Manifestion of Higgs and Goldstone modes in the hexagonal manganites

Structural phase transitions described by Mexican hat potentials should in principle exhibit aspects of Higgs and Goldstone physics. Here, we investigate the relationship between the phonons that soften at such structural phase transitions and the Higgs- and Goldstone-boson analogues associated with the crystallographic Mexican hat potential. We show that, with the exception of systems containing only one atom type, the usual Higgs and Goldstone modes are represented by a combination of several phonon modes, with the lowest energy phonons of the relevant symmetry having a substantial contribution. Using the hexagonal manganites as a model system, we calculate the temperature dependence of the frequencies of these phonons using a combination of density functional and Landau theories within the quasi-harmonic approximation, and predict that the Higgs-Goldstone coupling does not lead to a strong temperature dependence. We present Raman scattering data that support the predicted behavior, and show that it is consistent with the recently identified order-disorder nature of the transition.


I. INTRODUCTION
Phase transitions that break a symmetry spontaneously occur in a wide range of physical systems, from low-energy cold atoms, through magnetic, structural and superconducting transitions in condensed matter, to high-energy collisions at the Large Hadron Collider 2-10 . Perhaps the simplest and most-studied form of spontaneous symmetry breaking is that described by the 'φ 4 ' Lagrangian, which is used in the Landau-Ginzburg theory of phase transitions, as well as in the standard model of particle physics, where φ is a complex order parameter which is zero in the disordered phase at T > T C and acquires a nonzero value below T = T C . The energy density of such a Lagrangian has the so-called 'Mexican hat' potential with continuous U (1) symmetry (see Fig. 1) in which the ground-state value of the field, φ 0 , is degenerate in energy around the entire 360 • rim of the Mexican hat potential. This form was originally suggested by Landau to describe ferromagnets near the critical point [11][12][13] and has recently achieved notoriety following the discovery of the Higgs boson, whose formation it also describes. Perturbing the field φ around the ground state φ 0 gives two types of fluctuations -the Higgs and Goldstone modes -which correspond respectively to oscillations of the amplitude (towards and away from the peak of the hat) and phase (around the brim of the hat) of the broken continuous symmetry (Fig. 1).
Since the energy of the field is invariant with phase, the Goldstone mode is characteristically massless with zero frequency and a corresponding zero energy gap 14,15 . Many manifestations of the Goldstone mode are known in condensed-matter systems: For example a massless spin wave has been measured using neutron scattering in the prototypical Heisenberg ferromagnet EuS 16 , for which the Hamiltonian is invariant under the rotation of spins. Inelastic neutron scattering was also used to detect a gapless mode in a Bose Einstein condensate of spin-triplet states in TlCuCl 3 17 , consistent with theoretical predictions 18 . Polarized Raman scattering detected the development of a peak at zero frequency with divergent intensity at the structural phase transition in Cd 2 Re 2 O 7 pyrochlore, which has been associated with the Goldstone phonon 19 ; similar behavior has been predicted for Ruddeldsen-Popper-structure PbSr 2 Ti 2 O 7 20 . Finally, Goldstone modes also appear as sound waves at the normal-to-superfluid transition in 4 He 21 . In superconductors, the Goldstone mode gains mass by its interaction with an applied external field through the Anderson-Higgs-mechanism 22,23 , giving rise to the Meissner effect.

Higgs Goldstone
In contrast, the Higgs, or amplitude, mode is massive and harmonic 7 , with a finite energy excitation gap. It is harder to detect, because it can decay into Goldstone bosons 24 , although successful observations have been made in condensed matter systems for which the effective theory describing the system has a relativistic form. This is the case for the superconducting phase transition in 2H-NbSe 2 25 ; this provided the first experimental evidence of the Higgs mode 26 in its unusual Raman response, which was consistent with the occurrence of an amplitude mode of the CDW order parameter. Likewise in TlCuCl 3 the neutron spectroscopy measurements of the magnetic excitations revealed a Higgs mode that softened and vanished at the pressure-induced quantum phase transition from a sea of spin-singlet pairs to a longrange antiferromagnet 27 . Cold atoms in two-dimensional optical lattices have provided indirect measurement of the Higgs mode at the quantum phase transition between the superfluid and insulating phases, through observation of a finite-frequency response in the superfluid phase 28 , consistent with Monte Carlo simulations 29 and the scaling expected for a Higgs mode. Finally, the observation and manipulation of a Higgs mode has recently been demonstrated in a supersolid quantum gas 30 . A summary of experimental efforts to observe the Higgs mode in condensed matter systems can be found in Ref. 31.
