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arXiv:2205.04030 · Nature Communications

Reconciling scaling of the optical conductivity of cuprate superconductors with Planckian resistivity and specific heat

B. Michon, Christophe Berthod, Carl Willem Rischau, Amirreza Ataei, Lu Chen, Seiki Komiya, Shimpei Ono, Louis Taillefer, D. van der Marel, Antoine Georges

Published May 26, 2023• 59 citations
arXivHTMLPDF

Abstract

Abstract Materials tuned to a quantum critical point display universal scaling properties as a function of temperature T and frequency ω. A long-standing puzzle regarding cuprate superconductors has been the observed power-law dependence of optical conductivity with an exponent smaller than one, in contrast to T-linear dependence of the resistivity and ω-linear dependence of the optical scattering rate. Here, we present and analyze resistivity and optical conductivity of La2−xSrxCuO4 with x = 0.24. We demonstrate ℏω/kBT scaling of the optical data over a wide range of frequency and temperature, T-linear resistivity, and optical effective mass proportional to $$\sim \ln T$$ ~ ln T corroborating previous specific heat experiments. We show that a T, ω-linear scaling Ansatz for the inelastic scattering rate leads to a unified theoretical description of the experimental data, including the power-law of the optical conductivity. This theoretical framework provides new opportunities for describing the unique properties of quantum critical matter.

