Non-perturbative constraints from symmetry and chirality on Majorana zero modes and defect quantum numbers in 21D Naren Manjunath1Vladimir Calvera2and Maissam Barkeshli1

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Non-perturbative constraints from symmetry and chirality on Majorana

zero modes and defect quantum numbers in (2+1)D

Naren Manjunath,1Vladimir Calvera,2and Maissam Barkeshli1

1Department of Physics, Condensed Matter Theory Center, and Joint Quantum Institute,

University of Maryland, College Park, Maryland 20742, USA

2Department of Physics, Stanford University, Stanford, California 94305, USA

In (1+1)D topological phases, unpaired Majorana zero modes (MZMs) can arise only if the in-

ternal symmetry group Gfof the ground state splits as Gf=Gb×Zf

2, where Zf

2is generated by

fermion parity, (−1)F. In contrast, (2+1)D topological superconductors (TSC) can host unpaired

MZMs at defects even when Gfis not of the form Gb×Zf

2. In this paper we study how Gfto-

gether with the chiral central charge c−strongly constrain the existence of unpaired MZMs and the

quantum numbers of symmetry defects. Our results utilize a recent algebraic characterization of

(2+1)D invertible fermionic topological states, which provides a non-perturbative approach based

on topological quantum ﬁeld theory, beyond free fermions. We study physically relevant groups

such as U(1)foH, SU(2)f×H, U(2)foH, generic Abelian groups, as well as more general compact

Lie groups, antiunitary symmetries and crystalline symmetries. We present an algebraic formula

for the fermionic crystalline equivalence principle, which gives an equivalence between states with

crystalline and internal symmetries. In light of our theory, we discuss several previously proposed

realizations of unpaired MZMs in TSC materials such as Sr2RuO4, transition metal dichalcogenides

and iron superconductors, in which crystalline symmetries are often important; in some cases we

present additional predictions for the properties of these models.

