Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Helicity Modulus, Superfluidity, and Scaling in Isotropic Systems

Michael Ellis Fisher, Michael N. Barber, David Jasnow

Published Aug 1, 1973• 591 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.8.1111

Related papers

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.

Physical review. B, Condensed matter•Dec 1, 1981•1 citation

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.

Physical review. B, Solid state•Sep 1, 1977•42 citations

Critical behavior of the helicity modulus for the classical Heisenberg model

Robert Goodell Brown, Mikael Ciftan

The critical scaling of the helicity modulus of the classical O(3) $3d$ Heisenberg ferromagnet is studied directly. Monte Carlo methods that impose either an antiperiodic boundary condition or a finite twist of definite handedness across otherwise periodic boundaries in one lattice direction are used to measure scale-dependent enthalpy variations in a simple cubic lattice at the ferromagnetic critical temperature. Finite-size scaling is then used to determine the critical exponents ${v}_{E}$ and ${v}_{F}$ for helicity and, by evaluating three independent hyperscaling-linked pairs of $\ensuremath{\nu}$ and $\ensuremath{\alpha}$, to test hyperscaling for this model. It is observed that antiperiodic boundary conditions in particular constrain the lattice to have a nonzero topological charge, establishing a connection between topological charge and helicity in the model.

Physical Review B•Dec 14, 2006•12 citations

Criticality and superfluidity in a dilute Bose fluid

Peter B. Weichman, Mark Rasolt, Michael Ellis Fisher +1 more

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 ${s}^{4}$ 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 ${m}^{\mathrm{*}}$/m\ensuremath{\simeq}1.5, and an effective-scattering-length ratio ${a}^{\mathrm{*}}$/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.

Physical review. B, Condensed matter•Apr 1, 1986•84 citations

Helicity modulus for an ideal Bose fluid

Michael N. Barber

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.

Journal of Physics A Mathematical and General•Aug 1, 1977•7 citations

First-order transitions breaking O( n ) symmetry: Finite-size scaling

Michael Ellis Fisher, Vladimir Privman A

The finite-size rounding of a first-order transition is studied in systems representable as n-vector ferromagnets, so that O(n) symmetry (n\ensuremath{\ge}2) is broken at the bulk transition point. Both ``block,'' V=${L}^{d}$, and ``cylinder,'' ${L}^{d\mathrm{\ensuremath{-}}1}$\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}, geometries are considered for general dimensionality d. Explicit expressions are obtained for the scaling functions describing the rounded transitions and the crossover in shape. Spin-wave effects are shown to be of relative order 1/${L}^{d\mathrm{\ensuremath{-}}2}$, and are calculated in detail in the block case. For n=3 (and d=3) this provides an extension of N\'eel's phenomenological theory of superparamagnetism. The analysis for cylinders involves the formulation of a ``degeneracy kernel'' to describe the asymptotic rounding of first-order transitions and establishes a general relation between the helicity modulus (or ``spin-wave stiffness'' or ``superfluid density'') and the transfer operator spectrum. The relationship to finite-size scaling in the critical region is examined with emphasis on the extra scaling combination, ${\mathrm{Vt}}^{2\mathrm{\ensuremath{-}}\ensuremath{\alpha}}$, that is needed for d>${d}_{>}$=4. All the results found can be checked in the limit n\ensuremath{\rightarrow}\ensuremath{\infty} against exact results for spherical models (described elsewhere).

