Revival of Charge Density Waves and Charge Density Fluctuations in Cuprate High-Temperature Superconductors

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Review Revival of Charge Density Waves and Charge Density Fluctuations in Cuprate High-Temperature Superconductors

Carlo Di Castro

Dipartimento di Fisica, Università “Sapienza”. P.le A. Moro 2, I-00185 Rome, Italy; [email protected]

Received: 28 September 2020; Accepted: 28 October 2020; Published: 2 November 2020

Abstract: I present here a short memory of my scientific contacts with K.A. Müller starting from the Interlaken Conference (1988), Erice (1992 and 1993), and Cottbus (1994) on the initial studies on phase separation (PS) and charge inhomogeneity in cuprates carried out against the view of the majority of the scientific community at that time. Going over the years and passing through the charge density wave (CDW) instability of the correlated Fermi liquid (FL) and to the consequences of charge density fluctuations (CDFs), I end with a presentation of my current research activity on CDWs and the related two-dimensional charge density fluctuations (2D-CDFs). A scenario follows of the physics of cuprates, which includes the solution of the decades-long problem of the strange metal (SM) state.

Keywords: high-temperature superconductivity; cuprates; correlated Femi liquid; charge density wave; ﬂuctuation; strange metal

1. Introduction My scientific familiarity with Alex Müller goes back to the 1970s when I was involved in the research on critical phenomena. His contributions to this field could not be overlooked by any researcher active in the field. His research was fundamental in bringing the static and dynamical behavior of structural phase transition and ferroelectrics within the mainstream of critical phenomena [1]. Again, in the 1970s, he together with Harry Thomas presented a theory of structural phase transitions in which the lowering of symmetry comes from the Jahn–Teller effect of the ionic constituents [2]. That paper, in addition to showing the versatile nature and the usual deep understanding of the physics by Alex, is also the paper to which he was often referring when discussing his approach in searching high-temperature superconductivity. In 1986, his discovery with J. G. Bednorz [3] of high critical temperature superconductors (HTSCs) in barium-doped lanthanum copper oxides with the critical temperature TC in the thirty Kelvin range realized the chimera of experimental physicists for decades. Their paper was quickly followed by the discovery of various families of cuprate superconductors, with critical temperature values well above the liquefaction temperature of nitrogen. At the time, I was studying with C. Castellani another important problem in condensed matter—the generalization to correlated electrons of the metal–insulator transition resulting from the Anderson localization due to disorder. The problematic of interacting disordered electronic systems was still a very hot topic, but it was overshadowed by the new problematic of HTSCs. The importance of electron correlation for the study of these materials became immediately apparent, in particular under the input offered by P.W. Anderson [4]. Those who, like us, had been working on strongly correlated electron systems, could not escape the temptation to work on this new type of superconductor. Then, my adventure on HTSCs started and my friendly relations with Alex became direct and deeper.

Condens. Matter 2020, 5, 70; doi:10.3390/condmat5040070 www.mdpi.com/journal/condensedmatter Condens. Matter 2020, 5, x FOR PEER REVIEW 2 of 17 correlated electron systems, could not escape the temptation to work on this new type of superconductor.Condens. Matter 2020 ,Then,5, 70 my adventure on HTSCs started and my friendly relations with Alex became2 of 16 direct and deeper. It will be apparent how much our scientific community is indebted to Alex and how much our It will be apparent how much our scientific community is indebted to Alex and how much our research activity in this field, mine in particular, was inspired and supported by him. I therefore research activity in this field, mine in particular, was inspired and supported by him. I therefore consider it my duty and pleasure to dedicate this work to him. My aim is to retrace the line we consider it my duty and pleasure to dedicate this work to him. My aim is to retrace the line we followed followed over the years in Rome to produce a scenario of the physics of cuprates [5]. I proceed over the years in Rome to produce a scenario of the physics of cuprates, for a recent review see [5]. through the years to end up with two recent contributions [6,7] produced with S. Caprara, M. Grilli I proceed through the years to end up with two recent contributions [6,7] produced with S. Caprara, M. and G. Seibold in a collaboration with the experimental group of the Politecnico of Milano on Grilli and G. Seibold in a collaboration with the experimental group of the Politecnico of Milano on resonant X-ray scattering (RXS) (G. Ghiringhelli, L. Braicovich, R. Arpaia, among others) in which resonant X-ray scattering (RXS) (G. Ghiringhelli, L. Braicovich, R. Arpaia, among others) in which charge density fluctuations (CDFs) are thoroughly investigated leading to a deeper understanding of charge density fluctuations (CDFs) are thoroughly investigated leading to a deeper understanding of the physics of these systems. I have no claim of completeness; I only follow my line of reasoning for the physics of these systems. I have no claim of completeness; I only follow my line of reasoning for which no one else is responsible even though I am deeply indebted to Alex Müller for his continuous which no one else is responsible even though I am deeply indebted to Alex Müller for his continuous support in the first fifteen years of my research activity in this field and to all my collaborators support in the first fifteen years of my research activity in this field and to all my collaborators throughout the years. throughout the years. 2. Results 2. Results

2.1.2.1. General General Properties Properties LetLet me me recall recall the the general properties properties of of the cupr cupratesates by by referring to their generic phase diagram (Figure 1a). All these materials are made up of copper–oxygen planes (CuO2), a quasi-bidimensional (Figure1a). All these materials are made up of copper–oxygen planes (CuO 2), a quasi-bidimensional structurestructure intercalated intercalated with with layers layers of of rare rare earth earth (lanthanum, (lanthanum, yttriu yttrium,m, barium, barium, etc., etc., depending depending on the on various families of cuprates, Figure 1b). In their stoichiometric composition (e.g., La2CuO4), these the various families of cuprates, Figure1b). In their stoichiometric composition (e.g., La 2CuO4), materialsthese materials are antiferromagnetic are antiferromagnetic (AF) (AF)Mott Mott insulators insulators,, despite despite the the fact fact that that they they have have one one hole hole per per CuO2 unit cell in the copper–oxygen plane and should have been metals. Hence, there is the need for CuO2 unit cell in the copper–oxygen plane and should have been metals. Hence, there is the need for aa strong strong correlation correlation between charge carriers, holes in this case.

(a) (b)

Figure 1. (a) Generic phase diagram of cuprates of temperature T vs. doping p. Blue region: antiferromagnetic Figure(AF) phase. 1. (a Red) Generic region: phase superconductive diagram of phase. cupratesT* smeared of temperature gray line: pseudogapT vs. doping line. p. ( bBlue) Structure region: of

antiferromagneticcuprates, e.g., La2 (AF)xSrx CuOphase.2 (LSCO). Red region: superconductive phase. T* smeared gray line: pseudogap − line. (b) Structure of cuprates, e.g., La2−xSrxCuO2 (LSCO). Upon substitution of the rare earth with heterovalent dopants (e.g., strontium replacing lanthanum in LSCO),Upon thesubstitution number of holesof the increases rare earth in the with CuO 2heplaneterovalent and the dopants system becomes(e.g., strontium a metal, albeitreplacing a bad lanthanummetal. Nevertheless, in LSCO), the number system becomesof holes increases a superconductor, in the CuO with2 plane d-wave and the symmetry, system becomes when the a metal,temperature albeit isa lowered bad metal. below Nevertheless, a dome-shaped the doping-dependent system becomes critical a superconductor, temperature TC (p)with(red d-wave region symmetry,in Figure1a). whenTC reaches the temperature a maximum is at lowered the so-called below optimal a dome-shaped doping and doping-dependent decreases by lowering criticalp in temperaturethe underdoped TC(p) region (red region or increasing in Figurep in1a). the TC so-called reaches a overdoped maximum region, at the so-called where the optimal system doping is well anddescribed decreases by the by normal lowering Fermi p in liquid the underdoped (FL) theory. On region the contrary, or increasing the metallic p in the phase so-called in theunderdoped overdoped region,region andwhere above the optimalsystem is doping well described is not in line by withthe no ordinaryrmal Fermi metals liquid and (FL) the predictionstheory. On ofthe the contrary, normal theFermi metallic liquid phase theory. in Inthe the underdoped metal phase, region a strong and anisotropy above optimal is measured doping inis thenot electricalin line with conductivity ordinary that, in the CuO2 plane, exceeds the transversal conductivity by orders of magnitude. Moreover, the Condens. Matter 2020, 5, 70 3 of 16 resistance displays a linear in T dependence (strange metal) over a wide range of temperatures above the doping-dependent pseudogap line T*(p)[8]. T*(p) decreases with p and merges with TC around optimal doping. Below T*(p), a pseudogap is formed with an even stronger violation of Fermi liquid (FL) behavior (Figure1). The pseudogap formation is apparently connected [ 9] with the suppression of quasiparticle (QP) states at the Fermi surface (FS) along the so-called antinodal (0,0)–(0.5,0) direction, as shown in Figure 9, and the consequent formation of Fermi arcs in the nodal region. In classic superconductors, the necessary attraction for the formation of electron pairs is mediated by phonons, as indicated by the so-called isotope effect, whereby the critical temperature for the onset of superconductivity depends on the isotopic mass of the lattice ion. As a further anomaly of these systems, the isotope effect is either absent or anomalous, for example, having a strong effect on T*[10]. An international debate thus ensued among theoretical physicists about whether a strongly correlated hole (or electron) system, with an occupation of approximately one hole per lattice site of a quasi-two-dimensional system, can become a superconductor, what the new paired state would be, and by what mechanisms it might be formed. At the same time, another, perhaps even more stimulating theoretical problem, presented itself—understanding the anomalous metallic phase to which the superconductivity was linked. I embarked upon this twofold adventure together with the condensed matter theory group of Rome.

