# The Strange-Metal Behavior of Cuprates

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- In Section 2, we briefly describe a typical phase diagram of cuprates and introduce some of the ideas that have been put forward in the attempt to find a rationale behind the various phases that occur in it and their competition/interplay.
- In Section 3 and Section 4, we go through the main steps that led us to the formulation and elaboration of our model for the theoretical description of the strange-metal phase of high-temperature cuprate superconductors. In particular, an emphasis will be given to the role of the so-called charge density fluctuations that were identified in recent resonant X-ray scattering experiments on cuprates [15].
- Lastly, in Section 9, we make some concluding remarks and discuss the future perspectives of our work.

## 2. The Phase Diagram of Cuprates and the Quantum Critical Point Scenario

## 3. Charge Ordering in Cuprates

_{1.6−x}Nd

_{0.4S}r

_{x}CuO

_{4}[34], in the form of a static stripe-like charge order (i.e., a state characterized by a unidirectional charge density modulation), triggered by Nd co-doping, in the pseudogap phase [35,36]. Experiments showed that these stripe modulations are characterized by a strong interplay between charge and spin degrees of freedom. Nevertheless, stripe-like charge modulation seems to be a peculiarity of Nd co-doped La-based cuprates [37].

**q**

_{c}. From soft X-ray scattering experiments, the presence of a narrow peak around

**q**

_{c}≈ (0.31,0) r.l.u. (reciprocal lattice units) in the nearly-elastic portion of the spectrum is evident [15]. Of course, tetragonal symmetry of the lattice implies that similar peaks are observed at a star of wave vectors which are equivalent under the point group symmetries. Interesting phenomena involve charge density waves when an orthorhombic distortion occurs [27], but we shall concentrate on the tetragonal phase, although all our conclusions are essentially unchanged in the presence of a weak orthorhombic distortion.

**q**${}_{\mathrm{c}}$. This implies that low-temperature scattering can only occur between regions of the Fermi surface that are connected by

**q**${}_{\mathrm{c}}$, usually called hot spots. At the same time, as long as the charge density waves stay dynamical and the symmetry is not broken (i.e., one is still inside the FL phase), the hot spots are points where the scattering is strongest, but the Fermi surface is not spoiled [52]. Since the other regions of the Fermi surface are not involved in this scattering process, most of the quasiparticles retain their standard FL behavior [53]. It became evident that what was responsible for the strange-metal properties of cuprates had yet to be identified.

## 4. Charge Density Fluctuations

_{2}Cu

_{3}O

_{7−δ}and Nd

_{1+x}Ba

_{2−x}Cu

_{3}O

_{7−δ}films [14]. A clear peak in momentum space, centered at

**q**=

**q**

_{c}, has been identified in the quasi-elastic component of the inelastic X-ray scattering spectra. At sufficiently high temperatures, the peak appears broad and can be fitted with a simple Lorentzian profile. At lower temperature, two Lorentzian profiles are needed to fit the observed peak: a broad one, which is very similar to the one observed at high temperature, and a narrow one, which presents all the characteristics previously observed in several compounds and commonly attributed to charge density waves. The broad peak was attributed to collective excitations, called charge density fluctuations, that pervade the phase diagram of cuprates. The centers of the two Lorentzian peaks are very close to each other, meaning that the characteristic wave vectors of the two collective excitations are very similar, suggesting that they may have a common origin. Charge density fluctuations can be described as a sort of aborted charge density waves which, for some reason (e.g., locally enhanced disorder or other slightly less favorable conditions), fail to develop long-range correlations.

## 5. Role of the Landau Damping

_{1.6−x}Eu

_{0.4}Sr

_{x}CuO

_{4}and La

_{1.6−x}Nd

_{0.4}Sr

_{x}CuO

_{4}(we will carry out our analysis on these compounds, for which both resistivity and specific heat have been measured), detailed data are not available to extract the parameters we need. We will use the parameters fitted from the data related to Nd

_{1+x}Ba

_{2−x}Cu

_{3}O

_{7−δ}, namely m = 15 meV, $\overline{\nu}$ = 1.3 eV/(r.l.u.)

