The connection between nonzero density and spontaneous symmetry breaking for interacting scalars

Abstract

We consider U(1)-symmetric scalar quantum field theories at zero temperature. At nonzero charge densities, the ground state of these systems is usually assumed to be a superfluid phase, in which the global symmetry is spontaneously broken along with Lorentz boosts and time translations. We show that, in d > 2 spacetime dimensions, this expectation is always realized at one loop for arbitrary non-derivative interactions, confirming that the physically distinct phenomena of nonzero charge density and spontaneous symmetry breaking occur simultaneously in these systems. We quantify this result by deriving universal scaling relations for the symmetry breaking scale as a function of the charge density, at low and high density. Moreover, we show that the critical value of μ above which a nonzero density develops coincides with the pole mass in the unbroken, Poincaré invariant vacuum of the theory. The same conclusions hold non-perturbatively for an O(N) theory with quartic interactions in d = 3 and 4, at leading order in the 1/N expansion. We derive these results by computing analytically the zero-temperature, finite-μ one-loop effective potential, paying special attention to subtle points related to the iε terms. We check our results against the one-loop low-energy effective action for the superfluid phonons in λϕ4 theory in d = 4 previously derived by Joyce and ourselves, which we further generalize to arbitrary potential interactions and arbitrary dimensions. As a byproduct, we find analytically the one-loop scaling dimension of the lightest charge-n operator for the λϕ6 conformal superfluid in d = 3, at leading order in 1/n, reproducing a numerical result of Badel et al. For a λϕ4 superfluid in d = 4, we also reproduce the Lee-Huang-Yang relation and compute relativistic corrections to it. Finally, we discuss possible extensions of our results beyond perturbation theory.

Fermions at finite density in the path integral approach

“…Although we will not prove in full generality that the U(1) V symmetry is unbroken at finite density, we will provide evidence that the system can support a finite density phase with unbroken symmetry. This should be contrasted with the result discussed in [1] for the O(N ) vector model, where we showed that, for a system of N scalars, the breaking of the U(1) symmetry is instead inevitable at nonzero charge density, in the large-N limit.…”

“…In the presence of interactions, the fate of the internal U(1) symmetry in a state of finite charge density is less clear. For systems of interacting scalar fields, in [1] we have provided evidence that the ground state at finite chemical potential cannot develop a charge density unless it spontaneously breaks the U(1) symmetry (see also [2,3] for previous related discussions). We have proven this statement in perturbation theory at one loop for generic non-derivative scalar's self-interactions, and in a O(N ) vector model at large N .…”

“…To gain better control of these formal manipulations we would like to explicitly compute the partition function of a system of free fermions at finite µ and zero temperature in the path integral approach. In the case of free bosons at finite µ, the path integral with the iε prescription is convergent only for µ < m [1]. It is convenient to define D µ to be a formal µ-dependent covariant derivative.…”

“…A difficulty that one encounters when trying to address this question in full generality is that of properly accounting for the statistics of the fields [1]. This should enter non-trivially to encode the different properties of bosons and fermions at nonzero charge density.…”

“…In order to make progress in this direction, it seems instructive to reconsider the question about spontaneous symmetry breaking in fermionic QFTs at finite chemical potential and analyze the differences with respect to bosonic systems. In this spirit, our goal in this work is to revisit for fermionic fields some of the aspects that we addressed in [1] for scalar theories. In particular, we will pose the following question: given a fermionic theory at finite charge density, is the U(1) V symmetry realized linearly or spontaneously broken?…”