Stability Stability of of

2D Topological Insulators2D Topological Insulators

● Chiral and non-chiral Topological States of Matter

● Stability of edge states of 2D Topological Insulators: anomaly

● Partition function: electromagnetic and gravitational (discrete) responses

● Stability of TI with interacting & non-Abelian edges

Andrea Cappelli(INFN and Physics Dept., Florence)

with E. Randellini (Florence)

OutlineOutline

Z2

Topological States of MatterTopological States of Matter

● System with bulk gap but non-trivial at energies below the gap

● global effects and global degrees of freedom (edge states, g.s. degeneracy)

● described by topological field theory: Chern-Simons theory etc.

● quantum Hall effect is chiral (B field, chiral edge states)

● quantum spin Hall effect is non-chiral (edge states of both chiralities)

● other systems: Chern Insulators, Topological Insulators, Topological Superconductors, etc.

● Topological Band Insulators (free fermions) have been observed in 2 & 3 D

<< fever >>

● Q: systems of interacting fermions? (e.g. fractional insulators)

● A: use quantum Hall modelling and CFTs

● Q: but non-chiral edge states are stable?

● A: generically NO

● Q: stability with Time Reversal symmetry?

● A: YES/NO; there is a symmetry; if this is anomalous, they are stable

Questions/Answers

Chiral Topological StatesChiral Topological States

Quantum Hall effect● chiral edge states

● no Time-Reversal symmetry (TR)

● Laughlin's argument.

● spectral flow between charge sectors

● edge chiral anomaly = response of topological bulk to e.m. background

● chiral edge states cannot be gapped topological phase is stable

● anomalous response extended to other systems and anomalies in any D=1,2,3,......

chiral anomaly

f0g !©13

ª!©23

ª! f0g

¢Q =

Zdtdx @tJ

0R = º

ZF = º n

(S. Ryu, J. Moore, A. Ludwig '10)

©0

R

L¢Q

QR ! QR +¢Q = º;

©! ©+ ©0; H [© + ©0] = H [©]

©! ©+ n©0

Non-chiral topological statesNon-chiral topological states

Quantum Spin Hall Effect● take two Hall states of spins

● system is Time Reversal invariant:

● non-chiral CFT with symmetry

● adding flux pumps spin anomaly

● in Top. Insulators is explicity broken by Spin-Orbit Coupling etc.

● no currents

● but TR symmetry keeps symmetry Kramers theorem

º = 1

©0

¢S

U(1)Q £ U(1)S

U(1)S

©0¢Q = ¢Q" +¢Q# = 1¡ 1 = 0

(X-L Qi, S-C Zhang '08)

¢S = 12 ¡

¡¡ 12

¢= 1

¾H = ¾sH =·H = 0

U(1)S

T : Ãk" ! ák# ; Ãk# ! ¡Ã¡k"

Z2 (¡1)2S

Symmetry Protected Topological PhasesSymmetry Protected Topological Phases

● QSHE edge theory is used to describe Topological Insulator with Time-Reversal symmetry of

Main issue: stability of TI stability of non-chiral edge states

● e.g. TR symmetry forbids mass term in CFT with odd number of free fermions

classification (free fermions)

Trivial Insulator

QSHE Topological Insulator

no TRU(1)S

T : Hint: = m

ZÃy"Ã# + h:c: ! ¡Hint:

TR : Z2

(¡1)2SU(1)S ! Z2

Flux insertion argumentFlux insertion argument

● TR symmetry: &

● TR invariant points:

● Kane et al. define a TR-invariant polarization (bulk quantity) that:

- is topological and conserved by TR invariance

- is equal to parity of edge spin

- if there exits a pair of edge states degenerate by Kramers theorem

(Fu, Kane, Mele '05-06;Levin, Stern '10-13)

©02

¡©02

deg. with

H [© + ©0] = H [©]T H [©]T ¡1 = H [¡©]

(¡1)2S = ¡1

deg. with

©0

Kramers degeneracy

gapless edge statejexi

jexi

● topological phase is protected by TR symmetry if edge Kramers pair ( odd)

● spin parity is anomalous, discrete remnant of spin anomaly

©0

Conclusions

Kramers degeneracy

gapless edge statejexi

jexi

Nf9U(1)S ! Z2

Z2Fu-Kane argument is Laughlin's argument for anomaly:

● topological phase is protected by TR symmetry if edge Kramers pair ( odd)

● spin parity is anomalous, discrete remnant of spin anomaly

● Can we extend this to Topological Insulators with interacting fermions?

