We derive analytical expressions for the spectral moments of the dynamical response functions of the Hubbard model using the high-temperature series expansion. We consider generic dimension
d as well as the infinite-
d limit, arbitrary electron density
n, and both finite and infinite repulsion
U. We use moment-reconstruction methods to obtain the one-electron spectral function, the self-energy, and the optical conductivity. They are all smooth functions at high temperature and, at large
U, they are featureless with characteristic widths of the order of the lattice hopping parameter
t. In the infinite-
d limit, we compare the series expansion results with accurate numerical renormalization group and interaction expansion quantum Monte Carlo results. We find excellent agreement down to surprisingly low temperatures, throughout most of the bad-metal regime, which applies for $T\ensuremath{\gtrsim}(1\ensuremath{-}n)D$, the Brinkman-Rice scale. The resistivity increases linearly in
T at high temperature without saturation. This results from the
1/T behavior of the compressibility or kinetic energy, which play the role of the effective carrier number. In contrast, the scattering time (or diffusion constant) saturates at high
T. We find that $\ensuremath{\sigma}(n,T)\ensuremath{\approx}(1\ensuremath{-}n)\ensuremath{\sigma}(n=0,T)$ to a very good approximation for all
n, with $\ensuremath{\sigma}(n=0,T)\ensuremath{\propto}t/T$ at high temperatures. The saturation at small
n occurs due to a compensation between the density dependence of the effective number of carriers and that of the scattering time. The
T dependence of the resistivity displays a kneelike feature which signals a crossover to the intermediate-temperature regime where the diffusion constant (or scattering time) starts increasing with decreasing
T. At high temperatures, the thermopower obeys the Heikes formula, while the Wiedemann-Franz law is violated with the Lorenz number vanishing as
1/T2. The relevance of our calculations to experiments probing high-temperature transport in materials with strong electronic correlations or ultracold atomic gases in an optical lattice is briefly discussed.
Recent progress in experimental techniques has made it possible to extract detailed information on dynamics of carriers in a correlated electron material from its optical conductivity, [Formula: see text]. This review consists of three parts, addressing the following three aspects of optical response: (1) the role of momentum relaxation; (2) [Formula: see text] scaling of the optical conductivity of a Fermi-liquid metal, and (3) the optical conductivity of non-Fermi-liquid metals. In the first part (section 2), we analyze the interplay between the contributions to the conductivity from normal and umklapp electron-electron scattering. As a concrete example, we consider a two-band metal and show that although its optical conductivity is finite it does not obey the Drude formula. In the second part (sections 3 and 4), we re-visit the Gurzhi formula for the optical scattering rate, [Formula: see text], and show that a factor of [Formula: see text] is the manifestation of the 'first-Matsubara-frequency rule' for boson response, which states that [Formula: see text] must vanish upon analytic continuation to the first boson Matsubara frequency. However, recent experiments show that the coefficient b in the Gurzhi-like form, [Formula: see text], differs significantly from b = 4 in most of the cases. We suggest that the deviations from Gurzhi scaling may be due to the presence of elastic but energy-dependent scattering, which decreases the value of b below 4, with b = 1 corresponding to purely elastic scattering. In the third part (section 5), we consider the optical conductivity of metals near quantum phase transitions to nematic and spin-density-wave states. In the last case, we focus on 'composite' scattering processes, which give rise to a non-Fermi-liquid behavior of the optical conductivity at T = 0: [Formula: see text] at low frequencies and [Formula: see text] at higher frequencies. We also discuss [Formula: see text] scaling of the conductivity and show that [Formula: see text] in the same model scales in a non-Fermi-liquid way, as [Formula: see text].
We study dynamical scaling in the quantum-critical fan of the pseudogap-metal to Fermi-liquid transition of the two-dimensional Hubbard model. Using a four-patch dynamical cluster approximation with the numerical renormalization group as a cluster impurity solver, we access real-frequency dynamics over several decades at arbitrary temperatures. Close to the critical doping, the local spin and cluster-current susceptibility spectra exhibit
x=ω/T scaling of the form
χ′′(ω,T)∼tanh(x/2), and the cluster contribution to the optical conductivity obeys
Tσcl′(ω,T)∼tanh(x/2)/x, implying a
1/T cluster dc conductivity. In the scaling regime, the vertex contribution to the cluster optical response is much larger than the bubble contribution. We further find evidence for a marginal-Fermi-liquid nodal self-energy. This, together with the
1/T vertex contribution to the conductivity, implies strange-metal optical transport in the quantum critical region. Our results describe several qualitative aspects of several experimental observations.
Results for the optical conductivity and resistivity of the Hubbard model in infinite spatial dimensions are presented. At half-filling we observe a gradual crossover from a metal with a Drude peak at ω = 0 in the optical conductivity to an insulator as a function of U for temperatures above the antiferromagnetic phase transition. When doped, the "insulator" becomes a Fermi liquid with a corresponding temperature dependence of the optical conductivity and resistivity. We find a T 2 -coefficient in the low-temperature resistivity which suggests that the carriers in the system acquire a considerable mass-enhancement due to the strong local correlations. At high temperatures, a crossover into a semi-metallic regime takes place.
A bstract Semi-holography, originally proposed as a model for conducting lattice electrons coupled to a holographic critical sector, leads to an effective theory of non-Fermi liquids with only a few relevant interactions on the Fermi surface in the large N limit. A refined version of such theories has only two effective couplings, which give holographic and Fermi-liquid-like contributions to the self-energy, respectively. We show that a low co-dimension sub-manifold exists in the space of refined semi-holographic theories in which strange metallic behavior is manifested and which can be obtained just by tuning the ratio of the two couplings. On this sub-manifold, the product of the spectral function and the temperature is approximately independent of the critical exponent, the Fermi energy, and the temperature at all frequencies and near the Fermi surface when expressed in terms of suitably scaled momentum and frequency variables. This quasi-universal behavior leads to linear-in- T dc resistivity and Planckian dissipation over a large range of temperatures, and we also obtain T −3 scaling of the Hall conductivity at higher temperatures. The quasi-universal spectral function also fits well with photoemission spectroscopic data without varying the critical exponent with the doping. Combining with the results for optical conductivity, we construct a generalized version of Drude phenomenology for strange-metallic behavior, which satisfies non-trivial consistency tests. Finally, we discuss a possible dynamical mechanism for the fine-tuning of the ratio of the two couplings necessary to realize the strange metallic behavior in a typical state.
Understanding the origin of electron incoherence is believed to be the first step toward the resolution of the mysteries of the high-T
c cuprate superconductors. Such electron incoherence manifests itself most evidently in the non-Drude form of the optical absorption spectrum of the system. The spectral weight transfer related to such dissipative response, which is absent in conventional Fermi liquid metal, has direct consequence on the dc transport property of the system. However, a theoretical study of the optical conductivity of a strongly correlated model is a formidable task. Here we present an exact formula for the optical conductivity of the 2D Hubbard model from the low energy effective theory perspective. We show that the optical conductivity in Matsubara frequency of the 2D Hubbard model can be represented as the ensemble average of the optical conductivity of non-interacting systems in the background of fluctuating local moment. We find that such an ensemble average can be done exactly with a sign-problem-free Monte Carlo simulation if we assume the widely adopted Millis-Monien-Pines spin susceptibility for the fluctuating local moment. For thermal fluctuation of the local moment, our formula can be used to calculate directly the optical conductivity in real frequency which can be compared with the result of optical measurements in the cuprate superconductors.