Table Of ContentBulk-edge correspondence in topological pumping
Y. Hatsugai1,∗ and T. Fukui2,†
1Division of Physics, University of Tsukuba, Tsukuba 305-8571, Japan
2Department of Physics, Ibaraki University, Mito 310-8512, Japan
(Dated: July 22, 2016)
The topological pumping proposed in ’80s and recently realized by cold atom experiments[1–3]
is revisited from a view point of the bulk-edge correspondence. For a system with boundaries, a
6 new form of the pumped charge is derived by the Berry connection in the temporal gauge, that
1 corresponds to the shift of the center of mass (CM). Even with boundaries, the pumped charge is
0 carried by the bulk and its quantization is guaranteed by the discontinuities of the CM associated
2 with the edge states. This is a modified Laughlin argument based on the local U(1) invariance
although physicsbehind is quitedifferent.
l
u
J
1 Fundamentalstabilityofedgestates[4–6]inthequan- Many of the theoretical ideas for the topological phases
2 tum Hall effect (QHE) implies the phase is topological. andgaugestructuresofmatterarenowdirectlymeasured
The bulk is gapped but, with boundaries, there exists experimentallywhere quantumcoherenceandstructures
]
localized modes in the gap. Conversely non trivial topo- are under the ultimate control and one can handle or
l
al logicalphases are necessarilyassociatedwith edge states even synthesize dimensions [23, 24]. One of the most
h as generic localized modes near the boundaries of the clear and fundamental examples is a topological pump
- systemor impurities. It applies to any ofthe topological proposed long time ago [1], where the time is used as
s
e phases that include the fractional QH states, quantum a synthetic dimension and the topological effects in the
m spin Hall states as the topological insulators, the Hal- QHE in 2D is realized in a simple one dimensional sys-
dane phase of integer spin chains and so on. Generically tem. Even though the proposalis quite old as the QHE,
.
t
a emergence of the edge states is reduced to the feature of it took so long time until its experimental realization in
m the bulk, that is the bulk-edge correspondence [7]. cold atoms [2, 3], and intense studies have just began.
The phase is “topological”implies it is “hidden”[8], in Although the bulk-edge correspondence is fundamental
-
d the sense that one can hardly observe its character in a in topological phases, it has never been applied to this
n bulk, even though the phase is characterized by a topo- topological pump. Here we have first made the role of
o
logical invariant[9–11]. Further, they are mostly not the edge states clear in the topological pump. In this
c
physical observables. Even in the QHE, whether the ob- letter, a new expressionof the pumped charge is derived
[
served Hall conductance in experiments is of the bulk or by the Berryconnectionfor the systemwith boundaries.
2 of the edges is still controversial. The bulk topological In the adiabatic limit, the contribution of the the bulk
v
character is hidden but the edge states are real observ- and the edge are clearly separated. The pumped charge
7
ables. A typical example is a surface Dirac cone of the is carried by the bulk but its quantization is guaranteed
3
threedimensionaltopologicalinsulators,thatisobserved bythelocalityoftheedgestates. LocalU(1)gaugesym-
5
3 bytheangleresolvedphotoemissionspectroscopies. The metry as the charge conservation is crucially important
0 bulk-edge correspondence is thus a rigorous/conceptual but plays a role different from the QHE. Namely, in the
. bridge between the bulk and with boundaries, that was, QHE,theright-andleft-edgestatesarealwayspairedon
1
0 at first, rigorously shown for the QHE [7] and justified the Fermi surface, whereas in the pumping, they can be
6 today for the others as well [10–16]. separately observed at different times. Physically this is
1 Thisreversewayofthinkingis importantsinceonecan a fractionalizationof the electron into the massive Dirac
: realize why the edges are there from a universalpoint of fermions carrying half charge quantum.
