Anomalous softening of phonon-dispersion in cuprate superconductors
AAnomalous softening of phonon-dispersion in cuprate superconductors
Saheli Sarkar, Maxence Grandadam, and Catherine P´epin Institut de Physique Th´eorique, Universit´e Paris-Saclay, CEA, CNRS, F-91191 Gif-sur-Yvette, France.
A softening of phonon-dispersion has been observed experimentally in under-doped cuprate su-perconductors at the charge-density wave (CDW) ordering wave vector. Interestingly, the softeningoccurs below the superconducting (SC) transition temperature T c , in contrast to the metallic sys-tems, where the softening occurs usually below the CDW onset temperature T CDW . An understand-ing of the ‘anomalous’ nature of the phonon-softening and its connection to the pseudo-gap phasein under-doped cuprates remain open questions. Within a perturbative approach, we show that acomplex interplay among the ubiquitous CDW, SC orders and life-time of quasi-particles associatedto thermal fluctuations, can explain the anomalous phonon-softening below T c . Furthermore, ourformalism captures different characteristics of the low temperature phonon-softening depending onmaterial specificity. The ‘pseudo-gap’ phase [1–8] of the under-dopedhigh-temperature copper-oxide based superconductors(cuprates) remains incomprehensible even after decadesof research, by and large due to a complex interplayof several symmetry broken orders [9, 10]. A univer-sally present translational symmetry broken order in thecuprates is a charge-density wave (CDW) order [11–23].Since its discovery, the CDW order has become funda-mentally important due to growing evidences of its closerelation to the pseudo-gap phase, although a full knowl-edge about the CDW order and its relation to the pseudo-gap phase remains incomplete. One leading approach tounravel the relation, is to study the phonon-spectrumwhich couples to electronic degrees of freedom, thus leav-ing fingerprints associated to the electronic-structure.The phonon-spectrum has been largely studied inmetallic systems, where the the charge-correlations softenthe phonon-spectrum giving rise to the ‘Kohn-anomaly’[24]. In one dimensional metals [25–27] and in some tran-sitional metal dichalcogenides [28], this softening growstowards zero [Fig. 1] and a full phonon-softening occursat the CDW wave-vector (Q) below CDW ordering tem-perature T
CDW , reflecting the origin of CDW order inthem. With a similar outlook, the phonon-spectrum hasbeen measured even in cuprates using different experi-mental techniques, like inelastic x-ray scattering and in-elastic neutron scattering [17, 29–38]. All of these exper-iments have observed a partial phonon-softening [Fig. 1]associated to Q in several cuprates, only below the su-perconducting transition temperature T c , in stark con-trast to the metallic systems [27, 28, 39, 40]. The occur-rence of phonon-softening below T c is hence referred toas ‘anomalous’ phonon-softening.The anomalous phonon-softening indicates a close con-nection between the CDW and superconductivity inunder-doped cuprates. Such a connection between CDWand superconductivity have been widely discussed in var-ious theoretical studies [41–44]. Supporting evidences ofthis connection can also be found in several experiments[11, 13, 45, 46]. Notably, a recent proposal [47], based onthe fractionalization of a pair-density wave (PDW) or- FIG. 1. Schematic representation of a full softening in metalsand a partial softening in under-doped cuprates below T c . der [48, 49], advocates that for temperatures above T c ,a growing amount of fluctuations in CDW and supercon-ductivity arising from a connection between them, canprovide potential explanation to the pseudo-gap phase.While earlier studies [43, 50–52] discussed the role ofCDW, superconductivity and associated fluctuations onthe electronic-spectrum, their effect on the bosonic exci-tations, especially phonons, remain an outstanding ques-tion and perhaps can give a more complete understand-ing of the CDW orders in cuprates. In this letter, weincorporate simultaneous effects of CDW, superconduc-tivity and thermal fluctuations on the phonon-spectrum.In our model, we mimic the fluctuations by introduc-ing an inverse life-time of quasi-particles [52, 53] andtake its temperature dependence phenomenologically [53]based on earlier studies, which can capture various cru-cial aspects of the electronic spectrum in the pseudo-gapphase. We find that a strong phonon-softening occursonly below T c , thus explaining the anomalous nature ofthe phonon-softening seen in experiments. Additionally,we also show that at low temperatures, different temper-ature dependence of the superconducting (SC) gap andinverse life-time of quasi-particle give contrasting effectson the strength of the phonon-softening.We start with a total Hamiltonian H tot [54], given by a r X i v : . [ c ond - m a t . s up r- c on ] S e p H tot = H e + H ph + H e − ph , with, H e = X k,σ ξ k c † k,σ c k,σ + X k,σ ( χ k c † k + Q,σ c k,σ + h.c. ) (1)+ X k (∆ k c † k, ↑ c †− k, ↓ + h.c. ) ,H ph = X q ω q ( b † q b q + b †− q b − q ) ,H e − ph = ( g/ √ N ) X q X k,σ [ c † k + q,σ c k,σ ( b † q + b − q ) + h.c. ] , where H e is an effective mean-field Hamiltonian with SCand CDW orders. c † k,σ ( c k,σ ) is the creation (annihilation)operator for an electron with spin σ and momentum k , ξ k is the electronic dispersion, ∆ k is the SC order parame-ter and χ k is the CDW order parameter with modulationwave-vector Q. H ph is the Hamiltonian for free phononswith phonon creation operator b † q for wave-vector q andfrequency ω q . H e − ph is the Hamiltonian describingelectron-phonon interaction with strength g and N isthe number of lattice sites in the system. The Green’sfunction corresponding to H e is given by ˆ G − ( iω n , k ) =( iω n − ˆ H e ) and has a matrix form in the extended Nambubasis Ψ † k = (cid:16) c † k, ↑ , c − k, ↓ , c † k + Q, ↑ , c − k − Q, ↓ (cid:17) which is givenby, G − = iω n − ξ k − ∆ k − χ k − ∆ ∗ k iω n + ξ k χ k − χ ∗ k iω n − ξ k + Q − ∆ k + Q χ ∗ k − ∆ ∗ k + Q iω n + ξ k + Q , (2)where ω n is the Matsubara frequency. We use a band-structure for a prototype cuprate system [55] [see sup-plementary materials (SM) [56]]. We consider a d-wavesymmetric SC gap, given by ∆ k = (∆ max / k x ) − cos( k y )], where ∆ max denotes the maximum gap. Fol-lowing several theoretical studies [47, 57, 58] and ex-perimental evidences [20, 59], we consider a CDW or-der parameter with Q given by the axial wave-vectorconnecting two neighboring ‘hot-spots’, the points onFermi-surface which intersect the magnetic-brillouin zoneboundary [41]. Moreover, the CDW gap is taken to havea maximum ( χ max ) at the hot-spots, falling off exponen-tially away from the hot-spots [47].The modified electronic spectrum in the presence ofSC and CDW orders will re-normalize the free phononpropagator, D ( z, q ) = 2 ω q / ( z − ω q ). To analyze this,we begin by writing the imaginary time ( τ ) phonon prop-agators in matrix form in the ordered phase. The cor-responding matrix elements are given by D m,n ( q, τ ) = −hT φ q + mQ ( τ ) φ † q + nQ (0) i , where T is the time-orderingoperator [54], φ q is the phonon field operator given by b † q + b − q and m, n = ± . Noting that D ++ ≡ D −− := D ( z, q ) and D + − ≡ D − + := D ( z, q ), within a per- FIG. 2. (a), (b), (c) and (d) represent the Feynman diagramsfor the terms in the Dyson equations [Eq. (3)] involving theself-energies Π , Π , Π and Π respectively in the presenceof CDW and SC orders. turbative treatment of electron-phonon interaction, weevaluate the re-normalized phonon propagators D and D by using Dyson equations D ( z, q ) = D ( z, q + Q ) (cid:20) ( z, q ) D ( z, q )+ (3)Π ( z, q ) D ( z, q ) + Π ( z, q ) D ( z, q )+Π ( z, q ) D ( z, q ) (cid:21) ,D ( z, q ) = D ( z, q − Q ) (cid:20) Π ( z, q ) D ( z, q ) + Π ( z, q ) D ( z, q )+ Π ( z, q ) D ( z, q ) + Π ( z, q ) D ( z, q ) (cid:21) , where, Π , , , ( z, q ) represent the phonon self-energies.The leading contributions to the Dyson equations[Eqs. (3)] are shown in Fig. 2. Explicit expressions forΠ , , , ( z, q ) are presented in the SM [56].We obtain the new modes for phonon in the or-dered phase by decoupling Eq. (3), with the definition D ± ( z, q ) = D ( z, q ) ± D ( z, q ) and then solving D ± ( z, q )with the assumption that ω Q ± q ≈ ω Q for small q. Fi-nally, plugging in D ( z, q ), we obtain the solutions as, D ± ( z, q ) = 2 ω Q z − ω Q − ω Q Π ± ( z, q ) , (4)where Π + = Π +Π +Π +Π and Π − = Π +Π − Π − Π . The dispersion of the new phonon modes correspondto the values of z , for which denominator of Eq. (4) van-ishes. Subsequently, taking only q dependence in Π, thefrequency for each mode is given byΩ ± ( q ) = ω Q + 2 ω Q Π ± ( q ) . (5)These two new phonon modes in Eq. (5) with frequencyΩ ± signify branching of the free phonon near Q due topresence of CDW and SC orders. We find that the splitbetween Ω ± is proportional to the magnitude of the CDWorder. Also, we only plot Π ± as a function of ˜ q = q − Q , π ( q ̃ ) −0.4−0.3−0.2−0.10 q ̃ −1 −0.5 0 0.5 1 π - π + FIG. 3. Plots of the self-energy Π ± as a function of ˜ q = q − Q corresponding to the two re-normalized phonon modes Ω ± inthe presence of χ max = 0 .
05 and ∆ max = 0 .
05. Both Π ± exhibit a depletion around ˜ q = 0, implying a softening in thephonon-dispersion of the two new modes Ω ± around Q. as the modes Ω ± ( q ) can be easily identified from the cor-responding Π ± in Eq. (5). For depicting the strength ofthe phonon-softening, we look at Π ± (˜ q ) after subtract-ing Π ± (˜ q = − ± (˜ q ) de-creases strongly within a finite range around ˜ q = 0, witha minimum at ˜ q = 0, readily suggesting a softening ofphonon-frequency around Q. We also observe that, awayfrom ˜ q = 0, Π ± (˜ q ) goes towards zero, implying a sup-pression of phonon-softening away from Q. This suggeststhat the effect of CDW and SC orders on the phononare maximum at Q, and diminishes away from it. Ad-ditionally, we notice that the suppression of Π − is morethan the suppression of Π + and the ˜ q dependence of Π ± are extremely similar to each other. Hence, for a simplerpresentation, in the rest of the paper, we only plot Π − with ˜ q [relabeled as Π(˜ q )].So far, we obtain a phonon-softening in the presence ofSC and CDW orders. However, to address the anoma-lous phonon-softening in cuprates, we need to includefluctuation related effects, which are major constituentsgoverning the phase diagram of these systems. For exam-ple, such fluctuations can lead to quasi-particle scatteringwhich are known to have vital roles in Fermi-arc relatedphysics of the pseudo-gap phase [60–62]. The strength ofthe scattering depends on temperature; while it can belarge at high temperatures, a sudden reduction occursbelow T c , which can be attributed to a fractionalizationof a PDW [47]. To give an idea, the proposal of fraction-alization of a PDW order suggests that the fluctuationof a U(1) gauge field gives a constraint connecting SCand CDW. As a result, fractionalization of PDW occursat an energy scale associated to the pseudo-gap temper-ature T*, consequently fluctuations largely increase inthe system. However, below T c , the fluctuations quench,thus yielding a global phase coherence of CDW and SC −0.4−0.20−1 −0.5 0 0.5 1 Γ = 1.0 Γ = 0.38 Γ = 0.09 Γ = 0.02 (a) q ̃ π ( q ̃ ) −0.5−0.4−0.3−0.2−0.100 0.2 0.4 0.6 0.8 1 Δ max = 0.4 Δ max = 0.3 Δ max = 0.2 Δ max = 0.1 Δ max = 0.0 (b) Γ π ( q ̃ = ) FIG. 4. (a) The variation of Π(˜ q ) with ˜ q for four differentvalues of Γ with χ max = 0 . max = 0. The plots portraya suppression in phonon-softening with increase in Γ. (b)Plots of Π(˜ q = 0) with variation in Γ for five different valuesof ∆ max with χ max = 0 .
