Anomaly andSymmetry-Protected
Critical Phases
Masaki Oshikawa
ISSP, University of Tokyo

7-9 November 2018
Theoretical Physics Symposium 2018
Wolfgang-Pauli Centre, Hamburg, Germany
1This talk is based on discussion with many people,
in particular collaborations withFrank Pollmann (now at TU München)
Erez Berg (now at Weizmann Institute)
Ari M. Turner (now at Technion)
Shunsuke Furuya (now at RIKEN)
Yuan Yao (ISSP)
Chang-Tse Hsieh (Kavli IPMU & ISSP)

2

Hirosi and Me3

Komaba Campus,
University of Tokyo1987

4

Hirosi: obtained Master’s degree from Kyoto Univ.
and immediately appointed to “Joshu”
position at UTokyo in 1986, without Ph. D.
(very rare occurrence in high-energy theory)“Joshu” = Research Associate / Assistant Prof. /Wissenschaftlicher Assistent
I: entered UTokyo in April 1986, proceeded toDepartment of Physics in Summer 1987
Fall Semester 1987:
I was in Hirosi’s class for “Seminar”(problem-solving session) in Komaba Campus!
5Me (1990)
Hirosi (1987?)
6

@Aspen Centerfor Physics,
Summer 2005

MeAkiraFurusaki

7Hirosi

arXiv:1808.10466

8

ISSP Public Lectures
20149Largest turnout 〜400 people
in the history of ISSP Public Lectures10

Classiﬁcation of states of matter= distinction of diﬀerent phases
Phase diagram of a ferromagnet
ordered phaseTc

disordered phase

Critical Point(Curie Temp.)

simple model: (classical) Ising model

11

TAre liquid and gas diﬀerent?
Phase transition
can be “avoided”
by going beyond
the critical point
Liquid/gas are“essentiallyindistinguishable”
Figure fromSonntag R E, Borgnakke C, Van Wylen G J, “Fundamentals of Thermodynamics”
12What about solid?
Can we avoid the phase transition betweensolid/liquid at, e.g. higher pressures?
NO!
in solid, translation symmetry is
spontaneously broken,
while it is not in liquid/gas
SSB clearly distinguish different phases,
implying existence of phase transitions!13

Quantum Phase Transitions
Quantum fluctuations can drive the system at T=0
into different quantum phases, and cause quantum phasetransitions between quantum phases
Similarity (and in fact mathematical mapping inmany cases) to classical phase transition driven by thermal
fluctuations
gapped (off-critical)
E

continuum of
excited states
gap

ground state

gapless (critical)

What distinguishes different phases?
Different orders (or their absence) characterize eachphase
Ferromagnet: magnetic orderSuperfluid (3D): off-diagonal long-range order(order of U(1) phase of wavefunctions)etc.
“order”

Spontaneous Symmetry Breaking??
!15

However…………
Recently, it has been recognized that
there are many quantum phases that arebeyond understanding in terms of
conventional orders/spontaneous symmetry breaking“topological phases”
how to deﬁne them?
how to distinguish diﬀerent phases?

16

“Operational” definition of phases
A family of Hamiltonians H(g) parametrized by gSingularity in the ground state of H(g),as a function of g quantum phase transition
If the two gapped ground states are connected adiabatically,i.e. if there exists a path H(g) the two Hamiltonians without aquantum phase transition they belong to the same phase
Otherwise (if there is no adiabatic path connecting
the two) they belong to different phases, even if
there is no distinction in terms of SSB
H(1)
H(0)

!17

Topological Order in 1D
Any gapped ground state of a local 1D Hamiltonian isconnected to a trivial state adiabatically
Chen-Gu-Wen (2011)

Absence of (genuine)“topologically ordered phase” in 1D!
i.e. there is only one, trivial phase in 1D(in the absence of symmetries)
However, there can be more variety of phases
if some symmetries are imposed!18

Imposing Symmetries
For a gapped Hamiltonian with a symmetry1) the ground state is in a trivial phase2) the symmetry is spontaneously broken in
the ground state (SSB phase)3) the ground state cannot be adiabatically connected to atrivial (product) state, even if we break the symmetry(topological order, absent in 1D)4) the symmetry is unbroken, but the ground state cannot
be adiabatically connected to a trivial (product) state
as long as the symmetry is kept
!19

“SPT phase”
the symmetry is unbroken, but the gapped ground state can
NOT be adiabatically connected
to a trivial state,
if and only if the Hamiltonian respects the symmetry

then the ground state belongs to a
Symmetry-Protected Topological Phase(Generalization of “topological insulators”
of free fermions to interacting systems)!20

Haldane gap
Heisenberg antiferromagnetic chain

S=1/2, 3/2, 5/2........
“massless” = gapless, power-law decay of spincorrelations
S=1, 2, 3, .....
“massive” = non-zero gap, exponential decay of spincorrelations
Haldane conjecture (1981)!21

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”
22

“Proof” by large gauge invariance
Tx ||0i

0i = e

iPx0|

LSM, Affleck-Lieb, M.O……

0i|

adiabatic flux insertion
momentum unchanged

00i

Ux |

Φ

Tx (Ux |Large gaugetransformation

00i

02⇡i⌫
U x Tx |0 i) = e=e
momentum shift by 2⇡⌫ = 2⇡(Si(Px0 +2⇡⌫)m)

00i(Ux |00 i)

the “new” ground state has extra momentum π for half-odd-int S
23

Affleck-Lieb 1986
S: half-odd-integer→ gapless or2-fold g.s. degeneracy

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

24

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

Appendix….
25

?!
Perhaps refers to this paper

But no mention is actually made on
the generalization of LSM theorem?!
26

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”??

