PHYSICAL REVIEW X 12, 011024 (2022) - Leggett Modes Accompanying Crystallographic Phase Transitions
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PHYSICAL REVIEW X 12, 011024 (2022)
Leggett Modes Accompanying Crystallographic Phase Transitions
Quintin N. Meier ,1 Daniel Hickox-Young,2 Geneva Laurita ,3 Nicola A. Spaldin ,4
James M. Rondinelli ,2,5 and Michael R. Norman 6,*
1
Université Grenoble Alpes, CEA, LITEN, 17 rue des Martyrs, 38054 Grenoble, France
2
Department of Materials Science and Engineering,
Northwestern University, Evanston, Illinois 60208, USA
3
Department of Chemistry and Biochemistry, Bates College, Lewiston, Maine 04240, USA
4
Materials Theory, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerland
5
Northwestern-Argonne Institute of Science and Engineering (NAISE),
Northwestern University, Evanston, Illinois 60208, USA
6
Materials Science Division, Argonne National Laboratory, Lemont, Illinois 60439, USA
(Received 23 July 2021; revised 12 October 2021; accepted 9 December 2021; published 4 February 2022)
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 insight on structural phase transitions. Here, we show
that the Leggett mode, a collective mode observed in multiband 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
(antiphasons). 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 multiband superconductors as well. We illustrate the
existence and properties of these structural Leggett modes using the example of the pyrochlore relaxor
ferroelectric Cd2 Nb2 O7 .
DOI: 10.1103/PhysRevX.12.011024 Subject Areas: Condensed Matter Physics
I. INTRODUCTION Superconductors can exhibit other collective modes.
Spontaneous symmetry breaking is a ubiquitous phe- In particular, a relative phase mode of the order parameters
nomenon in physics, and is responsible for various collec- of the two bands of a two-band superconductor was
tive modes, most famously the well-known Goldstone and proposed by Leggett [5] and first realized in MgB2 [6].
Higgs modes. The former refers to fluctuations of the phase While the Goldstone mode is pushed to the plasma fre-
of an order parameter, the latter to fluctuations of its quency by coupling to the electromagnetic field, the Leggett
amplitude. These modes play a fundamental role in gauge mode maintains charge neutrality and so is not affected by
theories of particle physics, and are also important in the field [5]. Other modes are known as well, for instance,
condensed matter physics. The Goldstone-Higgs phenome- Carlson-Goldman modes in which the superconducting
non in high-energy physics is a relativistic generalization and normal condensates oscillate out of phase [7]. In
of the analogous behavior found in superconductors [1]. addition, in superfluid 3He a variety of clapping, flapping,
In superconductors, the Goldstone mode does not occur at and Higgs modes occur because of the high degeneracy of its
zero energy as in a neutral superfluid, but it is pushed to SOð3Þ × SOð3Þ × Uð1Þ order parameter space [8,9].
the plasma frequency by coupling to the electromagnetic Structural phase transitions are typically associated
field [2]. The superconducting Higgs mode is nontrivial to with soft phonons [10], and the concepts of Goldstone
and Higgs modes have provided new insight into these
observe since its energy tends to be located near the
transitions. A classic example is the pyrochlore Cd2 Re2 O7 .
quasiparticle continuum [3]. Nevertheless, it has been
Its structural phase transition arises from an instability
observed in several superconductors [4].
associated with a doubly degenerate Γ−3 phonon [11]. This
defines a Mexican-hat free-energy surface with the top of
*
norman@anl.gov the hat representing the high-temperature cubic phase and
the brim representing the lower symmetry distorted phase.
Published by the American Physical Society under the terms of In the Landau free energy, anisotropy terms that warp the
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to brim of the Mexican hat only arise at sixth order, suggesting
the author(s) and the published article’s title, journal citation, the existence of a low-energy Goldstone mode correspond-
and DOI. ing to oscillations along the brim. (Since they are not at zero
2160-3308=22=12(1)=011024(10) 011024-1 Published by the American Physical Society
QUINTIN N. MEIER et al. PHYS. REV. X 12, 011024 (2022)
energy because of the warping terms, these are sometimes force-constant matrix [26]. The advantage of Landau
referred to as pseudo-Goldstone modes). Subsequent theory is the reduction of the large force-constant matrix
Raman scattering experiments [12] exhibited strong evi- to this smaller one involving only the qi relevant to the
dence for this mode, and its existence has been recently phase transition [19].
