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.
Understanding the origin of electron incoherence is believed to be the first step toward the resolution of the mysteries of the high-Tc 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.
Exact formulas for the optical conductivity and the Hall conductivity of the two dimensional Hubbard model are derived in terms of an effective theory description of the local moment fluctuation in the system. In this framework, the quantum Monte Carlo simulation of the electromagnetic response of such a strongly correlated electron system becomes sign-problem-free in many physically relevant cases. In particular, it is sign-problem-free when we assume the widely used Millis-Monien-Pines form for the phenomenological susceptibility in the effective action of the fluctuating local moment, even though these local moments are now subjected to Landau damping as a result of their coupling to the itinerant quasiparticle on the fermi surface. This is true more generally when a φ4 term is included in the effective action and is thus not restricted to the Gaussian limit. Here we demonstrate the power of this framework by studying the effect of thermal fluctuation of the local moment on the optical conductivity σxx(ω) and the Hall conductivity σxy(ω) of the cuprate superconductors. Both σxx(ω) and σxy(ω) calculated are found to exhibit a two-component structure, with a Drude component at low energy and a mid-infrared component at higher energy. Depending on the relative importance of the hole pocket and the electron pocket on the reconstructed fermi surface and the coupling strength to the local moment, the Drude component in Imσxy(ω) can be either positive or negative.(full-length abstract can be found in the main text.)
The optical conductivity of the 2D t−J model and the origin of electron incoherence in the high-Tc cuprate superconductors: a variational study
Understanding the origin of electron incoherence is the first step toward a theoretical description of the non-Fermi liquid behavior of the high-Tc cuprate superconductors. Such electron incoherence manifests itself most evidently in the non-Drude behavior of the optical response of the system and the anomalous density fluctuation behavior in the long wave length limit. 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 in the normal state and the superfluid stiffness in the superconducting state. It is found that such electron incoherence remains significant even in the clean limit and at low temperature and thus must be attributed to the strong electron correlation effect in the cuprate superconductors. Here we study such an intrinsic effect in the 2D t−J model through the variational calculation of its optical conductivity σ(ω). We assume a resonating valence bond ground state as our starting point and find that a significant portion of the total optical spectral weight remains incoherent throughout the phase diagram. The optical absorption is found to extend all the way to an energy of the order of the bare band width. We find that both the total optical weight Kˉ and the integrated incoherent optical weight I increase monotonically with doping, with their ratio Rincoh=I/Kˉ decreasing monotonically with doping. Our results indicate that the majority part of electron incoherence in the 2D t−J model can be attributed to the electron fractionalization mechanism assumed in such a treatment. We also find that the Drude weight deduced from D=Kˉ−I scales linearly with hole doping, without any sign of a non-monotonic behavior in the overdoped regime.
A Monte Carlo maximum-entropy calculation of the optical conductivity of the infinite-dimensional Hubbard model is presented. We show that the optical conductivity displays the anomalies found in the cuprate superconductors, including a Drude width which grows linearly with temperature, a Drude weight which grows linearly with doping, and a temperature and doping-dependent mid-IR peak. These anomalies arise as a consequence of the dynamical generation of a quasiparticle band at the Fermi energy as T\ensuremath{\rightarrow}0, and are a generic property of the strongly correlated Hubbard model in all dimensions greater than one.
We study the Cu and O spectral density of states and the optical conductivity of CuO2 planes using an effective generalized one-band Hubbard model derived from the extended three-band Hubbard model. We solve exactly a square cluster of 10 unit cells and average the results over all possible boundary conditions, which leads to smooth functions of frequency. Upon doping, the Fermi energy jumps to Zhang-Rice states which are connected to the rest of the valence band (in contrast to an isolated band in the middle of the gap). The transfer of spectral weight depends on the parameters of the original three-band model not only through the one-band effective parameters but also through the relevant matrix elements. We discuss the evolution of the gap upon doping. The optical conductivity of the doped system shows a mid-infrared peak due to intraband transitions, a pseudogap, and a high-frequency part related to interband transitions. Its shape and integrated weight up to a given frequency (including the Drude weight) agree qualitatively with experiments in the cuprates for low to moderate doping levels, but significant deviations exist for doping x>0.3.
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].
On the basis of exact diagonalizations, a comparative study of two-particle optical and magnetic, as well as single-particle, excitations is presented for a two-dimensional (2D) multiorbital Hubbard model. For reasonable parameter sets appropriate for the cuprate superconductors, the single-particle excitations display strongly correlated states related to the Zhang-Rice Cu-O singlet construction. These states define the gap (to the upper Hubbard band) at half-filling and become partially occupied by doping holes in our 2\ifmmode\times\else\texttimes\fi{}2 unit-cell system. The optical results, which are the first quantitative calculations performed for realistic parameters of the three-band Hubbard model, clearly show three allowed optical transitions: (i) itinerant motion of the Cu-O singlets, having (for doping concentrations x\ensuremath{\ne}0) a spectral Drude distribution around \ensuremath{\omega}=0 with spectral weight proportional to x; (ii) unbinding of the O hole from the Cu spin in the singlet. This gives, in particular, a strong absorption peak due to singlet\ensuremath{\rightarrow}nonbonding oxygen transitions, again with relative weight \ensuremath{\sim}x.It is roughly centered at \ensuremath{\omega}\ensuremath{\sim}JKondo, where JKondo is the Kondo coupling and \ensuremath{\Delta} the bare Cu-O charge-transfer energy. It is this singlet unbinding that results in the by far dominant absorption structure between Drude and higher-energy Cu-O transitions and not the often discussed ``mid-infrared'' absorption due to transitions between different singlet states. (iii) Cu-O charge-transfer processes at energy \ensuremath{\sim}\ensuremath{\Delta}+Upd. They show a pronounced excitonic effect due to the p-d interaction Upd and have a reduced spectral weight shifted to higher energies for increased dopings. Findings (i)--(iii) are in general accordance with recent experimental data. Our study of the low-energy absorption is complemented with a numerical scaling analysis of the Drude weight in 1D, where, in particular, we find an interesting violation of Lenz's law for 4n-site Hubbard rings. Finally, the magnetic structure factor is calculated for the 2D case. For finite doping it contains a peak at 2JKondo, which should be detectable in experiment.