Thermodynamic geometry of one-dimensional spin one lattice models

Anurag Sahay, Riekshika<EMAIL_ADDRESS><EMAIL_ADDRESS>

Department of Physics, National Institute of Technology, Patna 800005, India

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 $\xi_{1}$ and $\xi_{2}$. 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.

##  1 Introduction

Thermodynamic geometry (TG), pioneered by Ruppeiner and other workers, has
been extensively used to probe a wide range of systems including fluids,
magnetic systems and several black hole solutions. Employing a metric based on
second moments of thermal fluctuations TG renders a Riemannian geometric
structure to the thermodynamic state space of the system, [1, 2]. The
thermodynamic metric quantifies in a co-ordinate independent way the classical
distinguishability of thermodynamic states, so that the easier it is for two
states to fluctuate into each other the shorter the separation between them
and vice versa. Not surprisingly then, near the critical point where
fluctuations diverge the separation between thermodynamic states shrinks and
the associated scalar curvature diverges.

Beyond this association, TG forges a surprising and a remarkable connection
between its geometric invariants which are calculated solely from
thermodynamics and the statistical mechanical description of the system. Thus,
in an early insight Ruppeiner conjectured that in the vicinity of the critical
point the state space scalar curvature $R$ is $equal$ to the correlation
volume $\xi^{d}$ upto a constant of order unity. In the asymptotically
critical region this conjecture could be further refined to an equality of the
scalar curvature with the inverse of the critical free energy $\psi_{s}$ upto
a universal constant $\kappa$ which depends only on the universal critical
exponents. This geometry-energy equation in the Riemannian state space is a
centre-piece of TG and is reminiscent of the Einstein’s equation which relates
space-time curvature to matter energy distribution. It will therefore be apt
to refer to the conjectured equality

$R=\kappa\,\frac{1}{\psi_{s}}$ (1)

with $\kappa$ a universal constant, as “the Ruppeiner equation”. The
correspondence of $R$ with the correlation length then follows, at least in
the near critical region, from the well known equality of the correlation
volume with the inverse of singular free energy density. The Ruppeiner
equation could be profitably used to calculate in a straightforward manner the
scaling function of the singular free energy, thus providing an alternative to
the more challenging calculation based on renormalization group analysis, [3,
4].

It turns out, the Ruppeiner equation can be thought of as the stronger form of
the conjecture relating geometry to thermodynamics, one which is exact $at$
the critical point. The conjecture relating curvature to correlation length as

$|R|\sim\xi^{d}$ (2)

(where $d$ is spatial dimension) on the other hand is found to be valid for
parameter values much beyond the critical point. We shall refer to it as the
weak end of Ruppeiner’s conjecture. It has been shown in this work as also
earlier elsewhere that the connection between curvature and correlation length
extends even to situations where there is no criticality. The weak conjecture
has found much use in recent times in calculating the phase coexistence curves
and the Widom lines for fluids and other systems, [5, 6, 7, 8]. Remarkably,
the scalar curvature packs even more information about the underlying
statistics of interactions via its signature. Thus, it is commonly believed
that a positive sign (in the convention used in [1]) of $R$ is indicative of
statistically repulsive interactions while a positive sign is suggestive of
statistically attractive interactions, [8, 9, 10, 11]. For example, the scalar
curvature has always been seen to diverge to negative infinity at criticality.
However, the issue of signature of $R$ is nuanced as suggested recently in
[12] and it still awaits a more fundamental resolution. In addition, scalar
curvature in three or higher dimensional parameter space has been investigated
for only a handful of thermodynamic systems [12, 13]. It is of interest to
pursue the geometry of higher dimensional parameter spaces since apart from
the scalar curvature the sectional curvatures could also signify underlying
physics. More generally $R$ is to be understood as a qualified measure of
interactions. While its behaviour is well understood broadly, a finer
understanding of the geometric curvature is still an ongoing work. In light of
this it is important to record the behaviour of $R$ for a range of parameter
values in a given system, something this work attempts to do.

Exactly solved models can serve as important testing grounds to verify the
conjectures of TG and also to further explore the features of $R$ inasmuch as
they offer an analytical control over the partition function and also possibly
the correlation length. One of the most important such models used to
successfully verify TG is the one dimensional Ising model which has a
(pseudo)critical point at zero temperature in zero magnetic field. The scalar
curvature earlier worked out numerically in [14] was later found to be a
surprisingly simple expression, [15]. For zero field, $R$ tends to exactly
twice $\xi$ as it nears the zero temperature pseudocritical point.
Furthermore, for the non critical case in the presence of a magnetic field the
curvature $R$ forms an umbrella over the correlation length and converges to
$2$ as $\xi$ decays to zero at low temperatures (see figs.(5) and (5)). Some
other cases where TG has been applied to exactly solved models are the Ising
model on a Bethe lattice, [16], Ising model on planar random graphs, [17], the
spherical model, [18] and the one dimensional Potts model, [19].

Ising model with lattice spins $S_{i}=\pm 1$ naturally generalizes to the spin
one model with lattice spins $S_{i}=0,\pm 1$. What might seem like an
innocuous addition of a degree of freedom at lattice sites results in a rich
and varied phase structure of the spin one model. This is because, apart from
the spin-spin quadratic coupling and the coupling of individual spins to the
magnetic field $H$, the spin one model admits the possibility of a non trivial
biquadratic coupling of quadrupoles $S_{i}^{2}$ to each other and their
coupling to an ordering field independent of the magnetic field. In effect,
the thermodynamics of a spin one model is governed by two order parameters,
the spin $\langle S_{i}\rangle$ and the quadrupole moment $\langle
S_{i}^{2}\rangle$. Both the order parameters, while kinematically coupled, are
separate stochastic variables and their interplay leads to a rich phase
structure with coexistence surfaces bordered by lines of critical points and
first order points which meet in one or more tricritical points. Naturally,
therefore, spin one lattice models have been extensively used to model the
behaviour of interacting systems with two types of ordering processes. One of
the most popular spin one modes, the Blume-Emery-Griffiths (BEG) model
originally formulated to study the phase behaviour
$\mbox{He}^{3}-\mbox{He}^{4}$ mixtures, has been widely used to model diverse
phenomena. As a model for the phase behaviour in Helium mixture it
successfully captures the phenomena of superfluid ordering as well as phase
separation depending on the relative concentration of the $\mbox{He}^{3}$
impurity, [20]. In addition the BEG model has been used in the context of
simple fluids to model condensation and solidification, [24], in binary alloys
to model ferromagnetism and phase separation, [25], and in microemulsions,
[26], to name a few. In the limit of zero biquadratic coupling, known as the
Blume-Capel model, [21, 22], it has been used to model phase behaviour in
magnetic systems wherein depending on the strength of crystal field splitting
the transition between a paramagnet and a magnetically ordered state changes
from first order to second order.

In this work we initiate the study of TG for classical higher spin lattice
models beginning with an investigation of the one dimensional spin one models,
namely the Blume Emery Griffiths model and its limiting cases of the Blume
Capel model and the Griffiths model,[23], where in the latter case the
quadratic spin coupling is set to zero. In addition to the advantage of being
exactly solved, the one dimensional spin one models retain much of the rich
phase behaviour of their higher dimensional counterparts. Thus, these models
continue to display a locus of pseudocritical points, a pseudotricritical
point and zero temperature phase coexistence, much of which is amenable to a
geometric treatment. Besides, the parameter space of BEG models is three
dimensional which provides an avenue to explore higher dimensional scalar
curvature and various sectional curvatures. In addition, the fact that there
are two order parameters in the model gives rise to the possibility of two
correlation lengths for some parameter values and it would be worthwhile
exploring if geometry encodes different correlation lengths. In this work we
hope to make good use of the given opportunity.

This paper is organised as follows. In section 2 we first discuss the
Hamiltonian of the Blume Emery Griffiths model and follow it up with a
discussion of its mean field phase structure. We then review the one
dimensional BEG model in the framework of the transfer matrix solution of its
partition function and subsequently review its phase structure. In section 3
we start with a tutorial introduction to the geometrical representation of
thermodynamic constraints and then argue our case for two hypersurface
geometries we believe are most relevant to our model. Finally, in section 4 we
present our main results for the one dimensional BEG model. In subsection
4.1.1 we obtain the singular part of free energy for the case of positive
spin-spin coupling and using TG we work out on the spin scaling function for
the one dimensional BEG model. In subsection 4.2 we obtain the geometry of the
Griffiths model, with a special emphasis on the three dimensional scalar
curvature. Finally in section 5 we summarize our main finding and scope of
work and point to some future directions.

##  2 The spin one model

The BEG model has the most general reflection symmetric Hamiltonian for a
classical spin one model with nearest neighbour interactions. The Hamiltonian
for the BEG model is written as, [20]

$\mathcal{H}_{beg}=-J\,\sum_{<ij>}\,S_{i}\,S_{j}-H\,\sum_{i}\,S_{i}-K\,\sum_{<ij>}\,S_{i}^{\,2}\,S_{j}^{\,2}+D\,\sum_{i}S_{i}^{\,2}$
(3)

The lattice spin variable $S_{i}$ is Ising like and can take up values $+1,-1$
and $0$. In addition to the bilinear coupling terms and a magnetic field that
couples to the magnetic moment, the Hamiltonian contains a biquadratic
coupling term of strength $K$ and a crystal field $D$ which couples to the
quadrupole moment. The coupling strengths $J$ and $K$ are positive in the
original BEG model, [20]. The $K=0$ limit is the Blume Capel (BC) model, [21,
22]. While the magnetic field term is not experimentally realizable in the
original context of the BEG model which refers to a mixture of
$\mbox{He}^{3}-\mbox{He}^{4}$, it plays its usual role in the BC limit which
refers to a magnetic system. The spin one models have two densities, namely
the mean magnetization and the mean quadrupole moment,

$M=\langle S_{i}\rangle\,\,\,\,;\,\,\,\,Q=\langle S_{i}^{2}\rangle$ (4)

Since the model is translationally invariant, there is no spatial variation in
the order parameters. In the lattice gas interpretation of the BEG model $M$
represents the superfluid order parameter while $Q$ is the concentration of
$\mbox{He}^{4}$. In a magnetic system $x=1-Q$ would measure the concentration
of non magnetic impurities. Similarly, apart from a small term, $D$ is the
difference in chemical potentials of $\mbox{He}^{3}$ and $\mbox{He}^{4}$,
namely $D\sim\mu_{3}-\mu_{4}$. For a positive $D$ larger concentrations of
$\mbox{He}^{3}$ would be energetically preferred. In the context of a magnetic
system, $D$ refers to a single ion anisotropy term which splits the single-
spin energy levels, with $S_{i}=0$ lower than the degenerate $S_{i}=\pm 1$
levels. It can also be thought of as an external field coupled to the order
parameter $Q$ (or $x$) analogous to $H$ which couples to $M$. Owing to the
interplay of the two order parameters via the interaction and field terms the
phase structure of the BEG and related models is rich, with the presence of
tricritical points, critical lines and a line of first order transitions, [20,
21, 22]. We now sketch the phase behaviour very briefly.

Figure 1: Schematic diagram of the mean field phase structure of the spin one
model in the $T-D-H$ plane. Here the ratio $K/J$ is small.

Fig.(1) above shows a mean field picture of the phase diagram of the spin one
model with the Hamiltonian given by eq.(3) and with the parameter ratio $K/J$
small which relates to the physial context of the BEG model, [20, 27]. ${\bf
A}$ is a coexistence surface in the $T-D$ plane where phases with positive and
negative $M$ coexist. For $D\to-\infty$ the $S=0$ state is completely
suppressed and the system can be mapped to a spin half Ising model. In the
$\mbox{He}^{3}-\mbox{He}^{4}$ context it would mean the presence of only the
$\mbox{He}^{4}$ state. In addition to the ${\bf A}$ surface there are
symmetrically placed wing like coexistence surfaces ${\bf B}$ and ${\bf
B^{\prime}}$ that extend into the $D-H$ plane for $D>0$. On each of these
surfaces two phases with different values of $Q$ coexist. The three
coexistence surfaces each terminate on the high temperature side in lines of
critical points, which intersect and terminate at the tricritical point TP.
The surfaces ${\bf A,B}$ and ${\bf B^{\prime}}$ intersect in the $T-D$ plane
along a line ${\bf F}$ of first order phase transitions which in turn
terminates at the tricritical point. Three phases coexist on ${\bf F}$ so that
it is a line of triple points. The tricritical point is different from the
critical point in that the critical exponents $\beta$ and $\delta$ are
different from their respective mean field values. The phase diagrams change
for higher values of the ratio $K/J$. Thus, for example, at $K/J=3.1$ there
are two more tricritical points symmetrically positioned on the critical
boundaries of the wings ${\bf B}$ and ${\bf B^{\prime}}$, [27]. Furthermore,
anti-ferromagnetic bilinear coupling or a repulsive biquadratic term which
render the ratio $K/J<0$, diversifies and further enriches the phase behaviour
in the spin one model, [30, 31]. In the limit $J=0$ which we call as the
Griffiths model, the BEG model exhibits phase coexistence as well as second
order transition in the quadrupolar order parameter, [23]. In this limit the
spin one model can be mapped onto the spin $\frac{1}{2}$ Ising model with a
temperature dependent magnetic field. We shall not be pursuing the geometry of
antiferromagnetic spin coupling or a repulsive biquadratic coupling in this
work. While we believe the geometry of these cases is very interesting, it is
more nuanced due to the presence of staggered spin and quadrupolar orders. We
shall return to these exciting cases in our future work.

