Symmetry-protected criticalphases and global anomaly
Croucher Advanced Institute“Topology in Condensed Matter and High-Energy Physics”
January 3-5, 2018 @ Chinese University of Hong Kong

Masaki Oshikawa(ISSP, UTokyo)

1

Lecture 1:
Anomaly and Condensed Matter Physics
Lecture II:
Symmetry-Protected Critical Phases
and Global Anomaly

2Dirac Fermion

massless(m=0)

3“Dirac cone”

Axial Symmetry and Current
Massless Dirac fermion Lagrangian densityµ¯
L = i @µ“vector” U(1) symmetryi✓V!e¯ ! e i✓V ¯
{

µ

,

5

}=0

5 2

charge current conservationµ@µ j = 0µµ¯j =in even space-time dimensions=1“axial” U(1) symmetry if m=0! ei✓A¯ ! ei✓A ¯axial current conservationµ 5@ jµ = 0
5
5
¯
j = µ
µ

4

U(1) Chiral Anomaly
Noether’s theorem (“classical”):
Massless Dirac fermion two conserved currents
However, one of these conservation laws is inevitably broken
in quantum theory through “regularization” of UV divergence
L = ¯i µ (@µAdler/Bell-Jackiw (1969)

iAµ )Anomalous non-conservationof axial current!
1
µ 5
µ⌫ ⇢
@ jµ =
✏
F
F
µ⌫⇢
16⇡ 2

µ 5

(in 3+1 dimensions)0⇡!
Decay of neutral pion5

Regularization/Renormalization“renormalization theory issimply a way to sweep the
difficulties of the divergencesof electrodynamics
under the rug.”
Richard Feynman, in
Nobel Lecture (1965)6

Modern Understanding of Renormalization
Field theory =universal long-distance behaviorof lattice model /condensed matter systems
Exact symmetry in thelattice model remains
exact in the long-distance limit→ no anomaly?
How can we understand anomaly
in this context?
7

Kenneth G. Wilson(1936-2013)

Chiral Anomaly in 1+1 Dim.
R,

L

Right mover, Left moverU(1) x U(1) symmetry
conserved individually?
However, one of the conservation law is brokenµ 5@ jµ =

1
µ⌫
✏µ⌫ F
2⇡

@
(nR
@t
8

nL ) / E

Chiral Anomaly

Chiral Symmetry on Lattice
If we can realize Dirac fermion on lattice withexact chiral symmetry, the chiral symmetry should persist
contradiction with chiral anomaly
In particular, if we can realize the right-moving and left-moving“Weyl fermion” individually on the lattice, the exact chiralsymmetry would follow
This suggests that, we cannot realize chiral symmetry
exactly in a lattice model,
and that we cannot realize an individual right-moving/left-moving“Weyl fermion” in a lattice

Nielsen-Ninomiya theorem
9Nielsen-Ninomiya in 1+1Dω

alwaysappear in pair!

Periodicity of the
momentum space（Brillouin zone）

10

Chiral Anomaly in 1+1D
ω

@
(nR
@t

nL ) / E

Acceleration of electrons by electric field!
universal in low-energy limit (topological quantization)
11

Same expectation (absence of single Weyl fermion on lattice)for 3+1D based on chiral anomaly
Proof is a bit more complicated12Chirality in 3+1 D
Momentum

Momentum

Spin

Spin

Right-handed

Left-handed（Helicity = Chirality if massless）
13

Weyl Fermion in 3+1D & Lattice
H(~p)|

p)i = ✏↵ (~p)|
↵ (~

p)i
↵ (~

band structure

Touching of two bands= Weyl point⇤

H(~p) ⇠ ✏(~
p )+

X

Vµ⌫

⌫

(p

µ,⌫

chirality = sgn det V

14

µ

⇤µ

p )

Vortex Linesha|p)i = 0↵ (~2 conditions (real part = imaginary part = 0)
3 parameters→ solution consists of curves in the momentum space“vortex line”
The direction of the vortex line is defined by“vorticity” i.e.
the winding of the complex phasearound the vortex line