^{2}and $\overline{\mathsf{\Omega}}$ = 30 meV [12], which are reasonable estimates.

**q**because of a diverging $\gamma $ at fixed $\xi $, rather than becoming small only near

**q**${}_{\mathrm{c}}$ because of a diverging $\xi $ (this is the standard slowing-down of critical phenomena). We will not provide a microscopic model for this mechanism, but we will limit ourselves to the phenomenological assumption that $\gamma $ depends both on temperature and doping, in such a way as to diverge at a certain doping ${p}^{*}$ (slightly) larger than the doping ${p}_{\mathrm{c}}$ for the QCP of charge density waves. We will show that this assumption is sufficient to explain two of the most peculiar aspects of the strange-metal phase: the aforementioned linear-in-temperature resistance and a divergent specific heat, which has been observed in several compounds. It is important to emphasize that in our scenario, ${p}^{*}$ should not be identified with ${p}_{\mathrm{c}}$. The distance between the two points is not universal and changes in the various cuprate families. For our scenario to be meaningful and coherent, we only need the two points to not be very far apart. We need a mechanism that brings charge density fluctuations to be characterized by a sufficiently low energy scale, so as to play a relevant role as scatterers for the electron quasiparticles, and this occurs if these fluctuations are not too far from a quantum critical point for charge ordering. At the same time, since they must be broad enough in momentum space to mediate an isotopic scattering, they cannot be too close to ${p}_{\mathrm{c}}$. A dynamical instability can occur at a doping $p*>{p}_{\mathrm{c}}$, such that the characteristic time scale of the fluctuations grows very large and the linear resistivity extends over a wide temperature range, down to very low temperature. This is achieved, e.g., if the parameter $\gamma $ grows large.

## 6. Singular Behavior of Specific Heat

_{1.6−x}Eu

_{0.4}Sr

_{x}CuO

_{4}and La

_{1.6−x}Nd

_{0.4}Sr

_{x}CuO

_{4}, highlighted the presence of a thermodynamic singularity at the QCP, signaled by the fact that the ratio ${C}_{V}/T$ seems to diverge as $log(1/T)$ at fixed $p={p}^{*}$ [11]. In principle, such behavior could be due to fermion quasiparticles if their chemical potential approaches a van Hove singularity, which we know to be present close to the critical doping level. Nevertheless, the presence of disorder and the weak coupling between lattice planes strongly smoothen the singularity; hence, it was argued that the fermion quasiparticles alone are not able to account for the singularity.

## 7. Methods

_{1.6−x}Nd

_{4}Sr

_{x}CuO

_{4}, while k

_{F}(ϕ), v

_{F}(ϕ) and Γ(ϕ), respectively, indicate the angular dependence of the Fermi momentum, Fermi velocity and scattering rate along the Fermi surface. Moreover, we have introduced

## 8. Effect of a Three-Dimensional Dispersion

## 9. Results and Conclusions

_{1.6−x}Nd

_{0.4}Sr

_{x}CuO

_{4}data, we need $\gamma $ to be of order 50 or 60 at low temperature. In order to fit the data relating to La

_{1.6−x}Eu

_{0.4}Sr

_{x}CuO

_{4}, it is sufficient that $\gamma $ is of order 20 or 30.

