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arXiv:1912.04603 · Physical Review A

Quench dynamics of entanglement spectra and topological superconducting phases in a long-range Hamiltonian

Kaixiang Su, Zheng‐Hang Sun, Heng Fan

Published Jun 11, 2020• 9 citations
arXivHTMLPDF

Abstract

We study the quench dynamics of entanglement spectra in the Kitaev chain with variable-range pairing quantified by power-law decay rate $\ensuremath{\alpha}$. Considering the post-quench Hamiltonians with flat bands, we demonstrate that the presence of entanglement-spectrum crossings during its dynamics is able to characterize the topological phase transitions (TPTs) in both short-range ($\ensuremath{\alpha}>1$) and long-range ($\ensuremath{\alpha}<1$) sectors. Properties of entanglement-spectrum dynamics are revealed for the quench protocols in the long-range sector or with $\ensuremath{\alpha}$ as the quench parameter. In particular, when the lowest upper-half entanglement-spectrum value of the initial Hamiltonian is smaller than the final one, the TPTs can also be diagnosed by the difference between the lowest two upper-half entanglement-spectrum values if the halfway winding number is not equal to that of the initial Hamiltonian. Moreover, we discuss the stability of characterizing the TPTs via entanglement-spectrum crossings against energy dispersion in the long-range model.

DOI
10.1103/physreva.101.063613

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We study the quench dynamics of entanglement spectra in the Kitaev chain with variable-range pairing quantified by power-law decay rate \ensuremathα\ensuremath{\alpha}\ensuremathα. Considering the post-quench Hamiltonians with flat bands, we demonstrate that the presence of entanglement-spectrum crossings during its dynamics is able to characterize the topological phase transitions (TPTs) in both short-range (\ensuremathα>1\ensuremath{\alpha}>1\ensuremathα>1) and long-range (\ensuremathα<1\ensuremath{\alpha}<1\ensuremathα<1) sectors. Properties of entanglement-spectrum dynamics are revealed for the quench protocols in the long-range sector or with \ensuremathα\ensuremath{\alpha}\ensuremathα as the quench parameter. In particular, when the lowest upper-half entanglement-spectrum value of the initial Hamiltonian is smaller than the final one, the TPTs can also be diagnosed by the difference between the lowest two upper-half entanglement-spectrum values if the halfway winding number is not equal to that of the initial Hamiltonian. Moreover, we discuss the stability of characterizing the TPTs via entanglement-spectrum crossings against energy dispersion in the long-range model.
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.
Long-range interactions exhibit surprising features which have been less explored so far. Here, studying a one-dimensional fermionic chain with long-range hopping and pairing, we discuss some general features associated to the presence of long-range entanglement. In particular, after determining the algebraic decays of the correlation functions, we prove that a long-range quantum mutual information exists if the exponent of the decay is not larger than one. Moreover, we show that the time evolution triggered by a quantum quench between short-range and long-range regions, can be characterized by dynamical quantum phase transitions without crossing any phase boundary. We show, also, that the adiabatic dynamics is dictated by the divergence of a topological length scale at the quantum critical point, clarifying the violation of the Kibble-Zurek mechanism for long-range systems.
We study the role of long-range interactions (more precisely, the long-range superconducting gap term) on the nonequilibrium dynamics considering a long-range ppp-wave superconducting chain in which the superconducting term decays with distance between two sites in a power-law fashion characterized by an exponent $\ensuremath{\alpha}$. We show that the Kibble-Zurek scaling exponent, dictating the power-law decay of the defect density in the final state reached following a slow (in comparison to the time scale associated with the minimum gap in the spectrum of the Hamiltonian) quenching of the chemical potential $\ensuremath{\mu}$ across a quantum critical point, depends nontrivially on the exponent $\ensuremath{\alpha}$ as long as $\ensuremath{\alpha}<2$; on the other hand, for $\ensuremath{\alpha}>2$, we find that the exponent saturates to the corresponding well-known value of 1/21/21/2 expected for the short-range model. Furthermore, studying the dynamical quantum phase transitions manifested in the nonanalyticities in the rate function of the return possibility I(t)I(t)I(t) in subsequent temporal evolution following a sudden change in $\ensuremath{\mu}$, we show the existence of a new region; in this region, we find three instants of cusp singularities in I(t)I(t)I(t) associated with a single sector of Fisher zeros. Notably, the width of this region shrinks as $\ensuremath{\alpha}$ increases and vanishes in the limit $\ensuremath{\alpha}\ensuremath{\rightarrow}2$, indicating that this special region is an artifact of the long-range nature of the Hamiltonian.
We explore the behavior of the disconnected entanglement entropy (DEE) across the topological phases of a long range interacting Kitaev chain where the long range interactions decay as a power law with an exponent α\alphaα. We show that while the DEE may not remain invariant deep within the topologically non-trivial phase when α<1\alpha<1α<1, it still shows a quantized discontinuous jump at the quantum critical point and can act as a strong marker for the detection of topological phase transition for the long range interacting systems. We also study the time evolution of the disconnected entanglement entropy after a sudden quench of chemical potential within same phase. In the short range limit of a finite chain, the DEE is expected to remain constant upto a critical time which diverges in the thermodynamic limit. However, we find that no such critical time exists in the presence of long range interactions (α<1\alpha<1α<1).
We explore the behavior of the disconnected entanglement entropy (DEE) across the topological phases of a long-range interacting Kitaev chain where the long-range interactions decay as a power law with an exponent $\ensuremath{\alpha}$. We show that while the DEE may not remain invariant deep within the topologically nontrivial phase when $\ensuremath{\alpha}<1$, it nevertheless shows a quantized discontinuous jump at the quantum critical point and can act as a strong marker for the detection of topological phase transition. We also study the time evolution of the DEE after a sudden quench of the chemical potential within the same phase. In the short-range limit of a finite chain, the DEE is expected to remain constant up to a critical time after the quench, which diverges in the thermodynamic limit. However, no such critical time is found to exist when the long-range interactions dominate (i.e., $\ensuremath{\alpha}<1$).
Abstract In this paper, we investigate how the entanglement spectrum relaxes to its steady-state values in one-dimensional quadratic systems after a quantum quench. In particular, we apply saddle-point expansion to dimerized chains and 1D p -wave superconductors. We found the entanglement spectrum to always exhibit a power-law relaxation superimposed with oscillations at certain characteristic angular frequencies. For dimerized chains, we found the exponent ν of the power-law decay to always be 3 / 2 . For 1D p -wave superconductors, however, we found that, depending on the initial and final Hamiltonian, the exponent ν can take its value from a limited list of values, the smallest possible of which is ν = 1 / 2 , which leads to a very slow convergence to its steady-state value.