Notably, no occurrences of the Higgs mode corresponding to structural phase transitions have been reported to date. Such an example would be convenient, since the order parameters in structural phase transitions are usually given by the positions of the atoms, which in turn can often be measured unambiguously and remain stable for long times. Indeed, a field theoretical treatment of both Higgs-and Goldstone phonons has recently been developed and would in principle be applicable to such a transition 32 .
Here we show, that for compounds containing multiple atomic species, an unambiguous association of specific single phonons with the Higgs and Goldstone modes can not in general be made, because the different atomic masses of the species cause the eigenvectors of the force constant and dynamical matrices to differ. Nevertheless, we show that phonon modes carrying substantial Higgs and Goldstone character can be identified, and demonstrate their existence in the hexagonal manganite family of improper ferroelectrics, which are unusual in that they have a structural phase transition whose energy landscape is described by a Mexican-hat-like potential 33,34 . By combining symmetry analysis, first-principles calculations, and phenomenological modeling, we analyse the potential and dynamical energy landscapes of two representative multiferroic hexagonal manganite materials, ErMnO 3 and InMnO 3 . We evaluate the signatures of Higgs-Goldstone coupling in the temperature dependence of the phonon frequencies and then use vibrational spectroscopy to verify the predicted behavior. Our work indicates that, while the hexagonal manganites should in principle provide a text-book manifestation of a crystallographic Higgs mode, in practice, subtleties associated with the order-disorder nature of the phase transition 1 lead to fundamental differences between the crystallographic Higgs excitations in the hexagonal manganites and those found in magnetic or cold-atom systems or particle physics.
A. Structural Higgs & Goldstone modes in multi-species crystalline materials

Identifying the structural Higgs and Goldstone modes
We begin by analyzing two aspects of structural phase transitions that are particularly problematic for the identification of any associated Higgs and Goldstone physics. The first is the difference between the force constant matrix eigenmodes, to which the Higgs and Goldstone modes belong, and the phonon eigenmodes, which are experimentally accessible. The second is the large number of degrees of freedom associated with atomic displacements in a solid, which leads in turn to a large number of often strongly coupled structural modes.
a. Force constant versus dynamical matrices. Since the Mexican hat discussed above is a potential energy surface, in the case of a crystallographic phase transition described by such a potential, the Higgs and Goldstone modes are the relevant eigenmodes of the force constant matrix, Here U is the internal energy and u i , u j are displacements of the i-th and j-th atoms from their positions in the zero-temperature structure. There is no obvious way to directly measure the eigenmodes of the force constant matrix, and therefore the phonon modes, which are readily accessible via vibrational spectroscopies, are often used as proxies. The phonon modes, however, are eigenmodes of the dynamical matrix, which is related to the force constant matrix by (see for example Ref. 35): where M i is the mass of the i-th atom. It is clear that the force constant and dynamical matrices have different eigenvalues and eigenvectors. As a result, the Higgs and Goldstone modes associated with a crystallographic phase transition do not correspond to single phonons, except in the special case that the system contains atoms of only one mass. b. Distinguishing the Higgs and Goldstone modes from the large number of structural eigenmodes. The second problem is the large number of degrees of freedom associated with the atoms in a solid. Expanding the total energy of a system of atoms around their ground-state positions in the zero-temperature structure, one obtains the total energy of the system, E, as the sum of its kinetic and potential energies: where i ∈ {1x, 1y, 1z, 2x, 2y, 2z, .., N x, N y, N z} can be a large number, leading to a large number of modes, even when periodic boundary conditions are used to constrain N to the number of atoms in the unit cell. Here is the harmonic force constant matrix at zero temperature and Λ ijk and Π ijkl are the anharmonic third-and fourth-order force constant matrices. At low temperatures, the amplitudes of the atomic displacements are small, the average atomic positions are unchanged from the zero temperature positions, and the phonon eigenmodes are obtained by diagonalizing the sum of the first two terms. In this limit, far from any structural phase transition, individual phonon modes can be clearly identified. As temperature is increased, however, the anharmonic couplings become relevant, the normal modes can no longer be separated into the zerotemperature phonons and the average positions of the atoms are changed. This behavior is well established as the origin of lattice expansion in conventional solids.

