# Leggett Modes Accompanying Crystallographic Phase Transitions

Quintin N. Meier  Université Grenoble Alpes, CEA, LITEN, 17 rue des Martyrs,
38054 Grenoble, France  Daniel Hickox-Young  Department of Materials Science
and Engineering, Northwestern University, Evanston, IL 60208, USA  Geneva
Laurita  Department of Chemistry and Biochemistry, Bates College, Lewiston, ME
04240, USA  Nicola A. Spaldin  Materials Theory, ETH Zurich, Wolfgang-Pauli-
Strasse 27, 8093 Zürich, Switzerland  James M. Rondinelli  Department of
Materials Science and Engineering, Northwestern University, Evanston, IL
60208, USA  Northwestern-Argonne Institute of Science and Engineering (NAISE),
Northwestern University, Evanston, IL 60208, USA  Michael R. Norman
<EMAIL_ADDRESS>Materials Science Division, Argonne National Laboratory,
Lemont, IL 60439, USA

###### Abstract

Higgs and Goldstone modes, well known in high energy physics, have been
realized in a number of condensed matter physics contexts, including
superconductivity and magnetism. The Goldstone-Higgs concept is also
applicable to and gives rise to new insights on structural phase transitions.
Here, we show that the Leggett mode, a collective mode observed in multi-band
superconductors, also has an analog in crystallographic phase transitions.
Such structural Leggett modes can occur in the phase channel as in the
original work of Leggett, Prog. Theor. Phys. 36, 901 (1966). That is, they are
antiphase Goldstone modes (anti-phasons). In addition, a new collective mode
can also occur in the amplitude channel, an out-of phase (antiphase) Higgs
mode, that should be observable in multi-band superconductors as well. We
illustrate the existence and properties of these structural Leggett modes
using the example of the pyrochlore relaxor ferroelectric, Cd2Nb2O7.

##  I Introduction

Spontaneous symmetry breaking is a ubiquitous phenomenon in physics, and is
responsible for various collective modes, most famously the well-known
Goldstone and Higgs modes. The former refers to fluctuations of the phase of
an order parameter, the latter to fluctuations of its amplitude. These modes
play a fundamental role in gauge theories of particle physics, and are also
important in condensed matter physics. The Goldstone-Higgs phenomenon in high
energy physics is a relativistic generalization of the analogous behavior
found in superconductors [1]. In superconductors, the Goldstone mode does not
occur at zero energy as in a neutral superfluid, but it is pushed to the
plasma frequency by coupling to the electromagnetic field [2]. The
superconducting Higgs mode is non-trivial to observe since its energy tends to
be located near the quasiparticle continuum [3]. Nevertheless, it has been
observed in several superconductors [4].

Superconductors can exhibit other collective modes. In particular, a relative
phase mode of the order parameters of the two bands of a two-band
superconductor was proposed by Leggett [5] and first realized in MgB2 [6].
While the Goldstone mode is pushed to the plasma frequency by coupling to the
electromagnetic field, the Leggett mode maintains charge neutrality and so is
not affected by the field [5]. Other modes are known as well, for instance
Carlson-Goldman modes in which the superconducting and normal condensates
oscillate out of phase [7]. In addition, in superfluid 3He a variety of
clapping, flapping, and Higgs modes occur because of the high degeneracy of
its SO(3) $\times$ SO(3) $\times$ U(1) order parameter space [8, 9].

Structural phase transitions are typically associated with soft phonons [10],
and the concepts of Goldstone and Higgs modes have provided new insights into
these transitions. A classic example is the pyrochlore Cd2Re2O7. Its
structural phase transition arises from an instability associated with a
doubly degenerate $\Gamma_{3}^{-}$ phonon [11]. This defines a Mexican-hat
free energy surface with the top of the hat representing the high temperature
cubic phase, and the brim representing the lower symmetry distorted phase. In
the Landau free energy, anisotropy terms that warp the brim of the Mexican hat
only arise at sixth order, suggesting the existence of a low energy Goldstone
mode corresponding to oscillations along the brim. (Since they are not at zero
energy because of the warping terms, these are sometimes referred to as
pseudo-Goldstone modes). Subsequent Raman scattering experiments [12]
exhibited strong evidence for this mode, and its existence has been recently
confirmed by diffuse scattering studies [13]. The Higgs mode is also evident
from the Raman data, though its interpretation has been challenged by recent
pump-probe measurements [14]. After these pioneering studies of Cd2Re2O7,
Higgs and Goldstone modes have been proposed in a variety of perovskite and
other complex oxides [15, 16, 17, 18, 19, 20]. Routes to detect them if they
are optically silent have also been identified [21].

Here, we investigate whether the Leggett mode can be realized in the
structural context, where it could give new insights into the associated
structural phase transitions. We begin by developing a minimal Landau model
that describes a structural phase transition that is accompanied by a new
collective mode, the antiphase Higgs mode (the amplitude analog of the Leggett
mode), that should also be observable in multi-band superconductors. We then
consider a Landau treatment of the ferroelectric transition in the pyrochlore
Cd2Nb2O7, where we show that both the antiphase Higgs mode and the Leggett
mode (the anti-phason) are possible. Finally, we discuss secondary modes that
can drive these Leggett modes by coupling to them in the context of pump-probe
studies. We conclude by discussing the relevance of our work to other
materials, in particular those that involve modes at non-zero wavevectors.

##  II Minimal Landau Model

A superconductor is described by an order parameter $\Delta e^{i\phi}$ with an
amplitude, $\Delta$, and a phase, $\phi$. In a two-band superconductor, one
can identify a collective mode, called the Leggett mode, that corresponds to
oscillations of the relative phase of the order parameters associated with the
two bands, $\phi_{1}-\phi_{2}$ [5]. This mode is an eigenvector of a secular
matrix whose eigenvalues are the collective mode energies [22]. In the
structural context, the corresponding secular matrix is the force-constant
matrix [19], with elements $\varphi_{ij}=\frac{\partial^{2}F}{\partial
u_{i}\partial u_{j}}$ where $u_{i}$ is the displacement of the $i$-th ion from
its high symmetry position and $F$ is the free energy. For a periodic system
with $N$ atoms in the unit cell, this is a $3N\times 3N$ matrix. For our
further analysis, we will use a reduced form of the force-constant matrix,
$\Phi_{ij}=\frac{\partial^{2}F}{\partial q_{i}\partial q_{j}}$, where $q_{i}$
are symmetry-adapted distortion modes [23, 24]. Each $q_{i}$ is a linear
combination of atomic displacements that transform like a particular group
representation of the high symmetry phase, and as such describe a collective
motion involving multiple ions. For structural phase transitions following
Landau theory [25], $F$ is formulated as a polynomial expansion in these
$q_{i}$. The distortions from the high-symmetry phase are given, in the
harmonic limit, by the eigenvectors of the force-constant matrix 111We note
that the force-constant matrix is related to, but not equal to, the dynamical
matrix determining the phonons, whose elements instead are
$\frac{\Phi_{ij}}{\sqrt{M_{i}M_{j}}}$ where $M_{i}$ is the mass of the $i$-th
ion [19]. As such, a given Higgs or Goldstone mode, defined in terms of the
eigenvectors of the force-constant matrix, can be an admixture of several
phonons of the appropriate symmetry [19]. . The advantage of Landau theory is
the reduction of the large force-constant matrix to this smaller one involving
only the $q_{i}$ relevant to the phase transition [19].

