Symmetry Protected Topological
Criticality
Linhao Li (ISSP @ U.tokyo)
Nov 9th 2022

arXiv:2204.03131
with Yunqin Zheng, Masaki Oshikawa

Gapped Quantum Matter
relatively well-understood

Symmetry
Enriched
Trivially
Topological
GappedPhase

Symmetry
Protected
Topological
Phase (SPT)‘t Hooft anomaly (Hep-th)

Integer QH (Cond-mat)Symmetry
Enriched
Topological
Phase (SET)LandauSymmetry
Breaking
Phase (SSB)

Symmetric TQFT (Hep-th)
Toric code (Cond-mat)Properties of Gapped SPT Phase
Systems belonging to a gapped SPT phase satisfy:• Global symmetry Γ is non-anomalous• Single ground state under periodic boundary condition(PBC).
• Degenerate ground states under open boundary condition(OBC).
• Ground state under Γ twisted boundary condition(TBC) carries Γ charge.

Examples: AKLT chain• Global symmetry is onsite SO(3) symmetry• Ground state degeneracy is 1 or 4 under PBC or OBC

Figure from Wiki

Gapless Quantum Mattermuch less well-understood
Gapless quantum matter typically appear in two situations:
1. Critical point/Critical region.
2. Goldstone boson/fermion.

Quantum Criticality andwe focus on it in this talk

What is the landscape of Quantum criticality?
Not know yet. But there are several examples in the past and we wish to organize them, analogous tothe classification of gapped SPT phase.
LandauTransition
Symmetry Protected Topological Criticality (SPTC)⊕ SPTLandscape of Quantum Criticality - A Dream

Symmetry
Enriched
TrivialTopologicalcriticality
Phase

Symmetry
Protected
Topological
Criticality(SPTC)

Symmetry
Enriched
Topological
Criticality(SETC)

LandauTransition⊕ SPT

Neel-VBS
Transition and etc

LandauTransition

Symmetry
Breakingat Criticality

Properties of SPTC
By minimally generalizing the properties of gapped SPT, we expect
SPTC should satisfy:• Global symmetry Γ is non-anomalous• Single ground state under PBC with a polynomial finite size gap.
• Degenerate ground states under OBC with a exponential decaying finitesize gap.
• Ground state under Γ TBC carries Γ charge.

• G SSB phase, decorate the G-defects
consistently.
Γ is anomaly-free• Proliferate the G-defect to restore theentire Γ symmetry• Fully fluctuate the decorated G-defects
to a gapped SPT phase.

[Chen, Lu, Vishwanath,1303.4301][Wang,Ning,Chen,2104.13233]

• G SSB phase, decorate the G-defectsconsistently.
Γ is anomaly-free• Fine-tune the fluctuation of thedecorated G-defects to criticality.
[Scadi,Parker,Vasseur,1705.01557][Li,Oshikawa,YZ,2204.03131]

[Scadi,Parker,Vasseur,1705.01557][Thorngren,Vishwanath,Verresen,2008.06638][Li,Oshikawa,YZ,2204.03131]

(&ℤ% ×ℤ% Weak SPTC in (1+1)d Spin chain(&ℤ% ×ℤ% Weak SPTC in (1+1)d Spin chain• Hibert space: two types of spin-1/2: 𝜏 and 𝜎• Symmetry operators:11𝑈& = + 𝜏 0 /-2/-.

𝑈( = + 𝜎-0%-2/• ℤ(% SSB Hamiltonian:0: :𝐻 = − ∑1-2/ 𝜏-.
7 + 𝜎- 𝜎-./8

• Two ground states:

Domain Wall and Fluctuation (Pre-decoration)<.= 0∏• Flipping a string of 𝜎 spin, by acting on the ground state with -2< 𝜎- ,creates one domain wall excitations at each end.

• Fluctuating the domain wall achieved by adding to Hamiltonian
1−ℎ ? 𝜎-0-2/

Domain Wall and Fluctuation (Pre-decoration)
00: :𝐻@ = − ∑1-2/(𝜏-.
7 + 𝜎- 𝜎-./ + ℎ𝜎- )8Domain Wall Decoration

If there is a ℤ(% domain wall, we stack a (0+1)d ℤ%& gapped SPT.

Domain Wall Decoration
If there is a ℤ(% domain wall, we stack a (0+1)d ℤ%& gapped SPT.

Between two adjacent and opposite σ spins and τ spin is ↓,we assign the wavefunction an additional − sign.

