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arXiv:2309.02686 · Physical review. B./Physical review. B

Dynamical relaxation behavior of an extended XY chain with a gapless phase following a quantum quench

Kaiyuan Cao, Yayun Hu, Peiqing Tong, Guangwen Yang

Published Jan 9, 2024• 8 citations
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Abstract

We investigate the dynamical relaxation behavior of the two-point correlation in extended XY models with a gapless phase after quenches from various initial states. Specifically, we study the XY chain with gapless phase induced by the following additional interactions: Dzyaloshinskii-Moriya interaction and the XZY-YZX type of three-site interaction. When quenching from the gapped phase, we observe that the additional interactions have no effect on the relaxation behavior. The relaxation behavior is $\ensuremath{\delta}{C}_{mn}(t)\ensuremath{\sim}{t}^{\ensuremath{-}3/2}$ and $\ensuremath{\sim}{t}^{\ensuremath{-}1/2}$ for the quench to the commensurate phase and the incommensurate phase, respectively. However, when quenching from the gapless phase, we demonstrate that the scaling behavior of $\ensuremath{\delta}{C}_{mn}(t)$ is changed to $\ensuremath{\sim}{t}^{\ensuremath{-}1}$ for the quench to the commensurate phase, and the decay of $\ensuremath{\delta}{C}_{mn}(t)$ follows $\ensuremath{\sim}{t}^{\ensuremath{-}1}$ or $\ensuremath{\sim}{t}^{\ensuremath{-}1/2}$ for the quench to the incommensurate phase depending on the parameters of prequench Hamiltonian. We also establish the dynamical phase diagrams based on the dynamical relaxation behavior of $\ensuremath{\delta}{C}_{mn}(t)$ in the extended XY models.

