Physics of Phonon-Polaritons in Amorphous Materials
Luigi Casella, Matteo Baggioli, Tatsuya Mori, Alessio Zaccone
IIFT-UAM/CSIC-20-55
Physics of Phonon-Polaritons in Amorphous Materials
Luigi Casella, ∗ Matteo Baggioli, † Tatsuya Mori, ‡ and Alessio Zaccone § Department of Physics "A. Pontremoli", University of Milan, via Celoria 16, 20133 Milan, Italy. Instituto de Fisica Teorica UAM/CSIC, c/Nicolas Cabrera 13-15,Universidad Autonoma de Madrid, Cantoblanco, 28049 Madrid, Spain. Division of Materials Science, University of Tsukuba,1-1-1 Tennodai, Tsukuba, Ibaraki 305- 8573, Japan Department of Physics "A. Pontremoli", University of Milan, via Celoria 16, 20133 Milan, Italy.Department of Chemical Engineering and Biotechnology,University of Cambridge, Philippa Fawcett Drive, CB30AS Cambridge, U.K.Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, CB30HE Cambridge, U.K.
The nature of bosonic excitations in disordered materials has remained elusive due to the difficultiesin defining key concepts such as quasi-particles in the presence of disorder. We report on theexperimental observation of phonon-polaritons in glasses, including a boson peak (BP), i.e. excessof THz modes over the Debye law. A theoretical framework based on the concept of diffusons isdeveloped to model the broadening linewidth of the polariton due to disorder-induced scattering.It is shown that the scaling of the BP frequency with the diffusion constant of the linewidthstrongly correlates with that of the Ioffe-Regel (IR) crossover frequency of the polariton. This resultdemonstrates the universality of the BP in the low-energy spectra of collective bosonic-like excitationsin glasses, well beyond the traditional case of acoustic phonons, and establishes the IR crossover asthe fundamental physical mechanism behind the BP.
Introduction . The low-energy vibrational spectra ofsolids provide direct insights into the complex many-bodyatomic dynamics of materials [1]. Understanding the vi-brational spectra is crucial for our understanding and tech-nological design of the optical, thermal and mechanicalproperties of solids. Substantial experimental and theoret-ical efforts have focused on the case of phonons in amor-phous materials, where phonons are well-defined quasi-particles only in the limit of long wavelengths. On shorterlength-scales, disorder dominates the vibrational excita-tions and gives rise to deviations from Debye’s quadraticlaw in the vibrational density of states (VDOS), result-ing in the boson peak in the Debye-normalized VDOSdetected originally in Raman scattering spectra [2]. Theorigin of the boson peak in phonon spectra remains con-troversial. A line of research has traditionally supportedthe identification of the BP with shifted and smeared vanHove (VH) singularities [3, 4]. However, recent studieshave pointed out that the boson peak may be largelyindependent, and in fact even unaffected by the loweredVH singularity. This is indicated by the co-existence ofthe BP with the lowest (transverse) VH singularity in thespectra of simple models systems [5–7].Another line of research points at the close link betweenBP and the Ioffe-Regel (IR) crossover between ballisticphonons and quasi-localized excitations [8–10], as theorigin of the BP. Among the theoretical frameworks, themost popular is the heterogeneous elasticity theory ofSchirmacher, Ruocco and co-workers [11–13] based onthe assumption of spatial correlations in the shear elasticmodulus, in agreement with simulations [13].Although growing consensus is emerging about the cru-cial role of randomness in driving the crossover fromballistic to quasi-localized excitations leading to a BP,
Figure 1. The destruction of the quasiparticles induced bydisorder-scattering. The dynamics of the system becomes to-tally incoherent and collective and its low frequency dynamicsis well described by hydrodynamics. a picture supported by random matrix theory [14–16],the physical origin of the BP remain controversial. Fur-thermore it remains to be established whether the BPis a distinctive feature of the phonon spectra or if it isa more general phenomenon common to all bosonic-likeexcitations in amorphous solids (e.g. excitons, plasmons,other polaritons).Here we provide an answer to these fundamentalquestions by reporting on the experimental observationof the BP in infrared absorption spectra of phonon-polaritons in a model glass, i.e. soda-lime silicates. Atheory of phonon-polaritons in