2.2. Phase Separation—Spin and Charge Order The question we asked ourselves was how could a system, whose interaction between charges is strongly repulsive, generate the pairing between charges in general without referring to special symmetry conditions to reduce the repulsion? We then referred to the phase separation (PS), as it occurs in simple fluids. In cuprates, the separation occurs between charge-rich metallic regions and charge-poor zones. The strong local repulsion between charge carriers reduces their mobility and favors the possibility of phase separation. Theoretically, all of the models with strong local repulsion that were introduced to represent the copper–oxygen planes showed PS in charge-rich regions and in charge-poor regions (see, e.g., [11]). Besides the PS in various other models, we found that whenever PS is present, pairing then occurs in a nearby region of phase space, thus enforcing a possible connection between the two phenomena [12–14]. Charge inhomogeneity appeared to be a bridge that connects the low doping region (in the presence of strong repulsion necessary for an insulating antiferromagnet), the intermediate doping region (with a special attraction for superconductivity), up to a correlated FL at higher doping. Long-range Coulomb interaction, however, forbids the macroscopic phase separation. The system chooses a compromise and segregates charge on a shorter scale, while keeping charge neutrality at a large distance giving rise to the so-called frustrated phase separation [14–16]. Emery and Kivelson [15,16] achieved the frustrated PS concept starting from the low doping region near the antiferromagnetic insulating phase, in which the few present charges are expelled from the antiferromagnetic substrate and align in stripes. Spin fluctuations are dominating in this case. Coming instead from the high doping region, correlated FL in the presence of long-range Coulomb interaction undergoes to a Pomeranchuk instability towards a charge density wave (CDW) state with finite modulation vector qc [17]. This instability occurs along a line of temperature as a function of doping TCDW(p), ending at T = 0 in a quantum critical point (QCP) nearby optimal doping. In this case, charge fluctuations are prominent. Both charge and spin modes may act as mediators of pairing [18,19]. Experimentally, spin and charge order have been observed in two different forms:

- Stripes with strong interplay between charge and spin degrees of freedom (Figure2a) observed in the LSCO family by neutron scattering since 1995 [20]. Charge and spin incommensurate modulation vectors qc = 2π/λc and qs = 2π/λs are strictly related (qs = qc/2) one to another and align along the Cu-O bond. Condens. Matter 2020, 5, x FOR PEER REVIEW 4 of 17 Condens. Matter 2020, 5, x FOR PEER REVIEW 4 of 17 Condens.modulation Matter 2020, 5vectors, 70 qc = 2π/λc and qs = 2π/λs are strictly related (qs = qc/2) one to another and4 align of 16 modulationalong the Cu-O vectors bond. qc = 2π/λc and qs = 2π/λs are strictly related (qs = qc/2) one to another and align along the Cu-O bond. However,However, both both spin spin and and charge charge modulation modulation were we observedre observed simultaneously simultaneously in co-doped in co-doped LSCO LSCO only only However,(e.g., La1.48 Ndboth0.4 Srspin0.12CuO and 4charge); otherwise, modulation there was we reno observed detection simultaneously of charge modulation in co-doped (only LSCOspin). (e.g., La1.48Nd0.4Sr0.12CuO4); otherwise, there was no detection of charge modulation (only spin). only (e.g., La1.48Nd0.4Sr0.12CuO4); otherwise, there was no detection of charge modulation (only spin).

-- ChargeCharge DensityDensity WaveWave (Figure(Figure2 b).2b). Charge Charge and and spin spin degrees degrees of of freedom freedom evolve evolve independently independently - Charge Density Wave (Figure 2b). Charge and spin degrees of freedom evolve independently oneone fromfrom anotheranother [[21,22].21,22]. CDWs, foreseen since 1995 [[17],17], werewere untiluntil 20112011 elusive,elusive, contrasted,contrasted, one from another [21,22]. CDWs, foreseen since 1995 [17], were until 2011 elusive, contrasted, andand conﬁrmedconfirmed onlyonly indirectly.indirectly. Due to the improvement of resonantresonant X-rayX-ray scatteringscattering (RXS),(RXS), and confirmed only indirectly. Due to the improvement of resonant X-ray scattering (RXS), CDWsCDWs areare now now ubiquitously ubiquitously observed observed (see, e.g.,(see, [21 e.g.,–28]) [21–28]) in cuprate in families, cuprate and families, most investigated and most CDWs are now ubiquitously observed (see, e.g., [21–28]) in cuprate families, and most amonginvestigated the orders among competing the orders with competing SC in countlesswith SC in papers countless following papers thefollowing pioneering the pioneering paper on investigated among the orders competing with SC in countless papers following the pioneering YBCOpaper on by GhiringhelliYBCO by Ghir etinghelli al. [23]. et al. [23]. paper on YBCO by Ghiringhelli et al. [23].

(a) (b) (a) (b)

Figure 2. (a) Stripes: charge and spin modulation vectors qc and qs are connected. (b) The different Figure 2. (a) Stripes: charge and spin modulation vectors qc and qs are connected. (b) The different modulationFiguremodulation 2. (a) of ofStripes: charge charge charge density density and waves waves spin (CDWs) modulation (CDWs) with respect withvectors respect to q stripesc and to q iss stripesare shown. connected. is In shown. this ( case,b) TheIn CDWs this different case, and SDWsmodulationCDWs evolve and ofSDWs independently charge evolve density independently one waves from another (CDWs) on (see,e fromwith e.g., respect [another21,22]). to(see, stripes e.g., is[21,22]). shown. In this case, CDWs and SDWs evolve independently one from another (see, e.g., [21,22]). AsAs shownshown inin FigureFigure3 3,, (i) (i) energy-integrated energy-integrated CDW CDW peaks peaks are are along along the the Cu-O Cu-O bond, bond, (0.31,0) (0.31,0) and and (0,031)(0,031)As inin shown CuOCuO22 planes;inplanes; Figure (ii)(ii) 3, thethe (i) widthenergy-integrated decreasesdecreases byby CDloweringloweringW peaks TT andand are thealong correlation the Cu-O lengthlength bond, increasesincreases (0.31,0) and butbut remains(0,031)remains in finitefinite CuO (less2(less planes; thanthan (ii) 16a16a the latticelattice width constant), constant),decreases sosoby that thloweringat thethe fluctuatingfluctuating T and the correlation CDWsCDWs tendtend length toto becomebecome increases criticalcritical but butremainsbut nevernever finite succeed,succeed, (less andthanand for for16a thisthis lattice reasonreason constant), theythey werewere so th calledcalledat the Quasi-CriticalfluctuatingQuasi-Critical CDWs dynamicaldynamical tend fluctuatingtofluctuating become CDWscriticalCDWs (QC-CDWs);but(QC-CDWs never succeed,); and and (iii) (iii) andthe the QC-CDWforQC-CDW this reasonpeak peak istheyis increasingincreasing were calledby by loweringlowering Quasi-CT Tritical downdown dynamical toto TcTc,, whilewhile fluctuating itit isis decreasingdecreasing CDWs for(forQC-CDWs TT << TC,, i.e., i.e.,); and the the (iii) incipientincipient the QC-CDW CDW CDW phase peak transition is increasing is preempted by lowering by the T SC down transition. to Tc, while it is decreasing for T < TC, i.e., the incipient CDW phase transition is preempted by the SC transition.

FigureFigure 3.3. AdaptedAdapted fromfrom [[23].23]. Energy-integratedEnergy-integrated resonantresonant X-ray scatteringscattering (RXS)(RXS) forfor underdopedunderdoped YBa2Cu3O6.6FigureYBa2Cu 3.3O Adapted6.6 (T (cT =c 61= from 61K) K)as [23]. asa function a functionEnergy-integrated of the of thein-plane in-plane reso momentumnant momentum X-ray vector scattering vector at various at (RXS) various temperatures. for temperatures. underdoped The TheYBaarrows arrows2Cu indicate3O6.6 indicate (Tc the= 61 increasing theK) as increasing a function or decreasing or of decreasing the in-plane of the of peak themomentum peakby lowering by loweringvector the at temperature various the temperature temperatures. above aboveor below The or belowarrowsTc, respectively.T indicatec, respectively. the increasing or decreasing of the peak by lowering the temperature above or below Tc, respectively. The QC-CDW onset line TQC-CDW(p) starts around optimal doping and initially follows the pseudogap line T*(p) (see Figure4).

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The QC-CDW onset line TQC-CDW(p) starts around optimal doping and initially follows the pseudogapCondens. Matter line2020 T, *(5,p 70) (see Figure 4). 5 of 16

Figure 4. AdaptedAdapted from from [21,22]. [21,22]. Phase Phase diagram diagram TT vs.vs. do dopingping for for YBCO. YBCO. Red Red line line:: Quasi-Critical-CDW Quasi-Critical-CDW

TQCCDW. Black. Black line: line: pseudogappseudogapT T*.*. Black Black dashed dashed line: line: superconductive superconductive critical critical temperature. temperature. Blue region Blue regionbelow thebelow superconductive the superconductive dome: sketched dome: 3Dsketched long-range 3D long-range CDW in the CDW presence in the of high presence magnetic of high ﬁeld. magneticThe two blue field. dots: The 3D two extrapolated blue dots: quantum3D extrapolat criticaled points quantum (QCPs). critical Red points dot: the (QCPs). extrapolated Red dot: QCP the of extrapolatedthe QC-CDW. QCP of the QC-CDW.

