Dynamics of quenched topological edge modes

1.2 MB·

by  P. D. Sacramento

January16,2017 1:14 WSPCProceedings-9inx6in dynmodes page1
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7
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Dynamics of quenched topological edge modes
0
2
P.D.Sacramento∗
n
a Departamento de F´ısica and CeFEMA,
J Instituto Superior T´ecnico, Universidade de Lisboa,

3 Av. Rovisco Pais, 1049-001 Lisboa, Portugal
1 ∗E-mail: [email protected]

]
n Acharacteristicfeatureoftopologicalsystemsisthepresenceofrobustgapless
o edge states. In this work the effect of time-dependent perturbations on the
c edge states is considered. Specifically we consider perturbations that can be
- understood as changes of the parameters of the Hamiltonian. These changes
r maybesuddenorcarriedoutatafixedrate. Ingeneral,theedgemodesdecay
p
in the thermodynamic limit, but for finite systems a revival time is found
u
that scales with the system size. The dynamics of fermionic edge modes and
s
. Majorana modes are compared. The effect of periodic perturbations is also
t
a referred allowing the appearance of edge modes out of a topologically trivial
m phase.
- Keywords:Topology;time-dependentperturbations.
d
n
o
c 1. Sudden quantum quenches
[
An example of a time-dependent transformation of the Hamiltonian is a

1
sudden change of its parameters. Let us consider an Hamiltonian defined
v
7 by an initial set of parameters ξ0 for times t < t0. The single-particle
6 eigenstates of the Hamiltonian are given by
5
3
H(ξ )|ψ (ξ )(cid:105)=E (ξ )|ψ (ξ )(cid:105), (1)
0 0 m0 0 m0 0 m0 0
.
1
where m are the quantum numbers. At time t=t a sudden transforma-
0 0 0
7 tion of the parameters is performed, ξ →ξ . The Hamiltonian eigenstates
0 1
1 transform to
:
v
Xi H(ξ1)|ψm1(ξ1)(cid:105)=Em1(ξ1)|ψm1(ξ1)(cid:105). (2)
r
a Afterthissuddenquenchthesystemwillevolveintimeundertheinfluence
of a different Hamiltonian. The time evolution of a single-particle state,
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with quantum number m , is given by
0
(cid:88)
|ψmI 0(t)(cid:105)= e−iEm1(ξ1)(t−t0)
m1
|ψ (ξ )(cid:105)(cid:104)ψ (ξ )|ψ (ξ )(cid:105) (3)
m1 1 m1 1 m0 0
for times t≥t . The survival probability of some initial state |ψ (ξ )(cid:105) is
0 m0 0
defined by
P (t)=|(cid:104)ψ (ξ )|ψI (t)(cid:105)|2. (4)
m0 m0 0 m0
Wewillbeinterestedinthefateofsingleparticlestatesafteraquantum
quenchacrossthephasediagram. Weconsiderasubspaceofoneexcitation
such that the total Hamiltonian is given by the ground state energy plus
oneexcitedstateandassumeweremainintheoneexcitationsubspaceafter
the quench. In this work only unitary evolution of single-particle states is
considered and effects of dissipation are neglected.
Wemayaswellconsiderfurtherquenchesdefinedinasequenceoftimes
and sets of parameters as t < t < t < t < ··· and ξ ,ξ ,ξ ,ξ ,···,
0 1 2 3 0 1 2 3
respectively. These intervals define regions as I(t ≤ t < t ),II(t ≤ t <
0 1 1
t ),III(t ≤ t < t ),···. The case of a single quench is clearly obtained
2 2 3
taking t →∞, and so on for further quenches (t =0 is chosen hereafter).
1 0
Consider now a case for which we have two quenches in succession.