It may be be noted that the most general nearest neighbour spin one
Hamiltonian can be written by adding an asymmetric exchange term between the
dipole and the quadrupole moments. This was found by Mukamel and Blume (MB)
[27, 28],

$\mathcal{H}_{mb}=\mathcal{H}_{beg}-\frac{L}{2}\sum_{<ij>}(S_{i}\,S_{j}^{\,2}+S_{i}^{\,2}\,S_{j})$
(5)

It can be seen that for $L>0$ neighbouring states with $S_{i}=1$ will be
energetically most preferred while those with $S_{i}=-1$ will be least
preferred thus breaking the up-down symmetry. Staggered quadrupolar order is
observed in this case for a range of values of $L$, [27, 28]. For the same
reason as above we set aside a geometric investigation of the MB model to a
future work.

###  2.1 The one dimensional spin one model.

The most general Hamiltonian of a spin one chain of N atoms with nearest
neighbour interaction is

$\mathcal{H}=-J\,\sum_{i}^{N}\,S_{i}S_{i+1}+K\,\sum_{i}^{N}S_{i}^{2}S_{i+1}^{2}-H\,\sum_{i}^{N}\,S_{i}-D\,\sum_{i}^{N}S_{i}^{2}-\frac{L}{2}\sum_{i}^{N}(S_{i}\,S_{i+1}^{\,2}+S_{i+1}^{\,2}\,S_{i})$
(6)

Here the coupling constants and fields have their usual meaning as in the
Hamiltonian, eq.(3). The one dimensional ring of $N$ spins is exactly solvable
in the large $N$ limit using the standard transfer matrix technique, [28]. The
transfer matrix for the Hamiltonian in eq.(6) is three dimensional,

$T=\left(\begin{array}[]{ccc}e^{(-D+H+J+K+L)\beta}&e^{(H-D)\beta/2}&e^{(-D-J+K)\beta}\\\
e^{(H-D)\beta/2}&1&e^{-(D+H)\beta/2}\\\
e^{(-D-J+K)\beta}&e^{-(D+H)\beta/2}&e^{(-D-H+J+K-L)\beta}\\\
\end{array}\right)$ (7)

This gives rise to three eigenvalues in general,
$\lambda_{1}>\lambda_{2}>\lambda_{3}$. In the limit of infinite $N$, i.e the
thermodynamic limit, the logarithm of the largest eigenvalue $\lambda_{1}$
becomes the free energy per spin. The free energy (Massieu function) for the
zero field BEG model, $L=H=0$ can be obtained in a closed form,

$\psi=\mbox{ln}\left[\frac{1}{2}e^{-\beta(D+J)}\left(e^{\beta D+\beta
J}+e^{2\beta J+\beta K}+e^{\beta K}+\sqrt{W}\right)\right]$ (8)

where

$W=\left(e^{\beta D+\beta J}+e^{2\beta J+\beta K}+e^{\beta
K}\right)^{2}-4\left(-2e^{\beta D+2\beta J}+e^{\beta D+\beta J+\beta
K}+e^{\beta D+3\beta J+\beta K}\right)$

while for the more general cases one can find it numerically by solving a
cubic equation for the eigenvalues. In the following the BEG model ($L=0$)
shall be our default model and we shall no more mention the Mukamel-Blume
model ($L\neq 0$).

While there is no finite temperature phase transition for the one dimensional
case, its thermal behaviour richly responds to the interplay of various
coupling strengths in the Hamiltonian. The phase diagram can be seen as a
limiting case of the mean field phase structure as represented in fig.(1). In
the limit of one dimension the critical points and the tricritical point will
lie at lower and lower temperature until they become respectively
pseudocritical points and the pseudotricritical point at $T=0$. In a sense the
whole phase diagram of fig.(1) flattens out onto the $H-D$ plane, [28, 27]. At
the pseudocritical point the eigenvalues $\lambda_{1}$ and $\lambda_{2}$
become asymptotically equal. At the pseudotricritical point all the three
eigenvalues become asymptotically equal.

The correlation length can be obtained in a standard manner via the ratio of
the largest and the next-to-largest eigenvalues. Interestingly, owing to the
fact that there are two correlation functions corresponding to the spin-spin
and the quadrupole-quadruple correlations, the spin one model admits the
possibility of two separate correlation lengths. Indeed, this possibility is
realized for the zero field BEG model with $H=L=0$ where due to increased
symmetry of the transfer matrix there are separate correlation lengths for
spin and quadrupole fluctuations, given respectively as,

$\displaystyle\xi_{1}^{-1}$ $\displaystyle=$
$\displaystyle\mbox{log}\frac{\lambda_{1}}{\lambda_{2}}$
$\displaystyle\xi_{2}^{-1}$ $\displaystyle=$
$\displaystyle\mbox{log}\frac{\lambda_{1}}{\lambda_{3}}$ (9)

For non-zero $H$ field there is only one correlation length $\xi_{1}$ for
correlations in both order parameters, [28]. It will be of interest to see how
far the state space geometry encodes the microscopic statistical interactions
of the chain. In particular one would like to find out if the state space
scalar curvature is representative of the two correlation lengths
$\xi_{1}\mbox{and}\xi_{2}$ and as to how successful it is in encoding the
pseudo critical and tricritical behaviours in the model. As reasoned out in
the previous section we shall be exploring two two-dimensional geometries for
the BEG model, namely the ones intrinsic to the $D$-surface and the
$H$-surface. We shall also comment briefly on the full three dimensional
geometry. Quite satisfyingly, we shall find that the two geometries faithfully
represent the underlying correlations in the two order parameters.

Let us first briefly discuss the phase structure for the one dimensional BEG
model. Our review closely follows the exposition in [28] and even the figures
3 and 3 are exactly similar to those in [28] down to the figure labels. In any
case the diagrams are easily obtainable from the transfer matrix.

Figure 2: Zero temperature phase diagram for the spin one model in the $H-D$
plane with $L=0,J>0,J+K>0$. Two phases coexist on the $a$ line and the $f$
lines in the $T=0$ place and they are a locus of pseudocritical points on
approaching from $T>0$. Point $c$ is a triple point at which three phases
coexist in the $T=0$ plane. On approaching from $T>0$ it shows
pseudotricritical behaviour. Adapted from [28].

Figure 3: Zero temperature phase diagram of the spin one model in the in the
$H-D$ plane with $L=J=0,K>0$. Infinite phases coexist on the line $d$ and no
pseudocritical behaviour is seen here. Two phases coexist along the $f$ lines
and pseudocritical behaviour is seen on approaching from $T>0$. The point $b$
is a triple point but not a pseudotricritical point. Adapted from [28].

As mentioned earlier, in this report we shall restrict ourselves to the case
of non-negative values of spin exchange and biquadratic exchange, namely
$K,J\geq 0$. Fig. (3) shows the phase diagram of the full one-dimensional BEG
model at $T=0$. The phase diagram remains the same in the Blume-Capel limit
$K=0$. Two phases with $Q=1$ and $M=\pm 1$ coexist on the $a$ line that is
positioned at zero magnetic field. Similarly, two phases $Q,M=0$ and
$Q=1,M=\pm 1$ coexist on the two symmetrically placed $f$ lines. There is a
discontinuous phase change on crossing these lines at zero temperature whereas
on approaching these lines from $T>0$ one sees pseudocritical behaviour. The
intersection of the $a$ line with the $f$ lines is the triple point $c$ at
which three phases coexist. On approaching the $c$ line from $T>0$
pseudotricritical behaviour is seen. Fig. (3) is the zero temperature phase
diagram of the BEG model in the limit of zero quadratic coupling, $J=0$. On
the $d$ line an infinite number of phases coexist in the limit of infinite
$N$. This is because, for $D<K$ on the $d$ line the $S=0$ spin is
energetically ruled out while the up and down spins are equally likely in the
absence of a discriminating quadratic coupling. In other words, the model
behaves like a collection of independent Ising spins, which also rules out any
pseudocritical behaviour in approaching the line from $T>0$. On the other
hand, the $f$ line continues to show two-phase coexistence just as for the
case $J>0$ as well as pseudocritical behaviour on approaching from $T>0$. The
point $b$ is a triple point but not a pseudotricritical point. The phase
diagram of the spin one model with the spin-spin coupling strength $J$ set to
zero is presented in fig.(3) which therefore represents the zero temperature
phase diagram of the one-dimensional Grifiths model.

Understandably, the zero temperature phase diagram in fig.(3) and fig.(3) is
straightforward, with straight line phase boundaries, because it is influenced
only by energy considerations in the absence of any thermal or quantum
fluctuation. The region to the right of the $f$ lines in both figures is the
one where the $D$ ‘field’ in the Hamiltonian, eq.(6) is greater than the
combined strength of the attractive couplings $J$ and $K$, so that the
energetically preferred state is the one with all spins set to zero.

##  3 The geometry of thermodynamic constraints: a tutorial introduction

In this section we explain our method of carving out the relevant state space
geometries when restrictions are applied on some thermodynamic fluctuations in
any system. The method of constructing state space hypersurfaces corresponding
to thermodynamic constraints was first discussed in the context of black hole
thermodynamics in [33]. While our reference system shall be the BEG model the
general method is applicable to any system with three or more thermodynamic
variables.

We first set up the equilibrium thermodynamic relations beginning with the
partition function. While basic and well known in the context of magnetic
systems the ensuing discussion is of fundamental value to the subsequent
construction of the model’s thermodynamic geometry. The canonical partition
function per site of the BEG model is a weighted sum over all possible spin
configurations,

$Q=\frac{1}{N}\sum_{\left\\{S_{i}\right\\}}\mbox{exp}^{-\beta\,(-J\,{\sum_{<ij>}}\,S_{i}\,S_{j}-K\,\sum_{<ij>}\,S_{i}^{\,2}\,S_{j}^{\,2}-H\,\sum_{i}\,S_{i}+D\,\sum_{i}S_{i}^{\,2})}$
(10)

The Massieu function per spin $\psi$, expressed as the log of the partition
function, can be taken as the starting point for thermodynamics and also for
the geometry. The state space Riemannian metric is conveniently defined as the
second deriative of the Massieu function with respect to its entropic
intensive variables. Note that throughout this text we shall interchangeably
use the terms “free energy” and Massieu function so that by both terms we mean
the log of the partition function. Indeed, “free entropy” would have been more
apt here and we hope to be forgiven for this abuse of terminology.

It shall be profitable in the following to think of the parameter $D$ as a
tunable external field, analogous to the magnetic field $H$. At the same time
the self interaction parameters $J,K$ could be thought of as fixed for a given
system. The macroscopic energy per site $E$ is obtained as the derivative of
the Massieu function

$\displaystyle-\left.\frac{\partial{\psi}}{\partial\beta}\right|_{H,D}$
$\displaystyle=$ $\displaystyle
E=\frac{1}{N}\langle{\mathcal{H}}_{beg}\rangle$ (11) $\displaystyle=$
$\displaystyle\frac{1}{N}\langle-J\,\sum_{<ij>}\,S_{i}\,S_{j}-K\,\sum_{<ij>}\,S_{i}^{\,2}\,S_{j}^{\,2}\,\rangle-\frac{1}{N}\langle
H\sum_{i}S_{i}\rangle+\frac{1}{N}\langle D\,\sum_{i}S_{i}^{2}\rangle$
$\displaystyle=$
$\displaystyle\langle\mathcal{F}_{1}\rangle+\langle\mathcal{F}_{2}\rangle+\langle\mathcal{F}_{3}\rangle$
$\displaystyle=$ $\displaystyle U-H\,M+D\,Q$

where

$\displaystyle U$ $\displaystyle=$
$\displaystyle\langle-J\,\sum_{<ij>}\,S_{i}\,S_{j}-K\,\sum_{<ij>}\,S_{i}^{\,2}\,S_{j}^{\,2}\,\rangle/N$
$\displaystyle,M$ $\displaystyle=$
$\displaystyle\langle\sum_{i}S_{i}\rangle/N\,\,\,\,\mbox{and}$ $\displaystyle
Q$ $\displaystyle=$ $\displaystyle\langle\sum_{i}S_{i}^{2}\rangle/N$ (12)

with $N$ being the total number of spins. Eq.(11) above shows a natural
breakup of the Hamiltonian into three stochastic variables
$\mathcal{F}_{1},\mathcal{F}_{2}\,\mbox{and}\,\mathcal{F}_{3}$ whose mean
values are related to the thermodynamic quantities, $U,M$ and $Q$. This can be
compared to the Hamiltonian of the spin half Ising model which breaks up into
two stochastic variables, [15]. The quantity $U$ in eq.(12) above could be
thought of as the“ internal” energy per site of the system comprising self
interaction terms in the Hamiltonian while $E$ is the “ total” energy per site
which includes the interaction energy between the spin system and the sources
of external fields [29]. The three correlated but independent fluctuations
render the geometry of the spin one model three dimensional.

The entropy per site $S$ is obtainable from the partition function via the
specific Massieu function

$\displaystyle\psi$ $\displaystyle=$ $\displaystyle S-\beta\,E$ (13)
$\displaystyle=$ $\displaystyle S-\beta\,U-\nu\,M-\mu\,Q$

where in the second equality we have used the entropic intensive variables
$\nu=-\beta H$ and $\mu=\beta D$.

