Fractional Topology in Interacting 1D Superconductors

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Frederick del Pozo∗, Lo¨ıc Herviou†, and Karyn Le Hur∗

∗Centre de Physique Th´eorique, ´

Ecole Polytechnique (IP Paris), 91128 Palaiseau, France and

†SB IPHYS, EPFL, Rte de la Sorge CH-1015 Lausanne

(Dated: March 17, 2023)

We investigate the topological phases of two one-dimensional (1D) interacting superconducting

wires and propose topological markers directly measurable from ground state correlation functions.

These quantities remain powerful tools in the presence of couplings and interactions. We show

with the density matrix renormalization group that the double critical Ising (DCI) phase discovered

in [1] is a fractional topological phase with gapless Majorana modes in the bulk, and a one-half

topological invariant per wire. Using both numerics and quantum ﬁeld theoretical methods, we

show that the phase diagram remains stable in the presence of an inter-wire hopping amplitude t⊥

at length scales below ∼1/t⊥. A large inter-wire hopping amplitude results in the emergence of two

integer topological phases, stable also at large interactions. They host one edge mode per boundary

shared between both wires. At large interactions, the two wires are described by Mott physics, with

the t⊥hopping amplitude resulting in a paramagnetic order.

I. INTRODUCTION

Quantum Mechanics gives rise to important phenom-

ena such as Bose-Einstein condensation and its charged

analog superconductivity, wherein a coherence builds up

between the constituents in the bulk of the material.

Such phases of matter are governed by quantum phase

transitions, wherein even at zero temperature quantum

ﬂuctuations become relevant enough to drive, or desta-

bilize, an underlying ordering. Already since the 70s,

it was found that such transitions can occur without

the inherent breaking of symmetries, and, instead, the

phases of matter are distinguished by winding numbers

or more generally by topological invariants [2, 3]. Whilst

these are usually found to be integer values [4], in the

presence of interactions fractional topological states of

matter have also been predicted as for example in the

fractional quantum hall eﬀect (FQHE) [5], associated

with the direct observation of fractional charges [6, 7].

Due to their robustness against perturbations and

impurities, topological materials have seen a growing

interest in recent years. Particularly the applications of

topological superconductors and insulators in quantum

circuits and computers [8] drive both experimental and

theoretical developments. The key reason these topolog-

ical materials are particularly promising is the existence

of exotic anyonic edge modes, which have been proposed

as candidates to build resilient, large-scale quantum

computers [9]. Since the seminal work by Kitaev [10],

one-dimensional spinless superconductors are predicted

to host elusive Majorana fermions [11] on their edges,

which are essentially the real and imaginary parts of

a complex Dirac fermion. Majoranas are as elusive as

they are sought for. Whilst their potential applications

to realizing low-error quantum computation [9, 12] have

stimulated vigorous research, the scientiﬁc community

has yet to reach a consensus on whether or not Majorana

edge modes have been detected experimentally. Yet,

there is recent hope that their discovery is on the horizon

[13], paving the way for further interesting applications,

for example with Majorana wire heterostructures. As

demonstrated in [9], networks of spinless p-wave super-

conducting nano-wires oﬀer a promising platform to

realise anyonic exchange statistics. Due to the proximity

of the wires in these heterostructures, hopping and

super-conductive pairing terms between diﬀerent wires

will naturally occur in realistic setups. Already two

coupled wires, i.e. ladders, are found to have a rich phase

diagram without interactions [14, 15]. Also in the quasi

two-dimensional limit, additional hopping and pairing

terms can have a substantial impact on the macroscopic

properties, as was demonstrated for example in [16]

for a quasi-two-dimensional grid of Majorana wires. It

was shown that in the presence of cross-wire couplings,

it is possible to design (p+ip) superconductivity by

threading appropriate ﬂuxes through each unit cell.