To illustrate the structural Leggett mode, we present a simple example in
which the phase transition involves only two modes, $\mathbf{q}$ and
$\mathbf{r}$, each from a different two-dimensional group representation. For
simplicity, we use a Cartesian basis, $\mathbf{q}~{}(q_{1},q_{2})$ and
$\mathbf{r}~{}(r_{1},r_{2})$. In a polar basis, $q_{i}=Q_{i}\cos(\phi_{i})$,
etc. In general, in the uncoupled case, each mode would have a different
transition temperature, $T_{\mathbf{q}}\neq T_{\mathbf{r}}$. We describe each
mode by a Mexican-hat potential, and include a biquadratic coupling between
them [27]. In this case, the free energy is given by

$\displaystyle F$
$\displaystyle=\dfrac{a_{\mathbf{q}}}{2}(q_{1}^{2}+q_{2}^{2})+\dfrac{b_{\mathbf{q}}}{4}(q_{1}^{2}+q_{2}^{2})^{2}+\dfrac{a_{\mathbf{r}}}{2}(r_{1}^{2}+r_{2}^{2})$
$\displaystyle+\dfrac{b_{\mathbf{r}}}{4}(r_{1}^{2}+r_{2}^{2})^{2}+\dfrac{c}{2}(q_{1}^{2}+q_{2}^{2})(r_{1}^{2}+r_{2}^{2})\,,$
(1)

where the temperature dependence is typically only included in the quadratic
terms $a_{\mathbf{q}}\equiv a_{\mathbf{q}0}(T-T_{\mathbf{q}})$ and
$a_{\mathbf{r}}\equiv a_{\mathbf{r}0}(T-T_{\mathbf{r}})$. The subspace of the
force-constant matrix of interest is now given by

$\Phi_{ij}(T)=\left.\dfrac{\partial^{2}F}{\partial\phi_{i}\partial\phi_{j}}\right|_{\phi_{i}=\left<\phi_{i}\right>,\phi_{j}=\left<\phi_{j}\right>}$
(2)

where $\phi_{i}\in\\{q_{1},q_{2},r_{1},r_{2}\\}$, and $\left<~{}\right>$
describes the thermodynamic average at a given temperature.

Choosing without loss of generality that $q_{1}=\left<q\right>$, $q_{2}=0$,
$r_{1}=\left<r\right>$, $r_{2}=0$, the force-constant matrix then becomes

$\Phi=\begin{bmatrix}H_{\mathbf{q}}&2c\left<q\right>\left<r\right>&0&0\\\
2c\left<q\right>\left<r\right>&H_{\mathbf{r}}&0&0\\\ 0&0&G_{\mathbf{q}}&0\\\
0&0&0&G_{\mathbf{r}}\end{bmatrix}\quad,$ (3)

with

$\displaystyle H_{\mathbf{q}}$
$\displaystyle=a_{\mathbf{q}0}(T-T_{\mathbf{q}})+3b_{\mathbf{q}}\left<q\right>^{2}+c\left<r\right>^{2}$
$\displaystyle G_{\mathbf{q}}$
$\displaystyle=a_{\mathbf{q}0}(T-T_{\mathbf{q}})+b_{\mathbf{q}}\left<q\right>^{2}+c\left<r\right>^{2}$
$\displaystyle H_{\mathbf{r}}$
$\displaystyle=a_{\mathbf{r}0}(T-T_{\mathbf{r}})+3b_{\mathbf{r}}\left<r\right>^{2}+c\left<q\right>^{2}$
$\displaystyle G_{\mathbf{r}}$
$\displaystyle=a_{\mathbf{r}0}(T-T_{\mathbf{r}})+b_{\mathbf{r}}\left<r\right>^{2}+c\left<q\right>^{2}\quad.$

Here, we use $H$ to indicate Higgs modes, that is amplitude modes with
symmetry $A_{1}$, and $G$ to indicate Goldstone modes, that is phase modes
whose symmetry depends on the underlying space groups involved.

Figure 1: Mode energies for the minimal Landau model with
$a_{\mathbf{q}}=a_{\mathbf{r}}$, $b_{\mathbf{q}}=b_{\mathbf{r}}$, and
$c=b_{\mathbf{q}}/2$, so that $T_{\mathbf{q}}=T_{\mathbf{r}}=T_{\mathbf{c}}$.
Units are such that $a_{\mathbf{q}0}/T_{\mathbf{q}}$ is set to unity. The
Goldstone modes (not shown) are at zero energy. Figure 2: Crystal structure of
cubic Cd2Nb2O7 shown to illustrate its two interpenetrating sublattices of
NbO6 octahedra and O′Cd4 tetrahedra with Cd (magenta), Nb (blue), and O
(orange) ions.

There are no off-diagonal terms coupling the Higgs and Goldstone sectors, but
in this simple example, the biquadratic coupling term couples the two Higgs
modes, $H_{\mathbf{q}}$ and $H_{\mathbf{r}}$. Because of this coupling, we
expect that a new collective mode, the antiphase Higgs (the amplitude analog
of the Leggett mode), can exist. To see this, note that the eigenvalues of the
force-constant matrix are the square of the collective mode frequencies 222The
$q_{i}$ are typically given in length units, meaning an effective mass is
needed to convert the force-constant eigenvalues to frequency units.. That is,
the eigenvalues of the upper $2\times 2$ block of the force-constant matrix
are

$\displaystyle\omega^{2}_{\pm}=\dfrac{1}{2}\left(H_{\mathbf{r}}+H_{\mathbf{q}}\pm\sqrt{(H_{\mathbf{r}}-H_{\mathbf{q}})^{2}+16c^{2}\left<q\right>^{2}\left<r\right>^{2}}\right)\,.$
(4)

The eigenvectors are

$(\mathbf{u},\mathbf{v})_{\pm}=\frac{1}{N_{\pm}}\left(1,\dfrac{\omega_{\pm}^{2}-H_{\mathbf{q}}}{2c\left<q\right>\left<r\right>}\right)\,,$
(5)

where $N_{\pm}$ is a normalization factor. For small $c\langle q\rangle\langle
r\rangle$ relative to $H_{\mathbf{q}}-H_{\mathbf{r}}$, we can think in terms
of separate Higgs modes that are weakly coupled. In the limit of large
coupling $c\langle q\rangle\langle r\rangle$, however, we obtain an in-phase
Higgs mode and an out-of-phase Higgs mode. The latter is the Higgs analog of
the Leggett phase mode. These are illustrated in Figure 1 for the simple case
where the $\mathbf{q}$ and $\mathbf{r}$ parameters are the same. For this
case, it is easy to show that the square of the Higgs mode energy is
$-2a_{\mathbf{q}}$ and that of the Leggett mode is
$-2a_{\mathbf{q}}\frac{b_{\mathbf{q}}-c}{b_{\mathbf{q}}+c}$.

For its realization, one finds from the above equations that even one-
dimensional group representations would produce this antiphase Higgs mode. To
determine the mode dispersions requires the addition of gradient terms that
typically lead to a quadratic dispersion about the ordering wavevector, unless
the mode energy at the ordering vector is at zero energy, in which case the
dispersion is linear in momentum instead.

Figure 3: Illustration of the Leggett phase mode for $Fdd2$ with (a) the free
energy surfaces drawn as nested spheres for $\Gamma_{4}^{-}$ and
$\Gamma_{5}^{-}$, whose radii are the amplitudes $Q_{4}$ and $Q_{5}$, and (b)
the corresponding Mexican-hat potentials when restricting to $Cc$. In (a) the
$Cc$ subspace is indicated by great circles on these spheres (which are in
reality warped given that $Q_{4}$ and $Q_{5}$ depend on the spherical angles).
The $Fdd2$ minimum in Cartesian coordinates is $(Q_{4},0,0)$ on the
$\Gamma_{4}^{-}$ sphere and $(0,Q_{5},0)$ on the $\Gamma_{5}^{-}$ sphere. In
(b) the vertical axis is energy, with the wavy line indicating the Landau
couplings between the two energy surfaces. In both panels, the Goldstone mode
corresponds to oscillations about the $Fdd2$ minima on each circle (these
minima are indicated by black dots) to one $Cc$ domain on one swing (a red
arrow on one surface, a blue arrow on the other) and to another $Cc$ domain on
the other swing (a blue arrow on one surface, a red arrow on the other). The
higher energy Leggett mode (anti-phason) instead involves oscillations
corresponding to antiphase $Cc$ domains, with one swing involving both red
arrows and the other swing involving both blue arrows. A similar picture
exists for the antiphase Higgs mode, with the arrows pointing along the radial
directions instead.

