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. B, Solid state

Two-point correlations near four dimensions

David R. Nelson

Published Aug 1, 1976• 39 citations
Publisher

Abstract

Renormalization-group recursion relations are used to calculate to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ the two-point correlation function near a critical point above ${T}_{c}$ and in zero field. This approach gives results which are valid for all values of the ratio of wave number to inverse correlation length.

DOI
10.1103/physrevb.14.1123

Related papers

Crossover scaling functions for two-point correlations near tricritical points

W. K. Theumann

The expectations of phenomenological crossover scaling theory are worked out for two-point correlation functions in zero field above the primary critical point. Renormalization-group recursion relations are then used to construct the two-point correlation function for the changeover from tricritical to critical-line behavior, above the tricritical point, to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$, for $3\ensuremath{\le}d<4$. An explicit expression is obtained for the normalized double-scaling function ${\stackrel{^}{\ensuremath{\Gamma}}}_{N}({q}^{2}{\ensuremath{\xi}}^{2}, \frac{g}{{t}^{\ensuremath{\phi}}})$, for the general scaling variable ${q}^{2}{\ensuremath{\xi}}^{2}$, and the crossover variable $w=\frac{g}{{t}^{\ensuremath{\phi}}}$, $q\ensuremath{\ll}{a}^{\ensuremath{-}1}$ being the wave vector; $\ensuremath{\xi}\ensuremath{\gg}a$ is a second-moment correlation length and $a$ the lattice spacing. In the tricritical regime, $w\ensuremath{\simeq}0$, ${\stackrel{^}{\ensuremath{\Gamma}}}_{N}$ (${x}^{2}, w$) has the Ornstein-Zernicke form and the changeover to the critical regime $w\ensuremath{\simeq}\ensuremath{\infty}$ is explicitly discussed. The limitation of the results, within our calculations to first order in $\ensuremath{\epsilon}$, are pointed out.

Physical review. B, Condensed matter•Dec 1, 1978•4 citations

Two-point correlation function near the coexistence surface

Jeffrey F. Nicoll

Differential renormalization-group equations for the one-particle-irreducible vertex functions are solved analytically to provide information about the scaling properties of the correlation function at small and moderate wave vectors everywhere in the critical region. For Ising-like spin systems, this extends previous work on the anomalous dimension crossover function to nonzero magnetization. The susceptibility-correlation-length relative-amplitude ratio is correctly given to $O(\ensuremath{\epsilon})$, $\ensuremath{\epsilon}=4\ensuremath{-}d$; moreover, a fully nonlinear crossover-amplitude function is obtained. For general $n$-component spins, the longitudinal part of the correlation function, ${\ensuremath{\Gamma}}_{2,L}$ is calculated to $O(\ensuremath{\epsilon})$ in crossover form for small and moderate $k$. The asymptotic singularity ${\ensuremath{\Gamma}}_{2,L}\ensuremath{\sim}{k}^{\ensuremath{\epsilon}}$ is automatically incorporated. The result is exact in the spherical limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$.

Physical review. B, Condensed matter•Feb 1, 1980•9 citations

Scaling function for two-point correlations. I. Expansion near four dimensions

Michael Ellis Fisher, Amnon Aharony A

A detailed calculation to order ${\ensuremath{\epsilon}}^{2}$ of the two-point correlation function in the zero-field critical region ($T\ensuremath{\ge}{T}_{c}$) is presented for an $n$-vector system in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions. Scaling behavior is verified over the full parameter range and the scaling function is obtained as a universal explicit convergent integral (independent of any cutoffs). Study of the scaling function as $T\ensuremath{\rightarrow}{T}_{c}$ in the finite ($q\ensuremath{\ne}0$) momentum limit, confirms the presence of terms varying as ${t}^{1\ensuremath{-}\ensuremath{\alpha}}$ and as $t$ [with $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$]; explicit evaluation of their amplitudes reliably determines the form of the scattering intensity at fixed $q$ and locates the corresponding maximum accurately.