Temperature dependence of the anharmonic modes
In this section, we use Landau theory to derive the temperature dependence of the Higgs and Goldstone modes in a system undergoing a structural phase transition described by a Mexican hat potential. We exploit the fact that in systems close to a structural phase transition the change of atomic positions associated with the anharmonicity of a single soft-mode coordinate dominates the response.
In our analysis, we avoid calculation of the full partition function of equation (4), by renormalizing the anharmonicities into a harmonic approximation around the new atomic positions at each temperature. We define the displacements around the new minimum-energy positions at each temperature by u * i = u i − ∆u i (T ) where ∆u i (T ) is the change of coordinate of atom i at temperature T from its initial position at T = 0. This allows us to define which describes the energy cost of small atomic displacements away from the minimum-energy atomic coordinates at temperature T . In this renormalized harmonic approximation the total free energy, F (T ), is given by where F 0 (T ) is the free energy of the minimum energy structure (u * i = 0) at the temperature T . Next, we use the Landau theory of phase transitions to analyze the finite-temperature force-constant matrix and determine the behavior of the soft mode. Diagonalizing the force constant matrix, Φ ij (T ), we obtain a set of eigenvectors φ 1 , ...φ 3N with eigenvalues (α 1 , .., α 3N ). For the case of a phase transition described by a Mexican hat potential, two of these eigenvectors (α 1 and α 2 say) are particularly relevant, since below T C they become the Goldstone and Higgs modes. The Goldstone mode has zero frequency, and therefore α 1 = 0, for all temperatures below the phase transition, and the Higgs mode softens, so that α 2 → 0, at the phase transition. We assume that the temperature evolution of the system is fully captured by the evolution of these two modes. We write the free energy in the usual Landau form for a broken continuous U (1) symmetry 36 in terms of the two-component order Here and (α 3 , .., α 3N ) are assumed to be independent of temperature. This approximation implies that the temperature dependence of the energy landscape is determined entirely by the anharmonicity in these two soft modes with no anharmonic coupling to other modes, and at any temperature, Φ ij (T ) is diagonalized by the same basis set, with the anharmonicities confined in the subspace (φ 1 ,φ 2 ) of the normal modes describing the order parameter. The inverse generalized susceptibility , that is the corresponding term of the diagonalized matrix of Φ ij . Next we calculate this inverse susceptibility above and below T C . Above T C , where the expectation values of φ 1 and φ 2 are equivalently zero, Making use of our assumption that α 3 , .., α 3N are temperature independent, and the entire temperature dependence of the free energy is contained in α 1 and α 2 , the susceptibility matrix above T C is by To obtain the solutions below T C , we find the expectation value of the order parameter, φ , by minimizing the free energy, F , (Eqn. 6) with respect to φ 1 and φ 2 . This yields two solutions, the trivial vacuum solution with φ 1 = φ 2 = 0 for −a(T )/b > 0, and a nontrivial solution describing a degenerate circle of vacua , which is the Mexican hat potential. Because of the U (1) symmetry we can choose φ 1 = 0 and φ 2 = φ without loss of generality, and expand around the low-symmetry vacuum ground state to obtain the excitation modes. This gives the massless Goldstone mode, corresponding to distortions along the φ 1 coordinate, δφ 1 , with and the massive Higgs mode, corresponding to distortions along the φ 2 coordinate, δφ 2 , with The modes fullfil The generalized susceptibilities of the Higgs and Goldstone modes are: and the whole susceptibility matrix below T C is where α 3 , .., α 3N are still temperature independent. We expect, therefore, a strong temperature dependence of the eigenvalue of the force-constant matrix corresponding to the Higgs mode on approaching the structural phase transition, with the remaining eigenvalues being largely temperature independent.