confirmed by diffuse scattering studies [13]. The Higgs To illustrate the structural Leggett mode, we present a
mode is also evident from the Raman data, though its simple example in which the phase transition involves only
interpretation has been challenged by recent pump-probe two modes, q and r, each from a different two-dimensional
measurements [14]. After these pioneering studies of group representation. For simplicity, we use a Cartesian
Cd2 Re2 O7 , Higgs and Goldstone modes have been pro- basis, qðq1 ; q2 Þ and rðr1 ; r2 Þ. In a polar basis, qi ¼
posed in a variety of perovskite and other complex oxides Qi cosðϕi Þ. In general, in the uncoupled case, each mode
[15–20]. Routes to detect them if they are optically silent would have a different transition temperature, T q ≠ T r . We
have also been identified [21]. describe each mode by a Mexican-hat potential and include
Here, we investigate whether the Leggett mode can be a biquadratic coupling between them [27]. In this case, the
realized in the structural context, where it could give new free energy is given by
insight into the associated structural phase transitions. We
begin by developing a minimal Landau model that aq 2 bq a
F¼ ðq1 þ q22 Þ þ ðq21 þ q22 Þ2 þ r ðr21 þ r22 Þ
describes a structural phase transition that is accompanied 2 4 2
by a new collective mode, the antiphase Higgs mode (the br 2 c
þ ðr1 þ r22 Þ2 þ ðq21 þ q22 Þðr21 þ r22 Þ; ð1Þ
amplitude analog of the Leggett mode) that should also be 4 2
observable in multiband superconductors. We then consider
a Landau treatment of the ferroelectric transition in the where the temperature dependence is typically only
pyrochlore Cd2 Nb2 O7 , where we show that both the included in the quadratic terms aq ≡ aq0 ðT − T q Þ and
antiphase Higgs mode and the Leggett mode (the anti- ar ≡ ar0 ðT − T r Þ. The subspace of the force-constant
phason) are possible. Finally, we discuss secondary modes matrix of interest is now given by
that can drive these Leggett modes by coupling to them
in the context of pump-probe studies. We conclude by ∂ 2 F
Φij ðTÞ ¼ ; ð2Þ
discussing the relevance of our work to other materials, in ∂ϕi ∂ϕj ϕi ¼hϕi i;ϕj ¼hϕj i
particular, those that involve modes at nonzero wave
vectors. where ϕi ∈ fq1 ; q2 ; r1 ; r2 g, and describes the thermo-
dynamic average at a given temperature.
II. MINIMAL LANDAU MODEL Choosing without loss of generality that q1 ¼ hqi,
q2 ¼ 0, r1 ¼ hri, r2 ¼ 0, the force-constant matrix then
A superconductor is described by an order parameter becomes
Δeiϕ with an amplitude Δ and a phase ϕ. In a two-band
2 3
superconductor, one can identify a collective mode, called Hq 2chqihri 0 0
the Leggett mode, that corresponds to oscillations of the 6 2chqihri 0 0 7
6 Hr 7
relative phase of the order parameters associated with the Φ¼6 7; ð3Þ
two bands ϕ1 − ϕ2 [5]. This mode is an eigenvector of a 4 0 0 Gq 0 5
secular matrix whose eigenvalues are the collective mode 0 0 0 Gr
energies [22]. In the structural context, the corresponding
secular matrix is the force-constant matrix [19], with with
elements φij ¼ ð∂ 2 F=∂ui ∂uj Þ, where ui is the displace-
ment of the ith ion from its high-symmetry position and F Hq ¼ aq0 ðT − T q Þ þ 3bq hqi2 þ chri2 ;
is the free energy. For a periodic system with N atoms in the Gq ¼ aq0 ðT − T q Þ þ bq hqi2 þ chri2 ;
unit cell, this is a 3N × 3N matrix. For our further analysis,
we use a reduced form of the force-constant matrix, H r ¼ ar0 ðT − T r Þ þ 3br hri2 þ chqi2 ;
Φij ¼ ð∂ 2 F=∂qi ∂qj Þ, where qi are symmetry-adapted Gr ¼ ar0 ðT − T r Þ þ br hri2 þ chqi2 :
distortion modes [23,24]. Each qi is a linear combination
of atomic displacements that transform like a particular Here, we use H to indicate Higgs modes, that is, amplitude
group representation of the high-symmetry phase, and as modes with symmetry A1, and G to indicate Goldstone
such describe a collective motion involving multiple modes, that is, phase modes whose symmetry depends on
ions. For structural phase transitions following Landau the underlying space groups involved.
theory [25], F is formulated as a polynomial expansion in There are no off-diagonal terms coupling the Higgs
these qi . The distortions from the high-symmetry phase are and Goldstone sectors, but in this simple example, the
given, in the harmonic limit, by the eigenvectors of the biquadratic coupling term couples the two Higgs modes
011024-2LEGGETT MODES ACCOMPANYING CRYSTALLOGRAPHIC PHASE … PHYS. REV. X 12, 011024 (2022)
Hq and Hr . Because of this coupling, we expect that a new is at zero energy, in which case the dispersion is linear in
collective mode, the antiphase Higgs (the amplitude analog momentum instead.
of the Leggett mode), can exist. To see this, note that the What about the antiphason, the out-of-phase mode in the
eigenvalues of the force-constant matrix are the square phase channel that would be the analog of the super-
of the collective mode frequencies [28]. That is, the conducting Leggett mode? To obtain this requires coupling
eigenvalues of the upper 2 × 2 block of the force-constant of the two Goldstone modes. One can show that such
matrix are coupling requires terms in Eq. (1) that are linear in the
two Goldstone variables. In the above example, this
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii
1h would be terms of the form q2 r2 times Higgs variables
ω2 ¼ H þ Hq ðHr − Hq Þ2 þ 16c2 hqi2 hri2 : ð4Þ (q1 ; r1 ; q21 ; r21 ; q1 r1 , etc.). Although these terms do not exist
2 r
in the above minimal model, they do exist in general.