What about the anti-phason, the out-of-phase mode in the phase channel that
would be the analog of the superconducting Leggett mode? To obtain this
requires coupling of the two Goldstone modes. One can show that such coupling
requires terms in Equation 1 that are linear in the two Goldstone variables.
In the above example, this would be terms of the form $q_{2}r_{2}$ times Higgs
variables ($q_{1},r_{1},q_{1}^{2},r_{1}^{2},q_{1}r_{1}$, etc.). Although these
terms do not exist in the above minimal model, they do exist in general.
Instead of extending our minimal model to the more general case, we instead
illustrate anti-phasons using the specific example of Cd2Nb2O7.

##  III Landau Theory of Cd2Nb2O7

The pyrochlore Cd2Nb2O7 is one of the few known stoichiometric materials that
exhibits relaxor ferroelectric behavior. Because it is stoichiometric, its
phonons, as observed by IR reflectivity and Raman scattering, are well
defined. The pyrochlore structure consists of a network of corner sharing NbO6
octahedra interpenetrated by CdO tetahedra (or CdO chains depending on how one
views it), as shown in Figure 2. Several structural transitions have been
observed as a function of temperature, where the symmetry lowers from the high
temperature cubic phase ($Fd\bar{3}m$). Of particular interest are the
ferroelastic transition at 204 K (which lowers the point group symmetry from
$m\bar{3}m$ to $mmm$, maintaining inversion) and the ferroelectric one at 196
K (which further lowers the point group symmetry to $mm2$) [29]. At much lower
temperatures, two other transitions have been reported that are consistent
with a monoclinic space group, probably $Cc$. Based on group-subgroup
relations and structural refinements, the accepted phase transition series
with decreasing temperature is $Fd\bar{3}m\rightarrow Imma\rightarrow
Ima2\rightarrow Cc$. The complexity of the phase diagram and the fact that the
polarization direction is easily reoriented by an external field [29] indicate
that the free energy landscape is soft, suggesting that this material is an
excellent hunting ground for new collective modes.

Density functional theory (DFT) calculations ($T=0$ K) show that $Fd\bar{3}m$
is unstable to both $\Gamma_{4}^{-}$ and $\Gamma_{5}^{-}$ distortions (the
former being polar in nature). $\Gamma_{4}^{-}$ and $\Gamma_{5}^{-}$ are both
primary order parameters for the transition from $Fd\bar{3}m$ to $Ima2$ 333The
speculated $Imma$ ferroelastic phase exists over a very limited range in
temperature and its nature is ambiguous; if Cd2Nb2O7 does experimentally adopt
$Imma$ symmetry, then it would be driven by a $\Gamma_{5}^{+}$ distortion
which is found not to occur in DFT studies [56, 34].. In the ferroelectric
phase, the minimum Landau subspace is $6\times 6$, given that both
$\Gamma_{4}^{-}$ and $\Gamma_{5}^{-}$ are three-dimensional group
representations, meaning Cd2Nb2O7 hosts a richer space of possibilities than
the minimal Landau model discussed earlier [31]. In addition, as we
demonstrate below, more coupling terms exist besides the biquadratic one, a
general result not specific to Cd2Nb2O7. Before turning to $Ima2$, we first
discuss the related $Fdd2$ space group which is simpler to present in a
Cartesian basis.

Table 1: Space groups generated on condensing the ($\Gamma_{4}^{-}$, $\Gamma_{5}^{-}$) order parameters along various crystallographic directions in Cd2Nb2O7 (generated from Ref. 32). $\Gamma_{4}^{-}$ | $\Gamma_{5}^{-}$ | Space Group
---|---|---
(a,0,0) | (0,b,0) | $Fdd2$
(a,a,0) | (0,b,-b) | $Ima2$
(a,a,a) | (b,b,b) | $R3$
(a,b,0) | (0,c,d) | $Cc$
(a,a,b) | (0,c,-c) | $Cm$
(a,a,0) | (-c,b,-b) | $C2$

###  III.1 $Fdd2$

The six-dimensional subspace formed by $\Gamma_{4}^{-}$ and $\Gamma_{5}^{-}$
defines two free energy surfaces (Figure 3a). These surfaces are three-
dimensional generalizations of two Mexican-hat potentials (Figure 3b), and can
be considered as nested warped spheres, one for $\Gamma_{4}^{-}$, the other
for $\Gamma_{5}^{-}$. Each representation condensing along the cubic axis of
its respective sphere gives rise to $Fdd2$. Interestingly, $Fdd2$ has been
identified as the local structure of Cd2Nb2O7 from diffuse scattering studies
[33] and is also the global space group upon sulfur doping [34]. But what is
evident from both the DFT and structural studies is that there are a number of
space groups which are close in energy and provide comparable descriptions of
the data. That is, the free energy landscape of these two coupled spheres is
relatively flat (i.e., the warping of the spheres is small). DFT calculations
indicate that structures where $\Gamma_{4}^{-}$ is dominant have a slightly
lower free energy than ones where $\Gamma_{5}^{-}$ is dominant [34], whereas
order parameter-like behavior of the distortion amplitudes as a function of
temperature has only been claimed for $\Gamma_{5}^{-}$ based on global
structural refinements [35]. Regardless, the net result is that one has two
coupled flat energy surfaces, one for $\Gamma_{4}^{-}$, the other for
$\Gamma_{5}^{-}$, which only differ from each other by a few meV per formula
unit [34].

In terms of order parameter directions (Table 1), the $Fdd2$ phase corresponds
to $(a,0,0)|(0,c,0)$ with $(a,0,0)$ being a point on the $\Gamma_{4}^{-}$
sphere and $(0,c,0)$ on the $\Gamma_{5}^{-}$ sphere, whereas $Ima2$
corresponds to $(a,a,0)|(0,c,-c)$ instead. Here, $a$ refers to the
$\Gamma_{4}^{-}$ order parameter and $c$ to the $\Gamma_{5}^{-}$ one. The
lower symmetry space group encompassing these two is $Cc$, corresponding to
$(a,b,0)|(0,c,d)$, which can be seen to correspond to particular great circles
on each sphere (analogous to the Mexican-hat brims of the minimal model). For
our purposes here, we will restrict our analysis to these two circles,
recognizing that there are also fluctuations orthogonal to them on the spheres
corresponding to rhombohedral ($R3$) and other monoclinic ($Cm$ and $C2$)
space groups, as well as to other domains of $Fdd2$ and $Ima2$ not represented
by these two particular circles (Table 2). That is, we consider a reduced
$4\times 4$ force-constant matrix. We note that for these flat free energy
surfaces, it is often useful to describe each order parameter with
spherical/polar coordinates, but for simplicity in the following derivations
we will use a Cartesian basis instead.

