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Physical review. B./Physical review. B

Quantum criticality and universality in the stationary state of the long-range Kitaev model

Akash Mitra, Sanku Paul, Shashi C. L. Srivastava

Published Mar 12, 2025• 3 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.111.104308

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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 \ensuremathα\ensuremath{\alpha}\ensuremathα. Using quantum information-theoretic measures, such as mutual information and logarithmic negativity, we show that irrespective of the values of \ensuremathα\ensuremath{\alpha}\ensuremathα, 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 \ensuremathα=0\ensuremath{\alpha}=0\ensuremathα=0 and \ensuremathα=2\ensuremath{\alpha}=2\ensuremathα=2. Therefore, information of the universality class can be inferred from the stationary state.
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 ααα. Using quantum information-theoretic measures, such as mutual information and logarithmic negativity, we show that, irrespective of the values of ααα, 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 non-critical points, signature of quantum criticality persists in the stationary state. Furthermore, the effective central charge, calculated from both mutual information and logarithmic negativity of stationary state following a critical-to-critical quench, agrees with the central charge of the corresponding ground states for both α=0α= 0α=0 and α=2α= 2α=2. Therefore, information of the universality class can be inferred from the stationary state.
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.
Entanglement exhibits universal behavior near the ground-state critical point where correlations are long ranged and the thermodynamic entropy is vanishing. On the other hand, a quantum quench imparts extensive energy and results in a build up of entropy, hence no critical behavior is expected at long times. In this work, we present a new paradigm in the quench dynamics of integrable spin chains which exhibit a ground-state order-disorder phase transition at a critical line. Specifically, we consider a quench along the critical line which displays a volume-law behavior of the entropy and exponentially decaying correlations; however, we show that quantum criticality is hidden in higher-order correlations and becomes manifest via measures such as the mutual information and logarithmic negativity. Furthermore, we showcase the scale invariance of the Rényi mutual information between disjoint regions as further evidence for genuine critical behavior. We attribute the emergent quantum criticality to the soft mode not getting excited in spite of the quench. Moreover, the results presented here are universal to models whose low-energy or long-wavelength dynamics are well described by a free-fermionic field theory. Our results are amenable to an experimental realization on different quantum simulator platforms, particularly the Rydberg simulators. Published by the American Physical Society 2024
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.
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.
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.