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arXiv:cond-mat/0304392 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Self-consistent equation for an interacting Bose gas

Philippe A. Martin, J. Piasecki

Published Jul 14, 2003• 14 citations
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Abstract

We consider interacting Bose gas in thermal equilibrium assuming a positive and bounded pair potential $V(r)$ such that $0<\ensuremath{\int}d\mathbf{r}V(r)=a<\ensuremath{\infty}.$ Expressing the partition function by the Feynman-Kac functional integral yields a classicallike polymer representation of the quantum gas. With the Mayer graph summation techniques, we demonstrate the existence of a self-consistent relation $\ensuremath{\rho}(\ensuremath{\mu})=F(\ensuremath{\mu}\ensuremath{-}a\ensuremath{\rho}(\ensuremath{\mu}))$ between the density $\ensuremath{\rho}$ and the chemical potential $\ensuremath{\mu},$ valid in the range of convergence of Mayer series. The function F is equal to the sum of all rooted multiply connected graphs. Using Kac's scaling ${V}_{\ensuremath{\gamma}}(\mathbf{r})={\ensuremath{\gamma}}^{3}V(\ensuremath{\gamma}r),$ we prove that in the mean-field limit $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}0,$ only the tree diagrams contribute and function F reduces to the free gas density. We also investigate how to extend the validity of the self-consistent relation beyond the convergence radius of the Mayer series (vicinity of Bose-Einstein condensation), and study the dominant corrections to the mean field. At the lowest order, the form of function F is shown to depend on a single polymer partition function for which we derive the lower and the upper bounds and on the resummation of ring diagrams which can be analytically performed.