We obtain the Landau free energy using the INVARIANTS routine [36] (accessible
electronically at Ref. 32), noting that it is important to specify the
appropriate $Cc$ domains for the subspace considered, specifically the order
parameter directions $(a,b,0)|(0,c,d)$. Considering terms to quartic order, we
obtain

$\displaystyle F$ $\displaystyle=$
$\displaystyle\frac{\alpha_{4}}{2}(a^{2}+b^{2})+\frac{\alpha_{5}}{2}(c^{2}+d^{2})+\frac{\beta_{4}}{4}(a^{2}+b^{2})^{2}$
$\displaystyle+\frac{\gamma_{4}}{4}(a^{4}+b^{4})+\frac{\beta_{5}}{4}(c^{2}+d^{2})^{2}+\frac{\gamma_{5}}{4}(c^{4}+d^{4})$
$\displaystyle+\frac{\delta}{2}(a^{2}+b^{2})(c^{2}+d^{2})+\epsilon_{1}(ab)(ad-
bc)$
$\displaystyle+\frac{\epsilon_{2}}{2}(a^{2}c^{2}+b^{2}d^{2})+\epsilon_{3}(abcd)+\epsilon_{4}(cd)(ad-
bc)\,.$

We note that in polar coordinates,
$a=Q_{4}\cos(\phi_{4}),b=Q_{4}\sin(\phi_{4}),c=Q_{5}\cos(\phi_{5}),d=Q_{5}\sin(\phi_{5})$,
where $Q_{i}$ are the amplitudes and $\phi_{i}$ are the phases.

We first consider expanding about an assumed $Fdd2$ free energy minimum, and
then consider $Ima2$ next. For $Fdd2$, fluctuations of $a$ and $c$ correspond
to Higgs modes ($A_{1}$ symmetry), fluctuations of $b$ and $d$ to Goldstone
modes ($B_{2}$ symmetry) (Table 2). Note these Goldstone modes are not at zero
energy because of the anisotropy terms $\gamma_{i}$, so they are formally
pseudo-Goldstone modes. In addition, the $\delta$ (biquadratic) and
$\epsilon_{i}$ terms will provide coupling between the two Higgs modes, and
the latter also between the two Goldstone modes. There is no coupling between
the Higgs and the Goldstone modes, so the 4 $\times$ 4 matrix reduces to two
$2\times 2$ blocks: one for the two Higgs modes, one for the two Goldstone
modes. This remains true when considering sixth-order terms in the free
energy, and also for the full $\Gamma_{4}^{-}\oplus\Gamma_{5}^{-}$ space. That
is, this larger $6\times 6$ matrix reduces to three $2\times 2$ blocks: one in
the Higgs sector and the other two in the Goldstone sector. That is, there are
no terms coupling the two Goldstone blocks.

Differentiating LABEL:eq:landauCNO with respect to $a,b,c,d$ and setting each
expression to zero determines these parameters via a set of coupled equations
[27]. One can see from these expressions that $b=d=0$ ($Fdd2$) is an allowed
solution to these coupled equations. Whether it is or is not depends on the
actual values of the Landau coefficients, but for purposes here we assume that
this is the case 444The $a,b,c,d$ coefficients can be determined from DFT
[57], or else extracted from an AMPLIMODES [23, 58, 59] or ISODISTORT [24, 32]
analysis of the experimental structural refinements..

The coefficients of the force-constant matrix for each of the modes are
determined by $\frac{\partial^{2}F}{\partial q_{i}\partial q_{j}}$ where
$q_{i}$ are $a,b,c,d$, evaluated at $a=\left<a\right>$, $c=\left<c\right>$,
$b=d=0$ The diagonal elements of the force-constant matrix will give the
uncoupled Higgs mode energies ($aa$ and $cc$ elements) and Goldstone mode
energies ($bb$ and $dd$ elements). Of interest here are the off-diagonal
terms. We find that in this case, there are two non-zero ones. The first is
the $ac$ one that couples the two Higgs modes

$\frac{\partial^{2}F}{\partial a\partial c}=2(\delta+\epsilon_{2})(ac)\,.$ (7)

The other is the $bd$ one that couples the two Goldstone modes

$\frac{\partial^{2}F}{\partial b\partial
d}=\epsilon_{1}(a^{2})+\epsilon_{3}(ac)-\epsilon_{4}(c^{2})\,.$ (8)

For each 2 $\times$ 2 block, we denote the diagonal elements corresponding to
the uncoupled mode energies as $\omega_{4}^{2}$ ($aa$ or $bb$) and
$\omega_{5}^{2}$ ($cc$ or $dd$). Denoting the off-diagonal element in each
block as $X$ ($ab$ or $cd$), one obtains as before coupled mode energies of
the form

$\omega_{\pm}^{2}=\frac{\omega_{4}^{2}+\omega_{5}^{2}}{2}\pm\sqrt{\frac{(\omega_{4}^{2}-\omega_{5}^{2})^{2}}{4}+X^{2}}\,.$
(9)

The two eigenvectors of this matrix either have the two components in-phase,
or out-of-phase, as in Equation 5. The in-phase modes then correspond to Higgs
and Goldstone modes, the out-of-phase modes to their Leggett analogs.

To understand the two phase modes, consider a given $Fdd2$ domain,
$(a,0,0)|(0,c,0)$. The Goldstone mode would correspond to oscillations along
the two circles on the free energy spheres towards one $Cc$ domain
$(a,b,0)|(0,c,d)$ (one swing) and towards another $Cc$ domain
$(a,-b,0)|(0,c,-d)$ (the other swing) as illustrated in Figure 3. The Leggett
analog corresponds instead to oscillations towards $(a,b,0)|(0,c,-d)$ and
$(a,-b,0)|(0,c,d)$. These can be thought of as antiphase domains of $Cc$ that
occur at a higher energy since they are penalized by the coupling terms in the
Landau free energy. As such, the Leggett phase mode occurs at a higher energy
than the Goldstone mode. We therefore denote the Leggett phase mode as an
anti-phason. A similar picture applies in the Higgs sector (oscillations along
the radial directions in Figure 3). That is, the amplitudes $Q_{4}$ and
$Q_{5}$ oscillate either in-phase or out-of phase, so we can denote the latter
as a Leggett amplitude mode, that is an antiphase Higgs mode. Considering the
full six-dimensional $\Gamma_{4}^{-}\oplus\Gamma_{5}^{-}$ space, there are
actually two ‘anti-phasons’ and one ‘antiphase Higgs’ mode, as alluded to
above.

###  III.2 $Ima2$

Although the $Fdd2$ space group is realized upon sulfur doping [34],
stoichiometric Cd2Re2O7 is thought to be $Ima2$ below the ferroelectric
transition. The $Ima2$ case is similar to the $Fdd2$. $Ima2$ corresponds to
$a=b$ and $d=-c$. These coordinates represent points on the two circles in
Figure 3 that are rotated by 45∘ relative to $Fdd2$. Therefore, to construct
Goldstone and Higgs variables in a Cartesian basis, it is convenient to
express $F$ using a 45∘ rotated coordinate frame instead, that is
$\tilde{a}=(a+b)/\sqrt{2},\,\tilde{b}=(a-b)/\sqrt{2},\,\tilde{c}=(c-d)/\sqrt{2},\,\tilde{d}=(c+d)/\sqrt{2}$,
so that $Ima2$ corresponds to $\tilde{b}=\tilde{d}=0$. The resulting Landau
free energy is:

$\displaystyle F$ $\displaystyle=$
$\displaystyle\frac{\alpha_{4}}{2}(\tilde{a}^{2}+\tilde{b}^{2})+\frac{\alpha_{5}}{2}(\tilde{c}^{2}+\tilde{d}^{2})+\frac{\beta_{4}}{4}(\tilde{a}^{2}+\tilde{b}^{2})^{2}$
$\displaystyle+\frac{\gamma_{4}}{8}(\tilde{a}^{4}+\tilde{b}^{4}+6\tilde{a}^{2}\tilde{b}^{2})+\frac{\beta_{5}}{4}(\tilde{c}^{2}+\tilde{d}^{2})^{2}$
$\displaystyle+\frac{\gamma_{5}}{8}(\tilde{c}^{4}+\tilde{d}^{4}+6\tilde{c}^{2}\tilde{d}^{2})+\frac{\delta}{2}(\tilde{a}^{2}+\tilde{b}^{2})(\tilde{c}^{2}+\tilde{d}^{2})$
$\displaystyle+\frac{\epsilon_{1}}{2}(\tilde{a}^{2}-\tilde{b}^{2})(\tilde{b}\tilde{d}-\tilde{a}\tilde{c})$
$\displaystyle+\frac{\epsilon_{2}}{4}[(\tilde{a}^{2}+\tilde{b}^{2})(\tilde{c}^{2}+\tilde{d}^{2})+4\tilde{a}\tilde{b}\tilde{c}\tilde{d}]$
$\displaystyle+\frac{\epsilon_{3}}{4}(\tilde{a}^{2}-\tilde{b}^{2})(\tilde{d}^{2}-\tilde{c}^{2})+\frac{\epsilon_{4}}{2}(\tilde{d}^{2}-\tilde{c}^{2})(\tilde{b}\tilde{d}-\tilde{a}\tilde{c})\,.$