Finally for this section, we analyze how this temperature dependence of the Higgs mode manifests in the phonon spectrum. In the case when all of the atoms in a system have the same mass, m, the eigenvectors of the force constants and dynamical matrices are the same, and the phonon frequencies of the Higgs and Goldstone modes, ω H and ω G , are given by: In a general multi-component system, however, each element of the force constant matrix must be divided by the product of the square roots of the relevant masses before diagonalization to extract the phonons, and in general the static eigenvectors of the force constant matrix do not correspond to specific single phonons. For the special case of the zero-frequency Goldstone mode, the atomic masses are not relevant, and as a result there is a zero frequency phonon for each zero frequency force constant mode, and the zero frequency eigenvectors of the forceconstant matrix are identical to the atomic displacements of the corresponding zero-frequency phonon mode. The static Higgs mode, however, is a linear combination of all dynamical phonon modes with the same irreducible representation, and the entire sub-space of phonon modes with the same symmetry as the Higgs mode should exhibit the strong temperature dependence that we derived above.

B. Structural phase transition in the hexagonal manganites
The multiferroic hexagonal manganites undergo a spontaneous symmetry-breaking structural phase transition at around 1200 K (around 500 K for InMnO 3 ) 37 between a high-temperature centrosymmetric P 6 3 /mmc phase and a ferroelectric P 6 3 cm structure. The primary order parameter is defined by trimerizing tilts of the MnO 5 trigonal bipyramids that form planes separating hexagonal planes of R ions, and is two-dimensional, with its amplitude set by the magnitude of the tilt, and its phase set by the tilt angle. The crystal structure and the distortion are illustrated in Fig 2(a). A combination of Landau theory and first-principles calculations [38][39][40] have shown that for small tilt amplitudes the energy is independent of the polyhedral tilt angle, and so near the phase transition the energy landscape can be described by a continuous Mexican hat potential with U (1) symmetry (See Fig. 2(b)). Thus, unusually for a crystallographic transition, the structural phase transition in the hexagonal manganites is described by a continuous primary two-dimensional order parameter (φ 1 , φ 2 ) leading to a energy landscape similar to that described in the previous section and as such might be expected to display Higgs and Goldstone modes. We note that the discreteness of the lattice manifests at larger amplitudes of the tilt mode through coupling to a secondary ferroelectric order parameter, P , which leads to a net shift of the rare earth ions relative to the manganese oxygen layers along the vertical axis 38 . This mode has shown to be irrelevant in the region of the phase transition 41 , a concept referred to as dangerous irrelevance 42 . The recent demonstration that the hexagonal manganites disorder continuously on all length scales close to T c 1 reinforces the continuous U (1) behavior in the region of the phase transition.
The coupling between P and the primary order parameter yields a low-temperature ground state with six minima around the brim of the hat reflecting the hexagonal symmetry so that the transition is described by an extended Landau free energy, that is conventionally written in the form 39 : For consistency with the hexagonal manganites literature, we use polar coordinates for the order parameter, with the amplitude Q = φ 2 1 + φ 2 2 and the phase θ = arctan(φ 1 /φ 2 ). The energy landscape of the primary order parameter corresponds to an almost perfect Mexican hat, while the secondary order parameter induces the minima in the brim 33 . We note that chemical tuning can be used to modify the coupling to the polar mode, and thus the height of the barriers around the brim of the hat, and in particular in the case of InMnO 3 these are close to zero and the brim is smoother 43 . The energy landscape for a minimized secondary order parameter is shown in figure 2 (b) for the case of ErMnO 3 , with perturbations of the phase, δθ, and the amplitude, δq, of the order parameter indicated. Perturbations of the amplitude conserve the space group symmetry, thus they belong to the irreducible representation A1. Perturbation of the order parameter angle change the space group symmetry from the ferroelectric P 6 3 cm to P 3c1, which corresponds to the irreducible representation B1 1 .