DOI
10.1103/physrevb.109.024303

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We investigate the dynamical relaxation behavior of the two-point correlation in extended XY models with a gapless phase after quenches from various initial states. Specifically, we study the XY chain with gapless phase induced by the following additional interactions: Dzyaloshinskii-Moriya interaction and the XZY-YZX type of three-site interaction. When quenching from the gapped phase, we observe that the additional interactions have no effect on the relaxation behavior. The relaxation behavior is \ensuremathδCmn(t)\ensuremath∼t\ensuremath−3/2\ensuremath{\delta}{C}_{mn}(t)\ensuremath{\sim}{t}^{\ensuremath{-}3/2}\ensuremathδCmn​(t)\ensuremath∼t\ensuremath−3/2 and \ensuremath∼t\ensuremath−1/2\ensuremath{\sim}{t}^{\ensuremath{-}1/2}\ensuremath∼t\ensuremath−1/2 for the quench to the commensurate phase and the incommensurate phase, respectively. However, when quenching from the gapless phase, we demonstrate that the scaling behavior of \ensuremathδCmn(t)\ensuremath{\delta}{C}_{mn}(t)\ensuremathδCmn​(t) is changed to \ensuremath∼t\ensuremath−1\ensuremath{\sim}{t}^{\ensuremath{-}1}\ensuremath∼t\ensuremath−1 for the quench to the commensurate phase, and the decay of \ensuremathδCmn(t)\ensuremath{\delta}{C}_{mn}(t)\ensuremathδCmn​(t) follows \ensuremath∼t\ensuremath−1\ensuremath{\sim}{t}^{\ensuremath{-}1}\ensuremath∼t\ensuremath−1 or \ensuremath∼t\ensuremath−1/2\ensuremath{\sim}{t}^{\ensuremath{-}1/2}\ensuremath∼t\ensuremath−1/2 for the quench to the incommensurate phase depending on the parameters of prequench Hamiltonian. We also establish the dynamical phase diagrams based on the dynamical relaxation behavior of \ensuremathδCmn(t)\ensuremath{\delta}{C}_{mn}(t)\ensuremathδCmn​(t) in the extended XY models.
We study the dynamical quantum phase transitions (DQPTs) in the XY chains with the Dzyaloshinskii–Moriya interaction and the XZY–YZX type of three-site interaction after a sudden quench. Both the models can be mapped to the spinless free fermion models by the Jordan–Wigner and Bogoliubov transformations with the form H = ∑ k ε k ( η k † η k − 1 2 ) , where the quasiparticle excitation spectra εk may be smaller than 0 for some k and are asymmetrical (εk ≠ ε –k ). It is found that the factors of Loschmidt echo equal 1 for some k corresponding to the quasiparticle excitation spectra of the pre-quench Hamiltonian satisfying εk ⋅ ε –k < 0, when the quench is from the gapless phase. By considering the quench from different ground states, we obtain the conditions for the occurrence of DQPTs for the general XY chains with gapless phase, and find that the DQPTs may not occur in the quench across the quantum phase transitions regardless of whether the quench is from the gapless phase to gapped phase or from the gapped phase to gapless phase. This is different from the DQPTs in the case of quench from the gapped phase to gapped phase, in which the DQPTs will always appear. Moreover, we analyze the different reasons for the absence of DQPTs in the quench from the gapless phase and the gapped phase. The conclusion can also be extended to the general quantum spin chains.
We study the universal real-time relaxation behaviors of a long-range quantum XY chain following a quench. Our research includes both the noncritical and critical quench. In the case of noncritical quench, i.e., neither the initial state nor the postquench Hamiltonian is at a critical point of equilibrium phase transition, a quench to the commensurate phase or incommensurate phase gives a scaling of t−3/2t^{-3/2}t−3/2 or t−1/2t^{-1/2}t−1/2, respectively, which is the same as the counterpart of the short-range XY model. However, for a quench to the boundary line between the commensurate and incommensurate phases, the scaling law t−μt^{-μ}t−μ may be different from the t−3/4t^{-3/4}t−3/4 law of the counterpart of the short-range model. More interestingly, the decaying exponent μμμ may depend on the choice of the parameters of the postquench Hamiltonian because of the different asymptotic behaviors of the energy spectrum. Furthermore, in certain cases, the scaling behavior may be outside the range of predictions made by the stationary phase approximation, because an inflection point emerges in the energy spectrum. For the critical quench, i.e., the initial state or the postquench Hamiltonian is at a critical point of equilibrium phase transition, the aforementioned scaling law t−μt^{-μ}t−μ may be changed because of the gap-closing property of the energy spectrum of the critical point.
Abstract We investigate the relaxation dynamics of the fermion two-point correlation function C mn ( t ) = 〈 ψ ( t ) ∣ c m † c n ∣ ψ ( t ) 〉 in the XY chain with alternating nearest-neighbor hopping interaction after a quench. We find that the deviation δ C mn (t) = C mn (t) − C mn (∞) decays with time following the power law behavior t −μ , where the exponent μ depends on whether the quench is to the commensurate phase (μ = 1) or incommensurate phase ( μ = 1 2 ). This decay of δ C mn (t) arises from the transient behavior of the double-excited quasiparticle occupations and the transitions between different excitation spectra. Furthermore, we find that the steady value C mn (∞) only involves the average fermion occupation numbers (i.e. the average excited single particle) over the time-evolved state, which are different from the ground state expectation values. We also observe nonanalytic singularities in the steady value C mn (∞) for the quench to the critical points of the quantum phase transitions (QPTs), suggesting its potential use as a signature of QPTs.
We study the universal real-time relaxation behaviors of a long-range Kitaev chain following a quench, and the results are compared to a short-range quantum XY chain. Our research includes both noncritical and critical quenches. In the case of noncritical quench, i.e., neither the initial state nor the postquench Hamiltonian is at a critical point of the equilibrium phase transition, a quench to the commensurate phase or incommensurate phase gives a scaling of ${t}^{\ensuremath{-}3/2}$ or ${t}^{\ensuremath{-}1/2}$, respectively, which is the same as the counterpart of the short-range quantum XY model. However, for a quench to the boundary line between the commensurate and incommensurate phases, the scaling law ${t}^{\ensuremath{-}\ensuremath{\mu}}$ may be different from the ${t}^{\ensuremath{-}3/4}$ law of the counterpart of the short-range quantum XY model. More interestingly, the decaying exponent $\ensuremath{\mu}$ may depend on the choice of the parameters of the postquench Hamiltonian because of the different asymptotic behaviors of the energy spectrum. Furthermore, in certain cases, the scaling behavior may be outside the range of predictions made by the stationary phase approximation, because an inflection point emerges in the energy spectrum. For the critical quench, i.e., the initial state or the postquench Hamiltonian is at a critical point of equilibrium phase transition, the aforementioned scaling law ${t}^{\ensuremath{-}\ensuremath{\mu}}$ may be changed because of the gap-closing property of the energy spectrum of the critical point.
The dynamical quantum phase transitions (DQPTs) in quantum spin chains with gapless phases after a sudden quench are studied. We mainly consider the general systems with asymmetrical quasiparticle excitation spectra and obtain the general expression of the Loschmidt echo as well as the general conditions for the occurrence of DQPTs. As two examples, we study the DQPTs in the \emph{XY} chains with Dzyaloshinskii-Moriya interaction and \emph{XZY-YZX} type of three-site interaction. It's found that the DQPTs may not occur in the quench across the quantum phase transitions regardless of whether the quench is from the gapless phase to gapped phase or from the gapped phase to gapless phase. This is different from the DQPTs in the case of quench from the gapped phase to gapped phase, in which the DQPTs will always appear. Besides, we also analyze the different reasons for the absence of DQPTs in the quench from the gapless phase and the gapped phase.
We study the universal dynamical relaxation behaviors of a quantum XY chain following a quench, paying special attention to the case in which the initial state is a critical ground state, or the postquenched Hamiltonian is at a critical point of equilibrium quantum phase transition, or both of them are critical. In such a ``critical quench,'' we find very interesting real-time dynamical scaling behaviors and we find crossover phenomena between them. For a quench from a noncritical point to a critical point, we find that, compared to the noncritical quench, the universal power-law scaling behavior does not change; however, there may be a crossover between the exponential decaying behavior and the power-law scaling. For a quench from a critical point to a noncritical point, the power-law scaling behaviors ${t}^{\ensuremath{-}3/2}$ and ${t}^{\ensuremath{-}3/4}$ in the noncritical quenches may be changed to ${t}^{\ensuremath{-}1}$ and ${t}^{\ensuremath{-}1/2}$, respectively. If the prequenched Hamiltonian is set to be a point that is close to but not exactly at a critical point, we find interesting crossover phenomena between different power-law scaling behaviors. We also study the quench from the vicinity of a multicritical point, we find crossover behaviors that are induced by a different mechanism, and we find another crossover exponent. All the results are related to the gap-closing properties of the energy spectrum of the critical points.