v
i view. Some of them are historically well-known such as Let us consider a many body topologicalpump by the
X
dangling bonds of semiconductors and some are new as time dependent one dimensional hamiltonian of lattice
r the chiral edge modes of photonic crystals [17, 18] and free fermions of L sites
a x
photon’s[19]. Thisanalogyevenextendstotheclassical
Newtonequation[20–22]. Itimpliestheuniversalfeature Lx
of the bulk-edge correspondence. H(θ,t)= −tθxc†j+1cj +h.c.+vj(t)c†jcj ,
Ultimate developments of recent quantum technology Xj (cid:2) (cid:3)
also trigger further studies of topological phases, such
as realization of the Hofstadter butterfly by a synthetic (tθx = txe−iθ/Lx,tx,ty ∈ R. The twist θ is introduced
gauge field in a artificial lattice using cold atoms [23]. to define the current operator as well as the Berry con-
nection for a generic many-body state). The time de-
pendent potential v (t) can be any if it is periodic as
j
v (t)=v (t)whereq isapositiveinteger. Forsimplic-
j+q j
∗ [email protected] ity, we choose as v (t) = −2t cos(2πt −2πφj), φ = p/q
† [email protected] with mutually primje p and qy. ThisTis equivalent to the
2
2D QHE under a periodic potential[6, 7, 9]. When map-
ping back to the QHE in 2D, the time as a synthetic
dimension corresponds to the momentum in y direction
(2πt →k [6]). Bywriting the oneparticle state as|ψi=
T y
ψ c†|0i, the wave function ψ satisfies the Harper
j j j j
Pequation,−tθψ −2t cos(2πt−2πφj)ψ −(tθ)∗ψ =
x j−1 y T j x j+1
E ψ . Thetime-reversal(TR)isbrokenbyafiniteθbut
ℓL j
is recovered by taking the θ → 0 limit after the calcula-
tion. The pumping period, T, controls the adiabaticity
of the pumping. For simplicity, we take T = 2π and
control the adiabaticity by the energy scale of tx and ty. FIG.1. (a)TheBerryconnection A(t) inthetemporalgauge
Twoboundaryconditionsarediscussed. Oneisthe open and (b) its lattice analogue [25].
boundary condition with edges, e: ψ = ψ = 0 and
0 Lx
the other is periodic one for the bulk b: ψ = ψ .
j+Lx j
The manybody eigenstate ofthe snapshothamiltonian point is general and applicable for both with/without
H(t) is givenby specifying a set of occupied one particle edges.
states {ℓ } as |αi= ( ψ c†|0i. Now let us consider a system with edges by imposing
UsingLthe current oQpℓeLraPtorj j,ℓL j the open boundary condition. In this case, the twist of
the hopping e−iθ/Lx is gauged out by the many body
1 t
J = (i xe−iθ/Lx) c† c +h.c=~−1∂ H(θ,t), gauge transformation
L ~ j+1 j θ
x X
j U(θ)= e−iθnj(j−j0)/Lx, j =L /2,
0 x
themeasuredcurrentatthetimet,δj =hG(t)|J|G(t)i− jY=1
hg(t)|J|g(t)i, is evaluated by the adiabatic approxima-
which operates for the fermion operator as Uc U† =
tion[1]assumingafinite energygapabovethe snapshot j
ground state |g(t)i, H(θ,t)|gi = |giE. The state |G(t)i eiθj/Lxcj andwehave|g(θ)i=U|g0iwhere|g0iisa snap
is a true many body state that obeys the time depen- shot ground state of the hamiltonian H(0,t). Then not-
dentSchr¨odingerequation,i~∂t|G(t)i=H(θ,t)|G(t)i. It ing that At = hg|∂tgi = hg0|∂tg0i is θ independent and
reads δj = −iB, B = ∂θAt −∂tAθ where B is a field Aθ =hg0|U†∂θU|g0i, the Berry connection in the tempo-
strength of the Berry connection A =hg|∂ gi, µ=θ,t. ral gauge for the system with edges is
µ µ
Notethathg(t)|J|g(t)i| =0duetotheTRinvariance.
θ=0 A(t),e =−i P(t)−P(0) , P(t)= x ρ (t)
Strianncesftohrmefiaetilodnst|gre′ing=th|gBieiiχs,gAau′g=eihngv′a|∂riagn′tif=orAthe+gia∂ugχe, θ (cid:2) (cid:3) Xj j j
let us take a temporal gauge bµy imposµing a gauµge cµon- where ρj(t)=hg0(t)|nj|g0(t)i, xj = jL−xj0 and P(t) is the
dition, A(t) = 0, which is quite useful for the discus- CM. Now we have the pumped charge as
t
sion of the pumping. Then the field strength is given as
∆Q =P(t )−P(t ).
