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arXiv:cond-mat/9507030 · Physical Review Letters

Universal Charge and Spin Response in Optimally Doped Antiferromagnets

Janez Jaklič, Peter Prelovšek

Published Aug 14, 1995• 32 citations
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Abstract

Charge and spin response in the planar $t\ensuremath{-}J$ model at finite temperatures are investigated numerically, in the regime corresponding to cuprates at intermediate doping. We show that the local spin correlation function is $T$ independent, leading to ${\ensuremath{\chi}}^{\ensuremath{'}\ensuremath{'}}(\ensuremath{\omega}<J)\ensuremath{\propto}\mathrm{tanh}(\ensuremath{\omega}/2T)$. The current correlation function $C(\ensuremath{\omega})$ appears to be $T$ as well as $\ensuremath{\omega}$ independent, and the optical conductivity $\ensuremath{\sigma}\ensuremath{\propto}({1\ensuremath{-}e}^{\ensuremath{-}\ensuremath{\omega}/T})/\ensuremath{\omega}$. Such anomalous response functions are brought in connection with the large entropy persisting at $T<J$.

DOI
10.1103/physrevlett.75.1340

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Charge and spin response in the planar t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model at finite temperatures are investigated numerically, in the regime corresponding to cuprates at intermediate doping. We show that the local spin correlation function is TTT independent, leading to \ensuremathχ\ensuremath′\ensuremath′(\ensuremathω<J)\ensuremath∝tanh(\ensuremathω/2T){\ensuremath{\chi}}^{\ensuremath{'}\ensuremath{'}}(\ensuremath{\omega}<J)\ensuremath{\propto}\mathrm{tanh}(\ensuremath{\omega}/2T)\ensuremathχ\ensuremath′\ensuremath′(\ensuremathω<J)\ensuremath∝tanh(\ensuremathω/2T). The current correlation function C(\ensuremathω)C(\ensuremath{\omega})C(\ensuremathω) appears to be TTT as well as \ensuremathω\ensuremath{\omega}\ensuremathω independent, and the optical conductivity \ensuremathσ\ensuremath∝(1\ensuremath−e\ensuremath−\ensuremathω/T)/\ensuremathω\ensuremath{\sigma}\ensuremath{\propto}({1\ensuremath{-}e}^{\ensuremath{-}\ensuremath{\omega}/T})/\ensuremath{\omega}\ensuremathσ\ensuremath∝(1\ensuremath−e\ensuremath−\ensuremathω/T)/\ensuremathω. Such anomalous response functions are brought in connection with the large entropy persisting at T<JT<JT<J.
Recent results for the finite-temperature static and dynamical properties of the planar t−Jt-Jt−J model, obtained by a novel numerical method for small correlated systems, are reviewed. Of particular interest are T>0T>0T>0 charge and spin response functions: optical conductivity σ(ω)σ(ω)σ(ω) and spin susceptibility χ(q⃗,ω)χ(\vec q,ω)χ(q​,ω), which show very universal features at intermediate doping. In spin dynamics the universality is best seen in the local spin correlation function, which appears to be TTT-independent and only weakly doping dependent. Results apply directly to the inelastic neutron scattering and to the NMR relaxation experiments in cuprates. Analysing σ(ω)σ(ω)σ(ω) we find that the current correlation function C(ω)C(ω)C(ω) is nearly TTT- and moreover ωωω-independent in the regime $T
We review recent results for the properties of doped antiferromagnets, obtained by the numerical analysis of the planar t-J model using the novel finite-temperature Lanczos method for small correlated systems. First we briefly summarize our present understanding of anomalous normal-state properties of cuprates, and present the electronic phase diagram, phenomenological scenarios and models proposed in this connection. The numerical method is then described in more detail. The following sections are devoted to various static and dynamical properties of the t-J model. Among the thermodynamic properties the chemical potential, entropy and the specific heat are evaluated. Discussing electrical properties the emphasis is on the planar optical conductivity and the d.c. resistivity. Magnetic properties involve the static and dynamical spin structure factor, as measured via the susceptibility measurements, the NMR relaxation and the neutron scattering, as well as the orbital current contribution. The analysis of electron spectral functions, studied by photoemission experiments, and their relation to the c-axis conductivity, follows. Finally we discuss density fluctuations, the electronic Raman scattering and the thermoelectric power. Whenever feasible a comparison with experimental data is performed. A general conclusion is that the t-J model captures well the main features of anomalous normal-state properties of cuprates, for a number of quantities the agreement is even a quantitative one. It is shown that several dynamical quantities exhibit at intermediate doping a novel universal behaviour consistent with a marginal Fermi-liquid concept, which seems to emerge from a large degeneracy of states and a frustration induced by doping the antiferro magnet.
The antiferromagnetic spin correlation function SQ{S}_{\mathbf{Q}}SQ​, the staggered spin susceptibility ${\ensuremath{\chi}}_{\mathbf{Q}}$, and the energy scale ${\ensuremath{\omega}}_{FL}={S}_{\mathbf{Q}}∕{\ensuremath{\chi}}_{\mathbf{Q}}$ are studied numerically within the $t\text{\ensuremath{-}}J$ model and the Hubbard model, as relevant to cuprates. It is shown that ${\ensuremath{\omega}}_{FL}$, related to the onset of the non-Fermi-liquid spin response at $T>{\ensuremath{\omega}}_{FL}$, is very low in the regime below the ``optimum'' hole doping ${c}_{h}<{c}_{h}^{*}\ensuremath{\sim}0.16$, while it shows a steep increase in the overdoped regime. A quantitative analysis of the NMR spin-spin relaxation rate 1∕T2G1∕{T}_{2G}1∕T2G​ for various cuprates reveals a similar behavior, indicating a pronounced crossover between a Fermi-liquid and a non-Fermi-liquid behavior as a function of doping.
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.
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