Physical review. B, Solid state•Oct 1, 1974•117 citations

Equations of state and renormalization-group recursion relations

Joseph Rudnick, David R. Nelson

Renormalization-group recursion relations are used to construct equations of state to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ for continuous spin models of critical behavior. The recursion relations map Hamiltonians out of the critical regime into regions of small correlation length, where Landau theory with fluctuation corrections is employed. This technique is used to discuss critical behavior along a $\ensuremath{\lambda}$ line of critical points, together with the effect of imposing an ordering field. An application of the method to tricritical phenomena is described briefly.

Physical review. B, Solid state•Mar 1, 1976•264 citations

Dynamics near the critical point: The hot renormalization group in quantum field theory

D. Boyanovsky, Hector J. de Vega

The perturbative approach to the description of long-wavelength excitations at high temperature breaks down near the critical point of a second order phase transition. We study the dynamics of these excitations in a relativistic scalar field theory at and near the critical point via a renormalization group approach at high temperature and an $\ensuremath{\epsilon}$ expansion in $d=5\ensuremath{-}\ensuremath{\epsilon}$ space-time dimensions. The long-wavelength physics is determined by a nontrivial fixed point of the renormalization group. At the critical point we find that the dispersion relation and width of quasiparticles of momentum p are ${\ensuremath{\omega}}_{p}\ensuremath{\sim}{p}^{z}$ and ${\ensuremath{\Gamma}}_{p}\ensuremath{\sim}(z\ensuremath{-}1){\ensuremath{\omega}}_{p},$ respectively, and the group velocity of quasiparticles ${v}_{g}\ensuremath{\sim}{p}^{z\ensuremath{-}1}$ vanishes in the long-wavelength limit at the critical point. Away from the critical point for $T\ensuremath{\gtrsim}{T}_{c}$ we find ${\ensuremath{\omega}}_{p}\ensuremath{\sim}{\ensuremath{\xi}}^{\ensuremath{-}z}[1+(p\ensuremath{\xi}{)}^{2z}{]}^{1/2}$ and ${\ensuremath{\Gamma}}_{p}\ensuremath{\sim}(z\ensuremath{-}1){\ensuremath{\omega}}_{p}(p\ensuremath{\xi}{)}^{2z}/[1+(p\ensuremath{\xi}{)}^{2z}]$ with $\ensuremath{\xi}$ the finite temperature correlation length $\ensuremath{\xi}\ensuremath{\propto}|T\ensuremath{-}{T}_{c}{|}^{\ensuremath{-}\ensuremath{\nu}}.$ The new dynamical exponent z results from anisotropic renormalization in the spatial and time directions. For a theory with $O(N)$ symmetry we find $z=1+\ensuremath{\epsilon}{(N+2)/(N+8)}^{2}+\mathcal{O}({\ensuremath{\epsilon}}^{2}).$ This dynamical critical exponent describes a new universality class for dynamical critical phenomena in quantum field theory. Critical slowing down, i.e., a vanishing width in the long-wavelength limit, and the validity of the quasiparticle picture emerge naturally from this analysis.

arXiv:hep-ph/0110012•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Apr 9, 2002•12 citations

Calculation of the Critical Exponent η via Renormalization-Group Recursion Formulas

Geoffrey R. Golner

This paper presents an extension of Wilson's renormalization-group calculation of Ising-model critical exponents to include calculation of the critical exponent $\ensuremath{\eta}$. New recursion formulas are derived using the simplest set of consistent approximations which allow a nonzero $\ensuremath{\eta}$. They are intended to demonstrate, qualitatively, how nonzero values for $\ensuremath{\eta}$ are consistent with the renormalization-group approach; they do not represent systematic, quantitative improvements to Wilson's earlier calculation of the exponents $\ensuremath{\nu}$ and $\ensuremath{\gamma}$. The equations are solved both by $\ensuremath{\epsilon}$ expansion about four dimensions and by numerical integration in three dimensions. To order ${\ensuremath{\epsilon}}^{2}$ we obtain $\ensuremath{\eta}=0.05{\ensuremath{\epsilon}}^{2}$. Numerical results in three dimension are $\ensuremath{\eta}=0.058$, $\ensuremath{\nu}=0.588$, and $\ensuremath{\gamma}=1.14$. The relation $\ensuremath{\gamma}=(2\ensuremath{-}\ensuremath{\eta})\ensuremath{\nu}$ is confirmed.

