Demagnetization artifact and ternary extensions in exactly solvable multicomponent spherical models of fluid criticality

Aqua and Fisher (Phys. Rev. E 79, 011118 (2009)) generalized the exactly solvable spherical model to multicomponent lattice gases, imposing one spherical constraint per species on interlaced sublattices that encode hard cores between species. For binary fluids with short-range attractive interactions they obtained exact free energies and correlation functions and showed that criticality occurs on a well-defined locus with one-component spherical exponents — but that, away from special symmetric compositions, a linear 'demagnetization' term in the equation of state suppresses the expected divergence of thermodynamic susceptibilities at criticality, an artifact unphysical for fluids. The later companion ionic spherical model (Phys. Rev. E 100, 052145 (2019)) showed that electroneutrality eliminates this artifact, but only for ionic systems with the solvent left implicit. The open problem, within the multicomponent per-species spherical framework these papers delimit, is to determine whether an exactly solvable construction exists for neutral mixtures at generic compositional asymmetry in which the susceptibilities diverge at criticality with standard one-component spherical exponents, or whether the artifact is intrinsic to per-species spherical constructions; and to pursue the ternary extension the source explicitly suggests — a neutral solvent plus positive and negative ions — whose critical behavior remains to be determined.

Background

The spherical model of Berlin and Kac (1952) replaces the discrete spins of a lattice ferromagnet by real variables subject to a single global constraint on the sum of their squares, which makes the partition function exactly evaluable by Gaussian integration. Stanley (1968) showed that it is the $n\to\infty$ limit of the $n$-component rotationally invariant spin models, so its order parameter is effectively an infinite-dimensional vector; its critical exponents are known exactly for all spatial dimensions $d$ (e.g. susceptibility exponent $\gamma=2/(d-2)$ for $2<d<4$, mean-field values above $d=4$, and no finite-temperature transition for $d\le 2$). This tractability has made the model a standard exactly solvable testing ground for critical phenomena, including lattice-gas descriptions of fluids, where the lattice gas maps onto a spin system with magnetization playing the role of density and magnetic field the role of chemical potential.

To describe fluid mixtures, Aqua and Fisher (2009) introduced an $S$-species lattice gas in which each species occupies its own sublattice of an interlaced lattice — the displacement between sublattices implements the hard-core repulsion between different species — and imposed one spherical constraint per species. The free energy and all pair correlation functions are then obtained exactly in terms of the $S\times S$ matrix of Fourier-transformed species interactions $\mathbf J(\mathbf k)$, with $\mathbf k$ in the Brillouin zone (the set of allowed lattice wavevectors); for binary systems the diagonalization of the resulting $2\times 2$ matrices yields two eigenmodes, expressed through the eigenvalues $\Lambda_\pm(\mathbf k;\lambda)$, in terms of which the density correlations decompose. For binary fluids with only hard cores and short-range overall attractive interactions, and for $d>2$, a suitable set of state-dependent mixing parameters $(\theta,\theta_m,\theta_h)$ — linear recombination coefficients relating physical fields and densities to mixed variables — is chosen by physical conditions (so that the mixed order parameter and a mixed Lagrange multiplier of the spherical constraints vanish at criticality); with this choice the usual critical properties of one-component spherical models are recovered, criticality being realized on a well-defined locus in the full phase diagram as the composition varies. The analysis assumes $d>2$ and smallness/positivity conditions on the intra-species asymmetry (equations (79), (A2) and (A7) of the source) under which the controlling eigenvalue $\Lambda_-(\mathbf k;\lambda)$ is minimized at $\mathbf k=\mathbf 0$; outside these sufficient conditions, minima at nonzero wavevector are not excluded.

