Abstract
We speed up thermal simulations of quantum many-body systems in both one- (1D) and two-dimensional (2D) models in an exponential way by iteratively projecting the thermal density matrix $\hat\rho=e^{-\beta \hat{H}}$ onto itself. We refer to this scheme of doubling $\beta$ in each step of the imaginary time evolution as the exponential tensor renormalization group (XTRG). This approach is in stark contrast to conventional Trotter-Suzuki-type methods which evolve $\hat\rho$ on a linear quasi-continuous grid in inverse temperature $\beta \equiv 1/T$. In general, XTRG can reach low temperatures exponentially fast, and thus not only saves computational time but also merits better accuracy due to significantly fewer truncation steps. 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 use our XTRG machinery to explore the thermal properties of Heisenberg models on 1D chains and 2D square and triangular lattices down to low temperatures approaching ground state properties. The entanglement properties, as well as the renormalization group flow of entanglement spectra in MPOs, are discussed, where logarithmic entropies (approximately $\ln\beta$) are shown in both spin chains and square lattice models with gapless towers of states. We also reveal that XTRG can be employed to accurately simulate the Heisenberg XXZ model on the square lattice which undergoes a thermal phase transition. We determine its critical temperature based on thermal physical observables, as well as entanglement measures. Overall, we demonstrate that XTRG provides an elegant, versatile, and highly competitive approach to explore thermal properties in both 1D and 2D quantum lattice models.
Highlights
Efficient simulations of interacting quantum manybody systems are crucial for a better understanding of correlated materials
We demonstrate that XTRG provides an elegant, versatile, and highly competitive approach to explore thermal properties, including finite-temperature thermal phase transitions as well as the different ordering tendencies at various temperature scales for frustrated systems
II, we introduce the XTRG scheme with symmetries implemented, as well as an improved series-expansion thermal tensor network (SETTN) method [42] based on a pointwise Taylor expansion algorithm that exploits the logarithmic temperature scale
Summary
Efficient simulations of interacting quantum manybody systems are crucial for a better understanding of correlated materials. By definition of the partition function, it acquires intrinsic PBC in the direction of temperature, which doubles the prefactor in entanglement entropy scaling, in agreement with the earlier arguments This logarithmic growth of entropy versus β provides a tight upper bound in efficient thermal simulations [32]. II, we introduce the XTRG scheme with symmetries implemented, as well as an improved series-expansion thermal tensor network (SETTN) method [42] based on a pointwise Taylor expansion algorithm that exploits the logarithmic temperature scale XTRG is employed to study the finite-temperature phase transition of the 2D Heisenberg XXZ model, where we demonstrate that XTRG can accurately pinpoint the critical temperature
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