amorphous materialsis presented, which clarifies the origin of the BP fromthe Ioffe-Regel (IR) crossover between the ballisticquasi-particle (coherent) propagation to a quasi-localizedregime dominated by disorder-induced scattering, wherethe quasi-particle loses its coherence and undergoes a r X i v : . [ c ond - m a t . d i s - nn ] A p r diffusive-like dynamics ( diffusons ) [7, 10, 17] (see Fig.1).These results show the that the BP is a truly universalfeature for all bosonic excitations in amorphous mate-rials, not just phonons. Also, the theoretical analysisshows that the BP is a genuine result of the Ioffe-Regelcrossover caused by disorder since the polariton spectraare not affected by VH singularities. Finally, the theoret-ical analysis shows that the BP does not originate fromthe flattening of the polariton dispersion relations. Theoretical model – We start by modelling the coupleddynamics of the optical phonon modes and the EM field.The first and fundamental step is the dynamical equationfor the relative atomic displacement (cid:126)u , ¨ (cid:126)u = − ω (cid:126)u + f ( ˙ (cid:126)u ) + b (cid:126)E. (1)The relative displacement field characterizes the relativemotion of the partially-charged particles and applies tooptical (and not acoustic) vibrational modes. The firstterm in the r.h.s. of Eq.(1) defines the characteristic fre-quency of the harmonic oscillator, which originates fromthe linear restoring force acting on the atoms. The secondterm is an effective damping contribution, where f is somefunction. The last term is a direct dipole coupling to theexternal electric field (cid:126)E due to the partial charge carriedby the atoms.In absence of damping, f ( ˙ (cid:126)u ) = 0 , this problem was con-sidered in [1, 18] where Eq.(1) was solved together withan equation for the polarization that contains the effectsof the relative displacement of the atoms, and with theMaxwell equations for the EM field. Notice that dampingis introduced in Eq.(1) only in the mechanical part of theequations but not in the EM sector. Upon identifying f ( ˙ (cid:126)u ) = Γ( k ) ˙ (cid:126)u , we obtain a quartic equation for the modes ω ( k ) which reads: ω ε ∞ + i ω Γ ( k ) ε ∞ − ω (cid:0) ω ε + k c (cid:1) − i Γ ( k ) k c ω + ω k c = 0 . (2)In the above equation, c is the speed of light, ω thecharacteristic frequency (the energy gap of the opticalmechanical mode), ε , ε ∞ the dielectric constants at zeroand infinite frequency, respectively. Finally, Γ( k ) definesthe linewidth, and hence lifetime τ , of the optical mode, τ − ∼ Γ . The details of the derivation of Eq.(2) areprovided in [19].Equation (2) can be viewed in two different ways.First, one can assume the momentum k ∈ R to be real,and the frequency to be complex. In this framework, themodes ω ( k ) = Re ω ( k ) + i Im ω ( k ) are usually referredto as quasinormal modes and the imaginary part of thefrequency determines their exponential decay in time ∼ e − Im ω ( k ) t . Alternatively, one could take the frequencyas real and the momentum to be complex. In this caseEq.(2) can be solved for k ( ω ) and the imaginary part of the momentum determines the exponential decay inspace ∼ e − Im k ( ω ) x – i.e. the penetration length. Thetwo scenarios are interchangeable. In the rest of themanuscript we will use the abbreviations Re k ≡ k (cid:48) and Im k ≡ k (cid:48)(cid:48) .By using the linear relation between the polarizationvector (cid:126)P and the electric field (cid:126)E , we can derive the di-electric function , ε ( ω, k ) = ε ∞ − ω ( ε − ε ∞ ) ω − ω + i ω Γ ( k ) (3). Furthermore, by studying the spatial exponential at-tenuation (Lambert-Beer) of the wave intensity I , I/I = e − α ( ω ) x = e − k (cid:48)(cid:48) ( ω ) x (4)we can define the absorbance coefficient α ( ω ) , which is theinverse of the penetration length [20]. Given a collectionof scattering centers, the mean free path is given by (cid:96) = ( σn ) − , where σ is the scattering cross-section and n the number density of scatterers. It can be shownthat dI/dx = − Inσ = − I/(cid:96) [20], leading to exponentialattenuation I ∼ exp( − x/(cid:96) ) . Upon comparing this withthe above equation, we thus obtain (cid:96) − = 2 k (cid:48)(cid:48) ( ω ) (5)a relation that will be useful also later on.An expression for the absorbance coefficient can bederived using the complex dielectric function for EM radi-ation in continuous media ε ( ω ) = c k /ω [21]. Solvingthis expression for k and taking