CONTENTS

I. Introduction 2

A. Summary of main results 3

1. Internal unitary symmetries 3

2. Internal antiunitary symmetries 6

3. Crystalline symmetries 6

4. Intrinsically interacting fermionic

phases 7

5. (1+1)D invertible fermionic phases 7

6. Organization of paper 7

II. Review of classiﬁcation of invertible fermionic

phases 8

A. Overview 8

B. Deﬁnition of (c−, n1, n2, ν3)8

1. c−8

2. n1and MZMs 9

3. ω2and fractional quantum numbers of ψ9

4. n2and fractional quantum numbers of

fermion parity ﬂuxes 9

5. O3obstruction to deﬁning n210

6. O4obstruction and ν310

C. Equations for c−=1

2mod 1 11

D. Equations for unitary symmetry with integer

c−11

E. Equations for antiunitary symmetries 12

1. Interpretation of n1=s112

2. A suﬃcient condition for a solution:

ω2=s1∪u113

III. Extending the constraints to crystalline

symmetries 13

A. Fermionic crystalline equivalence principle 13

B. Meaning of n1for crystalline symmetries 14

C. Remark on terminology 14

IV. Examples of nontrivial constraints 14

A. Constraints for (1+1)D invertible phases 14

B. Gf= U(1)f×Hand Gf= U(1)foH15

C. Gf= SU(2)f×H16

D. Constraints when Gfis abelian 16

E. Orthogonal groups Gf= O(n)f17

1. Gf= O(2)f18

2. Gf= O(4)f18

3. Gf= O(2n+ 1)f19

4. Gf= O(4n+ 2)f19

5. Gf= O(4n)f19

F. Unitary groups 19

1. Gf= SU(2n)f19

2. Gf= U(n)fand U(n)fo Z220

G. Symplectic groups Gf= Sp(n)f20

H. Constraints when Gbis a direct product 20

I. Gf=ZTf

4o Z221

J. Additional constraints from O4obstruction 21

V. Applications 21

A. Unpaired MZMs at half-quantum vortices in

a spinful p+ip SC 22

B. Unpaired MZMs at lattice dislocations 22

1. Layered p+ip SCs 22

2. MZMs at dislocations in a model for

Sr2RuO423

C. Unpaired MZMs at lattice disclinations and

corners 23

1. Inversion symmetry and monolayer

WTe223

2. Example with Gb=C4×ZT

224

D. MZMs in reﬂection symmetric systems 24

1. TSCs with mirror symmetry 24

arXiv:2210.02452v2 [cond-mat.str-el] 14 Mar 2023

2. MZMs in a mirror symmetric model for

Sr2RuO425

3. p+id superconductor 25

E. Magnetic translation symmetry 26

F. Unpaired MZMs in iron superconductors 27

VI. Discussion 27

VII. Acknowledgements 28

A. Results from group cohomology 29

1. Cup product deﬁnitions used in the main

text 29

2. Slant products 29

3. Pontryagin square 30

4. Spectral sequences 30

5. Bockstein homomorphisms 31

6. Comment about continuous groups 32

B. Algebraic formula for the fCEP 32

1. Notation 32

2. Checks 33

a. w233

b. w2

133

c. s1w133

3. Heuristic argument 34

C. Calculations when Gfis abelian 34

D. Calculations for charge conservation symmetry 35

E. O4obstruction calculations 35

1. Comments about the O4obstruction 35

2. Gf= Spin(N)f36

3. Proof that there is a solution with c−= 4

for suitable choices of Gb37

a. Intrincically interacting example with

c−= 4 37

4. Obstruction calculation for

Gf= SU(2)f×H38

5. Orthogonal groups Gf= O(2n)f,SO(2n)f38

a. Group cohomology of

PSO(4n+ 2),PO(4n+ 2) 39

b. Group cohomology of PSO(4n),PO(4n)40

c. Constraints for Gf= O(8n)f42

6. Group cohomology of SU(n)/Z243

7. Symplectic groups Gf= Sp(n)f45

8. Gf=Df

8◦Qf

8, Gb=Z4

246

9. Calculations for Section IV H 46

10. Calculations for Gf=Zn×ZT

2×Zf

247

11. Calculations for Gf=ZT

2n×Zf

247

F. Wreath products 48

G. Cohomology of B2A∼

=K(A, 2) 48

1. Justiﬁcation 48

References 49

I. INTRODUCTION

The possibility of topological superconductivity in

electronic systems has generated intense interest in con-

densed matter physics [1–9]. One major reason for this

interest is the possibility of realizing unpaired localized

Majorana zero modes (MZMs) and their potential ap-

plications for topological quantum computation.1In a

mean-ﬁeld treatment, in which phase ﬂuctuations are ig-

nored, topological superconductors (TSCs) can be mod-

eled in terms of gapped many-body states of fermions.

As such, they can be analyzed using theoretical tech-

niques that have been developed to characterize and clas-

sify gapped topological states of matter in general, e.g.

[10–22].

Most topological superconductors of interest, such as

the spinless p-wave superconductor in (1+1)D or the

p+ip superconductor in (2+1)D, are often modeled

in terms of a free fermion Bogoliubov-de-Gennes (BdG)

Hamiltonian. In a many-body context, these are exam-

ples of invertible fermionic topological states of matter

[17,20,22]. Invertible topological states are deﬁned by

the property that they do not host deconﬁned anyon ex-

citations and have a unique ground state on any spatial

manifold with a ﬁxed set of boundary conditions.2

In some cases topological superconductors host un-

paired localized MZMs at their 0-dimensional defects.

These defects may include boundaries of a (1+1)D sys-

tem, fermion parity vortices of a (2+1)D system, or

various other kinds of symmetry defects such as half-

quantum vortices, lattice disclinations and dislocations,

or corners of a system [23–36]. An important theoretical

question is to understand the fundamental constraints

on realizing unpaired localized MZMs. For example,

it is known that superconductivity is a crucial require-

ment, which translates into the statement that systems

with U(1) charge conservation symmetry preserved in the

ground state are forbidden from hosting unpaired MZMs.

However the constraints on unpaired MZMs are in gen-

eral signiﬁcantly stronger than simply requiring super-

conductivity.

