The O(N) model at large charge and thequartic/cubic equivalence
Jahmall BersiniRuđer Bošković Insitute (IRB)

Based on: O. Antipin, JB, F. Sannino, Z. W. Wang and C. Zhang, arXiv:2107.02528[hep-th]More on the cubic versus quartic interaction equivalence in the O(N) modelOutline

I will discuss the O(N) CFT in 4<d<6 dimensions and use thelarge-charge expansion to test a proposed dual description ofthis conformal theory.

The quartic O(N) modelConsider the massless quartic O(N) theory in d dimensions

Evidence for an infrared fixed point in 2<d<4.
Just below d=4, the O(N) CFT is weakly coupled:

Superfluid He4, Magnets, Superconductors, ..

The quartic O(N) modelConsider the massless quartic O(N) theory in d dimensions

Above d=4 the quartic interaction becomes RG irrelevant at the Gaussian
fixed point: free field theory in the IR.
In this range of dimensions, the model is usually studied in the 1/Nexpansion.
At large N the model flows to an interacting fixed point in the UV and thetheory is renormalizable order by order in the 1/N expansion.
(G. Parisi 1975)The O(N) model in 4<d<6The scaling dimension at the UV FP of the ΦiΦi operator isand violates the unitarity bound (d/2-1) when d>6.
The UV FP may be unitary in 4<d<6Non-SUSY 5D CFT
AdS/CFT: O(N) model in d dimensions/higher-spin theories on AdSd+1(I. R. Klebanov and A. M. Polyakov 2002)

The O(N) model in 4<d<6
However… there are rigorous results about the triviality of thequartic interaction in d>4.
(M. Aizeman 1981 - O. J. Rosten 2009 - R. Percacci and G. P. Vacca 2014)

The UV FP is metastable
The instability is realized by instantonic effects which give rise to complex
CFT data.
(S. Giombi, R. Huang, I. R. Klebanov, S. Pufu, G. Tarnopolsky 2020)

Instantons are suppressed at large N. When the imaginary part of the CFT
data is negligible the UV FP can be studied via the conformal bootstrap.
(J. B. Bae and S. J. Rey 2014 - S. M. Chester, S. S. Pufu and R. Yacoby 2015)

The cubic O(N) model
Conjecture: in 4<d<6 the UV fixed point of the quartic O(N) model
is equivalent to the IR fixed point of the cubic O(N) model below(L. Fei, S. Giombi, I. R. Klebanov 2014)

N+1 fields and cubic interactions. Weakly coupled just below d=6.
This model is usually investigated in d=6-ε via the ε-expansion.
(5 loops Padé)
The cubic FP exists above(M. Kompaniets and A. Pikelner 2021)

Matching of various scaling dimensions and OPE coefficients
considering the overlapping terms between the 1/N and the ε-expansion.

Fixed-charge operators
We consider the scaling dimension ΔQ of the lowest-lying operators
carrying total charge Q.
These operators have classical scaling dimension Q and transformin the Q-indices traceless symmetric O(N) representations.
.

These operators represent anisotropic perturbations in O(N)invariant systems. ΔQ defines a set of crossover (critical) exponents
measuring the stability of the system (e.g. magnets) against
anisotropic perturbations (e.g. crystal structure).

The large-charge sector
The contribution to ΔQ coming from the exponentiation of thediagrams with the leading charge-scaling at every loop ordermatches between the quartic and the cubic theory.

(G. Arias-Tamargo, D. Rodriguez-Gomez, J. G. Russo 2020)

Computation of ΔQ in the quartic model for arbitrary d in thedouble-scaling limit

F-1 is known.
(S. Giombi and J. Hyman 2020)

Complex scaling dimensions
At large charge the instantons contribution are suppressed as(G. Arias-Tamargo, D. Rodriguez-Gomez, J. G. Russo 2020)is the large-charge sector of the theory stable?

Complex scaling dimensions
At large charge the instantons contribution are suppressed as(G. Arias-Tamargo, D. Rodriguez-Gomez, J. G. Russo 2020)is the large-charge sector of the theory stable?

Above a critical value of the charge, the scaling dimensions
become complex.
(S. Giombi and J. Hyman 2020)

The critical charge
Above a critical value of the charge the scaling dimensions
become complex.
Numerical estimation of the critical charge at the leading orderin the double scaling limit(S. Giombi and J. Hyman 2020)

The cubic model at large
chargeWe consider the cubic model in d=6-ε and the double-scaling limit

We compute Δ-1 and Δ0 and compare with the known large N
results in the quartic theory.

Comparison
Cubic

We compute Δ-1 and Δ0.

Quartic

F-1 is known.
(S. Giombi and J. Hyman 2020)

The overlapping terms between the two expansions are
Match between Δ-1 and F-1

Match between Δ0 and F-1

Semiclassical computation
We use the state-operator correspondence on the cylinder andcompute ΔQ in a semiclassical expansion around the fixed-charge
ground state.

To simplify the computation, we use that ΔQ depends only on thetotal charge and not on the values taken by the individual O(N)charges. Thus we fix only one charge to Q.

The ground state
Introducing N/2 complex field variables as

The classical solution to the EOM reads

whereThe leading order Δ-1
The leading order Δ-1 of the semiclassical expansion is given by
the classical energy on the cylinder and reads

In the “small-charge” regime we obtain

The red terms match between the cubic and the quartic model.

The NLO Δ0

The red terms match between the cubic and the quartic model.
Considering a “large-charge” double-scaling limit it is possible tocompare an infinite series of terms between the two theories.

The large-charge regime

The above matches the quartic model result and illustrates theemergence of complex CFT data in the large-charge limit.

Complex CFT data
We can use our results to study the onset of complex CFTdynamics above a certain critical value of the charge Qc.

We compute Qc at the leading order the ε-expansion for thequartic model in d=4+ε and for the cubic model in d=6-ε.
Above Qc there are no real solutions to the saddle point equations.

Complex CFT data

Complex CFT data
Cubic model in d=6-εComplex CFT data
Cubic model in d=6-εQuartic model in d=4-εComplex CFT data
Cubic model in d=6-εQuartic model in d=4-ε

Numerical estimation in thequartic model at large N(S. Giombi and J. Hyman 2020)