The eigenvectors are Instead of extending our minimal model to the more
general case, we instead illustrate antiphasons using the
1 ω2 − Hq specific example of Cd2 Nb2 O7 .
ðu; vÞ ¼ 1; ; ð5Þ
N 2chqihri
III. LANDAU THEORY OF Cd2 Nb2 O7
where N is a normalization factor. For small chqihri
The pyrochlore Cd2 Nb2 O7 is one of the few known
relative to Hq − Hr , we can think in terms of separate
stoichiometric materials that exhibits relaxor ferroelectric
Higgs modes that are weakly coupled. In the limit of large
behavior. Because it is stoichiometric, its phonons, as
coupling chqihri, however, we obtain an in-phase Higgs
observed by IR reflectivity and Raman scattering, are well
mode and an out-of-phase Higgs mode. The latter is the
defined. The pyrochlore structure consists of a network of
Higgs analog of the Leggett phase mode. These are
corner sharing NbO6 octahedra interpenetrated by CdO
illustrated in Fig. 1 for the simple case where the q and
tetahedra (or CdO chains depending on how one views it),
r parameters are the same. For this case, it is easy to show
as shown in Fig. 2. Several structural transitions have been
that the square of the Higgs mode energy is −2aq and that
observed as a function of temperature, where the symmetry
of the Leggett mode is −2aq ðbq − cÞ=ðbq þ cÞ.
lowers from the high-temperature cubic phase (Fd3̄m). Of
For its realization, one finds from the above equations particular interest are the ferroelastic transition at 204 K
that even one-dimensional group representations would (which lowers the point group symmetry from m3̄m to
produce this antiphase Higgs mode. To determine the mode mmm, maintaining inversion) and the ferroelectric one at
dispersions requires the addition of gradient terms that 196 K (which further lowers the point group symmetry
typically lead to a quadratic dispersion about the ordering to mm2) [29]. At much lower temperatures, two other
wave vector, unless the mode energy at the ordering vector transitions have been reported that are consistent with a
monoclinic space group, probably Cc. Based on group-
1.5 subgroup relations and structural refinements, the accepted
phase transition series with decreasing temperature is
Soft Fd3̄m → Imma → Ima2 → Cc. The complexity of the
Higgs phase diagram and the fact that the polarization direction
is easily reoriented by an external field [29] indicate that
1 Leggett
Mode energy
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 ¼ 0 K)
0.5 show that Fd3̄m is unstable to both Γ−4 and Γ−5 distortions
(the former being polar in nature). Γ−4 and Γ−5 are both
primary order parameters for the transition from Fd3̄m to
Ima2 [30]. In the ferroelectric phase, the minimum Landau
0 subspace is 6 × 6, given that both Γ−4 and Γ−5 are three-
0 0.5 1 1.5 2 dimensional group representations, meaning Cd2 Nb2 O7
T/Tc hosts a richer space of possibilities than the minimal
Landau model discussed earlier [33]. In addition, as we
FIG. 1. Mode energies for the minimal Landau model with demonstrate below, more coupling terms exist besides the
aq ¼ ar , bq ¼ br , and c ¼ bq =2, so that T q ¼ T r ¼ T c . Units biquadratic one, a general result not specific to Cd2 Nb2 O7 .
are such that aq0 =T q is set to unity. The Goldstone modes (not Before turning to Ima2, we first discuss the related Fdd2
shown) are at zero energy. space group which is simpler to present in a Cartesian basis.
011024-3QUINTIN N. MEIER et al. PHYS. REV. X 12, 011024 (2022)
FIG. 2. Crystal structure of cubic Cd2 Nb2 O7 shown to illustrate its two interpenetrating sublattices of NbO6 octahedra and O0 Cd4
tetrahedra with Cd (magenta), Nb (blue), and O (orange) ions.
