Berry phase in quantum field
theoryPo-Shen Hsin
California Institute of Technology
With Anton Kapustin (Caltech) and Ryan Thorngren (Harvard CMSA)
2004.10758

Berry phase in quantum mechanics• Systems with continuous external parameters 𝜆, or slowly varying
degree of freedom.
• Start with unique ground state |𝜓0 (𝜆)⟩ with zero energy.
Adiabatically changing the parameter 𝜆 along a loop 𝛾 in the space ofparameter without closing energy gap:𝑖 ‫𝐴 𝛾ׯ‬[Berry]𝜓0 → 𝜓0 𝑒
where 𝐴 = 𝑖 𝜓0 (𝜆) 𝑑𝜆 𝜓0 (𝜆) is the Berry connection.
• Can be written using Berry curvature ‫׬ = 𝐴 𝛾ׯ‬Σ 𝑑𝐴 for Σ bounded by 𝛾Diabolical point in quantum mechanics• Spin ½ in magnetic field 𝐵: two eigenstates with energy ±𝜇 𝐵

𝐵 ∈ R3

𝐻(𝐵) = 𝜇𝜎 ⋅ 𝐵,
• 𝐵 = 0: degenerate zero energy states, codimension 3 in theparameter space R3 .
• 𝐵 ≠ 0: unique ground state with energy gap (i.e. trivially gapped)• The point 𝐵 = 0 has ground state differed from the surrounding
region: “diabolical point” in the parameter space. [Berry]• For generic Hamiltonian without symmetry, level-crossing occurs incodimension 3 (𝐻 spanned by 3 Pauli matrices) [von Neumann, Wigner]

B∈ R3
B=0

Describe “diabolical point” using Berry phase• Removing diabolical2 points from parameter space creates non-trivial
topology R3 \0 ∼ 𝑆𝜃,𝜙 . Detect diabolical point by the topology?
2• Ground state wavefunction does not depend continuously on 𝑆(𝜃,𝜙)
:
𝜃 𝑖𝜙
𝜃
𝜃
𝜃 −𝑖𝜙
0 𝑁 = sin 𝑒 , −cos , 0 𝑆 = sin , −cos 𝑒
.
2
2
2
2
Transition function at equator 0 𝑁 = 𝑒 𝑖𝜙 0 𝑆 [Berry], [Simon]
𝑈(1) Berry connection 𝐴𝐵 = 𝑖⟨0 𝜕𝐵 0⟩, 𝐴𝜃 = 0, 𝐴𝜙 = sin2 (𝜃/2)
Non-trivial Berry curvature 𝐹𝐵 = 𝑑𝐴𝐵 with quantized period
Berry number:ර 𝐹𝐵 /2𝜋 = 1 .
𝑆2

𝐹𝜃𝜙 =

1
sin 𝜃 𝑑𝜃𝑑𝜙 has monopole singularity at diabolical point.
2
2𝑟

B
𝑆2

Describe “diabolical point” using Berry phase• Berry number ≠ 0 ⇒ not the same phase in entire parameter space
Suppose the theory were trivially gapped in entire R3 , 𝑆 2 would becontractible, and the Berry number would be zero instead of one.

???

B

B

B

𝑆2

𝑆2

𝑆2

Nonzero Berry number‫ 𝐵𝐹 ׯ‬/2𝜋 ≠ 0Contractible:
Berry number ‫ 𝐵𝐹 ׯ‬/2𝜋 = 0
InconsistentPhasetransitionNon-contractible
Berry number ‫ 𝐵𝐹 ׯ‬/2𝜋 ≠ 0 possible
ConsistentDiabolical Points in Many Body Systems(Outline)• Critical points often sit at the end of first order phase transition lines
Critical point connected to 1st orderphase transition line

• There are also isolated critical points (diabolical points) in phase
diagram surrounded by gapped phase with non-degenerate vacuum.
Q1: What protects the stability of such diabolical critical points?

Usual argument: protected phase transition
by symmetry• Landau paradigm• symmetry protected topological (SPT) phase: symmetric phase thatcannot be connected to trivial phase without breaking symmetry.
Protects against symmetry preserving perturbations. SPT1 SPT2
• Q1: what about symmetry-breaking perturbations or systems without
symmetry?
• Ans: Berry phase. Generalization of SPT.