1𝜋𝑖
𝜋𝑖:::
𝑈CD = + exp(1 − 𝜎- )(1 − 𝜏 / ) exp(1 − 𝜏 : / )1 − 𝜎-./)-.
-.
44%%-2/Domain Wall Decoration• The Hamiltonian after the domain wall decoration
000 0:: :.
𝐻/ = 𝑈CD 𝐻@ 𝑈CD= − ∑1-2/(𝜎-: 𝜏-.
7 𝜎-./ + 𝜎- 𝜎-./ + ℎ𝜏 7 𝜎- 𝜏 7 )
-L-.
888

Trivializability Upon Stacking Gapped SPTs

It seems to imply that all the nontrivial topological properties of the ℤ%& ×ℤ(% weakSPTC inherit from gapped SPT.
As far as topological properties are concerned, it seems that there isn’t anything
new. Is it true?

Signatures under Periodic Boundary Condition
0:0 ::: :𝐻MNOP = − ∑1-2/(𝜎-: 𝜏-.
7 𝜎-./ + 𝜎- 𝜎-./ + 𝜏 7 𝜎- 𝜏 7 )-L-.
8• Ground state degeneracy is 1.
• ℤ%& ×ℤ(% charge of the ground state is (0, 0).

Same as the gapped SPT

88

Signatures under Twisted Boundary Condition
Use ℤ%& to twist the boundary condition of τ spins
ℤQ
0:
0 :
0 :
:
: :
1L/ :
8
𝐻MNOP
= − ∑1-2/(𝜎-: 𝜏-.
7 𝜎- 𝜏
77 𝜎-./ + 𝜎- 𝜎-./ ) − ∑-2/ 𝜏 7 𝜎- 𝜏 7 + 𝜏-L-.
1L
1.
88888

• Ground state degeneracy is 1.
• ℤ%& ×ℤ(% charge of the ground state is (0, 1).
Likewise, one can use ℤ(% to twist the boundary condition, and the ground statecharge is instead (1,0)
Same as the gapped SPT

Signatures under open Boundary Condition
00:0 ::: :
1
RSP
𝐻MNOP
= − ∑1-2/(𝜎-: 𝜏-.
7 𝜎-./ + 𝜎- 𝜎-./ ) − ∑-2% 𝜏 7 𝜎- 𝜏 7 − 𝜏
7-L-.
1L
8888

• Besides symmetry operators 𝑈& , 𝑈( , there are additional operators
localized on the boundary which commutes with the Hamiltonian.
𝜎/: , 𝜎1: 𝜏 0 / , 𝑈& , 𝑈(1.%• The dimension of irreducible representation is 2.
• Exact ground state degeneracy is 2.

....
....

This should be contrasted with GSD= 4 for gapped SPT

1𝐿

Summary of Signatures of ZA × ZG Weak SPTC

Stability of Signatures• Are the above signatures stable? Or just an artifact of the our model?
• Add ℤ%& ×ℤ(% symmetric perturbation, the exact boundary degeneracy under
OBC lifted by exponentially decaying gap.
0000
𝑉 = −ℎ ∑1-2%(𝜏-L7 𝜏 7 + 𝜎/ + 𝜎1 )-.
88

• Add the same perturbation, check the ground state charge under TBC.
1

𝑉 = −ℎ ? 𝜏 0 / 𝜏 0 /-L% -.%-2/

Stability of Signatures

Stability of Signatures• Exact ground state degeneracy under OBC lifted by an
exponential gap under generic symmetric perturbation• Non-trivial ground state charge under TBC is stable under asymmetric and small enough perturbation

Y
𝑍X Strong SPTC in (1 + 1)d Spin Chain

Symmetry Extension• Symmetry fits into nontrivial extension
aNontrivial extension means
Flatness of 𝑍Xgauge fieldgg

Domain Wall Decoration• We still start with a ℤ(% SSB phase, and decorated the ℤ(%domain wall by （0+1）d ℤ%& gapped SPT, as before

Domain wall decoration induces a nontrivial ℤ𝑮𝟐 anomaly.

Domain Wall Decoration• The anomaly needs to be compensated by the same anomaly

in the ℤ(% SSB phase.
• The entire system after domain wall decoration is anomaly free.
• One further fluctuate the domain walls to the critical point.