● Can we use modular non-invariance, i.e. discrete gravitational anomaly of the partition function as another probe?

©0

Conclusions

Kramers degeneracy

gapless edge state

Questions

(S. Ryu, S.-C. Zhang '12)

jexi

jexi

Nf9U(1)S ! Z2

Z2Fu-Kane argument is Laughlin's argument for anomaly:

Answers in this talkAnswers in this talk

● partition functions of edge states for QHE, QSHE and TI are completely understood

● we can study flux insertions and discuss stability

● modular transformations: e.m. & grav. responses

TI: classification extends to interacting & non-Abelian edges

(AC, Zemba '97; AC, Georgiev, Todorov '01; AC, Viola '11)

unstable, modular invariantstable, not modular invariant

spin-Hall conduct. = chiral Hall conduct. minimal fractional charge

¡1(¡1)2¢S = +1

(Levin, Stern)

2¢S =¾sHe¤

=º"

e¤

QHE partition functionQHE partition function

¯

R

Consider states of outer edge for described by CFT c = 1

K¸(¿; ³; p) = TrH(¸) [exp (i2¼¿L0 + i2¼³Q)]

edge energy & momentum

¿ = vi¯

2¼R+ t;

³ =¯

2¼(iVo + ¹) electric & chemical pot.

modular parameter

E » P » vL0R

sum of characters for representations of current algebra[U(1)

K¸+p = K¸

Partition function for one charge sector

theta function=1

´(¿)

X

n

exp

µi2¼

µ¿(np+ ¸)2

2p+ ³

np+ ¸

p

¶¶

Dedekind function

Q =¸

p+ n; n 2 Z

Modular trasformationsModular trasformations

unitary S matrix, completeness

odd-integer electron statistics

integer electron charge

flux insertion:

discrete coordinate changes respecting double periodicity

e.m. background changes: adds the weight adds a flux quantum

S : ¿ ! ¡1=¿T : ¿ ! ¿ + 1

V : ³ ! ³ + ¿

U : ³ ! ³ + 1 ei2¼Q

K¸(¿; ³; p) =1

´(¿)

X

n

exp

µi2¼

µ¿(np+ ¸)2

2p+ ³

np+ ¸

p

¶¶; Q =

¸

p+ Z

©! ©+ ©0;

V : K¸(¿; ³ + ¿) » K¸+1(¿; z) Q ! Q+ º©0

Partition Function of Topological InsulatorsPartition Function of Topological InsulatorsPartition function for a single edge:

- combine two chiralities

In fermionic systems there are always four sectors of the spectrum:

Ramond sector describes half-flux insertions:

Each sector has vacua for the would-be fractional charges

K"¸ K#¹

invariantZNS (¿; ³) =

pX

¸=1

K"¸ K#¡¸; S; T 2; U; V

¸ = 1; : : : ; p

add half flux

E.m. & gravitational responsesE.m. & gravitational responses

ZNS

ZgNS

ZR ZeRZ

gNSe.m.

grav.S

T

Modular invariant partition function of a single TI edge can be found

But: is consistent with TR symmetry?

invariantZIsing = ZNS + ZgNS + ZR + Z

eR; S; T; U; V12

V12

ZRZNS ZeR

ZIsing

Stability and modular (non)invarianceStability and modular (non)invariance

● fluxes are needed to create one electron in the same charge sector

● Flux argument: add fluxes and check if i.e. Kramers pair

● Spin parities of Neveu-Schwarz and Ramond ground states are different

spin-parity anomaly recovered

not consistent with TR symmetry

p2

ZIsing = ZNS + ZgNS + ZR + Z

eR

Z2

TR symmetry + modular invariance no anomaly trivial insulator

V p : K"¸ ! K"¸+p = K"¸; ¢Q" =p

p= 1

p

stable TI

TR symmetry + anomaly no modular invariance stable insulator

º =1

p

General stability analysisGeneral stability analysis

● Edge theory involves neutral excitations (possibly non-Abelian)

● electron is Abelian: “simple current” modular invariant

● fractional charge sectors parametrized by two integers

● minimal charge:

● Hall current:

● construct TI partition function as before

● Stability: add fluxes to create an excitation in the same charge sector

(A.C., Viola '11)

= {g.s.} + {1 el.} + {2 el.} +......

charged neutral

(k; p)

Kramers pairs if odd stable TIk

£®¸(¿; ³)

£®¸(¿; ³) =kX

a=1

K¸+ap (¿; k³; kp) ®¸+ap(¿; 0)

Analysis of modular invariance vs. stability can be extended to any edge CFT

even = unstable, TR-inv. modular invariant

odd = stable, modular vector Z =³ZNS; Z

gNS; ZR; ZeR´

ZIsing = ZNS+ ZgNS+ ZR+ Z

eR

Levin-Stern index fully general2¢S =º"

e¤; (¡1)2¢S = (¡1)k

k

k

Analysis of modular invariance vs. stability can be extended to any edge CFT

even = unstable, TR-inv. modular invariant

odd = stable, modular vector

Examples

● Jain-like TI

● (331) & Pfaffian TI

● Abelian TI

● Read-Rezayi TI

Z =³ZNS; Z

gNS; ZR; ZeR´

ZIsing = ZNS+ ZgNS+ ZR+ Z

eR

unstablestable

unstable

stable

stableunstable

º" =k

2nk + 1; e¤ =

1

2nk + 1; 2¢S =

º"

e¤= k

º" =1

2; e¤ =

1

4; 2¢S = 2

º" =3

7; e¤ =

1

7; 2¢S = 3K =

µ3 11 5

¶

º" =k

kM + 2; e¤ =

1

kM + 2; 2¢S = k

Levin-Stern index fully general2¢S =º"

e¤; (¡1)2¢S = (¡1)k

k

k

RemarksRemarks● general expression of partition function allows to extend Levin-Stern stability

criterium to any TI with interacting fermions

● neutral states are left invariant by flux insertions

● unprotected edge states do become fully gapped?

- Abelian states: yes, by careful analysis of possible TR-invariant interactions

- non-Abelian states: yes, use projection from “parent” Abelian states

e.g. (331) -> Pfaffian because [projection, TR-symm.]=0

(Levin, Stern; Neupert et al.; Y-M Lu, Vishwanath)

(A.C., to appear)

Z2 classification of TI protected by TR invariance

ConclusionsConclusions● spin parity anomaly characterizes Topological Insulators protected by

Time Reversal symmetry

● anomaly signalled by index

● Pfaffian TI is unstable

● To do:

- explicit form of interactions gapping the edge of non-Abelian states (done)

- stability of Topological Superconductors Ryu-Zhang stability criterium

- stability 3D systems and 2D-3D systems (cf. arxiv: 1306.3238, 3250, 3286)

Z2(cf. Ringel, Stern; Koch-Janusz, Ringel)

Gapping interactions for Pfaffian TIGapping interactions for Pfaffian TI

● Gapping interactions for Abelian states defined by matrix

● For (331) state, they can be written in terms of Weyl fermions fields

● Projection (331) Pfaffian, i.e. to identical layers→

neutral

charged field

● couples to fermion field, to charges modes, giving both mass

● Analysis extends to Read-Rezayi states and Ardonne-Schoutens NASS states

Topological SuperconductorsTopological Superconductors● stability of TS gapless non-chiral Majorana edge states

● free Majorana, they are neutral no flux insertion argument

● chiral spin parity no spin flip

classification (free fermions)

● unstable by non-trivial quartic interaction:

proposal: study partition function and gravitational response

● Standard invariant for any

Is it consistent with parity?

● Yes, for : mod. trasf. creates in R sector OK

● Ryu-Zhang: test modular invariance of subsector GSO projection

● general analysis of modular invariance + discrete symm. not understood yet

● many modular invariants are possible without charge matching

(Ryu, S-C-Zhang)

Z2 £ Z2Z

Nf = 8 Z! Z8

Nf

Z2 £ Z2Nf = 8 ST » V

12

(¡1)NR = (¡1)NL = 1

Nf = 8

ZIsing = ZNS+ZgNS+ZR+Z

eR

¢S" = ¢S# = 1

Z =X

N¸¹K"¸K

#¹

Nf

(¡1)2S"(¡1)2S#

= (¡1)N"(¡1)N#

(Kitaev, many people)