The theory for proximity-induced topological super-

conductivity (SC) focuses strongly on non-interacting

electron models [9]. However, realistic materials will

necessarily be exposed to both internal and external

interactions, which may give rise to previously unknown

transitions [17, 18]. For example, it was found in [19]

that due to the interplay between interactions and

inter-wire hopping amplitude a transition to a super-

conducting state can occur, for two chains of spinless

fermions without SC-pairing. Another fascinating eﬀect

often found in interacting systems is the fractionalization

of the underlying degrees of freedom, for example, the

fractionalization of charge in the FQHE [7] or also

in the case of quantum wires [20]. Another instance

of fractionalization was discovered in the case of two

interacting Kitaev wires [1], wherein the gapless double

critical Ising phase was found to host free Majoranas in

the bulk. Against the backdrop of topological materials

in modern technology, developing a deeper under-

standing of the DCI phase and its Majorana physics is

arXiv:2210.05024v3 [cond-mat.str-el] 16 Mar 2023

therefore relevant. Recent advances in unraveling the

phase diagram of two interacting Bloch spheres [21, 22]

revealed also the emergence of a fractional topological

phase at large interactions. In fact, for the single Kitaev

wire, there exists a well-established map onto the Bloch

sphere through the Bogoliubov-De Gennes representa-

tion [22, 23]. Thus these recent ﬁndings stimulate us to

investigate the DCI phase further, and in particular its

link to the fractional Bloch sphere phase in [21].

The extensions beyond non-interacting chains are

plentiful and there is a lot of ground to cover. In

this paper we focus in particular on deepening our

understanding of the DCI phase, its link to fractional

topology as well as the eﬀects of an inter-wire hopping

amplitude t⊥on the phase diagram found in [1]. We

ﬁrst propose in Sec. II topological invariants for the

one-dimensional superconducting Kitaev chains, which

by mapping onto the Bloch sphere are found to be

direct analogs of the TKNN invariant [3]. We show that

these quantities, deﬁned through two-point correlation

functions, can be extended to two or more wires and

that they remain powerful tools and topological markers

to study the phases of coupled wires also in the presence

of interactions. We evaluate these correlation functions

using Density Matrix Renormalization Group (DMRG)

calculations [24–26]. In Sec. III, we elaborate on

the C= 1/2 topological invariant(s) associated with

the critical theory of the Majorana fermions at the

topological quantum phase transition for one Kitaev

wire, and we establish a relation with the Bloch sphere.

In Sec. IV, we ﬁrst review brieﬂy the regions in the

phase diagram of two interacting superconducting wires

[1]. We present an approach to understanding the

underlying quantum ﬁeld theory (QFT) of both the

single and double critical Ising (DCI) phases [1] in terms

of chiral bulk modes, intimately related to the critical

theory of a single Kitaev wire. Then, in Sec. IV B,

we show with DMRG calculations that our topological

markers take on fractional values of C= 1/2 in the

DCI phase, which establishes a clear link between the

topological phases of two interacting Bloch spheres [21].

Furthermore, we argue that this also strengthens the

notion that the DCI phase is described by two c= 1/2

critical models per wire where now crefers to the central

charge [27–30] of the model.

Finally in Sec. V we also investigate both numerically

and analytically the phase diagram in [1] in the pres-

ence of an inter-wire hopping amplitude t⊥, similar to

the ladders studied in [14], to determine the stability of

the coupled-wire phases towards more experimentally re-

alistic setups. It is perhaps relevant to mention here that

the t⊥term does not have a simple correspondence on

the Bloch spheres’ model since mapping fermions onto

spins-1

2through the Jordan-Wigner transformation will

result in “strings” accompanying this operator in the

spin language. Therefore, it is worthwhile studying the

eﬀect of such a perturbation in the two-wires’ model.

The strength of the t⊥term compared to the Coulomb

interaction may be adjusted by ﬁxing the distance be-

tween the two wires (a hopping term is supposed to de-

cay exponentially with distance from a Wentzel-Kramers-

Brillouin picture). We ﬁnd that for perturbative values

of t⊥the DCI phase and phase diagram remain robust.

We study the phase diagram for larger values of the inter-

wires hopping term, as well as interactions. We also ad-

dress the case for prominent interactions between both

wires, which gives rise to Mott physics [1] in the |g|→∞

limit. Our results are ﬁnally summarized in VI. In Ap-

pendices, we present additional information on DMRG

and quantum ﬁeld theory.

