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arXiv:cond-mat/9612132 · Physical review. B, Condensed matter

Specific heat of the two-dimensional Hubbard model

Daniel Duffy, Adriana Moreo

Published May 15, 1997• 60 citations
arXivHTMLPDF

Abstract

Quantum Monte Carlo results for the specific heat c of the two-dimensional Hubbard model are presented. At half filling it was observed that c\ensuremath{\sim}${\mathrm{T}}^{2}$ at very low-temperatures. Two distinct features were also identified: a low-temperature peak related to the spin degrees of freedom and a higher-temperature broad peak related to the charge degrees of freedom. Away from half filling the spin-induced feature slowly disappears as a function of hole doping while the charge feature moves to lower temperature. A comparison with experimental results for the high-temperature cuprates is discussed.

DOI
10.1103/physrevb.55.12918

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Quantum Monte Carlo results for the specific heat c of the two-dimensional Hubbard model are presented. At half filling it was observed that c\ensuremath{\sim}T2{\mathrm{T}}^{2}T2 at very low-temperatures. Two distinct features were also identified: a low-temperature peak related to the spin degrees of freedom and a higher-temperature broad peak related to the charge degrees of freedom. Away from half filling the spin-induced feature slowly disappears as a function of hole doping while the charge feature moves to lower temperature. A comparison with experimental results for the high-temperature cuprates is discussed.
The Feynman path-integral quantum Monte Carlo (PI QMC) approach has been employed to study the specific heat C of the one-dimensional (1D) half-filled Hubbard chain as a function of temperature. For not too strong many-body interactions C exhibits a single maximum only. With increasing correlation strength the C curve shows an intermediate minimum separating a sharp low-temperature maximum and a broad high-temperature profile. The first peak is caused by spin excitations and the high-temperature values of C are determined by charge degrees of freedom. The contribution of both quantities to the net specific heat curve has been quantified in the present theoretical setup. The specific heat data are interpreted with the aid of calculated charge fluctuations, probabilities Pi(n) to find n = 2, 1, 0 electrons at the respective lattice sites, a spin density wave (SDW) order parameter and the electronic chemical potential μ.
Local moment formation driven by the on-site repulsion U is one of the most fundamental features in the Hubbard model. At the simplest level, the temperature dependence of the local moment is expected to have a single structure at $T\ensuremath{\sim}U,$ reflecting the suppression of the double occupancy. In this paper we show low-temperature quantum Monte Carlo data for half-filling which emphasize that the local moment also has a signature at a lower energy scale which previously had been thought to characterize only the temperatures below which moments on different sites begin to correlate locally. We discuss implications of these results for the structure of the specific heat, and connections to quasiparticle resonance and pseudogap formation in the density of states.
Recent experiments on cuprates show that as a function of doping, the normal-state specific heat sharply peaks at the doping ${\ensuremath{\delta}}^{*}$, where the pseudogap ends at low temperature. This finding is taken as the thermodynamic signature of a quantum critical point, whose nature has not yet been identified. Here we present calculations for the two-dimensional Hubbard model in the doped Mott insulator regime, which indicate that the specific heat anomaly can arise from the finite-temperature critical end point of a first-order transition between a pseudogap phase with dominant singlet correlations and a metal. As a function of doping at the temperature of the end point, the specific heat diverges. Upon increasing temperature, the peak becomes broader. The diverging correlation length is associated with uniform density fluctuations. No broken symmetries are needed. These anomalies also occur at half-filling as a function of interaction strength, and are relevant for organic superconductors and ultracold atoms.
Whether or not anomalies in the thermal conductivity from insulating cuprates can be attributed to antiferromagnetic order and magnons in a 2D Mott insulator remains an intriguing open question. To shed light on this issue, we investigate the thermal conductivity κ\kappaκ and specific heat cvc_vcv​ of the half-filled 2D single-band Hubbard model using the numerically exact determinant quantum Monte Carlo algorithm and maximum entropy analytic continuation. Both cvc_vcv​ and κ\kappaκ possess two peaks as a function of temperature, with scales related to the Hubbard interaction energy UUU and spin superexchange energy JJJ, respectively. At low temperatures where the charge degrees of freedom are gapped-out, our results for the contribution to both cvc_vcv​ and the Drude weight associated with κ\kappaκ from the kinetic energy agree well with spin-wave theory for the spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg model.
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.
Whether or not anomalies in the thermal conductivity in insulating cuprates can be attributed to antiferromagnetic order and magnons in a 2D Mott insulator remains an intriguing open question. To shed light on this issue, we investigate the thermal conductivity $\ensuremath{\kappa}$ and its relationship with the specific heat cv{c}_{v}cv​ in the half-filled 2D single-band Hubbard model, using the numerically exact determinant quantum Monte Carlo algorithm and maximum entropy analytic continuation. At low temperatures where the charge degrees of freedom are gapped-out and cv{c}_{v}cv​ exhibits a clear magnon peak, we observe that thermal conductivity $\ensuremath{\kappa}$ also tends to form a peak at similar temperatures. Reducing temperature further produces a sharp upturn in $\ensuremath{\kappa}$, associated with an increasing mean-free path. We identify this as the high-temperature side of the anomalous peak in insulating cuprates, where the mean-free path eventually is cut off by other scattering effects, including phonons, disorder, and physical size. Different scattering effects in our model are identified and analyzed in the thermal diffusivity.