A. Fdd2 where Γ−5 is dominant [31], whereas order parameterlike
The six-dimensional subspace formed by Γ−4 and Γ−5 behavior of the distortion amplitudes as a function of
defines two free-energy surfaces [Fig. 3(a)]. These surfaces temperature has only been claimed for Γ−5 based on global
are three-dimensional generalizations of two Mexican-hat structural refinements [35]. Regardless, the net result is that
potentials [Fig. 3(b)], and can be considered as nested one has two coupled flat energy surfaces, one for Γ−4 , the
warped spheres, one for Γ−4 , the other for Γ−5 . Each other for Γ−5 , which only differ from each other by a few
representation condensing along the cubic axis of its meV per formula unit [31].
respective sphere gives rise to Fdd2. Interestingly, Fdd2 In terms of order parameter directions (Table I), the
has been identified as the local structure of Cd2 Nb2 O7 from Fdd2 phase corresponds to ða; 0; 0Þjð0; c; 0Þ with ða; 0; 0Þ
diffuse scattering studies [34] and is also the global space being a point on the Γ−4 sphere and ð0; c; 0Þ on the Γ−5
group upon sulfur doping [31]. But what is evident from sphere, whereas Ima2 corresponds to ða; a; 0Þjð0; c; −cÞ
both the DFT and structural studies is that there are a instead. Here, a refers to the Γ−4 order parameter and c to
number of space groups which are close in energy and the Γ−5 one. The lower symmetry space group encompass-
provide comparable descriptions of the data. That is, the ing these two is Cc, corresponding to ða; b; 0Þjð0; c; dÞ,
free-energy landscape of these two coupled spheres is which can be seen to correspond to particular great circles
relatively flat (i.e., the warping of the spheres is small). on each sphere (analogous to the Mexican-hat brims of the
DFT calculations indicate that structures where Γ−4 is minimal model). For our purposes here, we restrict our
dominant have a slightly lower free energy than ones analysis to these two circles, recognizing that there are also
FIG. 3. Illustration of the Leggett phase mode for Fdd2 with (a) the free-energy surfaces drawn as nested spheres for Γ−4 and Γ−5 ,
whose radii are the amplitudes Q4 and Q5 , 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 Q4 and Q5 depend on the spherical angles).
The Fdd2 minimum in Cartesian coordinates is ðQ4 ; 0; 0Þ on the Γ−4 sphere and ð0; Q5 ; 0Þ on the Γ−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 (antiphason) 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.
011024-4LEGGETT MODES ACCOMPANYING CRYSTALLOGRAPHIC PHASE … PHYS. REV. X 12, 011024 (2022)
TABLE I. Space groups generated on condensing the (Γ−4 , Γ−5 ) TABLE II. Fluctuations about Fdd2 and Ima2 derived from
order parameters along various crystallographic directions in Table I. H denotes fluctuations indicated by the blue and red
Cd2 Nb2 O7 (generated from Ref. [36]). arrows along the specific great circles shown on the spheres in
Fig. 3. V denotes fluctuations along great circles orthogonal to
Γ−4 Γ−5 Space Group these circles. Note that the circles are turned 90° between the two
ða; 0; 0Þ ð0; b; 0Þ Fdd2 surfaces because of the different transformation properties of Γ−4
ða; a; 0Þ ð0; b; −bÞ Ima2 and Γ−5 . R denotes fluctuations along the radial directions of the
ða; a; aÞ ðb; b; bÞ R3 spheres. For Fdd2, the two Cc rows correspond to different Cc
ða; b; 0Þ ð0; c; dÞ Cc domains. Also, the reduction to C2 is not shown because it
ða; a; bÞ ð0; c; −cÞ Cm involves an Fdd2 → P1 → C2 path on the spheres in Fig. 3.
ða; a; 0Þ ð−c; b; −bÞ C2
Space group Symmetry Γ−4 Γ−5 Modes
Fdd2 → Fdd2 Γ1 (A1 ) R R Higgs, antiphase Higgs
fluctuations orthogonal to them on the spheres correspond- Fdd2 → Cc Γ4 (B2 ) H H Goldstone, antiphason
ing to rhombohedral (R3) and other monoclinic (Cm and Fdd2 → Cc Γ3 (A2 ) V V Goldstone, antiphason
C2) space groups, as well as to other domains of Fdd2 and Ima2 → Ima2 Γ1 (A1 ) R R Higgs, antiphase Higgs
Ima2 not represented by these two particular circles Ima2 → Cc Γ4 (B2 ) H H Goldstone, antiphason
(Table II). That is, we consider a reduced 4 × 4 force- Ima2 → Cm Γ3 (A2 ) V Goldstone
constant matrix. We note that for these flat free-energy Ima2 → C2 Γ2 (B1 ) V Goldstone
surfaces, it is often useful to describe each order parameter
with spherical or polar coordinates, but for simplicity in the
following derivations we will use a Cartesian basis instead. three 2 × 2 blocks: one in the Higgs sector and the other
We obtain the Landau free energy using the INVARIANTS two in the Goldstone sector. That is, there are no terms
routine [37] (accessible electronically [36]), noting that coupling the two Goldstone blocks.
it is important to specify the appropriate Cc domains for Differentiating Eq. (6) with respect to a, b, c, d and
the subspace considered, specifically the order parameter setting each expression to zero determines these parameters
directions ða; b; 0Þjð0; c; dÞ. Considering terms to quartic via a set of coupled equations [27]. One can see from these
order, we obtain expressions that b ¼ d ¼ 0 (Fdd2) is an allowed solution
to these coupled equations. Whether it is or is not depends
α4 2 α β on the actual values of the Landau coefficients, but for
F¼ ða þ b2 Þ þ 5 ðc2 þ d2 Þ þ 4 ða2 þ b2 Þ2
2 2 4 purposes here we assume that this is the case [38].
