Edge mode dynamics of quenched topological wires

0.85 MB·

by  P. D. Sacramento

Edge mode dynamics of quenched topological wires
P. D. Sacramento1
1 CeFEMA, Instituto Superior T´ecnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
(Dated: January 22, 2016)
The fermionic and Majorana edge mode dynamics of various topological systems is compared,
after a sudden global quench of the Hamiltonian parameters takes place. Attention is focused on
the regimes where the survival probability of an edge state has oscillations either due to critical or
off-critical quenches. The nature of the wave functions and the overlaps between the eigenstates of
different points in parameter space determine the various types of behaviors, and the distinction
6 duetotheMajorananatureoftheexcitationsplaysalesserrole. Performingasequenceofquenches
1 it is shown that the edge states, including Majorana modes, may be switched off and on. Also, the
0 generation of Majoranas due to quenching from a trivial phase is discussed.
2
PACSnumbers: 05.30.Rt,05.70.Ln,03.65.Vf
n
a
J
I. INTRODUCTION formed self-consistently56, showed the importance of the
1 topological properties in the evolution of the system57
2
and raised questions regarding the survival of the topo-
Quenching a quantum system raises interesting
logical order to the quench52,58–60. So the issue is not
] questions1, particularly when the evolution is unitary.
l resolved and is attracting considerable attention.
l When a sudden quench takes place the evolution is de-
a Ingeneral,quenchesthatleadthesystemfromatopo-
h termined by the overlaps between the eigenstates of the
logical phase to a trivial phase imply a decay of the gap-
- instantaneous Hamiltonians, prior and after the quench,
s less edge states and, in the reverse quenching, the topo-
expressed by a given change of a set of parameters on
e
logical states are not generated. However, since the sys-
m which the Hamiltonian depends.
tems have finite extent, a revival of the original states is
. An abrupt change of the state of an isolated quan- observed with a revival time that scales with the system
t
a tumsystemleadstoaunitarytimeevolutionand, there- size. Also, as will be shown here, Majorana edge states
m fore, the issue of thermalization has been addressed2,3.
maybegeneratedfromatrivialphaseunderappropriate
- In general, it is expected that correlation functions conditions.
d stabilize4–23. In the cases of soluble and integrable sys-
The behavior of edge states under an abrupt quantum
n
tems thermalization breaks down as one approaches an
o quench has been considered very recently in the context
integrable point. However, some sort of thermalization
c ofatwo-dimensionaltopologicalinsulator61,whereitwas
[ is predicted for which an equilibrium like distribution is foundthat,inthesuddentransitionfromthetopological
expectedintermsofageneralizedGibbsensemble,ofthe insulator to the trivial insulator phase, there is a col-
1 (infinitely) many conserved quantities24–43.
v lapse and revival of the edge states62. Similar results
6 An interesting case is the effect of a sudden quench of were obtained for the one-dimensional Kitaev model63,
7 theparametersofanHamiltonianwithtopologicalprop- also studying the signature of the Majoranas in the en-
4 ertiesand,particularly,achangeofparametersthatleads tanglement spectrum64. Their dynamical formation and
5 to a change of topological properties. Specificaly, how manipulation has been considered in65 and58. The ro-
0
thetopologicalpropertiesandtopologicaledgestatesre- bustness of edge states may also be studied in the con-
.
1 spond to such quenches and how robust they are. Topo- textofslowquenchesfromatopologicalphasetoatrivial
0 logical systems have attracted interest44,45 and, specif- phase.54,66,67.
6 ically, topological superconductors46 due to the predic- The effect of parity blocking on the dynamics of the
1
tionofMajoranafermions47–51. Ithasbeenshownbefore edge modes has been considered recently in which case
:
v that topological systems are quite robust to a quantum the dynamics is restricted if there is a change in fermion
Xi quench, as exemplified by the toric code model52,53. parityaccrossthequench68. Ontheotherhand,theMa-
It has also been shown recently54 that, in the case of joranazeromodesleadtosomeuniversalnon-equilibrium
r
a quenchesininfinitesizetopologicalsuperconductors,the signature in the Loschmidt echo with an universal expo-
Chern number can not be changed by a unitary evolu- nentassociatedwiththealgebraicdecay69. Also,thedy-
tion. The same result was shown more generally for any namics of the tunneling into non-equilibrium edge states
topological system, even though it is possible to change has been proposed as a possible signature of the exis-
the Bott index and topology if the system has a finite tence of these states70. Non-equilibrium situations also
size55. Therefore, in a finite system the Chern number may allow the transport of Majorana edges states using
may change54 and the response of the edge states to a extended gapless regions with a small but finite overlap
time dependent perturbation in finite systems may not with the Majoranas71. Their effect has also been consid-
be protected by topology. Furthermore, quenches in su- ered in72 and in73.
