Thermodynamics of the Hubbard Model on the Bethe Lattice

TL;DR

The paper analyzes the finite-temperature thermodynamics of the half-filled Hubbard model on a Bethe lattice with coordination $z=3$ using a fermionic thermal canonical tree tensor network. It reveals a finite-temperature transition from a paramagnetic to an antiferromagnetic phase with spontaneous SU(2) symmetry breaking but no Goldstone mode on the Bethe lattice, and demonstrates a pronounced separation of spin and charge energy scales evidenced by distinct susceptibilities, specific-heat behavior, and excitation gaps. A Pomeranchuk effect leads to non-monotonic double occupancy as temperature changes, tied to entropy considerations, while the entanglement spectrum shows parity-resolved signatures of spin–charge physics and confirms the symmetry-broken state. The results validate the Bethe lattice as a faithful, computationally efficient platform for capturing essential Hubbard-model physics in more than one dimension and point to future work extending beyond half-filling and exploring excited-state properties.

Abstract

We investigate the thermodynamic properties of the Hubbard model on the Bethe lattice with a coordination number of 3 using the thermal canonical tree tensor network method. Our findings reveal two distinct thermodynamic phases: a low-temperature antiferromagnetic phase, where spin SU(2) symmetry is broken, and a high-temperature paramagnetic phase. A key feature of the system is the separation of energy scales for charge and spin excitations, which is reflected in the temperature dependence of thermodynamic quantities and the disparity between spin and charge gaps extracted from their respective susceptibilities. At the critical point, both spin and charge susceptibilities exhibit singularities, suggesting that charge excitations are not fully decoupled from their spin counterparts. Additionally, the double occupancy number exhibits a non-monotonic temperature dependence, indicative of an entropy-driven Pomeranchuk effect. These results demonstrate that the loopless Bethe lattice effectively captures the essential physics of the Hubbard model while providing a computationally efficient framework for studying strongly correlated electronic systems.

Figures (10)

• Figure 1: (a) Phase diagram of the Hubbard model at half-filling. The dashed line represents the critical temperature of the Heisenberg model, $T_c^H = 1.36/U$, while the dotted line shows an exponential extrapolation to the small $U$ limit. Temperature dependence of staggered magnetization $M_s$ and specific heat $C_v$ (b-c), along with spin susceptibility $\chi_s$ and charge susceptibility $\chi_c$ (d-e), for the Hubbard model at $U = 4$ (middle panels) and $U = 8$ (right panels), respectively. The dotted lines indicate the antiferromagnetic phase transition temperatures $T_c$.
• Figure 2: Charge susceptibility $\chi_c$ as a function of temperature $T$ for the Hubbard model on the $z = 3$ Bethe-lattice with $U = 4, 4.5$, and $5$.
• Figure 3: Spin and charge excitation gaps, $\Delta_s$ and $\Delta_c$, as functions of $U$. The spin and charge gaps are obtained by fitting the exponential decay behaviors of the spin and charge susceptibilities at low temperatures with the formula $\chi_s \sim \exp(-\Delta_s/T)$ and $\chi_c \sim \exp(-\Delta_c/2T)$. The dashed line $\Delta_s^H$ represents the spin excitation gap of the Heisenberg model with an effective coupling constant $J = 4t^2/U$.
• Figure 4: Scaling behavior of the staggered magnetization $M_s$ as a function of the reduced temperature $t = \left(T_c - T\right)/T_c$ for the Hubbard model with $U=4$ and $U=8$. The dashed lines represent the fitting curves obtained using the formula $M_s = C t^{1/2}$ with $C$ a fitting parameter.
• Figure 5: Double occupancy $D$ as a function of temperature for the Hubbard model at $U = 4$ and $8$. The minima of $D$ indicate the onset temperature of the Pomeranchuk effect. Cross markers connected by the dashed line show how these minima vary with $U$ with a $\Delta U = 1$ step. Insets provide enlarged views of $D$ near the critical temperatures.