(t) [ta,tb] b a
B = −∂ A . Now we have a pumped charge between
t θ
the time period [t ,t ] as Note that this is only well defined for a system with
a b
boundaries. We also stress that the CM derived
tb tb tb here is distinct from the Zak phase for infinite systems.
∆Q = dtδj =i dt∂ A(t) =iA(t)(t) ,
[ta,tb] Z x Z t θ θ (cid:12) We assume P(t) is a CM measured for the snap shot
ta ta (cid:12)(cid:12)ta ground state in contact with a particle reservoir, thatis,
(cid:12)
where the integration over t has been carried out but it the system is specified by the chemical potential µ and
needs a special care, as discussed later. For any (regu- thetemperatureissufficientlylow. Itshouldbealsodis-
larly) gauge fixed many body state |gi, the state in the tinguishedfromtheCMofthetimedependentwavefunc-
temporal gauge is given by |g(t)(θ,t)i = |g(θ,t)ieiχ(θ,t), tion x hG(t)|n |G(t)i, which is recently observed in
j j j
with the phase factor, χ, that is path dependent and real exPperiments[2,3,26]. Eventhoughthefermienergy
t
is explicitly given as χ(θ,t) = i 0 dτAt(θ,τ) + is in the bulk gap, when the one particle energy of the
i θdϑA (ϑ,0). It surely satisfies the gRauge condition, edge state coincides to the fermi energy, the many body
R(0t) θ gap of the snap shot hamiltonian necessarily closes and
A (θ,t)=0 and one has
t the edge state becomes suddenly occupied/unoccupied
t (See Fig.2). This suddenchangeofthe snapshotground
A(θt)(θ,t)=Aθ(θ,t)−∂θZ dτAt(θ,τ)−Aθ(θ,0). state causes singularities (discontinuities) in P since the
0 edge state is spatially localized and its contribution to
This is gauge invariant: it is directly confirmed but is thenormalizedCMis±1/2inthelimitL →∞. Thisis
x
clearinadiscretizedform[25]asshowninFig.1. Substi- inevitable since topologically non trivial ground state is
tuting this into ∆Q above, we have a pumped charge in associatedwith the edge states passing throughthe gap.
a novel gauge invariant form. The discussion up to this Then labeling the gap closing time’s by t ’s (t < t ,
i i i+1
3
i=1,2,···) and dividing the period into the time inter- are always integers in the process of a unit flux penetra-
vals without the singularities (patch-working in the the tion,sincethe right/leftedgestatesarealwayspairedon
timedomain),thetotalpumpedchargeinasingleperiod the Fermi surface. This ±1/2 contribution can be un-
T is given as derstood as a fractionalization of electrons into massive
Dirac fermions[27, 28].
t−
∆Qe = i+1dt∂ P(t)= P(t− )−P(t+) Also counting the topological number with disconti-
Xi Zt+i t Xi (cid:2) i+1 i (cid:3) nuities here should be compared with the counting of
thesingularitiesoftheη-invariantfortheAtiyah-Patodi-
=− P(t+)−P(t−) =− ∆P(t ),
i i i Singerindex theorem[29, 30]. Here wehaveclarifiedthe
Xi (cid:2) (cid:3) Xi closeinter-relationbetweenthe topologicalnature ofthe
where t± ≡ t ±0 and ∆P(t ) ≡ P(t+)−P(t−) is a discontinuities and local U(1) gauge symmetry. This is
i i i i i not just a math but plays a crucial role in the recent
discontinuityoftheCMatt . Alsoweusedthe periodic-
i
experiments[2, 3].
ity of the CM in the adiabatic limit P(t+T) = P(t),
which follows from the periodicity of the hamiltonian As an example, we calculated the CM, P(t), numeri-
H(t + T) = H(t). This simple expression can be un- cally for φ = 2/7 and Lx = 1750 and Lx = 350. By the
derstood as a bulk-edge correspondence in time domain. clear finite size effects, the discontinuities deviates from
We put ”e” to mark the expression for the system with ±1/2 but approaches to the quantized values ±1/2 by
edges. Note here that P is only well defined with edges the limit Lx →∞. In this case, we have ∆Qe =−2.