perconducting systems with topological properties, per- While quenches from a topological to a trivial phase
2
lead to decay of the edge states, quenches within the At time t = 0 a sudden transformation of the parame-
same topological phase typically lead to a decrease of ters is performed, ξ → ξ . The eigenstates of the new
0 1
therevivalprobability,buttoafinitevalue. Aninterme- Hamiltonian are given by
diatecaseofaquenchbetweentwotopologicallydifferent
phasesandwithinthesamephaseisaquenchtoacritical H(ξ1)|ψm1(ξ1)(cid:105)=Em1(ξ1)|ψm1(ξ1)(cid:105). (2)
point or critical line, separating two different topologi-
Thetimeevolutionofasingle-particlestate,m ,isgiven
cal phases. At this point the spectrum becomes gapless. 0
by
In some appropriate conditions the survival probability
shows oscillations63. In this work particular attention (cid:88)
will be paid to these oscillations in various topological |ψmI 0(t)(cid:105)= e−iEm1(ξ1)t|ψm1(ξ1)(cid:105)(cid:104)ψm1(ξ1)|ψm0(ξ0)(cid:105)
systems. m1
(3)
We will consider sudden quantum quench in one-
for times t≥t (where t =0). The survival probability
dimensionalsystemsandfocusontheoscillationsatcrit- 0 0
of the initial state |ψ (ξ )(cid:105) is, as usual, defined by
ical quenches and non-critical quenches. The survival m0 0
probability will be studied taking into account finite size P (t)=|(cid:104)ψ (ξ )|ψI (t)(cid:105)|2 (4)
effects. A period doubling is observed as the initial state m0 m0 0 m0
of the quench is moved sufficiently far from the critical We may as well consider further quenches defined in
region. The dynamics of Majorana zero energy states is a sequence of times and sets of parameters as t < t <
0 1
compared with that of finite-energy excited states and t < t < ··· and ξ ,ξ ,ξ ,ξ ,···, respectively. These
2 3 0 1 2 3
the dynamics of Majoranas is also compared with that intervals define regions as I(t ≤ t < t ),II(t ≤ t <
0 1 1
offermioniczeroenergymodes. Theimportanceofover- t ),III(t ≤ t < t ),···. The case of a single quench is
2 2 3
laps and spectrum of the final eigenstates is shown. It clearly obtained taking t → ∞, and so on for further
1
is shown that an appropriate sequence of quenches may quenches.
lead to a on/off process of the existence of Majorana Consider now a case for which we have two quenches
states, with potential application in their manipulation in succession. In this case we have that
and storing of information. Also, it is shown that states
ofatrivialphasemayleadafteraquenchtoatopological |ψmII0(t)(cid:105) = e−iH(ξ2)t|ψmI 0(t1)(cid:105)
phase to Majorana states. (cid:88)
Universal single-frequency oscillations in the entangle- = e−iEm2(ξ2)t|ψm2(ξ2)(cid:105)(cid:104)ψm2(ξ2)|ψmI 0(t1)(cid:105)
mentspectrumoftwoKondoimpuritiescoupledtoleads m2
(cid:88)(cid:88)
have been discovered recently74. These are the result = e−iEm2(ξ2)te−iEm1(ξ1)t1|ψm2(ξ2)(cid:105)
of a off-critical quench accross the phase transition to a m2 m1
Kondo screened regime where all the spins are coupled (cid:104)ψ (ξ )|ψ (ξ )(cid:105)(cid:104)ψ (ξ )|ψ (ξ )(cid:105) (5)
m2 2 m1 1 m1 1 m0 0
to the impurity spins. The frequency scales with the in-
verse of the system size. In the systems considered here Choosingξ2 =ξ0 wegetthatfort1 ≤t<∞(t2 →∞)
theoscillationsingeneralareasuperpositionofdifferent the overlap with an initial state, n0, is given by
frequencies.
(cid:88)
The paper is organized as follows. In section II a dis- (cid:104)ψn0(ξ0)|ψmII0(t)(cid:105) = e−iEn0(ξ0)te−iEm1(ξ1)t1
cussionofthequantitiescalculatedafterasuddenquench m1
of the parameters of the Hamiltonian is presented. In (cid:104)ψ (ξ )|ψ (ξ )(cid:105)(cid:104)ψ (ξ )|ψ (ξ )(cid:105)
n0 0 m1 1 m1 1 m0 0
section III the topological models considered are briefly
(6)
discussed. In section IV the dynamics of the edge modes
ofKitaev’smodelisstudiedforthecaseofasinglequench Therefore, the probability to find a projection to an ini-
and the generation of Majorana modes due to a quench, tial state, n , given that the initial state is m is given
0 0
or sequence of quenches, is discussed. In section V the by
dynamics of the edge modes of some multiband systems
is discussed and the dynamics of Majorana modes and Pn0m0(t) = |(cid:104)ψn0(ξ0)|ψmII0(t)(cid:105)|2
fermionic zero energy modes are compared. In section (cid:88)
= | e−iEm1(ξ1)t1
VI the conclusions are presented.
m1
(cid:104)ψ (ξ )|ψ (ξ )(cid:105)(cid:104)ψ (ξ )|ψ (ξ )(cid:105)|2,
n0 0 m1 1 m1 1 m0 0
II. SINGLE-PARTICLE STATES AND (7)
QUANTUM QUENCHES
which is independent of time.