Let Gfbe the internal (equivalently, on-site) fermionic

symmetry group of the many-body ground state; here

fermionic refers to the fact that fermion parity, (−1)F,

is included as a symmetry in Gf.Gb=Gf/Zf

2is then

the symmetry group that acts on bosonic operators and

2is the group generated by fermion parity. In (1+1)D,

unpaired MZMs can only exist if Gfsplits as a direct

product: Gf=Zf

2×Gb. This is a signiﬁcantly stronger

constraint than simply the requirement of superconduc-

1Here ‘unpaired’ refers to having an odd number of MZMs, which

necessarily leads to topological degeneracies.

2The term invertible refers to the fact that the ground state |Ψi

possesses an “inverse” state |Ψ−1i, such that the |Ψi⊗|Ψ−1i

can be adiabatically connected to a trivial product state.

tivity in the ground state. The above constraint, for ex-

ample, immediately rules out spin singlet superconduc-

tors, which have SU(2)finternal symmetry.

As we discuss later, one can interpret the case where

Gfis not of the form Zf

2×Gbas the case where the

fermion carries fractional quantum numbers under Gb.

Therefore the requirement that Gf=Zf

2×Gbamounts

to the requirement that unpaired MZMs can only exist in

(1+1)D if the fermion does not carry fractional quantum

numbers under Gb.

Remarkably, in (2+1)D the constraints are quite dif-

ferent and far richer. In particular, it is possible that un-

paired MZMs can exist in systems where Gfis not of the

form Zf

2×Gb. That is, unpaired MZMs in (2+1)D invert-

ible topological phases are compatible with the fermion

carrying fractional quantum numbers under Gb. (2+1)D

topological states can also exhibit a non-trivial chiral cen-

tral charge c−, corresponding to the possibility of topo-

logically protected chiral edge states. There is a rich set

of constraints, involving c−and Gf, on when (2+1)D

topological superconductors can host unpaired localized

MZMs.

In addition to unpaired localized MZMs, symmetry de-

fects in (2+1)D systems can also carry non-trivial quan-

tum numbers of the Gfsymmetry group. These quan-

tum numbers must also satisfy a rich set of constraints

involving c−,Gf, and whether unpaired MZMs exist.

The main purpose of this paper is to study constraints

from symmetry Gfand chirality c−on unpaired MZMs

and defect quantum numbers in (2+1)D invertible topo-

logical states. Our results utilize a non-perturbative ap-

proach that relies primarily on the properties of symme-

try defects, and does not use explicit Hamiltonians. As

such, this approach applies to generic interacting many-

body systems of fermions, beyond the free fermion limit.

This approach is based on a recent complete characteri-

zation and classiﬁcation of invertible fermionic topologi-

cal phases in (2+1)D using the framework of G-crossed

braided tensor categories and Chern-Simons theory [20]

(see also [22]).

Our results emphasize that the possibility of unpaired

MZMs in a TSC is constrained not just by the symme-

try Gfbut also by the chiral central charge c−of the

system, when Gfis unitary. Such constraints are often

straightforwardly encoded in the properties of symme-

try defects; however, they may not always be apparent

from a Hamiltonian perspective. This approach is there-

fore a useful complementary tool in fully understanding

the physics of TSC systems. Symmetry groups Gfwith

anti-unitary components can often impose constraints on

when fermion parity vortices must carry a Kramers pair

of localized MZMs.

We note that Ref. [37] discusses constraints on (2+1)D

invertible fermionic phases using a method similar to

ours, but from the perspective of ‘enforced symmetry

breaking,’ i.e. that if Gfand c−are speciﬁed, and c−

is ﬁxed, invertible phases can sometimes only be realized

by breaking Gfdown to a subgroup. However, Ref. [37]

does not discuss speciﬁc results on unpaired MZMs in

topological superconductors.

A. Summary of main results

Our main results are summarized in Tables I,II and

III. In Table Iwe consider various physically relevant uni-

tary internal symmetry groups Gf, and list (i) the values

of c−that permit an invertible phase, and (ii) the values

of c−that additionally permit unpaired MZMs at sym-

metry defects. Note that c−can only be an integer or

a half-integer for invertible fermionic phases. In Table

II we present results for antiunitary internal symmetries.

