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

Time-Dependent Correlations in a Solvable Ferromagnetic Model

N. David Mermin

Published Apr 6, 1964• 20 citations
Publisher

Abstract

The exact equation of state, two-time correlation function, and linear response function, are calculated in the limit of infinite $N$ for a classical $N$-spin system with a ferromagnetic phase transition, in a particular nonuniform magnetic field. The correlation function can be analytically continued in temperature (or magneticfield strength) from the nonferromagnetic to the ferromagnetic region of the $T\ensuremath{-}H$ plane; the result of such continuation is not, however, the correlation function for the ferromagnetic region, but a function which grows exponentially in time. The frequency-dependent linear response function has a pole at zero frequency throughout the ferromagnetic region due to a broken symmetry; the corresponding function in the nonferromagnetic region developes a pole at zero frequency as the ferromagnetic region is approached, but when the function is continued in temperature (or field strength) into the ferromagnetic region, the pole detaches itself from the origin and moves up into the complex frequency plane, signifying an exponential growth in time of the linear response. The purpose of the model is to demonstrate that this kind of behavior does not contradict any general structural properties of equilibrium thermodynamic correlation or response functions. The possible general significance of such behavior for a theory of metastable states is discussed.

DOI
10.1103/physrev.134.a112

Related papers

Correlation function of random-field Ising chains: is it Lorentzian or not?

J. M. Luck, Theo M. Nieuwenhuizen

The two-point connected correlation function, or wavevector-dependent susceptibility, of the ferromagnetic Ising chain in a random field is calculated exactly at any temperature, for a two-parameter family of diluted symmetric exponential distributions of the magnetic fields. Thermodynamic properties of this model have been derived in a previous work by the authors. Besides the correlation function itself, the solution provides exact results for the (usual) susceptibility, the correlation length and the Edwards-Anderson parameter. The low-temperature regime is examined in full detail: the authors obtain in closed form the limit value at zero temperature of various quantities, and the first correction to this limit, which behaves linearly with temperature. The correlation length is discontinuous at zero temperature. They also derive the scaling form of the correlation function for small p, where the dilution p is the probability for a spin to be subjected to a non-zero random field. Even in this limit, the correlation function is more complicated than the simple Lorentzian predicted, e.g., by mean-field theory.

Journal of Physics A Mathematical and General•Jun 21, 1989•18 citations

Nonlinear-response theory of Ising-like systems

James H. Luscombe

The nonequilibrium response of a two-dimensional, paramagnetic system to a rapid and arbitrarily large change (quench) in a symmetry-breaking field is investigated. The complete time dependence of the conjugate order parameter is calculated with a real-space renormalization-group (RG) method. The nonlinear relaxation "rate" is found to be time dependent. Its approach to the corresponding prediction of linear-response theory is characterized. The nonlinear response is found to go over to linear response via an early, exponential regime followed by a crossover to an asymptotic, algebraic approach to linear response at long times. The initial transient decay is quench-strength dependent in contrast with the final behavior, which is largely "universal" and quench insensitive. We study a nearest-neighbor Ising model in a magnetic field in the paramagnetic, "disordered" phase. The dynamics are controlled by a kinetic Ising model with simultaneous spin-flip (adsorption-desorption) and spin-exchange (diffusion) kinetics. The model is applicable to non-equilibrium "dosing" experiments on adsorbed systems. Diffusion is shown not to affect the growth rate of the total order parameter (magnetization or coverage). General recursion relations for the temperature and magnetic field are derived for the entire thermodynamic plane. The RG analysis reproduces linear-response theory for arbitrary magnetic field and is consistent with known scaling laws concerning nonlinear relaxation. The time-dependent magnetization in the critical region has a power-law regime before crossing over into its final exponential decay.

