Tensor Network Annealing Algorithm for Two-Dimensional Thermal States

Abstract

Tensor network methods have become a powerful class of tools to capture strongly correlated matter, but methods to capture the experimentally ubiquitous family of models at finite temperature beyond one spatial dimension are largely lacking. We introduce a tensor network algorithm able to simulate thermal states of twodimensional quantum lattice systems in the thermodynamic limit. The method develops instances of projected entangled pair states and projected entangled pair operators for this purpose. It is the key feature of this algorithm to resemble the cooling down of the system from an infinite temperature state until it reaches the desired finitetemperature regime. As a benchmark we study the finite-temperature phase transition of the Ising model on an infinite square lattice, for which we obtain remarkable agreement with the exact solution. We then turn to study the finite-temperature Bose-Hubbard model in the limits of two (hard-core) and three bosonic modes per site. Our technique can be used to support the experimental study of actual effectively two-dimensional materials in the laboratory, as well as to benchmark optical lattice quantum simulators with ultra-cold atoms.

Thermodynamic properties of the Shastry-Sutherland model throughout the dimer-product phase

“…While the rank of the tensors used in the representation is determined by the physical problem, the amount of information they contain is determined by their bond dimension, D, which is used to control the accuracy of the ansatz. Although iPEPS were introduced originally for representing the ground states of local Hamiltonians, more recently several iPEPS methods have been developed for the representation of thermal states [36,[52][53][54][55][56][57][58][59][60][61][62]. Here we focus on the approaches discussed in Ref.…”

SU(3) fermions on the honeycomb lattice at 13 filling

“…While the finite color order of the dimer phase can only exist at zero temperature in two dimensions, the dimerization may in principle set in already at finite temperature (without coexisting color order), or occur simultaneously with the color order at zero temperature. Recently developed tensor network approaches for finitetemperature simulations [88][89][90][91][92][93] may provide further insights into the critical temperatures in the future.…”

Tensor Network Contractions

“…A TN operator is regarded as a mapping from the bra to the ket Hilbert space. Many algorithms explicitly employ the TN operator form, including the matrix product operator (MPO) for representing 1D many-body operators and mixed states, and for simulating 1D systems in and out of equilibrium [186,187,188,189,190,191,192,193,194,195,196], tensor product operator (also called projected entangled pair operators) in for higher-systems [140,141,143,197,198,199,200,201,202,203,204,205,206], and multiscale entangled renormalization ansatz [207,208,209].…”

“…The key concepts and ideas, such as environment, (simple, cluster, and full) update schemes, and the use of SVD, can be similarly applied to finite-size cases [236,238], the finite-temperature simulations [140,141,143,199,200,201,202,204,205], and real-time simulations [206,236] in two dimensions. The computational cost of the TN approaches is quite sensitive to the spatial dimensions of the system.…”