The Large Charge Sector of 3Dparity violating CFTs
Umang Mehta
umangmehta@uchicago.edu

with Luca Delacretaz and Gabriel Cuomo
arXiv: 2102.05046

Outline• EFT without parity1• Euler term at (∂ )• Vortices on the sphere• Spectrum and OPEs
• Anyon Super uids
• Derivation of the leading order EFT on a plane• Monopole operators• Vortices on the sphere as holes

fl𝒪• Beyond EFT - Chern-Simons terms, double scaling, rotons

Large U(1) charge EFT3213/2
32S∼
d x g | f | + (∂ )c1 ∫⇔CurrentsϕU(1)2| ∂ϕ | ∂μϕρϵμνρ∂ ϕ(1)𝒪U(1)ffaμϵμνρ fνρfμν /|f|Hellerman, Orlando, Re ert, Watanabe ’15
Monin, Pirtskhalava, Rattazzi, Seibold ’16
𝒪S ∼ c1 d x g | ∂ϕ | + (∂ )∫3

EFT without parity
0• JEuler integrated over space gives the1
3/2
μ33
S∼
d x g | f | + κ d x g aμJEuler∫c1 ∫Euler characteristic• κ ∈ ℤ is quantized• κ controls Hall viscosity

1 μνρ αβγ α1βγ
μβγ
JEuler =ϵ ϵ u ∇ν u ∇ρ u − ℛνρ()8π2• Euler term breaks one-form symmetryμνν
Equationofmotion:∇J=κJ
μ 1−form
•
Euler

αu =

ϵαβγ

fβγ

2|f|
Golkar, Roberts, Son ’14

EFT on the sphere• Integrated e.o.m. ⟹ 0 = κχS2 /2 = κneed κ vortices ≅ point charges•

Svortices = −κ∑∫

p=1 Xpa+…

⃗
Vortex
angularmomentum:J=•￼• Coulomb interaction: H = − ∑∑Q X p⃗ /22⃗⃗
log | X − X |pp′

Golkar, Roberts, Son ’14Spectrum: classical ground state• Energy minimization = Whyte’s problem - no general solution known
except in special cases κ = 1,2,3,4,5,6,12• Theorem: center of mass at sphere origin when κ ≠ 1• Main result: lightest operator has J = 0 for κ ≠ 1 and J = Q/2 for κ = 1with dimension

13/2
Δmin(Q) =Q + κ c1 Q log Q + …
c1• Inhomogeneous ground state, large degeneracy
Dragnev, Legg, Townsend ’02

see also Cuomo, de la Fuente, Monin, Pirtshkalava, Rattazzi, ’17

Spectrum: quantum ground state and vortex
excitations• Quantization ⟹ Landau levels on sphere(monopole harmonics)• Quantization lifts degeneracy, eg κ = 2Δmin(Q, J) = Δmin(Q) +J(J + 1)
c1
3/2
Q

• Much softer that phonon excitations withΔQ,J − Δmin = J(J + 1)/2
Hellerman, Orlando, Re ert, Watanabe ’15
Monin, Pirtskhalava, Rattazzi, Seibold ’16
Ja eris, Mukhametzhanov, Zhiboedov ’17
ff

ff

Wu, Yang ’76

• Kubo formula: ηH ∼ ⟨Q̄TxyTxxQ⟩. The stress tensor is linear in Goldstone
uctuations

T ∼ 1 + c1∂π + κ∂∂π• Therefore, we expect ⟨Q̄TQ′⟩odd ∼ ηH where Q′ = Q + single phonon
3/2Q
1/2⟨Q̄TQJ⟩even ∼ 1/4 Jc1ff

￼￼

Avron, Seiler, Zograf ’95
Bradlyn, Goldstein, Read ’12
ff

fl

OPE coeffs and Hall viscosity

1/4 3/2 3/2⟨Q̄TQJ⟩odd ∼ κc1 Q J
Hellerman, Orlando, Re ert, Watanabe ’15
Monin, Pirtskhalava, Rattazzi, Seibold ’16
Ja eris, Mukhametzhanov, Zhiboedov ’17

Candidate theories• U(N)k1,k2 or SU(N)k Chern-Simons matter theories?
• Global U(1) - topological or fermion number respectively• Seem to be a Fermi liquid at strict N = ∞, possibly super uid for nite N• Super uid for N = 1, k → ∞ - “anyon super uids”
Chen, Wilczek, Witten, Halperin ’89
Wiegmann ‘90
Abanov, Wiegmann ’99, ’20
fi

fl

fl

fl

Geracie, Goykhman, Son ’15
Minwalla, Mishra, Prabhakar ’20

Anyon Super uids - at spacek3μμνρ
S = d x ψ̄iγ Dμψ − ϵ aμ∂νaρ()∫4π1μμνρ
j=ϵf
GlobalcurrentU(1)νρ•
4πkb00Gausslaw:
n=
ψ̄γ
ψ
=kj=
ψ•
2πk+1k• k fully lled Landau levels
21flflfikHall
conductivity
σ=xy•2π⋮}bk

Leading order EFT• Integrate out fermions to getk
δS =ada4π ∫• Cancels CS term and generates higherorder terms

SEFT =

13/2
22e − b + … ∼ |f|()
∫∫
22πρ

kk+1k

3/2

• Read o c1 = 1/2πk

321⋮}

bkff

Nguyen, Gromov ’16

Monopole operators and double scaling• Charged operators are monopoles. Operator dimension from EFT (1 ≪ k ≪ Q) is given by2
3/2
Δmin(Q) = (kQ) + (1/k, k/Q)
3

• Opposite limit Q ≪ kfrom higher derivs
from higher loops2
3/2
Δmin(Q) = (kQ) + …
3• Double scaling limit: Q, k → ∞, Q/k xed - organize Dirac particles in Landau levels obtained fromthe monopole background at leading order

2 3 Q
Δmin(Q) = k
3 (k)
fi𝒪𝒪

3/2

1+[1( k )]Radicevic ’15
Chester, Iliesu, Mezei, Pufu ’17

• kQ Dirac fermions in sphere Landaulevels

k+1• Degeneracy Q + 2 | p | , p ∈ ℤ. k fully
lled LLs requires kQ + k(k + 1)

kfermions• k(k + 1) holes on the last one super uid vorticesfl• Identify κ = k(k + 1) ≫ 1
flfi

Anyon super uids on the sphere21⋮k(k + 1)(assumed k 2 ≃ κ ≲ Q)

Beyond EFT: Chern-Simons terms
2• What happens when k ≪ Q ≲ k ?
2• Number of lled LLs ∼ k − ⌊k /Q⌋• Integrating out ψ leaves residual Chern-Simons termk+1

kk−1

2

⌊k /Q⌋
SEFT′ = SEFT −ada4π ∫22• κ ≠ Nvortices ≃ k − ⌊k /Q⌋Q; coe cient ofsubleading term in Δmin(Q) changed from κ toNvortices

ffififl￼• Super uid phonons are gapped;
mphonon ∼ k/ρ ≪ 1/RS2; gap invisible on sphere21⋮

Beyond EFT: rotons?
• Particle-hole excitations set the cuto
scale for EFT (and EFT’) to ρ/k• Can compute at space spectrum

beyond EFT at weak coupling using abosonized Fermi surface approach “rotons”• Rotons on the sphere? What CFT

operators do they correspond to?

ωωc
4

3

2

1

0
0

q pF
2

4

6

8

10

12

14bff

fl

Du, UM, Son ’20

Thank you!