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arXiv:2204.05564 · Physical Review A

Loschmidt echo and momentum distribution in a Kitaev spin chain

Vimalesh Kumar Vimal, Harshawardhan Wanare, V. Subrahmanyam

Published Sep 28, 2022• 1 citation
arXivHTMLPDF

Abstract

We investigate the Loschmidt echo in a one-dimensional spin chain having Kitaev-type interaction in constant and kicked magnetic fields. The Loschmidt echo for the initial states having different magnon excitations shows long-time revivals for smaller chains and has short-time revival peaks for the longer chains. The system near the critical point shows peculiarly long-time revival peaks of the Loschmidt echo for relatively larger chains. The presence of a magnon in the initial state affects the Loschmidt echo revival peaks. The momentum distribution function exhibits maxima for a few momenta that are associated with the momentum of the magnon excitation present in the initial states. The probability maxima decay as $O(1/N)$ with the system size. For the Hamiltonian with kicked magnetic fields, the Loschmidt echo depends on the kick period. For a special kick period, the Loschmidt echo shows no evolution at all irrespective of the system size.

DOI
10.1103/physreva.106.032221

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We investigate the Loschmidt echo in a one-dimensional spin chain having Kitaev-type interaction in constant and kicked magnetic fields. The Loschmidt echo for the initial states having different magnon excitations shows long-time revivals for smaller chains and has short-time revival peaks for the longer chains. The system near the critical point shows peculiarly long-time revival peaks of the Loschmidt echo for relatively larger chains. The presence of a magnon in the initial state affects the Loschmidt echo revival peaks. The momentum distribution function exhibits maxima for a few momenta that are associated with the momentum of the magnon excitation present in the initial states. The probability maxima decay as O(1/N)O(1/N)O(1/N) with the system size. For the Hamiltonian with kicked magnetic fields, the Loschmidt echo depends on the kick period. For a special kick period, the Loschmidt echo shows no evolution at all irrespective of the system size.
We investigate the dynamics of a spin chain with Kitaev-type interaction in a transverse magnetic field, for both a constant field and a periodically kicked field. We investigate analytically the revivals in the dynamical behavior of the magnetization, and measures of local quantum correlations, the concurrence measure of the entanglement, and the quantum discord. The magnetization revivals are seen even for larger chain lengths, with reduced revival peaks, both near and away from the critical point at zero magnetic field. The quantum entanglement and correlation measures, contrary to the expected behavior, exhibit reduced quantum correlations in the critical region, compared to the region away from the critical point. The concurrence is larger than the quantum discord away from the critical region. The pair entanglement is near zero for all times in the critical region, though the quantum discord is nonzero, capturing the underlying diagonal correlations. The kicked magnetic field dynamics shows similar evolution for the magnetization and quantum correlations. For special values of the kicking period, the concurrence shows alternates between zero and nonzero values for successive kicks in the noncritical region.
Abstract We explore the dynamics of long-range Kitaev chain by varying pairing interaction exponent. It is well known that the distinctive features of the nonequilibrium dynamics of a closed quantum system are closely related to the equilibrium phase transitions. Specifically, the return probability (Loschmidt echo) of the system to its initial state, in the finite-size system, is expected to exhibit periodicity after a sudden quench to a quantum critical point. The time of the first revivals scales inversely with the maximum of the group velocity. We show that, contrary to expectations, the periodicity of the return probability breaks for a sudden quench to the non-trivial quantum critical point. Further, we find that the periodicity of the return probability scales inversely with the group velocity at the gap closing point for a quench to the trivial critical point of truly long-range pairing case. Also, analyzing the effect of averaging quenched disorder shows that the revivals in the short-range pairing cases are more robust against disorder than that of the long-range pairing case. We also study the effect of disorder on the non-analyticities of the rate function of the return probability which introduced as a witness of the dynamical phase transition. We show that the non-analyticities in the rate function of return probability are washed out in the presence of strong disorders.
Abstract We consider a local instantaneous quantum dynamical process (QDP) that disturbs the background unitary Hamiltonian dynamics of a spin chain. We consider both a non-unitary incoherent QDP and a coherent unitary QDP intervention of the dynamical evolution of the spin chain. To track the effect of QDP on the dynamics, we investigate the Loschmidt echo, which is quite sensitive to whether the background dynamics is integrable or not. For the integrable case, namely the Heisenberg model, the Loschmidt echo depends on the parameters corresponding to the QDP as well as the time of occurrence of QDP. The probability of reviving the system to its initial state is higher for non-unitary/incoherent QDPs occurring at large time intervals. In the case of a unitary/coherent QDP, some amount of the probability of reviving the state is lost. For the non-integrable background dynamics we consider a kicked Harper model. It exhibits a decaying behaviour when contrasted with integrable dynamics. The decay rate is slower when the corresponding classical Hamiltonian is non-chaotic.
We explore the dynamics of long-range Kitaev chain by varying pairing interaction exponent. It is well known that the distinctive features of the nonequilibrium dynamics of a closed quantum system are closely related to the equilibrium phase transitions. Specifically, the return probability (Loschmidt echo) of the system to its initial state, in the finite-size system, is expected to exhibit periodicity after a sudden quench to a quantum critical point. The time of the first revivals scales inversely with the maximum of the group velocity. We show that, contrary to expectations, the periodicity of the return probability breaks for a sudden quench to the non-trivial quantum critical point. Further, we find that the periodicity of the return probability scales inversely with the group velocity at the gap closing point for a quench to the trivial critical point of truly long-range pairing case. Also, analyzing the effect of averaging quenched disorder shows that the revivals in the short-range pairing cases are more robust against disorder than that of the long-range pairing case. We also study the effect of disorder on the non-analyticities of the rate function of the return probability which introduced as a witness of the dynamical phase transition. We show that the non-analyticities in the rate function of return probability are washed out in the presence of strong disorders. © 2020 IOP Publishing Ltd.
By introducing the Dzyaloshinsky–Moriya (DM) interaction, the Loschmidt Echo (LE) of a quantum system consisting of a central spin and its surrounding environment characterized by an XY spin chain was investigated. The analytical expression of the LE was obtained. The effects of magnetic field, DM interaction, spin chain size, and anisotropy parameter on the evolution behavior of the LE were discussed numerically. At the critical points of magnetic field, the LE presents an obvious decay following with local oscillation. The decaying and oscillation behavior of the LE can be tuned by the intensity of the DM interaction. With increasing the spin chain size, the decaying amplitude is increased and the following oscillation behavior vanishes for the spin chain with larger size. The effect of the anisotropy parameter on the LE presents different properties at both sides of the knee point. The effect of the DM interaction on the decay of the LE is also related to the value of anisotropy parameter.
In this paper, we study the Loschmidt Echo (LE) of a two-dimensional Kitaev model residing on a honeycomb lattice which is chosen to be an environment that is coupled globally to a central spin. The decay of the LE is highly influenced by the quantum criticality of the environmental spin model, e.g., it shows a sharp dip close to the anisotropic quantum critical point (AQCP) of its phase diagram. The early time decay and the collapse and revival as a function of time at AQCP also exhibit interesting scaling behavior with the system size, which is verified numerically. It has also been observed that the LE stays vanishingly small throughout the gapless phase of the model. The above study has also been extended to the 1D Kitaev model, i.e. when one of the interaction terms vanishes.