A. Sample preparation
InMnO 3 samples were prepared from a stoichiometric mixture of In 2 O 3 (99.9%) and Mn 2 O 3 placed in Au capsules and treated at 6 GPa in a belt-type high-pressure apparatus at 1373 K for 30 min (heating rate 110 K/min). After heat treatment, the samples were quenched to room temperature, and the pressure was slowly released. The resultant samples were black dense pellets. The energy balance between the polar P 6 3 cm and non-polar P3c1 structures is known to be sensitive to the details of the defect chemistry [43][44][45] and samples with the two phases were obtained by appropriate annealing treatment. ErMnO 3 samples were prepared using the PbO-PbF 2 flux method. The starting composition of 6.7g Er 2 0 3 , 7.7g MnCO 3 , 1g B 2 O 3 , 7g PbO, 56g PbF 2 and 3.3g PbO 2 was heated in a 50ml Pt crucible for 15 hours at ∼1280 • C then cooled at 1 • C per hour to produce thin platelets around 20mm 246,47

B. Raman spectroscopy
We performed Raman spectroscopy using a homemade spectrometer equipped with a liquid nitrogen cooled CCD camera and an Ar laser for the excitation with a wavelength of 514.5 nm. The Raman spectra were collected at the Stokes side of the elastic peak in the range from 50 to 850 cm −1 . Samples were mounted in a compact flow cryostat operating between 4 K and 300 K. The power of the laser was low enough to limit local heating of the sample. Of the six different irreducible representations that classify the phonon modes in the hexagonal manganites, only A1 and E2 are Raman active. We use two configurations of the polarization of the incoming and scattered photons: For the z(xx)z (polarization of the scattered photons is parallel to that of the incoming ones), both A1 and E2 modes are allowed by the Raman selection rules. For the z(xy)z configuration (scattered photons are polarized perpendicular to the incoming ones), only E2 modes can be observed. The relative angle of the two polarizers was calibrated using the selection rules for the 514 cm −1 phonon line of silicon.

C. Density functional calculations
For our first-principles calculations we used density functional theory as implemented in the abinit code 48,49 . We treated the exchange-correlation functional within the LDA+U approximation, with U and J values of 8 eV and 0.88 eV 50 , and the core electrons using the projector augmented wave (PAW) method 51 from the JTH pseudopotential table provided by the abinit pseudodojo 52 . We used a cutoff energy of 30 hartree and gammacentered k-point meshes of 8 × 8 × 2 for the 10-atom unit cells, and 6 × 6 × 2 for the 30-atom unit cells. Note that with these parameters InMnO 3 is ferroelectric; small adjustments in the parameters can stabilize the antiferroelectric state 53 . We obtained force constant matrices using the finite-displacement method provided in the phonopy package 54 . We calculated the Landau parameters for ErMnO 3 by displacing the atoms from their positions in the high-symmetry structure along the force constant eigenvectors. For our calculations within the quasiharmonic approximation, we calculated the phonons in 30 atom unit cells and computed the internal energies and phonon free energies as a function of in-plane and out-of-plane lattice parameters. We then interpolated between the calculated values to extract the minimum energy lattice parameters at each temperature.

III. THEORETICAL RESULTS
A. Density functional calculation of zero-kelvin energetics and lattice dynamics.
We begin by comparing the zero-temperature energetics of our two representative hexagonal manganites, ErMnO 3 and InMnO 3 . As stated above, the different chemistries of the two materials lead to quantitative differences in their Mexican-hat potentials, with a hat height of ∼ 500 (170) meV and a barrier in the brim of ∼ 200 (50) meV for ErMnO 3 (InMnO 3 ). The calculated Landau parameters from which these values were obtained are given in Table I  Next we calculate the phonon mode frequencies and eigenvectors for the two materials using density functional theory. Our calculated frequencies and the symmetries of each mode are listed in Table IV in Appendix B. We see that the lowest frequency A1 mode, which we expect to have the largest Higgs character, has almost the same frequency (∼130 cm −1 in both materials, reflecting the similar curvatures of their Mexican hat poten-tials in the brim of the hat in the direction towards and away from the peak. The lowest frequency B1 modes, which we expect to have the strongest Goldstone character, are strikingly different however, with the frequency in ErMnO 3 considerably higher than that in InMnO 3 . This is consistent with the larger barriers around the brim of the hat in the ErMnO 3 case. Note that even in the case of InMnO 3 , where the brim of the hat is very smooth, the frequency (∼ 65 cm −1 ) is still quite far from zero.
