution is obtained by using an extended Jordan and Wigner transformation to the pseudospin operators. The fidelity susceptibility, the concurrence, the block-block entanglement entropy, and the pseudospin correlation functions are calculated with antiperiodic boundary conditions. The QPT driven by the transverse-field only emerges at zero field and is of the second order. Several critical exponents obtained by finite-size scaling analysis are the same as those in the 1D transverse-field Ising model, suggesting the same universality class. A logarithmic divergence of the entanglement entropy of a block at the quantum critical point is also observed. From the calculated coefficient connected to the central charge of the conformal field theory, it is suggested that the block entanglement depends crucially on the detailed topological structure of a system.
The scaling behavior of the entanglement entropy in the two-dimensional random transverse field Ising model is numerically studied through the strong disordered renormalization group method. We find that the leading term of the entanglement entropy always linearly scales with the block size. However, besides this area law contribution, we find a subleading logarithmic correction at the quantum critical point. This correction is discussed from the point of view of an underlying percolation transition, both at finite and at zero temperature.