27

Haldane Phase and QPTgap“Haldanephase”

trivial phase(“large-D phase”)

quantum phase transition
No local order parameter, but is a distinct“Symmetry-Protected Topological Phase”
28

Haldane Phase as a SPT
In the presence of any one of the following symmetries, theHaldane phase is separated from a trivial (product) state by aquantum phase transition:i) time reversal symmetryii) dihedral (Z2×Z2) symmetry (π-rotation about x, y, and z axes)iii) lattice inversion symmetry about a bond center

Gu-Wen (2009)
Pollmann-Turner-Berg-MO (2010)!29

AKLT model/statee.g. S=1
Exact groundstate: (Affleck-Kennedy-Lieb-Tasaki 1987)

Singlet pair of two S=1/2’s -“valence bonds”

Symmetrization of two S=1/2’s

S=1✓non-zero gap, exponential decay of correlations(supporting the Haldane
30 conjecture)Why SPT?

Pollmann-Turner-Berg-MO (2009)

Open boundary condition : “edge state” of S=1/2
The ground state is doubly degenerate becauseof the edge spin (4-fold considering both ends)
This degeneracy is exact under time reversal(Kramers degeneracy):time reversal must be broken, or there must bea quantum phase transition to remove the degeneracy!
31

Quantum Phase Transitions
Quantum fluctuations can drive the system at T=0
into different quantum phases, and cause quantum phasetransitions between quantum phases
Similarity (and in fact mathematical mapping inmany cases) to classical phase transition driven by thermal
fluctuations
gapped (off-critical)
E

continuum ofexcited states
gap

ground state

gapless (critical)

Gapless Quantum Critical Point
Gapless excitations appear at quantum critical points
e.g. (quantum) transverse Ising modelordered phase

disordered phaseΓ
Quantum Critical Point

critical point = RG fixed point
relevant perturbation → gap
33

Gapless Critical Phases
However, quantum critical phases often appear in
cond-mat physics without any apparent fine-tuning- metallic systems- Dirac/Weyl semimetals- β-YbAlB4[Nakatsuji Group, ISSP]
34

Gapless Critical Phases- Kagome spin liquid (S=1/2 antiferromagnet):
Dirac spin liquid?

35

Gapless Critical Phases- Why are they stable?
- Classification/characterization of these phases
We do have some understanding based on CFT etc.
but we need more!
We can gain some insights from the recent developmentsin the classification of gapped topological phases…

36

Generalization of SPT phases?
I will attempt to extend the notion of“Symmetry-Protected (Topological) Phases”
to gapless phases
I will discuss an example in 1+1 dimensions(spin chains, effective CFT)
although the concept can be hopefully generalized
to higher dimensions
S. C. Furuya & M. O. Phys. Rev. Lett. 118, 021601 (2017)
Y. Yao, C.-T. Hsieh, & M. O. arXiv:1805.06885
37

Our Model
Spin-S antiferromagnetic chain with the global SU(2) andlattice translation symmetries

Lorentz invariance is expected;
when gapless, low-energy physics should be describedby a SU(2) symmetric CFT
SU(2)k Wess-Zumino-Witten theory
characterized by “level” k = 1, 2, 3, …
38

✓ ◆3/(k+2)
1r~0 · S~r i / ( 1)hSr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
39

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

SU(2)k WZWSU(2)k WZW

k: evenk: odd40

SU(2) WZW Theory
g: SU(2) matrix-valued ﬁeld

uniqueness of kΓWZ(modulo 2π)
k: integer

original space-time:surface of the sphere

B: (inside)sphere

RG has a nontrivial fixed point
if k≠0 → gapless critical phase41

Spin chain and WZW

Lattice translation symmetry
discrete Z2 symmetry
If there is the Z2 symmetry,
we should be able to consider a projection
to Z2-symmetric subspace?

42

Projection vs. Path Integral
periodic(imaginary)
time

antiperiodic

space

43

Modular Invariance
1+τ

τ01

Partition function of a consistent CFT must be
invariant under modular transformations
generated by44

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

45

Global Anomaly(’t Hooft 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

46

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) constraint” in CFT!
Global anomaly =“ingappability” in the presence of the symmetry(S. Ryu et al. on edge theory)47

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”48

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 state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
iskept50

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 k51

Summary & Outlook“Global Z2 anomaly” (’t Hooft anomaly) discovered by
Gepner and Witten in 1986 can be interpreted, in thecondensed matter / lattice context, an inheritance of
Lieb-Schultz-Mattis constraint in the microscopic modelto CFT as low-energy effective field theories“Symmetry-Protected Critical Phases”

Lieb-Schultz-Mattis type constraints in higher dimensions
and for various symmetries: corresponding anomalies?

!52

Kavli IPMU, Kashiwa, Japan(Institute for the Physics and Mathematics of the Universe)

Director: Hirosi Ooguri(October 2018〜)

ISSP (Institute for Solid State Physics)
High-energy physics and mathematics are closer tocondensed matter physics, than you might think…

We will see more
unexpected53 and fruitful encounters!