Let us consider an Hamiltonian defined by an initial We may now at some given finite time, t2, change the
set of parameters ξ0 for times t < 0. The single-particle parameters from ξ2 → ξ3. As before we may now find
eigenstates of the Hamiltonian are given by that for t2 ≤t<∞ the same probability is given by
H(ξ )|ψ (ξ )(cid:105)=E (ξ )|ψ (ξ )(cid:105). (1) P (t)=|(cid:104)ψ (ξ )|ψIII(t)(cid:105)|2 (8)
0 m0 0 m0 0 m0 0 n0m0 n0 0 m0
3
ofenergy. UsingaJordan-Wignertransformationdefined
by
Trivial phase cj = eiπ(cid:80)jl=−11S+(l)S−(l)S−(j)
c†j = S+(j)e−iπ(cid:80)jl=−11S+(l)S−(l) (11)
j=1 j=2 j=3 j=4
this model is equivalent to a spin-1/2 model
N
Topological phase H = −1(cid:88)(cid:2)(J −J )(cid:0)S+S+ +S−S− (cid:1)
4 X Y j j+1 j j+1
j=1
+ (J +J )(cid:0)S+S− +S−S+ (cid:1)(cid:3)
X Y j j+1 j j+1
N
(cid:88)
FIG. 1: (Color online) Phases of Kitaev model. At each − hzSjz (12)
lattice site, j, the two dots represent the two Majorana op- j=1
erators, γ and γ (real and imaginary parts of c ). The
j,1 j,2 j
The connection between the two models satisfies
linesrepresentthelinksbetweenMajoranaoperatorsatgiven
points in parameter space. Trivial phase with parameters: 1
t = (J +J )
∆ = 0,|µ| > 2t and topological phase with µ = 0,∆ = t. In 2 X Y
thetrivialphasethetwoMajoranasateachsitearelinkedin
1
the Hamiltonian and they constitute usual fermionic modes. ∆ = (J −J ) (13)
4 X Y
InthetopologicalphasetheMajoranasarelinkedatnearest-
neighbor sites and the first and last Majorana operators are and the chemical potential is the magnetic field along z.
decoupled and therefore have zero energy. In momentum space the model is simply written as
Hˆ = 1(cid:88)(cid:16)c†,c (cid:17)H (cid:18) ck (cid:19) (14)
where 2 k −k k c†
k −k
|ψIII(t)(cid:105)=e−iH(ξ3)t|ψII (t )(cid:105) (9) where
m0 m0 2
(cid:18) (cid:19)
(cid:15) −µ i∆sink
The probability is now a function of time, t. H = k
k −i∆sink −(cid:15) +µ
In this work only unitary evolution of single-particle k
states is considered and effects of dissipation are ne- (15)
glected.
with (cid:15) =−2tcosk.
k
Afermionoperatormaybewritenintermsoftwoher-
mitian operators in the following way
III. MODELS
1
c = (γ +iγ )
Variousmodelshavetopologicalpropertiesand,dueto j,σ 2 j,σ,1 j,σ,2
the bulk-edge correspondance, have gapless edge modes. c† = 1(γ −iγ ) (16)
Here we consider a few one-dimensional systems with j,σ 2 j,σ,1 j,σ,2
non-trivial topological properties.
Theindexσ representsinternaldegreesoffreedomofthe
fermionic operator, such as spin and/or sublattice index,
and the γ operators are hermitian and satisfy the anti-
A. One-band superconductor: the Kitaev model
commutation relations
The Kitaev one-dimensional superconductor with {γm,γn}=2δnm (17)
triplet p-wave pairing is described by the Hamiltonian75
In the case of the Kitaev model it is enough to consider
c = (γ +iγ )/2, since the fermions are spinless. In
N¯ j j,1 j,2
H = (cid:88)(cid:104)−t(cid:16)c†c +c† c (cid:17)+∆(cid:16)c c +c† c†(cid:17)(cid:105) terms of these hermitian (Majorana) operators we may
j j+1 j+1 j j j+1 j+1 j writethattheHamiltonianisgivenby,usingopenbound-
j=1
ary conditions,
N (cid:18) (cid:19)
(cid:88) 1
− µ c†c − (10) N−1
j j 2 i (cid:88)
H = [(−t+∆)γ γ +(t+∆)γ γ ]
j=1 2 j,1 j+1,2 j,2 j+1,1
j=1
where N¯ = N if we use periodic boundary conditions N
(and N +1=1) or N¯ =N −1 if we use open boundary − i (cid:88)µγ γ (18)
2 j,1 j,2
conditions. t is the hopping amplitude taken as the unit
j=1
4
ThechemicalpotentialterminvolvesallMajoranaop-
erators. Taking µ = 0 and selecting the special point
t=∆ the Hamiltonian reduces considerably to Topological phase
B
N(cid:88)−1 N(cid:88)−1 A
H(µ=0,t=∆)=it γ γ =−it γ γ
j,2 j+1,1 j+1,1 j,2
j=1 j=1 j=1 j=2 j=3 j=4
(19)
B
It is easily seen that the operators γ and γ are
1,1 N,2
A
missing from the Hamiltonian. Therefore there are two
Trivial phase
zero energy modes. Defining from these two Majorana
fermionsasingleusualfermionoperator(non-hermitian)
taking one of the Majorana operators as the real part
and the other as the imaginary part, its state may be ei-
FIG. 2: (Color online) Phases of Schockley model. The
ther occupied or empty with no cost in energy. Defining
symbols have the same meaning as in Fig. 1. Trivial phase
d = 1/2(γ +iγ ) and d = 1/2(γ +iγ ) we
j j,2 j+1,1 N N,2 1,1 with t =0 and topological phase with t =0. In the trivial
2 1
can write the Hamiltonian as
phase the Majorana fermions are coupled to form fermionic
modesatthesamesite. Inthetopologicalphasethelinksare
N−1
(cid:88) (cid:16) (cid:17) (cid:16) (cid:17) between nearest-neighbor sites and there are four Majorana
H =t 2d†d −1 +(cid:15) 2d† d −1 (20)
j j N N N operators decoupled at the edges. However, these give origin
j=1 tozero-energyfermionicmodes,oneateachendofthesystem.