In the presence of a sufficiently high magnetic field to suppress superconductivity, a static 3D In the presence of a sufficiently high magnetic field to suppress superconductivity, a static 3D transition is unveiled (blue region in Figure4), mainly in YBCO, by NMR [ 29,30], quantum-oscillation, transition is unveiled (blue region in Figure 4), mainly in YBCO, by NMR [29,30], quantum-oscillation, negative Hall, and FS reconstruction [31–38], and finally by thermodynamic 3D transition by pulsed negative Hall, and FS reconstruction [31–38], and finally by thermodynamic 3D transition by pulsed 0 ultrasound [39]. Zero field QC-CDW and 3D long-range charge order with two QCPs (pc’ < pc , pc p*) ultrasound [39]. Zero field QC-CDW and 3D long-range charge order with two QCPs (pc’ < pc ≠ ≈pc0 ≈ have the same in-plane incommensurate vector qc, suggesting a common origin. p*) have the same in-plane incommensurate vector qc, suggesting a common origin. 3. A Digression for Alex Müller and My Memories 3. A digression for Alex Müller and My Memories Actually, the community was and has been for decades strongly against the idea of phase Actually, the community was and has been for decades strongly against the idea of phase separation and of non-uniform charge in general. Most researchers were referring to spin fluctuations separation and of non-uniform charge in general. Most researchers were referring to spin fluctuations due to the vicinity of the AF-QCP in the low-doped Mott insulator [4,19,40–43]. Charge order has been due to the vicinity of the AF-QCP in the low-doped Mott insulator [4,19,40–43]. Charge order has at most considered as a byproduct of spin order. For instance, at the first big international conference been at most considered as a byproduct of spin order. For instance, at the first big international on HTSCs of 1988 held in Interlaken, following my remark on the possible interconnection between conference on HTSCs of 1988 held in Interlaken, following my remark on the possible interconnection different charge density values that could coexist in these materials [44], P.W. Anderson argued that between different charge density values that could coexist in these materials [44], P.W. Anderson “simple calculations” show it to be impossible. Alex Müller, instead, trusted us from the beginning, argued that “simple calculations” show it to be impossible. Alex Müller, instead, trusted us from the and gave a strong support to the community that was inquiring about charge inhomogeneity in beginning, and gave a strong support to the community that was inquiring about charge cuprates. He organized several conferences on PS (see, e.g., Erice in 1992 (Figures5 and6) and Cottbus inhomogeneity in cuprates. He organized several conferences on PS (see, e.g., Erice in 1992 (Figures in 1993). 5 and 6) and Cottbus in 1993). In the proceedings [45,46], our work carried out in Rome, the one by Emery and Kivelson, and the one by Sigmund, Hiznyakov, and Seibold on PS of that early period were summarized. A comprehensive presentation of the experimental situation was also given. The task of keeping the members of this community linked was carried out also by the three editors of this volume, in particular by Antonio Bianconi with the Superstripes organization. Alex often recalled the scientific contributions of Annette Bussman-Holder, Hugo Keller, and Antonio Bianconi (see, e.g., the distorted octahedrons of Antonio that, according to Alex, can be assigned to Jahn–Teller polarons, whereas the undistorted ones are located within a metallic cluster or stripe [47]).

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The idea of charge inhomogeneity corresponds, at least in part, to the idea of strong polarization that guided Müller in his discovery of high-temperature superconductivity. I have been corresponding with Alex extensively on the subject, for many years. I presented the “elogio” on the occasion of his Laurea Honoris Causa in Rome in November 1990 (Figure7). Furthermore, I was later called (1997) to Cottbus for the equivalent ceremony. In Cottbus, we experienced an episode that is unusual for meetings among physicists. After the ceremony, the neo-laureate with his wife as well as my wife and myself were taken to a party in a villa–castle located in the outskirts of Cottbus. We were proceeding in a Mercedes along a highway, when the car stopped; we then got into a seventeenth century-like chariot with the coachmen in “livrea” (livery). Cars were speeding around us, and we ﬁnally arrived in the villa, where we were received by a string quartet playing classical-style music, organized for Alex by the noble family who had recently repossessed the villa. Condens. Matter 2020, 5, x FOR PEER REVIEW 6 of 17 Condens. Matter 2020, 5, x FOR PEER REVIEW 6 of 17

Figure 5. Erice, 6–12 May 1992, Workshop on Phase Separation in cuprate Superconductors. FigureFigure 5. Erice, 6–12 May 1992, Workshop on Phase Separation in cupratecuprate Superconductors.

Figure 6. Erice, July 1995. E. Sigmund, C. Di Castro, and G. Benedek with K.A. Müller in the center, FigureFigure 6.6. Erice,Erice, JulyJuly 1995.1995. E. Sigmund, C. Di Castro, and G. G. Benedek Benedek with with K.A. K.A. Müller Müller in in the the center, center, looking at a gift for him, a book on rare cars, one of his hobbies. lookinglooking at at a a gift gift for for him, him, a a book book on rare cars, one of his hobbies. In the proceedings [45,46], our work carried out in Rome, the one by Emery and Kivelson, and In the proceedings [45,46], our work carried out in Rome, the one by Emery and Kivelson, and the one by Sigmund, Hiznyakov, and Seibold on PS of that early period were summarized. A the one by Sigmund, Hiznyakov, and Seibold on PS of that early period were summarized. A comprehensive presentation of the experimental situation was also given. The task of keeping the comprehensive presentation of the experimental situation was also given. The task of keeping the members of this community linked was carried out also by the three editors of this volume, in members of this community linked was carried out also by the three editors of this volume, in particular by Antonio Bianconi with the Superstripes organization. Alex often recalls the scientific particular by Antonio Bianconi with the Superstripes organization. Alex often recalls the scientific contributions of Annette Bussman-Holder, Hugo Keller, and Antonio Bianconi (see, e.g., the distorted contributions of Annette Bussman-Holder, Hugo Keller, and Antonio Bianconi (see, e.g., the distorted octahedrons of Antonio that, according to Alex, can be assigned to Jahn–Teller polarons, whereas the octahedrons of Antonio that, according to Alex, can be assigned to Jahn–Teller polarons, whereas the undistorted ones are located within a metallic cluster or stripe [47]). undistorted ones are located within a metallic cluster or stripe [47]). The idea of charge inhomogeneity corresponds, at least in part, to the idea of strong polarization The idea of charge inhomogeneity corresponds, at least in part, to the idea of strong polarization that guided Müller in his discovery of high-temperature superconductivity. that guided Müller in his discovery of high-temperature superconductivity. I have been corresponding with Alex extensively on the subject, for many years. I presented the I have been corresponding with Alex extensively on the subject, for many years. I presented the “elogio” on the occasion of his Laurea Honoris Causa in Rome in November 1990 (Figure 7). “elogio” on the occasion of his Laurea Honoris Causa in Rome in November 1990 (Figure 7). Furthermore, I was later called (1997) to Cottbus for the equivalent ceremony. Furthermore, I was later called (1997) to Cottbus for the equivalent ceremony.

Condens. Matter 2020, 5, 70 7 of 16 Condens. Matter 2020, 5, x FOR PEER REVIEW 7 of 17

Figure 7. K.A. Müller. Laurea Honoris Causa, Rome 1990.

4. CorrelatedIn Cottbus, Fermi we experienced Liquid and CDWan episode Fluctuations that is unusual for meetings among physicists. After the ceremony, the neo-laureate with his wife as well as my wife and myself were taken to a party in a 4.1. Correlated Fermi Liquid Instability Producing a Charge Density Wave State villa–castle located in the outskirts of Cottbus. We were proceeding in a Mercedes along a highway, whenLet the us car see stopped; how the scenariowe then presentedgot into a inseventeenth Section2 arises century-like from our chariot old and with new the stories. coachmen in “livrea”The (livery). proposal Cars of a were CDW speeding instability around in cuprates us, and is quitewe finally old [17 arrived] and it in is the based villa, on where the mechanism we were receivedof frustrated by a phasestring separationquartet playing discussed classical-style above. music, organized for Alex by the noble family who had recentlyAt high doping,repossessed as it hasthe beenvilla. already mentioned, strong correlations weaken the metallic character of the system giving rise to a correlated FL with a substantial enhancement of the QP mass m* and 4.a sizableCorrelated repulsive Fermi residual Liquid interaction. and CDW Fluctuations The presence of weak or moderate additional attraction due to phonons reduces this residual interaction and may provide a mechanism that induces an electronic PS with4.1. Correlated charge segregation Fermi Liquid on Instability a large scale, Producing i.e., a diverginga Charge Density compressibility Wave Stateκ , the static density-density responseLet us function see how at theq = scenario0: presented in Section 2 arises from our old and new stories. The proposal of a CDW instability in cuprates is quite old [17] and it is based on the mechanism κ = limχρρ(q, ω = 0) = lim< ρ(q, ω = 0)ρ( q, ω = 0) > (1) of frustrated phase separationq 0 discussed above.q 0 − → ∞ → → At high doping, as it has been already mentioned, strong correlations weaken the metallic Thecharacter FL expression of the system of thegiving compressibility rise to a correlatedκ in terms FL with of a the substantial static Γq enhancementand dynamic ofΓ ωthescattering QP mass m*amplitudes and a sizable is as follows: repulsive residual interaction. The presence of weak or moderate additional attraction due to phonons reduces this residual interaction and may provide a mechanism that κ = 2N(0)/(1 + 2N(0)Γω) = 2N(0)(1 2N(0)Γq) as q 0 (2) induces an electronic PS with charge segregation on a large− scale, i.e.,→ a ∞diverging→ compressibility 𝜅, the static density-density response function at q = 0: From equation (2), the Pomeranchuk condition for the instability of the FL reads as follows: κ=lim𝜒(𝑞, 𝜔 = 0) = lim 𝜌(𝑞, 𝜔 = 0)𝜌(−𝑞, 𝜔=0)>→∞ → → (1) 2N(0)Γω = 1, Γq (3) − → −∞ The FL expression of the compressibility κ in terms of the static Γq and dynamic Γω scattering Theamplitudes static scattering is as follows: amplitude diverges at q = 0. With a long-range Coulomb interaction, it would cost too much in energy to fully segregate the 𝜅 = 2𝑁(0)/(1 + 2𝑁(0)Γ) = 2𝑁(0)(1 − 2𝑁(0)Γ) → ∞ 𝑎𝑠 𝑞 →0 (2) charge and the system phase separates on a shorter scale. The Fermi liquid instability coming from the Fromoverdoped equation region (2), occurs the Pomeranchuk at ﬁnite modulation condition vector for q the= q cinstabilityleading to of an the incommensurate FL reads as follows: charge density wave state with wavelength λc = 2π/qc. qc is not related to nesting but is determined by the balance 2𝑁(0)Γ =−1, Γ →−∞ (3) between PS tendency and Coulomb energy cost [17]. The static scattering amplitude diverges at q = 0. With a long-range Coulomb interaction, it would cost too much in energy to fully segregate the charge and the system phase separates on a shorter scale. The Fermi liquid instability coming from the overdoped region occurs at finite modulation vector q = qc leading to an incommensurate charge