In this case we have that the evolution of the initial state with quantum
number m is
0
|ψII (t)(cid:105)=e−iH(ξ2)(t−t1)|ψI (t )(cid:105)
m0 m0 1
(cid:88)
= e−iEm2(ξ2)(t−t1)
m2
|ψ (ξ )(cid:105)(cid:104)ψ (ξ )|ψI (t )(cid:105)
m2 2 m2 2 m0 1
(cid:88)(cid:88)
= e−iEm2(ξ2)(t−t1)e−iEm1(ξ1)t1
m2 m1
|ψ (ξ )(cid:105)(cid:104)ψ (ξ )|ψ (ξ )(cid:105)(cid:104)ψ (ξ )|ψ (ξ )(cid:105)
m2 2 m2 2 m1 1 m1 1 m0 0
(5)
Choosingξ =ξ wegetthatfort ≤t<∞(t →∞)theoverlapwith
2 0 1 2
an initial state, n , is given by
0
(cid:88)
(cid:104)ψn0(ξ0)|ψmII0(t)(cid:105)= e−iEn0(ξ0)(t−t1)e−iEm1(ξ1)t1
m1
(cid:104)ψ (ξ )|ψ (ξ )(cid:105)(cid:104)ψ (ξ )|ψ (ξ )(cid:105)
n0 0 m1 1 m1 1 m0 0
(6)
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Therefore, the probability to find a projection to an initial state, n , given
0
that the initial state is m is given by
0
P (t)=|(cid:104)ψ (ξ )|ψII (t)(cid:105)|2
n0m0 n0 0 m0
(cid:88)
=| e−iEm1(ξ1)t1
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
(7)
which is independent of time.
We may now at some given finite time, t , change the parameters from
2
ξ →ξ . As before we find that for t ≤t<∞ the same probability as in
2 3 2
eq. (7) is given by
P (t)=|(cid:104)ψ (ξ )|ψIII(t)(cid:105)|2 (8)
n0m0 n0 0 m0
where
|ψIII(t)(cid:105)=e−iH(ξ3)(t−t2)|ψII (t )(cid:105) (9)
m0 m0 2
The probability is now a function of time.
2. Models
In this chapter we consider systems that are topologically non-trivial, such
as one or two-dimensional topological insulators or topological supercon-
ductors. The topological nature of these systems reveals itself both in the
topological nature of the groundstate of the infinite system and in the ap-
pearance of edge states if the system is finite (bulk-edge correspondance).
Different topological invariants may be defined such as winding numbers
for the one-dimensional examples considered here and the Chern number
forthetwo-dimensionalsuperconductorconsideredlater. Boththewinding
numbersandtheChernnumbermaybeunderstoodinvariousways1–3 and
typicallytheycountthenumberofedgemodesattheinterfacebetweenthe
topological system and the vacuum.
Some examples are the models considered in this section which display
bothtrivialandtopologicalphases. Thedynamicsoftheedgemodesofthe
topological phases after a quantum quench is considered in sections 3-5.
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2.1. One-band spinless superconductor: the 1D Kitaev
model
The Kitaev one-dimensional superconductor with triplet p-wave pairing is
described by the Hamiltonian4
N¯
(cid:88)(cid:104) (cid:16) (cid:17) (cid:16) (cid:17)(cid:105)
H = −t˜ c†c +c† c +∆ c c +c† c†
j j+1 j+1 j j j+1 j+1 j
j=1
N (cid:18) (cid:19)
(cid:88) 1
− µ c†c − (10)
j j 2
j=1
where N¯ = N if we use periodic boundary conditions (and N +1 = 1) or
N¯ = N −1 if we use open boundary conditions. Here N is the number
of sites. t˜is the hopping amplitude taken as the unit of energy, ∆ is the
pairing amplitude and µ the chemical potential. The operator c destroys
j
a spinless fermion at site j.