One can show that, as before, the only off-diagonal terms of the force-
constant matrix that survive are the $\tilde{a}\tilde{c}$ and
$\tilde{b}\tilde{d}$ terms. Thus, the force-constant matrix again reduces to
two $2\times 2$ blocks and results in a Higgs mode, a Goldstone mode, and the
two Leggett modes: an antiphase Higgs mode and an anti-phason. This is
straightforward to understand from a study of Figure 3 and Table 1 as we
summarize in Table 2.

Table 2: Fluctuations about $Fdd2$ and $Ima2$ derived from Table I. H denotes fluctuations indicated by the blue/red arrows along the specific great circles shown on the spheres in Figure 3. V denotes fluctuations along great circles orthogonal to these circles. Note that the circles are turned 90 degrees between the two surfaces because of the different transformation properties of $\Gamma_{4}^{-}$ and $\Gamma_{5}^{-}$. R denotes fluctuations along the radial directions of the spheres. For $Fdd2$, the two $Cc$ rows correspond to different $Cc$ domains. Also, the reduction to $C2$ is not shown because it involves an $Fdd2\rightarrow P1\rightarrow C2$ path on the spheres in Figure 3. Space group | Symmetry | $\Gamma_{4}^{-}$ | $\Gamma_{5}^{-}$ | Modes
---|---|---|---|---
$Fdd2\rightarrow Fdd2$ |  $\Gamma_{1}$ (A1) | R | R | Higgs, antiphase Higgs
$Fdd2\rightarrow Cc$ |  $\Gamma_{4}$ (B2) | H | H | Goldstone, anti-phason
$Fdd2\rightarrow Cc$ |  $\Gamma_{3}$ (A2) | V | V | Goldstone, anti-phason
$Ima2\rightarrow Ima2$ |  $\Gamma_{1}$ (A1) | R | R | Higgs, antiphase Higgs
$Ima2\rightarrow Cc$ |  $\Gamma_{4}$ (B2) | H | H | Goldstone, anti-phason
$Ima2\rightarrow Cm$ |  $\Gamma_{3}$ (A2) | V |  | Goldstone
$Ima2\rightarrow C2$ |  $\Gamma_{2}$ (B1) |  | V | Goldstone

We now connect these results to those on superconductors. Upon manipulation of
the corresponding dynamical matrix in the case of superconductivity [22], one
can show that the off-diagonal element for a charge-neutral two-band
superconductor is equal to the square root of the product of the two diagonal
elements. This results in one mode frequency that is zero (the Goldstone mode)
and one whose squared frequency is the sum of the two diagonal elements (the
Leggett mode). Besides this distinction, there is no qualitative difference
between the superconductivity and structural mode cases. In particular, for
both cases, the two components of the eigenvector do not have equal
amplitudes. In the superconductivity case, this is due to the difference in
the density of states of the two bands. In the structural case, it is because
the uncoupled mode frequencies for $\Gamma_{4}^{-}$ and $\Gamma_{5}^{-}$
differ.

The Leggett analog of the Higgs mode (the antiphase Higgs) has not been
explicitly described before. A related mode, however, was predicted recently
in time-reversal breaking superconductors [38]. It should be observable as
well in multi-band superconductors, and would correspond to out-of-phase
oscillations of the gap amplitudes of the two bands. We suggest that the
existence of this mode be searched for by appropriate experiments (Raman,
pump-probe) on two-band superconductors like MgB2. Below, we address its
observation in the structural case.

###  III.3 Secondary Order Parameters

The secondary order parameters present in Cd2Nb2O7 are even-parity ones with
symmetry $\Gamma_{3}^{+}$ and $\Gamma_{5}^{+}$. For $Cc$, this results in a
seven-dimensional space:
$\Gamma_{4}^{-}(a,b)\oplus\Gamma_{5}^{-}(c,d)\oplus\Gamma_{3}^{+}(e,f)\oplus\Gamma_{5}^{+}(g)$.
Expanding around $Fdd2$ ($b=d=g=0$) and $Ima2$ ($b=a,d=-c,f=0$), no couplings
exist between the primary Higgs and Goldstone modes (that is, the
$ab,ad,bc,cd$ terms in the force-constant matrix vanish at the $Fdd2$ and
$Ima2$ minima). This means that any coupling between these Higgs and Goldstone
modes are beyond the harmonic approximation, however higher-order coupling can
occur in non-equilibrium situations [19, 21]. At first glance, these secondary
modes shift the various primary mode frequencies that would be determined from
the smaller $4\times 4$ block. That is, one can in principle reduce this
$7\times 7$ matrix to an effective $4\times 4$ one by integrating out $e,f,g$
in the Landau free energy equations, as these secondary order parameters are
slaved to the primary ones.

But closer inspection of the form of the primary-secondary mode couplings
reveals new opportunities to study the Leggett modes. To see this, note that
in the $Cc$ subspace discussed above, one now has the following coupling terms
at cubic order since the secondary order parameters have even parity:

$\displaystyle F_{3}$ $\displaystyle=$
$\displaystyle\eta_{1}[\frac{1}{\sqrt{3}}(a^{2}+b^{2})e+(a^{2}-b^{2})f]$ (11)
$\displaystyle+\eta_{2}[\frac{1}{\sqrt{3}}(c^{2}+d^{2})e+(c^{2}-d^{2})f]$
$\displaystyle+\eta_{3}[(ac-bd)e-\frac{1}{\sqrt{3}}(ac+bd)f]$
$\displaystyle+\eta_{4}(abg)+\eta_{5}(cdg)+\eta_{6}[(ad-bc)g]\,.$

For both $Fdd2$ and $Ima2$, the $\eta_{3}$ term leads to both primary Higgs
and Goldstone mode couplings. For $Ima2$, the $\eta_{6}$ term also leads to
both primary Higgs and Goldstone couplings. We note that $\Gamma_{3}^{+}$ and
$\Gamma_{5}^{+}$ are Raman active modes in the cubic phase. Therefore, below
the ferroelectric transition, they can be used to drive the primary Leggett
modes via these two cubic terms, analogous to nonlinear phononics experiments
on perovskites [39] that have been studied theoretically using similar Landau-
like equations of motion [40, 41, 42]. Note that pump-probe experiments have
been instrumental in the study of Leggett modes in superconductors [43].

In the above analysis, we did not include strain (which typically renormalizes
the Landau coefficients), and gradient terms which need to be included to
address domain walls. One would expect that the collective modes could be
significantly modified by domain walls if they are spatially broad enough [44]
and present at a high enough density. These effects would be interesting to
study in future work.