Using the partial derivatives of $\psi$ with respect to $\beta$, $H$ and $D$
the differential of $\psi$ is obtained as

$d\psi(\beta,H,D)=-E\,d\beta+\beta\,M\,dH-\beta\,Q\,dD$ (14)

Eqs.(14) and the first equality in (13) lead to the first law for variations
in the total energy $E$,

$dS=\beta\,dE+\beta\,M\,dH-\beta\,Q\,dD$ (15)

On the other hand, considered as a function of the entropic intensive
variables $\beta$, $\nu$ and $\mu$ the differential of $\psi$ becomes

$d\psi(\beta,\nu,\mu)=-U\,d\beta\,-M\,d\nu-\,Q\,d\mu$ (16)

and the first law for variations in the internal energy $U$ is obtained as

$dS=\beta\,\,dU-\beta\,H\,dM+\beta\,D\,dQ$ (17)

In either case we of course find that $d^{2}S=-d^{2}\psi$. We shall always
take eq.(16) and eq.(17) as our starting point for thermodynamic geometry.

The fundamental thermodynamic equation of the spin one system in the entropy
representation is formally the function

$S=S(U,M,Q)$ (18)

from where all the thermodynamic quantities can be obtained by suitable
differentiation, in particular, the equations of state,

$\displaystyle T$ $\displaystyle=$ $\displaystyle T(U,M,Q)$ $\displaystyle H$
$\displaystyle=$ $\displaystyle H(U,M,Q)$ $\displaystyle D$ $\displaystyle=$
$\displaystyle D(U,M,Q).$ (19)

Since the number $N$ of latttice sites is held constant it has already been
absorbed in the definition of $M,Q,U$ and $S$ as per site thermodynamic
quantities. In a real magnetic system the volume $V$ could be an additional
parameter, with the background lattice serving to store energy. However, no
such consideration is made for the spin one lattice which is a model magnetic
system.

A spin one system in equilibrium with fixed $T,H$ and $D$ can most easily be
envisaged by considering it to be a small subsystem of a much larger spin
system which acts as a $reservoir$. This fixes the equilibrium densities of
the extensive quantities $U$, $M$ and $Q$ of the subsystem to be the same as
the reservoir values so that the global entropy of system plus reservoir is
maximized. However, at any instant, the respective fluctuating random
variables $\mathcal{F}_{1},\mathcal{F}_{2}$ and $\mathcal{F}_{3}$ in the
subsystem Hamitonian could take up values away from the global mean. Thus, in
its three dimensional thermodynamic state space $\Gamma$ with axes $U,M,$ and
$Q$ the subsystem would spontaneously fluctuate from its reservoir determined
equilibrium point. It is important to remember here that these are equilibrium
fluctuations so that even as the subsystem fluctuates away from the global
average it always satisfies the fundamental thermodynamic equation, eq.(18).

If all the fluctuations are unconstrained then the Riemannian metric of
fluctuations is three dimensional. We take the Massieu function as our
starting point and differentiate it twice with respect to the entropic
intensive variables $\beta,\nu,\mu$ to generate the metric, which we call the
$grand$ metric $g$. In our geometric notation we have
$x^{1}=\beta,x^{2}=\nu,x^{3}=\mu$, and for the extensive quantities
$X_{1}=U,X_{2}=M,X_{3}=Q$. Then the grand metric is the covariance matrix for
fluctuations in extensive variables,

$g_{\,ij}=\frac{\partial^{2}\psi}{\partial x^{i}\,\partial
x^{j}}=\langle\Delta X_{i}\,\Delta X_{j}\rangle$ (20)

A complimentary way of looking at fluctuations is via the three dimensional
parameter space $\mathcal{M}$ with axes $T,H$ and $D$ or, equivalently,
$\beta,\nu,\mu$. Thus, when the subsystem undergoes a spontaneous fluctuation
its $intrinsic$ intensive quantities, following their respective equations of
state in eq.(19), fluctuate about their mean values fixed by the constant
reservoir potentials. Therefore, while by construction the reservoir
potentials are constant the subsystem potentials can fluctuate. In this sense,
the intensive variables are thermodynamic quantities, inasmuch as they are
allowed to fluctuate, as opposed to the coupling strengths $J$ and $K$ which
are fixed parameters. Another instructive way of looking at intensive
fluctuations would be think of small spatial inhomogeneities in $T,H$ or $D$
on the scale of the subsystem size. This would in turn induce an “exchange” of
extensive quantities between the system and the reservoir as per the Le
Chatelier-Braun principle, in such a direction as to reduce the inhomogeneity,
[29]. *** It might be mentioned here that for the magnetic systems the
extensive quantities like the magnetization are unconstrainable, [29].
However, the formal fluctuation theory is equally applicable by consideration
of a large control subsystem far removed from the system of interest and of
size square root of the reservoir such that it renders the magnetization, etc
globally conserved by absorbing non-conserving fluctuations, [14]. The inverse
of the grand metric in eq.(20) above gives the equilibrium fluctuations in the
entropic intensive variables. We write it with raised indices, so that
following Einstein summation $g^{ik}g_{kj}=\delta^{1}_{j}$

$(g^{-1})_{ij}=g^{ij}=\langle\Delta x^{i}\,\Delta x^{j}\rangle$ (21)

From the above metric the fluctuations in the intensive variables $T,H,D$ can
be found by obtaining their differential relation with the entropic intensive
variables, [33],

$\displaystyle\Delta T$ $\displaystyle=$
$\displaystyle-\frac{1}{\beta^{2}}\,\Delta\beta$ $\displaystyle\Delta H$
$\displaystyle=$
$\displaystyle\frac{\nu}{\beta^{2}}\,\Delta\beta-\frac{1}{\beta}\,\Delta\nu$
$\displaystyle\Delta D$ $\displaystyle=$
$\displaystyle-\frac{\mu}{\beta^{2}}\,\Delta\beta+\frac{1}{\beta}\,\Delta\mu$

Using eq.(21) above the fluctuations in intensive variables can be calculated.
The variance of $T$ is

$\langle(\Delta\,T)^{2}\rangle=\frac{1}{\beta^{4}}\,g^{11}.$ (23)

The variance of $H$ is

$\langle(\Delta\,H)^{2}\rangle=\frac{\nu^{2}}{\beta^{4}}\,g^{11}-2\frac{\nu}{\beta^{3}}\,g^{12}+\frac{1}{\beta^{2}}\,g^{22}$
(24)

and that of $D$ is

$\langle(\Delta\,D)^{2}\rangle=\frac{\mu^{2}}{\beta^{4}}\,g^{11}-2\frac{\mu}{\beta^{3}}\,g^{13}+\frac{1}{\beta^{2}}\,g^{33}$
(25)

Similarly we could obtain the cross moments like $\langle\Delta T\Delta
H\rangle$, etc.

The word “grand” in the metric $g$ refers to the grand canonical ensemble
wherein the system is in full thermodynamic contact with the reservoir, with
all the extensive quantities fluctuating about the mean, [33]. On the other
hand, we could also consider $restricted$ equilibrium fluctuations wherein the
spin one system undergoes spontaneous motion only along limited directions in
its state space. Let us explain this in more detail. Normally a thermodynamic
constraint is understood in the context of some $process$ a system undergoes.
For example, a gas in a piston could undergo an isobaric expansion so that a
fixed pressure constrains the set of values of energy $E$, volume $V$ and
particle number $N$ accessible to the equilibrium system. In its usual
meaning, however, the isobaric constraint would imply a constant pressure only
in the mean. This is understood by a consideration of the isobaric constraint
$P(U,V,N)=constant$ as a hypersurface in the state space to which the
equilibrium states of the gas would remain confined. However, at any instant
the spontaneous equilibrium fluctuations could always take the system to
nearby points away from the constant $P$-hypersurface. We could now impose a
stronger “$P$-constraint” on the dynamics so that even the spontaneous
thermodynamic fluctuations of the gas remain restricted to the
$P$-hypersurface. We call it a $canonical$ constraint since it has the
connotation of a canonical ensemble wherein while some variable are exchanged
with the surrounding and hence can fluctuate about mean others are held
strictly constant within “walls”. Here, we expand the scope of the term to
include restriction on fluctuations in the intensive quantities also, as
discussed for the present example of a gas in a piston. For the spin one model
in principle one could restrict the full set fluctuations in as many ways as
one could slice the state space. Some of them however, while mathematically
admissible appear to be inaccessible or contrived experimentally. For example,
on an $M$-surface wherein the magnetic moment fluctuations are suppressed, the
fluctuations in $Q$ would have to peculiarly orchestrated. We shall return to
this in a moment.

As discussed in the preceding, the equilibrium fluctuations can be pictured
equally well in the state space $\Gamma$ or the parameter pace $\mathcal{M}$.
Since, as we shall see below, it is the intensive variables that are usually
the independent variables in calculations for the spin one problem our
reference space shall be the parameter space $\mathcal{M}$ by default and
unless otherwise stated the co-ordinates $x^{i}$ will refer to the entropic
intensive variables $\beta,\nu,\mu$. The fluctuation metric for a
thermodynamic system with a given canonical constraint is simply the metric
induced on the corresponding hypersurface from the ambient grand metric. Let
the co-ordinates for the parameter space be $x^{i}$ with $i=1..n$. Then a
general canonical constraint is given by setting some function $f$ of the co-
ordinates $x^{i}$ to be zero,

$f(x^{1},x^{2},..)=0$ (26)

This defines a co-dimension one hypersurface in the state space which can be
labelled by $n-1$ co-ordinates $y^{a}$ intrinsic to the hypersurface. We shall
call it the $f$-surface. The components of the grand metric along the
$f$-surface give the projection metric in the $n$ dimensional parameter space,

$g_{(f)\,ij}=g_{ij}-\,\frac{\partial_{i}\,f\,\partial_{j}\,f}{g^{kl}\partial_{k}f\,\partial_{l}f}$
(27)

where $\partial_{i}f$ is along the normal to the $f$-surface. In additon to
the $n$-dimensional projection metric we can also write down the
$n-1$-dimensional induced metric on the $f$-surface with its components along
the $n-1$ intrinsic co-ordinates $y^{a}$ labelling the hypersurface,

$h_{(f)\,ab}=g_{(f)\,i\,j}\frac{\partial x^{i}}{\partial
y^{a}}\,\frac{\partial x^{j}}{\partial y^{b}}=g_{\,i\,j}\frac{\partial
x^{i}}{\partial y^{a}}\,\frac{\partial x^{j}}{\partial y^{b}}$ (28)

The second equality stems from the fact that a tangent vector living on the
$f$-surface is orthogonal to the gradient vector $\partial_{i}f$. The two
equivalent metrics in eq.(27) and eq.(28) provide complimentary information on
fluctuations. Thus the induced metric $h_{f}$ gives the fluctuations in co-
ordinates intrinsic to the $f$-surface,

$h_{(f)\,ab}=\langle(\Delta Y_{a}\,\Delta Y_{b})\rangle_{f}$ (29)

while

$h_{(f)}^{\,ab}=\langle(\Delta y^{a}\,\Delta y^{b})\rangle_{f}$ (30)

Here $Y_{a}$ labels suitable extensive quantities conjugate to the local
coordinates defined on the hypersurface. On the other hand, the projection
metric directly reads off the $f$-constrained variance in the quantities
defined on the full parameter space,

$\displaystyle g_{(f)\,\,ij}$ $\displaystyle=$ $\displaystyle\langle(\Delta
X_{i}\,\Delta X_{j})\rangle_{f}$ $\displaystyle g_{f}^{\,ij}$ $\displaystyle=$
$\displaystyle\langle(\Delta x^{i}\,\Delta x^{j})\rangle_{f}$

We can immediately put to use the preceding results for the spin one model.
Let us first write down the geometry obtained by restricting fluctuations in
the intensive variable $D$. A constant $D$-hypersurface in the $\beta-\nu-\mu$
parameter space is the hyper-plane

$f(\beta,\nu,\mu)=\mu-\beta D=0$ (32)

A suitable choice of local co-ordinates on the $D$-surface is

$y^{1}=\beta\,\,\,\,\mbox{and}\,\,\,\,y^{2}=\nu.$ (33)

so that global co-ordinates relate to them on the $D$-surface as

$\beta(y^{1},y^{2})=y^{1},\,\,\nu(y^{1},y^{2})=y^{2},\,\,\mu(y^{1},y^{2})=D\,y^{1}$
(34)

Using eq.(32)-eq.(34) in eq.(28) the components of the induced metric $h_{D}$
on the $D$-surface are

$\displaystyle h_{(D)\,11}$ $\displaystyle=$ $\displaystyle
g_{11}+2D\,g_{13}+D^{2}g_{33}$ $\displaystyle h_{(D)\,12}$ $\displaystyle=$
$\displaystyle g_{12}+2D\,g_{32}$ $\displaystyle h_{(D)\,22}$ $\displaystyle=$
$\displaystyle g_{22}$ (35)

where the components $g_{ij}$ have their usual interpretation as variance of
extensive quantities, see eq.(20). Evidently, the results for the induced
metric in eq.(35) can be read off directly by simply setting $D$ to constant
in eq.(16) which gives

$d\psi=-U_{1}\,d\beta-M\,d\nu\,\,\,\,\,\,\,(D\,\,\mbox{constant}),$ (36)

where $U_{1}=U+QD$ is an enthalpy like term. By taking second derivatives of
$\psi$ with respect to $\beta$ and $\nu$ keeping $D$ constant, we obtain all
the components of $h_{D}$.

It can checked from eq.(LABEL:gf) and eq.(25) that on the $D$-surface the
fluctuations in $Q$ are somewhat suppressed in comparison to the unconstrained
case while the fluctuations in $M$ remain unaffected,

$\displaystyle\langle\,(\Delta Q)^{2}\,\rangle_{D}$ $\displaystyle=$
$\displaystyle\langle\,(\Delta
Q)^{2}\,\rangle-\frac{1}{\beta^{2}\,\langle\,(\Delta D)^{2}\,\rangle}$
$\displaystyle\langle\,(\Delta M)^{2}\,\rangle_{D}$ $\displaystyle=$
$\displaystyle\langle\,(\Delta M)^{2}\,\rangle$ (37)

Not surprisingly therefore, as we shall see in the next section, the scalar
curvature on the $Q$-surface is aligned more with the fluctuations in the
magnetization than the quadrupole moment.

