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Physical Review A

Application of Riemannian geometry to the thermodynamics of a simple fluctuating magnetic system

George Ruppeiner

Published Jul 1, 1981• 106 citations
Publisher

Abstract

It has been suggested previously by the author that the correlation length $\ensuremath{\xi}$ can be calculated from the Gaussian curvature in a Riemannian geometric model of thermodynamics. In the present paper, this hypothesis is tested for the one-dimensional Ising model. It is found that for ferromagnetic interactions ($J>0$) the thermodynamic ${\ensuremath{\xi}}_{G}$ is in excellent agreement with the correlation length $\ensuremath{\xi}$ which gives the range of the exponentially decaying spin-spin correlation function $G(r)$; ${\ensuremath{\xi}}_{G}$ and $\ensuremath{\xi}$ are never found to deviate by more than one lattice site. If $J=0$, there is no curvature, in accordance with expectations that curvature be a measure of interactions. If $J<0$, ${\ensuremath{\xi}}_{G}$ does not give the range of the envelope of the oscillatory $G(r)$, and hence, does not agree with usual definitions of the correlation length; however, ${\ensuremath{\xi}}_{G}$ can be interpreted satisfactorily as giving the average length, due to interactions, of clusters of aligned spins. The results in this paper support the contention that thermodynamics in general contains more information than had previously been thought.

DOI
10.1103/physreva.24.488

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It has been suggested previously by the author that the correlation length \ensuremathξ\ensuremath{\xi}\ensuremathξ can be calculated from the Gaussian curvature in a Riemannian geometric model of thermodynamics. In the present paper, this hypothesis is tested for the one-dimensional Ising model. It is found that for ferromagnetic interactions (J>0J>0J>0) the thermodynamic \ensuremathξG{\ensuremath{\xi}}_{G}\ensuremathξG​ is in excellent agreement with the correlation length \ensuremathξ\ensuremath{\xi}\ensuremathξ which gives the range of the exponentially decaying spin-spin correlation function G(r)G(r)G(r); \ensuremathξG{\ensuremath{\xi}}_{G}\ensuremathξG​ and \ensuremathξ\ensuremath{\xi}\ensuremathξ are never found to deviate by more than one lattice site. If J=0J=0J=0, there is no curvature, in accordance with expectations that curvature be a measure of interactions. If J<0J<0J<0, \ensuremathξG{\ensuremath{\xi}}_{G}\ensuremathξG​ does not give the range of the envelope of the oscillatory G(r)G(r)G(r), and hence, does not agree with usual definitions of the correlation length; however, \ensuremathξG{\ensuremath{\xi}}_{G}\ensuremathξG​ can be interpreted satisfactorily as giving the average length, due to interactions, of clusters of aligned spins. The results in this paper support the contention that thermodynamics in general contains more information than had previously been thought.
A Riemannian metrical structure of the parameter space has been introduced and investigated for magnetic systems described in the framework of quantum statistics. The introduced metric is based on the conception of the relative information. Two contrasting models have been investigated in detail: the one-dimensional Ising model, with short-range interactions, and the mean-field model of Kac, with long-range interactions. In the second case the metric tensor degenerates. The degeneration has been removed by adding the lattice energy to the original magnetic Hamiltonian. It turns out that in both cases the scalar curvature of parameter space tends toward plus infinity while approaching the critical points. The inverse of the scalar curvature, given by the second and third moments of stochastic variables, has been interpreted as a measure of the stability of the considered magnetic systems. The scalar curvature represents a joint part of fluctuations caused by the interactions of spins.
Microscopic models of realistic thermodynamic systems usually involve a number of parameters, not all of equal macroscopic relevance. We examine a decorated (1+3) Ising spin chain containing two microscopic parameters: a stiff parameter K mediating the long-range interactions, and a sloppy J operating within local spin groups. We show that K dominates the macroscopic behavior, with varying J having only a weak effect, except in regions where J brings about transitions between phases through its conditioning of the local spin groups with which K interacts. We calculate the heat capacity C(H), the magnetic susceptibility χ(T), and the thermodynamic curvature R. For large |J/K|, we identify four magnetic phases: ferromagnetic, antiferromagnetic, and two ferrimagnetic, according to the signs of K and J. We argue that for characterizing these phases, the strongest picture is offered by the thermodynamic geometric invariant R, proportional to the correlation length ξ. This picture has correspondences to other cases, such as fluids.
State space geometry is obtained for the one dimensional Blume Emery Griffiths model and the associated scalar curvature(s) investigated for various parameter regimes, including the Blume-Capel limit and the Griffiths model limit. For the one-dimensional case two complementary geometries with their associated curvatures RmR_mRm​ and RqR_qRq​ are found which are related to the fluctuations in the two order parameters, namely the magnetic moment and the quadrupole moment. An excellent agreement is obtained in significant regions of the parameter space between the two curvatures and the two corresponding correlation lengths ξ1ξ_1ξ1​ and ξ2ξ_2ξ2​. The three dimensional scalar curvature RgR_gRg​ is also found to efficiently encode interactions. The scaling function for the free energy near critical points and the tricritical point is obtained by making use of Ruppeiner's conjecture relating the inverse of the singular free energy to the thermodynamic scalar curvature.
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 spin-spin correlation functions of the random one-dimensional Ising model are calculated exactly and found to have a probability distribution. It is then possible to define two characteristic lengths, one to describe the behaviour of the mean and the other of the standard deviation of the probability distribution. The existence of these two characteristic lengths then suggests that the thermodynamic behaviour of the system is considerably more complex than for the corresponding nonrandom system.
We develop a geometric formulation of thermodynamic response in interacting spin systems and apply it to the two-dimensional Ising model. Treating inverse temperature and magnetic field as coordinates on a thermodynamic control manifold, we show that the mixed response field Ωβh=−N cov(m,e)Ω_{βh} = -N\,\mathrm{cov}(m,e)Ωβh​=−Ncov(m,e) arises naturally as a curvature-like quantity that measures correlations between magnetic and energetic fluctuations. Monte Carlo simulations reveal a strongly localized mixed-response ridge that emerges from the critical point and extends into the finite-field crossover regime. Analysis of the susceptibility, specific heat, and mixed-response maxima demonstrates distinct scaling behavior in the magnetic, energetic, and mixed fluctuation sectors. When represented in normalized response coordinates, trajectories obtained at different magnetic fields collapse onto a common curve, indicating that the evolution of the mixed response is strongly constrained by the susceptibility. This collapse suggests the emergence of a low-dimensional response manifold and points toward a geometric description of critical crossover based on relations among response functions rather than equilibrium states alone. The framework establishes a direct connection between fluctuation correlations, critical scaling, and geometric thermodynamic response.