Diabolic Points in Many Body Systems(Outline)• The symmetry protected topological phase has non-trivial boundary
physics constrained by bulk-boundary correspondence.
Example: edge of integer quantum Hall system has chiral current.
• Q2: Does Berry phase also have bulk-boundary correspondence? (Yes)• The possible symmetry protected topological phases are classified bygroup cohomology or cobordism groups. [Chen,Gu,Liu,Wen],[Kapustin],[Freed,Hopkins]…
• Q3: What classifies possible Berry phases? (Some cobordism group)• Applications: use Berry phase to study Néel / Valence Bond Solidtransition, and new tests for infrared Chern-Simons matter dualities

Berry Phase Protects Phase Transitions• Parameter space 𝑀 consists of trivially gapped phase (blue) withpossible diabolical loci (white) removed: non-trivial topology• Promote the parameters to be position-dependent background fields
𝜙: spacetime → 𝑀• Non-trivial topological term in effective action Seff 𝜙 forconfiguration 𝜙(𝑥) ∈ 𝑆 𝑟 ⊂ 𝑀 protects diabolical loci inside.
(otherwise sphere would be contractible in 𝑀 and the topological𝑆eff = 0
𝑆eff ≠ 0
term is trivial)
Phase transition𝑀Effective action in quantum mechanics• Let the parameter (magnetic field) to be periodic in time
1
𝐵: 𝑆time→ R3• Away from diabolical point (nonzero magnetic field), R3 \0 = 𝑆 2 :unique ground state with topological effective action
𝑆eff = ‫𝜏 ∗𝐵 ׬‬1𝜏1 is a 1-form on 𝑆 2 i.e. the Berry connection. Adiabatic 0 → 0 𝑒 𝑖𝑆eff
𝜏1 undergoes gauge transformation across coordinate patches on 𝑆 2 .
B𝑒 𝑖𝑆eff is gauge invariant, but the effective Lagrangian is not
𝑆2

Effective action in quantum mechanics as
Wess-Zumino-Witten term
• We can also write the effective action using the gauge invariant Berrycurvature.
• Introduce one-parameter family of background 𝐵(𝑡, 𝑠) defined on 21
manifold 𝑌 that bounds 𝑆time
𝑆eff = න 𝑑 𝐵∗ 𝜏1 = න 𝐵∗ 𝐻2
𝑌𝑌2• Berry curvature 𝐻2 = 𝑑𝜏1 is a 2-form on 𝑆 with quantized period• The action is an example of Wess-Zumino Witten (WZW) term. It is2characterized by non-trivial map 𝑆(𝑡,𝑠)→ 𝑆 2 ⊂ R3 whose degree is
the Berry number 𝜋2 𝑆 2 = Z.

Effective action in (d+1) Dimension Spacetime• Promote parameters to be spacetime-dependent background fieldspacetime
𝜙: 𝑋𝑑+1→𝑀
Trivially gapped with diabolical loci removed, which creates topology.
The Berry phase is the topological term in effective action
𝑆eff [𝜙] = ‫𝑑𝜏 ∗ 𝜙 ׬‬+1
where 𝜏𝑑+1 is a 𝑑 + 1 -form on parameter space 𝑀. 𝜏𝑑+1 → 𝜏𝑑+1 + 𝑑𝜆𝑑• 𝑆eff 𝜙 = ‫𝑑𝐻 ∗ 𝜙 𝑌׬‬+2 for 𝑌𝑑+2 bounds spacetime and 𝐻𝑑+2 = 𝑑𝜏𝑑+1 has
𝑑+2
quantized period. Such WZW term can arise from 𝜋𝑑+2 (𝑀).
• Family of lattice Hamiltonian systems: Berry phase is studied in [Kapustin, Spodyneiko]1

Include global symmetry (generalization of
SPT phase)• 𝑆eff [𝜙, 𝐴] with background gauge field 𝐴 for the symmetry• Example: Thouless pump for 𝑈(1) symmetry𝑆eff 𝜙, 𝐴 = ‫ 𝑑𝜏 ∗ 𝜙 ∧ 𝐴 𝑋׬‬, 𝜏𝑑 : closed 𝑑-form on parameter space𝑑+1• 𝑈(1) current for spacetime-dependent parameter: charge associated withsoliton configuration of 𝜙𝑗 =⋆ 𝜙 ∗ 𝜏𝑑 ,𝑄 = ර ⋆ 𝑗 = ර 𝜙 ∗ 𝜏𝑑
𝑆𝑑