2. The plots manifest a suppressionin phonon-softening with an increase in ∆ max . The effect of∆ max is strongest for low Γ, and weakest for high Γ. orders and increasing the life-time of quasi-particles.In order to study the evolution of the phonon-softeningwith temperature, we incorporate a finite inverse life-timeof quasi-particles, given by Γ, pertinent to the fluctuationrelated effects in the system. The self-energy in Matsub-ara frequency due to Γ can be written as Σ = i Γ sgn ( ω n )and the Green’s function in Eq. (2) will transform as G − i,j ( iω n , k ) → G − i,j ( iω n + Σ , k ) . (6)In the presence of Γ, the phonon-dispersion will be mod-ified by the real part of Π(˜ q ), again relabeled as Π(˜ q ).Detailed calculations are presented in the SM [56].To understand the collective effect of the SC gap andΓ on the phonon-softening, it is important to disentanglethe role played by Γ and the SC gap. Therefore, we startby studying the effect of Γ taking ∆ max = 0. Fig. 4(a)shows the variation of Π(˜ q ) as a function of ˜ q for four dif-ferent Γ with χ max = 0 .
2. We notice that for very smallvalue of Γ = 0 .
02, there is a significantly strong phonon-softening around ˜ q = 0. With increasing Γ, the phonon-softening starts to reduce and for a very large Γ = 1 . q = 0 is moststrongly affected by Γ. Therefore, for rest of the anal-ysis, we will concentrate on Π at ˜ q = 0 to quantify thephonon-softening.Now, we inspect the role of the SC order and the interplaybetween superconductivity and Γ. In Fig. 4(b), we plotthe variation of Π(˜ q = 0) with Γ, for five different ∆ max taking χ max = 0 .
2. We notice that ∆ max has a promi-nent effect when Γ is very small, as can be seen from thechange in Π(˜ q = 0) around Γ ∼ .
05. In this regime,∆ max weakens the softening of phonon. Similar effect onphonons in the SC phase has been indicated in conven-tional s-wave superconductors [63, 64]. With increasingΓ, for example around Γ ∼ .
3, the effect of ∆ max be-comes less significant. Finally, for very large Γ ’ . Δ max ( T ) Γ ( T ) Γ ( T ) Γ ( T ) Γ ( T ) T>T c T
BSCCO E n e r g y ( m e V ) T/T c FIG. 5. (a) Different sets of T-dependence for inverse life-time of quasiparticles denoted by Γ , Γ , Γ and Γ . TheT-dependence of the SC gap is denoted by ∆ max (T). In allcases, χ max = ∆ max . (b) The T-dependence of Π(˜ q = 0)for different parameter sets in (a). A large negative value ofΠ(˜ q = 0) in the regime T . T c implies a strong enhancementof phonon-softening, while Π(˜ q = 0) → > T c . (c)The variation of Π(˜ q ) with ˜ q at four different temperatures forparameter set Γ and ∆ max (T) shown in (a). (d) Schematicrepresentation of the experimental results of phonon-softeningat CDW wave-vector for YBCO and BSCCO, adopted fromRefs. [29, 31]. changing ∆ max has almost no effect. These results high-light two crucial points. First, both superconductivityand Γ suppress the phonon-softening. Second, the role of∆ max is prominent at low Γ, while negligible for large Γ.We have seen that the introduction of superconductiv-ity suppresses the phonon-softening, while experimentsobserve a seemingly opposite characteristic of enhance-ment of phonon-softening below T c . At this point,we should also notice that Γ suppresses the phonon-softening, as shown in Fig. 4(a). Moreover, Γ is expectedto increase with temperature due to increase in fluctua-tions, whereas ∆ max is expected to decrease with temper-ature, for example in a simple BCS type scenario. Thus,they behave in opposite manner with temperature.We consider temperature (T) dependence phenomeno-logically in ∆ max and Γ, similar to the T dependenceused in explaining spectral function in ARPES experi-ments [53]. The T-dependence of ∆ max and Γ are shownin Fig. 5(a). Below T c , ∆ max decreases with T, whereasremains approximately constant above T c . Moreover fol-lowing indications from Raman spectroscopy [45], χ max is taken to be equal to ∆ max . To illustrate how dif-ferent T-dependence of Γ and ∆ max can give different features in phonon-softening, we use four different typesof T-dependence for Γ, denoted by Γ , Γ , Γ and Γ inFig. 5(a). Note that, they differ in magnitudes comparedto ∆ max . In all these cases, Γ reduces significantly be-low T c , with the strongest fall in Γ and the weakest fallin Γ , but still remains finite even in the limit T → > T c , as suggested in someearlier works [66, 67].In Fig. 5(b), we plot Π(˜ q = 0) for the parameters inFig. 5(a). We start by closely inspecting the Γ case inFig. 5(b). We observe that the values of Π(˜ q = 0) areclose to zero for high temperatures (T (cid:29) T c ), implyingthat the phonon-softening is strongly suppressed. Re-markably, we observe that for temperatures T . T c , thevalues of Π(˜ q = 0) reduce sharply towards more negativevalues, which suggest that the phonon-softening enhancesstrongly. But surprisingly, towards further lower tem-peratures below T c , Π(˜ q = 0) enhances, which implies asuppression in phonon-softening. However, the phonon-softening below T c always remains stronger as comparedto T > T c . Very similar features have been observed inYBa Cu O y (YBCO) [29], as shown schematically inFig. 