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density save state with wavelength 𝜆 =2𝜋/𝑞. qc is not related to nesting but is determined by the Condens.balance Matter between2020, 5PS, 70 tendency and Coulomb energy cost [17]. 8 of 16 A strong scattering increase has been indeed measured in YBCO at optimal doping [48], as it is requiredA strong by Equation scattering (3) increase but this hastime been at a indeedfinite q. measured in YBCO at optimal doping [48], as it is required by Equation (3) but this time at a ﬁnite q. 4.2. Theoretical Realization 4.2. TheoreticalThe general Realization ideas of the previous section are implemented within a correlated FL theory describedThe general by a ideasHubbard–Holstein of the previous model section with are implemented large local repulsion within a correlated and a long-range FL theory describedCoulomb byinteraction a Hubbard–Holstein Vc(q). The values model of withthe hopping large local parameters repulsion ar ande derived a long-range from the Coulomb quasiparticle interaction band. VSlavec(q). TheBoson values technique of the hoppingwas used parameters to implement are derived the conditio fromn the of quasiparticle no double occupanc band. Slavey [17,49,50] Boson technique with the wasresult used that to implementstrong correlation the condition severely of noreduce doubles the occupancy kinetic [homogenizing17,49,50] with theterm. result This that technical strong correlationprocedure was severely indeed reduces shown the [51] kinetic to produce homogenizing a correlated term. FL with This a technical quasiparticle procedure dispersion was indeedhaving showna reduced [51] hopping to produce and a a correlated residual density–density FL with a quasiparticle interaction dispersion V(q) between having the a reduced quasiparticles: hopping and a residual density–density interaction V(q) between the quasiparticles: 𝑉(𝑞) =𝑉 (𝑞) + 𝑈(𝑞) −𝜆 (4)

V(q) = Vc(q) + U(q) λ (4) where U(q) is the short-range residual repulsion stemming− from large U of the one-band Hubbard wheremodelU and(q) isλ is the the short-range coupling of residual the Holstein repulsion e-ph stemming interaction from leading the largeto the repulsion thermodynamic of the one-band PS in the Hubbardabsence of model Vc. In and the λFL,is theV(q) coupling is screened of the in Holsteinthe random e-ph phase interaction approximation leading to via the the thermodynamic polarization due PS into the the absence repeated of particle–holeVc. In the FL, virtualV(q) is screenedprocesses in of the the random Lindhard phase polarization approximation bubble via Π the(q, polarizationω): due to the repeated particle–hole virtual processes of() the Lindhard polarization bubble Π(q, ω): 𝑉 = . (,)() (5) V(q) Vscr = . (5) The effects of strong correlations are contained1 + Π in(q the, ω) Vform(q) of V(q) and in the QP effective mass entering Π(q, ω). The dressed dynamical density–density correlation function 𝜒 = has The effects of strong correlations are contained in the form of V(q) and in the QP effective1+Π(𝑞,𝜔)𝑉(𝑞) mass entering the same denominator of Vscr. Π Π(q, ω). The dressed dynamical density–density correlation function χρρ = has the same Within the present approximation, the FL instability condition (Equation1+Π (2))(q,ω )towardsV(q) the CDW denominator of Vscr. is given by the vanishing of the denominator in Equation (5) at a finite modulation vector q = qc, Within the present approximation, the FL instability condition (Equation (2)) towards the CDW eventually marking a mean-field critical line as a function of doping p, 𝑇(𝑝) as it is sketched in is given by the vanishing of the denominator in Equation (5) at a finite modulation vector q = qc, Figure 8: 0 ( ) eventually marking a mean-field critical line as a function of doping p, TCDW p as it is sketched in Figure8: 1 + Π(𝑞, 𝜔)𝑉(𝑞) =0 (6) 1 + Π(q, ω)V(q) = 0 (6)

FigureFigure 8.8. CourtesyCourtesy ofof M.M. Grilli.Grilli. Sketch of the instabilityinstability condition of FL to formform CDW.CDW. DottedDotted line:line: CoulombCoulomb interaction.interaction. DashedDashed line:line: residualresidual repulsiverepulsive interaction. interaction.Red Redline: line: sumsum ofof thethe two.two. BlackBlack line:line: redred line line shiftedshifted by by thethe attractionattractionλ 𝜆.. Blue Blue line: line: inverseinverse LindhardLindhard polarizationpolarization bubble. bubble. TheThe FLFL instabilityinstability conditioncondition isis alsoalso shown.shown.

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U(q)U(q) is is less less repulsive repulsive and and the the hopping hopping largest largest along along the Cu-O the Cu-O direction direction ((0.5,0) ((0.5,0) and (0,0.5 and)). (0,0.5)).Mean- fieldMean-field CDW instability CDW instability of FL ofand FL modulation and modulation vector vector qc are q thereforec are therefore in this in direction this direction and anddetermine determine the hotthe regions hot regions of the of Fermi the Fermi surface surface (shown (shown in Figure in Figure 9) where9) where CDW CDWfluctuations fluctuations (CDFs) (CDFs) mediate mediate strong scatteringstrong scattering between between Fermi Fermi quasiparticles quasiparticles and andsuppress suppress quasiparticle quasiparticle states, states, forming forming the the so-called so-called Fermi arcs. We considerconsider thisthis asas thethe beginning beginning of of the the formation formation of of a pseudogapa pseudogap (see (see also also [9]); [9]); we we then then let letthe the mean-field mean-field CDW CDW critical critical line line start start at T at= 0T at= 0 the at samethe same doping dopingp* of p* the of extrapolated the pseudogap pseudogap line. line.

Figure 9. Red arrows: modulation vectors vectors q qcc ofof CDFs CDFs connecting two two branches of the quasiparticle Fermi surface (FS) in the antinodal regions of the FS. Blue line: nodal direction.

The formation formation of of gaped gaped electron electron states states near near th thee antinodal antinodal regions regions suggests suggests a pseudogap a pseudogap with with d- waved-wave symmetry symmetry as asthe the superconductive superconductive gap gap [9]. [ 9The]. The analysis analysis of the of thesupercondu superconductivective transition transition in the in presencethe presence of a ofdoping-dependent a doping-dependent normal normal state state pseudogap pseudogap with with d-wave d-wave symmetry symmetry [52] [52 accounts] accounts for for a a doping-dependent superconductive critical temperature Tc(p) and a Tc T* bifurcation near optimum doping-dependent superconductive critical temperature Tc(p) and a Tc −− T* bifurcation near optimum doping, within a d-Density-Wave scheme aa yearyear beforebefore aa moremore knownknown proposalproposal appearedappeared [[53].53]. Expanding in q-q and ω around the mean-field critical line T0 (p): Expanding in q-qc and ω around the mean-field critical line 𝑇CDW(𝑝): 𝜒 ∝𝐷 ≡𝑚1 +𝜈(𝑞−𝑞1 ) −𝑖𝜔𝛾,2 𝑚 =𝜈(𝜉0)2 ∝𝑇−𝑇0 (7) χ− D− m + ν(q qc) iωγ, m = ν(ξ )− T T (7) ρρ ∝ ≡ 0 − − 0 CDW ∝ − CDW where D in field theory is named the “propagator” of the fluctuation and m0 is its “bare” mass. whereEssentially,D in field Equation theory is named(7) represents the “propagator” a time-dependent of the fluctuation Landau–Ginzburg and m0 is(TDLG) its “bare” equation mass. in the GaussianEssentially, approximation: Equation (7) represents a time-dependent Landau–Ginzburg (TDLG) equation in the

Gaussian approximation:(,) =𝛾 ,𝐻𝜑= 𝑑𝑥𝑚 |𝜑| +𝜈|∇𝜑|, (7’) ( ) Z ∂ϕ x, t 1 δH ϕ 1 2 2 𝜑 = γ− , H ϕ = 𝛾 dx[m0 ϕ + ν ϕ ], (7’) where is the CDW field∂t (or fluctuationδϕ mode) and2 is the coefficient∇ of the Landau damping term 𝑖𝜔. whereFromϕ is an the inspection CDW field of (or Equation fluctuation (7), mode)it is natural and γ tois thedefine coe ffia characteristiccient of the Landau energy damping for each term q: iω. From an inspection of Equation (7), it is natural to define a characteristic energy for each q: 𝑚 𝜈(𝑞 −𝑞 ) 𝜔(𝑞) = + ~𝜏 (8) 𝛾 𝛾 2 m0 ν(q qc) 1 ωch(q) = + − τq− (8) which is minimum at q = qc and its inverse definesγ the relaxationγ ∼ time 𝜏 of the fluctuation mode 𝜑 to its equilibrium zero value. which is minimum at q = qc and its inverse defines the relaxation time τq of the fluctuation mode ϕq to 4.3.its equilibrium Correction to zero Mean value. Field due to Fluctuations

4.3. CorrectionIn systems to with Mean quasi-2D Field Due layered to Fluctuations structures such as cuprates fluctuations reduce strongly the mean-field critical temperature. The first correction to m0 due to fluctuations was calculated [54] in In systems with quasi-2D layered structures such as cuprates fluctuations reduce strongly the the mode–mode coupling of the CDFs, as shown diagrammatically in Figure 10. The loop of the mean-field critical temperature. The first correction to m due to fluctuations was calculated [54] in the fluctuation mode in the second diagram is a constant in0 momentum and gives a correction 𝛿𝑚 to the mean field m0.