Similarly, we can develop the geometry of the $H$-surface by projecting out
the grand metric along the hypersurface defined by $\nu+\beta H=0$, where $H$
is a constant. Following the same procedure, the components of the induced
metric $h_{H}$ along the local co-ordinates $y^{1}=\beta,y^{2}=\mu$ are,

$\displaystyle h_{(H)\,11}$ $\displaystyle=$ $\displaystyle
g_{11}-2Hg_{12}+H^{2}g_{22}$ $\displaystyle h_{(H)\,12}$ $\displaystyle=$
$\displaystyle g_{13}-Hg_{23}$ $\displaystyle h_{(H)\,22}$ $\displaystyle=$
$\displaystyle g_{33}$ (38)

Once again, the components of $h_{H}$ can be read off by setting $H$ to
constant in eq.(16),

$d\psi=-U_{2}\,d\beta-Q\,d\mu\,\,\,\,\,\,\,(H\,\,\mbox{constant}),$ (39)

Here $U_{2}=U-MH$ is an enthalpy like term. One can check from eq.(LABEL:gf)
and eq.(24) that on the $H$-hypersurface the $M$ fluctuations are weakened
while the $Q$ fluctuations remain unrestricted.

$\displaystyle\langle\,(\Delta M)^{2}\,\rangle_{H}$ $\displaystyle=$
$\displaystyle\langle\,(\Delta
M)^{2}\,\rangle-\frac{1}{\beta^{2}\,\langle\,(\Delta H)^{2}\,\rangle}$
$\displaystyle\langle\,(\Delta Q)^{2}\,\rangle_{H}$ $\displaystyle=$
$\displaystyle\langle\,(\Delta Q)^{2}\,\rangle$ (40)

Evidently, as our subsequent discussion shall bear out, the scalar curvature
on the $H$-surface encodes the fluctuations in $Q$ instead of $M$.

Apart from the two preceding examples, we can also obtain hypersurface
geometries corresponding to fixed $Q$ or fixed $M$. Using eq.(LABEL:gf) we can
obtain, after some algebra, projection metrics on the surfaces
$Q(\beta,\nu,\mu)=constant$ or $M(\beta,\nu,\mu)=constant$. Equivalently, we
can take partial Legendre transforms of the entropy. Starting from eq.(16) we
obtain ,

$d\psi_{q}=-U\,d\beta-M\,d\nu+\mu\,dQ$ (41)

where $\psi_{q}=\psi+\mu\,Q$. Similarly, we obtain

$d\psi_{m}=-U\,d\beta-Q\,d\mu+\nu\,dM$ (42)

where $\psi_{m}=\psi+\nu\,M$. The hypersurface metric can now be obtained by
successive partial differentiation of the Massieu functions $\psi_{q}$ or
$\psi_{m}$ by differentiating with respect to the entropic intensive variables
while holding $Q$ or $M$ constant. However, as alluded to earlier, we shall
not pursue these geometries here. The reason for this is that, owing to the
kinematic coupling between $M$ and $Q$, freezing out fluctuations in one of
them strongly constrains the fluctuations in the other. For example, using
eq.(LABEL:gf), the fluctuations in $M$ at constant $Q$ are

$\langle(\Delta\,M)^{2}\rangle_{Q}=\langle(\Delta\,M)^{2}\rangle-\frac{1}{||\partial
Q||}\left(\frac{\partial Q}{\partial\nu}\right)^{2}$ (43)

where $||\partial Q||=g^{ij}\partial_{i}Q\partial_{j}Q$. On the other hand, in
eq.(37) we see that the fluctuations in $M$ remain fully unconstrained on the
$D$-surface.

We note that the scalar curvatures for all the two dimensional geometries can
be obtained as sectional curvatures along their respective $f$-surfaces of the
three dimensional Riemann curvature tensor of the grand metric. We shall call
the scalar curvatures associated with the $H$-surface and the $D$-surface as
the curvatures $R_{q}$ and $R_{m}$ respectively. The full three dimensional
grand scalar curvature shall be denoted by $R_{g}$.

##  4 Geometry of the one dimensional spin one model.

We now discuss the behaviour of the one dimensional spin one model through its
state space geometry. Following our previous discussion, we shall be
investigating the scalar curvatures associated with two geometries, namely
that of the $D$-surface and the $H$-surface and the three dimensional scalar
curvature of the full fluctuation metric.

The free energy per spin can be doubly differentiated via eq.(36) to obtain
the fluctuation metric on the $D$-surface and, similarly, via eq.(39) to
obtain the metric on the $H$-surface. We remind ourselves that there are two
connections to be checked here. First, the $weak$ conjecture of Ruppeiner
which relates the state space scalar curvature to the correlation length over
a range not necessarily limited to the critical region, eq.(2). Second, the
$strong$ conjecture equating the scalar curvature to the singular part of the
free energy near criticality upto a universal constant of order unity, eq.(1).

Figure 4: Temperature plots of $-R$ and $\xi$ for the Ising model in zero
magnetic field with the coupling $J$ set to unity. Curvature goes as twice the
correlation length at low temperatures.

Figure 5: Temperature plots of $-R$ and $\xi$ for the Ising model in non-zero
magnetic field $H=0.1$ with the coupling $J=1$. Curvature maintains a steady
distance with the correlation length.

As a warm-up and to set the stage we do a lightning review of the geometry of
the $1D$ ferromagnetic Ising model first discussed in [14, 15]. The free
energy and the correlation function can be obtained via the transfer matrix in
the standard manner. The singular part of free energy near the pseudocritical
point $T=0,H=0$ in the $\beta-H$ plane has the well known scaling form, [35]

$\psi_{s}=e^{-2\,J\,\beta}\left[1+\left(\frac{H\,\beta}{e^{-2\,J\,\beta}}\right)^{2}\right]^{1/2}$
(44)

and the correlation length as

$\xi^{-1}=\log\left(\frac{e^{2\beta H}+W+1}{e^{2\beta H}-W+1}\right)$ (45)

where $W^{2}=-2e^{2\beta H}+e^{4\beta H}+4e^{2(H+2J)\beta}+1$.

while the scalar curvature is obtained as a simple expression, [15]

$R=-\cosh(\beta\,H)\frac{1}{\sqrt{\exp(-4J\,\beta)+\sinh^{2}(\beta\,H)}}-1$
(46)

Therefore, it can be easily checked that $R$ follows the Ruppeiner equation at
the pseudocritical point $H=0,\beta\to\infty$ with $\kappa=1$ and $-R\to 2\xi$
near the pseudocritial point. Satisfyingly, even for non zero magnetic field,
which takes the system away from criticality, the curvature $R$ closely
“covers” the correlation length with the former asymptoting to $-2$ as $\xi$
decays to zero for large $\beta$. This is shown in fig.(5) and fig.(5). Thus,
an exploration of the one dimensional Ising model reinforces both the strong
and the weak ends of Ruppeiner’s conjecture. Both the features will appear as
a common theme in the geometry of one dimensional the spin one model to be
discussed below. In addition, as we shall comment shortly, the Ruppeiner
equation also helps straightforwardly obtain the scaling form of the free
energy in eq.(44), [4].

###  4.1 $\bf{J>0}$

We first investigate the case of positive $J$. The zero temperature phase
diagram for this case is given in fig.(3). The lines marked $a$ in blue and
$f$ in red are sites of pseudocriticality and two-phase coexistence while the
point $b$ is a triple point as well as the pseudotrictritical point at zero
temperature. On line $a$ the two order parameters $M$ and $Q$ have independent
correlation lengths $\xi_{1}$ and $\xi_{2}$ as discussed earlier. While the
fluctuations in the magnetization diverge in the limit $T=0$ on the $a$ line,
the quadrupole fluctuations remain bounded and decay to zero on approaching
$T=0$. Zero temperature value of $Q$ saturates to unity on either side of and
also on the $a$ line. From several scalar curvature vs. correlation length
plots we shall see that $\xi_{1}$ and $\xi_{2}$ correspond well with the
scalar curvatures $R_{m}$ and $R_{q}$ respectively for much of the range of
parameter values. There are variations to the theme however, possibly
reflecting other attributes of $R$ apart from its association with the
correlation length that require further investigation. On the other hand some
of the unexpected behaviour could also be attributable to the geometry of the
surfaces on which we have calculated the two curvatures. We shall not make
much attempt in this work to place some of the “anomalous” behaviour of the
curvatures within the context of underlying statistical interactions. Indeed,
our analysis shall be necessarily limited at times, owing to the fact that a
fundamental understanding of many features of $R$, especially its sign, is
still a work in progress. In light of this it is important to undertake a
somewhat detailed survey of the state space geometry in order to record
patterns in variations, some of which could possibly have a bearing on a
future analysis. In the following subsections we shall sort our observations
of the scalar curvatures $R_{q}$, $R_{m}$ and $R_{g}$ for variations in
parameters $H,D,J,\mbox{and }K$. Before doing so, we verify Ruppeiner’s
stronger conjecture near criticality and show that it directly suggests the
form of the scaling function associated with the singular free energy.

####  4.1.1 Scaling form for the free energy

We now analyze the free energy near (pseudo)criticality with the aim of
connecting it to Ruppeiner’s conjecture relating the singular part of the free
energy to the scalar curvature, eq.(1). While the relation is not rigorously
proved it has a strong physical justification and there is ample verification
of it in several instances, see [1, 4] and references therein. One of the
contributions of this paper is to add one more instance of its validity. The
conjecture is expected to be exact in the limit of the critical point where it
will be called the Ruppeiner equation. For the two dimensional parameter space
the order unity universal constant $\kappa$ in eq.(1) works out to, [3],

$\kappa=\frac{(b-1)(2b-a)}{a(a-1)}$ (47)

where $a$ and $b$ are the universal critical exponents in the scaling form of
the singular free energy,

$\psi_{s}(\beta,h)=n_{1}\,|t|^{a}\,Y(\frac{n_{2}h}{|t|^{b}})$ (48)

In the expression for the singular free energy $t(\beta)$ is the reduced
temperature, $h$ the ordering field and $Y(z)$ is the scaling function which
depends only on the combination $z=h|t|^{-b}$. The constants $n_{1}$ and
$n_{2}$ are non-universal and system dependent. For a system with a finite
temperature critical point, like the Ising model in two or more dimensions the
reduced temperature is $t=(\beta_{c}-\beta)/\beta_{c}$ and the ordering field
is $h=H\beta$

For the three dimensional parameter space too the scalar curvature has been
shown to be proportional to the singular part of the free energy at
criticality, with the order unity proportionality constant $\kappa_{3}$ given
as, [3],

$\kappa_{3}=\frac{4a-4b-4c-4abc-a^{2}+4b^{2}+4c^{2}}{2(a-1)a}$ (49)

where the critical exponent $c$ is for the additional field $u$, apart from
$t$ and the ordering field $h$, in the expression,

$\psi_{s}(\beta,h,u)=n_{1}\,|t|^{a}\,Y(\frac{n_{2}h}{|t|^{b}},\frac{n_{3}u}{|t|^{c}})$
(50)

As it turns out, the two dimensional curvatures will be sufficient for our
purpose of calculating the spin scaling functions in this section. We shall
briefly revisit the constant $\kappa_{3}$ in eq.(49) when we discuss the three
dimensional curvature $R_{g}$ in later sections.

Returning to the proportionality constant $\kappa$ in eq.(47), it equals $1$
when either $a=b$ or $a=2\mbox{ and }b=0$. One dimensional models with short
range interaction have critical point at zero temperature so that the the
standard form of the reduced temperature will not apply. Instead, for example
in the case of the one dimensional ferromagnetic Ising model the reduced
temperature can taken to be $t=e^{-p\,J\beta}$ where $p$ is any positive
number. On setting $p=1$ the leading singular free energy becomes,[34]

$\psi_{s}(h,t)=e^{-2J\beta}Y\left(\frac{h}{e^{2J\beta}}\right)\hskip
28.45274pt\mbox{(1-D Ising model)}$ (51)

With $a=b=2$ above we get $\kappa=1$. Indeed, it would be the unity if $a$ and
$b$ were both multiplied by any positive constant. In a recent work, [4],
Ruppeiner worked out the form of the scaling function $Y(z)$ for the one
dimensional Ising model by using the geometric Ruppeiner equation. This was
done by first substituting the scaling form of the singular free energy
$\psi_{s}$, given in eq.(48), in the expression for $R$ which in two
dimensional parameter space can be written as, [1]

$R=-\frac{1}{2}\frac{\begin{vmatrix}\psi_{,11}&\psi_{,12}&\psi_{,22}\\\
\psi_{,111}&\psi_{,112}&\psi_{,122}\\\
\psi_{,112}&\psi_{,122}&\psi_{,222}\end{vmatrix}}{\begin{vmatrix}\psi_{,11}&\psi_{,12}\\\
\psi_{,21}&\psi_{,22}\end{vmatrix}^{2}}$ (52)

and then equating the expression to $\kappa$ times inverse of $\psi$ in
eq.(1). We can easily replace $\psi$ by $\psi_{s}$ in eq.(52) because near
criticality the leading behaviour of higher derivatives of the free energy is
determined by its singular term. The resulting equation is a third order
equation for the scaling function $Y(z)$ which contains only the variable $z$,
the function $Y$ and its derivatives and the constants $a$ and $b$, with all
dependencies on $\beta$ and $h$ dropping out. This is very much in keeping
with the conjectured exactness of the Ruppeiner equation near criticality so
that the only unknowns are the $universal$ exponents and the $universal$
scaling function. For the Ising model where the exponents $a=b$, the Ruppeiner
equation further simplifies to

$\frac{Y\left(Y\left(zY^{(3)}+2Y^{\prime\prime}\right)+z\left(zY^{\prime\prime
2}-Y^{\prime}\left(zY^{(3)}+2Y^{\prime\prime}\right)\right)\right)}{2\left(Y-zY^{\prime}\right)^{2}Y^{\prime\prime}}=\kappa$
(53)

which becomes independent of the critical exponents.