γ4 4 β γ The coefficients of the force-constant matrix for each of
þ ða þ b4 Þ þ ðc2 þ d2 Þ2 þ 5 ðc4 þ d4 Þ
5
the modes are determined by ð∂ 2 F=∂qi ∂qj Þ, where qi are
4 4 4
δ 2 a, b, c, d, evaluated at a ¼ hai, c ¼ hci, b ¼ d ¼ 0. The
2 2 2
þ ða þ b Þðc þ d Þ þ ϵ1 ðabÞðad − bcÞ diagonal elements of the force-constant matrix will give the
2
ϵ uncoupled Higgs mode energies (aa and cc elements) and
þ 2 ða2 c2 þ b2 d2 Þ þ ϵ3 ðabcdÞ þ ϵ4 ðcdÞðad − bcÞ: ð6Þ Goldstone mode energies (bb and dd elements). Of interest
2
here are the off-diagonal terms. We find that in this case
We note that in polar coordinates, a ¼ Q4 cosðϕ4 Þ, there are two nonzero ones. The first is the ac one that
b ¼ Q4 sinðϕ4 Þ, c ¼ Q5 cosðϕ5 Þ, d ¼ Q5 sinðϕ5 Þ, where couples the two Higgs modes:
Qi are the amplitudes and ϕi are the phases.
We first consider expanding about an assumed Fdd2 ∂ 2F
free-energy minimum, and then consider Ima2 next. For ¼ 2ðδ þ ϵ2 ÞðacÞ: ð7Þ
∂a∂c
Fdd2, fluctuations of a and c correspond to Higgs modes
(A1 symmetry), fluctuations of b and d to Goldstone modes The other is the bd one that couples the two Goldstone
(B2 symmetry) (Table II). Note these Goldstone modes are modes:
not at zero energy because of the anisotropy terms γ i , so
they are formally pseudo-Goldstone modes. In addition, the
δ (biquadratic) and ϵi terms will provide coupling between ∂2F
¼ ϵ1 ða2 Þ þ ϵ3 ðacÞ − ϵ4 ðc2 Þ: ð8Þ
the two Higgs modes, and the latter also between the two ∂b∂d
Goldstone modes. There is no coupling between the Higgs
and the Goldstone modes, so the 4 × 4 matrix reduces to For each 2 × 2 block, we denote the diagonal elements
two 2 × 2 blocks: one for the two Higgs modes, one for the corresponding to the uncoupled mode energies as ω24 (aa or
two Goldstone modes. This remains true when considering bb) and ω25 (cc or dd). Denoting the off-diagonal element in
sixth-order terms in the free energy, and also for the full each block as X (ab or cd), one obtains as before coupled
Γ−4 ⊕ Γ−5 space. That is, this larger 6 × 6 matrix reduces to mode energies of the form
011024-5QUINTIN N. MEIER et al. PHYS. REV. X 12, 011024 (2022)
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ω2 þ ω25 ðω24 − ω25 Þ2 along the radial directions in Fig. 3). That is, the amplitudes
ω2 ¼ 4 þ X2 : ð9Þ Q4 and Q5 oscillate either in phase or out of phase, so we can
2 4
denote the latter as a Leggett amplitude mode, that is, an
The two eigenvectors of this matrix have the two compo- antiphase Higgs mode. Considering the full six-dimensional
nents either in phase or out of phase, as in Eq. (5). The in- Γ−4 ⊕ Γ−5 space, there are actually two “antiphasons” and one
phase modes then correspond to Higgs and Goldstone “antiphase Higgs” mode, as alluded to above.
modes, the out-of-phase modes to their Leggett analogs.
To understand the two phase modes, consider a given B. Ima2
Fdd2 domain, ða; 0; 0Þjð0; c; 0Þ. The Goldstone mode
Although the Fdd2 space group is realized upon sulfur
would correspond to oscillations along the two circles
doping [31], stoichiometric Cd2 Re2 O7 is thought to be
on the free-energy spheres toward one Cc domain
Ima2 below the ferroelectric transition. The Ima2 case is
ða; b; 0Þjð0; c; dÞ (one swing) and toward another Cc domain
similar to the Fdd2. Ima2 corresponds to a ¼ b and
ða; −b; 0Þjð0; c; −dÞ (the other swing) as illustrated in Fig. 3.
d ¼ −c. These coordinates represent points on the two
The Leggett analog corresponds instead to oscillations
circles in Fig. 3 that are rotated by 45° relative to Fdd2.
