Hinge States of Second-order Topological Insulators as a Mach-Zehnder Interferometer Adam Yanis Chaou1Piet W. Brouwer1and Nicholas Sedlmayr2

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Hinge States of Second-order Topological Insulators as a Mach-Zehnder

Interferometer

Adam Yanis Chaou,1Piet W. Brouwer,1and Nicholas Sedlmayr2, ∗

1Dahlem Center for Complex Quantum Systems and Institut f¨ur Physik,

Freie Universit¨at Berlin, Arnimallee 14, D-14195 Berlin, Germany

2Institute of Physics, Maria Curie-Sk lodowska University,

Plac Marii Sk lodowskiej-Curie 1, PL-20031 Lublin, Poland

(Dated: January 25, 2023)

Three-dimensional higher-order topological insulators can have topologically protected chiral

modes propagating on their hinges. Hinges with two co-propagating chiral modes can serve as

a “beam splitter” between hinges with only a single chiral mode. Here we show how such a crystal,

with Ohmic contacts attached to its hinges, can be used to realize a Mach-Zehnder interferome-

ter. We present concrete calculations for a lattice model of a ﬁrst-order topological insulator in a

magnetic ﬁeld, which, for a suitable choice of parameters, is an extrinsic second-order topological

insulator with the required conﬁguration of chiral hinge modes.

I. INTRODUCTION

A Mach-Zehnder interferometer gives an interference

pattern which depends on the phase shift between two

paths taken by a beam-split signal. Optical Mach-

Zehnder interferometers have a long history [1]. Tak-

ing advantage of the absence of backscattering resulting

from the unidirectional electron motion in the edge states

of the two-dimensional integer quantized Hall eﬀect, a

Mach-Zehnder interferometer could also be realized in

a two-dimensional electron gas [2,3]. In this case, the

electronic equivalent of a “beam splitter” is formed by a

point contact, which allows for the controllable coupling

of modes at diﬀerent sample edges.

One-dimensional electron modes without backscatter-

ing also exist at the hinges of a three-dimensional second-

order topological insulator [4–13]. Signatures of such

hinge modes have been seen in pure Bismuth [14,15],

in Bi-based compounds [16,17], and in Fe-based super-

conductors [18,19]. Interference eﬀects involving pairs

of counter-propagating (“helical”) [20] or unidirectional

(“chiral”) [21,22] hinge modes were proposed theoret-

ically. In these proposals, the beam splitter is formed

by the point contact between an idealized single-channel

normal-metal lead and the topological insulator. In this

article, we show how a crystal hinge supporting multi-

ple chiral modes naturally forms a beam splitter between

adjacent hinges with only a single mode. This way, a

Mach-Zehnder interferometer can be realized with Ohmic

source and drain contacts placed over a crystal hinge.

The setup we consider is shown schematically in Fig.

1. It consists of a second-order topological insulator with

hinges that have one or two chiral modes, as indicated in

the ﬁgure. The crystal hinges with two co-propagating

chiral modes serve as beam splitters. Ohmic source and

drain contacts are placed at selected crystal edges with

a single chiral hinge mode, such that there are two paths

∗e-mail: sedlmayr@umcs.pl

Figure 1. (a) A schematic of the system considered in this ar-

ticle. It consists of a second-order topological insulator with

hinges that have one or two chiral states. Ohmic contacts,

which pairwise serve as source (S1, S2) and drain (D1, D2)

contacts, are attached to four of the hinges with one chiral

mode. A pair of interfering paths connecting contacts S1 and

D1 is indicated in blue. (b): Eﬀective network diagram in-

dicating the same interfering paths between contacts S1 and

D1.

connecting each pair of source and drain contacts along

the crystal hinges. By controlling the phase diﬀerence

between the interfering paths with a magnetic ﬁeld one

thereby obtains a Mach-Zehnder interferometer.

Like the chiral edge states of the integer quantized Hall

eﬀect, the hinge modes of a higher-order topological insu-

lator are topologically protected. One distinguishes “in-

trinsic” hinge modes, which are protected by the topol-

ogy of the bulk band structure and “extrinsic” modes, for

which the nontrivial topology resides in the surface band

structure, whereas the bulk may be topologically trivial

[12,23,24]. Whereas intrinsic higher-order phases re-

quire crystalline symmetries for their protection, extrin-

sic topological phases do not have additional symmetry

requirements. For the realization of an interferometer, all

that matters is the existence of the chiral hinge modes,

not where they derive their protection from. For that

reason, in this article we seek a (theoretical) realization

of a Mach-Zehnder interferometer in an extrinsic second-

order topological insulator.