Here c−must be zero. Results for crystalline symmetries

are given in Table III. We brieﬂy summarize these results

below, emphasizing that they hold even in the presence

of strong interactions. Applications to models of TSCs

in the prior literature (mostly involving crystalline sym-

metries) are discussed in Sec. V. Note we always consider

(2+1)D systems, unless otherwise speciﬁed.

Notational remarks: Below we often refer to the

groups O(n)f, SU(n)f, Sp(n)f,Hf, etc. The superscript

refers to the fact that a Z2subgroup of the center of these

groups is identiﬁed with fermion parity, Zf

2. We use the

notation kZfor the subgroup of the rational numbers

whose elements are kn where nis an integer and kis a

rational number, and Z+kfor the set with elements n+k,

where nis an integer. In particular when k=1

2we write

2Zand Z+1

2respectively.3The symbols ZT

2n,ZR

2n,ZRT

denote that the group Z2nis generated by a time-reversal

T, a spatial reﬂection R, or by the combination RT re-

spectively. Similarly, HTmeans that some element in H

is identiﬁed with T(as will be speciﬁed on a case-by-case

basis).

1. Internal unitary symmetries

We show that charge-conserving systems with Gf=

U(1)f×H(where His arbitrary) must have integer

c−, but cannot host unpaired MZMs at symmetry de-

fects (Section IV B). Thus, for example, Chern insulators

cannot host unpaired MZMs at symmetry defects. The

only way for a charge-conserving system to have unpaired

MZMs is if Hhas charge-conjugating elements, so that

the symmetry group instead becomes U(1)foH, where

odenotes the charge conjugation action. Here unpaired

MZMs can exist, but only when c−is odd; moreover, in

this case all defects associated to charge-conjugating ele-

ments must host unpaired MZMs. If c−is even, topologi-

cal insulators with the symmetry U(1)f×Hor U(1)foH

do exist, but do not host unpaired MZMs.

3This is diﬀerent from the symbol Z/2, which is sometimes used

for the group Z2∼

=Z/2Z.

Results for unitary symmetries

GfGbAllowed c−Values of c−supporting

unpaired MZMs

Physical model/other comments

Gb×Zf

2Gbunitary 1

2Z1

2ZUnpaired MZMs allowed at integer c−

only if Gbhas a Z2nor Zfactor

U(1)f×HU(1) ×HZ- Chern insulator

4q×HZ2q×HZ- Charge 4qsuperconductor

4q+2 ×H∼

=Zf

2×Z2q+1 ×

Z2q+1 ×HZ

2ZCharge 4q+ 2 superconductor

SU(2)f×HSO(3) ×H2Z- Spin-singlet superconductor

8nD4n=Z2no Z2Z2Z+ 1

2×AAbelian A1

2Z1

2Zor Z+1

2Zif Ahas a Z2nor Zfactor, and

Z+1

2otherwise

U(1)foH U(1) oHZ2Z+ 1 gsymmetry defect has unpaired MZM

iﬀ gacts as charge conjugation and c−

is odd

Fermion carries fractional

quantum numbers under

both GA

band GB

b×GB

b2Z(odd c−only allowed

if Eq. (56) holds)

Unitary compact Lie groups

O(n)fPO(n)gcd(n,16)

2Zgcd(n, 8)Z+gcd(n,16)

22c−identical layers of a spinless p+ip

SC have Gf= O(2c−)f; half-quantum

vortex in any 2 layers hosts unpaired

MZMs

SU(2n)fSU(2n)f/Zf

22gcd(4, n)Z-

U(n)fU(n)f/Zf

2gcd(n, 8)Z-nidentical layers of Chern insulator

U(n)fo Z2(U(n)f/Zf

2)o Z2gcd(n, 8)Z2Z+ 1 (only for odd n)

Sp(n)fPSp(n) gcd(n, 4)2Z-

TABLE I. Summary of mathematical results for internal unitary symmetry groups Gf.Gb=Gf/Zf

2is the symmetry acting

on bosonic operators and His an arbitrary group. The chiral central charge c−is in general either a half-integer or an integer.