_{1.6−x}Nd

_{0.4}Sr

_{x}CuO

_{4}and T

_{0}= 37 K, p

_{c}= 0.232, A = 0.117, B = 2.84 for La

_{1.6−x}Eu

_{0.4}Sr

_{x}CuO

_{4}, while resistivity data allowed us to find ${g}^{2}=0.045$ and ${\mathsf{\Gamma}}_{0}=13.7$ meV for La

_{1.6−x}Nd

_{0.4}Sr

_{x}CuO

_{4}and g

^{2}= 0.0415 and Γ

_{0}= 12.3 meV, for La

_{1.6−x}Eu

_{0.4}Sr

_{x}CuO

_{4}. The resistivity trend has been reproduced correctly down to the lowest temperatures for which data are available. The overall amplification factor resulting from the specific heat fit is 30 for La

_{1.6−x}Nd

_{0.4}Sr

_{x}CuO

_{4}and 11 for La

_{1.6−x}Eu

_{0.4}Sr

_{x}CuO

_{4}. We believe that a possible origin for this amplification may lie in an overestimation of the real number of collective charge density fluctuation degrees of freedom contributing to the specific heat. It may be possible that charge density fluctuations live, e.g., on a coarse-grained lattice with larger effective spacing [13].

_{1.6−x}Nd

_{0.4}Sr

_{x}CuO

_{4}and 11 for La

_{1.6−x}Eu

_{0.4}Sr

_{x}CuO

_{4}compounds at different temperatures.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FL | Fermi Liquid |