Finally, in anticipation of differences in the Higgs-Goldstone coupling caused by the different Mexican hats, we calculate the phonon-phonon couplings between the low frequency A1 and B1 modes in the two materials. Our results are presented in Table A of appendix A, with the form of the coupling given by For the lowest-lying A1 and B2 modes, we observe that the lowest-order coupling term, f , is lower in ErMnO 3 than in InMnO 3 , as expected from the bigger ripples in the Mexican hat in the ErMnO 3 compound.
B. Landau theory calculation of the temperature dependence of the Higgs-and Goldstone-like phonon modes.
Next we use Landau theory based on our calculated density functional theory parameters to calculate the explicit temperature dependence of the phonon frequencies.
We begin by extending the Landau theory framework that we developed in section I A 2 to include, in addition to the two-component primary order parameter, the secondary order parameter that is relevant in the hexagonal manganites. We then calculate explicitly the temperature dependence of the phonons in both ErMnO 3 and InMnO 3 using the coefficients of Table I. We proceed by expanding the free energy of Eqn. (??) in terms of small perturbations of the primary (treating each component separately) and secondary order parameters around the minimum energy positions, θ = 0 + δθ, Q =Q + δq, P =P + δp. HereQ is the expectation value of Q, given by the solution of the equation ∂F ∂Q θ=0,P =P =0. Correspondingly,P is the expectation value of P , which we obtain from the solution of ∂F ∂P Q=Q,θ=0 = 0, yielding We obtain the following effective susceptibilities for the perturbations of each component: We then calculate the phonon frequencies at each temperature by replacing the calculated zero-temperature susceptibilities by the response functions (20)- (22) in the force constant matrix, and assuming the usual linear evolution, a(T ) = a 0 (T −T c )/T c , for the temperature dependence of the a parameter of the soft mode. Our calculated phonon frequencies as a function of temperature are shown in Fig. 3 (a) for InMnO 3 and (b) for ErMnO 3 . Modes of A1 symmetry (and therefore Higgs character) are indicated in blue, B1 symmetry (Goldstone) modes in red, and polar modes in green. The frequencies of all the other phonons are temperature independent by construction in our approximation.
We begin by comparing, in Table II, the frequencies obtained from our Landau theory approach in the zerokelvin limit with those calculated for the fully relaxed cell using density functional theory. We find that the Landau theory frequencies underestimate the DFT values by around 30%. This is mostly a result of our neglecting the weak coupling of the order parameter modes to two additional modes, as described in Ref. 38. This additional coupling would harden the relevant phonon modes in the Landau description.
Next we discuss the temperature dependence of the modes, beginning with the softest mode. First, we note that this mode, which softens on approaching T C from above, is doubly degenerate above T C due to the equivalence of the order parameter directions in the highsymmetry phase. The degeneracy is lifted and the mode splits into two below the phase transition, an amplitude mode with A1 symmetry, which can be regarded as the primary Higgs mode of the structural transition (shown in blue) and a phase mode with B1 symmetry which represents the primary Goldstone mode (shown in red). The Goldstone mode in InMnO 3 retains a lower frequency down to zero kelvin than that in ErMnO 3 , reflecting the smaller barriers in the brim of the hat in the InMnO 3 case. The Higgs modes have a similar temperature dependence and zero-kelvin frequency in both compounds.
We note, as expected, the occurrence of additional Higgslike and Goldstone-like modes at higher frequencies, indicated with dashed lines in Fig. 3. These modes have the same symmetry as the soft modes, and are a consequence of the mixing of eigenmodes caused by the transformation from the force-constant to dynamical matrices. This mixing is stronger in ErMnO 3 than in InMnO 3 , because of the larger mass of Er, resulting in a stronger temperature dependence of the phonons in ErMnO 3 . The temperature evolutions of the phonons corresponding to the polar modes are plotted in green. These are independent of temperature above T C , but we find that their frequencies increase below the phase transition, as the increase in magnitude of the primary order parameter stabilizes the polar mode. We see that the frequency of the polar mode increases more in ErMnO 3 than InMnO 3 , consistent with the larger coupling g in the Q 3 P cos 3θ term of the Landau free energy for ErMnO 3 .