A series expansion of $Y(z)$ in the equation above, keeping in mind that it
must be an even function of $z$ for the ferromagnetic Ising model, produces a
solution which matches exactly with its known form, namely

$Y(z)=n_{1}\sqrt{1+(n_{2}z)^{2}},$ (54)

with $n_{1}=n_{2}=1$ for the Ising model. In addition, the series expansion of
the left hand side of eq.(53) shows that the proportionality constant $\kappa$
is unity for the Ising model, which is consistent with eq.(47) for $a=b$.
While it is relatively straightforward to obtain the scaling function for the
specific case of the one dimensional ferromagnetic Ising model ( see for
example [35]), in general it is a difficult problem in statistical mechanics
as emphasized in [4]. Indeed, it is remarkable that the scaling function can
be obtained directly by a $thermodynamic$ equation.

As we shall discuss in the sequel, for the one dimensional spin one model too
we have the exact relation $\kappa=1$ in the pseudocritical and the
pseudotricritical limit. Barring a few most of the checks are numerical. On
approaching the $a$ line the curvature $R_{m}$ follows the Ruppeiner equation
while $R_{q}$ remains finite, and on approaching the $f$ line and the
pseudotricritical point $c$ both $R_{m}$ and $R_{q}$ follow the Ruppeiner
equation. We recall here that $R_{m}$ and $R_{q}$ are both two dimensional
sectional curvatures in a three dimensional parameter space, with $\mu,\nu$
and $\beta$ as one possible set of thermodynamic coordinates. $R_{m}$ is the
intrinsic scalar curvature of the $D$-plane while $R_{q}$ is that of the
$H$-plane. The scaling function of a free energy with three scaling fields
would, in general, be $Y(z_{1},z_{2})$, with $z_{1}=h/t^{a}$ and
$z_{2}=u/t^{c}$. Here, $h$ and $u$ are some linear combination of $\mu$, $\nu$
and $\beta$. For the case where both $h$ and $u$ are relevant so that $b,c>0$,
the scaling function $Y$ becomes a genuine two parameter function and the
Ruppeiner equation connecting the sectional curvatures with $\psi$ will result
in a partial differential equation for $Y$. However, if one of the scaling
fields, say $u$, is irrelevant with $c<0$ then the dependence on $u$ drops out
near criticality and the scaling function becomes a one parameter function.
Surely, with the $a$ and $f$ lines each being a locus of pseudocritical points
the directions along the respective lines will not be relevant, so that the
scaling functions near these lines will be one parameter functions $Y(z)$.
Therefore, with a proper definition of the scaling field $h$, we expect the
previously discussed differential equation for $Y$ to be valid in the vicinity
of these lines. Clearly, on the $a$ line $Y(z)$ will be an even function of
$z$ which should be proportional to $\nu=H\beta$ on the line. It is not a
priori clear whether the scaling function is symmetric about the $f$ line. As
our numerical studies seem to suggest, probably there are different scaling
functions $Y_{\pm}(z)$ above and below the $f$ line. Nevertheless, we can
treat each of the functions as symmetric in the scaling field $h$. At least
our numerical checks seem to confirm this expectation in retrospect.

Of the two possibilities consistent with $\kappa=1$, the values $a=2,b=0$ are
easily ruled out because that would mean practically no dependence of the
singular free energy term on the scaling field $h$ at very low temperatures,
which is certainly not the case as can be easily checked. With $a=b$ equations
(53) and (54) apply to the spin one model as well. With more coupling terms in
its Hamiltonian the reduced temperature $t$ for the spin one model is not
always equal to $e^{-\beta J}$ but is of the more general form $e^{-\beta X}$
where $X$ is a linear combination of the coupling strengths in the
Hamiltonian. Moreover, the form of $X$ changes depending on the relative
strength of the coupling terms. The scaling form of the leading singular term
in the neighbourhood of the $a$ line and in the vicinity of the $f$ lines
outside the “$f$-wings”(i.e, to the left of $f$ line) can be written as

$\displaystyle\psi_{s}$ $\displaystyle=$ $\displaystyle
n\,t\,Y\left(\frac{h}{n\,t}\right)$ (55) $\displaystyle=$ $\displaystyle
n\,t\left[1+\left(\frac{h}{n\,t}\right)^{2}\right]^{\frac{1}{2}}\hskip
22.76228pt\mbox{( 1-D spin one model, outside $f$ wing)}$

where the value of the constant $n$ and also the form of the reduced
temperature $t$ depend upon the relative strength of the coupling terms. Note
that our definition of the reduced temperature is such that the exponents
$a=b=1$. In the vicinity of the pseudocritical $a$ line and the
pseudotricritical point $c$ of fig.(3) the ordering field $h=\beta H$, while
around the upper and lower $f$ lines it is $h=(H-D+J+K)\beta$ and
$h=(H+D-J-K)\beta$ respectively.

As mentioned earlier, the scaling form inside the “$f$-wings” (to the right)
is different. Our numerical checks suggest that it is of the following form,

$\displaystyle\psi_{s}$ $\displaystyle=$ $\displaystyle
n\,t\left[1+\left(\frac{h}{n\,t}\right)^{2}\right]^{-\frac{1}{2}}\hskip
22.76228pt\mbox{( 1-D spin one model, inside $f$ wing)}$ (56)
$\displaystyle\sim$ $\displaystyle\frac{n^{2}t^{2}}{h}\hskip 28.45274pt(h>>t)$

Indeed, this scaling form does not satisfy the differential equation (53) for
$Y(z)$ with $\kappa=1$. Nonetheless, our checks seem to suggest this form of
the scaling. We defer a detailed investigation of this observation to a future
investigation.

We first consider the $a$ line of fig.(3) for which $D<J+K$. We recall that
the $a$ line is a locus of pseudocritical points for the magnetization but not
for the quadrupolar order. Satisfyingly, it is possible to separate the
singular and regular parts of the free energy (or the Massieu function) near
the $a$ line which we shall explain now. Recalling the expression for zero
field Massieu function in eq.(8) we rewrite it keeping in mind that $D<J+K$ to
get an expression of the form

$\psi=(J+K-D)\beta+\mbox{ln}\frac{1}{2}\left[1+e^{(D-J-K)\beta}+e^{-2J}+\sqrt{1+w}\right]$
(57)

where

$w=e^{2(D-J-K)\beta}+8e^{(D-2J-2K)\beta}+e^{-4J}+2e^{-2J}-2e^{(D-3J-K)\beta}-2e^{(D-J-K)\beta}$
(58)

The logarithmic term in eq.(57) goes to zero as $\beta\to\infty$. This is
because with $D<J+K$ all the exponential terms within the logarithm, including
those comprising $w$, are less than one and approach zero as $\beta$ becomes
larger. Therefore, we can consider the first term in eq.(57) as the regular
part $\psi_{r}$ of the free energy near criticality and expect the singular
part to be contained in the logarithm. The dominant part of the singular free
energy must be that term in the logarithm which is the slowest to decay to
zero as $\beta$ becomes larger. In order to filter out such terms we express
the log term for large $\beta$ as follows,

$\psi-\psi_{r}=\mbox{ln}\,\left[1+\frac{1}{2}e^{(D-J-K)\beta}+\frac{1}{2}e^{-2J}+\frac{1}{2}(\sqrt{1+w}-1)\right]$
(59)

and keep just the linear term in its expansion. However we must look at all
the terms in the expansion of $\sqrt{1+w}$. This is because it leads to a
systematic cancellation of all the terms containing $e^{(D-J-K)\beta}$ or any
of its higher powers which are inadvertently generated on expanding the square
root term. The correlation length $\xi_{1}$ can also be analysed similarly on
the $a$ line. In the limit of large $\beta$ it goes as the inverse of the
singular free energy consistent with hyperscaling near the critical point.
Finally, the leading singular free energy and the correlation length for
magnetization fluctuations are obtained in the limit of the pseudocritical
point as

$\displaystyle\psi_{s}$ $\displaystyle=$
$\displaystyle\,\frac{1}{2}\xi_{1}^{-1}\,=e^{-2J\beta}\hskip
54.06006pt(D<2K,D<J+K,H=0)$ (60) $\displaystyle=$
$\displaystyle\frac{3}{4}\,\xi_{1}^{-1}=\,3\,e^{-2J\beta}\hskip
45.52458pt(D=2K,D<J+K,H=0)$ $\displaystyle=$
$\displaystyle\xi_{1}^{-1}\,\,\,\,\,=2\,e^{-(2J+2K-D)\beta}\hskip
28.45274pt(2K<D<J+K,H=0)$

For all cases where $K\geq J$ the second and third possibilities will not
arise. On the other hand, in the Blume-Capel limit $K=0$ we get different
scaling behaviour for $D<0$, $D=0$ and $D>0$. The full scaling form for the
free energy in the vicinity of the $a$ line is now easily written in the form
of eq.(55) by choosing the values of $n=1,3$ and $2$ respectively for the
above three cases in eq.(60), the reduced temperature $t$ as the exponential
term on the $r.h.s$ in each case and the ordering field as $h=H\beta$.

We now consider on the zero field line the point $D=J+K$ which is a triple as
well as pseudotricritical point. As can be checked from its expression in
eq.(57) the free energy now goes to zero in the zero temperature limit. Its
expression simplifies to

$\psi=\ln\left[1+\frac{1}{2}e^{-2J\beta}+\frac{1}{2}e^{-2J\beta}\sqrt{1+8e^{(3J-K)\beta}}\right]$
(61)

After a simple expansion of the logarithm we take the dominant part of the
free energy as the one which is the slowest to decay as the pseudotricritical
point is approached from non-zero temperatures. Depending on whether $K$ is
less than, equal to or greater than $3J$ we have three cases,

$\displaystyle\psi_{s}$ $\displaystyle=$
$\displaystyle\sqrt{2}\,e^{-(J+K)\beta/2}\hskip 28.45274pt(K<3J,\,D=J+K,H=0)$
(62) $\displaystyle=$ $\displaystyle 2\,e^{-2J\beta}\hskip
62.59596pt(K=3J,\,D=J+K,H=0)$ $\displaystyle=$ $\displaystyle
e^{-2J\beta}\hskip 68.28644pt(K>3J,\,D=J+K,H=0)$

At the pseudotricritical point both the correlation functions $\xi_{1}$ and
$\xi_{2}$ diverge with the following asymptotic relation to the singular free
energy,

$\displaystyle\xi_{1}$ $\displaystyle=$ $\displaystyle
2\,\xi_{2}\,\,\,=\psi_{s}^{-1}\hskip 48.36958pt(K<3J,D=J+K,H=0)$ (63)
$\displaystyle=$
$\displaystyle\xi_{2}\,\,\,\,\,\,\,=\frac{2}{3}\psi_{s}^{-1}\hskip
45.52458pt(K=3J,D=J+K,H=0)$ $\displaystyle=$ $\displaystyle
2\,\xi_{2}\,\,\,=\psi_{s}^{-1}\hskip 54.06006pt(K>3J,D=J+K,H=0)$

Once again, the full scaling form for the free energy in the vicinity of the
pseudotricritical point is expressed straightforwardly in the form of eq.(55)
by choosing the values of $n=\sqrt{2},2$ and $1$ respectively for the three
cases in eq.(62), the reduced temperature $t$ as the exponential term on the
$r.h.s$ in each case and the ordering field as $h=H\beta$.

Finally, we consider the $f$ line on which $D=|H|+J+K$. With non-zero $H$ it
is easier to work with the numerical solutions for the eigenvalues of the
transfer matrix. The $f$ line is a pseudocritical point for both the magnetic
order and the quadrupolar order when approached from non-zero temperatures.
For non zero field both the order parameters have the same correlation length
$\xi_{1}$, [28]. Just as for the pseudotricritical point the free energy goes
to zero on the $f$ line and its singular part is double of the inverse
correlation length

$\psi_{s}=\frac{1}{2}\,\xi_{1}^{-1}=e^{-(J+K)\beta/2}\hskip
28.45274pt(D=|H|+J+K,\mbox{$f$ line})$ (64)

The full scaling form of the free energy can be written in a manner similar to
the previous cases. We write the critical free energy explicitly for the
region in the vicinity of the positive $f$ line, to its left, as

$\psi_{s}=e^{-(J+K)\beta/2}\left[1+\left(\frac{H-D+J+K}{e^{-(J+K)\beta/2}}\right)^{2}\right]^{1/2}$
(65)

and, to its right as

$\psi_{s}=e^{-(J+K)\beta/2}\left[1+\left(\frac{H-D+J+K}{e^{-(J+K)\beta/2}}\right)^{2}\right]^{-1/2}$
(66)

Figure 6: Plots of the ratio of the singular part of free energy to its three
different scaling forms depending on different reference points. In all sub-
figures the erference point $P$ at which the free energy is calculated has
$J=0.3,K=0.2,H=0.01,D=0.49$ and it lies outside the $f$ “wings”. In $(a)$ the
scaling form refers to the $a$ line , in $(b)$ the scaling form refers to the
pseudotricritical point and in $(c)$ to the $f$ line.