II. TOPOLOGICAL MARKERS FROM

CORRELATION FUNCTIONS

The aim of this section is to deﬁne topological invari-

ants for Kitaev p-wave spin-polarized superconducting

wires [10], and express these quantities in terms of real-

space correlation functions. First, we remind the reader

of the Kitaev wire [10], and associated Bogoliubov-De

Gennes formalism, which due to the particle-hole sym-

metry of the Kitaev wire allows a direct mapping onto a

single Bloch sphere [22, 23, 31]. From this, a deﬁnition

of a Chern number `a la [21] leads directly to the desired

real-space correlation functions. Then, by similar consid-

erations, we argue that in the case of two weakly inter-

acting wires in the sense of [1], a mapping on two-coupled

Bloch spheres remains sensible [22]. Whilst the quanti-

zation of the invariants is no longer guaranteed, we show

numerically using DMRG that these Chern numbers re-

main sensible markers to characterize and investigate the

topological phases in the presence of interactions.

A. Review of the Kitaev wire and its topological

phase diagram

In this article we investigate the topological phases

of (interacting and coupled) Kitaev wires of spinless

fermions. The Kitaev wire is deﬁned by the following

Hamiltonian [8, 10]

HK=−µX

nj−tX

c†

jcj+1 + h.c.

+4eiϕ X

c†

jc†

j+1 + h.c..

(1)

Here µis the chemical potential acting globally on both

wires, tis the hopping amplitude, ∆eiϕ the supercon-

ducting pairing-strength with phase ϕand nc,j =c†

jcj.

The single wire has two topologically distinct extended

phases. A topological phase transition occurs at the criti-

cal chemical potentials µ=±2t. In his seminal work [10],

Kitaev developed an intuitive picture of these phases, by

decomposing the original c-fermions into their Majorana

constituents [23]

cj=1

2(γA,j +iγB,j ).(2)

The Hamiltonian (1), for ϕ= 0, takes the following form

when expressed in terms of the Majorana fermions

HK=X

it−∆

2γB,j γA,j+1 +it+ ∆

2γB,j+1γA,j

−iµ

γA,j γB,j .

(3)

The two topologically distinct phases can then be under-

stood in the two patterns of Majoranas which emerge,

depending on the chemical potential. For a pictorial rep-

resentation of the two distinct patterns, see ﬁgure 1. In

the trivial phases with |µ|>2t, the on-site pairing of

Majoranas is favoured, thus in the t= ∆ = 0 limit we

ﬁnd the upper (trivial) pattern in (2). In the topological

phase |µ|<2t, the nearest-neighbour pairing will domi-

nate, such that for t= ∆ and µ= 0 the lower (topolog-

ical) pattern in (2) emerges. This can be understood in

terms of the following non-local fermions

dR,j =1

2(γA,j+1 +iγB,j ).(4)

For open boundary conditions (OBCs) this results in

zero-energy “dangling edge modes” which make the

ground state doubly degenerate.

FIG. 1. The two distinct patterns of Majorana fermions in

the trivial (upper) and topological (lower) phases of a Kitaev

wire. We refer to Ref. [8] for a more in-depth discussion and

review of the topological phases of the Kitaev wire.

Whilst the emergence of edge modes and ground state

degeneracy are both hallmarks of a topological transi-

tion, it is the deﬁnition of global and robust invariant

which makes the direct link to topology in the mathe-

matical sense. This link is perhaps best understood when

considering the seminal work in [3], where the Zvalued

TKNN invariant (or First Chern number) is deﬁned from

the Berry connection [32] an

j(k) = ihn, k|∂

∂kj|n, kiof the

eigenstates of each band [23, 33]

c1=X

n’th ﬁlled band ZT2

dk2∂an

∂kx−∂an

∂ky.(5)