𝑆𝑑

• Family of lattice Hamiltonian systems with 𝑈(1) symmetry: Thouless pumpinvariant studied in [Kapustin, Spodyneiko]2• We will give example of diabolical points protected by higher-degree
currents generating higher-form symmetry [Gaiotto,Kapustin,Seiberg,Willett],[Wen]Example: Free Fermions in (1+1)d• One Dirac fermion with complex mass 𝑀 = 𝑚𝑒 𝑖𝛼 ∈ R2
ത 𝑖𝛾01 𝛼 𝜓 = 𝑀𝜓ത+ 𝜓− + h. c.
𝑚𝜓𝑒MThe theory has 𝑈 1 𝑉 symmetry 𝜓 → 𝑈𝜓.
• 𝑀 ≠ 0: gapped with a unique ground state.
• 𝑀 = 0: codimension 2 gapless diabolic point.
• 𝑀 = 0 : mixed anomaly for 𝑈 1 𝐴 − 𝑈 1 𝑉 , but 𝑈 1 𝐴 absent for 𝑀 ≠ 0
Q: Can we add interaction to gap out the system preserving only 𝑈 1 𝑉
symmetry? (No. Diabolic point protected by Thouless pump invariant.)
Q: Is there family of trivially gapped interfaces depend continuously on theparameters? (No, example of bulk-boundary correspondence for Berry phase)Diabolic Point Protected by Thouless Pump
• Denote the 𝑈 1 𝑉 background gauge field by 𝐴.
For 𝑀 > 0 the effective action for massive fermion has the Thouless
𝑀 = 𝑚𝑒 𝑖𝛼pump invariant that protects the diabolic point [Goldstone, Wilczek]𝑑𝛼1න𝛼𝑑𝐴 = ‫𝑗 ⋆ 𝐴 ׬‬
𝑆eff = න 𝐴
=
2𝜋 2𝜋
1
1
‫𝑑 ׯ = 𝛼𝑑 ׯ‬arg 𝑀 = 1: the loop is not contractible. The gapless
2𝜋
2𝜋point at the origin cannot be completely removed.

1• 𝑈 1 𝑉 current, 𝑗 = ⋆ 𝑑𝛼. For 𝛼 = 𝛼(𝑡) periodic in time and winds
2𝜋1

origin 𝑁 times: pumps charge Δ𝑄 = ‫= 𝑗 ⋆ׯ‬2𝜋

𝛼 𝑇 −𝛼 0= 𝑁.

[Thouless]Perturbation cannot remove the diabolicalpoint• Consider bosonization description of the free fermion in (1+1)d withperiodic scalars 𝜙, 𝜃. 𝑈 1 𝑉 symmetry 𝜃 → 𝜃 + 𝛼.
• Fermion mass corresponds to 𝜆𝑥 sin𝜙 + 𝜆𝑦 cos𝜙, 𝑀 = 𝜆𝑥 + 𝑖𝜆𝑦𝜆𝑦𝜆𝑥 = 0: charge conjugation
symmetry Z2 : 𝜙, 𝜃 → (−𝜙, −𝜃)𝜆𝑥Perturbation cannot remove the diabolicalpoint• Decrease radius pass through the self-dual point, new relevant
perturbations cos2𝜙, sin2𝜙. Deformation by cos2𝜙 changes thephase diagram
First order phase transition: broken
Z2 : 𝜙, 𝜃 → (−𝜙, −𝜃)• Perturbation can only deform the diabolical point to be diabolical loci,and it persists in the phase diagram

Family of Interfaces
𝜶 𝒙 = −∞ = 𝟎

𝜶 𝒙 = ∞ = 𝜶𝟎𝑥

• Interface labelled by 𝛼0 has charge
Δ𝑞 = න⋆ 𝑗 = ‫𝛼𝑑 ׬‬/2𝜋 = 𝛼0 /2𝜋 ,(Δ𝑞 = 1 for 𝛼0 = 2𝜋)• For 𝛼(𝑥) = 𝛼0 𝜃 𝑥 , bound state 𝜓 = 𝜓 0 𝑒 −𝛽|𝑥| for 𝛽 > 0. 𝛽 =𝑚 sin(𝛼0 /2), energy 𝐸 = 𝑚 cos(𝛼0 /2)
Single normalizable zero mode at 𝛼0 = 𝜋 [Jackiw, Rebbi]
Non-normalizable mode at 𝛼0 = 0,2𝜋, merges with the bulk modes• The existence of single zero mode and non-normalizable modes canalso be found in smooth interfaces in this theory with Thouless pump