Figure 4: Decomposition of the $Ima2$ structural refinement of Cd2Nb2O7 from
Ref. 35 using ISODISTORT [24, 32]. Shown is the overall $\Gamma_{5}^{-}$
($T_{2u}$) amplitude previously plotted in [35] along with various
$\Gamma_{4}^{-}$ ($T_{1u}$) and $\Gamma_{5}^{-}$ decompositions involving
either Nb or Cd ion displacements from their cubic positions, noting that for
each ion, $\Gamma_{4}^{-}$ has two submodes $\Gamma_{3}^{-}$ ($E_{u}$) and
$\Gamma_{2}^{-}$ (A2u) whereas $\Gamma_{5}^{-}$ has only one
($\Gamma_{3}^{-}$). Curves are proportional to $|T_{c}-T|^{\beta}$, where
$T_{c}=196$ K is the ferroelectric transition temperature and $\beta=0.27$ is
the order parameter exponent.

###  III.4 Submodes and Phonons for Cd2Nb2O7

In the real crystal, there are multiple force-constant matrix eigenmodes for
each symmetry, typically referred to in the literature as ‘submodes’. The full
matrix has a size of $66\times 66$. Restricting the symmetry to $Cc$, the
matrix reduces to $33\times 33$. Further restriction to $\Gamma_{4}^{-}$ and
$\Gamma_{5}^{-}$, it reduces to 24 $\times$ 24\. In the Landau treatment, each
$q_{i}$ is a particular sum of these submodes with the appropriate group
symmetry that is gotten by reducing this larger matrix to the smaller $4\times
4$ matrix. Still, one can ask the question whether this simplification is
supported by the data or not. To address this, in Figure 4, we plot the
temperature dependence of several relevant submode amplitudes based on the
$Ima2$ crystal structure refinements of Malcherek et al. [35]. We find that
the Cd and Nb displacements for both $\Gamma_{4}^{-}$ and $\Gamma_{5}^{-}$
symmetries scale with the overall $\Gamma_{5}^{-}$ order parameter amplitude
that was previously shown in Ref. [35]. This indicates that this simpler
Landau description is valid.

We now turn to the phonons. Since the ions involved (Cd, Nb, O) have different
masses, there is no one-to-one correspondence between the force-constant modes
and the phonons (unless the mode involved only one ion type). As a
consequence, a given force-constant mode could involve more than one phonon.
However, it has been shown that there is a strong correspondence between
single phonons and the Higgs and Goldstone modes in YMnO3 [19]. Therefore it
is probable that such modes for Cd2Nb2O7 can also be associated with specific
phonons.

To gain further insight, we turn to experimental Raman and infrared (IR) data.
In the $Ima2$ phase, all phonons are in principle Raman active. Raman data on
Cd2Nb2O7 find several low lying $A_{1}$ and $B_{2}$ modes below the
ferroelectric transition, three of which soften as the structural phase
transition is approached from below [45]. As the amplitude modes have $A_{1}$
symmetry and the phase modes have $B_{2}$ symmetry, there should be a
correlation between our collective modes and the data. That is, it is possible
that the two lowest lying $A_{1}$ Raman modes corresponds to the Higgs and
antiphase Higgs modes. Presumably the lowest lying $B_{2}$ mode is the
Goldstone mode, with the anti-phason corresponding to a higher energy $B_{2}$
mode that has yet to be studied.

The IR data for Cd2Nb2O7 are complicated given the presence of seven optic
modes of $\Gamma_{4}^{-}$ symmetry in the cubic phase, which further split in
the ferroelectric phase. Two of the lower lying IR modes have been interpreted
as being coupled (both above and below the transition) due to their
temperature dependencies [46]. One of these, referred to as a ‘central’ mode
and speculated to be due to hopping of Cd ions among equivalent locations
displaced from their cubic positions, is not thought to be of $\Gamma_{4}^{-}$
symmetry given that seven other IR modes of this symmetry are already seen.
Its coupling with a higher energy second mode, thought to be a Nb displacement
mode of $\Gamma_{4}^{-}$ symmetry, drives this central mode to soften at the
ferroelectric transition [46]. Whether this central mode is an $A_{1}$
symmetry mode (as typical for an order-disorder transition), or instead a
$\Gamma_{5}^{-}$ mode that becomes IR active due to coupling to the second
$\Gamma_{4}^{-}$ mode, is worth investigating. If the latter, this would be
consistent with the theory we offer above of two coupled modes of
$\Gamma_{4}^{-}$ and $\Gamma_{5}^{-}$ symmetry. Moreover, as discussed above,
pump-probe studies of Cd2Nb2O7 would be instrumental in probing for possible
collective modes: Higgs, Goldstone, and their Leggett analogs.

##  IV Searching for and Observing Leggett Modes

Although we considered the specific example of Cd2Nb2O7, the above analysis
should be applicable whenever more than one primary order parameter is
involved in a phase transition. As such, Leggett modes should exist in the
structural context, and could be identified from a DFT analysis of the force-
constant and dynamical (phonon) matrices in comparison to experimental data.
We plan to report on this in a future paper that will provide a detailed DFT
study of Cd2Nb2O7 [47]. In general, analysis of Raman and IR data, including
the symmetry of the modes and their temperature dependence, should be helpful
in elucidating the presence of Leggett modes. Specifics, including identifying
the out-of-phase behavior characteristic of the Leggett modes, could be
resolved by pump-probe studies of the phase of the oscillations as a function
of time. Moreover, the equations of motion associated with nonlinear phononics
involve the same cubic and quartic coupling terms invoked in this paper that
provide for the existence of the Leggett modes to begin with. Driving specific
modes would then be instrumental in identifying the various collective modes
via their couplings to the driven mode.

As for materials, obviously those phase transitions involving more than one
primary group representation would be obvious targets for study, including
those exhibiting improper and hybrid-improper ferroelectric transitions [48,
49, 50, 51, 52, 53]. Although the latter typically involve order parameters
with non-zero wavevector, the Landau coupling terms in the free energy are
similar and so the considerations presented here should be valid there as
well. The existence of low lying modes would be aided by having flat energy
surfaces with low transition barriers as considered here. In that context,
Cd2Nb2O7 consists of corner sharing NbO6 octahedra in an open framework
interpenetrated by CdO tetrahedra that are weakly coupled to the octahedra,
implying floppy low energy modes. Going from a cubic phase to a lower symmetry
orthorhombic or monoclinic phase is also useful in order to optimize the
number of coupling terms in the Landau free energy. This is particularly
pervasive in pyrochlores and spinels. Cage-like structures as in skutterudites
with their associated floppy modes would also be a good place to search. Much
of the work concerning Goldstone and Higgs modes has been done in the context
of perovskites, which also involve corner-sharing octahedra, and a number of
them exhibit improper ferroelectric transitions as referenced above. Similar
considerations to ours would also apply to ferroelastic transitions. However,
a large coupling to strain usually leads to large warping terms (see e.g. in
the case of ferroelastic WO3 [54]). A locally flat energy landscape can
sometimes be recovered at Ising-type domain walls (see e.g. for LiNbO3 [55]).
In the same way, locally-confined Leggett modes could be observed.

##  V Conclusion

We demonstrated via a Landau analysis that structural Leggett modes should
exist in the context of displacive transitions when more than one multi-
dimensional group representation is involved. These collective modes exist not
only in the phase channel as in the original work of Leggett, but also in the
amplitude channel, representing a new collective mode, the antiphase Higgs,
that should be observable in multi-band superconductors as well. We studied in
detail the specific case of the relaxor ferroelectric pyrochlore Cd2Nb2O7,
which we believe is a promising material to search for such modes. We believe
there should be a large group of materials where such modes could exist, and
advocate in particular that pump-probe studies would be the most illuminating
way to identify and characterize these modes. The study of such modes should
give new insights into their associated crystallographic phase transitions.

###### Acknowledgements.

Work at Argonne National Laboratory was supported by the Materials Sciences
and Engineering Division, Basic Energy Sciences, Office of Science, U.S. Dept.
of Energy. Q.N.M. was supported by the Swiss National Science Foundation under
project number P2EZP2_191872. Work at ETH Zurich was funded by the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation program project HERO grant (No. 810451). Work at Northwestern
University was supported by the National Science Foundation (NSF) Grant No.
DMR-2011208. G.L. gratefully acknowledges support for this work from Bates
College, and from NSF through DMR-1904980.

