Spinning charged operators in 3dCFTs
Gabriel Cuomo
based on 1711.02108 with A. De La Fuente, A. Monin, D. Pirtskhalava and R. Rattazzi& on a work in progress with Z. Komargodski

Focus Week on Quantum Mechanical Systems at Large Quantum Number 2021 - 31st August 2021

Overview
1. Introduction
2. The conformal superfluid EFT for large charge operators
3. Charged spinning operators and rotating superfluids in 3d CFT
4. Summary

Introduction

Universality of CFT data at large quantum numbers• Large spin J
Alday Maldacena 2007, Komargodski Zhiboedov 2012,
Fitzpatrick Kaplan Poland Simmons-Duffin 2012, Caron-Huot 2017,…

• Large internal charge Q
Hellerman Orlando Reffert Watanabe 2015, Monin Pirtskhalava Rattazzi Seibold 2016,…

• Large scaling dimension Δ
Lashkari Dymarsky Liu 2016, Cardy Maloney Maxfield 2017, Delacretaz 2020,…

Why universality at large quantum numbers?
State-operator map

Why universality at large quantum numbers?
State-operator map

Ex.: Alday-Maldacena “EFT’’ for large spin double-trace operators• To leading order, free moving particles

Alday Maldacena 2007• Corrections due to “Yukawa” potential propagating at distanceΔχ ∼ log J in AdS3 × S

d−3ℝ×Sd−1

𝒪Large charge expansion in CFT• d-dimensional CFT with internal symmetry G• {QI}: global Cartan charges of G (+ spin)• Lowest dimension operator at given charges Q :{QI}(x)• For | QI | ≫ 1 compute semiclassically

Hellerman Orlando Reffert Watanabe 2015, Monin Pirtskhalava Rattazzi Seibold 2016

The conformal superfluid EFTfor large charge operators

Large charge stateConsider a CFT w. U(1) symmetry. Properties of | Q⟩:• charge density j0 ∼ Q/Rd−1∝μd−1• scale separation μ ≫ 1/R suggests EFT description• condensed matter phase: nonlinearly realizes spacetime symmetries
low energy hydrodynamic modes are Goldstone bosons
Nicolis Penco Piazza Rattazzi 2015

The conformal superfluid
Summary: SO(d + 1,1) × U(1) ⟶ SO(d) × D̄ with D̄ = D + μQ

• 1 Goldstone mode χ(x) = μt + π(x)1d−1Q“radial”
modes:
genericallygappedatμ∼•R(But moduli in SCFTs: Hellerman Maeda Orlando Reffert Watanabe 2017-2021)

Can write ℒ (χ) systematically in a derivative expansion ∂/μ ∝ E/Q

1d−1

Simple prediction for the ground state energy:

Hellerman Orlando Reffert Watanabe 2015, Monin 2016, GC 2020

Sound speed of the phonon fluctuations fixed by conformal invariance:

Non-trivial informations about the spectrum

Hellerman Orlando Reffert Watanabe 2015

Charged spinning operators in 3dCFTs and rotating superfluids

Adding spin to the superfluid
A natural question is what happens to the large charge ground state energy as weincrease the spin, e.g., in the 3d O(2) model. We may try to look at phonon states:
1 phonon with spin=Jn phonons with spin=J/n

… but experiments shows that spinning superfluids develop vortices!

Bose-Einstein condensates in a magnetic trap develop an increasing number of vortices
as the angular velocity is increasedVortex EFT and particle-vortex duality
It is convenient to introduce a dual gauge field:

The U(1) current provides the explicit relation:

Cutoff: Λ ∼Q /R ∼

B . B = monopole field.

Vortices=charged particles

Horn Nicolis Penco 2015

• Effective vortex mass: mp = γp B ∼ Q• Physically: Landau levels (LLs) separated by ωL = B/mp ∼ cutoff• Integrate out all LLs but the first

EFT for the lowest LL• In practice treat mass term as higher derivative term (leading single derivative
kinetic term from the monopole connection)

Vortices=charged particles

Horn Nicolis Penco 2015

• Effective vortex mass: mp = γp B ∼ Q• Physically: Landau levels (LLs) separated by ωL = B/mp ∼ cutoff• Integrate out all LLs but the first

EFT for the lowest LL• In practice treat mass term as higher derivative term (leading single derivative
kinetic term from the monopole connection)

Classical analysis: electrostatics on the sphere
To leading order, a simple electrostatic problem:

Gauss law implies ∑ qp = 0.
p

GC de la Fuente Monin Pirtskhalava Rattazzi 2017

Classical analysis: electrostatics on the sphere
To leading order, a simple electrostatic problem:

Gauss law implies ∑ qp = 0.
p

GC de la Fuente Monin Pirtskhalava Rattazzi 2017

Results: vortex-antivortex pair

Quantization: fuzzy sphere
Quantization from SU(2) algebra:•i
Rp are non-commuting coordinates• Quantum corrected energy for a vortex-antivortex pair

Results: vortex distribution
• J≥Q

⟹

nv > 1• J≫Q⟹

nv ≫ 1: approximate by smooth distribution ρ

Minimize Δ at fixed J :Constant angular velocity (~ rigid body).