[Keil,Kobes]

Boundary-Bulk Correspondence for interface• The family of interfaces cannot be described by purely (0+1)dquantum mechanics for all 𝛼0 . Suppose otherwise, effective action
න𝑞 𝛼0 𝐴𝑡 𝑑𝑡
𝑞 𝛼0 ∈ Z is vacuum charge: jump only at gapless points and singlevalued (sum of all jumps when 𝛼0 varies from 0 to 2𝜋 is Δ𝑞 = 0)
• Thouless pump: Δ𝑞 = 1 when 𝛼0 varies from 0 to 2𝜋. Contradiction.
• Reason: 𝑞 can also jump at delocalized point. Δ𝑞 from interface
gapless points = −Δ𝑞 from non-normalizable modes escaped to bulk
spatial infinity (Thouless pump)General Bulk-Boundary Correspondence• Consider family of bulk with Berry phase in (d+1) dimension. Namely,
the phase diagram has diabolical loci but otherwise trivially gappedwith nontrivial topological term in the effective action• If bulk system with Berry phase has a boundary, then the gap mustclose on the boundary for some parameter. This arises at boundary
diabolical points (loci).
• Boundary-bulk ``Anomaly inflow’’:
Total Berry number of boundary diabolical points= Total Berry number in the bulk.
(Similar to the Nielsen-Ninomiya theorem)

General Bulk-Boundary Correspondence• Consider a diabolic point (red) enclosed by the ball 𝐵𝑟 ⊂ 𝑀 describedby bulk Berry phase.
• Consider a family of interfaces (special case: boundary) where theparameter interpolates between the North and South pole. Theinterpolation is a curve in 𝐵𝑟 connecting the two points• The interface whose curve hits the red point has diabolic point on theinterface𝐵𝑟 ⊂ 𝑀

General Bulk-Boundary Correspondence• What happens when the family ofinterfaces
varies
away
from
the
diabolic
point? Restrict curves to lie on 𝑆 𝑟−1
Bulk Berry phase implies that as the family of interfaces swipes around 𝑆 𝑟−1
there is additional phase, so the family cannot depend continuously on theparameters• Represent the parameter on interface as intersection (green) of the curvewith the equator
𝑆 𝑟−1
𝑀interface = 𝑆 𝑟−2
𝑆 𝑟−2• Compute interface Berry phase: introduce 1-parameter family of 𝑟−2
background to count how many times green dot swipes around 𝑆 .
As green dot swipes 𝑆 𝑟−2 once, the blue curve swipes around 𝑆 𝑟−1 once
Boundary Berry number = Bulk Berry number

