Stability of 2D Topological Insulatorstheory.fi.infn.it/cappelli/talks/scgp.pdf · Stability of 2D...
Transcript of Stability of 2D Topological Insulatorstheory.fi.infn.it/cappelli/talks/scgp.pdf · Stability of 2D...
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)