The Chern number is calculated on the ground state

by summing over the nlowest ﬁlled bands |n, ki. It

is quantized and can only take integer values, and is

a direct analogue of the Gauss-Bonnet theorem for

surfaces: The expression in brackets deﬁnes a curvature

2-form or tensor Ωkx,kyon the Brillouin zone which is a

2-torus T2due to periodic boundary conditions (PBCs)

in both directions. The TKNN-invariant deﬁned above

is not suitable for distinguishing all classes of topological

systems. For example, the Chern number (5) is not

invariant under time-reversal-symmetry (TRS), and

hence vanishes in such systems. Instead, for example in

the two-dimensional quantum spin Hall phase, Kane and

Mele discovered [34] that the appropriate topological

number is a Z2invariant. This has been generalized also

to other systems [35], and is now often referred to as the

Fu-Kane-Mele invariant. For non-interacting systems

the diﬀerent possible topological phases have been

classiﬁed by dimension and symmetries [36] in various

“periodic tables”, cf. Altand-Zirnbauer classiﬁcation

[37] or Kitaev [38]. The Kitaev wire described by (1) is

characterized by a Z2invariant.

In momentum space and the Bogoliubov-De Gennes

Hamiltonian representation (i.e. spinless Nambu ba-

sis) deﬁned through ψ†

k=c†

k, c−k, the single-particle

Hamiltonian in (1) can be written as the 2 ×2 Matrix

HK=X

ψ†

kk∆k

∆∗

k−kψk=X

ψ†

kHkψk,(6)

where we deﬁne k=−µ

2+tcos (ka)and SC pairing

∆k=i4eiϕ sin (ka). We choose the Fourier transform

on the N-site chain with x=ja with the convention

cj=1

√NX

k∈BZ

eikxck.(7)

The two distinct topological phases can then be de-

termined from the signs of the kinetic term kat the

high-symmetry points k= 0 and k=π/a [23], la-

beled by δak=0,π respectively. The trivial phases result

in δ0=δπ=±1. The topological ones instead have

δ0=−δπ=±1, depending on the signs of tand µ. This

deﬁnes a Z2-invariant ν, which in this case is introduced

in the well-known way [10, 23]

ν= (−1)δ0δπ.(8)

Long-range hopping and pairing terms may result in

higher topological invariants [39], which are then the

winding numbers m. However, as we consider only

nearest-neighbor hopping and pairing terms, cf. eq. (1),

the winding numbers are restricted to 0, ±1 (in agree-

ment with the structure of Majorana fermions at the

edges). The Z2number in (8) is thus enough to clas-

sify the system.

B. Chern number from real-space correlation

functions

We show in the following how to deﬁne a (particle)

Chern number C= 0,1, which is directly measurable

from real-space correlation functions of the wire.

Central to our argument is the duality between the mo-

mentum space Hamiltonian in (6) and a Bloch sphere in-

teracting with an external ﬁeld [22, 31], cf. the discussion

in appendix A. From the momentum space Hamiltonian

(6), the single-particle (Bogoliubov-De Gennes) Hamil-

tonian Hkis a complex 2 ×2 matrix. By introducing

(pseudo-)spin ~

Skand ~

dkvector

S=





c†

kc†

−k+c−kck

−ic†

kc†

−k−c−kck

c†

kck−c−kc†

−k





,~

dk=

−∆k+∆∗

∆k−∆∗

−k



,(9)

we may write the single-particle Hamiltonian reminiscent

of a spin in a magnetic ﬁeld [23]

Hk=−~

dk·~

Sk.(10)

Here ~

dkacts like a “magnetic ﬁeld” on the pseudo-

spin ~

Sk,cf. discussion in A. Together with the super-

conducting phase ϕ, which remains a free parameter, we

map the vector ~

dkonto the two-sphere, interpreting the

momentum label ka and super-conducting phase ϕas the

angular coordinates ϑand ˜ϕin (A1). Through the deﬁni-

tions in Appendix A on the sphere this results in [22, 31]

the identiﬁcations

cos (ϑk) = 2tcos(ka) + µ

E(ka),(11)

sin (ϑk)e−i˜ϕ=−i∆eiϕ2 sin(ka)

E(ka)

where E(ka) = q(µ+ 2tcos (ka))2+ 442sin (ka)2.

To be more precise, we introduce the ~

d-vector as

d=|~

d|(cos ϑsin ˜ϕ, sin ϑsin ˜ϕ, cos ϑ) with the energetic

correspondence E(ka)=2|~

d|. There is a correspondence

between the two eigenstates of the spin-1

2particle and

the deﬁnitions of the quasiparticles in the wire model.