5(d). In Fig. 5(c), we present the the full ˜ q depen-dence of Π at four different temperatures for the caseΓ . We observe that away from ˜ q = 0, phonon-softeningis less sensitive to the variation of temperature. Similarfeature has been found in experiments [29, 31].Next, we closely investigate the Γ case in Fig. 5(b)for T . T c . Very interestingly, the features for T . T c possess marked differences from Γ case. We noticea smoother enhancement in phonon-softening just belowT c (T ∼ T c ), while the enhancement is more rapid andsharper for Γ case. In particular, towards lower tem-peratures (T → . Analogous features in phonon-softeninghave been also observed in Bi Sr CaCu O y (BSCCO)[31], schematically presented in Fig. 5(d). To demon-strate the different features in phonon-softening resultingfrom an intricate interplay between SC gap and Γ belowT c , we plot results for two more cases Γ and Γ , shownin Fig. 5(b). Below T c , for Γ , phonon-softening sharplyenhances than for Γ as T → c , can successfully describe the‘anomalous’ phonon-softening. A reduction in the fluctu-ations below T c can be motivated from a recent proposalbased on fractionalization of a PDW order [47]. More-over, we also found that the features of phonon-softeningat low temperatures depend on an intricate interplay be-tween SC order and fluctuations. In this work, we con-sidered the strength of electron-phonon coupling to bemomentum (k)-independent. However, the formalism inthis work, can easily be extended to include k-dependentelectron-phonon coupling. We expect, in such a scenario,the phonon-softening will still occur at Q only below T c ,but the softening will have a different wave-vector de-pendence around Q. We believe our results can find ap-plications in many two-dimensional materials where aninterplay between CDW and SC orders plays an impor-tant role and thus opening much broader prospects of ourwork.We acknowledge A. Banerjee and Y. Sidis for valu-able discussions. This work has received financial sup-port from the ERC, under grant agreement AdG-694651-CHAMPAGNE. [1] H. Alloul, T. Ohno, and P. Mendels, Phys. Rev. Lett. , 1700 (1989).[2] W. W. Warren, R. E. Walstedt, G. F. Brennert, R. J.Cava, R. Tycko, R. F. Bell, and G. Dabbagh, Phys.Rev. Lett. , 1193 (1989).[3] C. Berthier, M. Julien, M. Horvati´c, and Y. Berthier,Journal de Physique I , 2205 (1996).[4] D. S. Marshall, D. S. Dessau, A. G. Loeser, C.-H. Park,A. Y. Matsuura, J. N. Eckstein, I. Bozovic, P. Fournier,A. Kapitulnik, W. E. Spicer, and Z.-X. Shen, Phys. Rev.Lett. , 4841 (1996).[5] J. M. Harris, P. J. White, Z.-X. Shen, H. Ikeda,R. Yoshizaki, H. Eisaki, S. Uchida, W. D. Si, J. W. Xiong,Z.-X. Zhao, and D. S. Dessau, Phys. Rev. Lett. , 143(1997).[6] C. Renner, B. Revaz, J.-Y. Genoud, K. Kadowaki, andO. Fischer, Phys. Rev. Lett. , 149 (1998).[7] A. Ino, T. Mizokawa, K. Kobayashi, A. Fujimori,T. Sasagawa, T. Kimura, K. Kishio, K. Tamasaku,H. Eisaki, and S. Uchida, Phys. Rev. Lett. , 2124(1998).[8] F. Ronning, T. Sasagawa, Y. Kohsaka, K. M. Shen,A. Damascelli, C. Kim, T. Yoshida, N. P. Armitage, D. H.Lu, D. L. Feng, L. L. Miller, H. Takagi, and Z.-X. Shen,Phys. Rev. B , 165101 (2003).[9] E. Fradkin, S. A. Kivelson, and J. M. Tranquada, Rev.Mod. Phys. , 457 (2015).[10] C. P´epin, D. Chakraborty, M. Grandadam, andS. Sarkar, Annual Review of Condensed Matter Physics , 301 (2020).[11] J. E. Hoffman, E. W. Hudson, K. M. Lang, V. Madhavan,H. Eisaki, S. Uchida, and J. C. Davis, Science , 466(2002).[12] N. Doiron-Leyraud, C. Proust, D. LeBoeuf, J. Levallois,J.-B. Bonnemaison, R. Liang, D. A. Bonn, W. N. Hardy,and L. Taillefer, Nature , 565 (2007).[13] G. Ghiringhelli, M. Le Tacon, M. Minola, S. Blanco-Canosa, C. Mazzoli, N. B. Brookes, G. M. De Luca,A. Frano, D. G. Hawthorn, F. He, T. Loew, M. M. Sala,D. C. Peets, M. Salluzzo, E. Schierle, R. Sutarto, G. A.Sawatzky, E. Weschke, B. Keimer, and L. Braicovich, Science , 821 (2012).[14] H.-H. Wu, M. Buchholz, C. Trabant, C. Chang, A. Ko-marek, F. Heigl, M. Zimmermann, M. Cwik, F. Naka-mura, M. Braden, et al. , Nature communications , 1023(2012).[15] A. J. Achkar, R. Sutarto, X. Mao, F. He, A. Frano,S. Blanco-Canosa, M. Le Tacon, G. Ghiringhelli,L. Braicovich, M. Minola, M. Moretti Sala, C. Mazzoli,R. Liang, D. A. Bonn, W. N. Hardy, B. Keimer, G. A.Sawatzky, and D. G. Hawthorn, Phys. Rev. Lett. ,167001 (2012).[16] E. Blackburn, J. Chang, M. H¨ucker, A. T. Holmes,N. B. Christensen, R. Liang, D. A. Bonn, W. N. Hardy,U. R¨utt, O. Gutowski, M. v. Zimmermann, E. M. For-gan, and S. M. Hayden, Phys. Rev. Lett. , 137004(2013).[17] E. Blackburn, J. Chang, A. H. Said, B. M. Leu, R. Liang,D. A. Bonn, W. N. Hardy, E. M. Forgan, and S. M.Hayden, Phys. Rev. B , 054506 (2013).[18] S. Blanco-Canosa, A. Frano, T. Loew, Y. Lu, J. Porras,G. Ghiringhelli, M. Minola, C. Mazzoli, L. Braicovich,E. Schierle, E. Weschke, M. Le Tacon, and B. Keimer,Phys. Rev. Lett. , 187001 (2013).[19] T. P. Croft, C. Lester, M. S. Senn, A. Bombardi, andS. M. Hayden, Phys. Rev. B , 224513 (2014).[20] E. H. da Silva Neto, P. Aynajian, A. Frano, R. Comin,E. Schierle, E. Weschke, A. Gyenis, J. Wen, J. Schnee-loch, Z. Xu, S. Ono, G. Gu, M. Le Tacon, and A. Yaz-dani, Science , 393 (2014).[21] K. Matsuba, S. Yoshizawa, Y. Mochizuki, T. Mochiku,K. Hirata, and N. Nishida, Journal of the Physical So-ciety of Japan , 063704 (2007).[22] K. Fujita, C. K. Kim, I. Lee, J. Lee, M. Hamidian, I. A.Firmo, S. Mukhopadhyay, H. Eisaki, S. Uchida, M. J.Lawler, E. A. Kim, and J. C. Davis, Science , 612(2014).