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arXiv:cond-mat/9603081 · Physical Review Letters

Thermodynamic Properties of the Planar t − J Model

Janez Jaklič, Peter Prelovšek

Published Jul 29, 1996• 66 citations
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Abstract

Several thermodynamic quantities within the planar $t\ensuremath{-}J$ model are calculated using the $T>0$ Lanczos method on clusters of up to 26 sites. The hole density ${c}_{h}(\ensuremath{\mu},T)$ shows a non-Fermi-liquid behavior as a function of $T$ and suggests a transition from small to large Fermi surfaces at ${c}_{h}\ensuremath{\sim}0.15$. The specific heat reveals a maximum at the exchange energy scale up to ${c}_{h}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.2$, where it becomes almost $T$ independent for $T\ensuremath{\gtrsim}0.15t$. At constant $T$ the entropy has maximum for ${c}_{h}\ensuremath{\sim}0.15$ with large values at low $T$, consistent with experiments on cuprates. In the underdoped regime the spin susceptibility ${\ensuremath{\chi}}_{0}(T)$ exhibits a maximum at finite ${T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}T}^{*}$, with ${T}^{*}$ decreasing with doping and disappearing for ${c}_{h}>0.15$.

DOI
10.1103/physrevlett.77.892

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Several thermodynamic quantities within the planar t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J model are calculated using the T>0T>0T>0 Lanczos method on clusters of up to 26 sites. The hole density ch(\ensuremathμ,T){c}_{h}(\ensuremath{\mu},T)ch​(\ensuremathμ,T) shows a non-Fermi-liquid behavior as a function of TTT and suggests a transition from small to large Fermi surfaces at ch\ensuremath∼0.15{c}_{h}\ensuremath{\sim}0.15ch​\ensuremath∼0.15. The specific heat reveals a maximum at the exchange energy scale up to ch=0.2{c}_{h}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.2ch​=0.2, where it becomes almost TTT independent for T\ensuremath≳0.15tT\ensuremath{\gtrsim}0.15tT\ensuremath≳0.15t. At constant TTT the entropy has maximum for ch\ensuremath∼0.15{c}_{h}\ensuremath{\sim}0.15ch​\ensuremath∼0.15 with large values at low TTT, consistent with experiments on cuprates. In the underdoped regime the spin susceptibility \ensuremathχ0(T){\ensuremath{\chi}}_{0}(T)\ensuremathχ0​(T) exhibits a maximum at finite T=T∗{T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}T}^{*}T=T∗, with T∗{T}^{*}T∗ decreasing with doping and disappearing for ch>0.15{c}_{h}>0.15ch​>0.15.
The thermodynamic properties of specific heat, entropy, chemical potential, spin susceptibility ${\ensuremath{\chi}}_{s},$ and charge susceptibility ${\ensuremath{\chi}}_{c}$ are studied as functions of temperature and doping within the two-dimensional Hubbard model with various U/t=4−−12.U/t=4--12.U/t=4−−12. Quantities are calculated using the finite-temperature Lanczos method with additional phase averaging for a system of $4\ifmmode\times\else\texttimes\fi{}4$ sites. Results show that the entropy at low T reaches a maximum near half-filling at the electron density $n\ensuremath{\sim}1\ifmmode\pm\else\textpm\fi{}0.15$ in the whole regime of studied U/t.U/t.U/t. The pseudogap in ${\ensuremath{\chi}}_{s}(T)$ becomes clearly pronounced for U/t> 8U/t>~8U/t> 8 while ${\ensuremath{\chi}}_{c}$ shows a maximum close to half-filling. The relation of results to those within the $t\ensuremath{-}J$ model and to experiments is discussed.
Some thermodynamic properties of the high-Tc cuprates have been studied using t-J-U model at different doping (<h>) by an exact method on a tilted square cluster. Low temperature specific heat curves show single-peak structure where the peak-height decreases with increasing <h>. The system becomes more disordered with decreasing hole occupancy. The extrapolation of all reciprocal susceptibility (χ-1) curves intercept at a negative temperature ΘN which suggests an antiferromagnetic correlation in the system.
