# Energy Landscape analysis of metal-insulator transitions: theory and
application to Ca2RuO4, $R$NiO3 and their heterostructures

Alexandru B. Georgescu  Northwestern University, Department of Materials
Science and Engineering, Evanston, Illinois, 60208  Center for Computational
Quantum Physics, Flatiron Institute, 162 5th Avenute, New York, NY 10010
<EMAIL_ADDRESS>Andrew J. Millis  Center for
Computational Quantum Physics, Flatiron Institute, 162 5th Avenute, New York,
NY 10010  Department of Physics, Columbia University, 538 West 120th Street,
New York, New York 10027

###### Abstract

We present a general methodology that enables the disentanglement of the
electronic and lattice contributions to the metal-insulator transition by
building an energy landscape from numerical solutions of the equation of
state. The methodology works with any electronic structure method that
provides electronic expectation values at given atomic positions. Applying the
theory to rare-earth perovskite nickelates ($R$NiO3) and Ruddlesden-Popper
calcium ruthenates (Ca2RuO4) in bulk, heterostructure and epitaxially strained
thin film forms using equation of state results from density functional plus
dynamical mean field calculations we show that the electron-lattice coupling
is an essential driver of the transition from the metallic to the insulating
state in these materials.

##  I Introduction

The relative importance of electronic and lattice effects in driving phase
transitions in quantum materials is a subject of long-standing interest and
controversy. One issue of particular focus is the metal to insulator
transition (MIT), which in many materials is accompanied by a lattice
distortion [1]. Multiple studies have addressed various aspects of the
interplay between electronic and lattice contributions to the metal-insulator
transition both theoretically and experimentally [2, 3, 4, 5, 6, 7, 5, 8, 9,
10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 17, 20, 21, 22, 23, 24, 25, 26, 27,
28, 29, 30, 31, 14, 32, 33, 34, 35, 31, 36, 37, 38, 39, 12, 40, 41, 42, 2, 43,
44, 3, 45, 46, 45, 46, 39, 47, 48]. Machine learning approaches are
discovering new features from comparison of MIT and non-MIT materials [49,
50]. Yet, the relative importance of electronic and lattice effects in
producing a transition has remained unclear, in part because of the lack of an
appropriate theoretical framework and computational schema for evaluating the
needed quantities. This paper aims to rectify both issues.

Figure 1: Energy landscape for NdNiO3 as a function of electronic charge
disproportionation $\Delta N$ and lattice distortion $Q$ as obtained from the
energy functional methods introduced in this paper using data from Ref. [3].

Metal insulator transition phenomena are properly analysed by constructing an
energy landscape in the space of lattice distortions and electronic orders
such as that shown in Fig. 1. We show how to build these energy landscapes
from numerical data of the kind now available from modern quantum many-body
methods and we apply the theory to bulk, thin film and superlattice
heterostructures of $R$NiO3 and Ca2RuO4, two material families that exhibit
metal-insulator transitions but have different local physics.

Our work was inspired by the pioneering work of Fisher and collaborators[51],
who combined elegant experimental measurements with an insightful free energy
analysis to show that the observed second order ‘nematic’ transition in an
iron pnictide material had an intrinsically electronic origin. Our work may be
viewed as a generalization of the ideas of Fisher et. al to first order
transitions, where a more global view of the energy landscape is required, and
as an extension of the insightful equation of state analysis of Peil, Hampel,
Ederer and Georges [4] to the computation and interpretation of the full free
energy landscape. We make substantial use of concepts put forward in analyses
of the metal-insulator transition in epitaxially strained Ca2RuO4 [7] and in
nickelate heterostructures [3].

The rest of this paper is organized as follows: the Framework section (II)
shows how we build total energy landscapes. In sections III and IV we use the
formalism to analyse the metal-insulator transitions occurring in two
representative classes of materials: the rare earth nickelate perovskite RNiO3
compounds, where the transition involves a symmetry breaking lattice
distortion and two sublattice electronic “charge order” and Ca2RuO4 where the
transition is isostructural (no structural symmetry breaking) and involves
electronic “orbital order”. Section V presents some qualitative considerations
related to the thermally driven transitions. Section VI is a summary and
conclusion, assessing the work and outlining future directions.

##  II Framework

Following Refs. [7, 4, 3] we consider a free energy $F(\Delta N,Q)$ that
depends on an electronic order parameter, $\Delta N$, and a lattice
distortion, $Q$. The precise definitions of $\Delta N$ and $Q$ will depend on
the specifics of the system. We choose $(Q,\Delta N)=(0,0)$ as the equilibrium
configuration of the metallic phase. The insulating state then corresponds to
a configuration $(\Delta N,Q)\neq(0,0)$ and the question of the existence of a
purely electronic transition relates to the properties of $F$ as a function of
$\Delta N$ at $Q=0$.

A second order transition corresponds to a linear instability of the $(\Delta
N,Q)=(0,0)$ state, in other words to a change in sign of the smallest
eigenvalue of the Hessian matrix $\partial^{2}F/\partial^{2}\Delta N,Q$
evaluated at $\Delta N=Q=0$. In general the eigenvector associated with the
negative eigenvalue will have components along both $\Delta N$ and $Q$,
indicating that the electronic and lattice orders are coupled. A purely
electronically driven transition would correspond to a change in sign of
$\partial^{2}F/\partial^{2}\Delta N$, a lattice-driven transition would
correspond to a change in sign of $\partial^{2}F/\partial^{2}Q$, and in a
mixed situation neither ‘pure’ derivative changes sign but the lowest
eigenvalue of the Hessian matrix does change sign. In the case of the nematic
transition in pnictides, Fisher and co-workers were able [51] to
experimentally determine $\partial^{2}F(Q=0)/\partial\Delta N^{2}$ and show
that it changed sign at a temperature only slightly lower than the observed
nematic transition temperature, thereby establishing that a purely
electronically driven nematic transition exists in these materials, and is
simply slightly enhanced by coupling to the lattice.

