Selected topics in quantum many-body physics : integrable and open quantum systems, the variational reduced density matrix method and topological insulators
The many-body problem is central and ubiquitous in quantum physics. Its applications are far reaching, varying from the design of new drugs to the research on novel nanotechnologies or building quantum computers. In this thesis we have studied three relevant topics of quantum many-bodyphysics: the description of new exactly solvable models that describe the evolution of quantum systems in contact with the environment, the advance in the variational reduced density matrix method to compute ground states of closed quantum systems and the characterization of the topological phases and currents in topological insulators...
The many-body problem is central and ubiquitous in quantum physics. Its applications are far reaching, varying from the design of new drugs to the research on novel nanotechnologies or building quantum computers. In this thesis we have studied three relevant topics of quantum many-bodyphysics: the description of new exactly solvable models that describe the evolution of quantum systems in contact with the environment, the advance in the variational reduced density matrix method to compute ground states of closed quantum systems and the characterization of the topological phases and currents in topological insulators...
(English) In recent years, the evolution of quantum technologies has resulted in an unprecedented control over individual quantum particles and many-body systems. This remarkable progress has given rise to a new era, marked by the convergence of classical and quantum methodologies to investigate fundamental questions concerning the nature of quantum matter, improving our understanding of the role of entanglement in solid-state systems or the mechanisms behind high-energy physics. From analog quantum simulators to digital quantum computers, these advancements hold potential across diverse domains. This thesis explores the interplay between classical and quantum methods in understanding topological phases of matter. We concentrate on three distinct directions: non-conventional topological superconductors, interaction-induced topological phases in ultracold atom quantum simulators, and applications of variational quantum algorithms. Each trajectory relies on the combination of different techniques with the aim of understanding and characterizing topological phenomena in different settings. Exploring non-conventional topological superconductors involves extending the paradigmatic Kitaev chain model by incorporating additional terms in the Hamiltonian such as long-range interactions and quasi-periodic potentials. This investigation is relevant to better understand the impact of real-world conditions on the both the topological and localization properties of systems hosting non-local Majorana modes, which are promising candidates for topological quantum computation. In the realm of interacting systems, we explore the realization of interaction-induced topological phases in systems of ultracold atoms in optical lattices, both in one and two dimensions. The remarkable control and versatility of such platforms enable the simulation of both theoretical topological models and strongly correlated physics. Notably, the interplay between interactions and topology can give rise to intriguing phenomena, such as delocalized fractional charges and gapless topological phases,challenging existing intuition. We employ advanced numerical methods based on tensor networks to benchmark the experimental proposals that open the door to the realization and detection of novel many-body phases of matter, including topological quantum critical points and a higher-order topological Peierls insulator in Bose-Hubbard models with long-range interactions. Variational quantum algorithms, conversely, have the potential to efficiently tackle a wide range of problems, including ground state search, phase classification or accessing topological invariants. Despite current limitations in trainability and scalability, these hybrid classical-quantum algorithms provide practical insights into current quantum hardware capabilities and can inspire future architectures. We explore the application of variational quantum algorithms to shed light on topological phenomena, raising questions about their ability to discern topological phase transitions and compute topological invariants in situations where classical approaches fail. This thesis presents a comprehensive exploration of distinct approaches to topological quantum matter by leveraging quantum technologies and quantum-inspired classical algorithms. Our results not only advance our understanding of quantum systems but also pave the way for the realization and discovery of novel physics extending to quantum information processing, materials science, and beyond.
