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Physical Review A

Evolution of entanglement towards criticality in the XY model in a quantum phase transition induced by a quench

Banasri Basu, Pratul Bandyopadhyay, Priyadarshi Majumdar

Published Aug 19, 2015• 2 citations
Publisher

Abstract

We have studied here the evolution of entanglement towards criticality in the $\mathit{XY}$ model when quantum phase transition (QPT) is induced by a quench. To this end, we have utilized the relation of the Berry phase acquired by a spin state when it traverses a closed path at criticality with the concurrence measuring the entanglement for a pair of nearest-neighbor spins. It has been argued that when the spin state is rotated towards the generation of the closed path, the segments of noncyclic paths with open ends can be mapped onto cyclic paths enclosing flux lines associated with $q$-deformed monopoles. This essentially gives rise to the sequence of Berry phases attained by the quasispin states belonging to the quantum group ${\mathrm{SU}}_{q}(2)$ when rotated in a closed path. As the standard Berry phase at criticality is obtained when the time-dependent deformation parameter $q$ attains the value 1, this formulation helps us to analyze the evolution of entanglement towards criticality when QPT is induced by a quench. From this analysis we also formulate a scaling law for the entanglement entropy for a block of $L$ spins with the rest at off-criticality. As in QPT off-critical points correspond to thermal fluctuations, this formulation represents the evolution of entanglement towards criticality with temperature as $T\ensuremath{\rightarrow}0$.

DOI
10.1103/physreva.92.022343

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We have studied here the evolution of entanglement towards criticality in the XY\mathit{XY}XY model when quantum phase transition (QPT) is induced by a quench. To this end, we have utilized the relation of the Berry phase acquired by a spin state when it traverses a closed path at criticality with the concurrence measuring the entanglement for a pair of nearest-neighbor spins. It has been argued that when the spin state is rotated towards the generation of the closed path, the segments of noncyclic paths with open ends can be mapped onto cyclic paths enclosing flux lines associated with qqq-deformed monopoles. This essentially gives rise to the sequence of Berry phases attained by the quasispin states belonging to the quantum group SUq(2){\mathrm{SU}}_{q}(2)SUq​(2) when rotated in a closed path. As the standard Berry phase at criticality is obtained when the time-dependent deformation parameter qqq attains the value 1, this formulation helps us to analyze the evolution of entanglement towards criticality when QPT is induced by a quench. From this analysis we also formulate a scaling law for the entanglement entropy for a block of LLL spins with the rest at off-criticality. As in QPT off-critical points correspond to thermal fluctuations, this formulation represents the evolution of entanglement towards criticality with temperature as T\ensuremath→0T\ensuremath{\rightarrow}0T\ensuremath→0.
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.
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.
In this paper the entanglement and quantum phase transition of the anisotropic spin-1/21/21/2 XY\mathit{XY}XY model are studied by using the quantum renormalization-group method. By solving the renormalization equations, we get the trivial and nontrivial fixed points, which correspond to the phase of the system and the critical point, respectively. The concurrence between two blocks are calculated and it is found that when the number of iterations of the renormalization tends to infinity, the concurrence develops two saturated values that are associated with two different phases, i.e., Ising-like and spin-fluid phases. We also investigate the first derivative of the concurrence and find that there exists nonanalytic behaviors at the quantum critical point, which are directly associated with the divergence of the correlation length. To gain further insight, the scaling behaviors of the system are analyzed and it is shown that the maximum value of the first derivative of the concurrence reaches infinity and the critical point is approached as the size of the system increases.
We use the quantum renormalization-group (QRG) method to study the entanglement and quantum phase transition (QPT) in the one-dimensional spin-1/2 Heisenberg—Ising model [Lieb E, Schultz T and Mattis D 1961 Ann. Phys. (N.Y.) 16 407]. We find the quantum phase boundary of this model by investigating the evolution of concurrence in terms of QRG iterations. We also investigate the scaling behavior of the system close to the quantum critical point, which shows that the minimum value of the first derivative of concurrence and the position of the minimum scale with an exponent of the system size. Also, the first derivative of concurrence between two blocks diverges at the quantum critical point, which is directly associated with the divergence of the correlation length.
We study the quantum entanglement and quantum phase transition (QPT) of the anisotropic spin-1/21/21/2 XYXYXY model with staggered Dzyaloshinskii-Moriya (DM) interaction by means of the quantum renormalization group method. The scaling of coupling constants and the critical points of the system are obtained. It is found that when the number of renormalization group iterations tends to infinity, the system exhibit a QPT between the spin-fluid and N\'eel phases which correspond with two saturated values of the concurrence for a given value of the strength of DM interaction. The DM interaction can enhance the entanglement and influence the QPT of the system. To gain further insight, the first derivative of the entanglement exhibit a nonanalytic behavior at the critical point and it directly associates with the divergence of the correlation length. This shows that the correlation length exponent is closely related to the critical exponent, i.e., the scaling behaviors of the system.