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arXiv:2011.06491

Thermodynamic geometry of one-dimensional spin one lattice models

Anurag Sahay, Riekshika Sanwari

Published Nov 12, 2020• 1 citation
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Abstract

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 $R_m$ and $R_q$ 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 $ξ_1$ and $ξ_2$. The three dimensional scalar curvature $R_g$ 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.

DOI
10.48550/arxiv.2011.06491

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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.
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.
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.
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.
A model for the critical relaxation in the one-dimensional Ising-type S=1 spin system is presented. This model is equivalent to the Blume-Emery-Griffiths model and exhibits two simple critical points and one tricritical point. The kinetic behavior is studied using the real-space renormalization-group approach. In the two critical points we find that the critical slowing down is described by the dynamic exponent z, z=2. In each point this exponent belongs to the critical order parameter, while the second order parameter relaxes faster, with z=1 or 0. At the critical point the two order parameters relax with the same z, z=1.
A Kadanoff lower-bound renormalization transformation with three variational parameters has been applied to the Blume-Capel model by Burkhardt. We report some additional results obtained in a particular one-parameter subspace of the three variational parameters. In this subspace the transformation treats the three possible Blume-Capel ground-state configurations with all the spins in the same state on an equal footing and preserves the three-state permutational symmetry of the Potts model. There are fixed points associated with first-order, critical, and tricritical transitions. The variational parameters maximizing the free energy at the critical and tricritical fixed points differ by only a few percent. The critical exponents associated with the different transitions and various points on the critical surfaces are calculated for d=2,3d=2,3d=2,3 dimensions.