toward ða; b; 0Þjð0; c; −dÞ and ða; −b; 0Þjð0; c; dÞ. These
Therefore, to construct Goldstone and Higgs variables in a
can be thought of as antiphase domains of Cc that occur at a
higher energy since they are penalized by the coupling terms Cartesian basis, it is convenient to express F using a p 45°
ffiffiffi
in the Landau free energy. As such, the Leggett phase mode rotated coordinate ffiffiffi is, ã ¼ ða þ p
pffiffiffi frame instead,pthat ffiffiffi 2,
bÞ=
occurs at a higher energy than the Goldstone mode. We b̃ ¼ ða − bÞ= 2, c̃ ¼ ðc − dÞ= 2, d̃ ¼ ðc þ dÞ= 2, so
therefore denote the Leggett phase mode as an antiphason. that Ima2 corresponds to b̃ ¼ d̃ ¼ 0. The resulting
A similar picture applies in the Higgs sector (oscillations Landau free energy is
α4 2 α β γ β
F¼ ðã þ b̃2 Þ þ 5 ðc̃2 þ d̃2 Þ þ 4 ðã2 þ b̃2 Þ2 þ 4 ðã4 þ b̃4 þ 6ã2 b̃2 Þ þ 5 ðc̃2 þ d̃2 Þ2
2 2 4 8 4
γ5 4 δ ϵ 1
þ ðc̃ þ d̃4 þ 6c̃2 d̃2 Þ þ ðã2 þ b̃2 Þðc̃2 þ d̃2 Þ þ ðã2 − b̃2 Þðb̃ d̃ −ã c̃Þ
8 2 2
ϵ2 2 ϵ3 2 ϵ
þ ½ðã þ b̃ Þðc̃ þ d̃ Þ þ 4ã b̃ c̃ d̃ þ ðã − b̃ Þðd̃2 − c̃2 Þ þ 4 ðd̃2 − c̃2 Þðb̃ d̃ −ã c̃Þ:
2 2 2 2
ð10Þ
4 4 2
One can show that, as before, the only off-diagonal terms of The Leggett analog of the Higgs mode (the antiphase
the force-constant matrix that survive are the ã c̃ and b̃ d̃ Higgs) has not been explicitly described before. A related
terms. Thus, the force-constant matrix again reduces to two mode, however, was predicted recently in time-reversal
2 × 2 blocks and results in a Higgs mode, a Goldstone breaking superconductors [42]. It should be observable as
mode, and the two Leggett modes: an antiphase Higgs well in multiband superconductors, and would correspond
mode and an antiphason. This is straightforward to under- to out-of-phase oscillations of the gap amplitudes of the
stand from a study of Fig. 3 and Table I as we summarize two bands. We suggest that the existence of this mode be
in Table II. searched for by appropriate experiments (Raman, pump-
We now connect these results to those on superconduc- probe) on two-band superconductors like MgB2 . Below, we
tors. Upon manipulation of the corresponding dynamical address its observation in the structural case.
matrix in the case of superconductivity [22], one can show
that the off-diagonal element for a charge-neutral two-band
C. Secondary order parameters
superconductor is equal to the square root of the product
of the two diagonal elements. This results in one mode The secondary order parameters present in Cd2 Nb2 O7
frequency that is zero (the Goldstone mode) and one whose are even-parity ones with symmetry Γþ þ
3 and Γ5 . For Cc,
squared frequency is the sum of the two diagonal elements this results in a seven-dimensional space: Γ−4 ða; bÞ ⊕
(the Leggett mode). Besides this distinction, there is no Γ−5 ðc; dÞ ⊕ Γþ þ
3 ðe; fÞ ⊕ Γ5 ðgÞ. Expanding around Fdd2
qualitative difference between the superconductivity and (b ¼ d ¼ g ¼ 0) and Ima2 (b ¼ a; d ¼ −c; f ¼ 0), no
structural mode cases. In particular, for both cases, the two couplings exist between the primary Higgs and
components of the eigenvector do not have equal ampli- Goldstone modes (that is, the ab; ad; bc; cd terms in the
tudes. In the superconductivity case, this is due to the force-constant matrix vanish at the Fdd2 and Ima2
difference in the density of states of the two bands. In the minima). This means that any coupling between these
structural case, it is because the uncoupled mode frequen- Higgs and Goldstone modes is beyond the harmonic
cies for Γ−4 and Γ−5 differ. approximation; however, higher-order coupling can occur
011024-6LEGGETT MODES ACCOMPANYING CRYSTALLOGRAPHIC PHASE … PHYS. REV. X 12, 011024 (2022)
in nonequilibrium situations [19,21]. At first glance, these Γ5¯
Γ5¯ Nb (Γ3¯ )
secondary modes shift the various primary mode frequen- 0.3
Γ5¯ Cd (Γ3¯ )
cies that would be determined from the smaller 4 × 4 block.
