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arXiv:1711.01164 · Physical review. E

Entanglement entropy of a three-spin-interacting spin chain with a time-reversal-breaking impurity at one boundary

Tanay Nag, Atanu Rajak

Published Apr 9, 2018• 1 citation
arXivHTMLPDF

Abstract

We investigate the effect of a time-reversal-breaking impurity term (of strength λ_{d}) on both the equilibrium and nonequilibrium critical properties of entanglement entropy (EE) in a three-spin-interacting transverse Ising model, which can be mapped to a p-wave superconducting chain with next-nearest-neighbor hopping and interaction. Importantly, we find that the logarithmic scaling of the EE with block size remains unaffected by the application of the impurity term, although, the coefficient (i.e., central charge) varies logarithmically with the impurity strength for a lower range of λ_{d} and eventually saturates with an exponential damping factor [∼exp(-λ_{d})] for the phase boundaries shared with the phase containing two Majorana edge modes. On the other hand, it receives a linear correction in term of λ_{d} for an another phase boundary. Finally, we focus to study the effect of the impurity in the time evolution of the EE for the critical quenching case where the impurity term is applied only to the final Hamiltonian. Interestingly, it has been shown that for all the phase boundaries, contrary to the equilibrium case, the saturation value of the EE increases logarithmically with the strength of impurity in a certain regime of λ_{d} and finally, for higher values of λ_{d}, it increases very slowly dictated by an exponential damping factor. The impurity-induced behavior of EE might bear some deep underlying connection to thermalization.