C. Effect of change in lattice parameters on the phonon mode frequencies
Finally for this section, we calculate how the change in lattice parameters with temperature affects the phonon frequencies in ErMnO 3 and InMnO 3 , with the goal of isolating any mode softening due to thermal expansion from the mode softening due to approaching the phase transition discussed above.
The hexagonal manganites are known experimentally to have an unusual lattice response to temperature, with the in-plane lattice parameter a increasing with temperature as expected, but the out-of-plane c lattice parameter decreasing with increasing temperature 56 .
We begin by demonstrating that this unusual evolution can be captured within the quasi-harmonic approximation, in which the total free energy as a function of the lattice parameters, F tot (a, c) is obtained from the sum of the internal energy U (a, c) plus the phonon free energy F phonons : where a and c are the in-plane and out-of-plane lattice parameters. U (a, c) is obtained by relaxing all internal degrees of freedom for the P 6 3 cm structure for the set of given lattice parameters, and the phonon free energy is calculated using the partition function for harmonic phonons: Using this approach, we calculate the temperature dependence of the lattice parameters, which we present in Figure 4 for ErMnO 3 . The excellent agreement with experiment 1 suggests that the quasi-harmonic population of phonons with increasing temperature is the dominant contribution to the thermal evolution of the lattice parameters. We then approximate the temperature dependence of the phonon frequencies, by calculating the eigenmodes of the dynamical matrix at the a, c lattice parameters for the corresponding temperature. We deliberately omit anharmonic interactions and phonon populations in this step, in order to isolate specifically the effect of the change in lattice parameters. We show our results for the Raman-active A1 and E2 phonons in Figure 4. We find that in this limit, most modes, in particular the A1 and B1 (not shown) phonons relevant to the Higgs-Goldstone coupling are largely temperature independent. Therefore we can exclude that any measured temperature dependence of the Higgs and Goldstone modes is a result of the change in lattice parameters with temperature.

A. Raman spectroscopy
In Fig. 5 the Raman spectra are displayed of (a) ErMnO 3 with the P 6 3 cm symmetry, of (b) InMnO 3 in the non-ferroelectric P3c1 state, and of (c) the ferroelectric P 63cm variant of InMnO 3 at 10 K for the parallel (red) and perpendicular (black) polarizations of incoming and scattered photons. Our ErMnO 3 data ( Fig. 5(a)) are in excellent agreement with previously published results 57 , showing all the previously reported A1 and E2 Raman active modes with the expected relative intensities and positions. The extinction of the A1 modes in the perpendicular configuration confirms the selection rules for the P 6 3 cm space group, all expected Raman active modes are present and they are narrow, thus confirming the high quality of the ErMnO 3 single crystal used in the present study. For both the P 6 3 cm and P 3c1 InMnO 3 crystals we observe in Fig. 5(b) and (c) the extinction of the mode at 680cm −1 and of the shoulder at 280cm −1 for the perpendicular polarizer configuration, which indicates that these modes belong to the A1 representation. The peaks at ∼ 140, 280 and 330 cm −1 persist for perpendicular polarization and therefore have E2 symmetry. The small crystal size leads to broad peaks and difficulty in umambiguously assigning the remaining peak frequencies, although the peaks at around 450 and 600 cm −1 are likely of A1 symmetry. a) Raman spectra collected at 10K on single crystal of (a) ErMnO3, (b) InMnO3 (P3c1) and (c) InMnO3 (P63cm). The (red and black lines) show the intensity of Raman scattering in the parallel and perpendicular configurations respectively. The polarization selection rule for the ErMnO3 sample (P 63cm symmetry) is clearly manifested.
In Fig. 6(a-c) we show the detailed temperature dependence of the Raman spectra of all three crystals. For sake of clarity the curves have been shifted vertically proportional to their temperatures. By fitting the curves with Lorentzian functions, we extracted the temperature dependence of the phonon frequencies, shown in Fig. 6(d-f) for (d) the ErMnO 3 sample, (e) the P3c1 InMnO 3 sample and (f) the P 6 3 cm InMnO 3 sample.