The transitions of interest in this paper are first order, characterized by
the appearance of a new free energy minimum at which $\Delta N$ and $Q$ are
both different from zero (see e.g. Fig. 1). Study of first order transitions
requires global knowledge of the free energy. One may say that the transition
is electronically driven if $F(\Delta N,Q=0)$ has an extremum at a $\Delta
N\neq 0$ with $\partial^{2}F(\Delta N,0)/\partial\Delta N^{2}>0$ and with
energy lower than $F(0,0)$. One may say that the transition is lattice-
assisted if $F(\Delta N,0)$ has a local minimum at a $\Delta N\neq 0$ but that
the lattice coupling is required to make this minimum the ground states.
Finally if $F(\Delta N,Q=0)$ has only one minimum, at $\Delta N=0$ but
exhibits a stable minimum with $\Delta N\neq 0$ at a $Q\neq 0$ the we may say
that the transition is not electronically driven. The energy functional shown
in Fig. 1 shows a transition that is not electronically driven because the
global minima are at a $Q,\Delta N\neq 0$ but along the line $Q=0$ the
functional has only one minimum, at the origin. It is important to emphasize
that this distinction is separate from the question of the importance of
correlation physics. It is common in the literature to distinguish strongly
from weakly correlated materials according to whether or not current
implementations of density functional theory capture the physics; in current
language whether a density functional calculation of $F_{el}$ suffices; even
if the transition is not electronically driven in the sense given above,
beyond density functional correlations might be needed to obtain an
$F_{el}(\Delta N,Q)$ with an insulating minimum at the physical $Q$.

The global knowledge of $F$ required to decide this issue has been challenging
to obtain. Available quantum many-body methods can with reasonable
computational cost obtain reasonable estimates of the optimal $\Delta N$ that
minimizes $F$ at fixed atomic positions ($Q$), but finding the dependence of
$F$ on $\Delta N$ at fixed $Q$ or on $Q$ at fixed $\Delta N$ is difficult,
both because it is not straightforward to control $\Delta N$ independently of
$Q$ and because calculations of energies are very expensive.

Here we build on the observation that the metal-insulator transitions of
greatest current experimental interest share two features that greatly
simplify an analysis. First, the lattice energetics is to good approximation
harmonic [7, 3, 4], as shown from calculations [3] and by the observation that
phonon frequencies change only very slightly ($\sim 1\%$) across the metal-
insulator transition. Second, and closely related to the first point, the
coupling between the electronic order parameter and lattice modes is linear
and is well determined by standard theoretical methods [3, 7, 4], as are the
lattice energetics. This means that we may write, schematically, [4, 3, 7]:

$F(\Delta N,Q)=\frac{1}{2}KQ^{2}-\frac{1}{2}gQ\Delta N+F_{el}(\Delta N)$ (1)

Requiring stationarity of $F$ with respect to variations in $\Delta N$ and $Q$
yields the equations of state [4, 3]

$KQ=\frac{1}{2}g\Delta N$ (2)

and

$\frac{\partial F_{el}(\Delta N)}{\partial N}=\frac{1}{2}gQ$ (3)

Because $g$ and $K$ are known, Eq. 2 can be solved. Substituting the solution
back into Eq. 1 produces:

$\bar{F}(\Delta N)=-\frac{1}{8K}\left(g\Delta N\right)^{2}+F_{el}(\Delta N)$
(4)

From this point of view the key question is the magnitude of the ’lattice
stabilization energy’ $\frac{1}{8K}\left(g\Delta N\right)^{2}$: is it large
enough to create a new extremum at a $\Delta N\neq 0$? Is it large enough to
ensure that the extremum at $\Delta N=0$ is unstable? Answering this question
requires knowledge of $F_{el}$.

Peil, Hampel, Ederer and Georges observed [4] that existing many-body methods
such as the dynamical mean field method enable computation of electronic
properties at fixed lattice configuration $Q$, in effect solving Eq. 3. Here
we take one further step: the ansatz for the electronic energy functional in
Eq. 1 means that the computed $\Delta N(Q)$ determines $dF_{el}(\Delta
N)/d\Delta N$; the derivative can then be integrated, thereby constructing $F$
itself. In practice we perform the integration by fitting the $\Delta N(Q)$
results to a polynomial, which is analytically integrated. While this method
requires a choice of polynomial form, we empirically find that the uncertainty
thereby introduced is small, of the order of a few meV, comparable to the
error in performing an explicit energy calculation. This approach is
computationally inexpensive, relatively unaffected by noise, and the
analytically specified functional forms provide additional insight, in
particular enabling straightforward examination of the global energy
landscape.

We use this approach to analyse the metal-insulator transitions occurring in
the rare-earth nickelates and their heterostructures, and in bulk and
epitaxially strained Ca2RuO4. We find that in all of these situations the
metal-insulator transition is, in the sense defined above, not electronically
driven.

##  III Bulk Rare-Earth Nickelates

###  III.1 Materials and notation

Figure 2: Schematic of the electronic and structural disproportionation in the
insulating state in rare-earth nickelates: NiO6 octahedra disproportionate
electronically and structurally in a checkerboard pattern. In the low-energy
model of Ref. [4], the Ni ions corresponding to the larger octahedra have
1e${}^{-}+\frac{\Delta N}{2}$ electrons in their eg shell, while those
corresponding to the smaller one have 1e${}^{-}-\frac{\Delta N}{2}$. The
difference in average bond lengths defines $Q$ for the nickelate materials.

The perovskite rare-earth nickelates have chemical formula $R$NiO3; when
synthesized in bulk form the materials with $R$=Lu, Y, Sm, Nd, Pr exhibit a
first order transition from a high temperature metal to a low temperature
insulator while LaNiO3 remains metallic down to lowest temperature. The
material properties depend systematically on the choice of rare-earth ion,
with LuNiO3 exhibiting the highest metal insulator transition temperature
$T_{MIT}=600K$, SmNiO3 having $T_{MIT}\approx 400K$, NdNiO3 having a
$T_{MIT}\approx 240K$ and PrNiO3 having a $T_{MIT}\approx 130K$[52, 53, 11].
The materials may also be grown in heterostructure form [5, 3, 2, 54, 55, 44,
18, 12, 11], with a small number of layers of $R$NiO3 sandwiched between
layers of other materials. The metal-insulator transition temperature in the
heterostructures differs from that in bulk.

