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Journal of Physics A Mathematical and General

Helicity modulus for an ideal Bose fluid

Michael N. Barber

Published Aug 1, 1977• 7 citations
Publisher

Abstract

The concept of the helicity modulus Y d (T), introduced by Fisher, Barber and Jasnow (see Phys. Rev., vol.A8, p.1111 (1973)), is applied to the ideal Bose gas in d dimensions. Above the critical temperature T c,d , Y is found to vanish identically, while for T<T c,d , Y d (T)=(h(cross) 2 rho /m)(1-(T/T c,d ) 1 2d/), where rho is the total density. The relevance of these results to more general theories of superfluidity is discussed.

DOI
10.1088/0305-4470/10/8/012

Related papers

Helicity Modulus, Superfluidity, and Scaling in Isotropic Systems

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The ordered state of a $d$-dimensional isotropic system with an $n$-vector ($n\ensuremath{\ge}2$) order parameter is considered. By the imposition of suitable boundary conditions it is shown how to define explicitly a helicity modulus $\ensuremath{\Upsilon}(T)$ which measures the free-energy increment associated with "twisting" the direction of the order parameter. For a Bose system the superfluid density is seen to be ${\ensuremath{\rho}}_{s}(T)={(\frac{m}{\ensuremath{\hbar}})}^{2}\ensuremath{\Upsilon}(T)$. A critical exponent $v$ is defined by $\ensuremath{\Upsilon}(T)\ensuremath{\sim}{|T\ensuremath{-}{T}_{c}|}^{v}$ as $T\ensuremath{\rightarrow}{T}_{c}$; for an ideal Bose gas and spherical model ($n\ensuremath{\rightarrow}\ensuremath{\infty}$), $v=1$ is an exact result for all $d>2$. The difficulties of defining a correlation length in the ordered phase are discussed. A full scaling theory of the correlations avoids these problems and may be linked to a phenomenological hydrodynamic approach, to clarify and rederive Josephson's relation $v=2\ensuremath{\beta}\ensuremath{-}\ensuremath{\eta}\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}\ensuremath{-}2\ensuremath{\nu}$. This reduces to $v=(d\ensuremath{-}2)\ensuremath{\nu}$ (used by some authors with $d=3$), only if one accepts $d$-dependent, "hyperscaling" relations such as $d\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}$; however, both these latter relations fail for the ideal Bose gas when $d>4$. An alternative derivation of the formula $v=2\ensuremath{-}\ensuremath{\alpha}\ensuremath{-}2\ensuremath{\nu}$ is based on the scaling theory for systems with a large but finite dimension.

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The critical properties of the phase transition from a normal gas to a BEC (superfluid) of a harmonically confined Bose gas are analyzed using a proposed model for the equation of state of the underlying homogenous Bose fluid. The equation of state is given in terms of the homogenous particle density as a function of the chemical potential and temperature of the gas. On the normal side of the fluid it is assumed to be an ideal Bose gas while in the superfluid state we propose a function that, both, asymptotically reaches the Thomas-Fermi solution of a weakly interacting Bose gas at large densities and low temperatures and, at the transition, matches the critical properties of the ideal Bose gas. With this model we obtain the {\it global} thermodynamics of the harmonically confined gas which shows that, at the critical transition of the confined gas, the thermodynamic susceptibilities, such as the isothermal compressibility, do not diverge, in contrast to the behavior of the uniform gas. The singularity of the critical transition in the confined system appears now at the derivatives of the susceptibilities. We discuss how these properties can be experimentally tested.