[23] T. Machida, Y. Kohsaka, K. Matsuoka, K. Iwaya,T. Hanaguri, and T. Tamegai, Nature communications , 11747 (2016).[24] E. J. Woll and W. Kohn, Phys. Rev. , 1693 (1962).[25] B. Renker, H. Rietschel, L. Pintschovius, W. Gl¨aser,P. Br¨uesch, D. Kuse, and M. J. Rice, Phys. Rev. Lett. , 1144 (1973).[26] K. Carneiro, G. Shirane, S. A. Werner, and S. Kaiser,Phys. Rev. B , 4258 (1976).[27] J. P. Pouget, B. Hennion, C. Escribe-Filippini, andM. Sato, Phys. Rev. B , 8421 (1991).[28] J. Wilson, F. Di Salvo, and S. Mahajan, Advances inPhysics , 1171 (2001).[29] M. Le Tacon, A. Bosak, S. M. Souliou, G. Dellea,T. Loew, R. Heid, K.-P. Bohnen, G. Ghiringhelli,M. Krisch, and B. Keimer, Nat. Phys. , 52 (2014).[30] H. Miao, D. Ishikawa, R. Heid, M. Le Tacon, G. Fabbris,D. Meyers, G. D. Gu, A. Q. R. Baron, and M. P. M.Dean, Phys. Rev. X , 011008 (2018).[31] W. Lee, K. Zhou, M. Hepting, J. Li, A. Nag, A. Walters,M. Garcia-Fernandez, H. Robarts, M. Hashimoto, H. Lu, et al. , arXiv preprint arXiv:2007.02464 (2020).[32] R. J. McQueeney, Y. Petrov, T. Egami, M. Yethiraj,G. Shirane, and Y. Endoh, Phys. Rev. Lett. , 628(1999).[33] H. Uchiyama, A. Q. R. Baron, S. Tsutsui, Y. Tanaka,W.-Z. Hu, A. Yamamoto, S. Tajima, and Y. Endoh,Phys. Rev. Lett. , 197005 (2004). [34] D. Reznik, T. Fukuda, D. Lamago, A. Baron, S. Tsut-sui, M. Fujita, and K. Yamada, Journal of Physics andChemistry of Solids , 3103 (2008).[35] J. Graf, M. d’Astuto, C. Jozwiak, D. R. Garcia, N. L.Saini, M. Krisch, K. Ikeuchi, A. Q. R. Baron, H. Eisaki,and A. Lanzara, Phys. Rev. Lett. , 227002 (2008).[36] M. d’Astuto, G. Dhalenne, J. Graf, M. Hoesch, P. Giura,M. Krisch, P. Berthet, A. Lanzara, and A. Shukla, Phys.Rev. B , 140511 (2008).[37] A. Q. Baron, J. P. Sutter, S. Tsutsui, H. Uchiyama,T. Masui, S. Tajima, R. Heid, and K.-P. Bohnen, Journalof Physics and Chemistry of Solids , 3100 (2008).[38] M. Raichle, D. Reznik, D. Lamago, R. Heid, Y. Li,M. Bakr, C. Ulrich, V. Hinkov, K. Hradil, C. T. Lin,and B. Keimer, Phys. Rev. Lett. , 177004 (2011).[39] J. Lorenzo, R. Currat, P. Monceau, B. Hennion,H. Berger, and F. Levy, Journal of Physics: CondensedMatter , 5039 (1998).[40] H. Requardt, J. E. Lorenzo, P. Monceau, R. Currat, andM. Krisch, Phys. Rev. B , 214303 (2002).[41] K. B. Efetov, H. Meier, and C. P´epin, Nat. Phys. , 442(2013).[42] L. E. Hayward, D. G. Hawthorn, R. G. Melko, andS. Sachdev, Science , 1336 (2014).[43] Y. Wang, D. F. Agterberg, and A. Chubukov, Phys. Rev.Lett. , 197001 (2015).[44] D. Chakraborty, C. Morice, and C. P´epin, Physical Re-view B , 214501 (2018).[45] B. Loret, N. Auvray, Y. Gallais, M. Cazayous, A. For-get, D. Colson, M.-H. Julien, I. Paul, M. Civelli, andA. Sacuto, Nature Physics , 1 (2019).[46] J. Chang, E. Blackburn, A. T. Holmes, N. B. Chris-tensen, J. Larsen, J. Mesot, R. Liang, D. A. Bonn, W. N.Hardy, A. Watenphul, M. v. Zimmermann, E. M. Forgan,and S. M. Hayden, Nat. Phys. , 871 (2012).[47] D. Chakraborty, M. Grandadam, M. H. Hamidian,J. C. S. Davis, Y. Sidis, and C. P´epin, Phys. Rev. B , 224511 (2019).[48] S. D. Edkins, A. Kostin, K. Fujita, A. P. Mackenzie,H. Eisaki, S. Uchida, S. Sachdev, M. J. Lawler, E.-A.Kim, J. S. Davis, et al. , Science , 976 (2019).[49] M. Hamidian, S. Edkins, S. H. Joo, A. Kostin, H. Eisaki, S. Uchida, M. Lawler, E.-A. Kim, A. Mackenzie, K. Fu-jita, et al. , Nature , 343 (2016).[50] M. Norman, H. Ding, M. Randeria, J. Campuzano,T. Yokoya, T. Takeuchi, T. Takahashi, T. Mochiku,K. Kadowaki, P. Guptasarma, et al. , Nature , 157(1998).[51] M. R. Norman and C. P´epin, Rep. Prog. Phys. , 1547(2003).[52] M. Grandadam, D. Chakraborty, X. Montiel, andC. P´epin, arXiv preprint arXiv:2002.12622 (2020).[53] M. R. Norman, M. Randeria, H. Ding, and J. C. Cam-puzano, Phys. Rev. B , R11093 (1998).[54] P. Lee, T. Rice, and P. Anderson, Solid State Commu-nications , 1001 (1993).[55] C. Berthod, I. Maggio-Aprile, J. Bru´er, A. Erb, andC. Renner, Phys. Rev. Lett. , 237001 (2017).[56] S. Sarkar, M. Grandadam, and C. P´epin, Supplementarymaterials.[57] Y. Wang and A. Chubukov, Physical Review B , 2113(2014).[58] D. Chowdhury and S. Sachdev, Phys. Rev. B , 134516(2014).[59] R. Comin, A. Frano, M. M. Yee, Y. Yoshida, H. Eisaki,E. Schierle, E. Weschke, R. Sutarto, F. He, A. Soumya-narayanan, Y. He, M. Le Tacon, I. S. Elfimov, J. E. Hoff-man, G. A. Sawatzky, B. Keimer, and A. Damascelli,Science , 390 (2014).[60] M. R. Norman, M. Randeria, H. Ding, and J. C. Cam-puzano, Phys. Rev. B , 615 (1995).[61] M. R. Norman, A. Kanigel, M. Randeria, U. Chatterjee,and J. C. Campuzano, Phys. Rev. B , 174501 (2007).[62] E. G. Dalla Torre, Y. He, D. Benjamin, and E. Demler,New Journal of Physics , 022001 (2015).[63] J. D. Axe and G. Shirane, Phys. Rev. B , 1965 (1973).[64] D. Reznik, Physica C: Superconductivity , 75 (2012).[65] A. Chubukov, M. Norman, A. Millis, and E. Abrahams,Physical Review B , 180501 (2007).[66] A. Kanigel, M. Norman, M. Randeria, U. Chatterjee,S. Souma, A. Kaminski, H. Fretwell, S. Rosenkranz,M. Shi, T. Sato, et al. , Nature Physics , 447 (2006).[67] C. M. Varma, P. B. Littlewood, S. Schmitt-Rink,E. Abrahams, and A. E. Ruckenstein, Phys. Rev. Lett. , 1996 (1989). upplementary materials forAnomalous softening of phonon-dispersion in cuprate superconductors Saheli Sarkar, Maxence Grandadam, and Catherine P´epin Institut de Physique Th´eorique, Universit´e Paris-Saclay, CEA, CNRS, F-91191 Gif-sur-Yvette, France.
