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arXiv:2401.00118

The optical conductivity of the 2D $t-J$ model and the origin of electron incoherence in the high-T$_{c}$ cuprate superconductors: a variational study

Jianhua Yang, Tao Li

Published Dec 30, 2023• 0 citations
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Abstract

Understanding the origin of electron incoherence is the first step toward a theoretical description of the non-Fermi liquid behavior of the high-T$_{c}$ 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 $\bar{K}$ and the integrated incoherent optical weight $I$ increase monotonically with doping, with their ratio $R_{incoh}=I/\bar{K}$ 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 $D=\bar{K}-I$ scales linearly with hole doping, without any sign of a non-monotonic behavior in the overdoped regime.

DOI
10.48550/arxiv.2401.00118

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The optical conductivity of the 2D t−Jt-Jt−J model and the origin of electron incoherence in the high-Tc_{c}c​ cuprate superconductors: a variational study
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Understanding the origin of electron incoherence is believed to be the first step toward the resolution of the mysteries of the high-Tc_{c}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.
A major puzzle in the study of the cuprate superconductivity is the origin of the pseudogap end point. Intriguingly, such a critical doping is also where the superfluid density of the system reaches its maximum. A non-monotonic doping dependence of the superfluid density is rather unusual since the Drude weight of the cuprate system is found to increase monotonically with the doping concentration. It is generally believed that such a peculiar behavior should be attributed to both the strongly correlated nature of the cuprate system and the disorder effect. In this work, we develop a variational theory for the zero temperature superfluid density of the disordered t−Jt-Jt−J model. This is achieved in two steps. First, we perform an unrestricted variational optimization of an RVB variational ground state for the disordered t−Jt-Jt−J model. Second, we construct the variational state that describes the paramagnetic current response on such an RVB state. The zero temperature superfluid density ρs(0)ρ_{s}(0)ρs​(0) is then extracted from the curvature of the variational ground state energy of the system as a function of the external electromagnetic field. We find that ρs(0)ρ_{s}(0)ρs​(0) computed in this way is remarkably robust against the disorder effect. More specifically, we find that ρs(0)ρ_{s}(0)ρs​(0) is a monotonically increasing function of doping concentration xxx and scales linearly with the total optical weight. This is consistent with the observation in the underdoped cuprates but is strongly at odd with the behavior in the overdoped cuprates. The strong contrast between the disorder effect in the underdoped and the overdoped regime lends strong support to our previous proposal that there exist a Mott transition between a doped-Mott-insulating metal in the underdoped regime and a fermi-liquid-like metal in the overdoped regime around the pseudogap end point.
The optical conductivity and Drude weight of a spin-fermion model for cuprates are studied as a function of electronic density and temperature using Monte Carlo techniques. This model develops stripes and robust D-wave pairing correlations upon hole doping, and it has the advantage that it can be numerically simulated without sign problems. Both static and dynamical information can be obtained. In this work, it was possible to analyze up to $12\ifmmode\times\else\texttimes\fi{}12$ site clusters at low temperatures ranging between 0.01t0.01t0.01t and 0.1t0.1t0.1t (between 50 K and 500 K for a hopping amplitude $t\ensuremath{\sim}0.5\mathrm{eV}).$ As the temperature is reduced, spectral weight is transferred from high to low frequencies in agreement with the behavior observed experimentally. Varying the hole density, the Drude weight has a maximum at the optimal doping for the model, i.e., at the density where the pairing correlations are stronger. It was also observed that the inverse of the Drude weight, roughly proportional to the resistivity, decreases linearly with the temperature at optimal doping, and it is abruptly reduced when robust pairing correlations develop upon further reducing the temperature. The behavior and general form of the optical conductivity are found to be in good agreement with experimental results for the cuprates. Our results also establish the spin-fermion model for cuprates as a reasonable alternative to the $t\ensuremath{-}J$ model, which is much more difficult to study accurately.
An understanding of the anomalous charge dynamics in the high-Tc{T}_{c}Tc​ cuprates is obtained based on a model study of doped Mott insulators. The high-temperature optical conductivity is found to generally have a two-component structure: a Drude-like part followed by a mid-infrared band. The scattering rate associated with the Drude part exhibits a linear-temperature dependence over a wide range of high temperature, while the Drude term gets progressively suppressed below a characteristic energy of magnetic origin as the system enters the pseudogap phase. The high-energy optical conductivity shows a resonancelike feature in an underdoped case and continuously evolves into a $1∕\ensuremath{\omega}$ tail at higher doping, indicating that they share the same physical origin. In particular, such a high-energy component is closely correlated with the $\ensuremath{\omega}$-peak structure of the density-density correlation function at different momenta, in systematic consistency with exact diagonalization results based on the $t\text{\ensuremath{-}}J$ model. The underlying physics is attributed to the high-energy spin-charge separation in the model, in which the ``mode coupling'' responsible for the anomalous charge properties is not between the electrons and some collective mode but rather between charge carriers, holons, and a topological gauge field controlled by spin dynamics, as the consequence of the strong short-range electron-electron Coulomb repulsion in the doped Mott insulator.
The optical conductivity and Drude weight of a spin-fermion model for cuprates are studied as a function of electronic density and temperature using Monte Carlo techniques. This model develops stripes and robust D-wave pairing correlations upon hole doping, and it has the advantage that it can be numerically simulated without sign problems. Both static and dynamical information can be obtained. In this work, it was possible to analyze up to 12312 site clusters at low temperatures ranging between 0.01t and 0.1t ~between 50 K and 500 K for a hopping amplitude t;0.5 eV). As the temperature is reduced, spectral weight is transferred from high to low frequencies in agreement with the behavior observed experimentally. Varying the hole density, the Drude weight has a maximum at the optimal doping for the model, i.e., at the density where the pairing correlations are stronger. It was also observed that the inverse of the Drude weight, roughly proportional to the resistivity, decreases linearly with the temperature at optimal doping, and it is abruptly reduced when robust pairing correlations develop upon further reducing the temperature. The behavior and general form of the optical conductivity are found to be in good agreement with experimental results for the cuprates. Our results also establish the spin-fermion model for cuprates as a reasonable alternative to the t-J model, which is much more difficult to study accurately.
Optical Conductivity σ(ω)σ(ω)σ(ω) and Resistivity ρdcρ_{dc}ρdc​ of a Hole Doped Spin-Fermion Model for Cuprates
The optical conductivity and Drude weight of a Spin-Fermion model for cuprates are studied as a function of electronic density and temperature. This model develops stripes and robust D-wave pairing correlation upon hole doping, and it has the advantage that it can be numerically simulated without sign problems. Both static and dynamical information can be obtained. In this work it was possible to analyze up to 12x12 site clusters at low temperatures ranging between 0.01t and 0.1t (between 50K and 500K for a hopping amplitude t∼0.5eVt \sim 0.5 eVt∼0.5eV). As the temperature is reduced, spectral weight is transferred from high to low frequencies in agreement with the behavior observed experimentally. Varying the hole density, the Drude weight has a maximum at the optimal doping for the model, i.e., at the density where the pairing correlations are stronger. It was also observed that the inverse of the Drude weight, roughly proportional to the resistivity, decreases linearly with the temperature at optimal doping, and it is abruptly reduced when robust pairing correlations develop upon further reducing the temperature. The behavior and general form of the optical conductivity are found to be in good agreement with experimental results for the cuprates. Our results also establish the Spin-Fermion model for cuprates as a reasonable alternative to the t-J model, which is much more difficult to study accurately.