QCP | Quantum Critical Point |

r.l.u. | reciprocal lattice units |

## References

- Damascelli, A.; Hussain, Z.; Shen, Z.X. Angle-resolved photoemission studies of the cuprate superconductors. Rev. Mod. Phys.
**2003**, 75, 473–541. [Google Scholar] [CrossRef] [Green Version] - Almasan, C.; Maple, M.B. Chemistry of High Temperature Superconductors; World Scientific: Singapore, 1991. [Google Scholar]
- Bednorz, J.G.; Müller, K.A. Possible high-T
_{c}superconductivity in the Ba-La-Cu-O system. Z. Für Phys. Condens. Matter**1986**, 64, 189–193. [Google Scholar] [CrossRef] - Wolf, S.A.; Kresin, V.Z. Novel Superconductivity; Springer: Berlin/Heidelberg, Germany, 1987. [Google Scholar]
- Wu, M.K.; Ashburn, J.R.; Torng, C.; Hor, P.H.; Meng, R.L.; Gao, L.; Huang, Z.J.; Wang, Y.Q.; Chu, A. Superconductivity at 93 K in a New Mixed-Phase Y-Ba-Cu-O Compound System at Ambient Pressure. Phys. Rev. Lett.
**1987**, 58, 908. [Google Scholar] [CrossRef] [Green Version] - Schilling, A.; Cantoni, M.; Guo, J.D.; Ott, H.R. Superconductivity above 130 K in the Hg-Ba-Ca-Cu-O system. Nat. Phys.
**1993**, 363, 56–58. [Google Scholar] [CrossRef] - Dagotto, E. Correlated electrons in high-temperature superconductors. Rev. Mod. Phys.
**1994**, 66, 763–840. [Google Scholar] [CrossRef] - Giraldo-Gallo, P.; Galvis, J.A.; Stegen, Z.; Modic, K.A.; Balakirev, F.F.; Betts, J.B.; Lian, X.; Moir, C.; Riggs, S.C.; Wu, J.; et al. Scale-invariant magnetoresistance in a cuprate superconductor. Science
**2018**, 361, 479–481. [Google Scholar] [CrossRef] [PubMed] [Green Version] - 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. [Google Scholar] [CrossRef] [Green Version] - Hussey, N.E.; Cooper, R.A.; Xu, X.; Wang, Y.; Mouzopoulou, I.; Vignolle, B.; Proust, C. Dichotomy in the T-linear resistivity in hole-doped cuprates. Philos. Trans. R. Soc. Math. Phys. Eng. Sci.
**2011**, 369, 1626–1639. [Google Scholar] [CrossRef] [Green Version] - Michon, B.; Girod, C.; Badoux, S.; Kačmarčík, J.; Ma, Q.; Dragomir, M.; Dabkowska, H.A.; Gaulin, B.D.; Zhou, J.S.; Pyon, S.; et al. Thermodynamic signatures of quantum criticality in cuprate superconductors. Nature
**2019**, 567, 218–222. [Google Scholar] [CrossRef] [Green Version] - 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. Commun. Phys.
**2021**, 4, 7. [Google Scholar] [CrossRef] - Caprara, S.; Di Castro, C.; Mirarchi, G.; Seibold, G.; Grilli, M. Dissipation-driven strange metal behavior. Commun. Phys.
**2022**, 5, 10. [Google Scholar] [CrossRef] - Arpaia, R.; Caprara, S.; Fumagalli, R.; De Vecchi, G.; Peng, Y.Y.; Andersson, E.; Betto, D.; De Luca, G.M.; Brookes, N.B.; Lombardi, F.; et al. Dynamical charge density fluctuations pervading the phase diagram of a Cu-based high-T
_{c}superconductor. Science**2019**, 365, 906–910. [Google Scholar] [CrossRef] [Green Version] - 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)Ba
_{2}Cu_{3}O_{6+x}. Science**2012**, 337, 821–825. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Vaknin, D.; Sinha, S.K.; Moncton, D.E.; Johnston, D.C.; Newsam, J.M.; Safinya, C.R.; King, H.E., Jr. Antiferromagnetism in La
_{2}CuO_{4−y}. Phys. Rev. Lett.**1987**, 58, 2802. [Google Scholar] [CrossRef] [PubMed] - Chakravarty, S.; Halperin, B.I.; Nelson, D.R. Low-temperature behavior of two-dimensional quantum antiferromagnets. Phys. Rev. Lett.
**1988**, 60, 1057. [Google Scholar] [CrossRef] - Lee, P.A.; Nagaosa, N.; Wen, X. Doping a Mott Insulator: Physics of High Temperature Superconductivity. Rev. Mod. Phys.
**2006**, 78, 17–85. [Google Scholar] [CrossRef] - Ding, H.; Campuzano, J.C.; Norman, M.R.; Randeria, M.; Yokoya, T.; Takahashi, T.; Takeuchi, T.; Mochiku, T.; Kadowaki, K.; Guptasarma, P.; et al. ARPES study of the superconducting gap and pseudogap in Bi
_{2}Sr_{2}CaCu_{2}O_{8+d}. J. Phys. Chem. Solids**1998**, 59, 1888. [Google Scholar] [CrossRef] [Green Version] - Zajcewa, I.; Chrobak, M.; Maćkosz, K.; Jurczyszyn, M.; Minikayev, R.; Abaloszew, A.; Cieplak, M.Z. Pseudogap behavior in single-crystal Bi