The 'caution' of the title is that, except on certain symmetric submanifolds, the near-critical equation of state contains a linear term $\mathfrak{n}\check m$ in the mixed order parameter $\check m$ — the 'demagnetization' term, by analogy with the demagnetizing field that, in magnetostatics, opposes the magnetization of an ellipsoidal ferromagnet and caps its susceptibility. This term suppresses the normal divergence of the thermodynamic susceptibilities (in fluid language, compressibilities) at criticality and induces a linear dependence of the chemical potential and pressure on the total density through the two-phase region; both features are unphysical for fluids. A mean-field analysis of an XY (two-component-vector) spin system below its critical temperature traces the effect to the vectorial character of the spherical order parameter combined with compositional asymmetry, and the authors note that no simple demagnetization description of the familiar magnetostatic type should be expected.

In a companion line of work (Phys. Rev. Lett. 92, 135702 (2004); J. Phys. A 37, L241 (2004); Phys. Rev. E 100, 052145 (2019)), the same framework was applied to charged fluids, where Coulomb interactions add $1/k^2$ singularities to the interaction matrix. The 2019 binary ionic spherical model (positive and negative ions with Coulomb plus short-range nonionic interactions, solvent implicit) established that electroneutrality — the requirement of zero total charge, which enforces vanishing of the composition-asymmetry variable $\bar m^\dagger$ — eliminates the demagnetization artifact even when the short-range interactions are asymmetric: the equation of state keeps its standard spherical form and exponents, and the model yields exact results on charge–density correlation coupling, the divergence of the charge correlation length in asymmetric systems, violation of the Stillinger–Lovett second-moment sum rule (an exact criterion distinguishing conducting from dielectric screening behavior) at criticality, and algebraic rather than exponential charge screening when semiclassical quantum effects are included. That resolution, however, rests on a symmetry available only to ionic systems. For neutral binary mixtures at generic fixed composition, the 2009 paper reports the artifact as an intrinsically unphysical result requiring caution in interpreting the models, rather than posing a remediation task; and for systems of three or more species — needed, as that paper emphasizes, to represent an electrolyte solution as a neutral solvent plus positive and negative ions — it suggests that 'ternary fluid models … could be instructive'. Neither the neutral-mixture artifact nor the ternary extension has been addressed in the literature since.

Problem Statement

The problem, as delimited by Aqua and Fisher's multicomponent spherical models and their ionic companion, is to make exactly solvable lattice models of multicomponent fluid criticality physically faithful at generic composition and to extend them to the mixtures they were motivated by. Its two linked components share one objective — an exactly solvable, physically standard description of criticality in fluid mixtures:

(1) Artifact removal or intrinsicness for neutral binary mixtures. For a two-species lattice gas with hard cores and short-range overall attractive interactions at generic compositional asymmetry (where the composition-asymmetry variable $\bar m^\dagger$ does not vanish), either (a) construct, within the multicomponent per-species spherical framework, an exactly solvable model — a modified multicomponent spherical constraint structure or an alternative treatment of the existing construction — in which the thermodynamic susceptibilities diverge at criticality with the standard one-component spherical critical exponents, i.e. the linear demagnetization term $\mathfrak{n}\check{m}$ is absent; or (b) prove that within this same class of per-species spherical constructions — also called mean-spherical models, in which the spherical constraint is imposed on thermal averages rather than per configuration, and including the modified constraint structures and alternative treatments that branch (a) admits — this suppression of susceptibility divergence at generic composition is unavoidable.

(2) Ternary extension. The source explicitly suggests extending the framework to $S\ge 3$ species — in particular a neutral solvent species plus positive and negative ions, the minimal tractable model of an electrolyte solution, which it calls instructive — as a direction worth pursuing. The critical behavior of such ternary per-species spherical models, in particular whether demagnetization-like linear terms arise, persist, or are mitigated by the additional species, remains to be determined. The question concerns the ternary case $S=3$ — a neutral solvent plus positive and negative ions — namely whether its thermodynamic susceptibilities diverge at criticality with the standard one-component spherical critical exponents (equivalently, whether the demagnetization-like linear term arises, persists, or is mitigated by the additional species), rather than general $S\ge 3$.