Physical review. B, Condensed matter•May 1, 1984•13 citations

A theory of the time-dependent properties of Heisenberg spin chains at infinite temperature

Stephen William Lovesey, Ewald Balcar

A coupled-mode theory for the spin response function is used to calculate time-dependent properties of the classical spin chain at infinite temperature. The behaviour of the spin autocorrelation function is consistent with computer simulation data, in that the exponent of the time tail decreases on moving from intermediate to asymptotic time regions. Results for the spin response function and memory function in the two time regions are obtained by a combination of asymptotic analysis and numerical studies. For a large wavevector, near the zone boundary, the spin response function displays a very pronounced oscillatory time dependence. However, there is next to no structure in the corresponding spectral function, in accord with simulation data. The asymptotic analysis shows that, in the infinite-temperature limit, coupled-mode theory does not comply with usual hydrodynamic assumptions, a view that has also emerged from analysis of some of the computer simulation data.

Journal of Physics Condensed Matter•Feb 7, 1994•16 citations

Infinite-range Ising ferromagnet in a time-dependent transverse magnetic field: Quench and ac dynamics near the quantum critical point

Arnab Das, K. Sengupta, Diptiman Sen +1 more

We study an infinite-range ferromagnetic Ising model in the presence of a transverse magnetic field, which exhibits a quantum paramagnetic-ferromagnetic phase transition at a critical value of the transverse field. In the thermodynamic limit, the low-temperature properties of this model are dominated by the behavior of a single large classical spin governed by an anisotropic Hamiltonian. Using this property, we study the quench and ac dynamics of the model both numerically and analytically, and develop a correspondence between the classical phase-space dynamics of a single spin and the quantum dynamics of the infinite-range ferromagnetic Ising model. In particular, we compare the behavior of the equal-time order-parameter correlation function both near to and away from the quantum critical point in the presence of a quench or ac transverse field. We explicitly demonstrate that a clear signature of the quantum critical point can be obtained by studying the ac dynamics of the system even in the classical limit. We discuss possible realizations of our model in experimental systems.

arXiv:cond-mat/0606137•Physical Review B•Oct 24, 2006•115 citations

Statistical Mechanics of Nearest Neighbor Systems II. General Theory and Application to Two-Dimensional Ferromagnets

Elliott W. Montroll

This is a continuation of an effort to reduce to the solution of a characteristic value problem the rigorous calculation of thermodynamic properties of systems in which the intermolecular forces are sufficiently short ranged so that practically the entire potential energy of the system results from interactions between nearest neighbors. The partition function of such a system can be expressed in terms of the largest characteristic value of a linear operator equation and finally as the ratio of partition functions of systems with relatively few particles whose potential energy functions differ somewhat from those in the original system. A method of evaluating grand partition functions and one of introducing interactions between more distant neighbors are discussed. The general theory is applied to the calculation of magnetization, internal energy, and specific heat of two-dimensional ferromagnets on the basis of the Ising model. There seems to exist a λ-point phase transition in the change from ferromagnetic to nonferromagnetic states.

The Journal of Chemical Physics•Jan 1, 1942•40 citations

Scaling laws and correlation functions near magnetic critical points

Robert Hecht

"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."