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Condens. Matter 2020, 5, x FOR PEER REVIEW 10 of 17 mode–mode coupling of the CDFs, as shown diagrammatically in Figure 10. The loop of the ﬂuctuation mode in the second diagram is a constant in momentum and gives a correction δm to the mean ﬁeld m0.

Figure 10. First correction to the CDF propagator at first order in the coupling u (blue dot). The blue Figure 10. First correction to the CDF propagator at first order in the coupling u (blue dot). The blue dashed loop in the second diagram is the Hartree self-energy term, constant in q, and gives the first dashed loop in the second diagram is the Hartree self-energy term, constant in q, and gives the first correction to m0. correction to m0. In the same paper [54], stimulated by discussion with Alex Müller, the anomalous isotope effect inIn cupratesthat paper [10 [54],] was stimulated also considered. by discussion Even though with Alex phonons Müller, do notthe directlyanomalous pair isotope the fermionic effect in quasiparticles,cuprates [10] they was nevertheless also considered. act indirectly Even onthough various phonons quantities do as Tnot* and directly have a doping-dependentpair the fermionic effquasiparticles,ect on Tc. they nevertheless act indirectly on various quantities as T* and have a doping- c dependentThe expressions effect on of T the. full mass m in terms of the corrected TCDW and of the corrected correlation CDW lengthTheξCDW expressionsderived by of the the loop full mass integral, m in shown terms inof Figurethe corrected 10, are T as follows: and of the corrected correlation length 𝜉 derived by the loop integral, shown in Figure 10, are as follows: 2 m(T) = m0 + uδm T TCDW(p), m = νξCDW− (9) 𝑚(𝑇) =𝑚 +𝑢𝛿𝑚∝𝑇−𝑇∝ − (𝑝), m=𝜈𝜉 (9) mm(T()T must) must therefore therefore vanish vanish at at criticality. criticality. m m should should replace replacem 0min0 in the the Equation Equation (8) (8) of ofω ch𝜔.. ThereThere are are three three cases cases according according to to the the value value of of the the loop loop integration. integration. - - InIn two two dimensions dimension (2D),(2D), the the density density of of states statesN( N(ωω)) is is constant constant and andδ mδmis is ln-divergent. ln-divergent. It It is is then then impossibleimpossible to to satisfy satisfy the the criticality criticality condition condition of vanishingof vanishingm(T) m(T)and and there there is no is transitionno transition at finite at finiteT, asT in, as general in general for a continuousfor a continuous symmetry. symmetry. A quantum A quantum critical pointcritical and point charge and order charge are possibleorder are onlypossible at T = only0. In at 2D T and= 0. finiteIn 2DT ,and dynamical finite T charge, dynamical fluctuations charge (CDFs)fluctuations only can (CDFs) be present. only can be - Thepresent. drastic role of fluctuations can be reduced by cutting their long-range effect in the loop integral - The drastic role of fluctuations can be reduced by cutting( )their = m long-range= ν 2 ) effect in the loop with an infrared cut-off ωmin (possibly of the order of ωch qc γ γ ξCDW− to simulate the integral with an infrared cut-off 𝜔 (possibly of the order of 𝜔 (𝑞 )= = 𝜉 ) to quasi-2D structure of cuprates. Lowering T at fixed doping gives rise to a quasi-critical dynamical CDWsimulate (QC-CDW), the quasi-2D which structure can be observed of cuprates. with Lowering sensitive experiments T at fixed doping like RXS. gives rise to a quasi- - Incritical three dimensiondynamical CDW (3D), (QC-CDW),N(ω) vanishes which as ω cangoes be toobserved zero and with makes sensitiveδm finite experiments also at finitelike RXS.T. - ByIn lowering three dimensionT further, (3D), the QC-CDWN(ω) vanishes may as cross ω goes over to to zero a long-range and makes 3D-CDW, δm finite which also at is finite however T. By hiddenlowering by superconductivity.T further, the QC-CDW may cross over to a long-range 3D-CDW, which is however hidden by superconductivity. 4.4. Experimental Observation of the Scenario Presented Above 4.4. Experimental Observation of the Scenario Presented Above RXS experiments may test directly the dissipative part of the density–density dynamical response and canRXS therefore experiments be analyzed may withintest directly the theoretical the dissipative scenario part presented of the above. density–density In this case, dynamical the RXS intensityresponseI andis can proportional therefore tobe the analyzedImχ%% withinImD, times the th theeoretical occupancy scenario CDFs presented given by above. the Bose In function this case, RXS ∝ b(theω/T) RXSas follows: intensity IRXS is proportional to the 𝐼𝑚𝜒∝ ImD, times the occupancy CDFs given by the Bose 𝜔 ω ω ω/γ function b( /T) as follows:IRXS = AImD(q, ω)b( ) Ab( ) , (10) T ∼ T ( )2 + ( )2 𝜔 𝜔 ωch 𝜔/𝛾 ω 𝐼 = 𝐴𝐼𝑚𝐷(𝑞,) 𝜔 𝑏( )~Ab( ) , (10) where A is a constant and the characteristic frequency𝑇 ωch 𝑇is specified(𝜔) +(𝜔) by Equation (8) with m instead of m0. where A is a constant and the characteristic frequency 𝜔 is specified by Equation (8) with m instead If ωch(qc) never vanishes, its finite value determines the dynamical nature of the fluctuations that of m0. do not become critical. This is visualized in the experiment by a shift of the peak of high-resolution If 𝜔 (𝑞 ) never vanishes, its finite value determines the dynamical nature of the fluctuations experiments. In Equation (10), b( ω ) can be classically approximated as T/ω and a T-factor appears in that do not become critical. ThisT is visualized in the experiment by a shift of the peak of high- IRXS, if kBT > ωch(qc). resolution experiments. In Equation (10), b( ) can be classically approximated as 𝑇/𝜔 and a T- A thorough RXS analysis in (Y, Nd)Ba2Cu3O7-(YBCO, NBCO) at various doping values (0.11–0.19) andfactor temperatures appears in 20 IRXS<, Tif <𝑘270𝑇>𝜔 K was(𝑞 performed). in [6]. AfterA thorough subtracting RXS the analysis background, in (Y, Nd)Ba the experimental2Cu3O7-(YBCO, results NBCO) for theat various density–density doping values part of (0.11– the response0.19) and can temperatures be summarized 20 < asT < follows: 270 K was performed in [6]. After subtracting the background, the experimental results for the density–density part of the response can be summarized as follows:

Condens. Matter 2020, 5, x FOR PEER REVIEW 11 of 17

At high T, the energy integrated RXS quasi-elastic intensity can be fitted by assuming a single profile. At lower T instead, two peaks are necessary. One peak (NP) has a narrow width at half maximum (FWHM ~ ξ−1). The other, instead, is very broad with very short correlation length (BP). In the underdoped YBCO sample at p = 0.14, the two peaks are centered at slightly different values of q. Both values, however, are around 0.3 r.l.u., a value proper of the modulation vector of Condens.CDW in Matter the YBCO2020, 5, 70family at this doping. Moreover, the two vectors become equal ( 𝑞 =𝑞 =11 0. of 32 16 r.l.u.) at optimal doping p=017. Both peaks are therefore related to the same phenomenon of CDW. The NP has the following properties: (i) a dome-shaped onset temperature as a function of dopingAt T highQC-CDWT,(p) the as energy shown integrated in Figure RXS4; (ii) quasi-elastic the peak narrows intensity and cthean correlation be fitted by length assuming increasesa single as T profiledecreases. At down lower toT Tinstead,c; and (iii)two this peaks orderingare necessary. competesOne with peak superconductivity (NP) has a narrow widthas Tc atis halfreached maximum (see 1 (FWHMFigure 11). ~ ξ− ). The other, instead, is very broad with very short correlation length (BP). InThe the NP underdoped presents all YBCO the characteristics sample at p = of0.14, the the quasi-critical two peaks aredynamical centered CDW at slightly discussed different above values for oftheq .quasi-2D Both values, layered however, structure. are around 0.3 r.l.u., a value proper of the modulation vector of CDW in NP BP the YBCOBy extrapolating family at this to doping.zero the Moreover,linear in T the dependence two vectors of becomeNP-FWHM equal before (qc =theqc competition= 0. 32 r.l.u.) with at optimalSC is reached, doping the p= 017.long-rangeBoth peaks 3D are onset therefore temperature related to theT3D(p) same was phenomenon determined of CDW. at which NP-FWHM wouldThe vanish NP has and the ξ followingCDW would properties: diverge in (i) the a dome-shaped absence of superconductivity onset temperature (Figure as a function 11). The of dopingvalues TdeterminedQC-CDW(p) asin shownthis way in Figureare in good4; (ii) theagreement peak narrows with the and values the correlation detected length in the increases presence as of T decreasesa strong downmagnetic to T cfield.; and (iii) this ordering competes with superconductivity as Tc is reached (see Figure 11).