Example: particle on a circle• Particle on circle 𝑥 ∼ 𝑥 + 2𝜋 with 𝑈(1) symmetry and parameter 𝛼
1 2
1
𝑥ሶ +
𝛼 𝑥,ሶ
𝛼 ∼ 𝛼 + 2𝜋,
𝑈 1 :𝑥 → 𝑥 + 𝑓
2
2𝜋
1
1
2
• Background 𝑈(1) gauge field 𝐴: 𝑥ሶ − 𝐴 + 𝛼 𝑥ሶ − 𝐴
22𝜋• 𝛼 is no longer periodic, violated by ‫𝐴 ׬‬. ``an anomaly’’ [Córdova,Freed,Lam,Seiberg]• The theory lives on the boundary of the bulk with Berry phase𝑑𝛼bulk𝑆eff = න 𝐴
2𝜋• Bulk-boundary correspondence: level-crossing occurs on boundary atsome 𝛼 (in fact, 𝛼 = 𝜋)Berry phase v.s. Anomaly: Two Free Diracfermions in (1+1)d• Mass 𝜓ത 𝑀0 + 𝑖𝛾 01 𝑀𝑖 𝜎 𝑖 𝜓, 𝑀𝐴 = 𝑀0 , 𝑀1 , 𝑀2 , 𝑀3 ∈ R4
𝑀 > 0: gapped with a unique ground state. R4 − {0}~𝑆 3
𝑀 = 0 codimension-4 gapless diabolical point.
• Gapless point 𝑀 = 0 is protected by ‘t Hooft anomaly for 𝑆𝑝𝑖𝑛(4)symmetry only against symmetry-preserving perturbation• Gapless point also protected by Berry phase, also against symmetrybreaking perturbations• Effective action: one parameter family of background 𝑌(𝑡,𝑥,𝑠) with WessZumino term 𝜋3 𝑆 3 = Z. 𝑆eff = ‫𝜔׬‬2 , 𝐻 = d𝜔2 the volume form of 𝑆 3
𝐻 = 1/(6𝜋 𝑀 4 ) 𝜖 𝐴𝐵𝐶𝐷 𝑀𝐴 𝑑𝑀𝐵 𝑑𝑀𝐶 𝑑𝑀𝐷[Abanov, Wiegman]Free Fermion in (2+1)d: No Perturbative
Anomaly• Two Dirac fermions with mass term
ത 𝑖 𝜓,
𝑚𝑛𝑖 𝜓𝜎𝑛𝑖 ∈ 𝑆 2 , ∑𝑛𝑖2 = 1
Gapped for 𝑚 > 0, gapless for 𝑚 = 0 (diabolical point of codimension 3)• 𝑚 = 0 has 𝑆𝑈(2) symmetry, protected by mixed parity- 𝑆𝑈(2) anomaly
Also protected by Thouless pump invariant: skyrmion current 𝜋2 𝑆 2 = Z
1 𝜇𝜈𝜌 𝑖𝑗𝑘
𝜇𝑆eff 𝑚𝑛𝑖 , 𝐴 = ‫= 𝑗 𝜇𝐴 ׬‬𝜖 𝜖 න𝐴𝜇 𝑛𝑖 𝜕𝜈 𝑛𝑗 𝜕𝜌 𝑛𝑘 [Abanov, Wiegman]
8𝜋• Berry phase has infinite order i.e. remains nontrivial in any N-copy systems
Parity anomaly has order 2: no anomaly by taking 2 copies, only robust
against 𝑆𝑈(2) sym preserving perturbation• In addition, the effective action also has 𝜃 = 𝜋 Hopf term 𝜋3 𝑆 2 = Z

Web of diabolical points related by gauging
U(1) symmetry• Starting from a system with diabolic loci protected by Thouless pumpinvariant, can we obtain new system protected by Thouless pump invariant?
[Kapustin, Strassler], [Witten]
Gauging the 𝑈(1) symmetry
11
new𝑆eff 𝜙, 𝐴 → gauging 𝑈(1) → 𝑆eff 𝜙, 𝐵 = 𝑆eff 𝜙, 𝑎 +
𝑎𝑑𝑎 −
𝑎𝑑𝐵
4𝜋
2𝜋
New 𝑈(1) magnetic symmetry 𝑗 =⋆ 𝑑𝑎/2𝜋.
1
new
∗
∗
𝑆eff 𝜙, 𝐴 = න𝐴𝜙 𝜏2 ,𝑆eff 𝜙, 𝐵 = න𝐵𝜙 𝜏2 + 𝐻 𝜙 −𝐵𝑑𝐵
4𝜋• Two free Dirac fermions in (2+1)d ⇒ Interacting 𝑈 1 1 with two fermions(realizes phase transition betw 𝑆1 sigma model and 𝑈 1 2 ) protected by
the same Thouless pump invariant𝑈(1) Gauge Theory with 2 Scalars in (2+1)d• 𝑈(1) gauge theory
withtwo
Wilson-Fisher
scalars
deformedby𝑆𝑈(2)triplet mass 𝑚2 𝑛𝑖 ∈ R3
𝑉 𝜙 = 𝑚 2 𝑛𝑖 𝜙 + 𝜎 𝑖 𝜙 + 𝜆 𝜙 + 𝜙 2• 𝑚 ≠ 0: 𝑆𝑈(2) symmetry explicitly broken. 𝑈(1) gauge field is Higgsed
𝑛1 𝑑𝑛2 − 𝑛2 𝑑𝑛1
1 𝑖𝑗𝑘𝑎=⇒ 𝑑𝑎 = − 𝜖 𝑛𝑖 𝑑𝑛𝑗 𝑑𝑛𝑘
2(1 − 𝑛3 )2Monopole ⇒ skyrmion 𝜋2 𝑆 2 = Z
1The 𝑈(1) magnetic symmetry 𝑗 = − ⋆ 𝑑𝑎 has Thouless pump
2𝜋
1𝑆eff 𝑚2 𝑛𝑖 , 𝐴 =𝜖 𝜇𝜈𝜌 𝜖 𝑖𝑗𝑘 න𝐴𝜇 𝑛𝑖 𝜕𝜈 𝑛𝑗 𝜕𝜌 𝑛𝑘
8𝜋
The phase transition 𝑚 = 0 is protected by the Thouless pump invariantApplication: Deconfined Quantum Criticality• Deconfined quantum critical point: scalar QED3 without ChernSimons term is believed to describe a deconfined quantum criticalpoint between the Néel state and the valence bound solid (VBS).
[Senthil,Vishwanath,Balents,Sachdev,Fisher]