In momentum space the model is written as
Hˆ = 1(cid:88)(cid:16)c†,c (cid:17)H (cid:18) ck (cid:19) (11)
2 k −k k c†
k −k
where
(cid:18) (cid:19)
(cid:15) −µ i∆sink
H = k
k −i∆sink −(cid:15) +µ
k
(12)
with (cid:15) =−2t˜cosk. Here c is the Fourier transform of c .
k k j
In general, a fermion operator may be writen in terms of two hermitian
operators, γ ,γ , in the following way
1 2
1
c = (γ +iγ )
j,σ 2 j,σ,1 j,σ,2
1
c† = (γ −iγ ) (13)
j,σ 2 j,σ,1 j,σ,2
The index σ represents internal degrees of freedom of the fermionic oper-
ator, such as spin and/or sublattice index, the γ operators are hermitian
and satisfy a Clifford algebra
{γ ,γ }=2δ . (14)
m n nm
InthecaseoftheKitaevmodelitisenoughtoconsiderc =(γ +iγ )/2,
j j,1 j,2
sincethefermionsarespinless. Intermsofthesehermitian(Majorana)op-
eratorswemaywritethattheHamiltonianisgivenby,usingopenboundary
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2
I
1
III III
0
D
II
-1
-2
-3 -2 -1 0 1 2 3
m
Fig.1. Phasediagramof1D Kitaevmodel. InphasesIandIIthereareedgemodes.
conditions,
N−1
H = i (cid:88) (cid:2)(−t˜+∆)γ γ +(t˜+∆)γ γ (cid:3)
2 j,1 j+1,2 j,2 j+1,1
j=1
N
i (cid:88)
− µγ γ (15)
2 j,1 j,2
j=1
Taking µ = 0 and selecting the special point t˜= ∆ the Hamiltonian
simplifies considerably to
N−1 N−1
(cid:88) (cid:88)
H(µ=0,t˜=∆)=it˜ γ γ =−it˜ γ γ (16)
j,2 j+1,1 j+1,1 j,2
j=1 j=1
Note that the operators γ and γ are missing from the Hamiltonian.
1,1 N,2
Therefore thereare two zeroenergy modes. Defining fromthese twoMajo-
rana fermions a single usual fermion operator (non-hermitian), taking one
of the Majorana operators as the real part and the other as the imaginary
part, its state may be either occupied or empty with no cost in energy.
Defining d =1/2(γ +iγ ) and d =1/2(γ +iγ ) we can write
j j,2 j+1,1 N N,2 1,1
the Hamiltonian as
N−1
(cid:88) (cid:16) (cid:17) (cid:16) (cid:17)
H =t˜ 2d†d −1 +(cid:15) 2d† d −1 (17)
j j N N N
j=1
with (cid:15) = 0. Therefore the fermionic mode d does not appear in the
N N
Hamiltonian and the state may be occuppied or empty (d† d = 1,0, re-
N N
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spectively) with no energy cost. These two states are therefore degenerate
inenergyandareperfectlylocalizedattheedgesofthechainasδ-function
peaks (with exponential accuracy as the system size grows).
ThephasediagramoftheKitaevmodelshowsthreetypesofphases(see
Fig. 1): twotopologicalphasesinwhichtherearegaplessedgemodes,ifthe
system is finite, and two trivial phases with no edge modes. In the various
phases the bulk of the system is gapped and at the transition lines the gap
closes, allowingthepossibilityofachangeoftopology. Thetransitionlines
are located at ∆=0,|µ|≤2t˜and at |µ|=2t˜and any ∆.
2.2. Multiband system: 1D Two-band Shockley model
The Shockley model is a model of a dimerized system of spinless fermions
with alternating nearest-neighbor hoppings, given by the Hamiltonian (see
for instance3)
N
H =(cid:88)ψ†(j)(cid:2)Uψ(j)+Vψ(j−1)+V†ψ(j+1)(cid:1) (18)
j=1
wherethe2×2matricesU andV andthespinorψrepresentingtwoorbitals
at site j that are hybridized by the matrices U and V are given by
(cid:18) 0 t∗(cid:19) (cid:18)0t∗(cid:19) (cid:18)c (cid:19)
U = 1 ;V = 2 ;ψ(j)= j,A . (19)
t 0 0 0 c
1 j,B
t and t are hoppings and c (c ) destroy spinless fermions at site j
1 2 j,A j,B
belonging to sublattice A (B), respectively.