15

Weyl (band-touching) Points
Two bands degenerate at the Weyl points
there is always a solution for ha| (~p)i = 0by considering a linear superposition of two states
Weyl point:source/sink of “vortex line”
in 3D momentum space
Each vortex line should havean “origin” and “endpoint”
Weyl points always appear
in pair of opposite chiralities
16

Dirac Fermions in 2+1D

Generic “band-touching” situation in CM
m<0

m=0

17

m>0

Parity anomaly in 2+1D
Field-theory calculation gives

Dirac mass

Pauli-Villars masse2
(sgnm + sgnM )xy =2h“parity anomaly”
Massless Dirac fermion somehow has non-zero Hall conductivity(breaking the time-reversal symmetry “spontaneously”)
This implies that one cannot realize a single massless Diracfermion in time-reversal invariant 2+1 dimensional lattice
model(Dirac fermions always appear in pairs)— distinct from, but similar to Nielsen-Ninomiya theorem
18
in even space-time dimensions

“Dimensional Reduction”
m<0

m=0

m>0

19

2-dimensional momentum space+ 1 external parameter
2+1 D massless Dirac fermion
at the critical point

3-dimensional
momentum space
3+1D Weyl fermion

2e= “Chern Number”
h20“total vorticity”

Dimensional Reduction & Chern Number

21

Chern number (2+1D)= total vorticity appearing
in a 2D cross section of
3D momentum spacee2(sgnm + sgnM )xy =2hFigure: Nielsen-Ninomiya (1981)
1X
Chern Number =
sgnmj
2 j

e2
“Chern Number”
xy =hM.O. 199422

“Corollary”
If a single massless Dirac fermion is realized ina time-reversal invariant lattice model in 2+1D,
xy = 0by symmetry
Now, with a perturbation which opens a mass gap,
2

1e
sgnm
xy =
2 h
which would contradict the TKNN quantization
By contradiction, a single massless Dirac fermion in 2+1D
CANNOT be realized in a TR-invariant lattice model in 2+1D
23

Graphenetime-reversal
invariant,
two Dirac points
at K and K’24

25

Anomaly & “No-Go” Theorems
Anomaly in quantum field theory implies“no-go” theorem for lattice model
Chiral anomaly → Absence of chiral Dirac fermion on lattice(Nielsen-Ninomiya theorem foreven space-time dimensions)
Parity anomaly → Absence of single massless Dirac fermion on
time-reversal invariant lattice model in 2+1D
Any “loophole” to realize them on lattice?

26

Chiral Fermion in Condensed Matter

Chiral (Weyl) fermion
in 1+1D as edge state27

How did we avoid N-N theorem?
Nielsen-Ninomiya theorem applies to 1+1D system:
it does not apply in the limit of “infinite width”

width

For a finite width strip, Nielsen-Ninomiya theorem still applies,
and there indeed is a pair of left/right-moving Weyl fermion
which are spatially separated at the opposite edges
28

What about anomaly?

current flow through bulk

width

Anomalous field theory may be realized asan edge/surface state of higher-dimensional lattice modelthe “bulk” provides sink of anomalous current
29

(Princeton University Group)30

Anomaly & “No-Go” Theorems
Anomaly in quantum field theory implies“no-go” theorem for lattice modelin the SAME DIMENSION
Chiral anomaly → Absence of chiral Dirac fermion on lattice
but 1+1D may be realized asa chiral edge state of QHE in 2+1D
Parity anomaly → Absence of single massless Dirac fermion on
time-reversal invariant lattice model in 2+1D
but may be realized as a surface state ofa TR-invariant topological insulator in 3+1D
anomalous field theory may be realized at the edge/surface!
31

Anomaly in Interacting Systems
So far, I have discussed only non-interacting fermionsHowever, anomalies are believed to persist even in the presenceof interactions“Anomalous field theory may be only realized
at the edge of a topological phase”*
Conversely, “anomalous field theory realized at the edge
implies a topological phase in the (higher dimensional) bulk”*
should be still valid in interacting systems!
*: many caveats!
32

Chiral Superfluid
Cooper pair withdefinite angular momentum lz=νpairing amplitude
Superconducting phaseof Sr2RuO4?