•Jan 1, 1967•0 citations
The exact equation of state, two-time correlation function, and linear response function, are calculated in the limit of infinite NNN for a classical NNN-spin system with a ferromagnetic phase transition, in a particular nonuniform magnetic field. The correlation function can be analytically continued in temperature (or magneticfield strength) from the nonferromagnetic to the ferromagnetic region of the T\ensuremath−HT\ensuremath{-}HT\ensuremath−H plane; the result of such continuation is not, however, the correlation function for the ferromagnetic region, but a function which grows exponentially in time. The frequency-dependent linear response function has a pole at zero frequency throughout the ferromagnetic region due to a broken symmetry; the corresponding function in the nonferromagnetic region developes a pole at zero frequency as the ferromagnetic region is approached, but when the function is continued in temperature (or field strength) into the ferromagnetic region, the pole detaches itself from the origin and moves up into the complex frequency plane, signifying an exponential growth in time of the linear response. The purpose of the model is to demonstrate that this kind of behavior does not contradict any general structural properties of equilibrium thermodynamic correlation or response functions. The possible general significance of such behavior for a theory of metastable states is discussed.
The two-point connected correlation function, or wavevector-dependent susceptibility, of the ferromagnetic Ising chain in a random field is calculated exactly at any temperature, for a two-parameter family of diluted symmetric exponential distributions of the magnetic fields. Thermodynamic properties of this model have been derived in a previous work by the authors. Besides the correlation function itself, the solution provides exact results for the (usual) susceptibility, the correlation length and the Edwards-Anderson parameter. The low-temperature regime is examined in full detail: the authors obtain in closed form the limit value at zero temperature of various quantities, and the first correction to this limit, which behaves linearly with temperature. The correlation length is discontinuous at zero temperature. They also derive the scaling form of the correlation function for small p, where the dilution p is the probability for a spin to be subjected to a non-zero random field. Even in this limit, the correlation function is more complicated than the simple Lorentzian predicted, e.g., by mean-field theory.
The nonequilibrium response of a two-dimensional, paramagnetic system to a rapid and arbitrarily large change (quench) in a symmetry-breaking field is investigated. The complete time dependence of the conjugate order parameter is calculated with a real-space renormalization-group (RG) method. The nonlinear relaxation "rate" is found to be time dependent. Its approach to the corresponding prediction of linear-response theory is characterized. The nonlinear response is found to go over to linear response via an early, exponential regime followed by a crossover to an asymptotic, algebraic approach to linear response at long times. The initial transient decay is quench-strength dependent in contrast with the final behavior, which is largely "universal" and quench insensitive. We study a nearest-neighbor Ising model in a magnetic field in the paramagnetic, "disordered" phase. The dynamics are controlled by a kinetic Ising model with simultaneous spin-flip (adsorption-desorption) and spin-exchange (diffusion) kinetics. The model is applicable to non-equilibrium "dosing" experiments on adsorbed systems. Diffusion is shown not to affect the growth rate of the total order parameter (magnetization or coverage). General recursion relations for the temperature and magnetic field are derived for the entire thermodynamic plane. The RG analysis reproduces linear-response theory for arbitrary magnetic field and is consistent with known scaling laws concerning nonlinear relaxation. The time-dependent magnetization in the critical region has a power-law regime before crossing over into its final exponential decay.
A coupled-mode theory for the spin response function is used to calculate time-dependent properties of the classical spin chain at infinite temperature. The behaviour of the spin autocorrelation function is consistent with computer simulation data, in that the exponent of the time tail decreases on moving from intermediate to asymptotic time regions. Results for the spin response function and memory function in the two time regions are obtained by a combination of asymptotic analysis and numerical studies. For a large wavevector, near the zone boundary, the spin response function displays a very pronounced oscillatory time dependence. However, there is next to no structure in the corresponding spectral function, in accord with simulation data. The asymptotic analysis shows that, in the infinite-temperature limit, coupled-mode theory does not comply with usual hydrodynamic assumptions, a view that has also emerged from analysis of some of the computer simulation data.
We study an infinite-range ferromagnetic Ising model in the presence of a transverse magnetic field, which exhibits a quantum paramagnetic-ferromagnetic phase transition at a critical value of the transverse field. In the thermodynamic limit, the low-temperature properties of this model are dominated by the behavior of a single large classical spin governed by an anisotropic Hamiltonian. Using this property, we study the quench and ac dynamics of the model both numerically and analytically, and develop a correspondence between the classical phase-space dynamics of a single spin and the quantum dynamics of the infinite-range ferromagnetic Ising model. In particular, we compare the behavior of the equal-time order-parameter correlation function both near to and away from the quantum critical point in the presence of a quench or ac transverse field. We explicitly demonstrate that a clear signature of the quantum critical point can be obtained by studying the ac dynamics of the system even in the classical limit. We discuss possible realizations of our model in experimental systems.
This is a continuation of an effort to reduce to the solution of a characteristic value problem the rigorous calculation of thermodynamic properties of systems in which the intermolecular forces are sufficiently short ranged so that practically the entire potential energy of the system results from interactions between nearest neighbors. The partition function of such a system can be expressed in terms of the largest characteristic value of a linear operator equation and finally as the ratio of partition functions of systems with relatively few particles whose potential energy functions differ somewhat from those in the original system. A method of evaluating grand partition functions and one of introducing interactions between more distant neighbors are discussed. The general theory is applied to the calculation of magnetization, internal energy, and specific heat of two-dimensional ferromagnets on the basis of the Ising model. There seems to exist a λ-point phase transition in the change from ferromagnetic to nonferromagnetic states.
"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."