_{2}Sr_{2}CaCu_{2}O_{8+δ}probed by Cu NMR. Phys. Rev. B**1998**, 58, 015009. [Google Scholar] - Billinge, S.J.L.; Gutmann, M.; Božin, E.S. Structural Response to Local Charge Order in Underdoped but Superconducting La
_{2ࢤx}(Sr,Ba)_{x}CuO_{4}. Int. J. Mod. Phys. B**1998**, 59, 1888. [Google Scholar] [CrossRef] [Green Version] - Saini, N.L.; Avila, J.; Bianconi, A.; Lanzara, A.; Asensio, M.C.; Tajima, S.; Gu, G.D.; Koshizuka, N. Topology of the Pseudogap and Shadow Bands in Bi
_{2}Sr_{2}CaCu_{2}O_{8+δ}at Optimum Doping. Phys. Rev. Lett.**2003**, 17, 3467–3470. [Google Scholar] - 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 La
_{1.875}Ba_{0.125}CuO_{4}without spin-charge locking. Proc. Natl. Acad. Sci. USA**2017**, 114, 12430. [Google Scholar] [CrossRef] [PubMed] [Green Version] - 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. X
**2019**, 9, 031042. [Google Scholar] [CrossRef] [Green Version] - Valla, T.; Fedorov, A.V.; Johnson, P.D.; Wells, B.O.; Hulbert, S.L.; Li, Q.; Gu, G.D.; Koshizuka, N. Evidence for Quantum Critical Behavior in the Optimally Doped Cuprate Bi
_{2}Sr_{2}CaCu_{2}O_{8+δ}. Science**1999**, 285, 2110–2113. [Google Scholar] [CrossRef] [PubMed] - Chen, S.D.; Hashimoto, M.; He, Y.; Song, D.; Xu, K.J.; He, J.F.; Devereaux, T.P.; Eisaki, H.; Lu, D.H.; Zaanen, J.; et al. Incoherent strange metal sharply bounded by a critical doping in Bi2212. Science
**2019**, 266, 1099–1102. [Google Scholar] [CrossRef] - Wahlberg, E.; Arpaia, R.; Seibold, G.; Rossi, M.; Fumagalli, R.; Trabaldo, E.; Brookes, N.B.; Braicovich, L.; Caprara, S.; Gran, U.; et al. Restored strange metal phase through suppression of charge density waves in underdoped YBa
_{2}Cu_{3}O_{7-δ}. Science**2021**, 373, 1506–1510. [Google Scholar] [CrossRef] - Castellani, C.; Di Castro, C.; Grilli, M. Singular Quasiparticle Scattering in the Proximity of Charge Instabilities. Phys. Rev. Lett.
**1995**, 75, 4650. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hertz, J.A. Quantum critical phenomena. Phys. Rev. B
**1976**, 14, 1165. [Google Scholar] [CrossRef] - Millis, A.J. Effect of a nonzero temperature on quantum critical points in itinerant fermion systems. Phys. Rev. B
**1993**, 48, 7183. [Google Scholar] [CrossRef] - Perali, A.; Castellani, C.; Di Castro, C.; Grilli, M. d-wave superconductivity near charge instabilities. Phys. Rev. B
**1996**, 54, 16216. [Google Scholar] [CrossRef] [Green Version] - Gerber, S.; Jang, H.; Nojiri, H.; Matsuzawa, S.; Yasumura, H.; Bonn, D.A.; Liang, R.; Hardy, W.N.; Islam, Z.; Mehta, A.S.; et al. Three-dimensional charge density wave order in YBa
_{2}Cu_{3}O_{6.67}at high magnetic fields. Science**2015**, 350, 949–952. [Google Scholar] [CrossRef] [Green Version] - 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 YBa
_{2}Cu_{3}O_{y}. Nature**2011**, 477, 191–194. [Google Scholar] [CrossRef] [PubMed] - Tranquada, J.M.; Sternlieb, B.J.; Axe, J.D.; Nakamura, Y.; Uchida, S. Evidence for stripe correlations of spins and holes in copper oxide. Nature
**1995**, 375, 561. [Google Scholar] [CrossRef] - Tranquada, J.M.; Axe, J.D.; Ichikawa, N.; Nakamura, Y.; Uchida, S.; Nachumi, B. Neutron-scattering study of stripe-phase order of holes and spins in La
_{1.48}Nd_{0.4}Sr_{0.12}CuO_{4}. Phys. Rev. B**1996**, 54, 7489–7499. [Google Scholar] [CrossRef] - Bianconi, A.; Bianconi, G.; Caprara, S.; Di Castro, D.; Oyanagi, H.; Saini, N.L. The stripe critical point for cuprates. J. Phys. Condens. Matter.