Except for the La compound, where the symmetry is slightly different, the high
temperature metallic state forms in a Pbnm structure which for present
purposes may be regarded as a pseudo-cubic lattice of Ni ions with an O ion at
the midpoint of each Ni-Ni bond. The insulating state is characterized by a
two-sublattice bond disproportionation, with one Ni sublattice having a short
mean Ni-O bond length and the other by a long one, represented qualitatively
in Fig. 2. The difference in mean Ni-O bond lengths defines the lattice mode
$Q$ appropriate to this transition.

In the low T insulating state the two Ni sublattices differ in electronic
configuration. This difference has been characterized in various ways in the
literature. [56, 57, 58, 30, 4, 4, 39]. We follow Peil and collaborators [4]
and define the electronic change $\Delta N$ as the difference in the
occupation of the eg states obtained from a narrow window Wannier fit of the
bands crossing the Fermi level.

###  III.2 Results, bulk nickelates

In Fig. 3 we present results for three different bulk-form rare earth
nickelates, (LuNiO3, SmNiO3, PrNiO3) as a function of $Q$, digitized from the
calculations of $\Delta N$ presented in Ref. [4] . In all compounds, two
regimes are found, a low $Q$ regime where $\Delta N$ is proportional to $Q$
and a high $Q$ regime where $\Delta N$ is large but weakly dependent on $Q$;
the two regimes are separated by a discontinuity.

Figure 3: Points: $\Delta N(Q)$ values used for the three $R$NiO3 materials as
obtained from DFT+DMFT calculations from Ref. [4]; Light lines: fits of the
points to Eq. 6. Also shown is the $Q(\Delta N)$ line obtained from the $Q$
equation of state Eq. 2. Note that K/g has the same value for all three
materials.

To determine the free energy model to which the results should be fit we
observe that the two sublattice nature of the insulating state means that the
free energy must be invariant under simultaneous inversion
$Q\leftrightarrow-Q$, $\Delta N\leftrightarrow-\Delta N$, so in particular
$F_{el}(\Delta N)$ must be even in $\Delta N$. Motivated by the observed first
order nature of the transition we assume that $F_{el}$ is a $6^{th}$ order
polynomial in $\Delta N$:

$F_{el}(\Delta N)=\frac{1}{2}\chi_{0}^{-1}\Delta N^{2}+\frac{1}{4}\beta\Delta
N^{4}+\frac{1}{6}\gamma\Delta N^{6}$ (5)

Solving Eq. 2 and eliminating $Q$ from the full free energy then changes the
quadratic term $\chi_{0}^{-1}\rightarrow\chi_{0}^{-1}-\frac{g^{2}}{4K}$. Note
that the shift in the quadratic term involves the parameter $g^{2}/K$, which
varies slightly across the $R$NiO3 series although as noted in Ref. [4] the
ratio $g/K$ is almost material-independent. The K for the Lu, Sm, Pr materials
were reported in reference [4] to be 39.29, 41.45 and 44.47 eV/ Å2, while the
g are 3.75, 4.02 and 4.24 ev/ Å.

Eq. 5 implies that the equation of state Eq. 3 takes the explicit form

$0=-gQ+\chi_{0}^{-1}\Delta N+\beta\Delta N^{3}+\gamma\Delta N^{5}$ (6)

We fit the points in Fig. 3 to Eq. 6, using the $g$ values presented in Ref.
[4], obtaining the light solid curves shown in Fig 3 and the fit parameters
given Table 1. It is important to note that the DMFT equation of state shown
in Fig. 3 has two branches, with the solution discontinuously changing from
one to the other as $Q$ is varied. In our fits we only use the data points
shown as symbols in Fig. 3; these points lie outside the region of
multistability (i.e. where two $\Delta N$ solutions exist within DMFT for the
same Q). However, our theoretical energy function reproduces correctly
behavior that was not part of the original fit, for example the locally stable
large $\Delta N$ at $Q=0$ solution for LuNiO3 [4]. Including points within the
region of multistability does not lead to significant changes to our results.
The free energy we obtain here may be thought of as the free energy that would
be obtained form the two component Landau theory of Ref. [4] if the metal-
insulator variable is integrated out.

$R$NiO3 |  $\chi_{0}^{-1}$(eV) |  $\beta$(eV) |  $\gamma$(eV) |  $-\frac{1}{4}\frac{g^{2}}{k}$(eV)
---|---|---|---|---
LuNiO3 | 0.0596 | -0.1314 | 0.0618 | -0.0895
SmNiO3 | 0.0945 | -0.1303 | 0.0667 | -0.0975
PrNiO3 | 0.1749 | -0.1713 | 0.0804 | -0.1011
Table 1: Parameters that characterize the electronic and total energy of the
three $R$NiO3 materials, as extracted by fitting the data from Ref [4] to Eq.
6. Figure 4: Left panel: electronic energy; central panel: total energy of
three RNiO3 compounds as function of electronic order parameter $\Delta N$.
Right panel: total energy as function of lattice displacement $Q$.

The left panel of Fig. 4 shows the purely electronic energy $F_{el}$
corresponding to the fit parameters given in Table 1. We see that for all of
the materials the purely electronic theory is characterized by a locally and
globally stable metallic minimum at $Q=0$. Only for LuNiO3 does a second,
insulating extremum exist and even in this case it is not energetically
favored. The central panel presents the total free energy as a function of
$\Delta N$ obtained by minimization over the lattice degrees of freedom $Q$.
We see that inclusion of the lattice coupling is necessary to stabilize the
$\Delta N,Q\neq 0$ solution: in other words, it is the electron-lattice
coupling that drives the transition.

Table 1 shows that as one moves across the RNiO3 series from $R$=Lu (most
insulating) to $R$=Pr (least insulating), the linear response electronic
susceptibility of the metallic state decreases, consistent with the empirical
observation that the Lu material is the strongest insulator and the Pr
material is the weakest. We see also that the nonlinear terms in the energy
are the same for the Lu and Sm compounds, which are the two strong insulators,
but are different in the Pr compound, which is close to the metal-insulator
transition. To understand composition-dependent changes one must consider more
than just the variation in linear response susceptibilities. Results to be
presented below for superlattices further confirm this point.

Figure 5: Fits and total energy for PrNiO3 based on DMFT data from Ref [4];
green lines correspond to the optimal polynomial fit, red lines to $4^{th}$
and $6^{th}$ kept constant to the values extracted for SmNiO3, and only the
quadratic term fitted to the data.