Figure 11. FromFrom [6]. [6]. Width Width vs. vs. TT inin (H,0) (H,0) scan scan of of energy energy integrated integrated quasi-elastic quasi-elastic peaks peaks for for the the optimally optimally doped YBCOYBCO sample:sample NPfor (bluethe NP squares) (blue squares) and BP (red and dots). BP (red The dots). extrapolation The extrapolation of the NP width,of the NP marked width, by

amarked dash-dotted by a dash-dotted line, tends toline, zero tends at the to hypotheticalzero at the hypothetical T3D and towards T3D and the towards BP width the atBP higher widthT. at higher T. The NP presents all the characteristics of the quasi-critical dynamical CDW discussed above for the quasi-2DThe new layeredfeature of structure. the experiments is the very broad BP (FWHM~ ξ−1 ~0.16–0.18 r.l.u.), by now observedBy extrapolating in many other to families zero the [55–60] linear in. ItT hasdependence many peculiar of NP-FWHM properties before (Figure the 11), competition as follows: with(i) it SCis present is reached, up tothe the long-rangehighest measured 3D onset T well temperature above T*(p)T3D; (ii)(p) thewas width determined (FWHM) at and which the NP-FWHM correlation wouldlength are vanish almost and constantξCDW would in T; the diverge corresponding in the absence charge of superconductivityfluctuations are then (Figure non-critical 11). The and values do not determinedshow any tendency in this to way become are incritical good at agreement any finite withT; and the (iii) values the detectedcharacteristic in the energy presence 𝜔 of has a strong been magneticmeasured field.by fitting the dynamical high-resolution spectra on the OP90 (and UD60) samples at T = 90 1 K, 150The K, new 250 feature K and of q=q thec. experimentsAt T= 150 K is and the very 250 broadK, only BP (FWHM~the BP is presentξ− ~0.16–0.18 (no NP), r.l.u.), 𝜔ch by ≈ 15 now meV observed of the inorder many of otherT* showing families the [55 dynamical–60]. It has nature many of peculiar the fluctuations properties of (Figure the BP, 11 which), as follows: can be (i)considered it is present as upthe to2D the dynamical highest charge measured fluctuationsT well above(CDFs)T *(non-criticalp); (ii) the at width any finite (FWHM) T. and the correlation length are almost At constant the lower in Ttemperature,; the corresponding T = 90, the charge NP is fluctuations also present are and then ωchnon-critical has a finite and but do smaller not show value, any tendencyreinforcing to becomethe idea critical of the at quasi-critical any finite T; andnature (iii) of the the characteristic CDW of the NP energy. ωch has been measured by fittingThe the extrapolation dynamical of high-resolution NP-FWHM from spectra T > T onC to the higher OP90 T (and(Figure UD60) 11) suggests samples that at T the= 90 NP K, emerges 150 K, 250 K and q = qc. At T = 150 K and 250 K, only the BP is present (no NP), ωch 15 meV of the order from the BP. It is not clear at the moment whether the two kinds of excitation coexist≈ or are spatially ofseparated,T* showing giving the rise dynamical to an inhomo naturegeneous of the fluctuations landscape, as of found the BP, in which [61]. can be considered as the 2D dynamical charge fluctuations (CDFs) non-critical at any finite T. At the lower temperature, T = 90, the NP is also present and ωch has a finite but smaller value, reinforcing the idea of the quasi-critical nature of the CDW of the NP. The extrapolation of NP-FWHM from T > TC to higher T (Figure 11) suggests that the NP emerges from the BP. It is not clear at the moment whether the two kinds of excitation coexist or are spatially separated, giving rise to an inhomogeneous landscape, as found in [61]. Condens. Matter 2020, 5, 70 12 of 16

As a ﬁrst conclusion, we can say that the three cases envisaged by the theory in Section 4.3. were found in the experiments with a continuous evolution from pure 2D dynamical CDFs at high T and all doping, to a coexistence with quasi-critical CDWs below TQC-CDW, to the static 3D CDW hindered by superconductivity and measured in a high magnetic ﬁeld.

4.5. Strange Metallic Behavior Above T* The very broad spectrum (BP) of CDFs allows us [7] to approach the old standing problem of strange metal (SM) of cuprates, in particular the anomalous linear in T resistivity ρ(T) T from T > T* up to the ∼ highest attained temperatures. One of the prerequisites identified in the marginal FL phenomenological theory [62] to reproduce the strange metal is the presence of a strong isotropic scattering channel for the Fermi quasiparticles with all states on the Fermi surface nearly equally affected. This was accomplished by assuming 2D-CDFs of the BP as mediators of scattering among the Fermi QPs. The width of the CDFs is such that a Fermi QP of one branch of the FS can always find a CDF that scatters it onto another region of FS. The scattering among the Fermi quasiparticles is then almost uniform along the entire Fermi surface. I refer to the original article for the explicit calculation of the following quantities for an optimally doped (Tc = 93 K) NBCO sample and an overdoped (Tc = 83 K) YBCO sample: (i) the fermion self-energy, i.e., how the Fermi QPs are dressed by CDFs, using the same set of parameters that was obtained by fitting the BP response; (ii) the zero frequency quasiparticle scattering rate along the Fermi surface from the imaginary part of the self-energy; and (iii) finally, the resistivity within the standard Boltzmann equation approach using the calculated scattering. In both cases, the strange metal requirements are satisfied and linearity in T of the resistivity for T > T* is quantitatively matched. I only consider here the following heuristic simple argument. Since the scattering is almost uniform over the entire Fermi surface, the scattering integral in the Boltzmann equation can be well approximated with an effective scattering time of a single scattering event. Its inverse then determines the effective characteristic frequency of the process, which depends on the characteristic frequency of the scatterers given by Equation (8) at q = qc. In analogy with a disordered electron system, the full inverse scattering time must be proportional to the one of the single event multiplied by the population of the ω scatterers, i.e., the Bose function. As long as kT > }ωch, the semiclassical approximation of b( T ) is valid and a linear T dependence is introduced:

} = a(υ, ξ, ...)kT (11) τ where a is a dimensionless doping-dependent function of the parameters of the theory. In the present argument a can be considered as a ﬁtting parameter, independent from T since the BP correlation length does not depend on T in any relevant way (Figure 11).

5. Conclusions The following Figure 12 gives a graphic representation of the scenario described above. The phase diagram visualizes the continuous evolution from a pure 2D dynamical CDF at high T and all doping, to a coexistence with quasi-critical CDWs below TQC-CDW, to the static 3D-CDW hindered by superconductivity and measured in a high magnetic ﬁeld. Finally, the dynamical CDFs (BP), characterized by the same parameters used to ﬁt RXS experiments, were chosen as mediators of isotropic scattering among Fermi quasiparticles and provide an understanding of the long-standing problem of strange metallic state, matching the resistivity data, in the whole range from room temperature down to T*[7]. Some open problems deserve further investigation: In Section 4.4., it was already mentioned that the present knowledge of the relation between the quasi-critical CDW (NP) and the dynamical CDF (BP) does not allow to distinguish between a spatial separation or a coexistence of the two. Condens. Matter 2020, 5, 70 13 of 16

Condens.At temperatures Matter 2020, 5, x TFOR< T*,PEERthe REVIEW role on scattering of the pseudogap state and of other intertwined 13 of 17 incipient orders (CDWs, Cooper pairing, etc.) must be clariﬁed.

FigureFigure 12. 12.Adapted Adapted from from [6 [6].]. CDW CDW in in the thephase phase diagramdiagram T vs. doping doping p p of of YBCO. YBCO. The The green green zone zone is is thethe antiferromagnetic antiferromagnetic phase. phase. The reddishThe reddish zone representszone represents the 2D chargethe 2D density charge dynamical density dynamical ﬂuctuations associatedfluctuations with associated the BP (actually with the the BP experiments (actually the inexperiments [6] were performed in [6] were for performed 0.11 < p for< 0.19). 0.11 < Inp < the 0.19). cone aboveIn the optimal cone above doping optimal (between dopingT *(between and the T Fermi* and the liquid), Fermi CDFs liquid), produce CDFs produce the strange the strange metal andmetal the linearand in theT linearresistivity. in T resistivity. Light blue Light zone: blue the quasi-criticalzone: the quasi-critical CDWs associated CDWs associated with the with NP. Darkthe NP. blue Dark zone: theblue static zone: 3D long-range the static order3D long-range hidden in theorder absence hidden of ain magnetic the absence ﬁeld byof thea magnetic superconductive field by dome.the superconductive dome.

Investigation is also required to see whether CDFs with a reduced value of ωch can also account for the so-calledThe phase Planckian diagram of or Figure better 12 for visualizes the linear the behavior continuous observed evolution at low from temperatures a pure 2D dynamical and special dopingCDF inat high the presence T and all of doping, high magnetic to a coexistence fields to with suppress quasi-critical superconductivity CDWs below (see, TQC-CDW e.g.,, to [63 the]). static 3D-CDW hindered by superconductivity and measured in a high magnetic field. Finally, the dynamical CDFs (BP), characterized by the same parameters used to fit RXS Funding: This research received no external funding. experiments, were chosen as mediators of isotropic scattering among Fermi quasiparticles and Acknowledgments:provide an understandingI wish to warmly of the thanklong-standing Claudio Castellani, problem of Marco strange Grilli, metallic Sergio Caprara,state, matching Lara Benfatto, the Roberto Raimondi, undergraduates, doctoral students, and post-doctoral researchers who worked with us in resistivity data, in the whole range from room temperature down to T* [7]. Rome from time to time, particularly José Lorenzana (now permanently in Rome), Walter Metzner, Goetz Seibold, and moreSome recently open Giacomo problems Ghiringhelli, deserve further Lucio Braicovich investigation: Riccardo Arpaia (Chalmers University), and the whole Milano PolitecnicoIn Section Group4.4., it for was the already present mentioned collaboration. that the present knowledge of the relation between Conflictsthe quasi-critical of Interest: CDWThe author (NP)declares and the no dynamical conflict of CDF interest. (BP) does not allow to distinguish between a spatial separation or coexistence of the two. ReferencesAt temperatures and Note T < T*, the role on scattering of the pseudogap state and of other intertwined incipient orders (CDWs, Cooper pairing, etc.) must be clarified. 1. The importance of A.K. Müller’s contribution in this field is stated e.g., by the quotations he received in the Investigation is also required to see whether CDFs with a reduced value of ωch can also account forfundamental the so-called paper Planckian by Hohenberg, or better P.C.; for the Halperin, linear B.be Theoryhavior ofobserved dynamic at critical low temperatures phenomena. Rev. and Mod.special Phys. doping1977, 49in, the 435. presence [CrossRef of] high magnetic fields to suppress superconductivity (see, e.g., [63]). 2. Thomas, H.; Müller, K.A. Theory of a Structural Phase Transition Induced by the Jahn-Teller Effect. Funding:Phys. Rev. This Lett. research1972, received28, 800. [noCrossRef external] funding. 3. Bednorz, J.G.; Müller, K.A. Possible high-Tc superconductivity in the Ba-La-Cu-O system. Z. Phys. B Acknowledgments: I wish to warmly thank Claudio Castellani, Marco Grilli, Sergio Caprara, Lara Benfatto, Roberto1986, 64Raimondi,, 189–193. undergraduates, [CrossRef] doctoral students, and post-doctoral researchers who worked with us in 4. RomeAnderson, from time P.W. to The time, Resonating particularly Valence José Bond Lorenzan Statea in (now La2CuO4 permanently and Superconductivity. in Rome), WalterScience Metzner,1987 , Goetz235, 119. Seibold,[CrossRef and] more [PubMed recently] Giacomo Ghiringhelli, Lucio Braicovich Riccardo Arpaia (Chalmers University), and 5. theCaprara, whole Milano S. The Politecnico Ancient Romans’ Group for Route the present to Charge collaboration. Density Waves in Cuprates. Condens. Matter. 2019, 4, 60. Conflicts[CrossRef of ]Interest: The author declares no conflict of interest.

