Disordered Kitaev chain with long-range pairing: Loschmidt echo revivals anddynamical phase transitions

Utkarsh Mishra,1, ∗ R. Jafari,2, 3, 4, † and Alireza Akbari1, 5, 6, 2, ‡1Asia Pacific Center for Theoretical Physics (APCTP), Pohang, Gyeongbuk, 790-784, Korea

2Department of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran3Beijing Computational Science Research Center, Beijing 100094, China

4Department of Physics, University of Gothenburg, SE 412 96 Gothenburg, Sweden5Department of Physics, POSTECH, Pohang, Gyeongbuk 790-784, Korea

6Max Planck POSTECH/Korea Research Initiative (MPK), Gyeongbuk 376-73, Korea(Dated: June 4, 2019)

We explore the dynamics of long-range Kitaev chain by varying pairing interaction exponent, α.It is well known that distinctive characteristics on the nonequilibrium dynamics of a closed quantumsystem are closely related to the equilibrium phase transitions. Specifically, the return probabilityof the system to its initial state (Loschmidt echo), in the finite size system, is expected to exhibitvery nice periodicity after a sudden quench to a quantum critical point. Where the periodicity ofthe revivals scales inversely with the maximum of the group velocity. We show that, contrary toexpectations, the periodicity of the return probability breaks for a sudden quench to the non-trivialquantum critical point. Further, We find that, the periodicity of return probability scales inverselywith the group velocity at the gap closing point for a quench to the trivial critical point of trulylong-range pairing case, α < 1. In addition, analyzing the effect of averaging quenched disordershows that the revivals in the short range pairing cases are more robust against disorder than thatof the long rang pairing case. We also study the effect of disorder on the non-analyticities of ratefunction of the return probability which introduced as a witness of the dynamical phase transition.We exhibit that, the non-analyticities in the rate function of return probability are washed out inthe presence of strong disorders.

I. INTRODUCTION

Recent remarkable advancement of the experimentalstudies of ultracold atoms, trapped in optical lattices[1, 2], provide a new framework for studying the nonequi-librium dynamics of isolated quantum systems, in par-ticular from the viewpoint of quantum quenches [3, 4].Quenching a quantum system to/across the critical pointraises striking questions, especially, when the time evo-lution is unitary [3, 4]. An abrupt change of the state ofa closed quantum system leads to a unitary time evolu-tion. When a sudden quench happens, the evolution isdetermined by the Loschmidt echo (LE) [5], modulus ofoverlaps between the eigenstates of the pre-quenched andpost-quenched Hamiltonians expressed by a given changeof parameters on which the Hamiltonian depends. Fora sudden quench to a quantum critical point, finite-sizecase studies reveal that the LE of several models, withshort range interaction, exhibits a periodic revival struc-ture, formed by brief deviation from its mean value [6–13], which can be used as a dynamical witness of quantumcriticality [6, 7, 14–19].

In addition, the nonanalyticities in the rate functionof the Loschmidt echo (return probability), when thequench is performed across an equilibrium quantum crit-

∗utkarsh.mishra@apctp.org†jafari@iasbs.ac.ir‡alireza@apctp.org

ical point, has been lately used to introduce the notionof dynamical phase transitions (DPTs) [20–25].

Very recently, the studies of the connection betweenquenching dynamics and quantum phase transition [5, 12,20], the topological order [26–28] and also the dynamicsof an edge state [29–32], have attracted the attentionof the scientists. Specifically, searching the robustnessand response of the topological edge states to quantumquenches [28, 31, 33–35]. The behavior of edge statesunder a sudden quantum quench has been investigated intwo-dimensional topological insulator [36], where it wasshown that, in the abrupt transition from the topologicalinsulator to the trivial insulator phase, there is a collapseand revival of the edge states [30, 34]. Similar resultshave been obtained for the one-dimensional Kitaev model[31, 32, 37, 38].

