Computation of thermal entropy for the doped Hubbard Model

Abstract

We develop a highly efficient framework for computing the thermal entropy in the doped Fermi-Hubbard model within the grand-canonical ensemble. The framework comprises four calculation schemes that express the entropy as path integrals in the parameter space of temperature, interaction strength, and chemical potential. The integrands involve only fundamental observables, including the total energy, fermion density, and double occupancy, which are readily accessible in a wide range of theoretical and numerical methods. We further derive useful Maxwell relations connecting the entropy to other quantities, and present practical formulas for directly evaluating the grand potential. As an application, we compute the entropy of the doped Hubbard model in two and three dimensions, using the numerically unbiased auxiliary-field quantum Monte Carlo method. The test results show excellent agreement across the different schemes and quantitatively verify the Maxwell relations, confirming the reliability of the framework. In two dimensions, we further benchmark our entropy results in physically relevant parameter regimes against diagrammatic Monte Carlo calculations and observe excellent quantitative consistency between the two approaches. By providing an efficient and broadly applicable route for entropy evaluation, our work facilitates the thermodynamic characterization of complex correlated states in the doped Hubbard model.

Figures (11)

• Figure 1: Illustration of four calculation schemes for the thermal entropy along different paths in the parameter space of the doped Hubbard model, consisting of interaction strength $U/t$, temperature $T/t$, and fermion filling $n$. The red dashed line shows the varying-$T$ path with fixed $U/t=6$ and $n=0.875$. The blue dashed line plots the varying-$T$ path with fixed $U/t=6$ and $\mu=\mu_0$. The orange dashed line represents the varying-$\mu$ path with fixed $U/t=6$ and $T/t=0.3$. The green dashed line denotes the varying-$U$ path with fixed $T/t=0.3$ and $n=0.875$. The value $\mu_0\simeq 1.23656$ is chosen such that $n(\mu_0)=0.875$ at $(U/t=6,T/t=0.3)$ for $L=4$ system in the 3D Hubbard model. This reference point is marked by the red star.
• Figure 2: Entropy density $\boldsymbol{s}$ as a function of $T/t$ for the 3D Hubbard model with $U=0$ from $L=4$ system. Panel (a) plots the results (blue squares) for fixed $n=0.875$ evaluated from Eq. (\ref{['eq:fixednT']}), while panel (b) shows the fixed-$\mu$ results (red circles) with $\mu/t=0.3$ obtained from Eq. (\ref{['eq:fixedmT']}). In both panels, the black dashed line denotes the exact results computed via Eq. (\ref{['eq:U0Exact']}). In (b), the blue squares are obtained from Eq. (\ref{['eq:fixedmT']}) using the incorrect energy $e_{\rm in}$ rather than $e_{\rm tot}$, yielding intentionally erroneous results.
• Figure 3: Test results for the varying-$T$ path with fixed $U/t$ and $n=0.875$. Panels (a) and (b) plot the energy density $e_{\mathrm{in}}/t$ versus inverse temperature $\beta t$ within $0\le \beta t\le 2.0$, and versus temperature $T/t$ within $0.30\le T/t\le 0.50$, respectively. The $e_{\rm in}/t$ data in (b) have been vertically offset (except for $U/t=6$) to fit into the plot, and the shifts are marked with the same colors as the data. Panel (c) shows the entropy density $\boldsymbol{s}$ as a function of $T/t$, computed via Eq. (\ref{['eq:fixednT']}) with $T_0/t=0.5$ and $\beta_0 t=2.0$. The calculations are performed on an $L=4$ system for the 3D Hubbard model with $U/t=+6,+4,+2,-2,-4,-6$. The black dashed line in (c) plots $\boldsymbol{s}_n(\infty) = \ln 4 - n \ln n - (2 - n) \ln (2 - n)$ as the entropy result at $T/t=\infty$.
• Figure 4: Test results for the varying-$T$ path with fixed $U/t$ and $\mu/t$. Panels (a) and (b) plot the energy density $e_{\mathrm{tot}}/t$ versus inverse temperature $\beta t$ within $0\le \beta t\le 2.0$, and versus temperature $T/t$ within $0.30\le T/t\le 0.50$, respectively. The $e_{\rm tot}/t$ data in (b) have been vertically offset (except for $U/t=6$) to fit into the plot, and the shifts are marked with the same colors as the data. Panel (c) shows the entropy density $\boldsymbol{s}$ as a function of $T/t$, computed via Eq. (\ref{['eq:fixedmT']}) with $T_0/t=0.5$ and $\beta_0 t=2.0$. The calculations are performed on an $L=4$ system for the 3D Hubbard model with $U/t=+6,+4,+2,-2,-4,-6$, at fixed chemical potential $\mu = \mu_0$, chosen such that $n(\mu_0) = 0.875$ at $T/t = 0.3$ for all values of $U/t$. The black dashed line in (c) plots $\boldsymbol{s}_{\mu}(\infty) = \ln 4$ as the entropy result at $T/t=\infty$.
• Figure 5: Comparison between the fixed-$n$ and fixed-$\mu$ calculations. Panels (a) and (b) show the results of entropy density $\boldsymbol{s}$ and two types of specific heat [$C_n$ and $C_{\mu}$, see Eq. (\ref{['eq:SpecHeat']})] versus $T/t$, for $U/t=2$. In the fixed-$n$ path, the filling is maintained at $n=0.875$, while for the fixed-$\mu$ path, the chemical potential is set to $\mu=\mu_0$, chosen such that $n(\mu_0)=0.875$ at $T/t=0.3$. The inset in (a) plots the fermion density as a function of $T/t$ for both cases. Panels (c) and (d) similarly show the results of $\boldsymbol{s}$, and $C_n$ and $C_{\mu}$, for $U/t=6$. The special point $(T/t=0.3,n=0.875)$ is marked by the black star in (a) and (c).