Results: vortex distribution
• J≥Q

⟹

nv > 1• J≫Q⟹

nv ≫ 1: approximate by smooth distribution ρ

Minimize Δ at fixed J :Constant angular velocity (~ rigid body). Can we go beyond this regime?

Let’s look again at experiments…

When the angular velocity exceeds the speed of sound in BECs the vortex lattice
becomes unstable towards the formation of a coherent “giant” vortex annulus.
Theory: Fischer Baym 2003, Fetter Jackson Stringari 2005
Experiment: Guo Dubessy de Herve Kumar Badr Perrin Longchambon Perrin 2019
Non-technical review: Sophia Chen - Physics 2020

A giant vortex in the O(2) model
For J/Q ∈ ℤ a natural candidate for the “giant vortex” profile isGC Komargodski in progressA giant vortex in the O(2) model
For J/Q ∈ ℤ a natural candidate for the “giant vortex” profile isFor J ≫ Q

3/2:

GC Komargodski in progressThree physically distinct regions• Away from the equator the centrifugal potential
2ℓV(θ) ∼gaps all excitations .
2sin θ

Three physically distinct regions• Away from the equator the centrifugal potential
2ℓV(θ) ∼gaps all excitations .
2sin θ

• The charge is localized around the equator,
where the derivative expansion is controlled by
322the density: (∂χ) ≫ ∂ (∂χ) ⟹ Q ≫ J .

Three physically distinct regions• Away from the equator the centrifugal potential
2ℓV(θ) ∼gaps all excitations .
2sin θ• The charge is localized around the equator,
where the derivative expansion is controlled by
322the density: (∂χ) ≫ ∂ (∂χ) ⟹ Q ≫ J .
222 2• Near sin θ ∼ ℓ /R μ the charge density rapidly

decreases and the gap of radial modes is controlled by
2 1/3
1/3
1/6
the steepness of the profile [∂θ(∂χ) ] ∼ J /Q .

Technically similar to rotating effective strings and NRCFTs in harmonic trap.
Hellerman Swanson 2013, Hellerman Swanson 2020

𝒪Results: the giant vortex energy
For Q

3/2

2

≪ J ≪ Q we can compute the energy of the ground state:• In the limit J ≫ Q

3/2

the result approaches the expectation for ϕ∂

J/Q

ϕ…∂

J/Qϕ

• The (Q /J )32 ncorrections arise from the LO action and depend on the same
3/2
parameter α controlling the homogeneous superfluid energy ΔJ=0 = αQ + …

• The parameter βb parametrizes energy corrections from the edge of the profile andit is reproduced in the EFT by a boundary operatorb ∼ δ ((∂χ) ) [∂θ(∂χ)]
25/3

The spectrum of fluctuations
3Q
2The quadratic action becomes tractable expanding in δ =:226π cJ

2• The y coordinate is related to the original one as cos θ = δ 2 − δ sin yπ±y=Regularity
implies
at∂
π
=0y
•
2

• To this order EOMs depend only on (∂t + ∂ϕ) and ∂y:32

In the limit Q /J → 0 the spectrum of fluctuations is then given by• Expected spectrum of a (free) multi-trace!
−223• Expansion holds for m, n ≪ δ ∼ J /Q ≪ Q3• Corrections in Q /J

2

lift the apparent degeneracy, e.g. the gap of the next-tolowest dimensional state with the same spin J of the giant vortex is:Summary of results

SummaryThe lowest dimensional operator at fixed Q ≫ 1 and J in the O(2) model corresponds to• 0 ≤ J ≪ Q : homogeneous superfluid +1 phonon•Q ≪ J ≤ Q : vortex-antivortex pair

• Q≪J≪Q
• Q

3/2
2

3/2

: regular vortex distribution2≪ J ≪ Q : giant vortex state• Q ≪ J : Alday-Maldacena multi-traceSome open questions• Giant-vortex for J/Q ∉ ℕ ?
• Supperadiant transition for J ∼ Q
23/2

at weak coupling• Can we approach J → Q from the Alday-Maldacena description systematically?
• What about 4d?

THANK YOU!