andcannotbedefinedforasystemwithperiodicbound-
ary condition. Depending on the position of the edge
statethatcutsthefermienergyandthewayitdoes,this
discontinuity is determined as
−1/2 Right: become unoccupied
+1/2 Right: become occupied
∆P(t )=
i +1/2 Left: become unoccupied
−1/2 Left: become occupied,
where“Right”impliestheedgestateislocalizednearthe
boundary j = L and the “Left” is for the edge states
x
near the boundary j ∼0.
Since the pair of the discontinuity coincides to the
winding of the corresponding edge state energy around
the “hole” (that corresponds to the energy gap) on the
Riemann surface [6], total discontinuity is given by the
winding number I of the edge states
M
∆P(t )=−I , ∆Qe =I ,
i M M
X
i
where we assume that the fermi energy is in the M-th
energy gap from below. This algebraic definition of the
windingnumberisonlypossiblefortheHarperequation,
that is, only for special form of the v (t). However the
j
relation is generically justified by defining the winding
number I as the number of paired edge states with
M
suitable sign depending on the direction of the crossing
ofthespectralflowwiththefermienergy. Thisisamod-
ified Laughlin argument [5] which is widely used for the
topologicalnumber of edge states for various topological
phases[6, 10, 14]. Since the total number of particles is
FIG. 2. (a) One particle spectrum E (t) and the
conserved, the gapless times t ’s that correspond to the ℓL
i fermi energy (blue line): tx = ty = 1, φ = 2/7
(dis)appearance of the edge state are paired (irrespec- and Lx = 1750 (350). (b) Enlarged spectrum near
tive to the position). It guarantees the quantization of the fermi energy. (c) The CM, P(t), with edges by
the total pumped charge ∆Qe as an integer since even numerical calculation. ∆P(ti) = −0.499503(−0.4954),
number of additions of ±1/2 is an integer. It is a conse- 0.496542(0.487094), 0.498412(0.494115), 0.497435(0.491092),
quenceofthelocalU(1)gaugesymmetryastheLaughlin 0.497435(0.491092), 0.498412(0.494115), 0.496542(0.487094),
argument, but has sharp contrast to the QHE in which and −0.499503(−0.4954) for i = 1,··· ,8. Pi∆P(ti) =
1.98577(1.9538).
contribution of the edge states to the Hall conductance
4
Note that although the total pumped charge is gov-
erned by the discontinuity ∆P(t ) due to the edge
i i
states, pumped charge is nPot carried by singularities
caused by the edge states. The charge is still carried by
the bulk as we explain below. This is the bulk-edge cor-
respondence in the topological pumping. As one can see
in Fig.2, the charge is pumped in the intervals between
the singularities (red lines), which is the bulk contribu-
tion. Eventhoughthesystemhasboundaries,theeffects
oftheedgesarenegligibleforthebulkstatesincetheone
particle state of the bulk is extended and the amplitude
near the boundaries is vanishing in the limit L →∞.
x FIG.3. Pumpedchargebetweenthetimeinterval[0,t]. The
Although the CM is ill-defined for bulk (both for red line is by the bulk (Bloch states) ∆Qb . The shifted
a periodic/infinite system), the pumped charge is well- pumpedcharge∆Qe[0,t]’sareshown forLx[0=,t]140(blue)and
defined as discussed by Thouless[1]. As for an infinite Lx = 1750 (black) lines. The parameters are the same as
system, the one particle state is givenby the Blochstate Fig.2.
ψj,ℓL ∝eikxjuj(kx).