Figure 7: Plots of ratio of the free energy to its scaling forms for points
inside the $f$ ‘wings’. The parameters $J=0.3$ and $K=0.2$ for all sub-
figures. In $(a)$ the reference point at $D=0.5001$ and $H=0$ is referred to
the vicinity of the pseudotrictritical point at $D=0.5,H=0$. In $(b)$ and
$(c)$ the reference point at $D=0.5101$ and $H=0.01$ is referred,
respectively, to the neighbourhood of the pseudocritical $f$-line and the
pseudotricritical point.

In fig.(6) we present for a fixed point three plots of the ratio of the
singular part of the free energy to its three different scaling forms obtained
by a consideration of the point as existing in three intersecting
neighbourhoods. The reference point which we shall label as $P$ lies slightly
to the left of and above the pseudotricritical point $c$ (see fig.(3) for
reference). For all the three sub-figures all the parameter values are the
same. In fig.6(a) the point $P$ is considered in the neighbourhood of the $a$
line from a pseudocritical point directly below it. The singular free energy
at $P$ is then taken to be $\psi-\psi_{r}$ as expressed in eq.(59) and the
scaling form is ascertained from eqs.(60) and (55). In fig.6(b) the point $P$
is considered as belonging to the neighbourhood of the pseudotricritical point
$c$ and the appropriate scaling form chosen accordingly following eqs.(62) and
(55). Finally, in fig.6(c) the point $P$ is consireded in the vicinity of a
pseudocritical point on the $f$ line at the same height as $P$, with the
scaled free energy given in eq.(65). In all three cases the ratios approach
unity at higher $\beta$ which is consistent with the respective scaling
expressions for the free energy.

The uniformity in scaling discussed above is not present in the neighbourhood
to the right of the pseudotricritical point and the $f$ line, in the region
which is inside the $f$ “wings” (refer to fig.(3)) . As stated earlier, within
the wing region the scaling function follows eq.(56). In fig.7(a) we plot for
a point to the right of the $c$ point on the zero field line the ratio of the
free energy to its scaling form, which is obtained from eq.(62) and (56). Thus
this point is considered as being in the vicinity of the pseudotricritical
ponit $c$. In fig.7(b) we consider a point in the wing region with a small non
zero $H$. Considering it to be in the vicinity of a point on the $f$ line at
the same height we plot the ratio of the free energy to its scaling form as
given in eq.(66). For both sub-figures (a) and (b) the ratio approaches unity
thus affirming that these points do belong to their considered neighbourhoods.
However, if now we consider the point of sub-figure 7.(b) to be in the
neighbourhood of the pseudotricritical point and plot the ratio, as in
fig.7(c), we see that it does not converge to unity but to $0.5$. Therefore,
our observations seem to suggest that only the zero field line constitutes a
genuine neighborhood of the pseudotricritical point in the wing region. We can
say that while the $a$ line, the $f$ line and the $c$ point share their
neighbourhoods on the left hand side of the pseudotricritical point, the $f$
line and the $c$ point do not share their neighbourhoods on the right hand
side. Presumably, this difference has to do with the presence of
pseudocritical $a$ line on the left side but not on the right. In any case,
the correctness of eq.(66) stands verified.

Having utilized the strong conjecture of Ruppeiner to obtain the scaling
function from thermodynamic geometry we turn our attention now to the
investigation of the thermodynamic curvature vis-a-vis the correlation length,
a relation which encompasses both the strong and the weak aspects of the
conjecture. We shall examine the curvatures $R_{q}$ and $R_{m}$ for different
parameter ranges and also comment on the three dimensional scalar curvature
$R_{g}$. The main focus of our investigation will be the parameter values
corresponding to the pseudocritical $a$ and $f$ lines and the
pseudotricritical $c$ point and we shall observe the curvature, correlation
length or the free energy as we approach these points from high temperatures.
We shall also discuss the geometry when the magnetic field is non-zero.

####  4.1.2 Geometry of the zero field case, $H=0$

##### 4.1.2.1 ${\bf{J>K}}$

This case is the most relevant to the BEG model inasmuch as it refers to
$\mbox{He}^{3}-\mbox{He}^{4}$ mixtures since for such mixtures $K$ is much
less than $J$. Similarly, it is also relevant to the Blume Capel model for
which $K=0$. The $a$ line, the pseudotricritical point $c$ and the part of of
$x$ axis to its right in fig.(3) constitute the zero field line. The $a$ line
is a pseudocritical line with respect to $M$ fluctuations but not for $Q$
fluctuations.

First we consider $R_{q}$ and $\xi_{2}$ on the whole of the zero field line.
We note that we can obtain a closed form expression for $R_{q}$ by setting $H$
to zero in the thermodynamic metric defined on the constant $H$ surface,
eq.(39). While the zero field expression for $R_{q}$ is too large to be shown
here, we obtain much smaller expressions for some special cases which we shall
present on appropriate occasions. It is seen that for values of $D<K-J$ the
magnitude of $R_{q}$ maintains an approximately constant distance with the
correlation length $\xi_{2}$, with $R_{q}$ converging to $-1$ and the latter
to zero as $T$ tends to zero.

Figure 8:  Plots of $-R_{q}$ and $\xi_{2}$ against $\beta$ for different
ranges of $D$ values, with $H=0$, $D<J+K$ and $K<J$. In all sub-figures
$J=0.05,K=0.03$. In $(a)$ $D=-0.029<K-J$, in $(b)$ $D=-0.02=K-J$ and in $(c)$
$D=0.05>K-J$. In $(a)$ and $(b)$ the curvature remains almost parallel to the
decaying correlation length while in $(c)$ it diverges away after running
parallel for a while.

At $D=K-J$ the curvature $R_{q}$ converges to a value slightly less than $-1$
which depends only on the ratio $K/J$, with the minimum value of $-1.25$ in
the Blume-Capel limit $K=0$ irrespective of the value of $J$. In figs. 8.(a)
and 8.(b) where we plot together the variation in the magnitude of $R_{q}$ and
$\xi_{2}$ with respect to $\beta$, it can be seen that they maintain a
separation of approximately one lattice unit for $D\leq K-J$. We also note
that the curvature-correlation length correspondence for the $H$-surface has
no connection to criticality here.

For $K-J<D<K+J$ the curvature $R_{q}$ does not match with $\xi_{2}$ for the
full range of temperature. While it parallels with $\xi_{2}$ for small values
of $\beta$ (high temperature) it soon enough diverges away to more and more
negative values. The negative divergence however is always less than or equal
to the rate of divergence of $\xi_{1}$ or, equivalently, the negative
divergence of $R_{m}$ as discussed below. While the apparently anomalous
divergence could possibly provide some clue to the nature of underlying
statistical interactions, we have already stated that we shall refrain from
hypothesizing and defer such questions to a future investigation.

At the pseudotricritical point $D=J+K$, the negative of $R_{q}$ and $\xi_{2}$
both remain parallel and diverge asymptotically as $e^{(J+K)\beta/2}$, which
is in line with eq.(62). It can be checked that asymptotically $-R_{q}\sim
2\,\xi_{2}$, which is consistent with eq.(63) with $R_{q}\to\psi^{-1}$ in the
pseudotricritical limit. Significantly, even at temperatures much farther from
zero $R_{q}$ is already approximately twice $\xi_{2}$ as can be seen in fig.
9.(a).

Beyond the tricritical point, for $J+K<D<2J$, the magnitude of $R_{q}$ and
$\xi_{2}$ again run parallel to each other with the former converging to $-2$
and the latter decaying to zero as $T$ tends to zero. This is shown in
fig.9(b). For $D=2J$ the curvature $R_{q}$ converges to a value between $-2$
and $-4$ with the latter value fixed for the Blume-Capel limit irrespective of
$J$. For $D>2J$ $R_{q}$ behaves anomalously and diverges to negative infinity
after following the correlation length $\xi_{2}$ for small $\beta$, as shown
in fig.9(c). The asymptotic rate of divergence is $e^{(D-2J)\beta}$.

Figure 9:  Plots of $-R_{q}$ and $\xi_{2}$ against $\beta$ for different
ranges of $D$ values, with $H=0$, $D\geq J+K$ and $K<J$. In all sub-figures
$J=0.05,K=0.03$. In sub-figure $(a)$, with $D=0.08=K+J$(pseudotricritical
point), the ratio $-R_{q}/\xi_{2}$ is plotted against the inverse temperature.
In $(b)$ with $D=0.09<2J$ and in $(c)$ with $D=0.11>2J$ both $R_{q}$ and
$\xi_{2}$ are plotted against $\beta$.

In the Blume-Capel limit $K=0$ the tricritical point expression for $R_{q}$
simplifies to

$R_{q}=\frac{\mathcal{N}_{1}}{\mathcal{D}_{1}}\hskip 14.45377pt(D=J,H=0,K=0)$
(67)

where $\mathcal{N}_{1}$ and $\mathcal{D}_{1}$ are given in eq.(76) of the
Appendix. The high temperature limit of $R_{q}$ can be expressed in terms of
the ratio $\alpha=K/J$, irrespective of $D$,

$R_{q}=\frac{-4\alpha^{4}+27\alpha^{2}-243}{2\left(\alpha^{2}+9\right)^{2}}\hskip
72.26999pt(\beta\to 0)$ (68)

For the Blume-Capel case $R_{q}\to-3/2$ as $T\to\infty$.

Figure 10: In all sub-figures $J=0.05,K=0.04$.$(a)$ Plot of $R_{m}/\xi_{1}$
vs. $\beta$ with $D=0.02$.$(b)$ Plots of $-R_{g}\,(\psi-\psi_{r})$ and
$-R_{m}\,(\psi-\psi_{r})$ vs. $\beta$ for $D=0.082$. $(c)$ Similar as $(b)$
but with $D=J+K=0.09$.

We now consider $R_{m}$ and the three dimensional scalar curvature $R_{g}$ on
the zero field line. Closed form expressions are not available for these
curvatures since the eigenvalues of the transfer matrix with non-zero $H$ are
to be obtained by solving a cubic equation. The curvature $R_{m}$ the
correlation length $\xi_{1}$ run parallel to each other upto the
pseudotricritical point, i.e, for $D\leq J+K$. The ratio of the curvature to
correlation length is fully consistent with eq.(60) for all range of values of
the parameter $D$, with $R_{m}\to\psi_{s}^{-1}$ towards zero temperature. In
fig.(10)(a) we observe that the ratio $R_{m}/\xi_{1}$ soon enough approaches
the value of $2$ which, for the chosen parameter values in the figure, is
consistent with eq.(60). The full curvature $R_{g}$ is seen to closely
parallel the curvature $R_{m}$. In fig. (10)(b) and (c) we plot together the
products of $R_{m}$ and $R_{g}$ with $\psi_{s}$ for different values of $D$,
with the latter sub-figure at the pseudotricritical value $D=J+K$. We note
here that while the two dimensional scalar curvature $R_{m}$ is seen to follow
the Ruppeiner equation with $\kappa=1$ consistent with eq.(47)same it is not
the same with the three dimensional curvature $R_{g}$. While $R_{g}$ still
follows the Ruppeiner equation at criticality in that it goes as the inverse
of singular free energy, the proportionality constant $\kappa$ does not appear
to be a universal number, though it is still of order one. There are patterns
to the variation. For example, at the pseudotricritical point for $K<3J$ the
constant $\kappa_{3}=3.5$ while for $K\geq 3J$ it is equal to $3$. In this
work we do not pursue issues of whether or not the proportionality
$\kappa_{3}$ follows eq.(49) and shall take it up in the future. Beyond the
pseudotricritical point while the correlation length $\xi_{1}$ decays to zero
at low temperatures $R_{m}$ does not match with $\xi_{1}$ and diverges in the
negative direction.

##### 4.1.2.2 ${\bf{J=K}}$

Figure 11: In all sub-figures $J=K=0.03$ and $H=0$. $(a)$ and $(b)$ are plots
of $R_{q}$ and $\xi_{2}$ vs. $\beta$ for $D=0.05$ and $D=0.07$ respectively .
In $(c)$ plots of $-R_{q}\,\psi$ and $-R_{m}\,\psi$ are shown at the
pseudotrictritical point for $D=J+K=0.06$

For the case when the spin coupling strength is equal to the quadrupole
coupling, the curvature $R_{q}$ shows a very good correspondence with
$\xi_{2}$ for all values of $D$. For $D<J+K$ the curvature $R_{q}$ runs
parallel to $\xi_{2}$ and asymptotes to $-1$ as the correlation length tends
to zero for low temperatures. Similarly, for $D>J+K$ it asymptotes to $-3/2$
as $\xi_{2}$ tends to zero. For the pseudotricritical point $D=2J$, $-R_{q}$
and $\xi_{2}$ diverge to infinity as $e^{J\,\beta}$ with $R_{q}\to 2\,\xi_{2}$
at low temperatures, consistent with eq.(62) and eq.(63). For $J=K$ the zero
field expression for $R_{q}$ reduces considerably and it can be expressed as
the fraction

$R=\frac{\mathcal{N}_{2}}{\mathcal{D}_{2}}$ (69)

where the numerator and denominator are given in the appendix, eq.(77).

$R_{m}$ corresponds well with $\xi_{1}$ upto the pseudotricritical point. For
$D<2J$ the curvature $R_{m}$ equals twice $\xi_{1}$ for low temperatures while
for the pseudotricritical point $D=2J$ it equals $\xi_{1}$ at low temperatures
in line with eq.(63). For $D<2J$ the asymptotic divergence of $\xi_{1}$ and
$R_{m}$ is $e^{2J\beta}$, while at the pseudotricritical point it is
$e^{J\beta}$, all of which follows eq.(62). Once again, beyond the
pseudotricritical point $R_{m}$ does not match with $\xi_{1}$ and diverges in
the negative direction in a manner similar to the $J>K$ case.