Here, ˜ϕrepresents the azimuthal angle on the sphere.

Fixing µ= 0 and t= ∆ we observe that ϑk=ka (and

e−i˜ϕ=−ieiϕ). On the sphere, ϑk∈[0; π] such that

this will eﬀectively correspond to a half Brillouin zone

on the lattice due to the particle-hole symmetry (PHS)

of the Kitaev model. Conversely if µt= ∆, then

ϑk= const..

The 2 ×2 Hamiltonian is diagonalized by the Bogoli-

ubov quasi-particles η†

k=u∗

kc†

k+v∗

kc−ksimilarly as in the

Bardeen Cooper and Schrieﬀer model [40]. For a general

phase ϕ, we can e.g. deﬁne the quasiparticle operator

η†associated to an occupied quasiparticle state and to

the lowest energy eigenstate of the spin-1

2in (A2). The

Bogoliubov de Gennes (BdG) transformation then diag-

onalizes the Hamiltonian, yielding two quasi-particles η±

corresponding to the upper and lower band. As particle-

hole symmetry relates both, the label ±can be dropped

and the BdG transformation deﬁned through the lowest

energy eigenvector of (6) with Hamiltonian −E(ka)η†

kηk.

This results in

η†

k= cos (ϑk/2) c†

k+ie−iϕ sin (ϑk/2) c−k.(12)

The |BCSiground state can be deﬁned for the ﬁlled

energy states as ηkη†

k|BCSi= 0. It has the following

explicit expression [1]

|BCSi=δµ<−2t+ (1 −δµ<−2t)c†

0×

k< π

k>0sin (ϑk/2) −ie+iϕ cos (ϑk/2) c†

kc†

−k

×δµ<2t+ (1 −δµ<2t)c†

π|0i.

(13)

The points k= 0 and ka =∓πrequire some care where

the pairing function goes to zero. We have adjusted

the δfunctions to correspond to ﬁlled or empty states

according to the value of the chemical potential in

agreement with the matrix (6). From the Bogoliubov-De

Gennes quasi-particle basis, the link to the Bloch sphere

eigenvector |GSiin Appendix A for each klabel is made

by taking c†

k−→ | ↑i and c−k−→ | ↓i [22, 31].

As was ﬁrst shown in [21] for (coupled-) Bloch spheres,

the quantized Chern number in (5) can be written from

the polarizations of the spin at the two poles of the Bloch

sphere:

C=1

2(hSz(ϑ= 0)i−hSz(ϑ=π)i).(14)

For two spheres the well-deﬁnedness of partial Chern

numbers [41] was demonstrated for a wide range of in-

teractions, and resulted in the discovery of a fractional

geometric phase at comparably large interactions [21].

By mapping (k, ϕ) onto the Bloch sphere (two-sphere),

we can deﬁne a Chern number C`a la [21] by simply

evaluating the z−Spin operator at the “poles” k= 0 and

ka =π

C=1

2(hSz

ka=0i−hSz

ka=πi).(15)

Acting with Szon the BCS ground state we explicitly

ﬁnd hSz

ki= cos (ϑk), which is the Ehrenfest theorem for

a spin half particle in a radial magnetic ﬁeld [22, 31].

For µtand thus ϑk=const., we immediately ﬁnd

C= 0. In the µ= 0 we have ϑk=ka, and hence

C=1

2(cos (0) −cos (π)) = 1. This splitting results

in two bands, the (quasi-)particles and holes, such

that Ctot =Cp+Ch. From the PHS it also follows

that Cp=−Ch, which can be readily seen by the

formula Ch=1

2(hSz

ka=πi−hSz

ka=2πi) and the fact that

ka = 2πis identiﬁed with k= 0. Thus the total Chern

number Ctot = 0, in agreement with the literature.

We also recover the same Z2invariant as deﬁned in

equation (8) [22], by taking the product hSz

ka=0i·hSz

ka=πi.

We now show that the particle Chern number Cde-

ﬁned in (15) can recast as a physically and experimen-

tally sensible quantity, as it can be expressed in terms of

real-space correlation functions of the wire.

