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arXiv:2206.09688 · Physical review. B./Physical review. B

Correlations, long-range entanglement, and dynamics in long-range Kitaev chains

Gianluca Francica, Luca Dell’Anna

Published Oct 14, 2022• 23 citations
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Abstract

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.

DOI
10.1103/physrevb.106.155126

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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 propose and analyze a generalization of the Kitaev chain for fermions with long-range p-wave pairing, which decays with distance as a power law with exponent α. Using the integrability of the model, we demonstrate the existence of two types of gapped regimes, where correlation functions decay exponentially at short range and algebraically at long range (α > 1) or purely algebraically (α < 1). Most interestingly, along the critical lines, long-range pairing is found to break conformal symmetry for sufficiently small α. This is accompanied by a violation of the area law for the entanglement entropy in large parts of the phase diagram in the presence of a gap and can be detected via the dynamics of entanglement following a quench. Some of these features may be relevant for current experiments with cold atomic ions.
Abstract The presence of algebraically decaying long-range interactions may alter the critical as well as topological behaviour of a quantum many-body systems. However, when the interaction decays at a faster rate, the short-range behaviour is expected to be retrieved. Similarly, the long-range nature of interactions has a prominent signature on the out of equilibrium dynamics of these systems, e.g. in the growth of the entanglement entropy following a quench, the propagation of mutual information and non-equilibrium phase transitions. In this review, we summarize the results of long-range interacting classical and quantum Ising chains mentioning some recent results. Thereafter, we focus on the recent developments on the integrable long-range Kitaev chain emphasising the role of long-range superconducting pairing term on determining its topological phase diagram and out of equilibrium dynamics and also incorporate relevant discussions on the corresponding Ising version.
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., $\ensuremath{\alpha}<1$).
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 investigate the signature of quantum criticality in the long-time stationary state of the long-range Kitaev chain by performing various quench protocols. In this model, the pairing interaction decays with distance according to a power law with exponent $\ensuremath{\alpha}$. Using quantum information-theoretic measures, such as mutual information and logarithmic negativity, we show that irrespective of the values of $\ensuremath{\alpha}$, critical-to-critical quench displays quantum criticality even in the stationary state. Remarkably, in the presence of long-range pairing interactions, where fermionic correlators decay algebraically even at noncritical points, the signature of quantum criticality persists in the stationary state. Furthermore, the effective central charge, calculated from both mutual information and logarithmic negativity of the stationary state following a critical-to-critical quench, agrees with the central charge of the corresponding ground states for both $\ensuremath{\alpha}=0$ and $\ensuremath{\alpha}=2$. Therefore, information of the universality class can be inferred from the stationary state.