Néel: broken 𝑆𝑂(3) spin symmetry
𝑆𝑂 3 vector 𝑁𝑖 Néel field ~𝑆𝑂 3 vector mass 𝑚2 𝑛𝑖
VBS: broken Z4 lattice rotation symmetry

𝑈(1) with two scalars: 𝑆𝑂 3 × 𝑈(1) symmetry

Application: Deconfined Quantum Criticality• The transition is protected by ‘t Hooft anomaly for 𝑆𝑂 3 × 𝑈(1)symmetry against symmetry preserving perturbations.
[Benini,Hsin,Seiberg],[Wang,Nahum,Metliski,Xu,Senthil],[Komargodski,Sharon,Thorngren,Zhou]……

• Symmetry of the action is[Benini,Hsin,Seiberg]
𝑈 1 gauge × 𝑆𝑈 2 global /Z2
Faithful flavor symmetry is 𝑆𝑂 3
𝑆𝑂 3 bundles that are not 𝑆𝑈 2 bundles: changes the quantizationof the flux of 𝑈 1 gauge . Anomaly for the 𝑈(1) magnetic symmetry𝑑𝐴𝜋න𝑤2 (𝑆𝑂(3))
2𝜋Application: Deconfined Quantum Criticality• The anomaly has order 2• For lattice that does not respect Z2 ⊂ 𝑈(1) symmetry: the anomaly
vanishes and does not offer protection to the phase transition.
• For honeycomb lattice with Z3 symmetry there could be intermediate triviallygapped phase [Jian, Zaletel], [Po,Watanabe,Jian,Zaletel]• Here we show if there is a Z𝑁 ⊂ 𝑈 1 symmetry for any 𝑁 ≠ 1 there
is non-trivial transition protected by the Thouless pump invariant.
Does not need 𝑆𝑂 3 symmetry. Protection even on honeycomb
lattice where 𝑁 = 3.

Generalization with Chern-Simons term andhigher rank gauge group• Chern-Simons matter theory: level 𝑘 gives 𝜃 = 𝑘𝜋 Hopf term for theparameter field associated with 𝜋3 𝑆 2 = Z [Wilczek,Zee]• Magnetic charge corresponds to Skyrmion number
𝑑𝑎
1 𝑖𝑗𝑘
ර
=−
𝜖 ර𝑛𝑖 𝑑𝑛𝑗 𝑑𝑛𝑘
2𝜋
4𝜋
𝜃
𝑘
Skyrmion has spin = : agrees with the spin of the monopole
2𝜋
2[Wilczek,Zee]• The discussion can be generalized to 𝑈(𝑁) gauge theory

Application: New Test for Boson/Fermion
Duality• 𝑈 𝑁 1 with Wilson-Fisher scalars are conjectured to flow to free Dirac fermionsin the infrared [Hsin,Seiberg]2𝜓 ↔ 𝑈 𝑁 1 + 2𝜙,
𝑁≥2• New consistency check: 𝑆𝑈(2) adjoint mass deformation on both sides
Constant mass: no information.
Novelty: promote mass to be spacetime-dependent.
• produces the same Berry phase effective action (𝜃 = 𝜋 Hopf term and theThouless pump invariant)• The effective action cannot be removed by adding local counterterm, must match
across duality (not well-define at 𝑚 = 0 where 𝑛𝑖 are ill-defined)• New test for web of dualities related by gauging or RG flow, e.g.
𝜓 ↔𝑈 1 1+𝜙[Seiberg,Senthil,Wang,Witten],[Karch,Tong]…