Mode amplitude (Å)
Γ4¯ Nb (Γ2¯ )
That is, one can in principle reduce this 7 × 7 matrix to an 0.24
Γ4¯ Nb (Γ3¯ )
effective 4 × 4 matrix by integrating out e, f, g in the
0.18
Landau free-energy equations, as these secondary order
parameters are slaved to the primary ones. 0.12
But closer inspection of the form of the primary-
secondary mode couplings reveals new opportunities to 0.06
study the Leggett modes. To see this, note that in the Cc
subspace discussed above, one now has the following 0
coupling terms at cubic order since the secondary order 100 120 140 160 180 200
parameters have even parity: Temperature (K)
FIG. 4. Decomposition of the Ima2 structural refinement of
1 2 2 2 2
F3 ¼ η1 pffiffiffi ða þ b Þe þ ða − b Þf Cd2 Nb2 O7 from Ref. [35] using ISODISTORT [24,36]. Shown is
3 the overall Γ−5 (T 2u ) amplitude previously plotted in Ref. [35]
along with various Γ−4 (T 1u ) and Γ−5 decompositions involving
1 2 2 2 2
þ η2 pffiffiffi ðc þ d Þe þ ðc − d Þf either Nb or Cd ion displacements from their cubic positions,
3 noting that for each ion, Γ−4 has two submodes Γ−3 (Eu ) and Γ−2
1 (A2u ), whereas Γ−5 has only one (Γ−3 ). Curves are proportional to
þ η3 ðac − bdÞe − pffiffiffi ðac þ bdÞf
3 jT c − Tjβ , where T c ¼ 196 K is the ferroelectric transition
temperature and β ¼ 0.27 is the order parameter exponent.
þ η4 ðabgÞ þ η5 ðcdgÞ þ η6 ½ðad − bcÞg: ð11Þ
For both Fdd2 and Ima2, the η3 term leads to both primary data or not. To address this, in Fig. 4 we plot the temper-
Higgs and Goldstone mode couplings. For Ima2, the η6 ature dependence of several relevant submode amplitudes
term also leads to both primary Higgs and Goldstone based on the Ima2 crystal structure refinements of
couplings. We note that Γþ þ
3 and Γ5 are Raman active modes
Malcherek et al. [35]. We find that the Cd and Nb
in the cubic phase. Therefore, below the ferroelectric displacements for both Γ−4 and Γ−5 symmetries scale with
transition, they can be used to drive the primary Leggett the overall Γ−5 order parameter amplitude that was pre-
modes via these two cubic terms, analogous to nonlinear viously shown in Ref. [35]. This indicates that this simpler
phononics experiments on perovskites [43] that have been Landau description is valid.
studied theoretically using similar Landau-like equations We now turn to the phonons. Since the ions involved
of motion [44–46]. Note that pump-probe experiments (Cd, Nb, O) have different masses, there is no one-to-one
have been instrumental in the study of Leggett modes in correspondence between the force-constant modes and the
superconductors [47]. phonons (unless the mode involved only one ion type). As a
In the above analysis, we did not include strain (which consequence, a given force-constant mode could involve
typically renormalizes the Landau coefficients) and gra- more than one phonon. However, it has been shown that
dient terms, which need to be included to address domain there is a strong correspondence between single phonons
walls. One would expect that the collective modes could be and the Higgs and Goldstone modes in YMnO3 [19].
significantly modified by domain walls if they are spatially Therefore, it is probable that such modes for Cd2 Nb2 O7 can
broad enough [48] and present at a high enough density. also be associated with specific phonons.
These effects would be interesting to study in future work. To gain further insight, we turn to experimental Raman
and infrared data. In the Ima2 phase, all phonons are in
principle Raman active. Raman data on Cd2 Nb2 O7 find
D. Submodes and phonons for Cd2 Nb2 O7
several low lying A1 and B2 modes below the ferroelectric
In the real crystal, there are multiple force-constant transition, three of which soften as the structural phase
matrix eigenmodes for each symmetry, typically referred transition is approached from below [49]. As the amplitude
to in the literature as “submodes.” The full matrix has a size modes have A1 symmetry and the phase modes have B2
of 66 × 66. Restricting the symmetry to Cc, the matrix symmetry, there should be a correlation between our
reduces to 33 × 33. With further restriction to Γ−4 and Γ−5 , it collective modes and the data. That is, it is possible that
reduces to 24 × 24. In the Landau treatment, each qi is a the two lowest lying A1 Raman modes correspond to the
particular sum of these submodes with the appropriate Higgs and antiphase Higgs modes. Presumably the lowest
group symmetry that is gotten by reducing this larger lying B2 mode is the Goldstone mode, with the antiphason
matrix to the smaller 4 × 4 matrix. Still, one can ask the corresponding to a higher energy B2 mode that has yet to be
question whether this simplification is supported by the studied.
