Thermal Tensor Network Simulations of Fermions with a Fixed Filling

TL;DR

The paper resolves the challenge of fixing particle number in finite-temperature tensor-network simulations by introducing fixed-$N$ tanTRG, which adaptively tunes the imaginary-time chemical potential $\mu(\tau)$ within the TDVP framework to stabilize the target filling $\langle N\rangle = N_{\rm target}$ during cooling in the grand-canonical ensemble. It builds a geometric, tangent-space MPO formalism with bilayer MPO representations and projectors, enabling gradient-based imaginary-time evolution and efficient computation of thermodynamic quantities. The method is validated on a non-interacting spinless fermion chain, showing excellent agreement with exact results, and then applied to the hole-doped square-lattice Hubbard model to reveal temperature-driven charge/spin stripe formation and multiple characteristic temperature scales. The approach offers an efficient, reliable tool for finite-temperature studies of correlated fermions, with potential generalization to other conserved quantities and tensor-network frameworks, and practical impact for investigating high-temperature superconductivity and related phenomena.

Abstract

Numerical simulations of strongly correlated fermions at finite temperature are essential for studying high-temperature superconductivity and other quantum many-body phenomena. The recently developed tangent-space tensor renormalization group (tanTRG) provides an efficient and accurate framework by representing thermal density operators as matrix product operators. However, the particle number generally varies during the cooling process. The conventional strategy of fine-tuning chemical potentials to reach a target filling is computationally demanding. Here we propose a fixed-$N$ tanTRG algorithm that stabilizes the average particle number by adaptively tuning the chemical potential within the imaginary-time evolution. We benchmark its accuracy on exactly solvable free fermions, and further apply it to the square-lattice Hubbard model. For hole-doped cases, we study the temperature evolution of charge and spin correlations, identifying several characteristic temperature scales for stripe formation. Our results establish fixed-$N$ tanTRG as an efficient and reliable tool for finite-temperature studies of correlated fermion systems.

Figures (7)

• Figure 1: (a) Bilayer MPO representation of the thermal density operator, which can be viewed as the reduced density operator of the purified state $\ket{\rho(\beta/2)}$ after tracing out the auxiliary indices. The triangles indicate the directions of canonical forms. (b) Tensor network used to compute the expectation value of an on-site observable located at the canonical center. (c) Illustration of the Choi isomorphism, which maps an MPO to an MPS with enlarged local Hilbert spaces.
• Figure 2: (a,b) Tensor network illustration of the tangent space projectors $P_i^{\rm 1s}$ and $P_i^{\rm b}$, respectively. The vertical line represents the identity operator acting on the $i$-th site. (c) Local renormalized state at the canonical center $\ket{\rho_i^{\rm 1s}}$, and the corresponding 1-site effective Hamiltonian $H_i^{\rm 1s}$ after the tangent space projection. (d) Similar to (c), but in the bond-canonical form.
• Figure 3: Benchmark of thermodynamic quantities for a spinless fermion chain of length 64 with open boundary conditions (OBC), where the filling is fixed at $n_{\rm target} = 3/4$ across all temperatures. (a-e) Temperature dependence of the chemical potential $\mu$, energy $E$, free energy $F$, entropy $S$, and specific heat $C_N$ and $C_\mu$, respectively, normalized by system size $N_{\rm s} = 64$. (f) Deviation of the actual filling and the target value $n_{\rm target} = 3/4$. The gray dashed line indicates the tolerance $10^{-6}$ that triggers the TEBD correction. (g-j) Absolute errors of the thermodynamic quantities shown in panels (b-e), compared with the exact solution.
• Figure 4: Benchmark of the EoS for an OBC spinless fermion chain of length 64, compared with the exact solution in TDL (red lines). (a) $n-\mu$ relation at several temperatures. Each marker corresponds to a target charge density obtained using the fixed-$N$ tanTRG algorithm. (b-c) Charge susceptibility $\chi_c$ as a function of $n$ and $\mu$, respectively. The lines are obtained via numerical differentiation, $\chi_c = (\partial n/\partial \mu)_T$, while the markers are obtained from charge fluctuations $\chi_c = \left(\expval{N^2} - \expval{N}^2\right) / (N_{\rm s}T)$. The bond dimension used is $D = 1024$, yielding fully converged results.
• Figure 5: Benchmark of thermodynamic quantities of the $\delta = 1/12$ hole-doped Hubbard model on 4$\times$24 cylinder, with typical parameters $U = 8$ and $U = 12$. (a) Chemical potential $\mu$ required to maintain the target doping, obtained from fixed-$N$ tanTRG and DQMC. (b) Per-site energy $E/N_{\rm s}$. Ground state energies $E_g/N_{\textrm{s}}$ from DMRG are indicated by horizontal dashed lines. The inset shows the convergence with bond dimension $D$ at $T =1/64$ (left) and $T = 0.5$ (right), where the offset $E_\infty$ is estimated by linearly extrapolating $1/D \rightarrow 0$. The Trotter step length in DQMC is set as $\delta\tau = 0.0125$ in main panel, whose effect is also examined at $T = 0.5$. (c) Specific heat $C_N/N_{\rm s}$ evaluated from numerical differentiation $(\partial S/\partial \ln T)_{\expval{N}}$. The data for $U=12$ is vertically offset by 0.3 for clarity.