THE MODEL AND PARAMETERS
In this section, we discuss the model describing the cuprate in the presence of charge-density wave (CDW) andsuperconducting (SC) orders. To analyze the phonon-dispersion in the presence of CDW and SC orders, we start witha total Hamiltonian H tot , which incorporates an effective mean-field electronic Hamiltonian ( H e ) describing CDWand SC orders, Hamiltonian for free phonons ( H ph ) and electron-phonon interaction Hamiltonian ( H e − ph ) as givenin Eq. (1) of the main text. The inverse Green’s function matrix ˆ G − ( iω n , k ) = ( iω n − ˆ H e ) corresponding to theHamiltonian H e in extended Nambu basis Ψ † k = (cid:16) c † k, ↑ , c − k, ↓ , c † k + Q, ↑ , c − k − Q, ↓ (cid:17) is given by, G − ( iω n , k ) = iω n − ξ k − ∆ k − χ k − ∆ ∗ k iω n + ξ k χ k − χ ∗ k iω n − ξ k + Q − ∆ k + Q χ ∗ k − ∆ ∗ k + Q iω n + ξ k + Q , (S1)where, ξ k is the electronic dispersion dispersion, given by, ξ k = 2 t [cos k x +cos k y ]+4 t cos( k x ) cos( k y )+2 t (cos(2 k x )+cos(2 k y )) − µ , with t = − .
25 meV, t = 34 .
75 meV, t = −
11 meV and µ = −
89 meV. In this paper, all energyscales are expressed in units of t . ∆ k is the SC order parameter and χ k is the CDW order parameter with finitewave-vector Q . ω n is the Matsubara frequency. Ψ is the Nambu spinor, c † k, ↑ is the creation operator for an electronicstate with wave-vector k and up spin and c − k, ↓ is the annihilation operator for an electronic state with wave-vector -kand down spin. The Fermi-surface for the electronic dispersion ξ k , ‘hot-spots’ and the CDW wave vectors (Q) parallelto the crystallographic axes are shown in Fig. S1. FIG. S1: Fermi surface for a prototype cuprate band structure. The solid black curves represent the Fermi-surface associatedto the dispersion ξ k . The dashed black lines represent the magnetic Brillouin zone boundary of the system. The intersection ofthe magnetic Brillouin zone boundary and the Fermi-surface are marked by red dots, representing the ‘hot-spots’ on the Fermi-surface. The CDW modulation wave vector Q, indicated by the arrows are considered to be parallel to the crystallographicaxes are shown in the figure. DYSON EQUATIONS AND CALCULATION OF THE SELF-ENERGY Π In this section ,we present the derivation of the phonon propagators, re-normalized due to CDW and SC orders.The free phonon is given by the propagator D ( z, q ) = 2 ω q / ( z − ω q ), where ω q is the frequency of the phonon mode a r X i v : . [ c ond - m a t . s up r- c on ] S e p for wave-vector q and z is a complex frequency (Im z > D m,n ( q, τ ) = −hT φ q + mQ ( τ ) φ † q + nQ (0) i , where T is the time-ordering operator, and m, n = ± . By notingthat D ++ ≡ D −− := D ( z, q ) and D + − ≡ D − + := D ( z, q ), within a perturbative approach for the electron-phononinteraction in Hamiltonian of Eq. (1) in the main text, the Dyson’s equations involving self-energies in the presenceof SC and CDW orders will give the modified phonon propagators D , D and can be written as, D ( z, q ) = D ( z, q + Q ) (cid:20) ( z, q ) D ( z, q ) + Π ( z, q ) D ( z, q ) + Π ( z, q ) D ( z, q ) + Π ( z, q ) D ( z, q ) (cid:21) , (S2) D ( z, q ) = D ( z, q − Q ) (cid:20) Π ( z, q ) D ( z, q ) + Π ( z, q ) D ( z, q ) + Π ( z, q ) D ( z, q ) + Π ( z, q ) D ( z, q ) (cid:21) . The self-energies Π , Π , Π , Π , Π , Π , Π and Π in Eq. S2, are given by,Π ( ω, q ) = g N X k,iω n [ G ( k, iω n ) G ( k + q, iω n + i(cid:15) n ) + ( k → k − q )] (S3)Π ( ω, q ) = g N X k,iω n [ G ( k, iω n ) G ( k + q, iω n + i(cid:15) n ) + ( k → k − q )]Π ( ω, q ) = g N X k,iω n [ G ( k, iω n ) G ( k + q, iω n + i(cid:15) n ) + ( k → k − q )]Π ( ω, q ) = g N X k,iω n [ G ( k, iω n ) G ( k + q, iω n + i(cid:15) n ) + ( k → k − q )]Π ( ω, q ) = g N X k,iω n [ G ( k, iω n ) G ( k + q, iω n + i(cid:15) n ) + ( k → k − q )]Π ( ω, q ) = g N X k,iω n [ G ( k, iω n ) G ( k + q, iω n + i(cid:15) n ) + ( k → k − q )]Π ( ω, q ) = g N X k,iω n [ G ( k, iω n ) G ( k + q, iω n + i(cid:15) n ) + ( k → k − q )]Π ( ω, q ) = g N X k,iω n [ G ( k, iω n ) G ( k + q, iω