the imaginary part (seeEq. (4)), we find: α ( ω ) = ωc (cid:112) | ε ( ω ) | − Re ε ( ω ) ) . (6)At this point, it is crucial to specify the nature of thelinewidth Γ( k ) , which is neglected in standard treatments[18, 22]. The linewidth encodes the effects of the disorderon the propagation of the polariton. The basic idea isthat disorder can be represented as a large number of"defects", each acting on the polariton quasi-particle as a(elastic [11]) scattering center. On scales larger than thedefects average separation, the result of a great numberof scattering events is the diffusion of momentum throughthe system. This effective description is based on the ideaof diffusons [17, 23], a concept which proved useful inexplaining the anomalies in thermal transport observedexperimentally in glasses [24]. A diffusive linewidth forphonons can indeed explain the ubiquitous appearance ofa boson peak in the vibrational density of states (VDOS)of glasses [7, 25] and even the presence of a linear in T term in the specific heat at low T [26]. Moreover, thediffusive nature of the linewidth is supported by RandomMatrix Theory [16, 27].Following [7, 9, 27], we take the linewidth to be ofdiffusive form Γ( k ) = D k (7)which follows from an effective hydrodynamic treatment[28] for quasi-particle excitations or simply from diffusionof momentum in the governing dynamic equation for thedisplacement field [7]. This expression is supported byexperiments and simulations [9, 10, 29–31] and is validonly at relatively large k , while it is expected to breakdown at larger momenta where hydrodynamics is nolonger a good approximation.In Fig.10, in the Supplemental material [19], we showthe Debye-normalized absorbance coefficient obtainedfrom this model for a wide range of values of the diffusioncoefficient D . The absorbance coefficient is directly pro-portional, up to a linear growing function of the frequencydenoted as C ( ω ) [32], to the VDOS. As a consequence,an excess in α ( ω ) /ω corresponds to a boson peak inthe normalized VDOS g ( ω ) /ω . We observe that the BPmoves to lower energies by increasing the diffusion con-stant D and it becomes sharper. In the inset, we show thedispersion relation of the corresponding phonon-polaritonmodes obtained from Eq.(2) and we compare it with theBP frequency ω BP . This dynamics and the underlyingphysics mechanism behind will be discussed in detail later. Comparison with experimental data – The linewidthceases to display a hydrodynamic diffusive behaviour asin Eq.(7) at large momenta approaching the molecularsize. We observe that the high-frequency part ofthe experimental spectra is well-fitted by a constantdamping coefficient: Γ( k ) = γ , which corresponds to aLangevin friction term in Eq.(2), as expected in localmolecular-level dynamics in glassy environment. Indeed,on small length-scales, we cannot coarse grain the effectsof disorder into a hydrodynamic description but we haveto consider the high-frequency microscopic dynamics [14]producing a momentum-independent relaxation time τ − ∼ γ .In order to have a good description of the experimentaldata across the entire range of momenta, we will considera linewidth which interpolates from the diffusive form (7)at low k to the Langevin constant damping γ at large k .More specifically, we use an interpolating model of theform: Γ( k ) = γ D k γ + D k (8)which retrieves the two limits. We test our theoreti-cal model using experimental measures of the infrared-absorbance spectra on a soda-lime glass sample using twodifferent THz time-domain spectrometers that can cover a wide frequency range between 0.3 – 5 THz (see Supple-mental Material [19]). The excellent agreement betweenthe theoretical predictions and the experimental data isshown in Fig.2. The diffusons behaviour at low k is keyto obtain a good agreement at low frequencies and wechecked that it cannot be described with simple dampedharmonic oscillator (DHO) model as shown in [19]. Itis also to be noted that the decay at large frequencies isdominated by the constant damping term γ . oooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooTheoryo Experiments ω [ THz ] α ( ω ) / ω [ c m - T H z - ] Figure 2. The comparison between the theoretical model basedon the diffusive linewidth function Eq.(8) (black line) and theexperimental data for a soda lime glass (orange cirlces). Thefit gives: D = 13 . THz − /c , γ = 69 . THz − , (cid:15) = 70 . , (cid:15) ∞ = 61 . , ω = 29 . THz.