with (cid:15) = 0. Therefore the fermionic mode d does
N N
not appear in the Hamiltonian and the state may be oc-
cuppied or empty (d† d = 1,0, respectively) with no B. Multiband system: Two-band Schockley model
N N
energycost. Thesetwostatesarethereforedegeneratein
energy. Solving the Bogoliubov-de Gennes (BdG) equa-
TheShockleymodelisamodelofadimerizedsystemof
tions of the Kitaev Hamiltonian using open boundary
spinless fermions with alternating nearest-neighbor hop-
conditions leads to two zero energy modes that at the
pings, given by the Hamiltonian (see for instance78)
special point are perfectly localized at the edges of the
chain as δ-function peaks (with exponential accuracy).
ThephasediagramoftheKitaevmodelhasthreetypes
N
ofphases: twotopologicalphasesinwhichtherearegap- H =(cid:88)ψ†(j)(cid:2)Uψ(j)+Vψ(j−1)+V†ψ(j+1)(cid:1) (21)
less edge modes if the system is finite and two trivial
j=1
phaseswithnoedgemodes. Inthevariousphasesthesys-
tem is gapped and at the transition lines the gap closes,
allowing the possibility of a change of topology. The where the 2×2 matrices U and V are given by
transition lines are located at ∆=0 and at |µ|=2t. In
terms of the spin model the transition line ∆=0 is just (cid:18) 0 t∗ (cid:19) (cid:18)0 t∗ (cid:19)
the isotropic XY model which is known to be gapless. U = 1 ;V = 2 (22)
t 0 0 0
Positive values of ∆ imply that the exchange interaction 1
is preferred along the X direction and if ∆ < 0 the pre-
ferred direction is along Y. The three phases of the Ki- and the spinor ψ represents two orbitals that are hy-
taevmodelhaveadirectcorrespondancewiththephases bridized by the matrices U and V
in the spin model: the topological phases have as duals
twophaseswithorderingalongtheX orY directionsand
(cid:18) (cid:19)
thetrivialphasehasasdualatrivialparamagneticphase ψ(j)= ψa(j) . (23)
in the spin model. There is therefore a duality between ψb(j)
the topological properties in the fermionic Kitaev model
and Landau-like ordering in the dual spin model as a re-
sult of the exact Jordan-Wigner non-canonical non-local We may as well define Majorana operators as
transformation76,77.
Thestructureofthephasescanbeunderstoodinterms
1
oftheMajoranarepresentationofthefermionicoperators c = (γ +iγ )
j,A 2 j,A,1 j,A,2
andtwophasesareillustratedinFig. 1. Thestructureis
1
particularly clear at these special points but due to the c = (γ +iγ ) (24)
j,B 2 j,B,1 j,B,2
topologically protected nature of the Hamiltonian (BDI
class)theirnatureisnotchangedaslongasthegapdoes
not close. Taking t∗ =t ,t∗ =t , the Hamiltonian may be written
1 1 2 2
5
as
h=-1, D=0
N
H = it1 (cid:88)(γ γ +γ γ ) B
2 j,A,1 j,B,2 j,B,1 j,A,2
j=1 A
N
+ t2 (cid:88)(γ γ +γ γ ) j=1 j=2 j=3 j=4
4 j,A,1 j−1,B,1 j,A,2 j−1,B,2
B
j=2
N A
+ it2 (cid:88)(γ γ −iγ γ ) h=0, D= t
4 j,A,1 j−1,B,2 j,A,2 j−1,B,1
j=2
N−1
+ t2 (cid:88) (γ γ +γ γ )
4 j,B,1 j+1,A,1 j,B,2 j+1,A,2 FIG. 3: (Color online) Phases of SSH-Kitaev model. The
j=1
symbolshavethesamemeaningasinFig. 1. When∆=0the
N−1
+ it2 (cid:88) (γ γ −iγ γ ) (25) modelreducestotheSSHmodelandfornegativeηthemodel
4 j,B,1 j+1,A,2 j,B,2 j+1,A,1 istopologicallynon-trivialwithedgestatesrepresentedbythe
j=1 decoupled Majorana operators. As in the Schockley model
since at each end site there are two decoupled Majoranas,
thesecombinetoformedgefermionicmodes. Thisconstitutes
Taking t = 0 we find that the Majorana fermions
1 phase SSH2 with η = −1,∆ = 0 and two edge modes. If
γ ,γ ,γ ,γ do not contribute and are
1,A,1 1,A,2 N,B,1 N,B,2 superconductivityispresent,andthereisnodimerizationη=
zero energy modes.