In fig.11(a) the curvature $R_{q}$ and $\xi_{2}$ are plotted for a value of
$D<J+K$ and in fig.11.(b) for $D>J+K$. For both cases there is a very good
correspondence between curvature and correlation length as discussed above. In
fig.11.(c) the product of the magnitude of $R_{m}$ and $R_{q}$ with the free
energy is plotted for the pseudotricritical value of $D$. The two curvatures
are seen to approach each other and the product quickly reaches a value of
$\kappa=1$ following the Ruppeiner equation. Furthermore, for the three
dimensional curvature the product $R_{g}\psi$ approaches $3.5$ as mentioned
earlier.

##### 4.1.2.3 ${\bf{J<K}}$

We now survey the geometry for the case when the quadrupole coupling strength
is greater than the dipole coupling. For values of $D\leq K-J$ the curvature
$R_{q}$ asymptotes to $-1$ towards zero temperature. However, contrary to the
previous cases, $-R_{q}$ does not rise to more positive values around the
regions of maxima of $\xi_{2}$. Instead, the negative curvature dips towards
the negative direction around the maxima of $\xi_{2}$, only to asymptote to
$1$ at lower temperatures. The nearer $D$ gets to $K-J$ the longer is the
range of temperature for which the negative of curvature dips and stays to
less positive values, even crossing the $y$-axis for $D$ close to $K-J$,
before finally rising to positive values and approaching $1$ for very low
temperatures. In fig.12.(a) is shown a plot of $-R_{q}$ and $\xi_{2}$ $vs.$
$\beta$ for for $D<K-J$. There is a pronounced dip in the negative curvature
before if finally converges to one. At $D=K-J$ the negative curvature $-R_{q}$
no longer rises to positive values at low temperatures but asymptotes to a
negative value depending only on the ratio $K/J$. For example for $K/J=8/3$
the curvature $-R_{q}$ goes to $-19/36$ at zero temperature, as shown in fig.
12.(b). For $K-J<D<K+J$, the $-R_{q}$ does not follow the correlation length
$\xi_{2}$ and diverges to negative infinity as $e^{(D-K+J)\beta}$, as shown in
fig. 12.(c). This is always slower than the divergence to infinity of
$\xi_{1}$ which goes as $e^{2J\beta}$ following eq.(9).

Figure 12:  Plot of $R_{q}$ and $\xi_{2}$ $vs.$ $\beta$ with $(a)$
$J=0.03,K=0.1,D=0.02<K-J,H=0$, $(b$) with $J=0.03,K=0.08,D=K-J=0.05,H=0$ and
$(c)$ with $J=0.03,K=0.08,D=0.051>K-J,H=0$.

At the pseudotricritical point $D=J+K$ the curvature $R_{q}$ behaves
differently compared to the previous two cases $J>K$ and $J=K$. Here, it is
also important to note the values of the quadrupole moment $Q$ and its
fluctuation $\langle(\Delta Q)^{2}\rangle$ the reason for which will soon be
clear. For $K\leq 3J$ $R_{q}$ diverges to negative infinity as
$e^{(J+K)\beta/2}$ and follows the Ruppeiner equation with $\kappa=1$. For
$K<3J$ the quadrupole moment $Q$ saturates to $1/2$ at zero temperature and
its fluctuation $\langle(\Delta Q)^{2}\rangle$ diverges to infinity as
$e^{(J+K)\beta/2}$. At $K=3J$ the quadrupole moment equals to $2/3$ in the
zero temperature limit. For $3J<K<5K$ the curvature continues to diverge to
minus infinity at the same rate as the correlation length $\xi_{2}$ which now
diverges as $e^{2J\beta}$. However, now at low temperatures -$R_{q}\,\psi=a$
where $a$ is less than $1$ and it reduces substantially as $K$ approaches
$5J$.

Figure 13: Zero field plots with $D$ at pseudocritical values $D=J+K$ and
$K>J$. $(a)$ Plot of the products of the singular free energy $\psi$ with
$R_{q}$ and $R_{m}$, for $J=0.03,K=0.14<5J,D=0.17$. $(b)$ Plot of $R_{q}$ and
$<(\Delta Q)^{2}>$ $vs$. $\beta$ with $(b)$ $J=0.03,K=0.15=5J,D=0.18$ and
$(c)$ $J=0.03,K=0.16>5J,D=0.19$

For example as $K$ increases from $0.149$ to $0.1499$, for $J=0.03$, $a$
decreases almost ten times from $\frac{299}{14400}$ to $\frac{2999}{1440000}$.
Thus, while $R_{q}$ still diverges to negative infinity in this parameter
range, it does not strictly follow the Ruppeiner equation. This can be seen in
fig. 13(a) where $R_{m}$ clearly follows the Ruppeiner equation but not
$R_{q}$. Meanwhile, for all values of $K>3J$ the quadrupole moment tends to
$1$ and its fluctuation goes as $e^{(5J-K)\beta}$. Therefore, as $K$
approaches $5J$ the divergence in quadrupole fluctuations kepdf flattening.
This might appear curious since the quadrupole-quadrupole correlaton length
$\xi_{2}$ grows steadily as $e^{2J\beta}$. At $K=5J$, the quadrupole
fluctuation completely flattens and approaches the fixed value of $4$. Exactly
paralleling this situation the curvature $R_{q}$ too stops diverging and
asymptotes to a fixed negative value of $-79/4$ at $K=5J$, as seen in fig.
13(b). The correlation length continues to diverge at the same rate as before.
For $K>5J$ the quadrupole fluctuation decays to zero in the zero temperature
limit. Now the curvature $R_{q}$ diverges to $positive$ infinity at the same
rate as the correlation length, namely as $e^{2J\beta}$. This is shown in fig.
13(c). This is interesting because in general the scalar curvature diverges to
negative infinity at the critical point. For $D>J+K$ the curvature $R_{q}$
diverges to positive infinity as $e^{(D-2J)\beta}$ after briefly following the
correlation length.

The apparent inconsistency of the quadrupole correlation length diverging to
infinity and at the same time the quadrupole fluctuations decaying to zero at
the pseudotricritical point is understood by examining the expression for the
quadrupole fluctuation,[28],

$\frac{\partial^{2}\psi}{\partial\mu^{2}}=\frac{1}{N}\langle(\
\sum_{i=1}^{N}\Delta
Q_{i})^{2}\rangle=\frac{Q(1-Q)}{1-\lambda_{3}/\lambda_{1}}$ (70)

so that while at the pseudotricritical point $\lambda_{3}\to\lambda_{1}$ which
implies $\xi_{2}\to\infty$, the numerator could compete with the zero in the
denominator if $Q\to 1$. This is more pronounced for $K>5J$ when the
quadrupole fluctuation now decays to zero while the correlation length
continues to diverge as before. It is significant that the curvature $R_{q}$
encodes this peculiar behaviour of quadrupole fluctuations.

The curvature $R_{m}$ follows the Ruppeiner equation for all $K>J$ and for all
$D\leq K+J$. The asymptotic ratio between $R_{m}$ and the singular free energy
(and also with $\xi_{1}$) is reached much before criticality thus again
emphasizing the weak conjecture of Ruppeiner. Beyond the pseudotricritical
point $R_{m}$ behaves exactly as in the previous cases in that it diverges at
zero temperature to negative infinity even as $\xi_{1}$ decays to zero.

####  4.1.3 Geometry of non-zero H: the $f$ line and beyond

For non-zero magnetic field there is only one correlation length, namely
$\xi_{1}$, for both the spin and the quadrupole fluctuations. We note that we
cannot obtain closed form expressions for the scalar curvatures in this case.
The non zero field region comprises the $f$ line and its vicinity, the regions
to its left and right and the vicinity of the pseudotricritical point. We
shall survey each region one by one.len1a

As stated earlier, the $f$ line is the boundary at $T=0$ such that to its
right, the energetically preferred state is the one with $S=0$ for all spins
while to its left the lowest energy state has all the spins aligned with $S=1$
for positive magnetic field. On the $f$ line both the curvatures $R_{q}$ and
$R_{m}$ diverge and follow the Ruppeiner equation with $\kappa=1$ as shown in
fig.(16). As is apparent from the figure, both curvatures merge into each
other at low temperatures which is consistent with the fact there is only one
correlation length for non-zero field. The scaling of scalar curvatures is
$e^{(J+K)\beta/2}$ which follows from eq.(64).

Figure 14: A diagram showing trends in the curvatures $R_{m}$ and $R_{q}$ in
the $H-D$ plane with $J=0.1,K=0.09$. See text for figure description.

In order to explore further the geometry of non-zero $H$ we refer to fig.(14)
which records trends in the curvatures $R_{m}$ and $R_{q}$. The values of
parameters $J$ and $K$ are fixed as $0.1$ and $0.09$ respectively. Turning our
attention first to the vicinity of the red coloured $f$ line we notice that
starting from small non-zero values of $H$ onwards there is a wedge shaped
band around the $f$ line, bordered by blue dots on the left and green squares
on the right. Within this band it is observed that the two curvatures $R_{m}$
and $R_{q}$ overlap very strongly at low temperatures and moreover they show a
very good correspondence with the decaying correlation length. Further, the
band is characterized by an additional high temperature local maximum (or a
“hump”) in both the $R_{m}\,vs.\,\beta$ and $R_{q}\,vs.\,\beta$ plots, see
fig.(16). While in this work we shall not seek to understand the physical
reason for the hump within the “f-band” we comment that the geometry retains a
possible memory of the pseudocritical $f$ line within the band. We note that
smaller values of $H$ (about $0.05$ in the figure), namely for points close to
the pseudotricritical point, the f-band shrinks to nearly zero and the “hump”
feature disappears too, suggesting a unique neighbourhood of the tricritical
point.

Figure 15: Plot of the products $R_{m}\psi$ and $R_{q}\psi$ $vs.$ $\beta$ on
the $f$-line. The parameters $J=0.03,K=0.1,H=0.01,D=H+J+K=0.14$.

Figure 16: A representative plot of $R_{m}$, $R_{q}$ and $\xi_{1}$ in the $f$
band, with parameter values $J=0.1,K=0.09,D=0.45$ and $H=0.25$. The hump in
the curvatures and their substantial overlap is noticeable.

Moving leftwards from the band towards decreasing values of $D$, one
encounters a succession of points at which one or the other of $R_{q}$ or
$R_{m}$ diverge to positive or negative values. Of course, in all this the
correlation length $\xi_{1}$ always decays to zero. While the overlap is
excellent within the $f$-band the two curvatures begin to separate on moving
leftwards but still follow each other till the curve marked by green coloured
“tilde” shaped symbols is reached. From this point leftwards $R_{q}$ diverges
to negative infinity until on the extreme left the curve marked by the blue
coloured “caret” symbols is reached from whence the curvature $R_{q}$ begins
to diverge towards positive infinity. On the other hand, on crossing the curve
marked by the purple coloured “plus” symbols the curvature $R_{m}$ diverges
towards positive infinity followed by its divergence towards negative infinity
on crossing the curve marked by red coloured “asterisk” symbols. There occur
two regions in which the direction of divergence of both the curvatures is in
opposite direction. The first region being between the “plus” and the
“asterisk” in which $R_{q}$ and $R_{m}$ diverge towards negative and positive
infinity respectively. The other region being beyond the curve marked by caret
symbols to the left of which the direction of divergence of both the
curvatures are reversed. To the right of $f$-band, for $D$ greater than but
close enough to $J+K+H$, $R_{m},R_{q}$ and the correlation length $\xi_{1}$
have a good match till the curve marked by pink coloured “open circles” is
reached. To the right of the pink curve both the curvatures diverge to
negative infinity. The three dimensional curvature $R_{g}$ more or less
follows the curvature $R_{m}$ so we do not describe it separately here.

Admittedly, we do not understand the physics behind the several sign changes
of the curvatures here but given that the sign of the scalar curvature is
commonly associated with the underlying statistical interactions it is
important to record patterns in its variation. Further analysis shall be the
subject of a future investigation.

###  4.2  J=0 case, the one dimensional Griffiths model

The zero temperature phase diagram for the one-dimensional Griffiths model,
namely the BEG model in the limit $J=0$, was described earlier in sec.(4) and
represented in fig.(3). It was mentioned that the whole of the $x$-axis is
non-critical in this case and that the $f$ line with $D=K+H$ is the locus of
pseudocritical points at zero temperature. The absence of the spin coupling
strength $J$ simplifies the transfer matrix in eq.(7) enough to render closed
form expressions for two eigenvalues, with the third eigenvalue identically
zero.