We begin by analyzing the ErMnO 3 spectrum, which shows all the A1 and E2 Raman active modes reported previously in the literature 57 with the expected relative intensities and positions. Moreover, we observe a new small peak below 80cm −1 that was not resolved in the 10K spectrum and increases in intensity on increasing the temperature, likely due to the anharmonicity of the potential energy surface. We also observe generally a general softening of all the modes as the temperature is increased. The frequency of the lowest-frequency A1 mode, which has the strongest Higgs character, reduces by ∼10 cm −1 between 10 and 300 K, with the higher energy A1 modes reducing in frequency by a similar amount. Since the ferroelectric phase transition in ErMnO 3 occurs at ∼1200K, the 300K limit of our experiment corresponds to T −T C T C = −0.75, which we see from Figure 3 (b) corresponds to a predicted Landau theory drop in frequency of around 10 cm −1 ,consistent with the experiment. The high T C of ErMnO 3 means that definitive experimental confirmation of Higgs behavior in ErMnO 3 would require measurement of the phonon frequencies to higher temperature than is available in our setup. The E1 mode at 250 cm −1 shows a particularly strong broadening and redshift; we suggest that this corresponds to a shear mode which we find in our quasi-harmomic calculations to be particularly sensitive to the change in lattice parameters.
Next we analyze the InMnO 3 spectra. Our first observation is that, despite their different ground-state crystal structures, the Raman spectra of the two InMnO 3 crystals are almost identical, confirming the similarity in the shapes of their Mexican hat potentials. In both InMnO 3 cases, the main Higgs excitation associated with the lowest frequency mode is lower in frequency and softens more rapidly with increasing temperature than in ErMnO 3 , consistent with the lower Curie temperature of ∼500 K. Once again we find a good agreement with the Landau theory prediction, with the calculated drop in frequency between zero and 300 K (corresponding to T −T C T C = −0.4) of around 30 cm −1 compared with the measured value of∼ 15 cm −1 . Interestingly, for InMnO 3 the E2 modes show a much weaker temperature dependence than in ErMnO 3 , with the mode at 135 cm −1 largely temperature independent and the mode at 225 cm −1 even hardening upon increasing the temperature. This is likely the result of the In-O covalency, which is known to lead to an anomalous c lattice parameter for InMnO 3 compared to other members of the hexagonal manganite series 53 , also causing markedly different changes in lattice parameters with thermal expansion 58 .

V. SUMMARY
In summary, we have analyzed the role of phonons as Higgs and Goldstone modes at the structural phase transitions in crystalline materials, especially focusing on the case of the hexagonal manganites. We showed that, in materials containing atoms of more than one mass, the static Higgs and Goldstone modes only map uniquely onto single phonon modes at T C , where both the Higgs and Goldstone frequencies are zero. Below T C , the different masses of the ions cause a softening of several phonon modes with the same symmetry as the static soft-mode distortion. Nevertheless, in both ErMnO 3 and InMnO 3 , our Landau theory analysis identified one primary A1 phonon corresponding to the Higgs mode, and one main B1 phonon corresponding to the Goldstone-like mode. Using Raman spectroscopy, we showed that the lowest A1 modes in both ErMnO 3 and InMnO 3 indeed have a red shift in the frequency on warming. For InMnO 3 , in which the temperature range measured is substantial with respect to the Curie temperature, the magnitude of the shift is similar to that predicted by the Landau theory, suggesting that the phase transition is well described within a standard displacive picture. A defini- tive comparison for ErMnO 3 , which has a much higher T C , will require Raman measurements to higher temperature. To motivate such measurements, we anticipate that ErMnO 3 might show intriguing deviations from the behavior that we calculated within Landau theory, since it will likely display similar strong order-disorder behavior to that recently identified in the related YMnO 3 1 . In a order-disorder transition, the softening of the phonon branches is limited, as observed in inelastic neutron scattering measurements for YMnO 3 59-61 and should be replaced by the emergence of a central peak (which has not yet been identified).