The present theory predicts that PrNiO3 is undistorted and metallic, although
close to the metal-insulator transition boundary, while the compound is
empirically distorted and insulating, although again with a very low
transition temperature. We believe this discrepancy arises because the
transition in PrNiO3 is from a paramagnetic metal to an antiferromagnetic
insulator, in contrast to the $Sm$ and $Lu$ materials where the transition is
between two paramagnetic phases. Inclusion of antiferromagnetism in the theory
will yield a lower energy for the insulating phase of PrNiO3 at the $U$, $J$
values used here while a slightly different choice of interaction parameters
would also push PrNiO3 to the insulating side of the phase diagram.

We may take the analysis further by substituting the $\Delta N(Q)$ obtained
from the solution of Eq. 6 into the full free energy expression to obtain the
free energy as a function of the lattice coordinate $Q$ shown in the right
panel of Fig. 4. This is analogous to the energy that is usually calculated by
electronic structure codes, which work at fixed atomic positions. The
discontinuities arise from the different branches of the $\Delta N(Q)$ curves.
The resulting free energy is, to a good approximation, the combination of two
parabolic energy curves, one corresponding to the metallic state, and one to
the insulating state. We see that in this theory the $\Delta N=0$, $Q=0$ state
is stable to lattice distortions for PrNiO3, marginally stable for SmNiO3 and
unstable for LuNiO3.

The analysis presented here is based on a fit of the computed $\Delta N(Q)$ to
a free energy model. We emphasize that this fit is in no way essential to our
method; one could simply numerically integrate a suitably dense set of $\Delta
N(Q)$ data. But for the procedure used here the question of the sensitivity of
the results to the choice of free energy arises. To quantify the uncertainties
we have refit the PrNiO3 data shown in Fig. 3, now constraining the $4^{th}$
and $6^{th}$ order coefficients to have the same values as found in SmNiO3.
Results are shown in Fig 5. We see from the left panel that this fit to the
data is not quite as good, especially in the small to intermediate $Q$ region.
The fitted energy (right panel) is very similar to that shown in Fig. 4, with
the value of $\Delta N$ at the minimum almost the same in the two cases and
the energy of insulating solution in the constrained fit is now slightly lower
than the energy of the metallic solution, placing paramagnetic PrNiO3 slightly
on the insulating side of the transition. These differences indicate that the
systematic uncertainties in the method are not large.

Figure 6: a) Schematic of superlattice consisting of NdNiO3 layered in between
the band insulator NdAlO3 b) Schematic view of two atomic layers of NdNiO3
sandwiched between NdAlO3 layers as described in reference [3]. Dashed line
with red slash: forbidden hopping path, indicating mechanism of dimensional
reduction. Zig-zag lines connecting O and Al: representation of increased
stiffness to distortions of the Al-O bonds relative to the Ni-O bonds.

###  III.3 Layered Structures of NdNiO3/NdAlO3

Figure 7: Electronic (left panel) and total energy (central panel) as
functions of electronic order parameter $\Delta N$ for bulk NdNiO3 and for
superlattices consisting of a bilayer of NdNiO3 alternating with NdAlO3 and a
monolayer of NdNiO3 alternating with 3 layers of NdAlO3 using data from Ref.
[3]. Right panel: total energy as function of lattice coordinate $Q$ after
minimizing over $\Delta N$. Note change of y-axis scale between left-most and
other two panels.

An important modern direction in quantum materials science is
heterostructuring: the ability to synthesize structures in which atomically
thin planes of one material alternate with atomically thin planes of another.
Previous experimental [5] and theoretical [3] work analysed systems of the
type shown schematically in Fig. 6 in which a few layers of metal-insulator
transition compound NdNiO3 were grown as a thin film sandwiched epitaxially
between layers of the wide-gap insulator NdAlO3. The few-layer systems had
significantly higher transition temperatures than did bulk NdNiO3. Theoretical
DFT+DMFT results are available [3] for three systems: bulk NdNiO3 and two
heterostructured systems with a 4 unit cell repeat distance: one layer of
NdNiO3 followed by three of NdAlO3 (”monolayer”), and two layers of NdNiO3
followed by two of NdAlO3 (”bilayer”). The $\Delta N(Q)$ results are shown in
Fig. 8

Two competing effects were found [3]: the electronic confinement of the
electrons in the NdNiO3 by the nearby NdAlO3 layers favors a higher electronic
disproportionation $\Delta N$, while the energy cost of imposing a lattice
distortion on the adjacent Al-O octahedra is equivalent to an increase in the
stiffness $K$ of the NiO6 lattice with a K=31.6914eV/$\AA^{2}$, for the bulk
material, K=35.6269ev/$\AA^{2}$ for the bilayer and K=40.4753eV/$\AA^{2}$ for
the monolayer; all energies per 10 atom unit cell. The values for g are those
quoted in [3]; we reproduce them here for convenience: $g=3.2212eV/\AA$ for
the bulk, $g=3.3465eV/\AA$ for the bilayer and $g=3.3780eV/\AA$ for the
monolayer. Performing an analysis similar to that presented in Sec. III leads
to the energy curves presented in Fig. 7 and to the fit parameters shown in
Table 2. The original data with the fit are presented for convenience in Fig 8

Figure 8: Data from [3] (points) and the polynomials (light lines) obtained by
fitting the data to the equation of state, Eq. 6

Turning first to the purely electronic correlation energy shown in the left
panel of Fig. 7, we observe that neither NdNiO3 nor the heterostructured
materials show a purely electronically driven insulating state; however, hints
of an inflection point, the precursor of the formation of a higher $\Delta N$
extremum, can be seen in bilayer and monolayer curves around $\Delta N=1$
reflecting the effect of quantum confinement in increasing the electron
correlations. Inclusion of the lattice energy produces a global, insulating
minimum in all three materials. The monolayer and bilayer show a significantly
larger energy difference between the insulating and the metallic states than
does bulk NdNiO3, in agreement with their higher $T_{MIT}$.