0,themodelreducestotheKitaevmodel. ThephaseK1with
In Fig. 2 the structure of the Hamiltonian terms is η = 0,∆ = t has two decoupled Majorana operators, one at
presented for two points in parameter phase that corre- each end, and therefore there is one Majorana mode at each
spond to the trivial and the topological phases. In the edge. The model interpolates between Majorana modes and
topologicalphasetherearedecoupledzero-energymodes fermionic modes as the parameters change. There is also a
trivialphasewithnozeroenergymodesdenotedSSH0which
thatarehoweverfermionicinnaturesincethedecoupled
is similar to the trivial phase of the Schockley model.
Majoranas are located at the two end sites, A and B,
respectively.
terms can be grouped in the matrix
 −µ −t(1+η) 0 −∆(1+η)
C. Multiband system: SSH model with triplet
−t(1+η) −µ ∆(1+η) 0
pairing Hj,j = 0 ∆(1+η) µ t(1+η) 
−∆(1+η) 0 t(1+η) µ
This model may be viewed as a dimerized Kitaev (27)
superconductor79. The dimerization is parametrized by Thenon-localtermstothenearest-neighborscanbewrit-
η and the superconductivity by ∆. ten as
This model is given by the Hamiltonian
 0 0 0 0
−t(1−η) 0 −∆(1−η) 0
H =−µ (cid:80) (cid:16)c† c +c† c (cid:17) Hj,j+1 = 0 0 0 0 (28)
j j,A j,A j,B j,B
∆(1−η) 0 t(1−η) 0
(cid:104)
−t (cid:80) (1+η)c† c +(1+η)c† c
j j,B j,A j,A j,B
and
(cid:105)
+ (1−η)c† c +(1−η)c† c
j+1,A j,B j,B j+1,A
0 −t(1−η) 0 ∆(1−η)
(cid:104)
+∆ (cid:80) (1+η)c† c† +(1+η)c c 0 0 0 0
j j,B j,A j,A j,B H =  (29)
j,j−1 0 −∆(1−η) 0 t(1−η) 
(cid:105)
+ (1−η)c†j+1,Ac†j,B +(1−η)cj,Bcj+1,A 0 0 0 0
(26)
In momentum space this model is given by an Hamil-
tonian matrix of the form
The model with no superconductivity (∆=0) is related
to the Schockley model taking t = t(1+η) and t =  −µ z(k) 0 w(k) 
1 2
t(1−η). The region of η >0 corresponds to t >t and z∗(k) −µ −w∗(k) 0
1 2 H =  (30)
vice-versaforη <0. TheHamiltonianinrealspacemixes k  0 −w(k) µ −z(k)
nearest-neighborsitesandalsohaslocalterms. Thelocal w∗(k) 0 −z∗(k) µ
6
sincetherearetwoMajoranaoperatorsmissingfromthe
Hamiltonian, γ and γ , one from each end. iii)
2 1,A,1 N,B,2
III I III An example of a trivial phase is the point η = 1 and
Oscillations ∆=0inwhichcasetherearenozeroenergyedgestates.
1
No oscillations In Fig. 3 the phases with edge modes are presented for
specialpointsinparameterspace. Thismodelprovidesa
Oscillations
D 0 No oscillations testinggroundforthecomparisonoffermionicandMajo-
ranaedgemodes. Also, insomeregimesitdisplaysfinite
-1 energymodesthatarelocalizedattheedgesofthechain,
II as obtained before in other multiband models80.
-2 In addition to direct measurements of tunneling den-
-3 -2 -1 0 1 2 3
m
sity of states of Majorana edge states, measurements of
the differential conductance at the interface between a
lead and a topological superconductor have been pro-
FIG. 4: (Color online) Phase diagram of Kitaev model. Re-
gions I and II are topologically non-trivial and region III is posedasawaytodetectMajoranamodes. Inparticular,
trivial. The points separate regions where critical quenches with a metallic lead one expects a zeo-bias peak in the
leadornottooscillationsinthesurvivalprobabilityofaMa- differentialconductance,ifzero-energymodesarepresent
jorana state of region I. inthesuperconductingside. InthepresenceofMajorana
modes one expects a vanishing conductance if the num-
ber of Majorana modes is even and a quantized value of
where this matrix acts on the spinors 2e2/h, ifthenumberofmodesisodd81,82. Thesemaybe
duetoedgechiralmodesor,inap-wavesuperconductor,
 cA(k)  associated with a vortex81. In the case of the dimerized
 cB(k)  SSH model here considered it has been shown79 that the
 c†A(−k)  (31) fermionic edge modes do not contribute to the conduc-
c† (−k) tance and, therefore, provides a method to distinguish
B
the various phases, since in the regime that is Kitaev-
and like the conductance is quantized, as expected, while in
the SSH regimes it vanishes. However, other types of
z(k) = −t(cid:2)(1+η)+(1−η)e−ik(cid:3) zero-energy states due for instance to disorder or tem-
w(k) = −∆(cid:2)(1+η)−(1−η)e−ik(cid:3) (32) peratureeffects,alsogiveorigintozero-biaspeaksinthe
conductance83. A possible way to clearly identify a zero
In terms of Majorana operators the Hamiltonian is mode as a Majorana has been proposed84 selecting a su-
written as perconducting lead with the prediction of peaks at the
valueofthegapoftheconventionalsuperconductorwith
µ(cid:88)N a quantized conductance of the form (4−π)2e2/h. In
H = − (2+iγ γ +iγ γ )
2 j,A,1 j,A,2 j,B,1 j,B,2 this work we will analyse possible distinctive signatures
j=1 of the edge modes in the dynamics following a sudden
N quantum quench.