011024-7QUINTIN N. MEIER et al. PHYS. REV. X 12, 011024 (2022)
The IR data for Cd2 Nb2 O7 are complicated given the considerations presented here should be valid there as well.
presence of seven optic modes of Γ−4 symmetry in the The existence of low lying modes would be aided by
cubic phase, which further split in the ferroelectric phase. having flat energy surfaces with low transition barriers as
Two of the lower lying IR modes have been interpreted as considered here. In that context, Cd2 Nb2 O7 consists of
being coupled (both above and below the transition) due corner sharing NbO6 octahedra in an open framework
to their temperature dependencies [50]. One of these, interpenetrated by CdO tetrahedra that are weakly coupled
referred to as a “central” mode and speculated to be due to to the octahedra, implying floppy low-energy modes.
hopping of Cd ions among equivalent locations displaced Going from a cubic phase to a lower symmetry ortho-
from their cubic positions, is not thought to be of Γ−4 rhombic or monoclinic phase is also useful in order to
symmetry given that seven other IR modes of this optimize the number of coupling terms in the Landau free
symmetry are already seen. Its coupling with a higher energy. This is particularly pervasive in pyrochlores and
energy second mode, thought to be a Nb displacement spinels. Cagelike structures as in skutterudites with their
mode of Γ−4 symmetry, drives this central mode to soften at associated floppy modes would also be a good place to
the ferroelectric transition [50]. Whether this central mode search. Much of the work concerning Goldstone and Higgs
is an A1 symmetry mode (as typical for an order-disorder modes has been done in the context of perovskites, which
transition), or instead a Γ−5 mode that becomes IR active also involve corner sharing octahedra, and a number of
due to coupling to the second Γ−4 mode, is worth inves- them exhibit improper ferroelectric transitions as refer-
tigating. If the latter, this would be consistent with the enced above. Similar considerations to ours would also
theory we offer above of two coupled modes of Γ−4 and Γ−5 apply to ferroelastic transitions. However, a large coupling
symmetry. Moreover, as discussed above, pump-probe to strain usually leads to large warping terms (see, e.g., in
studies of Cd2 Nb2 O7 would be instrumental in probing for the case of ferroelastic WO3 [58]). A locally flat energy
possible collective modes: Higgs, Goldstone, and their landscape can sometimes be recovered at Ising-type
Leggett analogs. domain walls (see, e.g., for LiNbO3 [59]). In the same
way, locally confined Leggett modes could be observed.
IV. SEARCHING FOR AND OBSERVING
LEGGETT MODES V. CONCLUSION
Although we considered the specific example of We demonstrate via a Landau analysis that structural
Cd2 Nb2 O7 , the above analysis should be applicable when- Leggett modes should exist in the context of displacive
ever more than one primary order parameter is involved in a transitions when more than one multidimensional group
phase transition. As such, Leggett modes should exist in representation is involved. These collective modes exist
the structural context, and could be identified from a DFT not only in the phase channel as in the original work of
analysis of the force-constant and dynamical (phonon) Leggett, but also in the amplitude channel, representing a
matrices in comparison to experimental data. We plan to new collective mode, the antiphase Higgs, that should be
report on this in a future paper that will provide a detailed observable in multiband superconductors as well. We
DFT study of Cd2 Nb2 O7 [51]. In general, analysis of studied in detail the specific case of the relaxor ferroelectric
Raman and IR data, including the symmetry of the modes pyrochlore Cd2 Nb2 O7 , which we believe is a promising
and their temperature dependence, should be helpful in material to search for such modes. We believe there should
elucidating the presence of Leggett modes. Specifics, be a large group of materials where such modes could
including identifying the out-of-phase behavior character- exist, and advocate in particular that pump-probe studies
istic of the Leggett modes, could be resolved by pump- would be the most illuminating way to identify and
probe studies of the phase of the oscillations as a function characterize these modes. The study of such modes should
of time. Moreover, the equations of motion associated with give new insight into their associated crystallographic
nonlinear phononics involve the same cubic and quartic phase transitions.
coupling terms invoked in this paper that provide for the
existence of the Leggett modes to begin with. Driving
ACKNOWLEDGMENTS
specific modes would then be instrumental in identifying
the various collective modes via their couplings to the Work at Argonne National Laboratory was supported by
driven mode. the Materials Sciences and Engineering Division, Basic
As for materials, obviously those phase transitions Energy Sciences, Office of Science, U.S. Department of
involving more than one primary group representation Energy. Q. N. M. was supported by the Swiss National
would be obvious targets for study, including those Science Foundation under Project No. P2EZP2_191872.
exhibiting improper and hybrid-improper ferroelectric Work at ETH Zurich was funded by the European Research
transitions [52–57]. Although the latter typically involve Council (ERC) under the European Union’s Horizon 2020
order parameters with nonzero wave vector, the Landau research and innovation program project HERO grant
coupling terms in the free energy are similar and so the (No. 810451). Work at Northwestern University was
011024-8LEGGETT MODES ACCOMPANYING CRYSTALLOGRAPHIC PHASE … PHYS. REV. X 12, 011024 (2022)
supported by the National Science Foundation (NSF) Grant Sn-Filled Skutterudite SnFe4 Sb12 , Phys. Rev. B 97, 024301
No. DMR-2011208. G. L. gratefully acknowledges support (2018).
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