DOI
10.1103/physreve.97.042108

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We investigate the effect of a time-reversal-breaking impurity term (of strength λ_{d}) on both the equilibrium and nonequilibrium critical properties of entanglement entropy (EE) in a three-spin-interacting transverse Ising model, which can be mapped to a p-wave superconducting chain with next-nearest-neighbor hopping and interaction. Importantly, we find that the logarithmic scaling of the EE with block size remains unaffected by the application of the impurity term, although, the coefficient (i.e., central charge) varies logarithmically with the impurity strength for a lower range of λ_{d} and eventually saturates with an exponential damping factor [∼exp(-λ_{d})] for the phase boundaries shared with the phase containing two Majorana edge modes. On the other hand, it receives a linear correction in term of λ_{d} for an another phase boundary. Finally, we focus to study the effect of the impurity in the time evolution of the EE for the critical quenching case where the impurity term is applied only to the final Hamiltonian. Interestingly, it has been shown that for all the phase boundaries, contrary to the equilibrium case, the saturation value of the EE increases logarithmically with the strength of impurity in a certain regime of λ_{d} and finally, for higher values of λ_{d}, it increases very slowly dictated by an exponential damping factor. The impurity-induced behavior of EE might bear some deep underlying connection to thermalization.
We investigate the dynamics of a one-dimensional p-wave superconductor with next-nearest-neighbor hopping and superconducting interaction derived from a three-spin interacting Ising model in transverse field by mapping to Majorana fermions. The next-nearest-neighbor hopping term leads to a new topological phase containing two zero-energy Majorana modes at each end of an open chain, compared to a nearest-neighbor p-wave superconducting chain. We study the Majorana survival probability (MSP) of a particular Majorana edge state when the initial Hamiltonian (H_{i}) is changed to the quantum critical as well as off-critical final Hamiltonian (H_{f}), which additionally contains an impurity term (H_{imp}) that breaks the time-reversal invariance. For the off-critical quenching inside the new topological phase with H_{f}=H_{i}+H_{imp}, and small impurity strength (λ_{d}), we observe a perfect oscillation of the MSP as a function of time with a single frequency (determined by the impurity strength λ_{d}) that can be analyzed from an equivalent two-level problem. On the other hand, the MSP shows a beating like structure with time for quenching to the phase boundary separating the topological phase (with two edge Majoranas at each edge) and the nontopological phase where the additional frequency is given by inverse of the system size. We attribute this behavior of the MSP to the modification of the energy levels of the final Hamiltonian due to the application of the impurity term.
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.
The quantum Ising chain of length, LLL, which is separated into two parts by localized or extended defects is considered at the critical point where scaling of the interface magnetization is nonuniversal. We measure the entanglement entropy between the two halves of the system in equilibrium, as well as after a quench, when the interaction at the interface is changed for time t>0t>0t>0. For the localized defect the increase in the entropy with log L\text{log}\text{ }Llog L or with log t\text{log}\text{ }tlog t involves the same effective central charge, which is a continuous function of the strength of the defect. On the contrary for the extended defect the equilibrium entropy is saturated but the nonequilibrium entropy has a logarithmic time dependence the prefactor of which depends on the strength of the defect.
We study the zero-temperature quenching dynamics of various extensions of the transverse Ising model (TIM) for when the transverse field is linearly quenched from to (or zero) at a finite and uniform rate. The rate of quenching is dictated by a characteristic scale given by τ. The density of kinks produced in these extended models while crossing the quantum critical points during the quenching process is calculated using a many-body generalization of the Landau–Zener transition theory. The density of kinks in the final state is found to decay as τ −1/2 . In the first model considered here, the transverse Ising Hamiltonian includes an additional ferromagnetic three-spin interaction term of strength J 3 . For J 3 <0.5, the kink density is found to increase monotonically with J 3 whereas it decreases with J 3 for J 3 >0.5. The point with J 3 = 0.5 and the transverse field h = −0.5 is multicritical, where the density shows a slower decay given by τ −1/6 . We also study the effect of ferromagnetic or antiferromagnetic next nearest neighbor (NNN) interactions on the dynamics of the TIM under the same quenching scheme. In a mean field approximation, the transverse Ising Hamiltonians with NNN interactions are identical to the three-spin Hamiltonian. The NNN interactions non-trivially modify the dynamical behavior; for example an antiferromagnetic NNN interaction results in a larger number of kinks in the final state in comparison to the case when the NNN interaction is ferromagnetic.
We consider a finite two dimensional XY spin model. The model consists of a set of seven localized spin- 12 particles in a two dimensional triangular lattice coupled through nearest neighbor exchange interaction in presence of an external magnetic field. We study the effect of a single impurity spin coupled to its nearest neighbor through an exchange interaction J′ on the pairwise entanglement between the different spins in the lattice. We found that when the spin is located at a border site the entanglement between the impurity and its nearest neighbors increases monotonically with J′ reaching a saturation value that depends on the magnetic field strength h. On the other hand the entanglement with the next nearest spins shows a critical behavior where the entanglement increases and then decreases rapidly until it vanishes. The critical coupling value at which the entanglement vanishes increases as h increases. Studying the entanglement between two spins excluding the impurity show that while the entanglement between two spins coupled to the impurity reach a saturation value as J′ increases, it vanishes for spins not coupled to the impurity. Furthermore studying the effect of a central impurity show that the entanglement between nearest neighbor and next nearest neighbor spins reaches an asymptotic value as J′ increases. Interestingly it was demonstrated that the impurity can be used as a switch to control the entanglement between different spins in the lattice turning it on and off.
In this paper, we investigate Quantum Phase Transitions (QPTs) in the long-range XXZ spin chain using two information-theoretic probes: The half-chain entanglement entropy and a witness of genuine multipartite nonlocality. Both quantities effectively identify the phase boundary, but exhibit distinct scaling behaviors: the entanglement entropy grows linearly with system size N, while the multipartite nonlocality increases exponentially, with [Formula: see text]. By independently tuning the power-law decay exponents of the transverse ([Formula: see text]) and longitudinal ([Formula: see text]) couplings, we uncover contrasting trends in the critical point [Formula: see text]: Increasing [Formula: see text] shifts it to larger values, while increasing [Formula: see text] shifts it to smaller ones. These monotonic and saturating dependencies are captured by an empirical exponential model, offering a compact and physically transparent description of the critical behavior. Our results highlight the crucial role of interaction range in driving criticality, with implications for engineered quantum systems and long-range entanglement control.