Entanglement spectroscopy of anomalous surface states

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Arjun Dey and David F. Mross

Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 7610001, Israel

We study entanglement spectra of gapped states on the surfaces of symmetry-protected topo-

logical phases. These surface states carry anomalies that do not allow them to be terminated by

a trivial state. Their entanglement spectra are dominated by non-universal features, which reﬂect

the underlying bulk. We introduce a modiﬁed type of entanglement spectra that incorporate the

anomaly and argue that they correspond to physical edge states between diﬀerent surface states. We

support these arguments by explicit analytical and numerical calculations for free and interacting

surfaces of three-dimensional topological insulators of electrons.

Introduction. Over the past years, entanglement has

surpassed the notion of correlations for characterizing

and identifying quantum many-body systems. Famously,

the entanglement entropy of two-dimensional gapped sys-

tems contains a universal contribution that is non-trivial

for topologically ordered states1,2. This contribution van-

ishes for symmetry-protected topological states (SPTs),

which do not host any fractional quasiparticles in the

bulk. Still, their non-trivial topological nature can be

deduced by resolving the entanglement entropy accord-

ing to symmetries3–5.

More reﬁned information about topological states can

be obtained from entanglement spectra (ES). In a semi-

nal work, Li and Haldane6showed that the ES of cer-

tain quantum Hall states and their energy spectra at

a physical edge are describable by the same conformal

ﬁeld theory. They argued that topological phases can

thus be identiﬁed by their ES. Subsequently, such an ES–

edge state correspondence has been proven for a broad

class of two-dimensional topological states7,8. Addition-

ally, the agreement of ES and physical edge spectra has

been conﬁrmed empirically for various other systems, in-

cluding topological insulators9,10,p-wave superﬂuids in

the continuum11 and on a lattice12,13, fractional quantum

Hall states14–16, spin chains17 and the Kitaev honeycomb

model18.

The conjectured ES–edge correspondence makes the

tacit assumption that a physical edge is possible. As

such, it does not directly apply to an essential and widely

studied class of condensed-matter systems: Surface states

of topological insulators or superconductors. Such states

cannot be ‘stripped’ from their host. Physically removing

a ﬁnite surface layer of any topological system exposes

its bulk, leading to the formation of a new surface state.

We expect that, similarly, real-space entanglement cuts

of such systems reveal the underlying bulk, and the ES

are dominated by bulk properties (cf. Fig. 1).

More formally, any state that can arise on the surface

of a given SPT carries the same anomaly. If the SPT

is non-trivial, this anomaly is incompatible with a trivial

vacuum state, and the surface state cannot be terminated

by a physical edge. Only boundaries between surface

states with the same anomaly are possible. Crucially,

the interface between two gapped surface phases hosts

topologically protected states, uniquely identifying one

Figure 1. In non-anomalous systems, a real-space entangle-

ment cut corresponds to an analogous physical cut, see panel

(a). Anomalous states are inextricably tied to a topologically

non-trivial bulk, which is exposed upon performing a cut, see

panel (b).

provided the other is known. In this paper, we show

how entanglement spectroscopy can determine these edge

states between an unknown state and an arbitrary free-

fermion state.

As a concrete system for our numerical calculations, we

use an electronic topological insulator (TI) with time-

reversal symmetry T2=−119–21 as the paradigmatic

example of a 3D SPT. When its two-dimensional sur-

face is symmetric and non-interacting, it hosts a sin-

gle two-component Dirac fermion22,23. This theory can-

not arise in strictly two-dimensional systems with the

same symmetries due to fermion doubling24–26. When

time-reversal symmetry is broken on the surface, the

Dirac fermions become massive and realize a surface

Hall conductance of σxy =1

27,28 (in units of e2

h). By

contrast, the Hall conductance of gapped free fermion

systems in strictly two-dimensions must be an integer.

Similarly, breaking charge conservation realizes a time-

reversal-invariant cousin of topological p+ip29–31 super-

conductors with Majorana modes in vortex cores. Fi-

nally, strong interactions may gap the surface while pre-

serving both symmetries by forming an anomalous topo-

arXiv:2210.00021v1 [cond-mat.mes-hall] 30 Sep 2022

Figure 2. The entanglement spectra of anomalous surface

states exhibit a large number of low-lying states (λi≈0.5),

that increases with the cutoﬀ. Panels (a) and (b) show the

spectra for a gapless and massive Dirac cone, respectively,

with cutoﬀ Λ = 10. In both cases, the number of states

within the shaded low-energy window scales linearly with the

cutoﬀ and is thus non-universal (c).

logical order32–36.

Entanglement spectra of anomalous surfaces.

