Journal of Physics A Mathematical and General

New logarithmic term in the superfluid density scaling in confined geometries

Vladimir Privman A

Published Aug 7, 1990• 8 citations

Abstract

A new logarithmic scaling term is identified in the finite-size form of the helicity modulus, equivalent to the superfluid density fraction, in three dimensions. Implications of the pattern of the scaling behaviour found in the analyses of experimental data are discussed.

DOI: 10.1088/0305-4470/23/15/003

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Helicity modulus for an ideal Bose fluid

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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.

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Journal of Physics A Mathematical and General

New logarithmic term in the superfluid density scaling in confined geometries

Vladimir Privman A

Published Aug 7, 1990• 8 citations

Publisher

Abstract

DOI: 10.1088/0305-4470/23/15/003

Related papers

Monte Carlo studies of the finite three-dimensionalXY-model helicity modulus

K. K. Mon

A new logarithmic term in the finite-size helicity modulus has been predicted recently for the three-dimensional classical XY-model universality class under nonperiodic boundary conditions. These theoretical ideas have important implications for the analysis of superfluid-density measurements in confined geometries. We present numerical results from Monte Carlo simulations of the helicity modulus for the three-dimensional classical XY model on a finite simple-cubic lattice, with pinned-surface-spin boundary conditions. Our data are consistent with the presence of an additional logarithmic term and an estimate for the amplitude is given.A new logarithmic term in the finite-size helicity modulus has been predicted recently for the three-dimensional classical XY-model universality class under nonperiodic boundary conditions. These theoretical ideas have important implications for the analysis of superfluid-density measurements in confined geometries. We present numerical results from Monte Carlo simulations of the helicity modulus for the three-dimensional classical XY model on a finite simple-cubic lattice, with pinned-surface-spin boundary conditions. Our data are consistent with the presence of an additional logarithmic term and an estimate for the amplitude is given.

Physical review. B, Condensed matterAug 1, 19912 citations

Renormalization-group calculation of superfluid density and helicity modulus

Miguel Calvo

The field-theoretical renormalization-group techniques previously developed for the description of the mestastable phases of isotropic, homogeneous, $d$ -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 field-theoretical renormalization-group techniques previously developed for the description of the mestastable phases of isotropic, homogeneous, $d$-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.

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Michael Ellis Fisher, Michael N. Barber, David Jasnow

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.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.

Physical Review AAug 1, 1973591 citations

Helicity modulus for an ideal Bose fluid

Michael N. Barber

Journal of Physics A Mathematical and GeneralAug 1, 19777 citations

Renormalization-group proof of Josephson's relation and ε expansion for the helicity modulus (superfluid density)

Joseph Rudnick, David Jasnow

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 $n$ -vector model with continuous spins; when $d<4$ 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.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 $n$-vector model with continuous spins; when $d<4$ 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.

Physical review. B, Solid stateSep 1, 197742 citations

The field-theoretical renormalization-group techniques previously developed for the description of the mestastable phases of isotropic, homogeneous,

d

-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 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.

n

-vector model with continuous spins; when

d<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.