DOI
10.1103/physreve.68.016113

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We consider interacting Bose gas in thermal equilibrium assuming a positive and bounded pair potential V(r)V(r)V(r) such that 0<\ensuremath∫drV(r)=a<\ensuremath∞.0<\ensuremath{\int}d\mathbf{r}V(r)=a<\ensuremath{\infty}.0<\ensuremath∫drV(r)=a<\ensuremath∞. Expressing the partition function by the Feynman-Kac functional integral yields a classicallike polymer representation of the quantum gas. With the Mayer graph summation techniques, we demonstrate the existence of a self-consistent relation \ensuremathρ(\ensuremathμ)=F(\ensuremathμ\ensuremath−a\ensuremathρ(\ensuremathμ))\ensuremath{\rho}(\ensuremath{\mu})=F(\ensuremath{\mu}\ensuremath{-}a\ensuremath{\rho}(\ensuremath{\mu}))\ensuremathρ(\ensuremathμ)=F(\ensuremathμ\ensuremath−a\ensuremathρ(\ensuremathμ)) between the density \ensuremathρ\ensuremath{\rho}\ensuremathρ and the chemical potential \ensuremathμ,\ensuremath{\mu},\ensuremathμ, valid in the range of convergence of Mayer series. The function F is equal to the sum of all rooted multiply connected graphs. Using Kac's scaling V\ensuremathγ(r)=\ensuremathγ3V(\ensuremathγr),{V}_{\ensuremath{\gamma}}(\mathbf{r})={\ensuremath{\gamma}}^{3}V(\ensuremath{\gamma}r),V\ensuremathγ​(r)=\ensuremathγ3V(\ensuremathγr), we prove that in the mean-field limit \ensuremathγ\ensuremath→0,\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}0,\ensuremathγ\ensuremath→​0, only the tree diagrams contribute and function F reduces to the free gas density. We also investigate how to extend the validity of the self-consistent relation beyond the convergence radius of the Mayer series (vicinity of Bose-Einstein condensation), and study the dominant corrections to the mean field. At the lowest order, the form of function F is shown to depend on a single polymer partition function for which we derive the lower and the upper bounds and on the resummation of ring diagrams which can be analytically performed.
We prove that the grand canonical Gibbs state of an interacting quantum Bose gas converges to the Gibbs measure of a nonlinear Schrödinger equation in the mean-field limit, where the density of the gas becomes large and the interaction strength is proportional to the inverse density. Our results hold in dimensions d ⩽ 3 d \leqslant 3 . For d > 1 d > 1 the Gibbs measure is supported on distributions of negative regularity and we have to renormalize the interaction. More precisely, we prove the convergence of the relative partition function and of the reduced density matrices in the L r L^r -norm with optimal exponent r r . Moreover, we prove the convergence in the L ∞ L^\infty -norm of Wick-ordered reduced density matrices, which allows us to control correlations of Wick-ordered particle densities as well as the asymptotic distribution of the particle number. Our proof is based on a functional integral representation of the grand canonical Gibbs state, in which convergence to the mean-field limit follows formally from an infinite-dimensional stationary phase argument for ill-defined non-Gaussian measures. We make this argument rigorous by introducing a white-noise-type auxiliary field, through which the functional integral is expressed in terms of propagators of heat equations driven by time-dependent periodic random potentials and can, in turn, be expressed as a gas of interacting Brownian loops and paths. When the gas is confined by an external trapping potential, we control the decay of the reduced density matrices using excursion probabilities of Brownian bridges.
We study a homogeneous Bose gas with purely repulsive forces. Using the Kac scaling of the binary potential we derive analytically the form of the thermodynamic functions of the gas for small but finite values of the scaling parameter in the low density regime. In this way we determine dominant corrections to the mean-field theory. It turns out that repulsive forces increase the pressure at fixed density and decrease the density at given chemical potential (the temperature is kept constant). They also flatten the Bose momentum distribution. However, the present analysis cannot be extended to the region where the mean-field theory predicts the appearance of condensate.
We present the description of thermodynamic properties of a Bose gas by considering the hierarchy of equations satisfied by imaginary-time Green functions. In the case of an ideal gas the appearance of the off-diagonal long-range order (presence of condensate) is derived in a particularly simple way. The simplicity of the analysis persists at the level of the mean-field theory. Using Kac's scaling of the repulsive binary potential we then determine asymptotic corrections to the mean-field theory for thermodynamic states far from the region of the mean-field condensation. Our predictions are in accordance with rigorous results obtained previously, whereas the Hartree-Fock approximation is shown to neglect a whole class of equally important contributions.
In this paper we present the construction of the equilibrium states at positive temperature in the presence of a condensation phase for a Gas of non relativistic Bose particles on an infinite space interacting through a localised two body interaction. We use methods of quantum field theory in the algebraic formulation to obtain this result and in order to prove convergence of the partition function and of the generating function of the correlation functions, we introduce an auxiliary stochastic Gaussian field which mediates the interaction of the Bose particles (Hubbard-Stratonovich transformation). The construction of the equilibrium state and of the partition function in the presence of the condensate, treating the auxiliary stochastic field as external potential, can be achieved using and adapting ideas and methods of Araki. Explicit formulas for the relative entropy of the equilibrium state with the external potential with respect to the equilibrium state of the free theory are obtained adapting known Feynman-Kac formulas for the propagators of the theory. If the two-body interaction is sufficiently weak, the proof of the convergence of the partition function after evaluation of the external stochastic field on a suitable Gaussian state can be given utilizing the properties of the relative entropy mentioned above. Limits where the localisation of the two-body interaction is removed are eventually discussed in combination of the limits of vanishing temperature and or in the weakly interacting regime.
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)(\mathbf{k}=0)(k=0) Hamiltonians and on the analysis of the associated microcanonical ensembles.
Quantum-field-theoretic descriptions of interacting condensed bosons have suffered from the lack of self-consistent approximation schemes satisfying Goldstone's theorem and dynamical conservation laws simultaneously. We present a procedure to construct such approximations systematically by using either an exact relation for the interaction energy or the Hugenholtz-Pines relation to express the thermodynamic potential in a Luttinger-Ward form. Inspection of the self-consistent perturbation expansion up to the third order with respect to the interaction shows that the two relations yield a unique identical result at each order, reproducing the conserving-gapless mean-field theory [T. Kita, J. Phys. Soc. Jpn. 74, 1891 (2005)] as the lowest-order approximation. The uniqueness implies that the series becomes exact when infinite terms are retained. We also derive useful expressions for the entropy and superfluid density in terms of Green's function and a set of real-time dynamical equations to describe thermalization of the condensate.