Layers |  $\chi_{0}^{-1}$(eV) |  $\beta$(eV) |  $\gamma$(eV) |  $-\frac{1}{4}\frac{g^{2}}{k}$(eV)
---|---|---|---|---
Bulk | 0.1049 | -0.1022 | 0.0631 | -0.0819
Bilayer | 0.0823 | -0.0879 | 0.0545 | -0.0789
Monolayer | 0.0818 | -0.0747 | 0.0377 | -0.0705
Table 2: Parameters that characterize the electronic and total energy of
NdNiO3 and NdNiO3/NdAlO3 heterostructures consisting of a single or two layers
of NdNiO3 from Ref [3].

The origin of the effect is surprising: the modest decrease in the magnitude
of $g^{2}/K$ as we pass from bulk to bilayer to monolayer reflects the
increase in lattice stiffness, opposing the order, and nearly counterbalances
the modest decrease in $\chi_{0}^{-1}$ reflecting the increased correlation
physics of the confined metallic state. The more important effect, however, is
the change in the magnitude of the higher order ($\Delta N^{4},\Delta N^{6}$)
terms shown in Table 2. The change can be seen in Fig. 8 from the variation of
the critical $Q$ below which the insulating state is not stable. In the
calculations presented in the previous subsection the higher order terms were
also found to be different in PrNiO3 (proximal to the metal-insulator
transition) than in LuNiO3 and SmNiO3 (farther from the transition point).

##  IV Ca2RuO4

###  IV.1 Materials and Notation

Ca2RuO4 forms in a slightly distorted version of the $n=1$ Ruddlesden-Popper
structure, a layered structure consisting of RuO2 planes separated by pairs of
CaO layers. Each Ru is six-fold coordinated by oxygen, forming RuO6 octahedra.
The relevant electronic states (actually Ru-O hybrids) may be thought of as Ru
$t_{2g}$ symmetry d-states ($d_{xy},d_{xz},d_{yz}$) with $4$ $t_{2g}$
electrons per Ru. Below $340K$ the material undergoes a first-order transition
from a high temperature metallic to a low temperature insulating phase. No
symmetry is broken at the transition: the two phases share the same crystal
structure but differ in the occupancies of the d orbitals and shape of the
RuO6 octahedra, and the relative occupancies of the d-levels. As sketched in
Fig. 9, the metallic state is characterized by an approximately equal
occupancy of the $t_{2g}$ orbitals while in the insulating state the $xy$
orbitals are doubly occupied and the $xz/yz$ orbitals each contain a single
electron. The appropriate electronic order parameter is the occupancy
difference between the $xy$ and $xz/yz$ orbitals

$\Delta N=n_{xy}-\frac{1}{2}\left(n_{xz}+n_{yz}\right).$ (7)

No crystal symmetry protects the orbital occupancies in the high temperature
metallic state, which is characterized by a $\Delta N=N_{H}$ close to, but not
exactly, zero.

The change in electron occupancy across the metal-insulator transition occurs
simultaneously with a decrease in the Ru-O apical bond length and an increase
in the in-plane Ru-O bond length (a flattening of the octahedra). At the
transition, there is a change in the lattice constants, and rearrangements of
other internal coordinates. The harmonic nature of the lattice Hamiltonian
means that most of the lattice modes may be integrated out, leaving an
effective theory involving two structural degrees of freedom. The two lattice
degrees of freedom are needed, because if the point symmetry of the Ru ion is
lower than cubic, then both the unit cell volume change and the relative
octahedral distortion couple linearly to the differential level occupancy
$\Delta N$.

Following Ref. [7] we write the two relevant structural degrees of freedom as
a variable $Q_{3}$ parametrizing a volume preserving change in the c-direction
Ru-O bonds relative to the average in-plane Ru-O bonds, and a change $Q_{0}$
in the octahedral volume and define the lattice variables such that in the
high-T metallic state $Q_{3}=Q_{0}=0$. In terms of changes $\delta x,\delta
y,\delta z$ to the three Ru-O bond lengths, defined as:

$Q_{0}=\frac{1}{\sqrt{3}}(\delta x+\delta y+\delta
z)\>~{}~{}Q_{3}=\frac{1}{\sqrt{6}}(2\delta z-\delta x-\delta x)$ (8)

Figure 9: Schematic of the electronic and structural disproportionation
characterizing the metal-insulator transition of Ca2RuO4. A) In the metallic
phase, all 3 t2g orbitals of Ca2RuO4 are approximately equally occupied;
$\Delta N\approx 0$. B) In the insulating phase, the xy orbital is doubly
occupied, while the yz and xz are singly occupied; $\Delta N=1$ C)
Representation of Q0 structural mode, in which all bond lengths change equally
D) Representation of structural mode $Q_{3}$ in which octahedral volume is
preserved and the change in the two in-plane Ru-O bond lengths are equal, and
opposite in sign to the change in the apical bond length. The electronic state
in B) is associated with a structural term of the form
$-(Q_{3}-\lambda_{0}Q_{0})$, with $\lambda_{0}>0$

We now build the free energy. We define the lattice distortion relative to the
high-T metallic state as the vector $\vec{Q}=(Q_{3},Q_{0})$; the lattice
restoring term $\mathbf{K}$ is a $2\times 2$ matrix with entries determined in
previous work[7] to be $K_{33}$=17.7, $K_{03}$=7.6, $K_{00}$=46.2 ev/$\AA^{2}$
per formula unit. We write the linear combination of the $\mathbf{Q}$ that
couples to the electronic disproportionation as
$\vec{\mathcal{F}}\cdot\vec{Q}$ with $\vec{\mathcal{F}}=F_{3}(1,-\lambda)$.
Ref. [7] finds F${}_{3}=2.8eV/\AA$ and $\lambda=0.45$.

$F(\Delta
N,Q)=\frac{1}{2}\mathbf{Q}^{T}\mathbf{K}\mathbf{Q}-\vec{\mathcal{F}}\cdot\vec{Q}(\Delta
N-\Delta N_{H})+F_{el}(\Delta N),$ (9)

Here $\Delta N_{H}$ is the value of $\Delta N$ in the high temperature
insulating phase; it is nearly, but not quite, zero.

For the purely electronic energy we chose the $4^{th}$ order form:

$F_{el}=\frac{c\Delta N^{4}}{4}+\frac{b\Delta N^{3}}{3}+\frac{a\Delta
N^{2}}{2}+\sigma_{0}\Delta N$ (10)

Odd powers occur because no symmetry is broken at the transition.