Any surface wave function of a free-fermion SPT may

be expressed as

|ΦiΛ=ˆ

ΦΛ(c†

E,i)|0i.(1)

Here ˆ

ΦΛis an operator-valued function, and c†

E,i creates

an electron in a single-particle eigenstate with energy E,

and additional quantum numbers i. The ‘empty state’

|0idenotes a Fermi sea ﬁlled up to the top of the va-

lence band at energy Λ. The symmetric, non-interacting

surface is thus represented by |Φi0

Λ=QiQµ

E=Λ c†

E,i |0i

with chemical potential µ. In general, |ΦiΛmust reduce

to |Φi0

Λfor cE→Λto describe a surface state that does

not hybridize with the bulk. The cutoﬀ Λ is inevitable

due to the anomalous nature of the surface state, but its

precise value does not aﬀect local observables.

To test our expectations for the ES of anomalous

states, we study the surface of a 3D TI, which hosts a

single Dirac cone governed by

HM=X

φ†

k[k·σ+Mσz]φk.(2)

Here, φ†

kcreates electrons with momentum k= (kx, ky),

σx, σy, σzare Pauli matrices, and Mis a time-reversal

symmetry-breaking mass. The un-normalized single-

particle eigenstates are

vk=√+M

√−Me−iϕk,uk=√−Meiϕk

−√+M,(3)

where =√k2+M2and k, ϕkare the magnitude and

polar angle of the two-dimensional momentum. The cor-

responding single-particle energies are Ev=−Eu=,

and the ground state of HMis |ΦMiΛ=Qkukφ†

k|0i. In

Appendix A, we show the spectrum of Eq. (2) for spher-

ical geometry.

We obtain the ES by decomposing the Hilbert space of

the surface into two parts, H=Hα⊗ Hβ. The Schmidt

decomposition

|Φi=Xne−λn/2|Φα,ni⊗|Φβ,ni,(4)

with |Φα(β),ni∈Hα(β)yields the ‘entanglement energies’

λi. We numerically compute these numbers for |ΦMiΛ

with α, β, the two hemispheres of a sphere with radius

Rand (Fig. 2). For local observables, we would expect

universal results when R−1MΛ. For the ES, we

instead ﬁnd that the number of low-lying λigrows lin-

early with the cutoﬀ Λ. In Fig. 2panel (c), we show the

number of pseudo energy states as a function of the cutoﬀ

for massless and massive Dirac fermions. We attribute

this cutoﬀ dependence and the large number of low-lying

states to the exposure of the underlying bulk (cf. Fig. 1).

As anticipated in the introduction, straightforward com-

putation of a surface state’s ES is not suitable for its

identiﬁcation.

Relative entanglement spectra. We adapt entan-

glement spectroscopy to SPT surfaces by drawing on in-

sights into physical surface spectra (cf. Appendix B).

Recall that edge-energy spectra are only meaningful for

two surface states with the same anomaly. However, their

boundary state can equivalently arise at the physical edge

of a speciﬁc non-anomalous phase, which is subject to the

standard ES-edge correspondence. Our strategy is thus

to construct non-anomalous wave functions that encode

the boundary between two surface states Aand B.

To obtain the desired wave functions, we begin with

a gapped free-fermion surface state A, whose particle-

and hole-like excitations are created by cA,†

+,i , cA,†

−,i . By

construction, Acorresponds to a non-anomalous wave

function of particles and holes, i.e., their trivial vacuum.

Next, we consider a puddle of any other surface state

embedded within A. It can be created locally from par-

ticle and hole excitations on top of the uniform Astate,

which does not introduce any anomaly. Likewise, any

surface state Bcan be encoded in a non-anomalous wave

function of the excitations cA

±,i. Such wave functions de-

pend on both Aand B; we thus refer to them as relative

wave functions from which we obtain the relative ES. We

now proceed by numerically calculating the relative ES of

various 3D TI surface states and comparing them with

the physical edge spectra. Subsequently, we elaborate

on the relative wave functions and provide an analytical

perspective on their ES.

Numerical results I: Free fermions. We compute

the relative ES described above for various surface states

of a spherical 3D TI. For |ΦAiΛ, we take states with a

magnetic or superconducting gap, i.e., the ground states

of Eq. (2) or of

HSC

∆s=H0+ ∆sX

[φk,↑φ−k,↓+ H.c.] .(5)

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摘要：

EntanglementspectroscopyofanomaloussurfacestatesArjunDeyandDavidF.MrossDepartmentofCondensedMatterPhysics,WeizmannInstituteofScience,Rehovot7610001,IsraelWestudyentanglementspectraofgappedstatesonthesurfacesofsymmetry-protectedtopo-logicalphases.Thesesurfacestatescarryanomaliesthatdonotallowthemtobe...

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