Exponential Thermal Tensor Network Approach for Quantum Lattice Models
Abstract
We speed up thermal simulations of quantum manybody systems in both one (1D) and twodimensional (2D) models in an exponential way by iteratively projecting the thermal density matrix ρ ^=e^{β H ^} onto itself. We refer to this scheme of doubling β in each step of the imaginary time evolution as the exponential tensor renormalization group (XTRG). This approach is in stark contrast to conventional TrotterSuzukitype methods which evolve ρ ^ on a linear quasicontinuous grid in inverse temperature β ≡1 /T . As an aside, the large steps in XTRG allow one to swiftly jump across finitetemperature phase transitions, i.e., without the need to resolve each singularly expensive phasetransition point right away, e.g., when interested in lowenergy behavior. A fine temperature resolution can be obtained, nevertheless, by using interleaved temperature grids. 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. For similar reasons, we also find that the series expansion thermal tensor network approach benefits in both efficiency and precision, from the logarithmic temperature scale setup. We work in an (effective) 1D setting exploiting matrix product operators (MPOs), which allows us to fully and uniquely implement nonAbelian 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 groundstate properties. The entanglement properties, as well as the renormalizationgroup flow of entanglement spectra in MPOs, are discussed, where logarithmic entropies (approximately ln β ) are shown in both spin chains and squarelattice models with gapless towers of states. We also reveal that XTRG can be employed to accurately simulate the Heisenberg X X Z 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, including finitetemperature thermal phase transitions as well as the different ordering tendencies at various temperature scales for frustrated systems.
 Publication:

Physical Review X
 Pub Date:
 July 2018
 DOI:
 10.1103/PhysRevX.8.031082
 arXiv:
 arXiv:1801.00142
 Bibcode:
 2018PhRvX...8c1082C
 Keywords:

 Condensed Matter  Strongly Correlated Electrons
 EPrint:
 17+10 pages