###  IV.2 Metal-insulator transition in bulk CaRuO4

Figure 10:  $\Delta N(Q)$ from [7] (heavy black points) and polynomial fit
(solid line) used for the equation of state.

Fig. 10 shows previously published results [7] for $\Delta N$ as a function of
the relevant combination of lattice distortions along with our fit to Eq. 10.
As in the previous section we have not used points in the coexistence region
for the fit. We find $\sigma_{0}=-0.009$, $a=0.70$, $b=-1.718$, $c=1.28$.

The higher curve (blue) in Fig. 11 shows the purely electronic free energy
resulting from the fit. We see that in the absence of lattice effects the
metallic, undistorted state is energetically favored and that there is not
even a metastable minimum corresponding to the insulating state although there
is a hint of an inflection point that is a precursor of a higher $\Delta N$
extremum. The lower trace (heavy black on-line) shows $\bar{F}$, the full free
energy after the lattice modes has been integrated out. We see that inclusion
of the lattice energetics qualitatively changes the free energy, strongly
favoring large $\Delta N$. Indeed we see that the energy does not have a
minimum in the physically allowed range $\Delta N\leq 1$; rather it is
minimized at the boundary $\Delta N=1$. We again conclude that the transition
in this compound is driven by the lattice contribution to the energetics.

Figure 11: $F_{el}(\Delta N)$ and $F_{total}(\Delta N)$ for Ca2RuO4.

###  IV.3 Epitaxially constrained Ca2RuO4

Ca2RuO4 has also been grown expitaxially on insulating substrates. The
epitaxial constraint means that the in-plane lattice parameters of the Ca2RuO4
are forced to be equal to the in-plane lattice parameters of the substrate
material. A priori, this constraint does not fix the values of internal
coordinates including the the Ru-O in-plane bond length of relevance here.
Density functional theory calculations, however, [7] indicate that in practice
the material adapts to the epitaxial constraint by adjusting the in-plane Ru-O
bond lengths, rather than by rotating the Ru-O6 octahedra, so that the
percentage change in the in-plane $Ru-O$ bond lengths is fixed by the
percentage change in the in-plane lattice parameters. The z-direction lattice
constants and Ru-O apical bond lengths are still of course free to relax. This
means that the theory of the previous section can be simply adapted to the
epitaxial case by a change of variables. We write:

$Q_{3}=\frac{1}{\sqrt{6}}(2z-x_{0}-y_{0})$ (11)

and

$Q_{0}=\frac{1}{3}(z+x_{0}+y_{0})$ (12)

with $x_{0}$ and $y_{0}$ fixed by the epitaxial constraint. It is also
convenient to define the zero of $z$ to be the value

$z_{0}=-(x_{0}+y_{0})\frac{K_{00}-K_{33}+\frac{K_{30}}{\sqrt{2}}}{2K_{33}+2\sqrt{2}K_{30}+K_{00}}$
(13)

that $z$ would take at $\Delta N=N_{H}$.

We obtain:

$F(\Delta
N,z)=\frac{K^{\prime}}{2}z^{2}+g^{\prime}\left(z-z^{\star}\right)\Delta
N+F_{el}(\Delta N)$ (14)

with

$\displaystyle K^{\prime}$ $\displaystyle=$
$\displaystyle\frac{2}{3}(K_{33}+\sqrt{2}K_{30}+\frac{1}{2}K_{00})$ (15)
$\displaystyle g^{\prime}$ $\displaystyle=$
$\displaystyle\frac{F_{3}(2-\sqrt{2}\lambda_{0})}{\sqrt{6}}$ (16)

and

$z^{\star}=(x_{0}+y_{0})\left(\frac{1+\sqrt{2}\lambda_{0}}{2-\sqrt{2}\lambda_{0}}+\frac{K_{00}-K_{33}+\frac{K_{30}}{\sqrt{2}}}{2K_{33}+2\sqrt{2}K_{30}+K_{00}}\right)$
(17)

Figure 12: Equilibrium $\Delta N$ and z versus epitaxial strain defined as the
difference in mean in-plane Ru-O bond length relative to the value in the high
temperature structure of Ca2RuO4 as imposed through x0+y0 where x0=y0 Figure
13: Phase diagram of epitaxially constrained Ca2RuO4 in plane of tensile
strain (defined as difference of mean in-plane Ru-O bond length from value in
the high temperature structure) in Å and coupling parameter $F_{3}$. Red
dashed line: value of $F_{3}$ at which metal insulator transition occurs in
bulk system. Arrows (red on line) indicate strain imposed by epitaxial growth
on (from left to right) LaAlO3, NSAT and NdGaO3; NdAlO3 corresponding to
tensile strain is not shown. Figure 14: Total energy plotted against orbital
disproportionation $\Delta N$ for materials grown on four substrates indicated
in the legend which provide different strains relative to the metallic phase
of Ca2RuO4. For tensile strain (NdAlO3), the material is always metallic and
the orbital polarization has the opposite sign from in the insulating state.

The epitaxial constraint thus has three effects: it increases the net
stiffness to lattice distortions by preventing relaxations of the lattice
coordinates associated with in-plane bond lengths, making the insulating phase
more difficult to obtain. Second, it reduces the electron lattice coupling
(Eq. 16) which also makes it more difficult to obtain an insulating phase.
Finally, it provides a term linearly proportional to $\Delta N$, which for
positive epitaxial strain favors the insulating phase. The left panel of Fig.
12 shows the change in apical Ru-O bond length as a function of epitaxial
strain; the right hand panel the occupancy difference $\Delta N$. The first
order transition is evident. Note that for an epitaxial strain matching the
in-plane lattice constant of the high temperature metallic phase, insufficient
lattice energy is available to stabilize the insulating phase. As the tensile
strain is increased, the octahedral deformation, which couples linearly to
$\Delta N$, increases, and beyond a critical value drives a first order
transition.