Domain Wall Decoration: The Algorithm• Start with a critical point with ℤ(% anomaly. (Levin-Gu model is awell-known example.)• Decorated the ℤ(% domain wall by ℤ%& (0+1)d gapped SPT .
(Realized by conjugating 𝑈CD .)Y𝑍X Strong SPTC
• Hamiltonian for 𝑍XY Strong SPTC
0:𝐻MNOP = − ∑1-2/(𝜎-: 𝜏-.
7 𝜎-./ + 𝜏8

• It is invariant under 𝑍XY symmetry,1

𝑈Y = + 𝜎-0 exp-2/𝜋𝑖(1 − 𝜏 0 / )-.
4%\:0 \0 :𝜎7 - 𝜏 7 + 𝜏 7 𝜎- 𝜏 7 )-L8-.8-L8-.8

Domain Wall Un-decoration
Let’s check that it comes from Levin-Gu model by decoration.
• Undecorated Hamiltonian:0.
𝑈CD 𝐻MNOP 𝑈CD= − ∑1-2/(𝜏-.
7 − 𝜎-L/ 𝜏800 0007 𝜎- 𝜏 7 𝜎-./ + 𝜎- )-L8-.80:− ∑1-2/(− 𝜎-L/𝜎-0 𝜎-./+ 𝜎-0 )Levin-Gu modelLow energy• Undecorated Symmetry

1𝜋𝑖0:.
𝑈CD 𝑈Y 𝑈CD = + 𝜎- exp(1 − 𝜎-: 𝜏 0 / 𝜎-./)-.
4%-2/
1𝜋𝑖:+ 𝜎-0 exp(1 − 𝜎-: 𝜎-./)Low energy
4-2/

Anomalous ℤ(%symmetry

Signatures under Periodic Boundary Condition• Ground state degeneracy is• 𝑍XY charge of the ground state isSignatures under Twisted Boundary Condition• Use 𝑍%& to twist the boundary condition of τ spins.
]8Q\:0 \0 :𝜎
7 - 𝜏 7 + 𝜏 7 𝜎- 𝜏 7 )-L8-.8-L8-.88\:
0 :0 \−𝜎1: 𝜏70 𝜎/: +𝜏1L𝜎𝜏+𝜏𝜎
7 1 77 1 𝜏71L: 0:
𝐻MNOP = − ∑1L/-2/ (𝜎- 𝜏-.7 𝜎-./ + 𝜏88888• Ground state degeneracy is 1.
• Relative 𝑍XY charge of the ground state is 2.

Signatures under Twisted Boundary Condition• Use 𝑍XY to twist the boundary condition.
]^_\:0 \0 :: 0:
𝐻MNOP = − ∑1L/(𝜎𝜏𝜎+𝜏𝜎𝜏+𝜏𝜎𝜏-.7 )7777-2/ - -. -./-L-L8-.8888\:
0 \0 :+𝜎1: 𝜏70 𝜎/: −𝜏1L
𝜎7 𝜎1 𝜏7 +𝜏
7 1 𝜏71L
88888• Ground state degeneracy is 2 or 4.
• Relative 𝑍%& charge of the ground state is 1.

Signatures under open Boundary Condition
RSP𝐻MNOP= − ∑1-2%( 𝜏\:0 \0 :1L/ : 0:∑𝜎𝜏+𝜏𝜎𝜏)+𝜎𝜏𝜎7777 7-2/ - -. -./-L8 - -.8-L8-.88• Besides symmetry operators 𝑈Y , there are additional operators localizedon the boundary which commutes with the Hamiltonian.
𝜎/: , 𝜎1: 𝜏 0 / , 𝑈Y
1.%

• The dimension of irreducible representation is 2.
• Exact ground state degeneracy is 2 or 4.

YComparing with 𝑍X Landau Transition• One can use these signatures to distinguish the 𝑍XY strong SPTC fromLandau transition:

Stability of Strong SPTCIs the 𝑍XY strong SPTC stable upon perturbing to a gapped phase witha single ground state?
• There is no nontrivial Z4 gapped SPT.
• Since Γ is not anomalous, it should be possible to deform the theory to atrivially gapped phase, if allow strong enough perturbation.
• Possible for arbitrarily small perturbation?

Stability of Strong SPTC
1

𝑉 = −ℎ ? 𝜎-0 + 𝜏 0 /-2/-.%

Stability of Strong SPTC
• For finite size, need to pass a critical strength to enter trivially gappedphase. ⇒ stable under perturbation.

• Weak SPTC: Signatures coincide with gapped SPT.
• Strong SPTC: Signatures distinct from gapped SPT.

Strong SPTC is more stable than weak SPTC.

Summary• There is a natural notion of SPT for quantum criticality – symmetry protected
topological criticality.
• Decorated defect construction is a powerful tool to systematically construct SPTC.
• Symmetry charge of the ground state under TBC is a physical observable to probe
the nontrivial SPTCs. But ground state degeneracy under OBC isn’t.
•Strong SPTC is more stable than weak SPTC against perturbing to gapped phasewith one ground state.

•Future directions: Classification? Continuous symmetry? Anomaly inflow picture?
Entanglement feature?

Thanks for listening!