Condens. Matter 2020, 5, 70 14 of 16

6. Arpaia, R.; Caprara, S.; Fumagalli, R.; De Vecch, G.; Peng, Y.Y.; Andersson, E.; Betto, D.; De Luca, G.M.; Brookes, N.P.; Lombardi, F.; et al. Dynamical charge density fluctuations pervading the phase diagram of a copper-based high Tc superconductor. Science 2019, 365, 906–9010. [CrossRef] 7. Seibold, G.; Arpaia, R.; Peng, Y.Y.; Fumagalli, R.; Braicovich, L.; Di Castro, C.; Grilli, M.; Ghiringhelli, G.; Caprara, S. Marginal Fermi Liquid behaviour from charge density fluctuations in Cuprates. arXiv 2019, arXiv:1905.10232. (to be published in Commun. Phys.) 8. Ando, Y.; Komiya, S.; Segawa, K.; Ono, S.; Kurita, Y. Electronic Phase Diagram of High-Tc Cuprate Superconductors from a Mapping of the In-Plane Resistivity Curvature. Phys. Rev. Lett. 2004, 93, 26700. [CrossRef] 9. Timusk, T.; Statt, B. The pseudogap in high temperature superconductors: An experimental survey. Rep. Prog. Phys. 1999, 62, 61–122. [CrossRef] 10. Rubio Temprano, D.; Mesot, J.; Janssen, S.; Conder, K.; Furrer, A.; Mutka, H.; Müller, K.A. Large Isotope Effect

on the Pseudogap in the High-Temperature Superconductor HoBa2Cu4O8. Phys. Rev. Lett. 2000, 84, 1990–1993. [CrossRef][PubMed] 11. Emery, V.J.; Kivelson, S.A.; Lin, H.Q. Phase separation in the t-J model. Phys. Rev. Lett. 1990, 64, 475. [CrossRef] [PubMed] 12. Cancrini, N.; Caprara, S.; Castellani, C.; Di Castro, C.; Grilli, M.; Raimondi, R. Phase separation and superconductivity in Kondo-like spin-hole coupled model. Europhys. Lett. 1991, 14, 597. [CrossRef] 13. Grilli, M.; Raimondi, R.; Castellani, C.; Di Castro, C.; Kotliar, G. Phase separation and superconductivity in the U=inﬁnite limit of the extended multiband Hubbard model. Int. J. Mod. Phys. B 1991, 5, 309. [CrossRef] 14. Raimondi, R.; Castellani, C.; Grilli, M.; Bang, Y.; Kotliar, G. Charge collective modes and dynamic pairing in the three-band Hubbard model. II. Strong-coupling limit. Phys. Rev. B 1993, 47, 3331. [CrossRef][PubMed] 15. Löw, U.; Emery, V.J.; Fabricius, K.; Kivelson, S.A. Study of an Ising model with competing long- and short-range interactions. Phys. Rev. Lett. 1994, 72, 1918. [CrossRef] 16. Emery, V.J.; Kivelson, S.A. Frustrated electronic phase separation and high-temperature superconductors. Phys. C Supercond. 1993, 209, 597–621. [CrossRef] 17. Castellani, C.; Di Castro, C.; Grilli, M. Singular Quasiparticle Scattering in the Proximity of Charge Instabilities. Phys. Rev. Lett. 1995, 75, 4650. [CrossRef][PubMed] 18. Perali, A.; Castellani, C.; Di Castro, C.; Grilli, M. D-wave superconductivity near charge instabilities. Phys. Rev. B 1996, 54, 16216. [CrossRef] 19. Monthoux, P.; Balatsky, A.V.; Pines, D. Weak-coupling theory of high-temperature superconductivity in the antiferromagnetically correlated copper oxides. Phys. Rev. B 1992, 46, 14803. [CrossRef] 20. Tranquada, J.M.; Sternlieb, B.J.; Axe, J.D.; Nakzmura, Y.; Uchida, S. Evidence for stripe correlations of spins and holes in copper oxide superconductors. Nature 1995, 375, 561. [CrossRef] 21. Blaco-Caosa, S.; Frano, A.; Schierle, E.; Porras, J.; Loew, T.; Minola, M.; Bluschke, M.; Weschke, E.; Keimer, B.; Le

Tacon, M. Resonant X-ray scattering study of charge-density- wave correlations in YBa2Cu3O6+x. Phys. Rev. B 2014, 90, 54513. [CrossRef] 22. Hücker, M.; Christensen, N.B.; Holmes, A.T.; Blackburn, E.; Forgan, E.M.; Liang, R.; Bonn, D.A.; Hardy, W.N.; Gutowski, O.; Zimmermann, M.V.; et al. Competing charge, spin, and superconducting orders in underdoped YBa2Cu3Oy. Phys. Rev. B 2014, 90, 54514. [CrossRef] 23. Ghiringhelli, G.; Le Tacon, M.; Minola, M.; Blanco-Canosa, S.; Mazzoli, C.; Brookes, N.B.; De Luca, G.M.; Frano, A.; Hawthorn, D.G.; He, F.; et al. Long-range incommensurate charge fluctuations in(Y,Nd)Ba2Cu3O6+x. Science 2012, 337, 821. [CrossRef][PubMed] 24. Comin, R.; Damascelli, A. Resonant X-Ray Scattering Studies of Charge Order in Cuprates. Annu. Rev. Condens. Matter Phys. 2016, 7, 369–405. [CrossRef] 25. Chang, J.; Blackburn, E.; Holmes, A.T.; Christensen, N.B.; Larsen, J.; Mesot, J.; Liang, R.; Bonn, D.A.; Hardy, W.N.; Watenphul, A.; et al. Direct observation of competition between superconductivity and charge density wave order in YBa2Cu3O6.67. Nat. Phys. 2012, 8, 871. [CrossRef] 26. Achkar, A.J.; Hawthorn, D.; Sutarto, R.; Mao, X.; He, F.; Frano, A.; Blanco-Canosa, S.; le Tacon, M.; Ghiringhelli, G.; Braicovich, L.; et al. Distinct charge orders in the planes and chains of ortho-III-ordered YBa2Cu3O6+_ superconductors identified by resonant elastic x-ray scattering. Phys. Rev. Lett. 2012, 109, 167001. [CrossRef] Condens. Matter 2020, 5, 70 15 of 16