The Origin of the Boson Peak – Our effective theo-retical model gives an accurate qualitative descriptionof the experimental data and it is able to reproduce theboson peak. We should now address the question of thefundamental physical origin of the BP in the polaritonspectra. Let us recall that the BP frequency is defined as ω BP : ddω α ( ω ) ω (cid:12)(cid:12)(cid:12) ω BP = 0 (9)and it corresponds to the maximum in the Debye-normalized absorbance spectra.A possible explanation for the BP could come from theflattening of the phonon-polariton band: ω flat : dωdk (cid:12)(cid:12)(cid:12) ω flat = 0 (10)which, similarly to the van Hove singularities in orderedcrystals, would produce a peak in the VDOS since g ( ω ) ∼ ( dω/dk ) − [33]. As one can readily verify inFig.8, the position of the BP does not correspond to theflattening of the lowest branch. Hence, the flattening ofthe polaritonic dispersion relations cannot satisfactorilyexplain the occurrence of the BP.From a different perspective, it is well-known that wavesin amorphous and disordered systems stop to propagateballistically at a certain frequency known as the Ioffe-Regel frequency ω IR [34]. Moreover, the correlation be-tween the Ioffe-Regel frequency and the BP frequency hasbeen observed and discussed in recent works [8–10].The Ioffe-Regel frequency is defined as the energy atwhich the mean free path of the wave (cid:96) becomes compa-rable to its wavelength λ [34], (cid:96) ( ω IR ) = λ ( ω IR ) (11)and its quasiparticle nature is lost. Upon combiningEq. (11) and Eq. (5) and k (cid:48) = 2 π/λ , we obtain ω IR : k (cid:48) ( ω IR ) = 4 π k (cid:48)(cid:48) ( ω IR ) (12)which provides a new operational quantitative definitionof the Ioffe-Regel frequency for a generic collectiveexcitation. ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● BPIR k'k'' ω π Figure 3. The experimental data for k ( ω ) in the soda limeglass (see fig.6 in the Supplementary Material [19]). The firstdashed line indicates the location of the IR frequency definedas (12); the second dashed line indicates the BP position whichcan be found from the absorbance data in Fig.2. We find that ω BP /ω IR ∼ . , confirming the results of Eq.(13). In Fig.3, we show the experimental data of k ( ω ) for thesoda lime glass (see more details in the SupplementalMaterial [19]). From there it is evident that the Ioffe-Regel crossover, defined using Eq.(12), is extremely closeto the BP frequency observed in the absorbance (Fig.2), ω BP /ω IR ∼ . . This represents a strong experimentalconfirmation of the intimate correlation between the BPand the IR frequencies in the phonon-polariton spectrumof glasses.In order to emphasize our point, we compare the BPfrequency ω BP and the Ioffe-Regel frequency ω IR for the theoretical model in Fig.4. We again observe that the twovalues strongly correlate: ω BP = C ω IR (13)where C is an O (1) constant prefactor. From thesoda lime glass experimental data we consistently find C ∼ . (see Fig.3).Importantly, we also find that both frequencies followan approximate power law scaling ω ∼ D − n with n ≈ . ,which strengthens the idea of correlation between thesetwo quantities. More broadly, we expect Eq.(13) to holdgenerally, up to a non-universal O (1) number – C – whichdepends on the microscopics of the system. BP FREQUENCY ω BP IR FREQUENCY ω IR ω Figure 4. The BP frequency ω BP and the IR frequency ω IR as a function of the diffusion constant D . The red and bluelines show the fit which at large D is consistent with a scaling ω ∼ D − n with n ≈ . . Conclusions – In summary, we reported on theexperimental observation and theoretical analysis ofphonon-polaritons in a model amorphous material. Thepolaritonic nature of the excitation cannot be reproducedby standard DHO fitting, but only using a diffusive ∼ k linewidth. We can confidently claim that the bosonpeak observed experimentally in the phonon-polaritonabsorption spectrum is controlled by the Ioffe-Regelcrossover from