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Physical review. B, Condensed matter

Finite-temperature conductivity in the planart-Jmodel

Janez Jaklič, Peter Prelovšek

Published Sep 1, 1994• 38 citations
Publisher

Abstract

Results are for finite-temperature optical conductivity \ensuremath{\sigma}(\ensuremath{\omega}) and dc resistivity \ensuremath{\rho}(T), obtained within the t-J model on a square lattice by a method based on the Lanczos diagonalization. Studying systems with N=16 or 18 sites and hole concentrations ${\mathit{c}}_{\mathit{h}}$\ensuremath{\le}0.25, a thermodynamic behavior is reached for quite low temperatures. \ensuremath{\sigma}(\ensuremath{\omega}) spectra reveal a slow non-Drude falloff at large frequencies and the emergence of a broad midinfrared peak at low doping. At optimum doping \ensuremath{\rho}(T) shows nearly linear variation with T. Both quantities agree well with experimental results in cuprates.

DOI
10.1103/physrevb.50.7129

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Results are for finite-temperature optical conductivity \ensuremath{\sigma}(\ensuremath{\omega}) and dc resistivity \ensuremath{\rho}(T), obtained within the t-J model on a square lattice by a method based on the Lanczos diagonalization. Studying systems with N=16 or 18 sites and hole concentrations ch{\mathit{c}}_{\mathit{h}}ch​\ensuremath{\le}0.25, a thermodynamic behavior is reached for quite low temperatures. \ensuremath{\sigma}(\ensuremath{\omega}) spectra reveal a slow non-Drude falloff at large frequencies and the emergence of a broad midinfrared peak at low doping. At optimum doping \ensuremath{\rho}(T) shows nearly linear variation with T. Both quantities agree well with experimental results in cuprates.
The finite-temperature optical conductivity \ensuremath{\sigma}(\ensuremath{\omega}) in the planar t-J model is analyzed using recently introduced numerical method based on the Lanczos diagonalization of small systems (up to 20 sites), as well as by analytical approaches, including the method of frequency moments and the retraceable-path approximation. Results for a dynamical mobility of a single hole at elevated temperatures T>t reveal a Gaussian-like \ensuremath{\mu}(\ensuremath{\omega}) spectra, however, with a nonanalytical behavior at low \ensuremath{\omega}. In the single-hole response a difference between the ferromagnetic (J=0) and the antiferromagnetic (J>0) polaron shows up at TJ. At larger dopings numerical results in studied systems are consistent with the thermodynamical behavior for T>T∗{\mathit{T}}^{\mathrm{*}}T∗\ensuremath{\ge}0.1t. $sigma ( omega )--- spectra show a non-Drude falloff at large frequencies. In particular for ``optimum'' doping nh{\mathit{n}}_{\mathit{h}}nh​\ensuremath{\sim}0.2 we obtain in the low-\ensuremath{\omega},T regime the relaxation rate ${\mathrm{\ensuremath{\tau}}}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\sim}2\ensuremath{\pi}\ensuremath{\lambda}(\ensuremath{\omega}+\ensuremath{\xi}T) with \ensuremath{\lambda}\ensuremath{\sim}0.09 and \ensuremath{\xi}\ensuremath{\sim}2.7, being consistent with the marginal Fermi-liquid concept and experiments. Within the same regime we reproduce the nearly linear variation of dc resistivity \ensuremath{\rho} with T. This behavior is weakly dependent on J, provided that Jt.
The optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ and the dc resistivity $\ensuremath{\rho}(T)$ within the extended $t\text{\ensuremath{-}}J$ model on a square lattice, as relevant to high-Tc{T}_{c}Tc​ cuprates, are reinvestigated using the exact-diagonalization method for small systems, improved by performing a twisted boundary condition averaging. The influence of the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$ is also considered. The behavior of results at intermediate doping is consistent with a marginal-Fermi-liquid scenario and in the case of ${t}^{\ensuremath{'}}=0$ for $\ensuremath{\omega}>T$ follows the power law $\ensuremath{\sigma}\ensuremath{\propto}{\ensuremath{\omega}}^{\ensuremath{-}\ensuremath{\nu}}$ with $\ensuremath{\nu}\ensuremath{\sim}0.65$ consistent with experiments. At low doping ch<0.1{c}_{h}<0.1ch​<0.1 for T<JT<JT<J $\ensuremath{\sigma}(\ensuremath{\omega})$ develops a shoulder at $\ensuremath{\omega}\ensuremath{\sim}{\ensuremath{\omega}}^{*}$, consistent with the observed midinfrared peak in experiments, accompanied by a shallow dip for $\ensuremath{\omega}<{\ensuremath{\omega}}^{*}$. This region is characterized by the resistivity saturation, whereas a more coherent transport appears at T<T∗T<{T}^{*}T<T∗ producing a more pronounced decrease in $\ensuremath{\rho}(T)$. The behavior of the normalized resistivity ${c}_{h}\ensuremath{\rho}(T)$ is within a factor of 2 quantitatively consistent with experiments in cuprates.
We calculate the optical conductivity \ensuremath{\sigma}(\ensuremath{\omega}) of the Hubbard and t-J models on 10- and 16-site lattices, respectively, using a Lanczos method. Results are presented for various values of the couplings U/t, J/t, and band fillings, using free- and periodic-boundary conditions. We discuss the behavior of the kinetic energy in the ground state, the distribution of spectral weight of \ensuremath{\sigma}(\ensuremath{\omega}) between low and high energies and the sum rule. Indications of a Drude-like peak at low energy in the one-hole subspace are observed in both models. Assuming that holes behave like independent (mass-renormalized) particles in the normal state of the superconductors, our results also suggest that the optical midinfrared broadband observed experimentally can be explained by hole excitations of strongly correlated systems.
The optical conductivity, \ensuremath{\sigma}(\ensuremath{\omega}), of the two-dimensional one-band Hubbard model is calculated at finite temperature using exact diagonalization techniques on finite clusters. The in-plane dc resistivity, ${\mathrm{\ensuremath{\rho}}}_{\mathit{a}\mathit{b}}$, is also evalauted. We find that at large U/t and temperature T, ${\mathrm{\ensuremath{\rho}}}_{\mathit{a}\mathit{b}}$ is approximately linear with temperature. We also observed that \ensuremath{\sigma}(\ensuremath{\omega}) displays charge excitations, a midinfrared (MIR) band, and a Drude peak. The combination of the Drude peak and the MIR oscillator strengths leads to a conductivity that decays slower than 1/${\mathrm{\ensuremath{\omega}}}^{2}$ at energies smaller than the insulator gap near half filling.
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.
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.