We may as well define Majorana operators as
1
c = (γ +iγ )
j,A 2 j,A,1 j,A,2
1
c = (γ +iγ ) (20)
j,B 2 j,B,1 j,B,2
Here A and B take the role of pseudospins. Taking t and t real, the
1 2
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Hamiltonian may be written as
N
H = it1 (cid:88)(γ γ +γ γ )
2 j,A,1 j,B,2 j,B,1 j,A,2
j=1
N
+ t2 (cid:88)(γ γ +γ γ )
4 j,A,1 j−1,B,1 j,A,2 j−1,B,2
j=2
N
+ it2 (cid:88)(γ γ −iγ γ )
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
j=1
N−1
+ it2 (cid:88) (γ γ −iγ γ ) (21)
4 j,B,1 j+1,A,2 j,B,2 j+1,A,1
j=1
Choosing t = 0 we find that the Majorana fermions γ , γ ,
1 1,A,1 1,A,2
γ and γ do not contribute and are zero energy modes. These
N,B,1 N,B,2
decoupled zero-energy modes are fermionic in nature, since the decoupled
Majoranas are located at the two end sites, A and B, respectively. This
point is characteristic of the topological phase as long as the bulk gap
does not vanish. In the trivial phase there are no decoupled Majorana
operators. As discussed for instance in Ref.3 the two types of phases may
also be distinguished by the winding number.
2.3. Multiband system: 1D SSH model with triplet pairing
This model may be viewed as a dimerized Kitaev superconductor5. The
dimerization is parametrized by η and the superconductivity by ∆.
This model is given by the Hamiltonian
(cid:16) (cid:17)
H =−µ(cid:80) c† c +c† c
j j,A j,A j,B j,B
(cid:104)
−t˜(cid:80) (1+η)c† c +(1+η)c† c
j j,B j,A j,A j,B
(cid:105)
+ (1−η)c† c +(1−η)c† c
j+1,A j,B j,B j+1,A
(cid:104)
+∆(cid:80) (1+η)c† c† +(1+η)c c
j j,B j,A j,A j,B
(cid:105)
+ (1−η)c† c† +(1−η)c c
j+1,A j,B j,B j+1,A
(22)
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1
K 1
0.5
SSH 2 SSH 0
0
D
-0.5
K 1
-1
-1 -0.5 0 0.5 1
h
Fig. 2. (Color online) Phase diagram of 1D SSH-Kitaev model for µ=0. The phase
SSH0 is trivial and has no edge modes. In the phases K1 there is one edge Majorana
modeateachedgeandinthephaseSSH2therearefermionicedgemodesateachedge
(t˜is the hopping, ∆ the pairing amplitude and µ the chemical potential).
The model with no superconductivity (∆ = 0) is related to the Shockley
modeltakingt =t˜(1+η)andt =t˜(1−η). Theregionofη >0corresponds
1 2
to t > t and vice-versa for η < 0. The Hamiltonian in real space mixes
1 2
nearest-neighbor sites and also has local terms.
In terms of Majorana operators the Hamiltonian is written as
N
µ(cid:88)
H =− (2+iγ γ +iγ γ )
2 j,A,1 j,A,2 j,B,1 j,B,2
j=1
it˜ (cid:88)N
− (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
− (1−η) (γ γ +γ γ )
2 j+1,A,1 j,B,2 j,B,1 j+1,A,2
j=1
N
i∆ (cid:88)
+ (1+η) (γ γ +γ γ )
2 j,A,1 j,B,2 j,A,2 j,B,1
j=1
N−1
i∆ (cid:88)
+ (1−η) (γ γ +γ γ )
2 j,B,1 j+1,A,2 j,B,2 j+1,A,1
j=1
(23)
Consider once again a vanishing chemical potential. Taking η = −1
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8
7
0
6
5
Mz4 -1 -1
3
1 C=-2 1
2
1 0 0 0 0
0
-6 -4 -2 0 2 4 6
e
F
Fig.3. (Coloronline)Phasediagramof2Dp-wavemodelasafunctionofthechemical
potentialandmagnetization. C istheChernnumberofeachphaseassociatedwiththe
numberofprotectedone-dimensionaledgemodesatfinitemagnetization.
and ∆ = 0 we have a state similar to the SSH or Shockley models
with two fermionic-like zero energy edge states, since the four operators
γ ,γ ;γ ,γ are missing from the Hamiltonian. If we select
1,A,1 1,A,2 N,B,1 N,B,2
η =0 and t=∆ is a Kitaev like state since there are two Majorana oper-
ators missing from the Hamiltonian, γ and γ , one from each end.