27. Tabis, W.; Li, Y.; Le Tacon, M.; Braicovich, L.; Kreyssig, A.; Minola, M.; Dellea, G.; Weschke, E.; Veit, M.J.; Ramazanoglu, M.; et al. Charge order and its connection with Fermi-liquid charge transport in a pristine high-Tc cuprate. Nat. Commun. 2014, 5, 5875. [CrossRef][PubMed] 28. Comin, R.; Frano, A.; Yee, M.M.; Yoshida, Y.; Eisaki, H.; Schierle, E.; Weschke, E.; Sutarto, R.; He, F.; Soumyanarayanan, A.; et al. Charge order driven by Fermi-arc instability in Bi2Sr2 xLaxCuO6+δ. Science − 2014, 343, 390. [CrossRef] 29. Wu, T.; Mayaffre, H.; Krämer, S.; Horvatić,M.; Berthier, C.; Hardy, W.N.; Liang, R.; Bonn, D.A.; Julien, M.-H. Magnetic-field-induced charge-stripe order in the high-temperature superconductor YBa2Cu3Oy. Nature 2011, 477, 191. [CrossRef] 30. Wu, T.; Mayaffre, H.; Krämer, S.; Horvatić, M.; Berthier, C.; Kuhn, P.L.; Reyes, A.P.; Liang, R.; Hardy, W.N.; Bonn, D.A.; et al. Emergence of charge order from the vortex state of a high-temperature superconductor. Nat. Commun. 2013, 4, 2113. [CrossRef] 31. Laliberté, F.; Frachet, F.; Benhabib, S.; Borgnic, B.; Loew, T.; Porras, J.; Le Tacon, M.; Keimer, B.; Wiedmann, S.; Proust, C.; et al. High field charge order across the phase diagram of YBa2Cu3Oy. NPJ Quantum Mater. 2018, 3, 11. [CrossRef] 32. Doiron-Leyraud, N.; Proust, C.; LeBoeuf, D.; Levallois, J.; Bonnemaison, J.; Liang, R.; Bonn, D.A.; Hardy, W.N. Taillefer Louis Quantum oscillations and the Fermi surface in an underdoped high-Tc superconductor. Nature 2007, 447, 565–568. [CrossRef] 33. Grissonnanche, G.; Cyr-Choinière, O.; Laliberté, T.; René de Cotret, S.; Juneau-Fecteau, A.; Dufour-Beauséjour, S.; Delage, M.È.; LeBoeuf, D.; Chang, J.; Ramshaw, B.J.; et al. Direct measurement of the upper critical field in cuprate superconductors. Nat. Commun. 2014, 5, 3280. [CrossRef] [PubMed] 34. LeBoeuf, D.; Doiron-Leyraud, N.; Vignolle, B.; Sutherland, M.; Ramshaw, B.J.; Levallois, J.; Daou, R.; Laliberté, F.; Cyr-Choinière, Q.; Chang, J.; et al. Lifshitz critical point in the cuprate superconductor YBa2Cu3Oy from high-field Hall effect measurements. Phys. Rev. B 2011, 83, 54506. [CrossRef] 35. Doiron-Leyraud, N.; Lepault, S.; Cyr-Choinière, O.; Vignolle, B.; Grissonnanche, G.; Laliberté, F.; Chang, J.; Barisic, N.; Chan, M.K.; Ji, L.; et al. Hall, Seebeck and Nernst coefficients of underdoped HgBa2CuO4+δ: Fermi-surface reconstruction in an archetypal cuprate superconductor. Phys. Rev. X 2013, 3, 21019. 36. Chang, J.; Daou, R.; Proust, C.; Leboeuf, D.; Doiron-Leyraud, N.; Laliberté, F.; Pingault, B.; Ramshaw, B.J.; Liang, R.; Bonn, D.A.; et al. Nernst and Seebeck coefficients of the cuprate superconductor YBa2Cu3O6.67: A study of fermi surface reconstruction. Phys. Rev. Lett. 2010, 104, 57005. [CrossRef] 37. Barišić,N.; Badoux, S.; Chan, M.K.; Dorow, C.; Tabis, W.; Vignolle, B.; Yu, G.; Béard, J.; Zhao, X.; Proust, C.; et al. Universal quantum oscillations in the underdoped cuprate superconductors. Nat. Phys. 2013, 9, 761. [CrossRef] 38. Grissonnanche, G.; Legros, A.; Badoux, S.; Lefrançois, E.; Zatko, V.; Lizaire, M.; Laliberté, F.; Gourgout, A.; Zhou, J.-S.; Pyon, S.; et al. Giant thermal Hall conductivity in the pseudogap phase of cuprate superconductors. Nature 2019, 571.[CrossRef] 39. Leboeuf, D.; Kramer, S.; Hardy, W.N.; Liang, R.; Bonn, D.A.; Proust, C. Thermodynamic phase diagram of static charge order in underdoped YBa2Cu3Oy. Nat. Phys. 2012, 9, 79–83. [CrossRef] 40. Sachdev, S.; Ye, J. Universal quantum-critical dynamics of two-dimensional antiferromagnets. Phys. Rev. Lett. 1992, 69, 2411. [CrossRef] 41. Sokol, A.; Pines, D. Toward a unified magnetic phase diagram of the cuprate superconductors. Phys. Rev. Lett. 1993, 71, 2813. [CrossRef][PubMed] 42. Monthoux, P.; Pines, D. Nearly antiferromagnetic Fermi-liquid description of magnetic scaling and spin-gap behavior. Phys. Rev. B 1994, 50, 16015. [CrossRef][PubMed] 43. Abanov, A.; Chubukov, A.; Schmalian, J. Quantum-critical theory of the spin-fermion model and its application to cuprates: Normal state analysis. Adv. Phys. 2003, 52, 119. [CrossRef] 44. Castellani, C.; Di Castro, C.; Grilli, M. Possible occurrence of band interplay in high Tc superconductors. In Proceedings of the International Conference on High-Temperature Superconductors and Materials and Mechanisms of Superconductivity Part II, Interlaken, Switzerland, 28 February–4 March 1988; Physica C 153–155. pp. 1659–1660. 45. Müller, K.A.; Benedeck, G. Phase Separation in Cuprate Superconductors. In Proceedings of the 3rd Workshop “The Science and Culture-Physics”, Erice, Italy, 6–12 May 1992; World Scientific: Singapore, 1993. Condens. Matter 2020, 5, 70 16 of 16

46. Müller, K.A.; Sigmund, E. Phase Separation in Cuprate Superconductors. In Proceedings of the Second International Workshop on “Phase Separation in Cuprate Superconductors”, Cottbus, Germany, 4–10 September 1993; Springer: Verlag, Germany, 1994. 47. Bianconi, A.; Saini, N.L.; Lanzara, A.; Missori, M.; Rossetti, T.; Oyanag, H.; Yamaguchi, H.; Oka, K.; Ito, T.

Determination of the Local Lattice Distortions in the CuO2 plane of La1.85Sr0.15CuO4. Phys. Rev. Lett. 1996, 76, 3412. [CrossRef] 48. Ramshaw, B.J.; Sebastian, S.E.; McDonald, R.D.; Day, J.; Tan, B.S.; Zhu, Z.; Betts, J.B.; Liang, R.; Bonn, D.A.; Hardy, W.N.; et al. Quasiparticle mass enhancement approaching optimal doping in a high-Tc superconductor. Science 2015, 348, 317–320. [CrossRef] 49. Becca, F.; Tarquini, M.; Grilli, M.; Di Castro, C. Charge-density waves and superconductivity as an alternative to phase separation in the inﬁnite-U Hubbard-Holstein model. Phys. Rev. B 1996, 54, 12443–12457. [CrossRef] 50. Castellani, C.; Di Castro, C.; Grilli, M. Non-Fermi Liquid behavior and d-wave superconductivity near the charge density wave quantum critical point. Zeit. Phys. 1997, 103, 137–144. [CrossRef] 51. Seibold, G.; Becca, F.; Bucci, F.; Castellani, C.; Di Castro, C.; Grilli, M. Spectral properties of incommensurate charge-density wave systems. Eur. Phys. J. B 2000, 13, 87. [CrossRef] 52. Benfatto, L.; Caprara, S.; Di Castro, C. Gap and pseudogap evolution within the charge-ordering scenario for superconducting cuprates. Eur. Phys. J. B 2000, 17, 95. [CrossRef] 53. Chakravarty, S.; Laughlin, R.B.; Morr Dirk, K.; Nayak, C. Hidden order in the cuprates. Phys. Rev. B 2001, 63, 94503. [CrossRef] 54. Andergassen, S.; Caprara, S.; Di Castro, C.; Grilli, M. Anomalous Isotopic Eﬀect Near the Charge-Ordering Quantum Criticality. Phys. Rev. Lett. 2001, 87, 5641. [CrossRef][PubMed] 55. Yu, B.; Tabis, W.; Bialo, I.; Yakhou, F.; Brookes, N.; Anderson, Z.; Tan, G.Y.; Yu, G.; Greven, M. Unusual dynamic charge-density-wave correlations in HgBa2CuO4+. arXiv 2019, arXiv:1907.10047. 56. Miao, H.; Fabbris, G.; Nelson, C.S.; Acevedo-Esteves, R.; Li, Y.; Gu, G.D.; Yilmaz, T.; Kaznatcheev, K.; Vescovo, E.; Oda, M.; et al. Discovery of Charge Density Waves in cuprate Superconductors up to the Critical Doping and Beyond. arXiv 2020, arXiv:2001.10294. 57. Lin, J.Q.; Miao, H.; Mazzone, D.G.; Gu, G.D.; Nag, A.; Walters, A.C.; Garcia-Fernandez, M.; Barbour, A.; Pelliciari, J.; Jarrige, I.; et al. Nature of the charge-density wave excitations in cuprates. arXiv 2020, arXiv:2001.10312. 58. Miao, H.; Lorenzana, J.; Seibold, G.; Peng, Y.Y.; Amorese, A.; Yakhou-Harris, F.; Kummer, K.; Brookes, N.B.; Konik, R.M.; Thampy, V.; et al. High-temperature charge density wave correlations in La1:875Ba0:125CuO4 without spin-charge locking. Proc. Natl. Acad. Sci USA 2017, 114, 12430. [CrossRef][PubMed] 59. Miao, H.; Fumagalli, R.; Rossi, M.; Lorenzana, J.; Seibold, G.; Yakhou-Harris, F.; Kummer, K.; Brookes, N.B.; Gu, G.D.; Braicovich, L.; et al. Formation of incommensurate Charge Density Waves in Cuprates. Phys. Rev. 2019, 9, 031042. [CrossRef] 60. Wang, Q.S.; Horio, M.; Von Arx, K.; Shen, Y.; Mukkattukavil, D.J.; Sassa, Y.; Ivashko, O.; Matt, C.E.; Pyon, S.;

Takayama, T.; et al. High-Temperature Charge-Stripe Correlations in La1.675Eu0.2Sr0.125CuO4. Phys. Rev. Lett. 2020, 124, 187002. [CrossRef] 61. Campi, G.; Bianconi, A.; Poccia, N.; Bianconi, G.; Barba, L.; Arrighetti, G.; Innocenti, D.; Karpinski, J.; Zhigadlo, N.D.; Kazakov, S.M.; et al. Inhomogeneity of charge-density-wave order and quenched disorder in a high-Tc superconductor. Nature 2015, 525, 359. [CrossRef] 62. Varma, C.M.; Littlewood, P.B.; Schmitt-Rink, S.; Abrahams, E.; Ruckenstein, A.E. Phenomenology of the normal state of Cu-O high-temperature superconductors. Phys. Rev. Lett. 1989, 63, 1996. [CrossRef] 63. Legros, A.; Benhabib, S.; Tabis, W.; Laliberté, F.; Dion, M.; Lizaire, M.; Vignolle, B.; Vignolles, D.; Raﬀy, H.; Li, Z.Z.; et al. Universal T-linear resistivity and Planckian dissipation in overdoped cuprates. Nat. Phys. 2019, 15, 142–147. [CrossRef]

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