DOI
10.1038/s41467-023-38762-5

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Abstract Materials tuned to a quantum critical point display universal scaling properties as a function of temperature T and frequency ω. A long-standing puzzle regarding cuprate superconductors has been the observed power-law dependence of optical conductivity with an exponent smaller than one, in contrast to T-linear dependence of the resistivity and ω-linear dependence of the optical scattering rate. Here, we present and analyze resistivity and optical conductivity of La2−xSrxCuO4 with x = 0.24. We demonstrate ℏω/kBT scaling of the optical data over a wide range of frequency and temperature, T-linear resistivity, and optical effective mass proportional to ∼ln⁡T\sim \ln T∼lnT ~ ln T corroborating previous specific heat experiments. We show that a T, ω-linear scaling Ansatz for the inelastic scattering rate leads to a unified theoretical description of the experimental data, including the power-law of the optical conductivity. This theoretical framework provides new opportunities for describing the unique properties of quantum critical matter.
The mysterious metallic phase showing perfect TTT-linear resistivity and a universal scattering rate 1/τ=αPkBT/ℏ1/τ= α_P k_B T /\hbar1/τ=αP​kB​T/ℏ with a universal prefactor αP∼1α_P \sim 1αP​∼1 and logarithmic-in-temperature singular specific heat coefficient, so-called Planckian metal phase was observed in various overdoped high-TcT_cTc​ cuprate superconductors over a finite range in doping. Here, we propose a microscopic mechanism for this exotic state based on quantum-critical bosonic charge Kondo fluctuations coupled to both spinon and a heavy conduction-electron Fermi surfaces within the heavy-fermion formulation of the slave-boson ttt-JJJ model. Using a controlled perturbative renormalization group (RG) analysis, we examine the competition between the pseudogap phase, characterized by Anderson's Resonating-Valence-Bond spin-liquid, and the Fermi-liquid state, characterized by the electron hoping (effective charge Kondo effect). We find a quantum-critical metallic phase with a universal Planckian ℏω/kBT\hbar ω/k_B Tℏω/kB​T scaling in scattering rate near a localized-delocalized (pseudogap-to-Fermi liquid) charge Kondo breakdown transition. Our results are in excellent agreement with the recent experimental observations on optical conductivity (without fine-tuning) in Nat. Commun. 14, 3033 (2023), universal doping-independent field-to-temperature scaling in magnetoresistance in Nature 595, 661 (2021), and the marginal Fermi-liquid spectral function observed in ARPES (Science 366, 1099 (2019)) as well as Hall coefficient in various overdoped cuprates in Nature 595, 661 (2021) and Annu. Rev. Condens. Matter Phys. 10, 409 (2019). Our mechanism offers a microscopic understanding of the quantum-critical Planckian metal phase observed in cuprates d-wave superconducting, and Fermi liquid phases.
Abstract The mysterious metallic phase showing T-linear resistivity and a universal scattering rate 1 / τ = α P k B T / ℏ with a universal prefactor α P ∼ 1 and logarithmic-in-temperature singular specific heat coefficient, the so-called ‘Planckian metal phase’ was observed in various overdoped high- T c cuprate superconductors over a finite range in doping. Revealing the mystery of the Planckian metal state is believed to be the key to understanding the mechanism for high- T c superconductivity. Here, we propose a generic microscopic mechanism for this state based on quantum-critical local bosonic charge Kondo fluctuations coupled to both spinon and a heavy conduction-electron Fermi surface within the heavy-fermion formulation of the slave-boson t–J model. By a controlled perturbative renormalization group analysis, we examine the competition between the pseudogap phase, characterized by Anderson’s Resonating-Valence-Bond spin-liquid, and the Fermi-liquid state, modeled by the electron hopping (effective charge Kondo effect). We find a quantum-critical metallic phase with a universal Planckian ℏ ω / k B T scaling in scattering rate near an extended localized-delocalized (pseudogap-to-Fermi liquid) charge-Kondo breakdown transition. The d-wave superconducting ground state emerges near the transition. Unprecedented qualitative and quantitative agreements are reached between our theoretical predictions and various experiments, including optical conductivity, universal doping-independent field-to-temperature scaling in magnetoresistance, specific heat coefficient, marginal Fermi-liquid spectral function observed in ARPES, and Fermi surface reconstruction observed in Hall coefficients in various overdoped cuprates. Our mechanism offers a microscopic understanding of the quantum-critical Planckian metal phase observed in cuprates and its link to the pseudogap, d-wave superconducting, and Fermi liquid phases. It offers a promising route for understanding how d-wave superconductivity emerges from such a strange metal phase in cuprates–one of the long-standing open problems in condensed matter physics since 1990s–as well as shows a broader implication for the Planckian strange metal states observed in other correlated unconventional superconductors.
The mysterious quantum-critical Planckian bad metal phase with perfect T-linear resistivity persisting beyond the quasi-particle limit and universal T-linear scattering rate has been observed in various high-Tc cuprate superconductors. Here, we develop a realistic theoretical approach to this phase in an analytically solvable large-N multi-channel Kondo lattice model, derived from a heavy-fermion formulated conventionaL t-J model, known for qualitatively describing cuprates. This phase is originated from critical charge Kondo fluctuations where disordered local bosonic charge fluctuations couple to spinon and heavy conduction-electron Fermi surfaces near a charge-Kondo-breakdown local quantum critical point associated with pseudogap-to-Fermi liquid transition. Our results show excellent agreement with experiments and offer broad implications for other unconventional superconductors.
We study the frequency and temperature dependence of the optical conductivity in the weakly coupled two-dimensional Hubbard model using a renormalized perturbative expansion. The perturbative expansion is based on the skeleton series for the current-current correlation function with a dressed Green`s function and the results are obtained directly on the real frequency axis using Algorithmic Matsubara Integration (AMI). The resulting conductivity shows a temperature-independent power law behaviour in the intermediate frequency regime. Moreover, the associated transport scattering time and renormalized mass exhibit a Planckian behaviour. We show that the self-energy of the Hubbard model, however, is distinct from existing Planckian models. The Planckian behaviour of the conductivity, observed in optimally doped cuprates for example, can thus be obtained from a different form of self-energy than the Planckian model, such as the weakly coupled Hubbard model at half-filling.
According to Zaanen's interpretation of Homes' empirical law~[Zaanen, {\it Nature} {\bf 430}, 512 (2004)], the superconducting transition temperatures in the cuprates are high because their metallic states are as viscous as quantum mechanics permits. Here, we show that Homes' law in fact implies three key points: (i) the resistivity is linear in temperature in the normal state near the transition temperature; (ii) the dimensionless coefficient of proportionality of the relaxation rate with temperature is of order unity -- the so-called universal Planckian relaxation rate; and (iii) the logarithmically broad applicability of this law arises from an unusually wide range of effective masses throughout the cuprate phase diagram. In fact, a universal Planckian relaxation rate implies Homes' law only if the mechanism of mass renormalization is independent of the Planckian relaxation.
We use the Kubo response functions to calculate the electrical and thermal conductivity and Seebeck coefficient at low temperatures and frequencies in the quantum-critical region for fermions on a lattice. The theory uses scattering of the fermions with the previously derived collective fluctuations due to topological defects of the quantum XY model coupled to fermions. The microscopic model is applicable to the fluctuations of the loop-current order in cuprates as well as to a class of quasi-two-dimensional heavy-fermion and other metallic antiferromagnets, and proposed recently also for the possible loop-current order in Moiré twisted bi-layer graphene and bilayer WSe2_22​. All these metals have a linear-in-temperature electrical resistivity in the quantum-critical region of their phase diagrams, often termed ``Planckian" resistivity. The solution of the Kubo equation for transport shows that vertex renormalizations to the external fields, beside those caused by Aslamazov-Larkin (A-L) processes, are absent. A-L appears as an Umklapp scattering matrix, which gives a temperature-independent multiplicative factor for the electrical resistivity but does not affect the thermal conductivity. We also show that the mass renormalization which gives a logarithmic enhancement of the marginal Fermi-liquid specific heat does not appear in the electrical resistivity and, more remarkably, in the thermal conductivity. On the other hand the mass renormalization ∝ln⁡ωc/T\propto \ln ω_c/T∝lnωc​/T appears in the Seebeck coefficient. We also discuss in detail the conservation laws which play a crucial role in all transport properties. We calculate exactly, the numerical coefficients of the transport properties for a circular Fermi surface. The leading temperature dependences is shown to remain the same for a general Fermi surface, but it is too messy to calculate the numerical coefficient.