A-phase of superfluid 3He

33

Chiral Majorana Edge State
Chiral p+ip superconductor in 2+1 D
has edge state which is chiral Majorana fermion in 1+1D
Chiral Majorana fermion is anomalous→ stable against perturbations(No backscattering = “ingappable”)
Stability of the edge state implies the topological nature of
the chiral p+ip superconductor in 2+1D
“topological superconductor”

34

Non-chiral edge state?
Nf copies of right-moving AND left-moving chiral Majorana fermionsequivalent to N = Nf/2 right/left-moving complex fermions
Can this be gapped by an edge perturbation?
Right-movers: from “↑ spin” Left-movers: from “↓ spin”
Symmetries: n↑ & n↓ separately conserved modulo 2
mass term

a
L

b
R

is forbidden by this symmetry

non-interacting system is stable for any Nf“Z classification”
35

Z2 × Z 2

Effect of Interactions?
Let us see if the edge theory is anomalous or not(anomaly should give a criterion applicable to interacting systems)[Ryu-Zhang, 2012]
Impose the Z2 × Z2 symmetry by “gauging”
or equivalently “orbifolding” (more on this later….)
1 + ( 1)n" 1 + ( 1)n#PGSO =
2
2
Zorb ⇠ TrA PGSO e

HA+ TrP PGSO e

Z++ + Z+ + Z + + Z
⇠
2
36

HP

2

(roughly)Modular Invariance
1+τ

τ

0

1

Partition function of a consistent CFT must be
invariant under modular transformations
generated by2

T : ⌧ ! ⌧ + 2 for fermions37

Single Complex Fermion
Z µ (⌧ ) = e

2⇡i µ

q

1/24

q

2

/2

1
Y

(1 + wq n 1/2 )(1 + w 1 q n 1/2 )

n=1

q=e

2⇡i⌧

w=e

, µ = 0, 1/2

0Z00 (⌧ + 1) = e i⇡/12 Z1/2
(⌧ )

2⇡iµ

q

Orbifold partition functionof the single complex fermion(Nf =2, N=1)
cannot be modular invariant0i⇡/12 0Z1/2 (⌧ + 1) = e
Z0 (⌧ )
1/2
i⇡/6 1/2
Z0 (⌧ + 1) = e
Z0 (⌧ )

1/2
i⇡/6 1/2
Z1/2 (⌧ + 1) = e
Z1/2 (⌧ )38

N=4 Complex Fermions
40i⇡/3Z0 (⌧ + 1) = e

⇣

⇣

⇣

0
Z1/2 (⌧ + 1)

1/2
Z0 (⌧ + 1)

1/2
Z1/2 (⌧ + 1)

⌘4

⌘4

⌘4

=e
=e

=e

⇣

0
Z1/2 (⌧ )40
Z0 (⌧ )

i⇡/3

2i⇡/3

2i⇡/3

⇣

⇣

⌘4

1/2
Z0 (⌧ )

1/2
Z1/2 (⌧ )⌘4⌘4

Chiral orbifold partition function is modular covariant if
N is an integral multiple of 4 (Nf is an integral multiple of 8)
Total (non-chiral) orbifold partition function
Zorb (⌧, ⌧¯) = Zorb (⌧ )Zorb (¯⌧ ) is then modular invariant
39Non-chiral edge state?
Nf copies of right-moving AND left-moving chiral Majorana fermions

Can this be gapped by an edge perturbation?
Right-movers: from “↑ spin” Left-movers: from “↓ spin”
Symmetries: n↑ & n↓ separately conserved modulo 2
non-interacting system is stable for any Nf

Z2 × Z 2“Z classification”

interacting system can be gapped only if Nf is an integral multiple of 8
“Z8 classification”cf.) Fidkowski-Kitaev 201040