These results can be applied to spatial symmetries, if we modify the deﬁnition of Gfin accordance with the fermionic crystalline

equivalence principle stated in Section III.

A simple example of such a charge conjugating sym-

metry is Gf= O(2)f=U(1)fo Z2, where H=Z2. In a

system of two identical layers of a spinless p+ip supercon-

ductor, Gfis the symmetry which permutes the fermions

in the two layers while keeping the superconducting pair-

ing term invariant. In this case, a defect of the O(2)f

reﬂections can host an unpaired MZM when c−is odd.

Such defects correspond to ‘half-quantum vortices’ in a

spinful p+ip TSC, which has previously been proposed

as a mean-ﬁeld model for the material Sr2RuO4.4

Interestingly, the example with U(1)foHsymmetry

also captures the well-known possibility of creating un-

paired MZMs by inducing a superconducting gap at an

interface between two Chern insulators with equal and

odd Chern numbers. Each endpoint of the supercon-

ducting interface can be viewed as a charge-conjugating

symmetry defect if the system has a particle-hole symme-

try in addition to U(1)f(see Sec. IV B and Fig. 2therein

4The nature of the order parameter in Sr2RuO4is however still

not understood in light of recent experiments [38–40].

for more details). The endpoint is a defect because, upon

encircling it, a particle must cross the interface and hence

transform into a hole.

Next, we consider spin rotation symmetry with Gf=

SU(2)f×H, where His any unitary symmetry (Section

IV C). We ﬁnd that c−must be even, and that unpaired

MZMs cannot exist for any H. This describes the situ-

ation in a spin-singlet superconductor. Thus, unless the

spin rotation symmetry is broken or it interacts nontriv-

ially with elements of H(i.e. not as a direct product),

unpaired MZMs are impossible to realize in (2+1)D spin-

singlet superconductors.

We show that if Gfis abelian, unpaired MZMs can

only exist if Gf=Zf

2×Gbfor some abelian Gb(Sec-

tion IV D). On the other hand, if Gfis abelian but not

isomorphic to Zf

2×Gb(for example Zf

4), then the sys-

tem cannot host unpaired MZMs, and can exist only at

integer values of c−. This extends previous results in

Refs. [18,41]. Two remarkable applications of this result

are to charge-2qsuperconductors and to superconductors

with an M-fold rotational point group symmetry, as we

discuss in subsequent subsections.

Results for antiunitary internal symmetries (c−= 0)

GfGbChoices of n1which imply un-

paired MZMs

Physical model/comments

Zn×ZT

2×Zf

2Zn×ZT

2n1= w1(only if 8 divides n) If n= 4, unpaired MZMs are (O4) obstructed

2n×ZT

2Zn×ZT

A×ZT

2×Zf

2A×ZT

2n1admits a lift to H1(A, Z8). Class BDI + Abelian symmetries (A).

U(1)f×HTU(1) ×HT- Generalization of Class AIII TI

ZTf

4o Z2ZT

2×Z2n1=s1+ w1Fermions with Z2eigenvalues +1,−1 form spinless p+ip

and p−ip SC layers respectively

ZTf

4oHZT

2×H n1=s1+ρ ρ(g)∈Z2is deﬁned by gT¯

g=T1+2ρ(g)

GfGbFermion parity ﬂux carries Ma-

jorana Kramers pair

Physical model/comments

Zn×ZTf

4Zn×ZT

2n1=s1Class DIII TSC

ZTf

4nZT

2nn1=s1(only if nis odd) When nis odd ZTf

4n∼

=ZTf

4×Zn

TABLE II. Constraints when the internal symmetry group Gfcontains antiunitary elements. s1:Gb→Z2speciﬁes which

elements of Gbare antiunitary. n1:Gb→Z2is a homomorphism, further explained in Sec. II B. If g∈Gbis a unitary

operation, n1(g) = 1(0) indicates that a gsymmetry defect will(will not) host unpaired MZMs (upper section of table). When

gis time-reversal, n1(g) = 1 instead implies that a fermion parity ﬂux carries a degenerate Kramers pair of MZMs (lower

section). If Znis unitary and nis even, w1is the nontrivial homomorphism from Zn→Z2. These results can also be applied to

crystalline symmetries using the fCEP; there, the interpretation of n1as indicating unpaired MZMs/Majorana Kramers pairs

may diﬀer.