This thesis is focused on the application and development of numerical methods for studying quantum many- body spin systems. Part I sets the stage for research projects. In chapter 1, I go over major milestones in the field of frustrated magnetism, how it fits into a larger category of condensed matter physics, and recent developments that provide context for the research projects presented in part II. Chapters 2 3 provide the reader with necessary concepts in physics of phase transitions and numerical methods that are often employed when studying ground states of quantum many body systems. Part II focuses on the research projects carried out during my PhD research. In chapter 4 I present a machine-learning inspired method for solving quantum many body problems. This method addresses issues of representation of variational wave functions and their optimization. In chapter 5, I present a numerical study on the embedding of an exactly solvable point in a phase diagram of an extended Heisenberg model on a kagome lattice. Chapter 6 studies a quantum Heisenberg model on the stuffed honeycomb lattice. Phase transitions in quantum many body systems manifest themselves as a reorganization of low energy spectra of the system. They come in a variety of forms ranging from symmetry breaking transitions, where a local order parameter is formed and to topological phase transitions where the change of the ground state can be attributed to non-local changes in the entanglement patterns. In chapter 2, I review classic models studied in the context of frustrated magnetism and briefly go over the formulation of Landau symmetry breaking theory and common order parameters. The breakdown of Landau symmetry-breaking theory for topological phase transition is discussed using the example of Resonance Valence Bond (RVB) state, which serves as a qualitative exemplification of spin liquid phases. I discuss means of detecting phase transitions with a focus on methods that are applicable in a numerical setting. Chapter 3 presents elements of the numerical machinery that the rest of the thesis employs and builds upon. I discuss traditional trade-offs and provide rough metrics used for analysis of strengths and drawbacks of different approaches. Then I focus on Exact Diagonalization (ED) and Variational Monte Carlo (VMC) methods that are used in later chapters. In chapter 4, I describe the application of machine learning techniques to the variational approach to the quantum many body problem. I introduce Computational Graph States (CGS), a new class of variational wave functions based on the dataflow programming paradigm. The use of computational graphs makes composing new wave functions easy due to its modular structure. Automatic differentiation allows us to obtain efficient derivatives with respect to variational parameters automatically. We also develop a novel optimization scheme, Supervised Wave-function Optimization. It combines ideas of supervised learning and imaginary time evolution. Our method resolves a number of technical challenges and allows seamless optimization of complicated variational models. During the exploration process we constructed a new class of system size invariant model wave functions that can be trained on one system size and generalized to other system sizes with high accuracy. In chapter 5, we study the phase diagram of the kagome antiferromagnet in the J2, Jz plane, which corresponds to second nearest neighbor interactions and spin anisotropy. We report an exactly solvable point with macroscopic degeneracy at J2 = 0, Jz = −0.5 and show that many phases emanate from around this point. In addition, we find a previously unknown transition within the spin-liquid region suggesting a possibility of two distinct spin liquids. In chapter 6, we consider a Heisenberg model on the stuffed honeycomb lattice which interpolated between the triangular and honeycomb lattices. Using exact diagonalization we compute the phase diagram of this model. We find that for this model, the triangular lattice J1-J2 spin liquid region is extended to a larger domain that has lower symmetries than that of the triangular lattice. We perform a variational study of projected BCS wave functions which we directly compare to exact ground states on 36 site cluster as well as consider variational energies from larger system sizes. We argue that construction of symmetric states is essential to obtain good variational energies and give evidence that Dirac spin-liquid state is likely to characterize the spin liquid phase.
This thesis is a collection of research work in quantum many-body physics that I have done during the past few years. The topics include: 1. a theory of anomalous magnetotransport from mass anisotropy of the fermions on a Fermi surface; 2. attempts to understand and build a model for magic-angle bilayer graphene; 3. phenomena of dimensional decoupling at various continuous Mott transitions; 4. possible field-induced neutral Fermi surface and QCD3-Chern-Simons quantum criticalities in Kitaev materials; 5. examples showing that the commonly-employed methods for detecting topological orders may generate spurious results; 6. a classification of three-dimensional symmetry enriched U(1) quantum spin liquids, and the applications of the ideas therein to study topological crystalline insulators.
Quantum many body physics is an exciting research area, involving novel phases of matter with fundamentally new properties, but is also notoriously hard due to complexity of interacting quantum systems. Some popular approaches involve approximation techniques and numerical simulations, which are known to fail in several important cases. In contrast, rigorous mathematical tools, such as exact solutions and operator inequalities, have a narrower range of applicability, but provide guaranteed results and insights into the underlying physical mechanisms. This research aims to develop new techniques in this direction and use them to explore novel phases of matter. My first direction is to construct toy models where exact solutions are possible. Such models are important as they prove that certain physical phenomena are theoretically possible in nature, and often lead to the discovery of new phases. Such insights are provided by three exactly solvable models I have discovered: (1) a family of 1D quantum spin models hosting free parastatistical quasiparticles (an exotic type of identical particles beyond fermions, bosons, and anyons), proving for the first time that parastatistics is theoretically possible as an emergent phenomenon; (2) a 3D classical Ising model whose phases are characterized by topological features of certain loop observables, suggesting existence of previously unknown classical phases and phase transitions with topological order parameters; and (3) a family of models with exact p-wave superconducting ground states demonstrating the existence of Majorana quasiparticles and non-Abelian statistics in particle number-conserving systems. My second direction is to derive rigorous bounds and exact constraints on physical observables, which are applicable to large families of quantum many-body systems. I present three directions of progress: (1) a method that dramatically improves the upper bounds on the speed of information propagation in locally-interacting systems, which significantly extends the scope of these bounds and enables new applications; (2) bounds on finite-size errors in numerical simulations of lattice systems, including quench dynamics and gapped ground states; and (3) a locality bound on gapped ground states of power-law interacting systems, which leads to a generalization of the aforementioned error bounds to such systems. These error bounds have important theoretical implications such as proving the existence of the thermodynamic limit and stability of phases, and are practically useful in determining the validity of finite-size numerical simulations.
