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Physical review. E

Thermodynamic geometry of spin-one lattice models. I. Spin and quadrupolar orders and critical scaling functions in one dimension

Riekshika Sanwari, Anurag Sahay

Published Mar 24, 2022• 8 citations
Publisher

Abstract

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.

DOI
10.1103/physreve.105.034134

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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.
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.
Four-dimensional state space geometry is worked out for the exactly solved one-dimensional spin-3/2 lattice with a Blume-Emery-Griffiths (BEG) Hamiltonian as well as a more general one with a term containing a non-zero field coupling to the octopole moments. The phase behaviour of the spin-3/2 chain is also explored extensively and novel phenomena suggesting anomalies in the hyperscaling relation and in the decay of fluctuations are reported for a range of parameter values. Using the method of constrained fluctuations worked out earlier in \cite{asknbads,riekan1} three sectional curvatures and a 3d3d3d curvature are obtained and shown to separately encode dipolar, quadrupolar and octopolar correlations both near and away from pseudo-criticality. In all instances of a seeming hyperscaling violation the 3d3d3d scalar curvature is found to encode the correlation length while the relevant 2d2d2d curvature equals the inverse of singular free energy. For parameter values where the order parameter fluctuation anomalously decays despite a divergence in correlation length the relevant scalar curvature undergoes a sign change to positive values, signalling a possible change in statistics.
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.
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.
A quantum Ising-like spin-1 system with both dipole and quadrupole interactions is analyzed. Such model is a generalized quantum version of the Blume-Emery-Griffiths model in an external anisotropic field. Models of this type are interesting candidates to describe phase transitions in liquid crystals. The phase diagram as well as the thermodynamic potentials of the model are constructed by a recently formulated improved version of mean field theory. The latter, which has the usual mean field theory as its zero-th order approximation, takes into account fluctuations up to the fourth-order. The phase diagram is found to exhibit phase transition manifolds of both the first and the second order, as well as multicritical lines. Four phases can be recognized: one ferromagnetic and three paramagnetic. The phases are identified by the requirements of minimum free energy constrained by the condition of self consistency of the order parameters p and q