References

1. T. H. Berlin and M. Kac, 'The Spherical Model of a Ferromagnet', Physical Review 86, 821–835 (1952). DOI: 10.1103/PhysRev.86.821, https://doi.org/10.1103/PhysRev.86.821
2. H. E. Stanley, 'Spherical Model as the Limit of Infinite Spin Dimensionality', Physical Review 176, 718–722 (1968). DOI: 10.1103/PhysRev.176.718, https://doi.org/10.1103/PhysRev.176.718
3. J.-N. Aqua and M. E. Fisher, 'Criticality in multicomponent spherical models: Results and cautions', Physical Review E 79, 011118 (2009). DOI: 10.1103/PhysRevE.79.011118, https://doi.org/10.1103/PhysRevE.79.011118 (preprint arXiv:0810.5539, https://arxiv.org/abs/0810.5539)
4. J.-N. Aqua and M. E. Fisher, 'Ionic Criticality: An Exactly Soluble Model', Physical Review Letters 92, 135702 (2004). DOI: 10.1103/PhysRevLett.92.135702, https://doi.org/10.1103/PhysRevLett.92.135702
5. J.-N. Aqua and M. E. Fisher, 'Charge and density fluctuations lock horns: ionic criticality with power-law forces', Journal of Physics A: Mathematical and General 37, L241–L248 (2004). DOI: 10.1088/0305-4470/37/24/L02, https://doi.org/10.1088/0305-4470/37/24/L02
6. J.-N. Aqua and M. E. Fisher, 'Critical charge and density coupling in ionic spherical models', Physical Review E 100, 052145 (2019). DOI: 10.1103/PhysRevE.100.052145, https://doi.org/10.1103/PhysRevE.100.052145 (open-access version: https://hal.science/hal-03970656)
7. W. de Castilho and S. R. Salinas, 'Spherical model with Dzyaloshinskii–Moriya interactions', Journal of Statistical Mechanics: Theory and Experiment 2022, 073106 (2022). DOI: 10.1088/1742-5468/ac7e3e, https://doi.org/10.1088/1742-5468/ac7e3e (preprint arXiv:2203.09970, https://arxiv.org/abs/2203.09970)
8. A. Alastuey and S. K. Das, 'Algebraic infection of charge correlations at the ionic critical point', Contributions to Plasma Physics 63, e202300016 (2023). DOI: 10.1002/ctpp.202300016, https://doi.org/10.1002/ctpp.202300016 (preprint arXiv:2111.13375, https://arxiv.org/abs/2111.13375)
9. S. K. Das, Y. C. Kim, and M. E. Fisher, 'When Is a Conductor Not Perfect? Sum Rules Fail Under Critical Fluctuations', Physical Review Letters 107, 215701 (2011). DOI: 10.1103/PhysRevLett.107.215701, https://doi.org/10.1103/PhysRevLett.107.215701
10. S. K. Das, Y. C. Kim, and M. E. Fisher, 'Near critical electrolytes: Are the charge-charge sum rules obeyed?', The Journal of Chemical Physics 137, 074902 (2012). DOI: 10.1063/1.4743957, https://doi.org/10.1063/1.4743957
11. O. Patsahan, I. Mryglod, and J.-M. Caillol, 'Ionic fluids: charge and density correlations near gas–liquid criticality', Journal of Physics: Condensed Matter 17, L251–L256 (2005). DOI: 10.1088/0953-8984/17/25/L02, https://doi.org/10.1088/0953-8984/17/25/L02
12. M. Dinpajooh, N. N. Intan, T. T. Duignan, E. Biasin, J. L. Fulton, S. M. Kathmann, G. K. Schenter, and C. J. Mundy, 'Beyond the Debye–Hückel limit: Toward a general theory for concentrated electrolytes', The Journal of Chemical Physics 161, 230901 (2024). DOI: 10.1063/5.0238708, https://doi.org/10.1063/5.0238708