Exponential Thermal Tensor Network Approach for Quantum Lattice Models

We speed up thermal simulations of quantum many-body systems in one- (1D) and two-dimensional (2D) models in an exponential way by iteratively projecting the thermal density matrix ρ≡e^-βH onto itself. We refer to this approach as the exponential tensor renormalization group (XTRG) [1]. It is in stark contrast to conventional Trotter-Suzuki-type methods which employ a linear quasi-continuous grid in inverse temperature β≡1/T. By avoiding Trotterization alltogether, XTRG can also deal with longer-range interactions in a straightforward algorithmic way. By construction, XTRG can reach exponentially low temperatures by a linear number of iterations, and thus not only saves computational time but also merits better accuracy due to significantly fewer truncation steps. More fine-grained temperature resolution can be achieved via simple interleaving of data sets. We work in an (effective) 1D setting exploiting matrix product operators (MPOs) which allows us to fully and uniquely implement non-Abelian and Abelian symmetries to greatly enhance numerical performance. We show exemplary XTRG results for Heisenberg models on 1D chains and 2D lattices with a finite temperature phase transition down to low temperatures approaching ground state properties. [1] Phys. Rev. X 8, 031082 (2018)