Experiments (Ref [59, 60]) have studied films gown epitaxially on substrates
NdAlO3, LaAlO3, NSAT and NdGaO3 corresponding to changes (relative to the
high-T state, and assuming the Ru-O bond length exactly tracks the in-plane
strain) in average in -plane Ru-O lattice parameters of $-0.04\AA$,
$0.008\AA$, $.03\AA$ and $0.046\AA$ respectively. Fig. 13 shows that the
theory, with no adjustable parameters, predicts that for the theoretically
predicted coupling $F_{3}=2.8eV/\AA$ the films grown on NdAlO3 and LaAlO3
should be metallic, while the films grown on NSAT and NdGaO3 should be
insulating. In the experiment the LaAlO3-strained material has a transition at
$T\approx 200K$ from a high temperature metal to a low temperature weak
insulator/bad metal, while the others are metallic and insulating as predicted
by the theory. The qualitative agreement is good; the quantitative
discrepancies may arise from a more subtle relation between Ru-O bond length
and in plane lattice constant than was assumed in Ref [7] or from small
systematic errors in the theory. The good correspondence between experiment
and theory show the utility of the methodology introduced here.

##  V Temperature Dependence

The methods presented here provide some insight into the physics of the
temperature-driven metal-insulator transition, indicating an important area
for future research. We see from Figs. 4 and 7 that for the nickelates the low
T energy landscape is characterized by two minima, with energy differences
ranging from $80$ meV (LuNiO3) to $25$ meV (NdNiO3). In the nickelates, as the
temperature increases a first order metal-insulator transition occurs. The
lattice distortion $Q$ exhibits negligible temperature dependence over the
entire insulating phase, indicating that the minima remain robust and the
transition is driven by an entropic effect that lifts up the free energy of
the large $\Delta N$ extremum relative to that of the $\Delta N=0$ extremum.

Figure 15: Upper left panel: electronic free energy as function of electronic disproportionation $\Delta N$ computed for NdNiO3 at temperatures from 290K to 1200K. Upper right panel: total free energy computed for bulk NdNiO3 for temperatures of T=290K, 387K, 580K, 773K, 967K, 1054K, 1131K. Lower left panel: total expanded view of total free energy for bulk NdNiO3 at temperatures of 1054K, 1080K, 1105K. Lower Right Panel: Optimal Q(T)=$\Delta N(T)\frac{g}{2K}$ Temperature |  $\chi_{0}^{-1}$(eV) |  $\beta$(eV) |  $\gamma$(eV)
---|---|---|---
387K | 0.0937 | -0.0781 | 0.0519
580K | 0.0811 | -0.0427 | 0.0346
773K | 0.0773 | -0.0197 | 0.0227
966K | 0.0806 | -0.0082 | 0.0172
Table 3: Parameters that characterize the temperature dependence of the
electronic energy as extracted by fitting the temperature dependent electronic
free energy data presented in Fig. 15

To investigate whether the entropic effect arises from the local correlations
captured by the dynamical mean field approximation, for bulk NdNiO3 we have
extended the calculations shown in Fig. 7 to much higher temperatures by
performing calculations with the same methodology and interaction parameters
as in [3] but changing the electronic temperature. Fig. 15 shows the results:
all of the parameters in the electronic free energy vary systematically with
temperature, decreasing in magnitude as the temperature is increased from
$\sim 290$K to $1000$K; the result is a very weakly first order transition at
$\approx 1080$K (the model actually passes very close to the tricritical point
at which the second and fourth order coefficients of the theory simultaneously
change sign). This behavior, while theoretically interesting, is inconsistent
with the observed strongly first order and much lower transition temperature
behavior, suggesting that either the single site dynamical mean field
approximation does not adequately describe the temperature dependence of the
electronic contribution to the free energy, or that an entropic effect
associated with the lattice degrees of freedom plays an important role. Indeed
the neglect of spatial fluctuations generically means that single-site DMFT
approximations overestimate transition temperatures [61, 62]. It should also
be noted that in this material antiferromagnetism plays an important role in
the ordering, and that a direct computation [39] of the free energy $F(\Delta
N,Q)$ using a different DMFT formalism places paramagnetic NdNiO3 on the
paramagnetic metal side of the transition and yields a stronger low T
temperature dependence, and shows for LuNiO3 a temperature dependence of
$F(\Delta N,Q)$ in the range $T<500K$ very similar to that found here for
NdNiO3.

Figure 16: $F_{total}(\Delta N)$ computed for bulk Ca2RuO4 for different
values of F3.

Ca2RuO4 presents different issues. A substantial temperature dependence is
observed [63] across the insulating phase. Since the electronic order
parameter $\Delta N$ in this material is fixed at its largest possible value
for a range of couplings, the temperature dependence must imply a temperature
dependence of the electron-lattice coupling $F_{3}$ as proposed for related
reasons in [7]. Within single-site dynamical mean field theory, there is no
theoretical justification for a strongly temperature-dependent coupling
parameter, but on the assumption that changes in electron-lattice coupling F3
are a proxy for changes in temperature we show in Fig. 16 the energy computed
for different values of $F_{3}$. We see that a first order transition occurs
at an $F_{3}\approx 1.58\approx 0.6$ of the $F_{3}$ we estimated from the low
temperature calculation. The phase diagram presented in Fig. 13 then indicates
that films grown on substrates with a tensile epitaxial constraint exhibit
significantly higher transition temperatures than found in bulk, roughly
consistent with experiment.

##  VI Discussion and Outlook

This paper has analysed metal-insulator transitions in quantum materials by
constructing an energy functional $F(\Delta N,Q)$ that depends on suitably
defined electronic ($\Delta N$) and lattice ($Q$) order parameters. The first
assumption enabling the construction is that the lattice energy is quadratic
in deviations from an equilibrium position at fixed value of the electron
order parameter, in other words that the physics controlling the relevant
atomic force constants arises from chemical bonds inside the solid that are
only weakly affected by the transition from metal to insulator. This and the
harmonic nature of the lattice response are found in density functional
calculations (see e.g. the supplementary material of Ref. [3]) and are further
supported by the observation that phonon frequencies do not change much across
the transitions of interest. The second assumption is that the coupling
between electronic and lattice degrees of freedom is linear, and may be
determined a-priori. This second assumption has been confirmed by explicit
calculations (see, for example, the supplementary information in [3]). Further
investigation of these assumptions, and identification of compounds in which
they break down, is an important topic for future research. One important
direction is a detailed comparison of results obtained along the lines
indicated here to direct computations of energies and free energies [57, 58,
64, 39]

Given these assumptions, a many-body calculation of electronic configuration
as a function of lattice distortion may be used to construct free energies,
essentially by integrating the equation of state. We accomplish the
integration by fitting the equation of state results to a polynomial which is
integrated analytically. This procedure avoids ambiguities associated with the
coexistence region of the first order transition.

We applied the methodology to two currently interesting families of compounds,
that exhibit first order metal-insulator transitions as the temperature is
decreased: the rare earth perovskite nickelates RNiO3, and Ca2RuO4. The
transition in the RNiO3 is a symmetry breaking transition leading to two
inequivalent Ni sites and an alternating pattern of long and short Ni-O bond
lengths. The transition in Ca2RuO4 is isosymmetric, with no broken crystal
symmetries. In both materials, we find that if the lattice coordinate is set
to the value appropriate to the high temperature metallic state, the resulting
energy $F(\Delta N,Q=0)$ has only one minimum, at $\Delta N=0$. It is only
after the coupling to the lattice is included that the total functional
$F(\Delta N,Q)$ acquires a minimum at a $\Delta N,Q\neq 0$. We therefore
conclude that lattice effects are essential in driving the metal-insulator
transition in both compounds, rather than being merely a minor consequence of
a fundamentally electronically driven transition, thereby settling a long-
standing controversy. It is important to note however that although the purely
electronic theory does not produce a metal-insulator transition, the
correlation contribution to the electronic energy is still important and
calculations on a beyond-DFT level are apparently required to obtain correct
energetics.

It is important to consider the limitations of our conclusions. First, while
the approach introduced here will work with any method that calculates the
electronic order parameter as a function of lattice distortion $\Delta N(Q)$,
in practice the information available in the literature comes from the density
functional plus dynamical mean field methodology. While this method
successfully produces results that are consistent with many experiments, its
precise quantitative accuracy is unknown. The method does require an
identification of correlated orbitals, it relies on the assumption that
density functional theory gives an adequate account of the energetics of the
uncorrelated orbitals, it requires specification of interaction parameters,
and its solution of the many body problem involves a stringent locality
assumption. Cross checking via other methods (as for example was done for
Ca2RuO4 in Ref. [65] ) would be desirable. Further, there are several variants
of the DMFT plus DFT methodology, differing the choice of energy window and
the representation of correlated states [66]; the dependence of our
conclusions on the implementation is an interesting question.

The specification of interaction parameters is of particular importance. As
shown e.g. in Refs. [49, 35], as the interaction parameters are increased in
magnitude, a transition can be generated in the purely electronic theory. The
precise statement made here is that with interaction parameters determined by
reproducing physical properties including gaps and effective masses , the
purely electronic theory does not exhibit a phase transition: electron-lattice
coupling plays an essential role in stabilizing the insulating states.

It will also be noted that while the theory presented here reproduces trends
and orders of magnitude very well, it has some quantitative deficiencies, for
example predicting that PrNiO3 is metallic but close to the phase boundary
when in fact it is insulating but close to the phase boundary, with the lowest
transition temperature of any bulk member of the $R$NiO3 nickelate family.
This deficiency is likely to be remedied by inclusion of magnetism in the
theory, since in PrNiO3 and NdNiO3, but not in the other compounds studied in
this paper, the metal-insulator transition is accompanied by a magnetic
transition. Similarly the phase diagram of epitaxially constrained Ca2RuO4
films is qualitatively correct but exhibits quantitative discrepancies with
experiment. We note that in this paper no attempt was made to adjust the
parameters from the literature or to fine-tune the fits to obtain better
agreement with experiment. Small changes, reflecting moderate parameter
uncertainties and modest systematic errors in the DFT+DMFT approach, would
likely cure these discrepancies.

The access to the free energy provided by our methods enables additional
insights. Issues relating to temperature dependence were discussed in Section
V. The energy landscape shown in Fig. 1 makes it clear that only one narrowly
defined path connects the metallic and insulating minima; this information may
be used to investigate the kinetics of order parameter nucleation if the
material is supercooled or superheated across the phase boundary. Further, the
differing roles of the electronic and lattice degrees of freedom in defining
the energy landscape make possible an analysis of the kinetics of state
evolution after excitation. For example, excitation might rapidly heat the
electrons (leading to changes in $F_{el}$) but only slowly heat the lattice
degrees of freedom (which would in any event respond more slowly). Extension
of the theory presented here to include the antiferromagnetism that occurs in
the rare earth nickelates may help understand recent ultrafast experiments
exploring the relation between antiferromagnetism, lattice distortions and
metallicity in NdNiO3 [67, 68].

The methods can be applied to quantify the importance of lattice effects in
other systems of current interest, including CaFeO3 [69, 70] which exhibits
charge ordering and lattice disproportionation, perovskite titanates and
vanadates, where lattice distortions couple to orbital ordering [71, 72, 73],
VO2 where a dimerization instability occurs [74], V2O3 where the metal-
insulator transition couples to the volume of the material [75], and the rare
earth manganites where electronic charge, orbital and magnetic ordering are
tightly coupled to lattice distortions [76, 77]. We also observe that while we
presented results based on dynamical mean field calculations, the method is
generic in nature and can easily be used with other electronic structure
methods, including DFT, DFT+U, Gutzwiller[78, 79], Auxiliary-Boson[80, 81, 82,
83] \- particularly with the advent of new codes that calculate energy and
naturally handle symmetry breaking[84, 85, 86, 87], DFT+U[88, 89], and quantum
Monte Carlo[90].

###### Acknowledgements.

We are indebted to Antoine Georges and Oleg Peil for inspiration, helpful
comments and advice at various stages of this work. Discussions with Jennifer
Fowlie, Claribel Dominguez, Bernat Mundet Bolos, Marta Gibert, and Jean-Marc
Triscone stimulated this project. We would also like to acknowledge
suggestions to improve this manuscript from Lauren Walters, Sophie Beck,
Alexander Hampel, Claribel Dominguez, as well as conversations with James
Rondinelli, Kyle Miller and Danilo Puggioni during the preparation of this
manuscript. ABG was partly supported by the Advanced Research Projects Agency-
Energy (ARPA-E), U.S. Department of Energy, under Award Number DE-AR0001209.
The views and opinions of authors expressed herein do not necessarily state or
reflect those of the United States Government or any agency thereof. The
Flatiron Institute is a division of the Simons Foundation.

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