Hypermetallic Polar Molecules for Precision Measurements

Laser cooling is a powerful method to control molecules for applications in precision measurement, as well as quantum information, many-body physics, and fundamental chemistry. However, many optically-active metal centers in valence states which are promising for these applications, especially precision measurement, are difficult to laser cool. In order to extend the control afforded by laser cooling to a wider array of promising atoms, we consider the use of small, hypermetallic molecules that contain multiple metal centers. We provide a detailed analysis of YbCCCa and YbCCAl as prototypical examples with different spin multiplicities, and consider their feasibility for precision measurements making use of the heavy Yb atom. We find that these molecules are linear and feature metal-centered valence electrons, and study the complex hybridization and spin structures that are relevant to photon cycling and laser cooling. Our findings suggest that this hypermetallic approach may be a versatile tool for experimental control of metal species that do not otherwise efficiently cycle photons, and could present a new polyatomic platform for state-of-the-art precision measurements

Towards High-Accuracy Simulations of Strongly Correlated Materials Using Tensor Networks

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

Exploring the magnetic properties of the largest single molecule magnets

The giant {Mn₇₀} and {Mn₈₄} wheels are the largest nuclearity single-molecule magnets synthesized to date, and understanding their magnetic properties poses a challenge to theory. Starting from first-principles calculations, we explore the magnetic properties and excitations in these wheels using effective spin Hamiltonians. We find that the unusual geometry of the superexchange pathways leads to weakly coupled {Mn₇} subunits carrying an effective S = 2 spin. The spectrum exhibits a hierarchy of energy scales and massive degeneracies, with the lowest-energy excitations arising from Heisenberg-ring-like excitations of the {Mn₇} subunits around the wheel. We further describe how weak longer-range couplings can select the precise spin ground-state of the Mn wheels out of the nearly degenerate ground-state band

On the generalization of the exponential basis for tensor network representations of long-range interactions in two and three dimensions

In one dimension (1D), a general decaying long-range interaction can be fit to a sum of exponential interactions e^(−λrij) with varying exponents λ, each of which can be represented by a simple matrix product operator with bond dimension D=3. Using this technique, efficient and accurate simulations of 1D quantum systems with long-range interactions can be performed using matrix product states. However, the extension of this construction to higher dimensions is not obvious. We report how to generalize the exponential basis to two and three dimensions by defining the basis functions as the Green's functions of the discretized Helmholtz equation for different Helmholtz parameters λ, a construction which is valid for lattices of any spatial dimension. Compact tensor network representations can then be found for the discretized Green's functions, by expressing them as correlation functions of auxiliary fermionic fields with nearest-neighbor interactions via Grassmann Gaussian integration. Interestingly, this analytic construction in three dimensions yields a D=4 tensor network representation of correlation functions which (asymptotically) decay as the inverse distance (r^(−1)_(ij)), thus generating the (screened) Coulomb potential on a cubic lattice. These techniques will be useful in tensor network simulations of realistic materials

Efficient representation of long-range interactions in tensor network algorithms

We describe a practical and efficient approach to represent physically realistic long-range interactions in two-dimensional tensor network algorithms via projected entangled-pair operators (PEPOs). We express the long-range interaction as a linear combination of correlation functions of an auxiliary system with only nearest-neighbor interactions. To obtain a smooth and radially isotropic interaction across all length scales, we map the physical lattice to an auxiliary lattice of expanded size. Our construction yields a long-range PEPO as a sum of ancillary PEPOs, each of small, constant bond dimension. This representation enables efficient numerical simulations with long-range interactions using projected entangled pair states.Comment: Main Document: 9 pages, 7 figures. Moved supplementary material into main text. Added more discussion of computational cost. Fixed minor errors in Figs 2c and 3