Specific heat and density anomaly in the Hubbard model

Abstract

Understanding thermal properties of materials is fundamental to technological applications and to discovering new phenomena. In particular, advances in experimental techniques such as cold-atom measurements allow the simulation of paradigmatic Hamiltonians with great control over model parameters, such as the Hubbard model. One aspect of this model which is not much explored is the behavior of the specific heat as a function of density. In this work, we perform Determinant Quantum Monte Carlo simulations of the Hubbard model interpolating between the square and triangular lattices to analyze the specific heat as the filling, interaction, and temperature of the system are changed. We found that, with strong correlations, the specific heat presents a three-maxima structure as a function of filling, with local minima between them. This effect can be explained by a decomposition of kinetic and potential contributions to the specific heat, demonstrating interesting phenomena away from the commonly studied half-filling regime. Moreover, by analyzing the kinetic contribution in momentum space we show that, connected to this specific heat behavior, there is a density anomaly detected through the thermal expansion coefficient. These momentum-space quantities are accessible using cold-atom experiments measurements at multiple temperatures. Finally, we map the location of these phenomena and connect the thermal expansion anomaly with the well-known Seebeck coefficient change of sign. Our results provide a new perspective to analyze this change of sign.

Figures (13)

• Figure 1: a) Specific heat at constant density $c_n$ as a function of lattice filling for $T = 1$ and various values of interaction $U$. b) and c) are the kinetic and potential energy contributions to the specific heat for the same temperature and interaction strengths. Results are for the $10 \times 10$ lattice.
• Figure 2: Specific heat and kinetic/potential contributions for $U = 10$ at temperatures a) $T = 2$, b) $T = 1$ and c) $T = 0.5$. The inset in b) is a $T \times U$ ($0 \leq U \leq 10$, $0\leq T \leq 2.5$) plot at half-filling with dashed lines showing the temperatures for which we display specific results in a), b), and c). Red circles and purple squares represent, respectively, the high and low $T$ peaks in the specific heat as a function of temperature, with data as a function of temperature of the inset extracted from reference paiva_signatures_2001. Results are for the $10 \times 10$ lattice.
• Figure 3: Specific heat as a function of lattice filling $n$ for $U = 10$ and different lattice sizes at a) $T = 1.0$ and b) $T = 0.5$.
• Figure 4: The momentum distribution of $\left( \frac{\partial n_{\bm{k}}}{\partial T} \right)_{\mu}$ in the first Brillouin zone for interactions $U = 2$ (upper panels) and $U = 10$ (lower panels), and fillings $n \approx 0.3$ (left panels), $n \approx 0.51$ (central panels) and $n \approx 0.83$ (right panels). The temperature is $T = 1$ and the lattice size is $24 \times 24$, with a cubic spline interpolation between points.
• Figure 5: The gradient of the momentum distribution of $\left( \frac{\partial n_{\bm{k}}}{\partial T} \right)_{\mu}$ in the first Brillouin zone for interactions $U = 2$ (upper panels) and $U = 10$ (lower panels), and fillings $n \approx 0.3$ (left panels), $n \approx 0.51$ (central panels) and $n \approx 0.83$ (right panels). The temperature is $T = 1$ and the lattice size is $24 \times 24$, with a cubic spline interpolation between points.