**2000**, 12, 10655. [Google Scholar] [CrossRef] [Green Version] - Di Castro, C. Revival of Charge Density Waves and Charge Density Fluctuations in Cuprate High-Temperature Superconductors. Condens. Matter.
**2020**, 5, 70. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef] [Green Version] - 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. [Google Scholar] [CrossRef] [PubMed] - 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. [Google Scholar] [CrossRef] - Andergassen, S.; Caprara, S.; Di Castro, C.; Grilli, M. Anomalous Isotopic Effect Near the Charge-Ordering Quantum Criticality. Phys. Rev. Lett.
**2001**, 87, 056401. [Google Scholar] [CrossRef] - Caprara, S.; Di Castro, C.; Seibold, G.; Grilli, M. Dynamical charge density waves rule the phase diagram of cuprates. Phys. Rev. B
**2017**, 95, 224511. [Google Scholar] [CrossRef] [Green Version] - Bianconi, A.; Saini, N.L.; Lanzara, A.; Missori, M.; Rossetti, T.; Oyanagi, H.; Yamaguchi, H.; Oka, K.; Ito, T. Determination of the Local Lattice Distortions in the CuO
_{2}Plane of La_{1.85}Sr_{0.15}CuO_{4}. Phys. Rev. Lett.**1996**, 76, 3412. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hoffman, J.E.; Hudson, E.W.; Lang, K.M.; Madhavan, V.; Eisaki, H.; Uchida, S.; Davis, J.C. A Four Unit Cell Periodic Pattern of Quasi-Particle States Surrounding Vortex Cores in Bi
_{2}Sr_{2}CaCu_{2}O_{8+δ}. Science**2002**, 295, 466–469. [Google Scholar] [CrossRef] [Green Version] - 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 YBa
_{2}Cu_{3}O_{6.67}. Nat. Phys.**2012**, 8, 871–876. [Google Scholar] [CrossRef] [Green Version] - 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 Bi
_{2}Sr_{2−x}La_{x}CuO_{6+δ}. Science**2013**, 343, 390. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Blanco-Canosa, 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 YBa
_{2}Cu_{3}O_{6+x}. Phys. Rev. B**2014**, 90, 054513. [Google Scholar] [CrossRef] [Green Version] - 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-T
_{c}superconductor. Nature**2015**, 525, 359–362. [Google Scholar] [CrossRef] - Comin, R.; Damascelli, A. Resonant X-Ray Scattering Studies of Charge Order in Cuprates. Annu. Rev. Condens. Matter Phys.
**2016**, 7, 369–405. [Google Scholar] [CrossRef] [Green Version] - Keimer, B.; Kivelson, S.A.; Norman, M.R.; Uchida, S.; Zaanen, J. From quantum matter to high-temperature superconductivity in copper oxides. Nature
**2015**, 518, 179–186. [Google Scholar] [CrossRef] - Zaanen, J. Why the temperature is high. Nature
**2004**, 430, 512–513. [Google Scholar] [CrossRef] [Green Version] - Caprara, S.; Sulpizi, M.; Bianconi, A.; Di Castro, C.; Grilli, M. Single-particle properties of a model for coexisting charge and spin quasi-critical fluctuations coupled to electrons. Phys. Rev. B
**1999**, 59, 14980. [Google Scholar] [CrossRef] [Green Version] - Hlubina, R.; Rice, T.M. Resistivity as a function of temperature for models with hot spots on the Fermi surface. Phys. Rev. B
**1995**, 51, 9253. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Castellani, C.; Di Castro, C.; Grilli, M. Non-Fermi-liquid behavior and d-wave superconductivity near the charge-density-wave quantum critical point. Z. Phys. B
**1996**, 103, 137. [Google Scholar] [CrossRef] [Green Version] - Hussey, N.E. Phenomenology of the normal state in-plane transport properties of high-T
_{c}cuprates. J. Phys. Condens. Matter**2008**, 20, 123201. [Google Scholar] [CrossRef] [Green Version] - Mazza, G.; Grilli, M.; Di Castro, C.; Caprara, S. Evidence for phonon-like charge and spin fluctuations from an analysis of angle-resolved photoemission spectra of La
_{2−x}Sr_{x}CuO_{4}superconductors. Phys. Rev. B**2013**, 87, 014511. [Google Scholar] [CrossRef] [Green Version] - Wagner, H.; Mermin, N.D. Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models. Phys. Rev. Lett.
**1966**, 17, 1133. [Google Scholar] - Walmsley, P.; Putzke, C.; Malone, L.; Guillamón, I.; Vignolles, D.; Proust, C.; Badoux, S.; Coldea, A.I.; Watson, M.D.; Kasahara, S.; et al. Quasiparticle Mass Enhancement Close to the Quantum Critical Point in BaFe
_{2}(As_{1−x}P_{x})_{2}. Phys. Rev. Lett.**2013**, 110, 257002. [Google Scholar] [CrossRef] [Green Version] - Stewart, G.R. Non-Fermi-liquid behavior in d- and f- electron metals. Rev. Mod. Phys.
**2001**, 73, 797. [Google Scholar] [CrossRef]

