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arXiv:1705.03770 · Physical review. B./Physical review. B

Probing the role of long-range interactions in the dynamics of a long-range Kitaev chain

Anirban Dutta, Amit Dutta

Published Sep 8, 2017• 107 citations
arXivHTMLPDF

Abstract

We study the role of long-range interactions (more precisely, the long-range superconducting gap term) on the nonequilibrium dynamics considering a long-range $p$-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 $1/2$ 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 $I(t)$ 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 $I(t)$ 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.

DOI
10.1103/physrevb.96.125113

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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.
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Abstract We discuss how nonlocality originates in long-range quantum systems and how it affects their dynamics at and out of equilibrium. We focus in particular on the Kitaev chains with long-range pairings and on the quantum Ising chain with long-range antiferromagnetic coupling (both having a power-law decay with exponent α ). By studying the dynamic correlation functions, we find that for every finite α two different behaviours can be identified, one typical of short-range systems and the other connected with locality violation. The latter behaviour is shown related also with the known power-law decay tails previously observed in the static correlation functions, and originated by modes—having in general energies far from the minima of the spectrum—where particular singularities develop as a consequence of the long-rangedness of the system. We refer to these modes as to ‘singular’ modes, and as to ‘singular dynamics’ to their dynamics. For the Kitaev model they are manifest, at finite α , in derivatives of the quasiparticle energy, the order of the derivatives at which the singularity occurs is increasing with α . The features of the singular modes and their physical consequences are clarified by studying an effective theory for them and by a critical comparison of the results from this theory with the lattice ones. Moreover, a numerical study of the effects of the singular modes on the time evolution after various types of global quenches is performed. We finally present and discuss the presence of singular modes and their consequences in interacting long-range systems by investigating in the long-range Ising quantum chain, both in the deep paramagnetic regime and at criticality, where they also play a central role for the breakdown of conformal invariance.
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.
We theoretically study the dynamics of a transverse-field Ising chain with power-law decaying interactions characterized by an exponent α, which can be experimentally realized in ion traps. We focus on two classes of emergent dynamical critical phenomena following a quantum quench from a ferromagnetic initial state: The first one manifests in the time-averaged order parameter, which vanishes at a critical transverse field. We argue that such a transition occurs only for long-range interactions α≤2. The second class corresponds to the emergence of time-periodic singularities in the return probability to the ground-state manifold which is obtained for all values of α and agrees with the order parameter transition for α≤2. We characterize how the two classes of nonequilibrium criticality correspond to each other and give a physical interpretation based on the symmetry of the time-evolved quantum states.
In this work, we investigate quenches in a free-fermion chain with long-range hopping, which decay with the distance with an exponent $\ensuremath{\nu}$ and has range DDD. By exploring the exact solution of the model, we found that the dynamic free energy is nonanalytical in the thermodynamic limit, whenever the sudden quench crosses the equilibrium quantum critical point. We were able to determine the nonanalyticities of dynamic free energy f(t)f(t)f(t) at some critical times tc{t}^{c}tc by solving nonlinear equations. We also show that the Yang-Lee-Fisher (YLF) zeros cross the real-time axis at those critical times. We found that the number of nontrivial critical times, Ns,{N}_{s},Ns​, depends on $\ensuremath{\nu}$ and DDD. In particular, we show that for small $\ensuremath{\nu}$ and large DDD the dynamic free energy presents nonanalyticities in any time interval $\mathrm{\ensuremath{\Delta}}t\ensuremath{\sim}1/D\ensuremath{\ll}1$, i.e., there are nonanalyticities at almost all times. For the spacial case $\ensuremath{\nu}=0$, we obtain the critical times in terms of a simple expression of the model parameters and also show that f(t)f(t)f(t) is nonanalytical even for finite system under antiperiodic boundary condition, when we consider some special values of quench parameters. We also show that, generically, the first derivative of the dynamic free energy is discontinuous at the critical time instant when the YLF zeros are nondegenerate. On the other hand, when they become degenerate, all derivatives of f(t)f(t)f(t) exist at the associated critical instant.