Computational inverse method for constructing spaces of quantum models from wave functions

Traditional computational methods for studying quantum many-body systems are "forward methods," which take quantum models, i.e., Hamiltonians, as input and produce ground states as output. However, such forward methods often limit one's perspective to a small fraction of the space of possible Hamiltonians. We introduce an alternative computational "inverse method," the Eigenstate-to-Hamiltonian Construction (EHC), that allows us to better understand the vast space of quantum models describing strongly correlated systems. EHC takes as input a wave function $|\psi_T\rangle$ and produces as output Hamiltonians for which $|\psi_T\rangle$ is an eigenstate. This is accomplished by computing the quantum covariance matrix, a quantum mechanical generalization of a classical covariance matrix. EHC is widely applicable to a number of models and in this work we consider seven different examples. Using the EHC method, we construct a parent Hamiltonian with a new type of antiferromagnetic ground state, a parent Hamiltonian with two different targeted degenerate ground states, and large classes of parent Hamiltonians with the same ground states as well-known quantum models, such as the Majumdar-Ghosh model, the XX chain, the Heisenberg chain, the Kitaev chain, and a 2D BdG model.Comment: 13 pages, 7 figures, 1 table; new example in results section; updated supplement; additional references; other minor change

Inverse methods in quantum many-body physics

The interactions of many quantum particles can give rise to fascinating emergent behavior and exotic phases of matter with no classical analogues. Examples include phases with topological properties, which can occur at low temperatures in frustrated magnets and certain superconductors, and many-body localized (MBL) phases that do not obey the laws of thermodynamics, which can occur in interacting disordered magnets. Traditionally, such quantum phases of matter have been studied using a "forward" approach, where a model for the phase is solved to understand the phase's properties. In this thesis, we explore an alternative "inverse" approach to the problem, where we find models from properties, and show how inverse methods and related tools can be used to efficiently study topological and MBL physics in a new way. In Chapter 1, we introduce the theoretical background necessary for understanding this thesis. First, we discuss the typical forward approach used to study quantum physics and some of its limitations. We introduce the alternative inverse approach that we take in this thesis and give some background on how methods for solving inverse problems have been highly successful in areas such as machine learning and classical physics. Next, we describe two types of topological phases of matter, quantum spin liquids with Wilson loops and topological superconductors with Majorana zero modes (MZMs). These phases have exotic properties, such as long-ranged entanglement and anyonic quasiparticles, that make them interesting to study and potentially useful in emerging technologies such as quantum computing. Finally, we provide an overview of the phenomenon of many-body localization, the failure of many quantum particles to thermalize -- equilibrate with their surroundings -- in the presence of strong interactions and disorder. We introduce the concept of thermalization and discuss how MBL systems defy thermalization. We also explain the various key signatures of MBL physics, such as low-entanglement of eigenstates and the existence of local integrals of motion known as local bits or l-bits. In Chapter 2, we discuss the main numerical techniques that we used to study quantum many-body systems. First, we discuss the exact but computationally expensive exact diagonalization (ED) method, which can be used to study small systems with few quantum spins. Next, we discuss the variational Monte Carlo (VMC) method, which can be used to compute properties of certain classes of variational wave functions by sampling a Markov chain. Then, we explain techniques for performing calculations with tensor networks, a class of quantum states defined through the contraction of many tensors. Finally, in addition to the state-based methods we just described, we also introduce operator-based methods that we be essential for our inverse approach and our study of MBL. In Chapter 3, we introduce the eigenstate-to-Hamiltonian construction (EHC) inverse method that finds Hamiltonians with desired eigenstates. We benchmark the method with many different input states in one and two-dimensions. In each case, we find that the EHC method can find many different Hamiltonians with the target state as an eigenstate, and in many cases a ground state. We show how EHC can be used to find new Hamiltonians with interesting ground states, find Hamiltonians with degenerate ground states, and expand the ground state phase diagrams of previously studied Hamiltonians. In Chapter 4, we introduce the symmetric Hamiltonian construction (SHC) inverse method that finds Hamiltonians with desired symmetries. We use SHC to study quantum spin liquids and topological superconductors. In particular, by providing Wilson loops as input to SHC, we find new types of spin liquid Hamiltonians with properties not seen in previously studied models and, by providing MZMs as input to SHC, we find a large class of superconductor Hamiltonians with tunable MZM physics. In Chapter 5, we develop a tool that allows us to study MBL physics in higher dimensions than was previously possible. While MBL has been clearly observed in one spatial dimension, it is a key open question whether MBL survives in two or three dimensions. Because of the numerical difficulty of studying two and three dimensional quantum systems, this problem has been largely unexplored. We develop an algorithm for finding approximate l-bits, local integral of motions and a key signature of MBL physics, in arbitrary dimensions. Using this algorithm, we observe a sharp change in the properties of l-bits versus disorder strength for four different models in one, two, and three dimensional spin systems. This provides the first evidence for the existence of a thermal to MBL transition in three dimensions. In Chapter 6, we present a method for constructing a large family of Hamiltonians with magnetically ordered "spiral colored" ground states. We demonstrate how these Hamiltonian and states can be arranged into many different geometrical patterns. We also show that with slight modification these Hamiltonians can be made to realize quantum many-body scars, a type of anomalous high-energy excited state that does not exhibit thermal properties as is typical for quantum systems that thermalize. In Chapter 7, we summarize our work and provide an outlook on paths forward