Anomaly in Interacting Systems
So far, I have discussed only non-interacting fermionsHowever, anomalies are believed to persist even in the presenceof interactions“Anomalous field theory may be only realized
at the edge of a topological phase”*
Conversely, “anomalous field theory realized at the edge
implies a topological phase in the (higher dimensional) bulk”*
should be still valid in interacting systems!
*: many caveats!
41SU(2) WZW theories
Lorentz-invariant critical point:
expect chiral SU(2) x SU(2) symmetry
Natural action with the SU(2) x SU(2) symmetry

g: SU(2) matrix-valued field
However, RG implies that this theory isalways massive (gapped) “asymptotic freedom”

42Wess-Zumino term

original space-time:surface of the sphereuniqueness of kΓWZ(modulo 2π)
k: integer
B: (inside) sphere

RG has a nontrivial fixed point
if k≠0 → gapless critical phase43

Kac-Moody algebraThis “includes” Virasoro algebra (conformal invariance)
and is very powerful — determines scaling dimensions(critical exponents) etc.

central charge

scaling dimension of spin-j field
44

Discrete Symmetry
WZW action is also invariant underg!

gDiscrete Z2 symmetry
Let us also consider gauging this Z2 symmetry byconsidering the Z2 orbifold….

45

Projection vs. Path Integral
periodic(imaginary)
time

antiperiodic

space

46

Orbifold Construction
The “projected” partition function Z+proj is notmodular invariant by itself — must be supplemented
by twisted sectors

The resulting partition function represents
the “Z2 orbifold” of the original SU(2)k WZW theory

47

Global Anomaly
The Z2 orbifold should be modular invariant
by construction — but this is NOT always the case!
The Z2 orbifold is modular invariant if k is even,but it is modular NON-invariant if k is odd
Gepner-Witten 1986

48

What does it mean?
SU(2) WZW with an odd level k has the global Z2 anomaly
it can be only realized at the edge of a 2+1D topological phase?
However, it is known thatexactly solvable SU(2)-invariant spin chain with S=2k
(Takhtajan-Babujan model)
realizes the SU(2)k WZW , with the Z2 symmetry,in the low-energy limit!
Why the “anomalous” field theory can be realized in thelattice model in the same dimension?
What is the implication of the anomaly in this case?
49

Types of Anomaly
ABJ anomaly- t’Hooft anomaly-

obstruction of gauging a global symmetry
orbifold construction = gauging the Z2 symmetry
the WZW theory is consistent for any integer k
if it is not gauged
SU(2)k with odd k has a t’Hooft anomaly concerning
the Z2 symmetry50

Anomaly in Interacting Systems
So far, I have discussed only non-interacting fermionsHowever, anomalies are believed to persist even in the presenceof interactions“Anomalous field theory may be only realized
at the edge of a topological phase”*
Conversely, “anomalous field theory realized at the edge
implies a topological phase in the (higher dimensional) bulk”*
should be still valid in interacting systems!
*: many caveats!
51

Realization of anomalous theory
If a global symmetry of the field theory is realized
as a “on-site” symmetry of a lattice model,
it can be gauged exactly
a field theory with ’t Hooft anomaly cannot be realized
on a lattice
However, the global symmetry of the field theory
may be realized in other manners…

52

Spin chain and WZW

Lattice translation symmetry
discrete Z2 symmetry
Translation symmetry of the lattice may not be gauged
Field theory with a ’t Hooft anomaly may havea lattice realization in the same dimension, if
the anomalous global symmetry corresponds tothe translation symmetry[Cho-Hsieh-Ryu 2017]53

What does this mean?
If the orbifold is modular invariant, we can consider
projection onto the symmetric sector, and open a gap
within that sector to obtain the unique ground state
However, if it is modular non-invariant (ie. k: odd),
we cannot open the gap to obtain a unique ground state
within the symmetric sector;
ground states in the symmetric/antisymmetric sectors
must be degenerate!
“Lieb-Schultz-Mattis (LSM) theorem” from field theory

54

Lieb-Schultz-Mattis theorem
For translation & SU(2) invariant spin chainsif S is integer: no constraint
if S is half-odd-integer:
the system must be gapless,OR the ground state is at least doubly degenerate
Lieb-Schultz-Mattis 1961 (S=1/2 chain at zero magnetization)
Affleck-Lieb 1986 (arbitrary S chain at zero magnetization)
MO-Yamanaka-Affleck 1997, MO 2000, Hastings 2004, etc etc.
more generally, “filling-enforced constraints”
55