Thouless Pump with Higher-Form Symmetry• Similar analysis applies to (3+1)d 𝑈(1) gauge theory with 2 scalars
1• The current 𝑗 = − ⋆ 𝑑𝑎 is a 2-form in (3+1)d: magnetic 𝑈(1) 12𝜋

form symmetry that transforms the ‘t Hooft line operators.
• Same computation implies that the phase transition is protected byThouless pump invariant
1 𝜇𝜈𝜌𝜎 𝑖𝑗𝑘
(2)
2
(2)
𝑆eff 𝑚 𝑛𝑖 , 𝐴=𝜖𝜖 න𝐴𝜇𝜈 𝑛𝑖 𝜕𝜌 𝑛𝑗 𝜕𝜎 𝑛𝑘
8𝜋(2)
𝐴𝜇𝜈 : background gauge field for the 𝑈(1) 1-form symmetry

Further Examples: Thouless Pump with
Gravitational Invariant• Free Weyl fermion in (3+1)d with complex mass 𝑀 = 𝑚𝑒 𝑖𝛼𝑀 = 𝑚 > 0: gapped with a unique ground state𝑀 = 0: codimension 2 gapless diabolical point• Diabolical point protected by effective action for 𝑚 > 01
12
න𝛼 Tr 𝑅 = −න𝑑𝛼 CSgrav
𝑆eff =
2384𝜋
2𝜋• The effective action cannot be removed by a well-defined function of
𝑀 since 𝛼 = arg 𝑀 is not well-defined at the origin. Berry curvature
has singularity.
• This is called an ``anomaly’’ in [Córdova,Freed,Lam,Seiberg]

Further Examples: Thouless Pump with
Gravitational Invariant
1
න𝑑𝛼 CSgrav
𝑆eff = −2𝜋• For 𝛼 = arg 𝑀 winds around the origin of 𝑥 1 - 𝑥 2 plane 𝑛 times, there𝑛3will be gapless chiral edge mode propagating along 𝑥 with 𝑐− = .
2

1• If turn on coupled to 𝑈(1) gauge field, additional term 2 ‫ 𝛼 ׬‬Tr 𝐹 2
8𝜋

Classification of Berry Phase• There is no stable diabolical locus (protected by Berry phase) without
symmetry in codimension 𝑚 > 𝑑 + 3. Intuitively, for region 𝐵𝑚 in
the parameter space contains the diabolic point, 𝜋𝑑+2 𝑆 𝑚−1 = 0 if
𝑚 > 𝑑 + 3.
• In general we propose the following classification for Berry phase:family of trivially gapped theories parametrized by 𝑀 are classified by
Ω𝑑+1
𝑀𝑠Where 𝑠 represents additional structure (e.g. 𝑆𝑂, 𝑆𝑝𝑖𝑛)• Including symmetry 𝐺: replace the argument by 𝑀 × 𝐵𝐺 possiblywith twist (𝐺 action on 𝑀). If trivial 𝑀: known classification for SPT[Kapustin], [Freed,Hopkins]

Berry phases of other degrees as Symmetries• Phase diagram can have topology not captured by the Berry phase in (d+1)
dimension. E.g. What about 𝜋𝑘 (𝑀) for 𝑘 ≠ 𝑑 + 2?
They can define symmetry defects• Example: 4d 𝑁 = 1 SU(𝑛) gauge theory with 𝜃 angle, 𝜃 ∼ 𝜃 + 2𝜋
Define interface such that 𝜃 changes by 2𝜋 across the interface‫𝜽𝒅 ׯ‬/𝟐𝝅 = 𝟏
𝜽𝜽 + 𝟐𝝅0-form symmetry defects associated with 𝜋1 𝑀 = 𝜋1 𝑆1 = Z.

Symmetry permutes 𝑛 vacua: spontaneously broken. ``vacuum-crossing’’[Sharon]Berry phases of other degrees as Symmetries• Similarly, 𝜋𝑘 (𝑀) can define symmetry defect of various codimensions
and they generate higher form symmetries. The symmetries may notact faithfully and may be spontaneously broken.
• It would be interesting to explore the implications of thesesymmetries.

Conclusion• We use Berry phase to study diabolical points in phase diagram for system
in general dimensions• We argue that the theory with Berry phase implies the gap closes on theboundary for some parameter• We discuss examples including free fermions and interacting gauge theory
with bosons or fermions.
• Proposal for classifying family of invertible theories: cobordism group forthe parameter space• Applications include new evidence of the stability of the deconfinedquantum critical point in Néel-VBS transition, and the infrared duality
between free fermions and 𝑈 𝑁 1 Chern-Simons matter theory withscalars.

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