ballistic quasi-particle propagation toincoherent diffusive-like excitations ( diffusons [17]).This identification, which is valid with high precision,suggests that the physical mechanism underlying theBP in the phonon-polariton spectra of glasses is dueto the quasi-localization of the excitations and tothe propagating-to-diffusive crossover á la Ioffe-Regel.Working with polaritons has the advantage that wecould clearly rule out the influence of dispersion relationband flattening on the peak, away from the influence ofpseudo-van Hove singularities, and hence this analysisprovides the first unambiguous demonstration thatboson peak and Ioffe-Regel crossover coincide, in anexperimental system. Crucially, our results suggestthat this mechanism for the BP may apply to any bosonic excitation in amorphous materials (such asexcitons, magnons, plasmons) [35], which opens up newopportunities for technological design and control ofoptical, electrical and thermal properties of materials bytailoring the disorder-induced effects. Aknowledgments – M.B. acknowledges the support ofthe Spanish MINECO’s “Centro de Excelencia SeveroOchoa” Programme under grant SEV-2012-0249. A.Z.acknowledges financial support from US Army ResearchLaboratory and US Army Research Office through con-tract nr. W911NF-19-2-0055. T.M. is grateful to Y.Matsuda for providing the glass sample and acknowl-edges JSPS KAKENHI Grant Nos. JP17K14318 andJP18H04476, and the Asahi Glass Foundation. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected][1] M. Born and K. Huang,Dynamical Theory of Crystal Lattices (Oxford Uni-versity Press, Oxford, 1954).[2] R. Shuker and R. W. Gammon, Phys. Rev. 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Supplementary Material
Experimental Methods
As a standard glass system exhibiting the BP in theinfrared spectrum, we selected a soda-lime glass which isa typical network glass former. The sample is purchasedfrom Central Glass Co., Ltd. We utilized two differentcommercial THz time-domain spectrometers to cover awide frequency range between 0.3 – 5 THz (0.3 - 1.2 THz:RT-10000, Tochigi Nikon Co.; 1.2 – 5 THz: TAS7500SU,Advantest Corp.) [36–41]. The measured THz waveformsincluding multiple reflections in the sample surfaces wereconverted to the frequency domain, and the obtainedcomplex transmission coefficient t was analyzed using thefollowing equation: t ( ω ) = t vs t sv e i ( n s − d ω/c − r sv ( ω ) e i n s d ω/c , (14)where: t ij = 2 n i n i + n j , r ij = n i − n j n i + n j (15)are the complex Fresnel’s transmission and reflection co-efficients, respectively, at the interface between regions i and j . The subscripts i and j stand for v and s in equation(14), representing the vacuum and sample, respectively. n i is the complex refractive index of region i , d is thethickness of sample, and c is the speed of light. Then, theabsorption coefficient α ( ω ) is obtained from the relation: α ( ω ) = 2 ω κ ( ω ) c , (16)where κ ( ω ) is the imaginary part of n s , i.e. the extinctioncoefficient. From the linear response theory for disorderedsystems [32], α ( ω ) and the vibrational density of states g ( ω ) are related through the infrared photon-phonon cou-pling coefficient C IR ( ω ) as following: α ( ω ) = C IR ( ω ) g ( ω ) (17)The BP appears in the spectrum of g ( ω ) /ω , thereforethe BP in the infrared spectrum appears in the plot of α ( ω ) /ω .Some experimental data for the dielectric constant areshown in Fig.6 together with the fits from the theoreticalmodel. Moreover, in Fig.5 we show the dispersion relationof the polariton as extracted from the experimental data. Theoretical model calibration on experimental data
As discussed in the main text, the damping mechanismis different depending on the frequency range we are
Figure 5. The dispersion relation of the Polariton in theSoda-lime glass extracted from the experimental data used inthis work. The yellow dashed line indicates the BP frequency ω BP / (2 π ) ∼ . THz.