Physical review. B, Solid state•Jul 1, 1973•36 citations
Renormalization-group recursion relations are used to calculate to first order in \ensuremathϵ=4\ensuremath−d\ensuremath{\epsilon}=4\ensuremath{-}d\ensuremathϵ=4\ensuremath−d the two-point correlation function near a critical point above Tc{T}_{c}Tc​ and in zero field. This approach gives results which are valid for all values of the ratio of wave number to inverse correlation length.
The expectations of phenomenological crossover scaling theory are worked out for two-point correlation functions in zero field above the primary critical point. Renormalization-group recursion relations are then used to construct the two-point correlation function for the changeover from tricritical to critical-line behavior, above the tricritical point, to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$, for $3\ensuremath{\le}d<4$. An explicit expression is obtained for the normalized double-scaling function ${\stackrel{^}{\ensuremath{\Gamma}}}_{N}({q}^{2}{\ensuremath{\xi}}^{2}, \frac{g}{{t}^{\ensuremath{\phi}}})$, for the general scaling variable ${q}^{2}{\ensuremath{\xi}}^{2}$, and the crossover variable $w=\frac{g}{{t}^{\ensuremath{\phi}}}$, $q\ensuremath{\ll}{a}^{\ensuremath{-}1}$ being the wave vector; $\ensuremath{\xi}\ensuremath{\gg}a$ is a second-moment correlation length and aaa the lattice spacing. In the tricritical regime, $w\ensuremath{\simeq}0$, ${\stackrel{^}{\ensuremath{\Gamma}}}_{N}$ (x2,w{x}^{2}, wx2,w) has the Ornstein-Zernicke form and the changeover to the critical regime $w\ensuremath{\simeq}\ensuremath{\infty}$ is explicitly discussed. The limitation of the results, within our calculations to first order in $\ensuremath{\epsilon}$, are pointed out.
Differential renormalization-group equations for the one-particle-irreducible vertex functions are solved analytically to provide information about the scaling properties of the correlation function at small and moderate wave vectors everywhere in the critical region. For Ising-like spin systems, this extends previous work on the anomalous dimension crossover function to nonzero magnetization. The susceptibility-correlation-length relative-amplitude ratio is correctly given to $O(\ensuremath{\epsilon})$, $\ensuremath{\epsilon}=4\ensuremath{-}d$; moreover, a fully nonlinear crossover-amplitude function is obtained. For general nnn-component spins, the longitudinal part of the correlation function, ${\ensuremath{\Gamma}}_{2,L}$ is calculated to $O(\ensuremath{\epsilon})$ in crossover form for small and moderate kkk. The asymptotic singularity ${\ensuremath{\Gamma}}_{2,L}\ensuremath{\sim}{k}^{\ensuremath{\epsilon}}$ is automatically incorporated. The result is exact in the spherical limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$.
A detailed calculation to order ${\ensuremath{\epsilon}}^{2}$ of the two-point correlation function in the zero-field critical region ($T\ensuremath{\ge}{T}_{c}$) is presented for an nnn-vector system in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions. Scaling behavior is verified over the full parameter range and the scaling function is obtained as a universal explicit convergent integral (independent of any cutoffs). Study of the scaling function as $T\ensuremath{\rightarrow}{T}_{c}$ in the finite ($q\ensuremath{\ne}0$) momentum limit, confirms the presence of terms varying as ${t}^{1\ensuremath{-}\ensuremath{\alpha}}$ and as ttt [with $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$]; explicit evaluation of their amplitudes reliably determines the form of the scattering intensity at fixed qqq and locates the corresponding maximum accurately.