Nowlet|g(t)ibeamanybodygroundstateofthesnap
shothamiltonianinthetemporalgauge. Assumingthe The pumped charge by bulk in a cycle is given by
fermi energy is in the M-th gap, one has the Chern number CM [1] of the Berry connection aµ
as ∆Qb = ∆Qb = C . Now we have established
[0,T] M
A(t),b =hg(t)|∂ g(t)i= ∆k dkx a(t), the physicalbulk-edge correspondencein the topological
θ θ Z 2π kx pumping as
0
a(ktx) =TrMA(ktx), A(ktx) =u†∂kxu, u=(u1,··· ,uM), ∆Qe =∆Qb
where the limit Lx → ∞ is taken and additional gauge that gives the relation between the two topological in-
conditiona(t) =Tr u†∂ u=0 is imposed. Also we put variants as was also shown in the QHE as
t M t
“b” to specify that it is purely from bulk. By using the
I =C .
gauge invariant form of the temporal gauge, M M
It guarantees quantization of the Chern number C ,
t M
a(ktx)(kx,t)=akx(kx,t)−∂kxZ dτat(kx,τ)−akx(kx,0), since IM due to the singularities of edge states is quan-
0 tized by definition.
Generically the bulk-edge correspondence is for the
the time derivative of A(t),b is written by the field
θ manybody state and the bulk and the edge can not be
strengthof the Bloch state, b=∂kxat−∂takx =−∂ta(ktx). separated. Withoutgappedbulk,edgestatesarenotwell
Nowthebulkcontributionofthepumpedchargebetween defined. We put a stress on that in the present pumping
the time period [ta,tb] is written as[1] ofthenoninteractingsystem,themanybodystateiscon-
structed from the one particle states which are classified
tb 1 tb ∆k
∆Qb =i dt∂ A(t),b = dt dk b(k ,t). into the bulk(unnormalizable) and the edge(localized).
[ta,tb] Zta t θ 2πiZta Z0 x x Then the contributions to the pumped charge are also
clearly separated (Fig.3).
It implies that i∂tA(θt),b is an effective CM of the bulk, To describe the topological pumping by the CM, one
even though the CM itself is not well defined. necessarily needs to use a system with edges in contact
To demonstrate the contribution of the pumping with withaparticlereservoir. Aswehaveestablishedhere,the
edgesisfromthebulk,letusdefineapumpedchargebe- pumpedchargeofthesystemwithedgesiscarriedbythe
tweenthetimeperiod[ta,tb]bycompensating(skipping) bulk not by the edge states and the CM observed in the
the singularities from the edges as cold atom experiments P (t) = x hG(t)|n |G(t)i
exp j j j
tb is described by the bulk. Still thePquantization of the
∆Qe =P(t )−P(t )− dτ ∆P(t )δ(τ −t ). pumped charge is governedby the edge states where the
[ta,tb] b a Zta Xi i i edge states induce discontinuities in the CM in the adi-
abaticlimit. Eventhoughthe CMshowssingularitiesin
This is a bulk contribution to the CM for the system
theadiabaticlimit,thesesingularitiescannotbeobserved
with edges. According to the consideration here, we ex-
in realistic experiments in cold atoms, since the appear-
pect this bulk contribution ∆Qe asymptotically ap-
[ta,tb] ance of the edge states at the fermi energy necessarily
proaches to ∆Qb in the large system. We have nu- implies the breakdown of the adiabaticity. As in an re-
[ta,tb]
mericallyevaluatedthemforφ=2/7andshowninFig.3. alistic experiment with finite speed pumping, the many
They show clear coincidence within numericalerrors,in- body wave function approximately remains unchanged
dicating the relation between the bulk and the edges. passing through the gapless point, that is, the bulk is
5
adiabatic but the edge states is well described by the cially for cases with high Chern numbers. We have also
sudden approximation. It causes excitation across the confirmedconsistencyoftheargumentbyadirectnumer-
gap, which eventually results in accumulation/depletion ical integration of the Liouville−von Neumann equation
of the surface charges after one cycle. These two condi- for the density matrix.
tions,asthebulkisadiabaticandtheedgeissudden,are We thank stimulating discussions on the topological
conflicting requirements, if one requires both limits rig- pump with Y. Takahashi, S. Nakajima and M. Lohse.
orously. Then in the experiment, it may not be easy to The workis supported in partby KAKENHI from JSPS
observe exact quantization of the pumped charge, espe- (Nos.26247064,25400388,16K13845).
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