$\displaystyle\lambda_{\pm}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{-(D+H)\beta}\left(e^{(D+H)\beta}+e^{(2H+K)\beta}+e^{\beta
K}\pm\sqrt{W}\right)$ (71)

where

$W=(e^{(D+H)\beta}+e^{(2H+K)\beta}+e^{K\beta})^{2}+4(e^{(D+H)\beta}+e^{(D+3H)\beta}-e^{(D+H+K)\beta}-e^{(D+3H+K)\beta})$
(72)

From the above equation we can calculate the free energy $\psi$ as log
$(\lambda_{+})$ and the correlation length $\xi$ as
$1/\,\mbox{log}\,(\lambda_{+}/\lambda_{-})$. Starting with the free energy
closed form expressions can be obtained for $R_{m}$, $R_{q}$ and $R_{g}$. We
note that while the numerical values of the curvatures in this section can be
obtained from those of the previous section on setting $J=0$, closed form
expressions were not available earlier for non-zero $H$. We can obtain the
scaling form of the free energy on the $f$ line by obtaining its approximate
expression for large values of $\beta$

$\psi\underset{\beta\to\infty}{\to}\frac{1}{2}e^{-2\,H\,\beta}+\frac{1}{2}\sqrt{4e^{2\beta
H}+4e^{4\beta H}+e^{\beta K}}$ (73)

The leading singular term of the free energy and its relation to the
correlation length on the $f$ line with $D=H+K$ can now be obtained as

$\displaystyle\psi_{s}$ $\displaystyle=$
$\displaystyle\,\xi^{-1}\,=e^{-2\,H\,\beta}\hskip 113.81102pt(H<K/4)$ (74)
$\displaystyle=$
$\displaystyle\frac{1+\sqrt{5}}{2\sqrt{5}}\,\xi^{-1}=\frac{1+\sqrt{5}}{2}\,e^{-2H\beta}\hskip
45.52458pt(H=K/4)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\xi^{-1}\,\,\,\,\,=2\,e^{-K\beta/2}\hskip
91.04872pt(H>K/4)$

It would also be instructive to know about the thermal fluctuations in
magnetization and quadrupole moment in view of their bearing upon the geometry
in a manner analogous to sec.(4.1.2.3). We present the quadrupole fluctuations
on the $f$ line $D=K+H$ for different ranges of $H$ values,

$\displaystyle\langle(\Delta Q)^{2}\rangle$ $\displaystyle=$ $\displaystyle
2\,e^{(6H-K)\beta}\hskip 99.58464pt(H<K/4)$ (75) $\displaystyle=$
$\displaystyle\frac{2}{5\sqrt{5}}\,e^{2H\beta}\hskip 99.58464pt(H=K/4)$
$\displaystyle=$ $\displaystyle\frac{1}{4}\,e^{K\beta/2}\hskip
113.81102pt(H>K/4)$

It is evident from the above equation that for $K/6<H<K/4$ the rate of growth
of quadrupole fluctuations is progressively slower than that of the
correlation length, with the former eventually completely flattening at
$K=6H$. For $H<K/6$ the quadrupole fluctuation decays to zero towards zero
temperature while the correlation length continues to grow. For the
magnetization fluctuations the rate remains the same as above excepting the
case $H<K/8$ for which the rate of decay of fluctuations becomes
$e^{-2H\,\beta}$.

Figure 17:  Temperature plots of curvatures and correlation length for the
pseudocritical $f$ line in the Griffiths model. In $(a)$
$H=0.12,D=0.42,K=0.30$, in $(b)$ $H=0.06,K=0.30,D=0.36$ and in $(c)$
$H=0.04,K=0.30,D=0.34$

We now survey the geometry of the Griffiths model, starting with the $f$ line.
For $H>K/4$ both $R_{m}$ and $R_{q}$ show a negative divergence and overlap
with each other at low temperatures. They follow the Ruppeiner equation with
$\kappa=1$. On the other hand the three dimensional curvature $R_{g}$ follows
the Ruppeiner equation with $\kappa_{3}=3/2$. The relation with the
correlation length can be worked out from eq.(74) and is obtained as
$R_{m},R_{q}\sim 2\xi$ and $R_{g}\sim 3\xi$. This is shown is fig.17(a) where
$R_{m}$ and $R_{q}$ are seen to completely overlap with each other. For
$K/4<H<K/6$ $R_{m}$ and $R_{q}$ still show a negative divergence at the same
rate as the correlation length (or $\psi^{-1}$), however the proportionality
constant $\kappa$ in the Ruppeiner equation is less than unity and steadily
decreases to zero as $H$ approaches $K/6$. Moreover, its strength only depends
on the ratio of $K/H$. For the grand curvature $R_{g}$ the constant
$\kappa_{3}=4$ now. This is shown in fig.17(b) where the correlation length is
now seen to lead the curvatures $R_{m}$ and $R_{q}$ which again overlap with
each other. We could say that the sectional curvatures $R_{m}$ and $R_{q}$ are
sensitive to the same statistical interactions which cause the fluctuations to
flatten out as $H$ lowers from $K/4$ to $K/6$. Finally, for $H<K/6$ the
quadrupole fluctuations start decaying to zero. At the same time the
curvatures $R_{m}$ and $R_{q}$ now turn positive and diverge to positive
infinity at the same rate as the correlation length. Though the two curvatures
follow each other they are not as close in value as the previous cases. The
poroportionality constant $\kappa$ now ranges from less than unity to much
larger values as $H$ approaches zero while $\kappa_{3}$ continues to remain
$4$. This is shown in fig.17(c) and is favourably compared with an analogous
behaviour of $R_{q}$ in fig.(13) in sec.(4.1.2.3). In both cases some of the
curvatures show a positive divergence at the same rate as the correlation
length. The sign change is concomitant with the decay to zero of quadratic
fluctuations. While it is unusual for the state space scalar curvature to
undergo a positive divergence at criticality, see [1] for instance, we believe
that the same statistics which causes the quadratic fluctuations to decay near
criticality also causes a sign change in the curvatures here. We shall analyse
this further in a future investigation.

The $H=0$ axis does not become critical as mentioned earlier. This is
reflected very well by the respective curvatures. Fig.18(a) is a
representative zero field plot with $D<K$. The curvature $R_{m}$ quickly
converges to a value of $1$ while $R_{q}$ converges to $-2$ and $R_{g}$ to
$-3.5$, with the correlation length decaying to zero. At the triple point
$D=K$ the curvatures asymptote to $R_{g}\to-7.5,R_{q}\to-4,R_{m}\to-1$ and the
correlation length now does not decay to zero but converges to 1/log($2$). For
$D>K$ we report a minor but interesting feature which is ably captured by the
geometry of the model. As shown in fig.18(b) the quadratic fluctuation after
initially decaying to zero undergoes a spike in fluctuation. Exactly in the
middle of this spike the magnetization fluctuation sharply drops from $1$ to
$0$. The curvature $R_{q}$ remains exactly twice the correlation length as can
be seen from fig.18(c). The curvature $R_{m}$ and $R_{g}$ on the other hand
jump to large positive values at the temperature at which the magnetization
fluctuation drops to near zero values (not shown in the figure).

Figure 18:  $(a)$ Scalar curvatures and correlation length for the Griifiths
model with $H=0,K=0.30,D=0.31$. Plots of $(b)$ spin and quadrupole fluctuation
moments and $(c)$ $2\xi$ and $-R_{q}$ for $H=0.06,K=0.30,D=0.31$

Figure 19: Temperature plots of scalar curvatures and the correlation length
for the Griffiths model. In all sub-figures, $H=0.12,K=0.30$ and $D$ values
are less than but progressively closer to the $f$-line at $D=K+H$. In $(a)$
$D=0.36$, in $(b)$ $D=0.41$ and in $(c)$ we have $D=0.419$.

We turn now to non-zero $H$ values in regions away from the $f$ line. In
fig.(19) we take a representative selection of plots for a fixed value of $H$
and different values of $D$ upto the $f$ line and to its left. In all the sub-
figures the curvatures $R_{m}$ and $R_{q}$ show differing behaviour perhaps
signalling the different nature of underlying statistics, a question we shall
be addressing in a future investigation. On the other hand the grand curvature
$R_{g}$ appears to always correspond well with the correlation length, albeit
at an elevated value. To the right of the $f$ line all the curvatures change
sign and diverge to positive infinity (not shown).

##  5 Conclusions

In this work we have undertaken an extensive analysis of the thermodynamic
geometry associated with the one dimensional BEG model and its limiting cases.
The BEG model which is extensively used to model interacting systems with two
competing ordering processes preserves much of the richness of its phase
structure in the one dimensional case. In addition, being exactly solvable in
one dimension it offers an excellent opportunity to comprehensively probe its
geometry.

The three dimensional state space of the BEG model has been systematically
sectioned into two co-dimension one hypersurfaces, namely constant $H$-surface
and the constant $D$-surface. The associated scalar curvatures $R_{q}$ and
$R_{m}$ are found to be relevant to the fluctuations in the quadrupolar order
parameter $Q$ and the magnetic moment $M$ respectively. For $H=0$ the spin and
quadrupolar order parameters have separate correlation lengths $\xi_{1}$ and
$\xi_{2}$ and, remarkably enough, it is seen that the two sectional curvatures
encode these separately. The curvatures $R_{q}$ and $R_{m}$ not only follow
the Ruppeiner equation near (pseudo)criticality with $\kappa=1$ they also
satisfy the weak conjecture of Ruppeiner showing a very good correspondence
with the correlation length away from criticality and even in non-critical
situations. Making use of the Ruppeiner equation we are also able to ascertain
the scaling form of the free energy of the BEG model near criticality and
pseudocriticality. The three dimensional scalar curvature $R_{g}$ also
efficiently encodes the interactions in the system, especially in the limit of
the Griffiths model where the bilinear spin coupling is set to zero. One of
the key messages in our work is that for higher dimensional parameter spaces
the relevant sectional curvatures contain essential information not found in
the full scalar curvature. Therefore, for higher dimensional parameter spaces,
in addition to the full scalar curvature the relevant hypersurface geometries
must also be explored to gain a fuller picture of underlying statistical
interactions.

While $R_{g}$ is proportional to the inverse of singular free energy near
criticality, the proportionality constant $\kappa_{3}$, though of order unity,
is seen to vary depending on the parameter range. This variation in the
proportionality constant does not seem to accord with its universal constant
value as worked out in [13] and also verified there for the mean field case.
We also extensively record the sign changes in the scalar curvatures. While in
some cases the sign changes are found to encode the peculiar nature of
fluctuations in other cases we are not able arrive at a more fundamental
understanding of the same. We hope to address these unresolved issues in a
future investigation.

##  6 Acknowledgements

We thank George Ruppeiner for fruitful and encouraging discussions during the
early stages of the work. AS thanks Ritu Sharma and Rishabh Jha for
discussions and DST, Govt. of India for support through grant no.
MTR/2017/001001.

$\displaystyle\mathcal{N}_{1}$ $\displaystyle=$
$\displaystyle-2e^{-J\,\beta}\,(37e^{2\beta J}+4e^{16\beta J}+5We^{\beta
J}+(6W+24)e^{15\beta J}+(24W+334)e^{14\beta J}+$
$\displaystyle(37W+74)e^{3\beta J}+(54W+106)e^{4\beta J}+(106W+464)e^{5\beta
J}+(274W+980)e^{13\beta J}+$ $\displaystyle(304W+1100)e^{7\beta
J}+(316W+480)e^{6\beta J}+(396W+1702)e^{12\beta J}+$
$\displaystyle(676W+1671)e^{8\beta J}+(703W+1684)e^{9\beta
J}+(733W+1868)e^{11\beta J}+$ $\displaystyle(740W+2589)e^{10\beta J}+5)$
$\displaystyle\mathcal{D}_{1}$ $\displaystyle=$ $\displaystyle W(6e^{2\beta
J}+2e^{7\beta J}+2We^{\beta J}+We^{6\beta J}+(4W+27)e^{5\beta J}+$
$\displaystyle(5W+4)e^{4\beta J}+(6W+13)e^{3\beta J}+2)^{2}$ $\displaystyle W$
$\displaystyle=$ $\displaystyle\sqrt{e^{-2\beta J}+8e^{\beta J}}$ (76)

$\displaystyle\mathcal{N}_{2}$ $\displaystyle=$
$\displaystyle\mathcal{A}\,\mathcal{B}$ $\displaystyle\mathcal{A}$
$\displaystyle=$ $\displaystyle-2e^{-\beta J}\left(e^{3\beta
D}+(W+6)e^{\beta(D+2J)}+3(W+2)e^{\beta D}+\right.$
$\displaystyle\left.(W+6)e^{2\beta D}+e^{6\beta J}+(W+3)e^{4\beta
J}+(2W+3)e^{2\beta J}+W+1\right)$ $\displaystyle\mathcal{B}$ $\displaystyle=$
$\displaystyle 6e^{\beta(5D+J)}+e^{\beta(D+9J)}+4e^{4\beta D+3\beta
J}+e^{3\beta D+5\beta J}+3e^{2\beta D+7\beta J}+$ $\displaystyle
14(W+2)e^{\beta(D+3J)}+10(W+3)e^{\beta(D+5J)}+5(W+5)e^{2\beta D+5\beta
J}+2(W+6)e^{\beta(D+7J)}+$
$\displaystyle(6W+9)e^{\beta(D+J)}+(6W+28)e^{\beta(4D+J)}+(6W+34)e^{3\beta(D+J)}+(13W+27)e^{\beta(2D+J)}+$
$\displaystyle(14W+41)e^{\beta(3D+J)}+(18W+49)e^{2\beta D+3\beta J}+e^{11\beta
J}+(W+1)e^{\beta J}+$ $\displaystyle(W+5)e^{9\beta J}+2(2W+5)e^{7\beta
J}+2(3W+5)e^{5\beta J}+(4W+5)e^{3\beta J}$ $\displaystyle\mathcal{D}_{2}$
$\displaystyle=$ $\displaystyle W\left(e^{\beta D}+e^{2\beta
J}+W+1\right)\left(2e^{3\beta
D}+e^{2\beta(D+J)}+(W+6)e^{\beta(D+2J)}+3(W+2)e^{\beta D}+\right.$
$\displaystyle\left.(2W+7)e^{2\beta D}+e^{6\beta J}+(W+3)e^{4\beta
J}+(2W+3)e^{2\beta J}+W+1\right)^{2}$ $\displaystyle W$ $\displaystyle=$
$\displaystyle\sqrt{6e^{\beta D}+e^{2\beta D}-2e^{\beta(D+2J)}+2e^{2\beta
J}+e^{4\beta J}+1}$ (77)

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