Renormalization-group recursion relations are used to construct equations of state to first order in $\ensuremath{\epsilon}=4\ensuremath{-}d$ for continuous spin models of critical behavior. The recursion relations map Hamiltonians out of the critical regime into regions of small correlation length, where Landau theory with fluctuation corrections is employed. This technique is used to discuss critical behavior along a $\ensuremath{\lambda}$ line of critical points, together with the effect of imposing an ordering field. An application of the method to tricritical phenomena is described briefly.
The perturbative approach to the description of long-wavelength excitations at high temperature breaks down near the critical point of a second order phase transition. We study the dynamics of these excitations in a relativistic scalar field theory at and near the critical point via a renormalization group approach at high temperature and an $\ensuremath{\epsilon}$ expansion in $d=5\ensuremath{-}\ensuremath{\epsilon}$ space-time dimensions. The long-wavelength physics is determined by a nontrivial fixed point of the renormalization group. At the critical point we find that the dispersion relation and width of quasiparticles of momentum p are ${\ensuremath{\omega}}_{p}\ensuremath{\sim}{p}^{z}$ and ${\ensuremath{\Gamma}}_{p}\ensuremath{\sim}(z\ensuremath{-}1){\ensuremath{\omega}}_{p},$ respectively, and the group velocity of quasiparticles ${v}_{g}\ensuremath{\sim}{p}^{z\ensuremath{-}1}$ vanishes in the long-wavelength limit at the critical point. Away from the critical point for $T\ensuremath{\gtrsim}{T}_{c}$ we find ${\ensuremath{\omega}}_{p}\ensuremath{\sim}{\ensuremath{\xi}}^{\ensuremath{-}z}[1+(p\ensuremath{\xi}{)}^{2z}{]}^{1/2}$ and ${\ensuremath{\Gamma}}_{p}\ensuremath{\sim}(z\ensuremath{-}1){\ensuremath{\omega}}_{p}(p\ensuremath{\xi}{)}^{2z}/[1+(p\ensuremath{\xi}{)}^{2z}]$ with $\ensuremath{\xi}$ the finite temperature correlation length $\ensuremath{\xi}\ensuremath{\propto}|T\ensuremath{-}{T}_{c}{|}^{\ensuremath{-}\ensuremath{\nu}}.$ The new dynamical exponent z results from anisotropic renormalization in the spatial and time directions. For a theory with O(N)O(N)O(N) symmetry we find $z=1+\ensuremath{\epsilon}{(N+2)/(N+8)}^{2}+\mathcal{O}({\ensuremath{\epsilon}}^{2}).$ This dynamical critical exponent describes a new universality class for dynamical critical phenomena in quantum field theory. Critical slowing down, i.e., a vanishing width in the long-wavelength limit, and the validity of the quasiparticle picture emerge naturally from this analysis.
This paper presents an extension of Wilson's renormalization-group calculation of Ising-model critical exponents to include calculation of the critical exponent $\ensuremath{\eta}$. New recursion formulas are derived using the simplest set of consistent approximations which allow a nonzero $\ensuremath{\eta}$. They are intended to demonstrate, qualitatively, how nonzero values for $\ensuremath{\eta}$ are consistent with the renormalization-group approach; they do not represent systematic, quantitative improvements to Wilson's earlier calculation of the exponents $\ensuremath{\nu}$ and $\ensuremath{\gamma}$. The equations are solved both by $\ensuremath{\epsilon}$ expansion about four dimensions and by numerical integration in three dimensions. To order ${\ensuremath{\epsilon}}^{2}$ we obtain $\ensuremath{\eta}=0.05{\ensuremath{\epsilon}}^{2}$. Numerical results in three dimension are $\ensuremath{\eta}=0.058$, $\ensuremath{\nu}=0.588$, and $\ensuremath{\gamma}=1.14$. The relation $\ensuremath{\gamma}=(2\ensuremath{-}\ensuremath{\eta})\ensuremath{\nu}$ is confirmed.