Lowering the temperature of two-dimensional fermionic tensor networks with cluster expansions

Abstract

Representing the time-evolution operator as a tensor network constitutes a key ingredient in several algorithms for studying quantum lattice systems at finite temperature or in a non-equilibrium setting. For a Hamiltonian composed of strictly short-ranged interactions, the Suzuki-Trotter decomposition is the main technique for obtaining such a representation. In [B.~Vanhecke, L.~Vanderstraeten and F.~Verstraete, Physical Review A, L020402 (2021)], an alternative strategy, the cluster expansion, was introduced. This approach naturally preserves internal and lattice symmetries and can more easily be extended to higher-order representations or longer-ranged interactions. We extend the cluster expansion to two-dimensional fermionic systems, and employ it to construct projected entangled-pair operator (PEPO) approximations of Gibbs states. We also discuss and benchmark different truncation schemes for multiplying layers of PEPOs together. Applying the resulting framework to a two-dimensional spinless fermion model with attractive interactions, we resolve a clear phase boundary at finite temperature.

Figures (5)

• Figure 1: (top) Magnetization of the quantum Ising model for $g = 2.5$ in function of the inverse temperature for both the second-order ST decomposition and second-order ($P=3$) cluster expansions (CE). We indicate an accurate estimate $\beta_c = 1 / 1.2736$ of the critical temperature, based on sign-problem-free Monte Carlo results QMC_Tc_Ising. (bottom) Correlation length for the same model in comparison with simple update for $\Delta \tau = 10^{-5}$ and $D = 4$.
• Figure 2: Fidelity between the exact cluster expansion PEPO of the free spinless fermion model at $\beta = 0.1$ and the PEPO resulting from truncating it to a lower bond dimension $D$ using the different truncation schemes of section \ref{['sec:truncation_schemes']} with a random initial guess (top panel) and the local truncation result as initial guess (bottom panel). Bond dimension $\chi = 20$ was used for the CTMRG environment in both the global and variational truncation scheme. Since the exact PEPO has bond dimensions $5$, truncating to $D = 5$ or higher would result in a fidelity of $1$ for all truncation schemes.
• Figure 3: Occupancy of the spinless fermion model for $V = -2.5$ where the update-PEPOs were truncated to different bond dimensions using the local truncation scheme. The subsequent time evolution was performed with the local truncation scheme and $D_1 = 5$. The dotted lines denote the predictions from QMC up to 1 standard deviation SF_QMC.
• Figure 4: Occupancy (top) and correlation length (bottom) of the spinless fermion model for $V = -3.0$ for different bond dimensions. The VUMPS algorithm used in the calculation of the observables requires only a single layer of the PEPO.
• Figure 5: Finite-temperature phase diagram of the spinless fermion model for different values of the interaction strength $V$. The QMC results are taken from Ref. SF_QMC.