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International Journal of Modern Physics B

Thermodynamics of the Planar Hubbard Model

J. Bonča, Peter Prelovšek

Published Aug 10, 2003• 3 citations
Publisher

Abstract

The thermodynamic properties: specific heat and entropy are studied as a function of temperature and doping within the two-dimensional Hubbard model with various U/t=4-12. Quantities are calculated using the finite-temperature Lanczos method with additional phase-averaging for a system of 4×4 sites. Results show that the entropy at low T reaches a maximum near half-filling at the electron density n~1±0.15 in the whole regime of studied U/t.

DOI
10.1142/s0217979203021034

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The thermodynamic properties: specific heat and entropy are studied as a function of temperature and doping within the two-dimensional Hubbard model with various U/t=4-12. Quantities are calculated using the finite-temperature Lanczos method with additional phase-averaging for a system of 4×4 sites. Results show that the entropy at low T reaches a maximum near half-filling at the electron density n~1±0.15 in the whole regime of studied U/t.
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.
Several thermodynamic quantities within the planar $t\ensuremath{-}J$ model are calculated using the T>0T>0T>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 TTT 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 ch=0.2{c}_{h}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0.2ch​=0.2, where it becomes almost TTT independent for $T\ensuremath{\gtrsim}0.15t$. At constant TTT the entropy has maximum for ${c}_{h}\ensuremath{\sim}0.15$ with large values at low TTT, consistent with experiments on cuprates. In the underdoped regime the spin susceptibility ${\ensuremath{\chi}}_{0}(T)$ 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.
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.
We study thermodynamic properties of the doped Hubbard model on the square lattice in the regime of strong charge and spin fluctuations at low temperatures near the metal-to-insulator crossover and obtain results with controlled accuracy using the diagrammatic Monte Carlo method directly in the thermodynamic limit. The behavior of the entropy reveals a non-Fermi-liquid state at sufficiently high interactions near half filling: A maximum in the entropy at nonzero doping develops as the coupling strength is increased, along with an inflection point, evidencing a metal to non-Fermi-liquid crossover. The specific heat exhibits additional distinctive features of a non-Fermi-liquid state. Measurements of the entropy can, therefore, be used as a probe of the state of the system in quantum simulation experiments with ultracold atoms in optical lattices.
We study the thermodynamics of the two-dimensional Hubbard model within the dynamical cluster approximation. We use continuous time quantum Monte Carlo as a cluster solver to avoid the systematic error which complicates the calculation of the entropy and potential energy (double occupancy). We find that at a critical filling, there is a pronounced peak in the entropy divided by temperature, S/TS/TS/T, and in the normalized double occupancy as a function of doping. At this filling, we find that specific heat divided by temperature, C/TC/TC/T, increases strongly with decreasing temperature and kinetic and potential energies vary like T2 ln T{T}^{2}\text{ }\text{ln}\text{ }TT2 ln T. These are all characteristics of quantum critical behavior.
The thermodynamic quantities of the 1D Hubbard model are investigated exactly by means of the Bethe Ansatz solution at finite temperatures. For an arbitrary electron concentration, the magnetic susceptibility, the charge susceptibility and the specific heat are calculated numerically as a function of temperature. On the basis of the obtained results, the characteristic behavior, which appears in the charge excitation near half-filling, is discussed, and is understood to be closely related to the gap formation of the Mott-type insulator.