DHO ModelExperiment ω [ THz ] α ( ω ) / ω [ c m - T H z - ] Figure 6. Absorbance (normalized by Debye law) in the THzrange. The yellow markers are the experimental data and theblue line is the fit with the constant damping model. looking at. At high energy (frequency/momentum), themicroscopic details of the disorder are relevant, and thedisorder-induced scattering is well approximated by aconstant damping term Γ( k ) = γ as in the Drude modelfor electron conduction or in the Langevin equation formolecular motion in dense environment. In this regime,the damping is basically provided by the microscopiccollisions in the localized motion of atoms. In Fig.6we show that this damping provides indeed a goodapproximation for the experimental data but only atlarge frequencies, much above the boson peak frequency ω BP . At low frequency, the experimental normalizedabsorbance turns down, while the DHO model with aconstant damping cannot reproduce such trend.As explained in the main text, at low frequencies thenature of the linewidth can be well approximated by thehydrodynamic expressions for diffusons: Γ( k ) = D k . (18)This mechanism comes from a coarse-grained descriptionfor which, on sufficiently large length scales, the effectsof the microscopic scattering events are encoded in aneffective ”diffusion” dynamics. In order to have controlover the full range of frequency, we build an interpolatingmodel: Γ( k ) = γ D k γ + D k (19)which smoothly crosses over between the two, low- k andhigh- k , regimes. Using this model, we are able to accu-rately fit the experimental data across the whole rangeof frequencies. This is emphasized in Fig.7 where the fullset of experimental data is shown and compared to ourtheory. Theory ϵ ' ( ω ) Experimental ϵ ' ( ω ) Theory ϵ '' ( ω ) Experimental ϵ '' ( ω ) ω [ THz ] Figure 7. Dielectric function of the soda lime glass in theTHz range. The theoretical model uses the diffusive linewidthmodel, Eq.(8) in the main text.
Derivation of the Phonon-Polariton DispersionRelations
In order to derive our main relation Eq.(2), we start bywriting down the system of coupled dynamical equationsfor the relative (partially-)charged-particle displacement (cid:126)u , the polarization vector (cid:126)P and the EM fields (cid:126)E, (cid:126)H . ¨ (cid:126)u = b (cid:126)u − Γ ( k ) ˙ (cid:126)u + b (cid:126)E(cid:126)P = b (cid:126)u + b (cid:126)E ∇ · ( (cid:126)E + 4 π (cid:126)P ) = 0 ∇ · (cid:126)H = 0 ∇ × (cid:126)E = − c ˙ (cid:126)H ∇ × (cid:126)H = c ( ˙ (cid:126)E + 4 π ˙ (cid:126)P ) (20)in which we importantly add an effective damping term Γ( k ) which encodes the effects of disorder and dissipation on the atomic motion. Going to Fourier space and identi-fying the coefficient b with the characteristic mechanicaloscillation frequency ω [1], the equations can be writtenas (cid:126)u = − b ω − ω + iω Γ ( k ) (cid:126)E(cid:126)P = (cid:18) b − b b ω − ω + i ω Γ ( k ) (cid:19) (cid:126)E(cid:126)k · (cid:126)E (cid:18) π b − πb b ω − ω + i ω Γ ( k ) (cid:19) = 0 (cid:126)k · (cid:126)H = 0 (cid:126)k × (cid:126)E = ωc (cid:126)H(cid:126)k × (cid:126)H = − ωc ( (cid:126)E + 4 π (cid:126)P ) (21)Using the known relation (cid:126)D = (cid:126)E + 4 π (cid:126) P = ε ( ω ) (cid:126)E , we cansubstitute the unknown parameters b , b , b in termsof the dielectric constant. We denote: ε ( ω = 0) ≡ ε , ε ( ω → ∞ ) ≡ ε ∞ (22)and using these definitions, the third equation in Eq.(21)can be re-written as: (cid:126)k · (cid:126)E (cid:18) ε ∞ − ω ( ε − ε ∞ ) ω − ω + i ω Γ ( k ) (cid:19) = 0 . (23)At this point, a comment is in order. The scalar product (cid:126)k · (cid:126)E distinguishes between two different types of modes: (cid:126)k · (cid:126)E (cid:54) = 0 → LO modes , (24) (cid:126)k · (cid:126)E = 0 → TO modes . (25)Starting from the LO modes and assuming (cid:126)k · (cid:126)E (cid:54) = 0 , theEq.(23) implies ε ∞ = ω ( ε − ε ∞ ) ω − ω + i ω Γ ( k ) ω LO + i ω LO Γ ( k ) − ω ε ε ∞ = 0 (26)where we have indicated the frequency of the mode ω = ω LO .Moving on to the TO modes and taking (cid:126)k and (cid:126)E or-thogonal, we can observe from Eq.(21) that the magneticfield (cid:126)H is orthogonal to both the vectors (cid:126)k, (cid:126)E . As a con-sequence, the fifth and sixth equations in Eq.(21) becomea coupled relation between the amplitudes: k E = ω T O c Hk H = ω T O c E (cid:18) ε ∞ − ω ( ε − ε ∞ ) ω T O − ω + i ω Γ ( k ) (cid:19) ⇒ k c = ω T O (cid:18) ε ∞ − ω ( ε − ε ∞ ) ω T O − ω + i ω Γ ( k ) (cid:19) (27)After simple algebraic manipulations, we finally obtainthe quartic equation (2) presented in the main text: ω ε ∞ + iω Γ ( k ) ε ∞ − ω (cid:0) ω ε + k c (cid:1) − i Γ ( k ) k c ω + ω k c = 0 (28)where for simplicity we have omitted the label T O whichstands for ”transverse optical”.