it (cid:88)
− (1+η) (γ γ +γ γ )
2 j,B,1 j,A,2 j,A,1 j,B,2
j=1
it N(cid:88)−1 IV. DYNAMICS OF EDGE MODES OF KITAEV
− (1−η) (γ γ +γ γ ) MODEL
2 j+1,A,1 j,B,2 j,B,1 j+1,A,2
j=1
N A. Single quench
i∆ (cid:88)
+ (1+η) (γ γ +γ γ )
2 j,A,1 j,B,2 j,A,2 j,B,1
j=1 In Fig. 4 we show the phase diagram of the Kitaev
N−1 model as a function of the chemical potential µ and ∆.
i∆ (cid:88)
+ 2 (1−η) (γj,B,1γj+1,A,2+γj,B,2γj+1,A,1) RegionsI andII aretopologicallynon-trivialandregions
j=1 III are trivial, as discussed above. Consider first µ = 0
(33) andquencheswhereonevaries∆,orafixed∆andchang-
ingµ. Thepointsseparatetheregionswhereonefindsor
Taking as before µ = 0 we have a couple of special does not find oscillations in the survival probability of a
points: i) η = −1 and ∆ = 0 we have a state simi- Majorana state of a topologically non-trivial phase after
lar to the SSH or Schockley models with two fermionic- aquenchtoacriticalpoint,foragivensystemsize. Inthe
like zero energy edge states, since the four operators caseofµ=0thecriticalpointislocatedatµ=0,∆=0
γ ,γ ;γ ,γ are missing from the Hamil- and in the second case there is a line of critical points at
1,A,1 1,A,2 N,B,1 N,B,2
tonian. ii) η = 0 and t = ∆ is a Kitaev like state µ=2t. For instance, in the quench from the topological
7
1 0.2 0.1
0.8 D=0.1 0.08 D=0.05
D=0.5
0.6 0.06
2 2 0.1 2
|y|0.4 |y| |y|0.04
0.2 0.02
0 50 100 150 200 0 50 100 150 200 0 50 100 150 200
x x x
0.01 50 50
D=0
D=0.01
0.02 0.008 40 40
0.006 30 30
2 2
|y|0.01 |y| 0.004 x20 20
0.002 10 10
0 50 100 150 200 50 100 150 2000 00 0.2 0.4 0.6 0.8 1 00 25 50 75 100
x x D 1/D
FIG.5: (Coloronline)ZeroenergymodewavefunctionsofKitaevmodelfordifferentpointsapproachingthecriticalpointat
µ=0. In the last panel we show the dependence of the decay length of the wave functions as a function of ∆ and 1/∆. The
resultsareforasystemsizeofN =200. Atthepoint∆=1thewavefunctionisstrictlylocalatoneedgeofthesystem. Each
Majorana state is perfectly localized with a decay length ξ =0. For ∆=0.5 the decay length is also very small and the two
Majorana modes (in black, left edge, and in red, right edge) are decoupled. For smaller values of ∆ the two Majorana modes
are coupled and each is peakd at both ends of the chain. For very small values of ∆ the modes get extended as one tends to
the gapless regime at ∆=0.
phase I to the critical point at µ = 0,∆ = 0, the point and as one increases the distance from the critical point
is located as N = 100,∆ = 0.34, N = 200,∆ = 0.18, the amplitude decreases considerably. The oscillations
N = 400,0.05 < ∆ < 0.1. The points at µ = 2 sepa- are quite smooth and clear until the amplitude has de-
rate two regions for which making a quench from region creased enough to reach zero. Beyond this point there
I to the critical line µ = 2, one may find oscillations, is a periodicity but no longer oscillations since there are
if the initial point is not very far from the critical line. increasing regions where P(t) basically vanishes. In this
In the vicinity of the two critical lines of points (around case it seems more like the revival times of non-critical
µ = 2t,∆ = 0), no matter how close the initial point is quenches, even though the curves are still smooth. Be-
to the critical line, one does not find oscillations. yondagivenvalueof∆ thereisaperiod doubling. Also,
d
after this period doubling the survival probability looses
In Fig. 5 is shown the absolute value squared of the
its regular periodic behavior and shows more oscillations
lowestenergyeigenvectorforN =200fordiferentvalues
of smaller periods and amplitude decays that are similar
of ∆ and keeping µ = 0. Note that if one is very close
to results previously found in quenches away from criti-
to the transition point the wave functions are not very
cal points54,63. In the second panel are shown quenches
localized. At ∆ = 0 the state is extended since the sys-
to the critical line µ=2 keeping ∆=0.5 and decreasing
tem is gapless. A similar behavior is observed for small
the chemical potential. The behavior is similar to the
values of ∆ and, as ∆ increases, the decay length de-
first panel. In the third panel is shown in greater detail
creasessignificantly. Thedecaylengthasafunctionof∆
thecrossovertoperioddoublingforthetransitiontothe
is shown in the last panel. The fit of the wave function
critical point. In the fourth panel we show the scaling
dependence with distance, x = j, from the edge of the
system, isofanexponentialform|ψ|2(2n+1)=ψ e−x/ξ of the point of crossover, ∆d, when the period doubling
0
(here x = 2n+1 since |ψ|2 oscillates), with ξ the decay takes place. It scales linearly with 1/N.
length.
