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Journal of Applied Physics

Scaling Laws for Dynamics Near the Critical Point

Pierre C. Hohenberg, Bertrand I. Halperin

Published Mar 1, 1970• 4 citations
Publisher

Abstract

The static scaling laws may be generalized to dynamic phenomena by making assumptions on the behavior of time-dependent correlation functions near the magnetic critical point. At any temperature different from Tc, the correlation functions are assumed to reflect the ``hydrodynamic'' behavior of the system, for sufficiently long wavelengths and low frequencies. As the critical temperature is approached, however, the range ξ of spatial correlations in the system diverges, and the domain of applicability of hydrodynamics is reduced to very small wavenumbers, satisfying the inequality kξ≪1. The dynamic scaling assumptions relate the behavior of the time-dependent correlation functions in the ``hydrodynamic'' (kξ≪1) and ``critical'' (kξ≫1) regions. It is thus possible to predict certain features of the dynamic behavior at Tc, from a knowledge of the behavior away from Tc. Application is made to the spin-correlation function of the isotropic Heisenberg antiferromagnet, which is directly measurable by inelastic neutron scattering. Detailed predictions are presented on the form of the cross section as a function of energy and momentum transfer, and of temperature near Tc.

DOI
10.1063/1.1658953

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Journal of Physics A Mathematical and General•Dec 1, 1983•18 citations
The static scaling laws may be generalized to dynamic phenomena by making assumptions on the behavior of time-dependent correlation functions near the magnetic critical point. At any temperature different from Tc, the correlation functions are assumed to reflect the hydrodynamic'' behavior of the system, for sufficiently long wavelengths and low frequencies. As the critical temperature is approached, however, the range ξ of spatial correlations in the system diverges, and the domain of applicability of hydrodynamics is reduced to very small wavenumbers, satisfying the inequality kξ≪1. The dynamic scaling assumptions relate the behavior of the time-dependent correlation functions in the hydrodynamic'' (kξ≪1) and ``critical'' (kξ≫1) regions. It is thus possible to predict certain features of the dynamic behavior at Tc, from a knowledge of the behavior away from Tc. Application is made to the spin-correlation function of the isotropic Heisenberg antiferromagnet, which is directly measurable by inelastic neutron scattering. Detailed predictions are presented on the form of the cross section as a function of energy and momentum transfer, and of temperature near Tc.
The usual static scaling laws are generalized to nonequilibrium phenomena by making assumptions on the behavior of time-dependent correlation functions near the critical point of second-order phase transitions. At any temperature different from Tc{T}_{c}Tc​, the correlation functions are assumed to reflect the hydrodynamic behavior of the system, for sufficiently long wavelengths and low frequencies. As the critical temperature is approached, however, the range of spatial correlations in the system diverges, and the domain of applicability of hydrodynamics is reduced to a vanishingly small region of wavelengths and frequencies. The dynamic-scaling assumptions lead to predictions for the behavior of the hydrodynamic parameters near Tc{T}_{c}Tc​, as well as for the form of the correlation functions for macroscopic distances and times, outside the hydrodynamic range. In particular, singularities are predicted to occur in the temperature dependence of transport coefficients, and anomalies are expected in the frequency spectrum of certain operators, which are observable by inelastic scattering of neutrons or light. A distinction is made between the restricted dynamic-scaling hypothesis, which refers to the order parameter only, and extended dynamic scaling, which applies to other operators and involves stronger assumptions. Applications are discussed to antiferromagnets, ferromagnets, the gas-liquid critical point, and the $\ensuremath{\lambda}$ transition in superfluid helium. Specific experiments are suggested to test the scaling assumptions, and existing experimental evidence is briefly reviewed. Finally, a comparison is made with other theories of dynamical behavior near critical points.
The Widom-Kadanoff scaling laws are generalized to dynamic phenomena, by making assumptions on the structure of time-dependent correlation functions near Tc{T}_{c}Tc​. The theory is applied to isotropic Heisenberg antiferromagnet and predictions are made, which can be tested by inelastic neutron-scattering and nmr measurements. The ferromagnet is also briefly discussed.