C. Measuring topology from correlation functions

The ﬁrst step is to represent the z-spin operators

ka=0,π in terms of the real-space (spinless-) fermionic

operators cjand c†

j. For this, we perform a Fourier trans-

form of the pseudo-spin Sz

kyielding

i≡1

√NX

eikxiSz

√N3X

k∈BZ;j,r

eik(xi+xj−xr)c†

jcr

−1

√N3X

k∈BZ;j,r

e−ik(xr−xj−xi)cjc†

(16)

Performing the sum over the momentum label k∈BZ

reduces the above expression to

i=1

√NX

jc†

jcj+i−cjc†

j+i.(17)

As can be seen, for i > 0 these operators are intrinsi-

cally related to the amplitudes of i’th neighbour hopping,

whilst for i= 0 it is found to be simply 1

NPj(2nj−1).

Since the momenta k= 0 and k=π/a are special in

the sense that eikx = 1 and eikx = (−1)xrespectively,

the “backwards” Fourier transform is especially simple

to perform, and results in

ka=0 =1

√NX

i, Sz

ka=π=1

√NX

(−1)iSz

i.(18)

From these two (independent) equations we deﬁne the

Chern number C, as well as its “dual” Cas

2DSz

k=0 −Sz

k=π

aE≡C=1

√N

N/2

i=0hSz

2i+1i

2DSz

k=0 +Sz

k=π

aE≡C=1

√N

N/2

i=0 hSz

2ii.

(19)

Here the Chern number Cis the relative polarization,

and Cthe absolute polarization,dual to Cby measuring

the degree of alignment of the spins. Equation (19)

deﬁnes topological markers in real space. This sets them

apart from other, momentum space topological numbers,

making them directly accessible numerically for example

with DMRG. Whilst other topological markers have

previously been deﬁned over non-local correlation func-

tions in real space [42, 43], the quantities deﬁned in (19)

repackage that information into a physically intuitive,

and conceptual observable. As an example, we now show

how to obtain the correct pattern of Majoranas shown

in ﬁgure 1 from Cand C.

For the single Kitaev wire, the ground state is sim-

ply given by the BCS wave function, from which it is

straightforward to calculate the real-space correlation

functions. In terms of the ϑkangles, the expectation

values of hc†

r+jcj+ h.c.ican be calculated directly and

one ﬁnds

hSz

j>0i=1

k∈BZ

cos (k·j) (1 −cos (ϑk))

hSz

j=0i=1

j∈BZ

(1 −cos (ϑk)) −1.

(20)

For the j= 0 case one needs to account for the anti-

commutation relations in (17). When the spectrum has

a gap, the correlations decay exponentially with a corre-

lation length ξ, and thus we only need of order ξterms

to obtain a robust topological marker. The following two

examples correspond to the extreme limit where ξis min-

imal: The two patterns presented in ﬁgure 1 above are

exact in the limits of t= ∆ = 0, and µ= 0 respectively

[10], for which ϑksimpliﬁes considerably. In the trivial

(µt) case ϑk=π, such that the only non-zero expec-

tation value in (20) is hSz

j=0i= 1. This is equivalent to

a density of fermions nc= 1 and the wire is ﬁlled with

on-site bound Majoranas for each lattice site, cf. ﬁgure 1.

For the topological case at µ= 0 we found that ϑk=ka,

thus only hSz

j=1i 6= 0. This implies on the one hand that

2nc−1 = 0, i.e. half-ﬁlling. In addition to the non-

local fermion (4), we introduce the left-binding fermions

dL,j =1

2(γA,j −iγB,j+1). Together with dR,j one ﬁnds

j=1 =1

jd†

R,j dR,j −d†

L,j dL,j =nR−nL(21)

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FractionalTopologyinInteracting1DSuperconductorsFrederickdelPozo,LocHerviouy,andKarynLeHurCentredePhysiqueTheorique,EcolePolytechnique(IPParis),91128Palaiseau,FranceandySBIPHYS,EPFL,RtedelaSorgeCH-1015Lausanne(Dated:March17,2023)Weinvestigatethetopologicalphasesoftwoone-dimensional(1D)interac...

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Fractional Topology in Interacting 1D Superconductors

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