The epsilon expansion meets semiclassics

Abstract

We study the scaling dimension ∆ φ n of the operator φ n where φ is the fundamental complex field of the U (1) model at the Wilson-Fisher fixed point in d = 4 − ε. Even for a perturbatively small fixed point coupling λ * , standard perturbation theory breaks down for sufficiently large λ * n. Treating λ * n as fixed for small λ * we show that ∆ φ n can be successfully computed through a semiclassical expansion around a non-trivial trajectory, resulting in We explicitly compute the first two orders in the expansion, ∆ −1 (λ * n) and ∆ 0 (λ * n). The result, when expanded at small λ * n, perfectly agrees with all available diagrammatic computations. The asymptotic at large λ * n reproduces instead the systematic large charge expansion, recently derived in CFT. Comparison with Monte Carlo simulations in d = 3 is compatible with the obvious limitations of taking ε = 1, but encouraging.

Correlation functions in scalar field theory at large charge

“…In the strong coupling regime -which overlaps with the regime of validity of the large charge effective theory-, one finds ∆ ∼λ 4−ǫ 3−ǫ + · · · , thus recovering the expected scaling ∆ ∼ n d d−1 , with the extra bonus of providing an analytic expression for the actual coefficients in the large charge expansion [5,6,7].…”

“…Alternatively, the double-scaling limit can be understood from a saddle-point calculation. This can be easily generalized to the general correlation function (5). We first introduce the scaled scalar field…”

“…This approach in fact uncovers a rich structure, as the sector of operators at fixed charge n is described by an effective theory depending on both n and g (recall that at the WF fixed point, g ∼ ǫ), in such a way that, depending on how the large charge limit is taken, a different behavior emerges. Some aspects of the large n expansion were recently investigated in [4,5,6,7], where the two-point function of the operators φ n ,φ n was computed at large n. In particular, a general way to organize the expansion was discussed in [5,6,7] (generalizing earlier work in [8,9,10]). The effective description of the large n sector naturally depends on n and g n =λ, in such a way that a 't Hooft-like double expansion emerges.…”

“…On the other hand, at weak coupling and in the large n limit, the dominant term is ∆ −1 (λ), which must admit a perturbative expansion for smallλ. This gives [4,5,6]…”

Feynman diagrams and the large charge expansion in 3 − ε dimensions

“…While the effective superfluid description should equally well apply to strongly and weakly coupled theories, in the latter case it is also possible to work directly in the full theory, bypassing the EFT construction, or, in fact, deriving it. This was recently illustrated in [13], by focusing on the two-point function of the charge n operator φ n in the U (1) invariant Wilson-Fisher fixed point in 4 − ε dimensions. For arbitrary n, the scaling dimension ∆ φ n was computed semiclassically by expanding the path integral around a non-trivial trajectory.…”