A particularly simple and controllable model of an ex-

trinsic second-order topological insulator was proposed

by Sitte et al. in Ref. [5]. It consists of a (ﬁrst-order)

arXiv:2210.02749v2 [cond-mat.mes-hall] 24 Jan 2023

Figure 2. (a): An extrinsic second-order topological insula-

tor with chiral hinge modes can be realized by placing a ﬁrst

order topological topological insulator in a uniform magnetic

ﬁeld B0. The direction of B0is chosen such that there is

a nonzero ﬂux through all crystal surfaces. As a result, the

Dirac-cone surface states form an integer quantized Hall eﬀect

with a ﬁlling fraction that depends on the magnetic ﬂux den-

sity and the electron density at the surface. The latter can be

controlled capacitively by a metal gate, one is shown explic-

itly on surface x+. (b): Labeling of surfaces and hinges. The

surfaces on the sides are labelled by x±and y±analogously

to z±, labelled here in red.

topological insulator placed in a magnetic ﬁeld B0at a

generic direction with respect to the crystal faces, see

Fig. 2. Without the magnetic ﬁeld, there are Dirac-cone

surface states at the crystal surfaces. The magnetic ﬁeld

B0gaps these out. This eﬀectively turns the crystal sur-

faces into two-dimensional quantized Hall systems with

a (half-integer) ﬁlling fraction that depends on the per-

pendicular component of the magnetic ﬁeld and the po-

sition of the Fermi level with respect to the Dirac point

of the surface band structure. The former can be con-

trolled by the applied magnetic ﬁeld, the latter by a gate

voltage applied locally at the surface. The number of

hinge states then follows as the diﬀerence of the ﬁlling

fractions of the two adjacent surfaces. To bring about

the interference pattern, one considers a small change

δBof the magnetic ﬁeld. If |δB|  B0,δBchanges the

phases which electrons pick up while propagating along

the hinges, while not aﬀecting the number of hinge states

and their properties.

The remainder of this article is organized as follows:

In Sec. II we present a simple lattice model of an extrin-

sic second-order topological insulator as discussed above

and establish that it has the phenomenology shown in

Fig. 1for a suitable choice of parameters. In Sec. III

we add Ohmic contacts to crystal edges with a single

chiral mode, as indicated in Fig. 1, and theoretically de-

scribe the resulting interferometer setup using scattering

theory. In Sec. IV we consider a two-terminal Aharonov-

Bohm interferometer based on the same model system.

We conclude in Sec. V. Further details and supporting

material can be found in the appendices.

II. LATTICE MODEL OF AN EXTRINSIC

SECOND-ORDER TOPOLOGICAL INSULATOR

We theoretically describe the extrinsic second-order

topological insulator using a four-band lattice model with

nearest-neighbor hopping [5]. It has the Hamiltonian

H=X

hi,ji

ˆc†

it(ri,rj)ˆcj+X

ˆc†

iu(ri)ˆci,(1)

where the indices iand jrun over all neighboring sites of

a three-dimensional simple cubic lattice, riand rjare the

corresponding position vectors, ˆci, ˆcjand ˆc†

i, ˆc†

jare four-

component spinor annihilation and creation operators,

and t(ri,rj) and u(ri) are 4 ×4 matrices. We consider a

lattice of size Lx×Ly×Lz, with surfaces perpendicular

to the coordinate axes, shown schematically in Fig. 2.

For the nearest-neighbor term twe take

t(r,r0) = −t

2σ3τ0+i

aσ1τ·(r−r0)

×eie(A(r)+A(r0))·(r−r0)/2~c,(2)

where ais the lattice constant, ta hopping amplitude

(with the dimension of energy), σαand τα,α= 1,2,3,

are Pauli matrices, and A(r) is the vector potential cor-

responding to the uniform applied magnetic ﬁeld B. The

on-site term uis

u(r) = (3 + m)tσ3τ0+V(r)σ0τ0,(3)

where mis a parameter governing the bulk band struc-

ture and V(r) a scalar potential, which is nonzero in the

vicinity of the crystal boundaries only.

Without applied magnetic ﬁeld, the system has time-

reversal symmetry. It is in a topological phase with gap-

less Dirac-cone surface states for −2<m<0. The sur-

face Dirac nodes are at zero energy if the scalar potential

Vis zero, but they may be pushed away from zero by

application of uniform potential at the surface. We take

a scalar potential of the form

V(r) = X

Vse−rs,⊥/ξs,(4)

where the summation index sruns over all six surfaces of

the crystal, Vsis a gate voltage at surface s,rs,⊥is the

distance to the surface s, and ξsa decay length. In our

calculations, we set m=−1 and ξs= 5athroughout.

A uniform magnetic ﬁeld at a direction such that there

is a ﬁnite ﬂux penetrating all six crystal surfaces, gaps out

the surface Dirac cones and eﬀectively turns the surfaces

into gapped quantized Hall eﬀects. The ﬁlling fractions

of the diﬀerent surfaces scan be tuned by varying the

surface gate voltages Vsof Eq. (4).

To establish that the model describes an extrinsic

second-order topological insulator [5] with the conﬁgu-

ration of hinge states shown in Fig. 1, we consider a sys-

tem that is inﬁnite along each one of the coordinate axes

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HingeStatesofSecond-orderTopologicalInsulatorsasaMach-ZehnderInterferometerAdamYanisChaou,1PietW.Brouwer,1andNicholasSedlmayr2,1DahlemCenterforComplexQuantumSystemsandInstitutfurPhysik,FreieUniversitatBerlin,Arnimallee14,D-14195Berlin,Germany2InstituteofPhysics,MariaCurie-SklodowskaUniversity,Pla...

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Hinge States of Second-order Topological Insulators as a Mach-Zehnder Interferometer Adam Yanis Chaou1Piet W. Brouwer1and Nicholas Sedlmayr2

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