**Figure 1.**Typical temperature vs. doping phase diagram of cuprates. At temperatures T larger than ${T}_{\mathrm{c}}$, the phase diagram appears to be divided into three main regions; however, it is important to keep in mind that this division is not necessarily neat and the partitioning lines might be understood as crossover temperatures between different regimes. The regions denoted with AF and SC indicate, respectively, the antiferromagnetic and superconducting phases. These regions are sharply delimited by the Néel temperature ${T}_{\mathrm{N}}$ and superconducting critical temperature ${T}_{\mathrm{c}}$, respectively.

**Figure 2.**Plot of ${C}_{V}^{b}/T$ as a function of ${\overline{\nu}}_{\perp}/{\overline{\nu}}_{\perp}^{\mathrm{max}}$ for $0<{\overline{\nu}}_{\perp}<{\overline{\nu}}_{\perp}^{\mathrm{max}}$. Parameter values are $m=$15 meV, $\overline{\nu}=$$1.3$ eV/(r.l.u.)${}^{2}$, $\gamma =1$, $\overline{\mathsf{\Omega}}=$30 meV and $d/a=11$.

**Figure 3.**Plot of the resistivity as a function of temperature. Parameter values, when set, are $m=$ 15 meV, $\overline{\nu}=$ $1.3$ eV/(r.l.u.)${}^{2}$, $\gamma =1$, $\overline{\mathsf{\Omega}}=$ 30 meV, ${g}^{2}=0.0415$ and ${\mathsf{\Gamma}}_{0}=$$12.3$ meV. Note that the resistivity value at $T=$0 K is greater than zero and it is the same for all curves, as this value is determined solely by ${\mathsf{\Gamma}}_{0}$.

**Figure 4.**Plot of the resistivity as a function of temperature for different values of $\gamma $. Parameter values are $m=$15 meV, $\overline{\nu}=$$1.3$ eV/(r.l.u.)${}^{2}$, $\overline{\mathsf{\Omega}}=$30 meV, ${g}^{2}=0.0415$ and ${\mathsf{\Gamma}}_{0}=$$12.3$ meV.

**Figure 5.**For each curve, we show the plot of the specific heat at $T=$1 K when a single parameter changes, leaving all the others fixed. Parameter values, when set, are $m=$15 meV, $\overline{\nu}=$$1.3$ eV/(r.l.u.)${}^{2}$, $\gamma =1$ and $\overline{\mathsf{\Omega}}=$30 meV.

**Figure 6.**Plot of the specific heat as a function of the doping level for different temperatures. Parameter values are $m=$15 meV, $\overline{\nu}=$$1.3$ eV/(r.l.u.)${}^{2}$, $\overline{\mathsf{\Omega}}=$30 meV.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mirarchi, G.; Seibold, G.; Di Castro, C.; Grilli, M.; Caprara, S.
The Strange-Metal Behavior of Cuprates. *Condens. Matter* **2022**, *7*, 29.
https://doi.org/10.3390/condmat7010029

**AMA Style**

Mirarchi G, Seibold G, Di Castro C, Grilli M, Caprara S.
The Strange-Metal Behavior of Cuprates. *Condensed Matter*. 2022; 7(1):29.
https://doi.org/10.3390/condmat7010029

**Chicago/Turabian Style**

Mirarchi, Giovanni, Götz Seibold, Carlo Di Castro, Marco Grilli, and Sergio Caprara.
2022. "The Strange-Metal Behavior of Cuprates" *Condensed Matter* 7, no. 1: 29.
https://doi.org/10.3390/condmat7010029