Inverse design of disordered stealthy hyperuniform spin chains

Positioned between crystalline solids and liquids, disordered many-particle systems which are stealthy and hyperuniform represent new states of matter that are endowed with novel physical and thermodynamic properties. Such stealthy and hyperuniform states are unique in that they are transparent to radiation for a range of wavenumbers around the origin. In this work, we employ recently developed inverse statistical-mechanical methods, which seek to obtain the optimal set of interactions that will spontaneously produce a targeted structure or configuration as a unique ground state, to investigate the spin-spin interaction potentials required to stabilize disordered stealthy hyperuniform one-dimensional (1D) Ising-like spin chains. By performing an exhaustive search over the spin configurations that can be enumerated on periodic 1D integer lattices containing $N=2,3,\ldots,36$ sites, we were able to identify and structurally characterize \textit{all} stealthy hyperuniform spin chains in this range of system sizes. Within this pool of stealthy hyperuniform spin configurations, we then utilized such inverse optimization techniques to demonstrate that stealthy hyperuniform spin chains can be realized as either unique or degenerate disordered ground states of radial long-ranged (relative to the spin chain length) spin-spin interactions. Such exotic ground states are distinctly different from spin glasses in both their inherent structural properties and the nature of the spin-spin interactions required to stabilize them. As such, the implications and significance of the existence of such disordered stealthy hyperuniform ground state spin systems warrants further study, including whether their bulk physical properties and excited states, like their many-particle system counterparts, are singularly remarkable, and can be experimentally realized.Comment: 11 pages, 9 figure

The Future of the Correlated Electron Problem

The understanding of material systems with strong electron-electron interactions is the central problem in modern condensed matter physics. Despite this, the essential physics of many of these materials is still not understood and we have no overall perspective on their properties. Moreover, we have very little ability to make predictions in this class of systems. In this manuscript we share our personal views of what the major open problems are in correlated electron systems and we discuss some possible routes to make progress in this rich and fascinating field. This manuscript is the result of the vigorous discussions and deliberations that took place at Johns Hopkins University during a three-day workshop January 27, 28, and 29, 2020 that brought together six senior scientists and 46 more junior scientists. Our hope, is that the topics we have presented will provide inspiration for others working in this field and motivation for the idea that significant progress can be made on very hard problems if we focus our collective energies.Comment: 55 pages, 19 figure

Designer stealthy disordered spin chains and their quantum behavior

The goal of this thesis is to study disordered stealthy hyperuniform spin chains on a one-dimensional integer lattice with classical and quantum interactions. Hyperuniform systems possess anomalously suppressed long-wavelength density fluctuations. Stealthy hyperuniform systems are transparent to radiation with wavelength greater than 2¿/K, where K is an exclusion radius in k-space [1]. Disordered hyperuniformity characterizes an interesting variety of physical systems, including large-scale structure in the Universe [2], the arrangement of avian photoreceptors [3], and disordered 2D photonic materials with complete photonic band gaps [4]. We enumerate all periodic stealthy hyperuniform one-dimensional spin systems up to a finite unit cell size. To study the enumerated stealthy configurations, we make use of a recently developed inverse statistical mechanics method for classical spin systems [5, 6] as well as quantum Monte Carlo simulation. The inverse method finds classical spin-spin interaction potentials that stabilize the stealthy configurations as classical ground states. By adding a transverse field to the designer Hamiltonian, we observe how the classically disordered spin states behave with additional quantum fluctuations and discover fascinating ordering behavior for an exemplary stealthy system

Polymorphism of Crystalline Molecular Donors for Solution-Processed Organic Photovoltaics

Using ab initio calculations and classical molecular dynamics simulations coupled to complementary experimental characterization, four molecular semiconductors were investigated in vacuum, solution, and crystalline form. Independently, the molecules can be described as nearly isostructural, yet in crystalline form, two distinct crystal systems are observed with characteristic molecular geometries. The minor structural variations provide a platform to investigate the subtlety of simple substitutions, with particular focus on polymorphism and rotational isomerism. Resolved crystal structures offer an exact description of intermolecular ordering in the solid state. This enables evaluation of molecular binding energy in various crystallographic configurations to fully rationalize observed crystal packing on a basis of first-principle calculations of intermolecular interactions