n + i(cid:15) n ) + ( k → k − q )] . With a further assumption of small q, we note that Π ≈ Π , Π ≈ Π , Π ≈ Π and Π ≈ Π , which gives the finalform of the Dyson’s equations as D ( z, q ) = D ( z, q + Q ) (cid:20) ( z, q ) D ( z, q ) + Π ( z, q ) D ( z, q ) + Π ( z, q ) D ( z, q ) + Π ( z, q ) D ( z, q ) (cid:21) , (S4) D ( z, q ) = D ( z, q − Q ) (cid:20) Π ( z, q ) D ( z, q ) + Π ( z, q ) D ( z, q ) + Π ( z, q ) D ( z, q ) + Π ( z, q ) D ( z, q ) (cid:21) . The corresponding Feynman-diagrams for the above self-energies in Eq. (S4) are shown in Fig. (2) of the main text.In Eq. (S4), we consider the strength of electron-phonon interaction, g to be k -independent and the number of latticesites in the system to be N. The Dyson’s equations from D and D can be decoupled to obtain the new re-normalizedphonon modes, by introducing D ± ( z, q ) = D ( z, q ) ± D ( z, q ) and then solve for D ± . The solution for the frequenciesof the new phonon modes are, Ω ± ( q ) = ω Q + 2 ω Q Π ± ( q ) , (S5)where, Ω ± ( q ) represent the re-normalized frequencies [also given in Eq. (5) of the main text], and Π + = Π + Π +Π + Π and Π − = Π + Π − Π − Π . The Green’s function matrix elements [ G ( i, j )] that are appearing in theself-energy expressions in Eq. (S4), are given by, G ( k, iω n ) = A ( iω n + E − k ) + A ( E − k − iω n ) + A ( E + k + iω n ) + A ( E + k − iω n ) , (S6) G ( k, iω n ) = A ( iω n + E − k ) + A ( E − k − iω n ) + A ( E + k + iω n ) + A ( E + k − iω n ) ,G ( k, iω n ) = A ( iω n + E − k ) + A ( E − k − iω n ) + A ( E + k + iω n ) + A ( E + k − iω n ) ,G ( k, iω n ) = A ( iω n + E − k ) + A ( E − k − iω n ) + A ( E + k + iω n ) + A ( E + k − iω n ) ,G ( k, iω n ) = A ( iω n + E − k ) + A ( E − k − iω n ) + A ( E + k + iω n ) + A ( E + k − iω n ) ,G ( k, iω n ) = A ( iω n + E − k ) + A ( E − k − iω n ) + A ( E + k + iω n ) + A ( E + k − iω n ) ,G ( k, iω n ) = A ( iω n + E − k ) + A ( E − k − iω n ) + A ( E + k + iω n ) + A ( E + k − iω n ) ,G ( k, iω n ) = A ( iω n + E − k ) + A ( E − k − iω n ) + A ( E + k + iω n ) + A ( E + k − iω n ) , where, E ± k is the re-normalized electronic dispersion and is given by, E ± k = ± √ q β k − η k (S7)with, β k = E k + E k + ∆ k + ∆ k + 2 χ k ,η k = [( E k + E k )( E k − E k ) + (∆ k + ∆ k )(∆ k − ∆ k )] + 4 χ k (cid:2) ( E k + E k ) + (∆ k − ∆ k ) (cid:3) , where we use the following simplified notations, ξ k = E k , ξ k + Q = E k , ∆ k = ∆ k and ∆ k + Q = ∆ k . In the righthand side of the Eq. (S6), the numerators are given by, A = ( E k − E − k )( E k − ( E − k ) + ∆ k ) − ( E k + E − k ) χ k E − k ( E − k − E + k )( E − k + E + k ) ,A = ( E k + E − k )( E k − ( E − k ) + ∆ k ) + ( − E k + E − k ) χ k E − k ( E − k − E + k )( E − k + E + k ) A = − ( E k − E + k )( E k − ( E + k ) + ∆ k ) + ( E k + E + k ) χ k E + k ( E − k − E + k )( E − k + E + k ) ,A = − ( E k + E + k )( E k − ( E + k ) + ∆ k ) + ( E k − E + k ) χ k E + k ( E − k − E + k )( E − k + E + k ) ,A = ( E k − E − k )( E k − ( E − k ) + ∆ k ) − ( E k + E − k ) χ k E − k ( E − k − E + k )( E − k + E + k ) ,A = ( E k + E − k )( E k − ( E − k ) + ∆ k ) + ( − E k + E − k ) χ k E − k ( E − k − E + k )( E − k + E + k ) ,A = − ( E k − E + k )( E k − ( E + k ) + ∆ k ) + ( E k + E + k ) χ k E + k ( E − k − E + k )( E − k + E + k ) A = − ( E k + E + k )( E k − ( E + k ) + ∆ k ) + ( E k − E + k ) χ k E + k ( E − k − E + k )( E − k + E + k ) , A = ∆ k ( E k − ( E − k ) + ∆ k ) + ∆ k χ k E − k ( E − k − E + k )( E − k + E + k ) ,A = ∆ k ( E k − ( E − k ) + ∆ k ) + ∆ k χ k E − k ( E − k − E + k )( E − k + E + k ) ,A = ( −
1) ∆ k ( E k − ( E + k ) + ∆ k ) + (∆ k ) χ k E + k ( E − k − E + k )( E − k + E + k ) ,A = ( −
1) ∆ k ( E k − ( E + k ) + ∆ k ) + (∆ k ) χ k E + k ( E − k − E + k )( E − k + E + k ) ,A = ∆ k ( E k − ( E − k ) + ∆ k ) + ∆ k χ k E − k ( E − k − E + k )( E − k + E + k ) ,A = ∆ k ( E k − ( E − k ) + ∆ k ) + ∆ k χ k E − k ( E − k − E + k )( E − k + E + k ) ,A = ( −
1) ∆ k ( E k − ( E + k ) + ∆ k ) + (∆ k ) χ k E + k ( E − k − E + k )( E − k + E + k ) ,A = ( −