InthefirstpanelofFig. 7areshownquenches,keeping
In Fig. 6 the survival probability as a function of time µ=0.5, to the critical line ∆=0. In this case there are
for various critical quenches is presented. In the first no oscillations. In the second panel the quench to the
panel are shown the oscillations of P(t) as one quenches criticallineµ=2fromtheinitialpointµ =1.9isshown
i
fromagivenvalueof∆tothecriticalpointµ=0,∆=0 for different values of ∆. For small values of ∆ there are
maintaining µ = 0. For small deviations of the initial nooscillationsandas∆increasestheoscillationsappear.
value of ∆ from the critical point, P(t) is close to 1 The results in this figure are for N =100.
8
m=0 D=0.5
1 1
D=0.01 m=1.99
0.8 0.05 0.8 1.95
0.1 1.9
0.2 1.8
0.6 0.5 0.6 1.5
P(t) 1.0 P(t) 1.0
0.4 0.4
0.2 0.2
0 0
0 50 100 150 200 250 0 50 100 150 200 250
time time
m=0
0.8 D=0.25
0.3 0.5
0.325
0.33
0.6 0.4
0.34
0.35
P(t)0.4 0.4 Dd0.3
0.2
0.2
0.1
0 0
0 50 100 150 200 0 0.005 0.01 0.015 0.02
time 1/N
FIG.6: (Coloronline)Inthefirsttwopanelsthesurvivalprobability,P(t),ofaMajoranamodeintheKitaevmodelisshown
as one approaches critical points. In the third panel the crossover to period doubling is shown as one approches the critical
region. In the fourth panel the linear dependence of the point of crossover, ∆ , on 1/N is shown.
d
The survival probability is determined by the vari- their energy values. In general, the overlaps have very
ous energies of the final Hamiltonian eigenstates and the different magnitudes to the two states and the period of
overlaps to the initial state. In Fig. 8 the overlaps be- oscillations shown in P(t) depends on their magnitudes.
tween the initial lowest energy state (Majorana mode) Addingsignificantcontributionsfromotherenergyeigen-
and all the final state eigenvectors are shown, as a func- states leads first to modulated oscillations and then to a
tionoftheirenergies,forthecasesofFig. 6,forN =200. complicated time dependence.
Ingeneral,theoverlapsarepeakedatthelowestenergies. Itwasarguedbefore63 thattheenergyspectrumofthe
There is a clear separation of regimes as one reaches the finalstateofthequenchisimportanttodetermineifthere
crossover region where the period doubling occurs. At are oscillations. In some cases there is indeed a regular
small values of ∆ the overlaps oscillate between finite spacingofthefinalstateenergiesandinothersno. Ifthe
values and zero values, as we move accross the energy regular spacing is observed we should find a dominant
eigenvalues. This is probably a parity effect distinguish- frequencybutalsomanyharmonics. However,theroleof
ingevenandoddnumberofsites. Thisisconfirmedinthe the overlaps is more significant because it clearly selects
third panel where the cases of N =100 and N =101 are which energy states actually contribute85.
compared. IfN =101theoscillationsintheoverlapsare
The origin of the period doubling is understood in the
absent. It can be noted that the overlaps are very sharp
following way. In Fig. 9 the time evolution of the Majo-
around the lowest energy states. As the crossover occurs
rana state is shown for a critical quench from the region
the overlaps are no longer zero at some energy eigenval-
of oscillations, close to the critical point, and a quench
ues and actually become very smooth. This means that
from a region where the period has doubled. In the first
thecontributionsfromthevariousenergystateschanges,
casethewavefunctionsateachedgeareseparatedintwo
the time behavior is affected and the clean oscillations
energy modes while for the second they are mixed. This
are no longer observed. In order to have clean oscilla-
is due to the long range correlations close to the critical
tions one needs contributions from few energy levels. A
pointthateffectivelydecreasethesystemsizeandleadto
perfect oscillation requires finite overlaps to two states
the coupling of the two edge modes. In the first case the
and the frequency of the oscillations is the difference in
time evolved states from each edge cross each other in a
9
m=0.5 m=0
1 0.1
D=0.1
D=0.01 N=200
0.8 0.05 0.08 00..1157
0.1 0.175
P(t)0.6 001...250 Overlaps00..0046 000...11289
0.4
0.02
0.2
0
-1 -0.5 0 0.5 1
Energy
0
0 100 200 300 400
time
D=0.5
1
m=1.9
m=1.99
1 0.8 1.95
1.9
D=0.1 1.8
0.8 0000....34575 Overlaps00..46 11..50
0.6
1.0
)
(t 0.2
P
0.4
0
-1.5 -1 -0.5 0 0.5 1 1.5
0.2 Energy
m=0
0
0 100 200 300
time 0.2
D=0.1
N=100
N=101
0.15
FIG.7: (Coloronline)Leftpanel: atfinitechemicalpotential
s
the critical quench to the line ∆=0 does not lead to oscilla- p
a
tionsofthesurvivalprobabilityoftheMajoranamode. Close erl 0.1
v
totheintersectionofthetwocriticallinesnearµ=2t,∆=0 O
there are no oscillations as well as shown in the right panel. 0.05
0
-1 -0.5 0 0.5 1
solitonic like behavior while in the second case there is a Energy
constructive interference when the peaks of the evolved
state meet at the center of the wire. Consistently with
FIG. 8: (Color online) Overlaps for the Kitaev model as a
the results for the overlaps, in this regime the energy
functionofenergy. Toppanelthecriticalpointisµ=0,∆=0
spectrum between states with high weight halves, and
and in the middle panel is µ=2,∆=0.5. In the third panel
the period doubles.
the even-odd effect in the overlaps is shown.