"Critical point phenomena in magnetic systems are studied with the aid of the scaling laws. These laws relate the critical indices which describe the behavior of thermodynamic and correlation functions near T. Kadanoff's derivation of the scaling 1aw is reviewed. An experimental survey of ferromagnetic and antiferromagnetic data near T is presented. Susceptibility, correlation length, magnectization, specific heat and a magnetic equation of state are discussed. It is found that the ""critical region"" can extend as far as [equation] although frequently it is restricted to E < 10-2 The critical indices S, y, v, ~ and ~, are close to the three dimensional ISing model values. Values for 0, y' and ~ are not known as conclusively. Specific heat behavior is discussed in terms -~ of a three-parameter fit, namely C = aE + b. Varying b allows one to estimate a possible range of a's. The experimental information is consistent with the scaling law relations 2 - alpha = 2 - alpha' = dV = d gamma / (2-pi) = gamma + 2B = B(d+1) The scaling law theory is extended to predict the temperature, field and spatial dependence of various spin correlation functions. These predictions are checked by analytic calculations for the two dimensional Ising model in zero field. Results are presented for the energy density-energy density and energy density-spin correlation functions in the limit E « 1, R » 1, but the product ER arbitrary. Results are also presented for a group of two, three and four spin correlations at cT. In each case the scaling law predictions are verified."
A single assumption is made about the dynamic correlation function, that close to the critical point and for large r and t, it is a generalized homogeneous function. After proving and using certain simple theorems on this class of functions, contact is made with previous formulations of dynamic and static scaling. It is found that 1. this single assumption is sufficient to obtain all previous results of both the dynamic and static scaling hypotheses, 2. simple expressions, which can be written down ‘by inspection’ for all critical point exponents, enable all the exponent relations to be derived immediately, 3. we have additional ways of plotting data in the form of ‘scaling functions’, 4. it becomes possible to easily and accurately calculate the scaling function for models (such as the Heisenberg model) for which no low‐temperature series expansions exist, and 5. the two‐exponent relations (e. g., dν=2−α) are not consequences of our approach unless we make an additional assumption, an assumption made implicitlyby many other workers. If we don't make this additional assumption, the two‐exponent relations are found to have correction terms which transform them into ordinary thermodynamic scaling laws.
We calculate the dynamic structure factor $S(\mathbit{k},\ensuremath{\omega})$ in the paramagnetic regime of quantum Heisenberg ferromagnets for temperatures TTT close to the critical temperature Tc{T}_{c}Tc​ using our recently developed functional renormalization group approach to quantum spin systems. In d=3d=3d=3 dimensions, we find that for small momenta $\mathbit{k}$ and frequencies $\ensuremath{\omega}$ the dynamic structure factor assumes the scaling form $S(\mathbit{k},\ensuremath{\omega})=(\ensuremath{\tau}TG(\mathbit{k})/\ensuremath{\pi})\mathrm{\ensuremath{\Phi}}(k\ensuremath{\xi},\ensuremath{\omega}\ensuremath{\tau})$, where $G(\mathbit{k})$ is the static spin-spin correlation function, $\ensuremath{\xi}$ is the correlation length, and the characteristic timescale $\ensuremath{\tau}$ is proportional to ${\ensuremath{\xi}}^{5/2}$. We explicitly calculate the dynamic scaling function $\mathrm{\ensuremath{\Phi}}(x,y)$ and find satisfactory agreement with neutron scattering experiments probing the critical spin dynamics in EuO and EuS. Precisely at the critical point where $\ensuremath{\xi}=\ensuremath{\infty}$, our result for the dynamic structure factor can be written as $S(\mathbit{k},\ensuremath{\omega})={(\ensuremath{\pi}{\ensuremath{\omega}}_{k})}^{\ensuremath{-}1}{T}_{c}G(\mathbit{k}){\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\omega}/{\ensuremath{\omega}}_{k})$, where ${\ensuremath{\omega}}_{k}\ensuremath{\propto}{k}^{5/2}$. We find that ${\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\nu})$ vanishes as ${\ensuremath{\nu}}^{\ensuremath{-}13/5}$ for large $\ensuremath{\nu}$, and as ${\ensuremath{\nu}}^{3/5}$ for small $\ensuremath{\nu}$. While the large-frequency behavior of ${\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\nu})$ is consistent with calculations based on mode-coupling theory and with perturbative renormalization group calculations to second order in $\ensuremath{\epsilon}=6\ensuremath{-}d$, our result for small frequencies disagrees with previous calculations. We argue that until now, neither experiments nor numerical simulations have been sufficiently accurate to determine the low-frequency behavior of ${\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\nu})$. We also calculate the low-temperature behavior of $S(\mathbit{k},\ensuremath{\omega})$ in one- and two-dimensional ferromagnets and find that it satisfies dynamic scaling with exponent z=2z=2z=2 and exhibits a pseudogap for small frequencies.
Critical spin wave dynamics in the dilute Heisenberg chain near the percolation threshold is treated by two complementary approaches. The first exploits the renormalisation group transformation of parameters under a length scaling achieved by decimation on the equations of motion of the random system, the second obtains from the contributions from chain segments of all possible sizes the average dynamic response using a continuum approach valid for small wavevector k and long percolation correlation length xi p. Both approaches yield identical dynamic scaling forms for the dynamic response, with dynamic exponent z=2, and details of the crossover of characteristic frequency between hydrodynamic and critical form as k xi p varies from 0 to infinity . A detailed expression for the scaling function for the dynamic response is also obtained.