Compensating strong coupling with large charge

Abstract

We study some (conformal) field theories with global symmetries in the sector where the value of the global charge Q is large. We find (as expected) that the low energy excitations of this sector are described by the general form of Goldstone's theorem in the non-relativistic regime. We also derive the unexpected result, first presented in [1], that the effective field theory describing such sector of fixed Q contains effective couplings λ eff ∼ λ b /Q a , where λ is the original coupling. Hence, large charge leads to weak coupling. In the last section of the paper we present an outline of how to compute anomalous dimensions of the O(n) model in this limit.

Abelian scalar theory at large global charge

“…where each k stands for an ordinary qft coherent state constructed on top of the condensated vacuum |v . For the purpose of discussing the quantum theory of those momentum fluctuations k, we follow a different path compared to the canonical quantization implemented in [1] or the coset construction invoked in [6]. We use the path integral formulation and realize that the fixed charge physics can be interpreted as the Fourier transform at level Q 0 of the grand-canonical ensemble at finite chemical potential iθ .…”

“…The naturalness condition on the coupling constants reminds us that these potentials are to be thought of as descending from some effective Wilsonian action with generic couplings dictated solely by the symmetries of the problem. In the present context the action associated to (1) sets the initial high scale , from which we srtart to observe our model. Since the behaviour of the Q 0 -leading outcome at each stage of our derivation does not change, we will additionally set λ i = 0 for 1 < i < N, in order to simplify further the calculations.…”

“…In [1] the authors argued that under certain (mild) assumptions a strongly coupled scalar theory can be found effectively at weak-coupling in the sector of fixed and large global charge Q 0 . This was motivated by the previous observation [2] that a perturbative expansion in the example of the three-dimensional O(2) model at the conformal fixed point exhibits 1/Q 0 as a good controlling parameter 1 , even when the original coupling is large.…”

“…Fixing ν ≤ n of the global charges implies then a spontaneous symmetry breaking at a non-trivial vacuum state |v , but also the appearance of Goldstone bosons with generalized dispersion relations (see also the discussion in [3] and [4]). Most interestingly, it was shown that by appropriately U(ν)-rotating in field space, we can always arrange, such that the vacuum expectation value (vev) is 1 In natural units, Q 0 is conveniently dimensionless.…”

“…and are thus identified with the generators of the Lie algebra of su (1,1). A relevant realization [8] of this algebra for the isotropic (an)harmonic oscillator problem is given by…”