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arXiv:cond-mat/0209243

Effective Field Theory for Goldstone Bosons in Nonrelativistic Superfluids

Jens O. Andersen

Published Sep 10, 2002• 6 citations
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Abstract

We consider nonrelativistic superfluids where the global U(1)-symmetry is spontaneously broken. At sufficiently long wavelengths, the relevant degree of freedom is the massless Goldstone mode and we construct an effective low energy theory for the Goldstone boson. The damping rate of collective excitations at low energy is calculated. In the case of a weakly interacting Bose gas, we recover the results by Beliaev, and by Hohenberg and Martin.

DOI
10.48550/arxiv.cond-mat/0209243

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We consider nonrelativistic superfluids where the global U(1)-symmetry is spontaneously broken. At sufficiently long wavelengths, the relevant degree of freedom is the massless Goldstone mode and we construct an effective low energy theory for the Goldstone boson. The damping rate of collective excitations at low energy is calculated. In the case of a weakly interacting Bose gas, we recover the results by Beliaev, and by Hohenberg and Martin.
Beliaev damping in a superfluid is the decay of a collective excitation into two lower-frequency collective excitations; it represents the only decay mode for a bosonic collective excitation in a superfluid at T=0T=0T=0. The standard treatment for this decay assumes a linear spectrum, which in turn implies that the final-state momenta must be collinear to the initial state. We extend this treatment, showing that the inclusion of a gradient term in the Hamiltonian yields a realistic spectrum for the bosonic excitations; we then derive a formula for the decay rate of such excitations and show that even moderate nonlinearities in the spectrum can yield substantial deviations from the standard result. We apply our result to an attractive Fermi gas in the BCS--Bose-Einstein condensate crossover: Here the low-energy bosonic collective excitations are density oscillations driven by the phase of the pairing order field. These collective excitations, which are gapless modes as a consequence of the Goldstone mechanism, have a spectrum that is well established both theoretically and experimentally and whose linewidth, we show, is determined at low temperatures by the Beliaev decay mechanism.
We study collective modes of superfluid Bose gases in optical lattices at commensurate fillings.We focus on the vicinity of the quantum phase transition to the Mott insulator, where there exists the Higgs amplitude mode in addition to the Nambu-Goldstone phase mode associated with the spontaneous U(1) symmetry breaking.We analyze finite-temperature effects on the damping of the collective modes by using an effective spin-1 model and field-theoretical methods based on the finite-temperature Green's function.We calculate the damping rates up to 1-loop order and evaluate them analytically and numerically.We show that the damping rate of the Higgs mode increases with increasing temperature but it remains underdamped up to the typical temperature achieved in experiments.Moreover, we find that the Nambu-Goldstone mode attenuates via a Landau damping process resulting from interactions with the Higgs mode and it can be overdamped at the typical temperature in a certain parameter region..............
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We discuss excitations in nonrelativistic field theories with spontaneous breaking of a continuous global symmetry. It is known that in such systems there are two types of Goldstone bosons (Type A and Type B) whose dispersion law is generically linear or quadratic, respectively. We show that Type B Goldstone bosons may have gapped partners which we call almost-Goldstone bosons. With some nondegeneracy assumption about the low-energy effective action, the total number of Goldstone and almost-Goldstone bosons adds up to the number of broken symmetry generators. We propose that deviations of the dispersion law of Goldstone bosons from linearity at small momenta may serve as a signature of small breaking of time-reversal symmetry.
Abstract It is generally accepted that phonons in a superfluid Bose gas are Goldstone bosons. This is justified by spontaneous symmetry breaking (SSB), which is usually defined as follows: the Hamiltonian of the system is invariant under the U ( 1 ) transformation Ψ ^ ( r , t ) → e i α Ψ ^ ( r , t ) , whereas the order parameter Ψ ( r , t ) is not. However, the strict definition of SSB is different: the Hamiltonian and the boundary conditions are invariant under a symmetry transformation, while the ground state is not. Based on the latter criterion, we study a finite system of spinless, weakly interacting bosons using three approaches: the standard Bogoliubov method, the particle-number-conserving Bogoliubov method, and the approach based on the exact ground-state wave function. Our results show that the answer to the question in the title is ‘no’. Thus, phonons in a real-world (finite) superfluid Bose gas are similar to sound in a classical gas: they are not Goldstone bosons, but quantized collective vibrational modes arising from the interaction between atoms. In the case of an infinite Bose gas, however, the picture becomes paradoxical: the ground state can be regarded as either infinitely degenerate or non-degenerate, making the phonon both similar to a Goldstone boson and different from it.
Inspired by the hunt for new phases of matter in quantum mixed states, it has recently been proposed that the equivalence of microcanonical and canonical ensembles in statistical mechanics is a manifestation of strong-to-weak spontaneous symmetry breaking (SWSSB) in an underlying many-body quantum description. Here, we build an effective field theory for SWSSB of a global U(1) symmetry; the answer exactly reproduces the Schwinger-Keldysh effective field theory of diffusion for the conserved charge. We conclude that hydrodynamics can be understood as a theory of superfluidity for the broken strong symmetry: a nonvanishing susceptibility is a measurable order parameter for SWSSB, the diffusion mode is the Goldstone boson of the spontaneously broken continuous symmetry, and a generalization of Goldstone's theorem implies that the diffusion mode is always long lived. This perspective provides a transparent physical explanation for the unusual reparameterization symmetries, which are a necessary ingredient of Schwinger-Keldysh effective field theories for normal fluids.