Current experimental progresses in realization of long-range interacting quantum models with tunable longrange interactions [39] has renewed the interest in study-ing the non-equilibrium dynamics of quantum systemswith infinite-range interactions [40–44]. Motivated bythe short-range one dimensional Kitaev chain [31, 32], along-range pairing version of an integrable p-wave super-conducting chain of fermions has been proposed, wherestrength of super conducting pairing between two sitesseparated by a distance r falling off in a power-law fash-ion as rα [44–56]. Despite numerous attempts to link thesignificant features of quantum phase transition (QPT)to the quench dynamics (LE), the general principle hasnot been established to connect the QPT to dynamics ofthe systems, specifically in the disordered and long range

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quantum systems.In this paper, we study the effects of long range interac-

tion and disorder on the LE of the ground state and edgestates of the long range pairing Kitaev (LRPK) modelwith open boundary condition. It should be mentionedthat the integrability of the LRPK model breaks in theopen boundary condition and also in the presence of dis-order. To the best of our knowledge, such contributionshave not been studied in previous works and can shedlight on several new effects to the subject.

We show that, in the clean truly LRPK chain, the re-vival time (periodicity of the revival) in the LE of thefinite size system scales exponentially with the power-law exponent, α, and inversely with the group velocityat the gap closing point. While in the models with shortrange interaction, as expected, the revivals time is con-trolled by the maximum group velocity[9, 10]. We alsoshow that a surprising result occurs for a quench to thenon-trivial critical point where the periodic revivals elim-inated and the LE oscillating randomly around its meanvalue. Moreover, in the presence of strong local disorderthe revivals washed out, and the first revivals in shortrange pairing cases are more robust than that of the longrange pairing cases. We further show that, the LE ofthe localized edge mode in the clean case, for a quenchto the critical point, exhibits periodic revivals which in-crease consequently with the increase of the power-lawexponent, α. These revivals are suppressed in the pres-ence of disorder and disappear for the long-range modelwhile few of them survives in the short-range model forthe same strength of disorders. Finally, studying thedynamical phase transition in the presence of disordershows that the strong disorder leads to the disappear-ance of singularity in the rate function of the LE (returnprobability) [20].

The paper is presented as follows: Sec. II describesthe model and its numerically obtained band structurefor the open chain. The scheme of global quenching isdescribed in Sec. III with the techniques to solve the dy-namics of the underlying Hamiltonian in the presenceand absence of disorder. Dynamics of edge state undersudden quenching is performed in Sec. IV and the effectof disorder is discussed. Scaling of revival time, obtainedfrom the dynamics of Loschmidt echo in the finite-sizesystem, is reported in Sec. V including its behavior inpresence of disorder and role of power-law pairing ex-ponent. In Sec. VI, we present the result on the dy-namical phase transition in presence of disorder for long-and the short-range limiting case of the power-law expo-nent. A discussion on the results is included in conclusionSec. VII.

II. THE MODEL

We consider long-range pairing Kitaev chain wherethe pairing interaction is not only present between thenearest-neighbor sites but at all other distant sites. The

Hamiltonian of the model, describing N−free fermions inone dimension lattice, is given by [47]

H =−N∑j=1

(w(a†jaj+1 +H.c.) + µ(nj −

12))

+ ∆2

N∑j=1

N−1∑`=1

d−α` (ajaj+` +H.c.).

(1)

Here, a†j(aj) is the fermioninc creation (annihilation) op-erator on site j, nj = a†jaj , w is the tunnelling rate, µis the chemical potential, and ∆ denotes the strength ofthe p-wave pairing. The summation index ` varies foreach site in the lattice with weightage dα` . For a closedchain, anti-periodic boundary conditions, aj+N = −ajmakes the effect of long-range paring term intact, withthe choice of d` = ` (d` = N − `) if ` ≤ N/2 (` ≥ N/2),respectively [47, 52]. For an open chain, d` = ` and wedrop terms containing aj>N .

A proposal to realize the Hamiltonian of long-rangepairing Kitaev chain in the experiment has been put for-ward recently [57]. For α→∞, one recovers the standardshort-range Kitaev model with pairing range limited to

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