Proof by large gauge invariance
Tx |

|

0i

0i = e

iPx0

|

0i

adiabatic flux insertion
momentum unchanged

|

0
0i

Ux |

Φ

Tx (Ux |LSM, Affleck-Lieb, M.O……

02⇡i⌫
U x Tx |0 i) = e=emomentum shift by

0i(Px +2⇡⌫)2⇡⌫ = 2⇡(S
56

0
0i

(Ux |
m)

Large gaugetransformation

0
0i

0
0 i)

Affleck-Lieb 1986
S: half-odd-integer→ gapless or
2-fold g.s. degeneracy

was a generalization of“Lieb-Schultz-Mattis Theorem”

57

Main Result of “LSM” paper:
S=1/2 XY chain is solvableby mapping to fermions
What about the LSM theorem?

Appendix….
58?!
Perhaps refers to this paper

But no mention is actually made on
the generalization of LSM theorem?!
59

Maybe….
LSM tried to generalize their theorem to general S,
but “failed” to prove it for integer S
So they scrapped the generalization and never published(until Affleck-Lieb paper 25 years ago)…. maybe missing the evidence of the “Haldane gap”??

60

Selection Rule
Perturb SU(2)k WZW with SU(2) and Z2-symmetricrelevant operators; suppose the RG flow reaches
SU(2)k’ WZW
if k is even, we should be able to consider
the projection onto Z2 symmetric sector;
the RG flow can be understood in terms of
the Z2 orbifold → k’ is also even
if k is odd, the IR fixed point should also have theglobal anomaly (otherwise contradicts with “LSM”)→ k’ is also odd“anomaly matching”
61

In terms of RG…
SU(2)k WZW← c- theorem← present work
SU(2)k’ WZW
SU(2)0 WZW is identified withgapped phase with a unique ground state62

“Symmetry Protected” gapless phases
SU(2) + Lorentz + lattice translation symmetries

SU(2)k WZWSU(2)k WZW

k: evenk: odd63

Spin Chains and WZWThere is a special integrable (Bethe-ansatz solvable)
spin chain model for any S,
Takhtajan-Babujian (TB) modele.g. for S=1:
Spin-S TB model is described by SU(2)2S WZW(k=2S even if S is integer, k odd if S is half-odd integer)
Other models can be regarded as
TB model + perturbations, so
k: even if S is integer, k:odd if S is half-odd integer
if the one-site translation symmetry
is
kept
64

Our ClaimIn the presence of the SU(2) andlattice translation (by one site) symmetries,- The system is gapped with a SSB ofthe translation symmetry (doubly degenerate GS)
OR - The system is gapless, described by
SU(2)k WZW with an odd k- The system is gapped (can be without SSB)
OR - The system is gapless, described by
SU(2)k WZW with an even k
65

Anomaly and LSM[Cho-Hsieh-Ryu 2017]

Nf Dirac fermions in 1+1D

U(1)×ZN symmetry’t Hooft anomaly classified by

Spinc cobordism group
✏N =
66

(

1
2

(N : odd)(N : even)Anomaly and LSM[Cho-Hsieh-Ryu 2017]

⌫a =

sR,a

sL,a
N

Chiral Anomaly, LSM
67

Summary
Anomaly:symmetry of Lagrangian violated in quantization
emergent symmetry in condensed matter/lattice model
Exact requirement of “anomalous” symmetry often leads to“no-go theorem” on lattice realization (Nielsen-Ninomiya etc.)
However, an anomalous field theory can generally be realized asedge/boundary of a higher-dimensional condensed matter/latticemodel
A field theory with ’t Hooft anomaly may be realized in the same
space-time dimensions, if the symmetry is not “on-site”
e.g. translation symmetry corresponds to ’t Hooft anomaly→ field-theory version of “Lieb-Schultz-Mattis theorem”
symmetry protection of gapless, critical phases
68