We study the thermodynamic properties of the 3D Hubbard model for temperatures down to the Néel temperature by using cluster dynamical mean-field theory. In particular, we calculate the energy, entropy, density, double occupancy, and nearest-neighbor spin correlations as a function of chemical potential, temperature, and repulsion strength. To make contact with cold-gas experiments, we also compute properties of the system subject to an external trap in the local density approximation. We find that an entropy per particle S/N ≈ 0.65(6) at U/t = 8 is sufficient to achieve a Néel state in the center of the trap, substantially higher than the entropy required in a homogeneous system. Precursors to antiferromagnetism can clearly be observed in nearest-neighbor spin correlators.
We study the anisotropic 3D Hubbard model with increased nearest-neighbor tunneling amplitudes along one direction using the dynamical cluster approximation and compare the results to a quantum simulation experiment of ultracold fermions in an optical lattice. We find that the short-range spin correlations are significantly enhanced in the direction with stronger tunneling amplitudes. Our results agree with the experimental observations and show that the experimental temperature is lower than the strong tunneling amplitude. We characterize the system by examining the spin correlations beyond neighboring sites and determine the distribution of density, entropy, and spin correlation in the trapped system. We furthermore investigate the dependence of the critical entropy at the Néel transition on anisotropy.
We study the thermodynamics of the three-dimensional Hubbard model at half filling on approach to the N\'eel transition by means of large-scale unbiased diagrammatic determinant Monte Carlo simulations. We obtain the transition temperature in the strongly correlated regime, as well as the temperature dependence of the energy, entropy, double occupancy, and nearest-neighbor spin correlation function. Our results improve the accuracy of previous unbiased studies and present accurate benchmarks in the ongoing effort to realize the antiferromagnetic state of matter with ultracold atoms in optical lattices.
We present results for the equation of state of the two-dimensional Hubbard model on an isotropic square lattice as obtained from a controlled and numerically exact large-cluster dynamical mean field simulation. Our results are obtained for large but finite systems and are extrapolated to infinite system size using a known finite-size scaling relation, and are supplemented by reliable error bars accounting for all sources of errors. We establish the importance of examining the decay of spatial spin correlations to determine whether a sufficiently large cluster has been used and with this in mind we present the energy, entropy, double occupancy, and nearest-neighbor spin correlations extrapolated to the thermodynamic limit. We discuss the implications of these calculations on pseudogap physics of the 2D Hubbard model away from half filling, where we find a strong behavioral shift in energy below a temperature
T∗ which becomes more pronounced for larger clusters. Finally, we provide reference calculations and tables for the equation of state for values of doping away from half filling which are of interest to cold-atom experiments.
We formulate a finite-temperature scheme of the variational cluster approximation (VCA) particularly suitable for an exact-diagonalization cluster solver. Based on the analytical properties of the single-particle Green's function matrices, we explicitly show the branch-cut structure of logarithm of the complex determinant functions appearing in the self-energy-functional theory (SFT) and whereby construct an efficient scheme for the finite-temperature VCA. We also derive the explicit formulas for entropy and specific heat within the framework of the SFT. We first apply the method to explore the antiferromagnetic order in a half-filled Hubbard model by calculating the entropy, specific heat, and single-particle excitation spectrum for different values of on-site Coulomb repulsion
U and temperature
T. We also calculate the
T dependence of the single-particle excitation spectrum in the strong coupling region, and discuss the overall similarities to and the fine differences from the spectrum obtained by the spin-density-wave mean-field theory at low temperatures and the Hubbard-I approximation at high temperatures. Moreover, we show a necessary and sufficient condition for the third law of thermodynamics in the SFT. On the basis of the thermodynamic properties, such as the entropy and the double occupancy, calculated via the
T and/or
U derivative of the grand potential, we obtain a crossover diagram in the
(U,T) plane, which separates a Slater-type insulator and a Mott-type insulator. Next, we demonstrate the finite-temperature scheme in the cluster-dynamical-impurity approximation (CDIA), i.e., the VCA with noninteracting bath orbitals attached to each cluster, and study the paramagnetic Mott metal-insulator transition in the half-filled Hubbard model. Formulating the finite-temperature CDIA, we first address a subtle issue regarding the treatment of the artificially introduced bath degrees of freedom which are absent in the originally considered Hubbard model. We then apply the finite-temperature CDIA to calculate the finite-temperature phase diagram in the
(U,T) plane. Metallic, insulating, coexistence, and crossover regions are distinguished from the bath-cluster hybridization-variational-parameter dependence of the grand-potential functional. We find that the Mott transition at low temperatures is discontinuous, and the coexistence region of the metallic and insulating states persists down to zero temperature. The result obtained here by the finite-temperature CDIA is complementary to the previously reported zero-temperature CDIA phase diagram.
Even though the Hubbard model is one of the most fundamental models of highly correlated electrons, analytical and numerical data describing its thermodynamics at nonzero magnetization are relatively scarce. We present a detailed investigation of the thermodynamic properties for the one-dimensional repulsive Hubbard model in the presence of an arbitrary magnetic field for all values of the filling fraction and temperatures as low as $T\ensuremath{\sim}0.005t.$ Our analysis is based on the system of integral equations derived in the quantum transfer matrix framework. We determine the critical exponents of the quantum phase transitions and also provide analytical derivations for some of the universal functions characterizing the thermodynamics in the vicinities of the quantum critical points. Extensive numerical data for the specific heat, susceptibility, compressibility, and entropy are reported. The experimentally relevant double occupancy presents an interesting doubly nonmonotonic temperature dependence at intermediate values of the interaction strength and also at large repulsion and magnetic fields close to the critical value. The susceptibility in zero magnetic field has a logarithmic singularity at low temperatures for all filling factors similar to the behavior of the same quantity in the spin-1/2 isotropic Heisenberg model. We determine the density profiles for a harmonically trapped system and show that while the total density profile seems to depend mainly on the value of chemical potential at the center of the trap the distribution of phases in the inhomogeneous system changes dramatically as we increase the magnetic field.
We present a comprehensive study of the thermodynamic properties of the three-dimensional fermionic Hubbard model, with application to cold fermionic atoms subject to an optical lattice and a trapping potential. Our study is focused on the temperature range of current experimental interest. We employ two theoretical methods---dynamical mean-field theory and high-temperature series---and perform comparative benchmarks to delimit their respective range of validity. Special attention is devoted to understand the implications that thermodynamic properties of this system have on cooling. Considering the distribution function of local occupancies in the inhomogeneous lattice, we show that, under adiabatic evolution, the variation of any observable (e.g., temperature) can be conveniently disentangled into two distinct contributions. The first contribution is due to the redistribution of atoms in the trap during the evolution, while the second one comes from the intrinsic change of the observable. Finally, we provide a simplified picture of a recently proposed cooling procedure, based on spatial entropy separation, by applying this method to an idealized model.