## References

  * Anderson [1963] P. W. Anderson, Plasmons, gauge invariance, and mass, Physical Review 130, 439 (1963).
  * Anderson [1958] P. W. Anderson, Random-phase approximation in the theory of superconductivity, Physical Review 112, 1900 (1958).
  * Schmid and Schön [1975] A. Schmid and G. Schön, Linearized kinetic equations and relaxation processes of a superconductor near $T_{c}$, Journal of Low Temperature Physics 20, 207 (1975).
  * Pekker and Varma [2015] D. Pekker and C. Varma, Amplitude/higgs modes in condensed matter physics, Annu. Rev. Condens. Matter Phys. 6, 269 (2015).
  * Leggett [1966] A. J. Leggett, Number-phase fluctuations in two-band superconductors, Progress of Theoretical Physics 36, 901 (1966).
  * Blumberg  _et al._ [2007] G. Blumberg, A. Mialitsin, B. S. Dennis, M. V. Klein, N. D. Zhigadlo, and J. Karpinski, Observation of Leggett’s Collective Mode in a Multiband MgB2 Superconductor, Physical Review Letters 99, 227002 (2007).
  * Carlson and Goldman [1975] R. V. Carlson and A. M. Goldman, Propagating order-parameter collective modes in superconducting films, Physical Review Letters 34, 11 (1975).
  * Wölfle [1976] P. Wölfle, Order-Parameter Collective Modes in ${}^{3}\textrm{He}-A$, Physical Review Letters 37, 1279 (1976).
  * Zavjalov  _et al._ [2016] V. Zavjalov, S. Autti, V. Eltsov, P. Heikkinen, and G. Volovik, Light higgs channel of the resonant decay of magnon condensate in superfluid 3 he-b, Nature communications 7, 1 (2016).
  * Scott [1974] J. F. Scott, Soft-mode spectroscopy: Experimental studies of structural phase transitions, Reviews of Modern Physics 46, 83 (1974).
  * Sergienko and Curnoe [2003] I. A. Sergienko and S. H. Curnoe, Structural Order Parameter in the Pyrochlore Superconductor Cd2Re2O7, Journal of the Physical Society of Japan 72, 1607 (2003).
  * Kendziora  _et al._ [2005] C. A. Kendziora, I. A. Sergienko, R. Jin, J. He, V. Keppens, B. C. Sales, and D. Mandrus, Goldstone-Mode Phonon Dynamics in the Pyrochlore Cd2Re2O7, Physical Review Letters 95, 125503 (2005).
  * Venderley  _et al._ [2021] J. Venderley, M. Matty, K. Mallayya, M. Krogstad, J. Ruff, G. Pleiss, V. Kishore, D. Mandrus, D. Phelan, L. Poudel, A. G. Wilson, K. Weinberger, P. Upreti, M. R. Norman, S. Rosenkranz, R. Osborn, and E.-A. Kim, Harnessing interpretable and unsupervised machine learning to address big data from modern x-ray diffraction (2021), arXiv:2008.03275 [cond-mat.str-el] .
  * Harter  _et al._ [2018] J. W. Harter, D. M. Kennes, H. Chu, A. de la Torre, Z. Y. Zhao, J.-Q. Yan, D. G. Mandrus, A. J. Millis, and D. Hsieh, Evidence of an improper displacive phase transition in cd2re2o7 via time-resolved coherent phonon spectroscopy, Physical Review Letters 120, 047601 (2018).
  * Nakhmanson and Naumov [2010] S. M. Nakhmanson and I. Naumov, Goldstone-like States in a Layered Perovskite with Frustrated Polarization: A First-Principles Investigation of PbSr2Ti2O7, Physical Review Letters 104, 097601 (2010).
  * Fu  _et al._ [2018] Y. Fu, X. He, L. Zhang, and D. J. Singh, Collective-Goldstone-mode-induced ultralow lattice thermal conductivity in Sn-filled skutterudite SnFe4Sb12, Physical Review B 97, 024301 (2018).
  * Marthinsen  _et al._ [2018] A. Marthinsen, S. M. Griffin, M. Moreau, T. Grande, T. Tybell, and S. M. Selbach, Goldstone-like phonon modes in a (111)-strained perovskite, Physical Review Materials 2, 014404 (2018).
  * Prosandeev  _et al._ [2020] S. Prosandeev, S. Prokhorenko, Y. Nahas, A. Al-Barakaty, L. Bellaiche, P. Gemeiner, D. Wang, A. A. Bokov, Z.-G. Ye, and B. Dkhil, Evidence for Goldstone-like and Higgs-like structural modes in the model PbMg1/3Nb2/3O3 relaxor ferroelectric, Physical Review B 102, 104110 (2020).
  * Meier  _et al._ [2020] Q. N. Meier, A. Stucky, J. Teyssier, S. M. Griffin, D. van der Marel, and N. A. Spaldin, Manifestation of structural higgs and goldstone modes in the hexagonal manganites, Physical Review B 102, 014102 (2020).
  * Zhang  _et al._ [2021] X. Zhang, Q.-J. Ye, and X.-Z. Li, Structural phase transition and goldstone-like mode in hexagonal BaMnO3, Physical Review B 103, 024101 (2021).
  * Juraschek  _et al._ [2020] D. M. Juraschek, Q. N. Meier, and P. Narang, Parametric excitation of an optically silent goldstone-like phonon mode, Physical Review Letters 124, 117401 (2020).
  * Sharapov  _et al._ [2002] S. Sharapov, V. Gusynin, and H. Beck, Effective action approach to the leggett’s mode in two-band superconductors, The European Physical Journal B 30, 45 (2002).
  * Orobengoa  _et al._ [2009] D. Orobengoa, C. Capillas, M. I. Aroyo, and J. M. Perez-Mato, AMPLIMODES: symmetry-mode analysis on the bilbao crystallographic server, Journal of Applied Crystallography 42, 820 (2009).
  * Campbell  _et al._ [2006] B. J. Campbell, H. T. Stokes, D. E. Tanner, and D. M. Hatch, ISODISPLACE: a web-based tool for exploring structural distortions, Journal of Applied Crystallography 39, 607 (2006).
  * Cowley [1980] R. Cowley, Structural phase transitions i. landau theory, Advances in Physics 29, 1 (1980).
  * Note [1] We note that the force-constant matrix is related to, but not equal to, the dynamical matrix determining the phonons, whose elements instead are $\frac{\Phi_{ij}}{\sqrt{M_{i}M_{j}}}$ where $M_{i}$ is the mass of the $i$-th ion [19]. As such, a given Higgs or Goldstone mode, defined in terms of the eigenvectors of the force-constant matrix, can be an admixture of several phonons of the appropriate symmetry [19].
  * Holakovský [1973] J. Holakovský, A new type of the ferroelectric phase transition, physica status solidi (b) 56, 615 (1973).
  * Note [2] The $q_{i}$ are typically given in length units, meaning an effective mass is needed to convert the force-constant eigenvalues to frequency units.
  * Ye  _et al._ [1991] Z. G. Ye, N. N. Kolpakova, J.-P. Rivera, and H. Schmid, Optical and electric investigations of the phase transitions in pyrochlore Cd2Nb2O7, Ferroelectrics 124, 275 (1991).
  * Note [3] The speculated $Imma$ ferroelastic phase exists over a very limited range in temperature and its nature is ambiguous; if Cd2Nb2O7 does experimentally adopt $Imma$ symmetry, then it would be driven by a $\Gamma_{5}^{+}$ distortion which is found not to occur in DFT studies [56, 34].
  * Talanov and Talanov [2021] M. V. Talanov and V. M. Talanov, Structural diversity of ordered pyrochlores, Chemistry of Materials 33, 2706 (2021).
  * [32] H. T. Stokes, D. M. Hatch, and B. J. Campbell, ISOTROPY Software Suite.
  * Malcherek [2011] T. Malcherek, The ferroelectric properties of Cd2Nb2O7: a Monte Carlo simulation study, Journal of Applied Crystallography 44, 585 (2011).
  * Laurita  _et al._ [2019] G. Laurita, D. Hickox-Young, S. Husremovic, J. Li, A. W. Sleight, R. Macaluso, J. M. Rondinelli, and M. A. Subramanian, Covalency-driven Structural Evolution in the Polar Pyrochlore Series Cd2Nb2O7-xSx, Chemistry of Materials 31, 7626 (2019).
  * Malcherek  _et al._ [2010] T. Malcherek, U. Bismayer, and C. Paulmann, The crystal structure of Cd2Nb2O7: symmetry mode analysis of the ferroelectric phase, Journal of Physics: Condensed Matter 22, 205401 (2010).
  * Hatch and Stokes [2003] D. M. Hatch and H. T. Stokes, INVARIANTS: program for obtaining a list of invariant polynomials of the order-parameter components associated with irreducible representations of a space group, Journal of Applied Crystallography 36, 951 (2003).
  * Note [4] The $a,b,c,d$ coefficients can be determined from DFT [57], or else extracted from an AMPLIMODES [23, 58, 59] or ISODISTORT [24, 32] analysis of the experimental structural refinements.
  * Poniatowski  _et al._ [2021] N. R. Poniatowski, J. B. Curtis, A. Yacoby, and P. Narang, Spectroscopic signatures of time-reversal symmetry breaking superconductivity, arXiv preprint arXiv:2103.05641 (2021).
  * Först  _et al._ [2011] M. Först, C. Manzoni, S. Kaiser, Y. Tomioka, Y. Tokura, R. Merlin, and A. Cavalleri, Nonlinear phononics as an ultrafast route to lattice control, Nature Physics 7, 854 (2011).
  * Subedi  _et al._ [2014] A. Subedi, A. Cavalleri, and A. Georges, Theory of nonlinear phononics for coherent light control of solids, Physical Review B 89, 220301 (2014).
  * Juraschek  _et al._ [2017] D. M. Juraschek, M. Fechner, and N. A. Spaldin, Ultrafast structure switching through nonlinear phononics, Physical Review Letters 118, 054101 (2017).
  * Gu and Rondinelli [2018] M. Gu and J. M. Rondinelli, Nonlinear phononic control and emergent magnetism in Mott insulating titanates, Physical Review B 98, 024102 (2018).
  * Giorgianni  _et al._ [2019] F. Giorgianni, T. Cea, C. Vicario, C. P. Hauri, W. K. Withanage, X. Xi, and L. Benfatto, Leggett mode controlled by light pulses, Nature Physics 15, 341 (2019).
  * Huang  _et al._ [2014] F.-T. Huang, X. Wang, S. M. Griffin, Y. Kumagai, O. Gindele, M.-W. Chu, Y. Horibe, N. A. Spaldin, and S.-W. Cheong, Duality of topological defects in hexagonal manganites, Phys. Rev. Lett. 113, 267602 (2014).
  * Taniguchi  _et al._ [2008] H. Taniguchi, T. Shimizu, H. Kawaji, T. Atake, M. Itoh, and M. Tachibana, Ferroelectric phase transition of Cd2Nb2O7 studied by Raman scattering, Physical Review B 77, 224104 (2008).
  * Buixaderas  _et al._ [2001] E. Buixaderas, S. Kamba, J. Petzelt, M. Savinov, and N. Kolpakova, Phase transitions sequence in pyrochlore Cd2Nb2O7 studied by IR reflectivity, The European Physical Journal B 19, 9 (2001).
  * Hickox-Young  _et al._ [2021] D. Hickox-Young, G. Laurita, Q. N. Meier, N. A. Spaldin, M. R. Norman, and J. M. Rondinelli, In Preparation (2021).
  * Bousquet  _et al._ [2008] E. Bousquet, M. Dawber, N. Stucki, C. Lichtensteiger, P. Hermet, S. Gariglio, J.-M. Triscone, and P. Ghosez, Improper ferroelectricity in perovskite oxide artificial superlattices, Nature 452, 732 (2008).
  * Xu  _et al._ [2015] B. Xu, D. Wang, H. J. Zhao, J. Íñiguez, X. M. Chen, and L. Bellaiche, Hybrid improper ferroelectricity in multiferroic superlattices: Finite-temperature properties and electric-field-driven switching of polarization and magnetization, Advanced Functional Materials 25, 3626 (2015).
  * Bellaiche and Íñiguez [2013] L. Bellaiche and J. Íñiguez, Universal collaborative couplings between oxygen-octahedral rotations and antiferroelectric distortions in perovskites, Phys. Rev. B 88, 014104 (2013).
  * Benedek and Fennie [2011] N. A. Benedek and C. J. Fennie, Hybrid improper ferroelectricity: A mechanism for controllable polarization-magnetization coupling, Phys. Rev. Lett. 106, 107204 (2011).
  * Mulder  _et al._ [2013] A. T. Mulder, N. A. Benedek, J. M. Rondinelli, and C. J. Fennie, Turning ABO3 Antiferroelectrics into Ferroelectrics: Design Rules for Practical Rotation-Driven Ferroelectricity in Double Perovskites and A3B2O7 Ruddlesden-Popper Compounds, Advanced Functional Materials 23, 4810 (2013).
  * Perez-Mato  _et al._ [2004] J. M. Perez-Mato, M. Aroyo, A. García, P. Blaha, K. Schwarz, J. Schweifer, and K. Parlinski, Competing structural instabilities in the ferroelectric Aurivillius compound $\mathrm{Sr}{\mathrm{Bi}}_{2}{\mathrm{Ta}}_{2}{\mathrm{O}}_{9}$, Phys. Rev. B 70, 214111 (2004).
  * Mascello  _et al._ [2020] N. Mascello, N. A. Spaldin, A. Narayan, and Q. N. Meier, Theoretical investigation of twin boundaries in WO3: Structure, properties, and implications for superconductivity, Phys. Rev. Research 2, 033460 (2020).
  * Mukherjee  _et al._ [2019] D. Mukherjee, S. Prokhorenko, L. Miao, K. Wang, E. Bousquet, V. Gopalan, and N. Alem, Atomic-scale measurement of polar entropy, Phys. Rev. B 100, 104102 (2019).
  * Fischer  _et al._ [2008] M. Fischer, T. Malcherek, U. Bismayer, P. Blaha, and K. Schwarz, Structure and stability of Cd2Nb2O7 and Cd2Ta2O7 explored by ab initio calculations, Physical Review B 78, 014108 (2008).
  * Artyukhin  _et al._ [2013] S. Artyukhin, K. T. Delaney, N. A. Spaldin, and M. Mostovoy, Landau theory of topological defects in multiferroic hexagonal manganites, Nature Materials 13, 42 (2013).
  * Aroyo  _et al._ [2006a] M. I. Aroyo, J. M. Perez-Mato, C. Capillas, E. Kroumova, S. Ivantchev, G. Madariaga, A. Kirov, and H. Wondratschek, Bilbao crystallographic server: I. databases and crystallographic computing programs, Zeitschrift für Kristallographie - Crystalline Materials 221, 15 (2006a).
  * Aroyo  _et al._ [2006b] M. I. Aroyo, A. Kirov, C. Capillas, J. M. Perez-Mato, and H. Wondratschek, Bilbao crystallographic server. II. representations of crystallographic point groups and space groups, Acta Crystallographica Section A Foundations of Crystallography 62, 115 (2006b).