1) ∆ k ( E k − ( E + k ) + ∆ k ) + (∆ k ) χ k E + k ( E − k − E + k )( E − k + E + k ) ,A = χ k [( E k − E − k )( − E k + E − k ) + ∆ k ∆ k + χ k ]2 E − k ( E − k − E + k )( E − k + E + k ) ,A = χ k [ − ( E k + E − k )( E k + E − k ) + ∆ k ∆ k + χ k ]2 E − k ( E − k − E + k )( E − k + E + k ) ,A = ( − χ k [( E k − E + k )( − E k + E + k ) + ∆ k ∆ k + χ k ]2 E + k ( E − k − E + k )( E − k + E + k ) ,A = χ k [( E k + E + k )( E k + E + k ) − ∆ k ∆ k − χ k ]2 E + k ( E − k − E + k )( E − k + E + k ) ,A = A ,A = A ,A = A ,A = A ,A = ( − χ k [( E k + E − k )(∆ k ) + E k ∆ k − E − k ∆ k ]2 E − k ( E − k − E + k )( E − k + E + k ) ,A = ( − χ k [( E k − E − k )(∆ k ) + E k ∆ k + E − k ∆ k ]2 E − k ( E − k − E + k )( E − k + E + k ) ,A = χ k [( E k + E + k )(∆ k ) + E k ∆ k − E + k ∆ k ]2 E + k ( E − k − E + k )( E − k + E + k ) ,A = χ k [( E k − E + k )(∆ k ) + E k ∆ k + E + k ∆ k ]2 E + k ( E − k − E + k )( E − k + E + k ) , A = ( − χ k [( E k + E − k )(∆ k ) + E k ∆ k − E − k ∆ k ]2 E − k ( E − k − E + k )( E − k + E + k ) ,A = ( − χ k [( E k − E − k )(∆ k ) + E k ∆ k + E − k ∆ k ]2 E − k ( E − k − E + k )( E − k + E + k ) ,A = χ k [( E k + E + k )(∆ k ) + E k ∆ k − E + k ∆ k ]2 E + k ( E − k − E + k )( E − k + E + k ) ,A = χ k [( E k − E + k )(∆ k ) + E k ∆ k + E + k ∆ k ]2 E + k ( E − k − E + k )( E − k + E + k ) . To evaluate the self-energies (Π) in Eq. (S3), we first perform the summation over the Matsubara frequency usinganalytic tool of contour integration. Next, we do the summation over k using numerical tools taking ∆ k = ∆ k + Q , g = 1 and N = 40000. The plot of the self-energy in this case is presented in Fig. (3) in the main text. CALCULATION OF SELF-ENERGY Π IN THE PRESENCE OF INVERSE LIFE-TIME Γ In this section, we present the self-energy calculation in the presence of finite inverse life-time (Γ) of the quasi-particles, associated to thermal fluctuations. Here, Green’s function elements become, G i,j ( iω n , k ) → G i,j ( iω n + i Γ sgn ( ω n ) , k ) . The self-energies Π in Eq. (S3) now have the following general structure: X k,iω n G ak ( iω n + i Γ sgn ( ω n )) G bk + q ( iω n + i Γ sgn ( ω n ) + i(cid:15) n ) , (S8)where, either of ‘a’ and ‘b’ symbolically represents the (i, j) th element of the Green’s function matrix. To evaluatethe Matsubara summation in this case, we need to use a contour avoiding the branch cuts defined by Im ( z ) = 0 and Im ( z + i(cid:15) n ) = 0 as shown in the Fig. S2. Using this contour, we arrive at the following integrations, FIG. S2: Contour for complex Matsubara frequency summation: Im ( z ) = 0 and Im ( z + i(cid:15) n ) = 0 denote the two branch-cutsin the complex plane. γ , γ and γ are the three contours of integration. I γ = X k (cid:20)Z ∞−∞ dω πi n F ( ω ) G ak ( ω + i Γ) G bk + q ( ω + (cid:15) + i Γ) (cid:21) ,I γ = − X k (cid:20)Z ∞−∞ dω πi n F ( ω ) G ak ( ω − (cid:15) − i Γ) G bk + q ( ω − i Γ) (cid:21) ,I γ = X k (cid:20)Z ∞−∞ dω πi n F ( ω ) (cid:8) G ak ( ω − (cid:15) − i Γ) G bk + q ( ω + i Γ) − G ak ( ω − i Γ) G bk + q ( ω + (cid:15) + i Γ) (cid:9)(cid:21) , (S9)where, n F ( ω ) = 1 / (exp βω + 1) is the Fermi-distribution function. Moreover β = 1 /k B T , where k B is the Boltzmannconstant.Next, in the limit T →
0, the integrals I γ and I γ in Eq. (S9) become, I γ = X k (cid:20)Z −∞ dω πi G ak ( ω + i Γ) G bk + q ( ω + (cid:15) + i Γ) (cid:21) I γ = − X k (cid:20)Z −∞ dω πi G ak ( ω − (cid:15) − i Γ) G bk + q ( ω − i Γ) (cid:21) . (S10)We replace ω → ω + (cid:15) in the first term of I γ and successively uselim (cid:15) → n F ( ω + (cid:15) ) − n F ( ω ) (cid:15) = − δ ( ω ) , where δ ( ω ) is a Dirac delta function with the property, R ∞−∞ dωf ( ω ) δ ( ω ) = f (0). Thus, I γ in Eq. (S9) becomes, I γ = X k − (cid:15) πi (cid:2) G ak ( i Γ) G bk + q ( (cid:15) + i Γ) (cid:3) . (S11)Finally, we evaluate the real frequency ( ω ) integration in Eq. (S10) for each of the four self-energies Π , Π , Π andΠ in Eq. (S3) by using, Z −∞ dω (cid:20) ω − x ) ∗ ( ω − y ) (cid:21) = log[ x ] − log[ y ] x − y , (S12)where, x, y ∈ C . The summation over kk