•Jul 8, 2016•0 citations
The concept of the helicity modulus Y d (T), introduced by Fisher, Barber and Jasnow (see Phys. Rev., vol.A8, p.1111 (1973)), is applied to the ideal Bose gas in d dimensions. Above the critical temperature T c,d , Y is found to vanish identically, while for T<T c,d , Y d (T)=(h(cross) 2 rho /m)(1-(T/T c,d ) 1 2d/), where rho is the total density. The relevance of these results to more general theories of superfluidity is discussed.
The ordered state of a ddd-dimensional isotropic system with an nnn-vector ($n\ensuremath{\ge}2$) order parameter is considered. By the imposition of suitable boundary conditions it is shown how to define explicitly a helicity modulus $\ensuremath{\Upsilon}(T)$ which measures the free-energy increment associated with "twisting" the direction of the order parameter. For a Bose system the superfluid density is seen to be ${\ensuremath{\rho}}_{s}(T)={(\frac{m}{\ensuremath{\hbar}})}^{2}\ensuremath{\Upsilon}(T)$. A critical exponent vvv is defined by $\ensuremath{\Upsilon}(T)\ensuremath{\sim}{|T\ensuremath{-}{T}_{c}|}^{v}$ as $T\ensuremath{\rightarrow}{T}_{c}$; for an ideal Bose gas and spherical model ($n\ensuremath{\rightarrow}\ensuremath{\infty}$), v=1v=1v=1 is an exact result for all d>2d>2d>2. The difficulties of defining a correlation length in the ordered phase are discussed. A full scaling theory of the correlations avoids these problems and may be linked to a phenomenological hydrodynamic approach, to clarify and rederive Josephson's relation $v=2\ensuremath{\beta}\ensuremath{-}\ensuremath{\eta}\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}\ensuremath{-}2\ensuremath{\nu}$. This reduces to $v=(d\ensuremath{-}2)\ensuremath{\nu}$ (used by some authors with d=3d=3d=3), only if one accepts ddd-dependent, "hyperscaling" relations such as $d\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}$; however, both these latter relations fail for the ideal Bose gas when d>4d>4d>4. An alternative derivation of the formula $v=2\ensuremath{-}\ensuremath{\alpha}\ensuremath{-}2\ensuremath{\nu}$ is based on the scaling theory for systems with a large but finite dimension.
The field-theoretical renormalization-group techniques previously developed for the description of the mestastable phases of isotropic, homogeneous, ddd-dimensional systems, which (close to their critical temperature) can be described in terms of the Landau-Ginzburg-Wilson functional integral with a two-component order parameter, and applied to obtain the scaling form of the helicity modulus, defined as a measure of the free-energy increment associated with a metastable, helical twisting of the order parameter. The general scaling form of this quantity is determined even in the presence of an external helical field, for arbitrary helix pitch and temperature. In the case of vanishing external field, this quantity represents the superfluid density and is a function of the temperature and the superfluid velocity (which corresponds to the inverse of the helix pitch). In the limit of small velocities this expression exhibits the well-known Josephson asymptotic form ${\ensuremath{\rho}}_{s}\ensuremath{\sim}{({T}_{c}\ensuremath{-}T)}^{\ensuremath{\nu}(d\ensuremath{-}2)}={({T}_{c}\ensuremath{-}T)}^{2\ensuremath{\beta}\ensuremath{-}\ensuremath{\nu}\ensuremath{\eta}}$. An explicit $\ensuremath{\epsilon}=4\ensuremath{-}d$ expansion for the helicity modulus is presented in the case of arbitrary external field to the one-loop approximation.
The crossover from ideal-Bose-gas behavior to nonclassical, XY-like criticality in a dilute, interacting Bose fluid is considered in detail with emphasis on the superfluid density, ${\ensuremath{\rho}}_{s}$(T), below the transition. A general discussion of crossover scaling in constrained systems shows that ${\ensuremath{\rho}}_{s}$ and other temperature-dependent thermodynamic functions at controlled overall density can be expressed in terms of universal scaling functions. The finite-temperature Green's function formalism is used to calculate the associated linear scaling fields and the crossover exponent exactly for all dimensionalities d>2, scaling predictions, including logarithmic singularities just at d=3, being checked via second-order computations. A mapping of the Bose problem onto the classical s4{s}^{4}s4 spin model is achieved which is asymptotically exact in the critical region. In this way, previously known renormalization-group results are used to calculate the scaling functions for ${\ensuremath{\rho}}_{s}$, for the off-diagonal susceptibility, etc., to first order in \ensuremath{\epsilon}=4-d. Good agreement is found with the experimental measurements by Reppy and co-workers of ${\ensuremath{\rho}}_{s}$(T) for helium absorbed at low coverages in Vycor glass (of pore size 50--80 A\r{}). An effective-mass ratio m∗{m}^{\mathrm{*}}m∗/m\ensuremath{\simeq}1.5, and an effective-scattering-length ratio a∗{a}^{\mathrm{*}}a∗/a\ensuremath{\simeq}1.3 are indicated for mobile helium in the Vycor (although the irregular character of Vycor has not, at this point, been fully allowed for in the theory). Other approaches in the literature to computing the critical behavior of quantum-mechanical systems are reviewed and related to the present analysis.
We show that Josephson's relation, $\ensuremath{\upsilon}=2\ensuremath{\beta}\ensuremath{-}\ensuremath{\eta}\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}\ensuremath{-}2\ensuremath{\nu}$, connecting the critical exponent $\ensuremath{\upsilon}$, describing the vanishing of the helicity modulus (superfluid density), to other thermodynamic and correlation exponents is an exact scaling law within the renormalization-group framework. We use the standard isotropic nnn-vector model with continuous spins; when d<4d<4d<4 hyperscaling presumably holds, and we have $\ensuremath{\upsilon}=(d\ensuremath{-}2)\ensuremath{\nu}$. Secondly, we derive the first two leading terms of the helicity modulus in an $\ensuremath{\epsilon}=4\ensuremath{-}d$ expansion; this allows second-order evaluation of universal amplitude ratios associated with the superfluid transition.
Following the approach of Barber and Fisher, we formulate a finite-size scaling theory for the Bose condensate. Using bulk results as input, we make a number of predictions for the behaviour of the condensate fraction f0(L, T) in an ideal Bose system confined to a hypercube, of side L, in d dimensions. A comparison is made with analytical results for a system in three dimensions under a variety of boundary conditions. While the standard temperature variable t[= (T – Tc)/Tc] is appropriate in the case of periodic and antiperiodic boundary conditions, the use of a shifted variable t[= t – ε(L), where ε(L) = O(L−1 In L)] is essential in the case of Neumann and Dirichlet boundary conditions. Nonetheless, in each case, the predictions of the scaling formulation are fully borne out. Finally, the formulation is extended (i) to include the so-called surface condensate, and (ii) to cover all temperature down to 0 K.
The critical properties of the phase transition from a normal gas to a BEC (superfluid) of a harmonically confined Bose gas are analyzed using a proposed model for the equation of state of the underlying homogenous Bose fluid. The equation of state is given in terms of the homogenous particle density as a function of the chemical potential and temperature of the gas. On the normal side of the fluid it is assumed to be an ideal Bose gas while in the superfluid state we propose a function that, both, asymptotically reaches the Thomas-Fermi solution of a weakly interacting Bose gas at large densities and low temperatures and, at the transition, matches the critical properties of the ideal Bose gas. With this model we obtain the {\it global} thermodynamics of the harmonically confined gas which shows that, at the critical transition of the confined gas, the thermodynamic susceptibilities, such as the isothermal compressibility, do not diverge, in contrast to the behavior of the uniform gas. The singularity of the critical transition in the confined system appears now at the derivatives of the susceptibilities. We discuss how these properties can be experimentally tested.