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Broken U(1) gauge symmetry and excitations in Bose liquids

W. Vincent Liu

Published Nov 24, 1997• 0 citations
OpenAlex

Abstract

We consider the low-energy collective excitations at finite temperature of Bose--Einstein condensed gases (and liquids as well). A most general model-independent effective Lagrangian is written down according to a prescription obtained from the breakdown of the global symmetry U(1). To show how the theory predicts easily, we derive the momentum and temperature dependence of the damping of excitations by means of power counting as an example.

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We consider the low-energy collective excitations at finite temperature of Bose--Einstein condensed gases (and liquids as well). A most general model-independent effective Lagrangian is written down according to a prescription obtained from the breakdown of the global symmetry U(1). To show how the theory predicts easily, we derive the momentum and temperature dependence of the damping of excitations by means of power counting as an example.
We discuss the collective excitations in a spatially inhomogeneous (cylindrically symmetric) Bose-Einstein condensation (BEC) at low temperature (T≪TλT \ll T_λT≪Tλ​). The main result is the dispersion relation for a (first) sound wave that is obtained by describing the perturbation as a Lagrangian coordinate. The dispersion curve is in good agreement with the Bogoliubov phonon spectrum ω=ckω=c kω=ck, where k=kθ=m/rk=k_θ=m/rk=kθ​=m/r, the wave number and c=[4πaρℏ2]1/2/Mc=[4 πa ρ\hbar^{2}]^{1/2}/Mc=[4πaρℏ2]1/2/M, the speed of first sound. Based on Bohm's quantum theory, a spontaneously broken gauge symmetry in a quantum fluid is discussed in terms of the quantum fluctuation-dissipation, from which it is shown that the symmetry breaking takes place at the free surface of BEC in an external field.
We adopt an effective-field-theory view to describe the low-energy excitations of trapped Bose gases, which allows a direct and systematic way to investigate the consequences of spontaneous symmetry breaking. The derivation of the effective Lagrangian can incorporate various approximations and can reproduce the results obtained by the standard hydrodynamic approach. Based on the effective Lagrangian, we calculate the energy spectrum and Matsubara Green's function of trapped one-dimensional Bose gases with $\ensuremath{\delta}$-function repulsive interaction allowing the comparison of various results obtained by different approaches. We also analytically calculate the finite-temperature correlation function of trapped two-dimensional (2D) Bose gases. The calculation scheme can be easily extended to higher dimensions. We find that particle interactions will always decrease the coherence length of the condensate in 2D, confirming recent numerical results. The validation of various asymptotic expressions are given.
The interplay between spontaneously broken gauge symmetries and Bose-Einstein condensation has long been controversially discussed in science, since the equation of motions are invariant under phase transformations. Within the present model it is illustrated that spontaneous symmetry breaking appears as a non-local process in position space, but within disjoint subspaces of the underlying Hilbert space. Numerical simulations show that it is the symmetry of the relative phase distribution between condensate and non-condensate quantum fields which is spontaneously broken when passing the critical temperature for Bose-Einstein condensation. Since the total number of gas particles remains constant over time, the global U(1)-gauge symmetry of the system is preserved.