The single-particle spectral functions A(k,\ensuremath{\omega}) and self-energies \ensuremath{\Sigma}(k,\ensuremath{\omega}) are calculated within the t-J model using the finite-temperature Lanczos method for small systems. A remarkable asymmetry between the electron and hole part is found. The hole (photoemission) spectra are overdamped, with Im\ensuremath{\Sigma}\ensuremath{\propto}\ensuremath{\omega} in a wide energy range, consistent with the marginal Fermi liquid scenario, and in good agreement with experiments on cuprates. In contrast, the quasiparticles in the electron part of the spectrum show weak damping.
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.
We study the t-J and the Hubbard models at zero temperature using exact-diagonalization techniques on \ensuremath{\surd}10 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}10 and 4\ifmmode\times\else\texttimes\fi{}4 sites clusters. Quantum Monte Carlo simulation results on larger lattices are also presented. All electronic fillings have been analyzed for the three models. We have measured equal-time correlation functions corresponding to various types of order (ranging from ``standard'' staggered spin order to more ``exotic'' possibilities like chiral order), as well as various dynamical properties of these models. Upper bounds for the critical hole doping (xc{\mathit{x}}_{\mathit{c}}xc​), where long-range antiferromagnetic order disappears, are presented. It was found that xc{\mathit{x}}_{\mathit{c}}xc​ is very small in agreement with experiments for the high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductors. For example, in the t-J model, ${\mathit{x}}_{\mathit{c}}$0.08 at J/t=0.4. However, short-distance spin correlations are important up to much higher dopings producing a sharp well-defined spin-wave-like peak in S(q=(\ensuremath{\pi},\ensuremath{\pi}),\ensuremath{\omega}). Regarding the possibility of phase separation in the Hubbard model, we have studied the behavior of the density of particles, 〈n〉, as a function of the chemical potential, using the Lanczos method on a 4\ifmmode\times\else\texttimes\fi{}4 Hubbard cluster, finding no indications of phase separation for any value of U/t. Then, we conclude that the t-J model at small J/t should not phase separate.In order to compare theoretical predictions with photoemission experiments, we evaluated the electronic density of states N(\ensuremath{\omega}), of the Hubbard and t-J models at several doping fractions. We found that upon doping the antiferromagnetic gap is filled for U\ensuremath{\sim}8t or smaller. The chemical potential moves across the insulating gap as one goes from electron to hole doping of the half-filled cluster, in agreement with x-ray absorption experiments but at variance with photoemission experiments. We have also calculated the optical conductivity, ${\mathrm{\ensuremath{\sigma}}}_{1}$(\ensuremath{\omega}), of the Hubbard and t-J models at all dopings on 4\ifmmode\times\else\texttimes\fi{}4 clusters. Results are compared with experiments and the weight of the Drude peak is presented as a function of couplings and dopings. Spectral weight found at small frequencies is associated with the mid-infrared band observed experimentally in ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$Srx{\mathrm{Sr}}_{\mathit{x}}Srx​CuO4{\mathrm{CuO}}_{4}CuO4​, and with the states filling the insulating gap in photoemission experiments. An overall good agreement with experiments in the normal state was found. Regarding the possibility of superconductivity n these models, we have studied s-, d-, and p-wave pairing correlations. Naively, the d-wave channel seems enhanced near half filling while the extended s-wave channel seems enhanced from half filling up to 40% doping.However, we found that the enhancement comes from short-distance effects and, thus, no numerical indications of superconductivity were found in these models. We emphasize the importance of analyzing the pairing correlations as a function of distance to distinguish between short- and long-distance effects in the susceptibilities. We also observed that spiral order is enhanced at small J/t and low doping. Uniform chiral rder is suppressed by dynamical holes while staggered chiral order may be enhanced, although with a small plaquette order parameter. We conclude with the observation that the simple one-band Hubbard model with intermediate values of U\ensuremath{\sim}8t may account for many of the ``anomalous'' properties of the normal state of the high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductors.