The usual symmetry-breaking procedures for Bose condensed systems, namely, the Bogoliubov prescription, the symmetry-breaking term added to the Hamiltonian, and the canonical shift transformation are unified into a single formalism. By taking into account the condensate reservoir as a source and sink of excited particles, exact Ward identities are solved in the shielded-potential approximation. A relationship between the condensate density
n0 and the superfluid density
nS is obtained in closed form. It is shown that the Bogoliubov prescription yields
n0\ensuremath{\simeq}
nS and
nU0\ensuremath{\ll}\ensuremath{\Vert}\ensuremath{\mu}\ensuremath{\Vert}, where n is the total density,
U0 the interaction constant, and \ensuremath{\mu} the chemical potential. On the other hand, for the canonical shift transformation one has
n0\ensuremath{\ll}
nS and
nU0\ensuremath{\gg}\ensuremath{\Vert}\ensuremath{\mu}\ensuremath{\Vert}. The latter, applied to superfluid
4He at saturated vapor pressure, gives excellent agreement between theory and experiment, without any adjustable parameter. The condensate density turns out to be strongly dependent on pressure as observed experimentally. The formalism provides in a natural way a consistent description of Bose systems in arbitrary D-dimensional space.
We investigate an ultracold and dilute Bose gas by taking into account a finite-range two-body interaction. The coupling constants of the resulting Lagrangian density are related to measurable scattering parameters by following the effective-field-theory approach. A perturbative scheme is then developed up to the Gaussian level, where both quantum and thermal fluctuations are crucially affected by finite-range corrections. In particular, the relation between spontaneous symmetry breaking and the onset of superfluidity is emphasized by recovering the renowned Landau’s equation for the superfluid density in terms of the condensate one.
A renormalizable parameter \ensuremath{\xi} (0\ensuremath{\le}\ensuremath{\xi}\ensuremath{\le}1) is introduced into the interacting Bose-system Hamiltonian such that the zero-mode amplitudes satisfy the commutation relation [
b0,${b}_{0}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$]=1-\ensuremath{\xi}. The different phases of the system correspond to specific values of \ensuremath{\xi}. The normal component (phase I) is obtained when \ensuremath{\xi}=0 and there are two ordered states according to the strength of the interaction: Phase II' (0<\ensuremath{\xi}\ensuremath{\ll}1) is found in the strongly interacting system and is characterized by a broken gauge symmetry without Bose-Einstein condensation, while phase II'' (\ensuremath{\xi}=1), in the weakly interacting system, displays Bose-Einstein condensation. The continuity equation for the particle density holds in both ordered phases to within an infinitesimal term. A gauge transformation on the field operators shows that \ensuremath{\xi} ''twists'' the phase angle related to the zero mode, and a response to this twisting field is defined in terms of the generalized force \ensuremath{\chi}(T) associated with \ensuremath{\xi}. For the ideal Bose gas \ensuremath{\chi}(T) equals the condensate density and hence is simply related to the helicity modulus introduced by Fisher et al. A renormalization-group transformation shows that the dense system transition (I-II') gives rise to a two-thirds correlation-length critical exponent, whereas a unity crossover exponent appears in the dilute system transition (I-II''), in agreement with the experimental findings.
Adding a gauge symmetry breaking field $\ensuremath{-}\ensuremath{\nu}\sqrt{V}({a}_{0}+{a}_{0}^{*})$ to the Hamiltonian of some simplified models of an interacting Bose gas we compute the condensate density and the symmetry breaking order parameter in the limit of infinite volume and prove Bogoliubov's asymptotic hypothesis ${\mathrm{lim}}_{V\ensuremath{\rightarrow}\ensuremath{\infty}}⟨{a}_{0}⟩∕\sqrt{V}=\mathrm{sgn}\phantom{\rule{0.2em}{0ex}}\ensuremath{\nu}\phantom{\rule{0.2em}{0ex}}{\mathrm{lim}}_{V\ensuremath{\rightarrow}\ensuremath{\infty}}\sqrt{⟨{a}_{0}^{*}{a}_{0}⟩∕V}$ where the averages are taken in the ground state or in thermal equilibrium states. Letting $\ensuremath{\nu}$ tend to zero in this equation we obtain that Bose-Einstein condensation occurs if and only if the gauge symmetry is spontaneously broken. The simplification consists in dropping the off-diagonal terms in the momentum representation of the pair interaction. The models include the mean field and the imperfect (Huang-Yang-Luttinger) Bose gas. An implication of the result is that the compressibility sum rule cannot hold true in the ground state of the one-dimensional mean-field Bose gas. Our method is based on a resolution of the Hamiltonian into a family of single-mode
(k=0) Hamiltonians and on the analysis of the associated microcanonical ensembles.
We derive several important features of a Bose-condensed system, based on a diagrammatic analysis at finite temperatures. Firstly, the symmetry-breaking term necessary to describe Bose condensation is shown to give a rigorous basis for the fictitious gap argument of Gavoret and Nozi\`eres (1964). This puts their well-known results on a firm footing. Secondly, in the presence of a moving condensate with velocity
vS, the current carried by the noncondensate atoms is shown to be $({\ensuremath{\rho}}_{S}\ensuremath{-}{\mathrm{mn}}_{o}){v}_{S}$. The results of this paper are direct consequences of various Ward identities which have been recently discussed for Bose-condensed systems.
We derive Ward-Takahashi identities for correlated Bose-Einstein condensates based on the expressions of the first-order variations
(δΨ,δG) due to perturbations obtained in the preceding paper [T. Kita, J. Phys. Soc. Jpn.
90, 024001 (2021)] for the condensate wave function
Ψ and Green's function
G. They enable us to obtain several exact results on the density and current correlation functions
Kνν′, and also express
Kνν′ in terms of low-energy Green's functions and vertices. The latter expressions open up the possibility of constructing theory of superfluid Bose liquids in the same way as that for fermions at low temperatures. The vertices are found to have different limits depending on which of frequency
ω and wavenumber
q is set equal to zero first.