Accurate and verifiable computation of the properties of real materials with strong electron correlation has been a long-standing challenge in the fields of chemistry, physics, and material science. Most existing algorithms suffer from either approximations that are too inaccurate, or fundamental computational complexity that is too high. In studies of simplified models of strongly-correlated materials, tensor network algorithms have demonstrated the potential to overcome these limitations. This thesis describes our research efforts to develop new algorithms for two-dimensional (2D) tensor networks that extend their range of applicability beyond simple models and toward simulations of realistic materials.
We begin by describing three algorithms for projected entangled-pair states (PEPS, a type of 2D tensor network) that address three of their major limitations: numerical stability, long-range interactions, and computational efficiency of operators. We first describe (Ch. 2) a technique for converting a PEPS into a canonical form. By generalizing the QR matrix factorization to entire columns of a PEPS, we approximately generate a PEPS with analogous properties to the well-studied canonical 1D tensor network. This connection enables enhanced numerical stability and ground state optimization protocols. Next, we describe (Ch. 3) a technique to efficiently represent physically realistic long-range interactions between particles in a 2D tensor network operator, a projected entangled-pair operator (PEPO). We express the long-range interaction as a linear combination of correlation functions of an auxiliary system with only nearest-neighbor interactions. This allows us to represent long-range pairwise interactions with linear scaling in the system size. The third algorithm we present (Ch. 4) is a method to rewrite the 2D PEPO in terms of a set of quasi-1D tensor network operators, by exploiting intrinsic redundancies in the PEPO representation. We also report an on-the-fly contraction algorithm using these operators that allows for a significant reduction in computational complexity, enabling larger scale simulations of more complex problems.
We then move on to describe (Ch. 5) an extensive study of a "synthetic 2D material"---a two-dimensional square array of ultracold Rydberg atoms---enabled by some of the new algorithms. We investigate the ground state quantum phases of this system in the bulk and on large finite arrays directly comparable to recent quantum simulation experiments. We find a greatly altered phase diagram compared to earlier numerical and experimental studies, and in particular, we uncover an unexpected entangled nematic phase that appears in the absence of geometric frustration.
Finally, we finish by describing (Ch. 6) a somewhat unrelated, but topically similar project in which we investigate the feasibility of laser cooling small molecules with two metal atoms to ultracold temperatures. We study in detail the properties of the molecules YbCCCa and YbCCAl for application in precision measurement experiments.</p
