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Physical Review A

Riemannian geometry and the thermodynamics of model magnetic systems

Henryk Janyszek, R. Mrugal a

Published Jun 1, 1989• 159 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.39.6515

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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.
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.
It is shown that the stability of ideal quantum gases can be measured by means of the Riemann scalar curvature R of the parameter space. The components of the metric tensor were assumed to be the second moments of energy and the number of particle fluctuations. As a result, R is a function of the second and third moments of those quantities. For bosons R is positive and increases monotonically from zero at the classical limit to positive infinity in the condensation region. A system is less stable if R is bigger and vice versa. For fermions R is negative and this means that Fermi gases are more stable than the ideal Bose and ideal classical systems.
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>0J>0J>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)G(r)G(r); ${\ensuremath{\xi}}_{G}$ and $\ensuremath{\xi}$ 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, ${\ensuremath{\xi}}_{G}$ 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, ${\ensuremath{\xi}}_{G}$ 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.
We continue our study of the thermodynamic geometry of the spin one model from paper I by probing the state space geometry of the Blume Emery Griffiths (BEG) model, and its limiting case of the Blume Capel model, in their mean field approximation. By accounting for the stochastic variables involved we construct from the thermodynamic state space two complimentary two-dimensional geometries with curvatures RmR_mRm​ and RqR_qRq​ which are shown to encode correlations in the model's two order parameters, namely, the magnetization mmm and the quadrupole moment qqq. The geometry is investigated in the zero as well as the non zero magnetic field region. We find that the relevant scalar curvatures diverge to negative infinity along the critical lines with the correct scaling and amplitude. We then probe the geometry of phase coexistence and find that the relevant curvatures predict the coexistence curve remarkably well via their respective RRR-crossing diagrams. We also briefly comment on the effectiveness of the geometric correlation length compared to the commonly used Ornstein-Zernicke type correlation length vis-a-vis their scaling properties.
We continue our study of the thermodynamic geometry of the spin one model from A. Sahay and R. Sanwari, Phys. Rev. E 105, 034134 (2022)10.1103/PhysRevE.105.034134. by probing the state space geometry of the Blume-Emery-Griffiths model and the Blume-Capel model in their mean-field approximation. By accounting for the stochastic variables involved we construct from the thermodynamic state space two complementary two-dimensional geometries with curvatures R_{m} and R_{q} which are shown to encode correlations in the model's two order parameters, the magnetization m and the quadrupole moment q. The geometry is investigated in the zero as well as the nonzero magnetic field region. We find that the relevant scalar curvatures diverge to negative infinity along the critical lines with the correct scaling and amplitude. We then probe the geometry of phase coexistence and find that the relevant curvatures predict the coexistence curve remarkably well via their respective R-crossing diagrams. We also briefly comment on the effectiveness of the geometric correlation length compared to the commonly used Ginzburg-Landau correlation length vis-à-vis their scaling properties.
State space Riemannian geometry is obtained for the one-dimensional Blume-Emery-Griffiths model and its Blume-Capel and Griffiths model limits, and its (pseudo)critical as well as noncritical parameter regimes are extensively investigated. Two codimension one geometries are obtained by taking suitable hypersurfaces in the three-dimensional state space manifold, and the induced thermal metrics are accordingly interpreted in terms of constrained fluctuations. The three-dimensional scalar curvature and the two two-dimensional curvatures are shown to be consistent with Ruppeiner's conjecture relating the inverse of the singular free energy to the thermodynamic scalar curvature. Moreover, they are found to be in an excellent agreement over a greater part of the noncritical region with the corresponding correlation lengths for the spin and the quadrupolar order parameters. The scaling function for the free energy near the pseudocritical and tricritical points is obtained thermodynamically by using Ruppeiner's conjecture. A connection is made between the sign change in the curvatures and the change in fluctuation patterns of the order parameters. In the accompanying paper we shall analyze the geometry of the spin-one model in its mean field approximation.