Physical review. B, Condensed matter•Jul 1, 1985•179 citations
The ordered state of a ddd-dimensional isotropic system with an nnn-vector (n\ensuremath≥2n\ensuremath{\ge}2n\ensuremath≥2) order parameter is considered. By the imposition of suitable boundary conditions it is shown how to define explicitly a helicity modulus \ensuremathΥ(T)\ensuremath{\Upsilon}(T)\ensuremathΥ(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 \ensuremathρs(T)=(m\ensuremathℏ)2\ensuremathΥ(T){\ensuremath{\rho}}_{s}(T)={(\frac{m}{\ensuremath{\hbar}})}^{2}\ensuremath{\Upsilon}(T)\ensuremathρs​(T)=(\ensuremathℏm​)2\ensuremathΥ(T). A critical exponent vvv is defined by \ensuremathΥ(T)\ensuremath∼∣T\ensuremath−Tc∣v\ensuremath{\Upsilon}(T)\ensuremath{\sim}{|T\ensuremath{-}{T}_{c}|}^{v}\ensuremathΥ(T)\ensuremath∼∣T\ensuremath−Tc​∣v as T\ensuremath→TcT\ensuremath{\rightarrow}{T}_{c}T\ensuremath→Tc​; for an ideal Bose gas and spherical model (n\ensuremath→\ensuremath∞n\ensuremath{\rightarrow}\ensuremath{\infty}n\ensuremath→\ensuremath∞), 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\ensuremathβ\ensuremath−\ensuremathη\ensuremathν=2\ensuremath−\ensuremathα\ensuremath−2\ensuremathνv=2\ensuremath{\beta}\ensuremath{-}\ensuremath{\eta}\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}\ensuremath{-}2\ensuremath{\nu}v=2\ensuremathβ\ensuremath−\ensuremathη\ensuremathν=2\ensuremath−\ensuremathα\ensuremath−2\ensuremathν. This reduces to v=(d\ensuremath−2)\ensuremathνv=(d\ensuremath{-}2)\ensuremath{\nu}v=(d\ensuremath−2)\ensuremathν (used by some authors with d=3d=3d=3), only if one accepts ddd-dependent, "hyperscaling" relations such as d\ensuremathν=2\ensuremath−\ensuremathαd\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}d\ensuremathν=2\ensuremath−\ensuremathα; 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−\ensuremathα\ensuremath−2\ensuremathνv=2\ensuremath{-}\ensuremath{\alpha}\ensuremath{-}2\ensuremath{\nu}v=2\ensuremath−\ensuremathα\ensuremath−2\ensuremathν 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.
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.
The critical scaling of the helicity modulus of the classical O(3) 3d3d3d Heisenberg ferromagnet is studied directly. Monte Carlo methods that impose either an antiperiodic boundary condition or a finite twist of definite handedness across otherwise periodic boundaries in one lattice direction are used to measure scale-dependent enthalpy variations in a simple cubic lattice at the ferromagnetic critical temperature. Finite-size scaling is then used to determine the critical exponents vE{v}_{E}vE​ and vF{v}_{F}vF​ for helicity and, by evaluating three independent hyperscaling-linked pairs of $\ensuremath{\nu}$ and $\ensuremath{\alpha}$, to test hyperscaling for this model. It is observed that antiperiodic boundary conditions in particular constrain the lattice to have a nonzero topological charge, establishing a connection between topological charge and helicity in the model.
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.
The finite-size rounding of a first-order transition is studied in systems representable as n-vector ferromagnets, so that O(n) symmetry (n\ensuremath{\ge}2) is broken at the bulk transition point. Both ``block,'' V=Ld{L}^{d}Ld, and ``cylinder,'' ${L}^{d\mathrm{\ensuremath{-}}1}$\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}, geometries are considered for general dimensionality d. Explicit expressions are obtained for the scaling functions describing the rounded transitions and the crossover in shape. Spin-wave effects are shown to be of relative order 1/${L}^{d\mathrm{\ensuremath{-}}2}$, and are calculated in detail in the block case. For n=3 (and d=3) this provides an extension of N\'eel's phenomenological theory of superparamagnetism. The analysis for cylinders involves the formulation of a ``degeneracy kernel'' to describe the asymptotic rounding of first-order transitions and establishes a general relation between the helicity modulus (or ``spin-wave stiffness'' or ``superfluid density'') and the transfer operator spectrum. The relationship to finite-size scaling in the critical region is examined with emphasis on the extra scaling combination, ${\mathrm{Vt}}^{2\mathrm{\ensuremath{-}}\ensuremath{\alpha}}$, that is needed for d>d>{d}_{>}d>​=4. All the results found can be checked in the limit n\ensuremath{\rightarrow}\ensuremath{\infty} against exact results for spherical models (described elsewhere).