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# Inverse methods in quantum many-body physics

#### Chertkov, Eli

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https://hdl.handle.net/2142/112964

## Description

- Title
- Inverse methods in quantum many-body physics
- Author(s)
- Chertkov, Eli
- Issue Date
- 2021-06-22
- Director of Research (if dissertation) or Advisor (if thesis)
- Clark, Bryan K
- Doctoral Committee Chair(s)
- Goldenfeld, Nigel
- Committee Member(s)
- Song, Jun
- Gadway, Bryce

- Department of Study
- Physics
- Discipline
- Physics
- Degree Granting Institution
- University of Illinois at Urbana-Champaign
- Degree Name
- Ph.D.
- Degree Level
- Dissertation
- Keyword(s)
- quantum many-body physics
- inverse methods
- Majorana zero modes
- quantum spin liquids
- many-body localization
- quantum many-body scars
- frustrated magnetism
- numerical methods

- Abstract
- "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."
- Graduation Semester
- 2021-08
- Type of Resource
- Thesis
- Permalink
- http://hdl.handle.net/2142/112964
- Copyright and License Information
- Copyright 2021 Eli Chertkov

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