The distribution of the overlaps may be parametrized
by the work distribution given by86
(cid:88) obtained performing a Fourier analysis of the time evo-
P(W)= P δ(W −W ) (34)
m m lution of the survival probability. This is shown in the
m
lowerpanel. Whileforsmallinitialvaluesof∆thedistri-
where butionisquitenarrowaroundlowfrequencies,itchanges
significanlyas∆grows,becomingquiteextended. Inthe
P = |(cid:104)ψ (ξ )|ψ (ξ )(cid:105)|2 Fourier decomposition the amplitudes, u , are for the
m m 1 m=0 0 n
frequencies with values ω = π(n−1)/N , where N is
W = E (ξ )−E (ξ ) (35) n t t
m m 1 m=0 0
the number of time points considered.
In Fig. 10a we show the work distribution associated It is also interesting to study the survival probabil-
with critical quenches for µ = 0 to the critical point ity of excited states, that in this problem are extended
µ=0,∆=0 starting from different initial points. Close states throughout the chain. This is shown in Fig. 11.
to the critical point the distribution is quite sharp but it Here we show the survival probability of different ini-
becomesverybroadas∆increases. Theδ-functionpeaks tial states, including the Majorana states, for two cases
have been broadened for clarity. A similar conclusion is of ∆ = 0.01,0.1. Close to the critical point the survival
10
2 0.5 0.3 100
2|y|0.02 0.02 80 D=D=00..011
|y| 0.02 0.02 D=0.5
2
|y| 0.02 0.02 ) 60
2 W
|y| 0.02 0.02 P( 40
2
|y| 0.02 0.02
2 20
|y| 0 0
20 40 60 80 100 20 40 60 80 100
x x
0
-2 -1 0 1 2
W
0.2
2
|y|0.005
2 1 2
|y|0.004 D=0.01
2 0.8 0
|y| 0.007 1 2.5 D=0.1
2
|y|0.005 0.09 0.06
2 P(t) D=0.2 u
2|y|0.004 0.07 0.02 n
|y| 0 D=0.5
20 40 60 80 100
x
0 0
0.40 100 200 300 4000.1
D=1
FIG. 9: Solitonic-like vs. constructive interference behavior 00 100 200 300 400 0 100 2w00 300 400
in the Kitaev model. Top panel the initial state is far from time n
the critical point (CP) and in the lower panel one is close to
the CP.
FIG.10: (Coloronline)Toppanel: Workdistributionforthe
Kitaev model. Lower panel: Fourier analysis for the Kitaev
model.
probabilityofmoststatesiscloseto1exceptnearthelow
energy modes. Further away from the critical point the
deviation of the survival probability from unity is larger
jorana states through a sudden quench starting from a
duetolargerorthogonalitybetweentheeigenstatesofthe
trivial phase. Even though, as stated above, in the ther-
originandfinalHamiltonians. Inthethirdpanelthelow
modynamic limit the topological properties can not be
energy region is enhanced showing the complex behavior
changedbyaunitarytransformation,asshowninFig. 12
as a function of time and eigenstate, n.
theprobabilitythatagiveninitialstateinatrivialphase
III may collapse to a Majorana of the final state Hamil-
tonian in phase I is finite. We have a finite although
B. Generation of Majorana states
small overlap to the Majorana state of the topological
phase. This probability is given by
While quenches, either abrupt or slow, in general
destabilize the edge states, topological phases can be in- |(cid:104)ψ (ξ )|ψI (t)(cid:105)|2 =|(cid:104)ψ (ξ )|ψ (ξ )(cid:105)|2 (36)
duced by periodically driving the Hamiltonian of a non- m1=0 1 m0 m1=0 1 m0 0
topological system. The periodic driving leads to new and is independent of time, as expected. Quenching to a
topological states87, and to a generalization of the bulk- state close to the transition line, the overlaps of several
edge correspondence, that reveals a richer structure88,89 (extended) states are considerable due to the spatial ex-
as compared with the equilibrium situation90, such as tent of the Majorana states. If the quench is deeper into
shown before in topological insulators87,91,92 and in the topological phase these become more localized and
topological superconductors, with the appearance of the overlap decreases. Interestingly the larger overlap is
Majorana fermions93–96. Their appearance in a one- found for higher energy, extended states.
dimensionalp-wavesuperconductorwasstudiedinRef.97 A sequence of quenches allows for the manipulation of
and in Ref.98; the case of intrinsic periodic modulation the states, as shown in eq. (9). A possibility to turn
was also considered99 and new phases may be induced off and on Majoranas can be trivialy seen in the follow-
and manipulated due to the presence of the periodic ing way. Consider starting from a state inside region I.
driving97,100,101 or inducing spontaneous currents102. Perform a critical quench to the line ∆ = 0 and then a
DuetothefinitenessofasystemwemaygenerateMa- quench back to the original state. Choosing appropri-