Effects of Disorder and Damping on thePhonon-Polariton Dispersion Relation
We start with the generic fourth-order equation whichwe derived in the previous section ω ε ∞ + iω Γ ( k ) ε ∞ − ω (cid:0) ω ε + k c (cid:1) − i Γ ( k ) k c ω + ω k c = 0 (29)where the disorder and damping effects are effectivelyencoded in the momentum dependent parameter Γ( k ) .Let us start by reminding the reader about the knownresults in absence of any damping mechanism, Γ( k ) = 0 ,which was derived in [18, 22]. In this simple case, thesolution can be written concisely as: ω = (cid:115) c k ± (cid:112) c k + 2 c k ω ( (cid:15) − (cid:15) ∞ ) + (cid:15) ω + (cid:15) ω (cid:15) ∞ (30)The two modes display the repulsion phenomenon whichis typical of the polariton dynamics and is due to theelectromagnetic interactions encoded in the non-trivialdieletric constant ( (cid:15) ∞ (cid:54) = (cid:15) ). This behaviour is verysimilar to the one is displayed in panel a) of Figure 8for a concrete choice of parameters with small damping.Obviously, this is an idealized situation in which all theeffects which originate from internal scattering events areneglected.As a first step forward, let us consider the situation inwhich the optical phonons have a finite and momentumindependent relaxation time: τ − = Γ( k = 0) = γ (31)which determines their lifetime and mean free path. Here,we take an effective field theory perspective and we donot discuss the microscopic origin of this relaxation time.Several are the physical mechanisms that can contribute tothis effect. Theoretically, this relaxation time implies thenon-conservation of momentum, which now dissipates ata rate γ , exactly as in the simple Drude model for electricconduction [33] or in the Langevin equation for Brownianmotion in liquids. This relaxation time approximationcan be formally derived using Boltzmann equation andkinetic theory [42] and it is valid only in the regime when τ is large enough. The dynamics of the low energy modes ck [ ω ] ck [ ω ] ck [ ω ] ck [ ω ] ck [ ω ] ck [ ω ] a ) b ) c ) d ) e ) f ) ck - - - - - - [ ω ] ck - - - - [ ω ] ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ck - 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- - - [ ω ] ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ck - - - - - - [ ω ] Figure 8. The dispersion relation of the excitations in thedamped model by changing the damping parameter γ =0 . , . , , . , . , , from panel a) to panel f). Top:
Thereal part
Re( ω ) in function of the momentum k . Bottom:
The imaginary part
Im( ω ) in function of the momentum k . is displayed in Fig. 8 upon increasing the relaxationrate γ ∈ [0 , from panel a) to panel f). For small γ (cid:28) ω , the gapless mode acquires a small damping Im( ω )( k = 0) (cid:54) = 0 which grows with γ . This mode isnot anymore a hydrodynamic mode. The other gappedmode does not acquire a finite damping and remainsdiffusive at low momentum. When the damping parameterbecomes comparable with the characteristic frequency ofthe gapped mode γ ∼ ω , the two modes attract eachother and they move closer as shown in panel b) of Fig.8.When γ ≥ ω , the dynamics is not anymore under-damped and the modes merge producing a complicatedpattern shown in the panels c) and d) of Fig.8. Finally,in the limit γ (cid:29) ω ( over-damped regime), the soundmode gets completely destroyed and it acquires a verylarge damping. Its lifetime becomes very short and itcompletely disappears from the dynamics (see panel e) inFig.8). As a consequence, the ”photon root” does not feel ck Re [ ω ] ck Re [ ω ] ck Re [ ω ] ck Re [ ω ] ck Re [ ω ] ck Re [ ω ] a ) b ) c ) d ) e ) f ) Figure 9. The dispersion relation of the excitations in thediffusive model by changing the diffusion constant D =0 . , . , . , . , , . Top:
The real part
Re( ω ) in func-tion of the momentum k . Bottom:
The imaginary part