Results for crystalline symmetries

GfGeﬀ

fAllowed c−c−which allows

unpaired MZMs

Interpretation

C2k×Zf

2Zf

4kZ- No unpaired MZMs

C2k+1 ×Zf

2Z2k+1 ×Zf

2Z- Unpaired MZMs only at fermion parity ﬂuxes (c−∈Z+1

4kZ2k×Zf

2Z1

2ZUnpaired MZMs at disclinations/corners with angle π/k or at

fermion parity ﬂuxes (eg. spinless p+ip SC)

Z2×Zf

2(translation) Z2×Zf

2Z1

2ZUnpaired MZMs at dislocations with Burgers vector along either

ˆxor ˆy

Z2×πZf

2Z2×πZf

2Z Z Gfhas πﬂux per unit cell; unpaired MZMs allowed at disloca-

tions with Burgers vector along ˆxor ˆy

2×Zf

2ZTf

40 0 Unpaired MZMs; Reﬂection axis can carry a Kitaev chain

ZRf

4ZT

2×Zf

20 -

ZRT

2×Zf

2Z2×Zf

2Z1

2ZFor any c−unpaired MZMs can be found at endpoints of the

reﬂection axis if it carries a generator of Class BDI TSCs

ZRTf

4Zf

4Z-

TABLE III. Constraints when the symmetry group Gfis spatial. CMdenotes spatial rotations of order M; we use Cf

2Mwhen a

2πspatial rotation equals (−1)F.ZR

2denotes the order-2 group generated by a unitary reﬂection symmetry while ZRT

2denotes

the order-2 group generated by the anti-unitary reﬂection RT.Geﬀ

fis the eﬀective internal symmetry group, determined

through the fermionic crystalline equivalence principle. n1:Gb→Z2is discussed in Sec. II B.

We obtain several results for compact Lie

groups. In particular, we study the orthogo-

nal families (O(n)f,SO(2n)f), unitary families

(U(n)fo Z2,U(n)f,SU(2n)f), and symplectic fam-

ilies Sp(n)f.

We ﬁnd a rich set of constraints for the family of groups

O(n)f(Section IV E). We show that in any invertible

phase, 2c−must be a multiple of gcd(n, 16). Moreover,

only a certain subset of these c−values is compatible

with unpaired MZMs. For example, when Gf= O(2)f,

invertible phases can exist for any integer c−, but un-

paired MZMs can only exist when c−is odd. When

Gf= O(4)f, invertible phases can exist for any even c−,

but unpaired MZMs can exist only when c−= 2 mod 4.

Note that these results also hold in the free fermion con-

text, where an O(n)fsymmetric phase can be obtained

by stacking nidentical layers of a spinless p+ip SC.

Our theory provides further information constraining a

parameter n2which ﬁxes the quantum numbers carried

by the fermion parity ﬂuxes (see Sec. IV E); this is not

apparent from free fermion constructions.

In deriving the above constraints, we calculated the co-

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Non-perturbativeconstraintsfromsymmetryandchiralityonMajoranazeromodesanddefectquantumnumbersin(2+1)DNarenManjunath,1VladimirCalvera,2andMaissamBarkeshli11DepartmentofPhysics,CondensedMatterTheoryCenter,andJointQuantumInstitute,UniversityofMaryland,CollegePark,Maryland20742,USA2DepartmentofPhysics,S...

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Non-perturbative constraints from symmetry and chirality on Majorana zero modes and defect quantum numbers in 21D Naren Manjunath1Vladimir Calvera2and Maissam Barkeshli1

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