1,A,1 N,B,2
An example of a trivial phase is the point η = 1 and ∆ = 0 in which case
therearenozeroenergyedgestates. InFig. 2thephasediagramisshown.
Thismodelprovidesatestinggroundforthecomparisonbetweenfermionic
and Majorana edge modes. In addition, in some regimes it displays finite
energymodesthatarelocalizedattheedgesofthechain,asobtainedbefore
in other multiband models6.
2.4. Two-dimensional spinfull triplet superconductor
Anotherinterestingcaseisthatofatwo-dimensionaltripletsuperconductor
with p-wave symmetry, spin-orbit coupling and a Zeeman term7. We write
the Hamiltonian for the bulk system in momentum space as
Hˆ = 1(cid:88)(cid:16)ψ†,ψ (cid:17)(cid:32)Hˆ0(k) ∆ˆ(k) (cid:33)(cid:18) ψk (cid:19) (24)
2 k −k ∆ˆ†(k) −HˆT(−k) ψ†
k 0 −k
(cid:16) (cid:17) (cid:16) (cid:17)
where ψ†,ψ = ψ† ,ψ† ,ψ ,ψ and
k −k k↑ k↓ −k↑ −k↓
Hˆ =(cid:15) σ −M σ +Hˆ . (25)
0 k 0 z z R
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Here, (cid:15) = −2t˜(cosk + cosk ) − ε is the kinetic part, t˜ denotes the
k x y F
hopping parameter set in the following as the energy scale (t˜= 1), k is a
wave vector in the xy plane, and we have taken the lattice constant to be
unity. Furthermore, M is the Zeeman splitting term responsible for the
z
magnetization, in t˜units. The Rashba spin-orbit term is written as
Hˆ =s·σ =α(sink σ −sink σ ) , (26)
R y x x y
where α is measured in the same units and s = α(sink ,−sink ,0). The
y x
matrices σ ,σ ,σ are the Pauli matrices acting on the spin sector, and σ
x y z 0
is the 2×2 identity. The pairing matrix reads
(cid:18) (cid:19)
−d +id d
∆ˆ =i(d·σ)σ = x y z . (27)
y d d +id
z x y
We consider here d =0. If the spin-orbit coupling is strong it is energeti-
z
cally favorable that the pairing is of the form d=ds.
The energy eigenvalues and eigenfunction may be obtained solving the
Bogoliubov-de Gennes equations
(cid:32)Hˆ (k) ∆ˆ(k) (cid:33)(cid:18)u (cid:19) (cid:18)u (cid:19)
0 n =(cid:15) n . (28)
∆ˆ†(k) −HˆT(−k) v k,n v
0 n n
The 4-component spinor can be written as
 u (k,↑) 
n
(cid:18) (cid:19)
un = un(k,↓) . (29)
vn vn(−k,↑)
v (−k,↓)
n
The superconductor we consider here is time-reversal invariant if the
Zeemantermisabsent. ThesystemthenbelongstothesymmetryclassDIII
wherethetopologicalinvariantisaZ index8. IftheZeemantermisfinite,
2
time reversal symmetry (TRS) is broken and the system belongs to the
symmetry class D. The topological invariant that characterizes this phase
is the first Chern number C, and the system is said to be a Z topological
superconductor. The phase diagram is shown in Fig. 3.
Due to the bulk-edge correspondence if the system is placed in a strip
geometry and the system is in a topologically non-trivial phase, there are
robust edge states, in a number of pairs given by the Chern number, if
time reversal symmetry is broken. There are also counterpropagating edge
states in the Z phases even though the Chern number vanishes, as in the
2
spin Hall effect. In these phases time reversal symmetry is preserved and