On the large charge sector in the critical O(N) model at large N

Abstract

We study operators in the rank-j totally symmetric representation of O(N) in the critical O(N) model in arbitrary dimension d, in the limit of large N and large charge j with j/N ≡ $$ \hat{j} $$ j ̂ fixed. The scaling dimensions of the operators in this limit may be obtained by a semiclassical saddle point calculation. Using the standard Hubbard-Stratonovich description of the critical O(N) model at large N, we solve the relevant saddle point equation and determine the scaling dimensions as a function of d and $$ \hat{j} $$ j ̂ , finding agreement with all existing results in various limits. In 4 < d < 6, we observe that the scaling dimension of the large charge operators becomes complex above a critical value of the ratio j/N, signaling an instability of the theory in that range of d. Finally, we also derive results for the correlation functions involving two “heavy” and one or two “light” operators. In particular, we determine the form of the “heavy-heavy-light” OPE coefficients as a function of the charges and d.

Convexity, large charge and the large-N phase diagram of the φ4 theory

“…This same result has been found in [12] (where Q fl has been identified as the critical value above which ∆(Q) becomes complex), and it is consistent with the conformal dimension of the Q-th power of a field of dimension 3/2 (see also [35]). It would be interesting to understand better the meaning of this expression.…”

“…Here we concentrate on the interval 4 < D < 6 in Euclidean space (see [31,32]). Imposing the convexity of the effective potential, we find that the standard φ 4 model is not ultraviolet (uv) complete if we require unitarity, consistently with recent observations [12,33,34]. We also present a possible completion in terms of a complexified effective potential, describing the flow to a non-unitary conformal field theory (cft) for which we compute the (complex) dimensions of the lowest operators of given charge, finding complete agreement with the existing literature [12,[33][34][35].…”

“…Imposing the convexity of the effective potential, we find that the standard φ 4 model is not ultraviolet (uv) complete if we require unitarity, consistently with recent observations [12,33,34]. We also present a possible completion in terms of a complexified effective potential, describing the flow to a non-unitary conformal field theory (cft) for which we compute the (complex) dimensions of the lowest operators of given charge, finding complete agreement with the existing literature [12,[33][34][35]. This is reminiscent of complex cfts obtained from the merging and annihilation of perturbative fixed points, corresponding to walking renormalization group (rg) flows and weakly first-order phase transitions [19,[36][37][38][39][40][41][42][43][44]; however, in the case at hand, it seems that the mechanism of the generation of complex dimensions is different since it is related to instabilities of the theory.…”

“…This is reminiscent of complex cfts obtained from the merging and annihilation of perturbative fixed points, corresponding to walking renormalization group (rg) flows and weakly first-order phase transitions [19,[36][37][38][39][40][41][42][43][44]; however, in the case at hand, it seems that the mechanism of the generation of complex dimensions is different since it is related to instabilities of the theory. Having access to the phase diagram, we also find a previously unknown metastable massive phase for the model on the cylinder R × S d , much in the spirit of [33], and we show that the cft remains unstable also in the small-charge regime in which the conformal dimensions are real [12,34,35,45,46].…”

“…However, it is nonetheless useful to introduce yet another control parameter in order to extract physical information from this semiclassical description. We choose to work in a sector of large global charge Q here [8], 1 working in the double-scaling limit N → ∞, Q → ∞ with Q/N fixed [10][11][12][13]. 2 In this work (and the companion paper [24]) we use this technology to derive the large-N effective potential without having to resort to Feynman-diagram techniques.…”