The study of open quantum systems forms one of the main problems of modern physics. This thesis aims at contributing to this field, and focuses on the investigation of non-Markovian dynamics of open quantum systems. Here, as a special case, quantum many-body systems provide for a rich field of research. With the ability to control them being of essential importance for the development of quantum technologies, the characterization of their dynamical properties has also drawn a lot of interest lately. One of the main methodological tasks in the study of open quantum systems is to reduce the infinite degrees of freedom of the total system. In order to achieve this, the thesis makes use of the tensor network method matrix product states and combines it with the picture of the quantum stochastic Schrödinger equation. However, when computing the dynamics of many-body systems of system size N, even the reduced system dynamics scale with 2N, making larger system sizes difficult to access. Therefore, especially when computing non-Markovian time-dynamics where the information backflow from the reservoir has to be taken into account, efficiently constructed algorithms are required. This thesis proposes these type of algorithms and demonstrates their application. They enable an access to larger system sizes, thus making it possible to investigate non-equilibrium steady-states of quantum many-body systems in a non-Markovian interaction for system sizes up to N=30. This thesis consists of three parts. The first part provides an introduction to the theory of open quantum systems and to the methodology for modeling them. The second part examines many-body system-reservoir interaction in the non-Markovian regime by investigating the case of fully coherent self-feedback. It uses the one-dimensional Heisenberg spin-1/2 chain as a paradigmatic and recently intensely studied many-body system. It is demonstrated that this gives rise to new conditions for the feedback phase and to highly non-trivial states within the system as well as to the possibility of partially characterizing the system non-invasively and storing excitation within it. Next, a novel approach to the boundary-driven Heisenberg chain is presented, where the chain is exposed to a coherent driving field, thus enabling a comparison to the well-studied incoherently driven model. The third part demonstrates the use of an external pump to control the feedback phase of a Lambda-type three-level system, where the external pump pulse is shown to be a new control parameter. The thesis concludes with the proposition to enhance the indistinguishability of two photons by feedback control.
The ability to precisely and coherently control atomic systems has improved dramatically in the last two decades, driving remarkable advancements in quantum computation and simulation. In recent years, atomic and atom-like systems have also been served as a platform to study topological phases of matter and non-equilibrium many-body physics. Integrated with rapid theoretical progress, the employment of these systems is expanding the realm of our understanding on a range of physical phenomena. In this dissertation, I draw on state-of-the-art experimental technology to develop several new ideas for controlling and applying atomic systems. In the first part of this dissertation, we propose several novel schemes to realize, detect, and probe topological phases in atomic and atom-like systems. We first theoretically study the intriguing properties of Hopf insulators, a peculiar type of topological insulators beyond the standard classification paradigm of topological phases. Using a solid-state quantum simulator, we report the first experimental observation of Hopf insulators. We demonstrate the Hopf fibration with fascinating topological links in the experiment, showing clear signals of topological phase transitions for the underlying Hamiltonian. Next, we propose a feasible experimental scheme to realize the chiral topological insulator in three dimensions. They are a type of topological insulators protected by the chiral symmetry and have thus far remained unobserved in experiment. We then introduce a method to directly measure topological invariants in cold-atom experiments. This detection scheme is general and applicable to probe of different topological insulators in any spatial dimension. In another study, we theoretically discover a new type of topological gapless rings, dubbed a Weyl exceptional ring, in three-dimensional dissipative cold atomic systems. In the second part of this dissertation, we focus on the application of atomic systems in quantum computation and simulation. Trapped atomic ions are one of the leading platforms to build a scalable, universal quantum computer. The common one-dimensional setup, however, greatly limits the system's scalability. By solving the critical problem of micromotion, we propose a two-dimensional architecture for scalable trapped-ion quantum computation. Hamiltonian tomography for many-body quantum systems is essential for benchmarking quantum computation and simulation. By employing dynamical decoupling, we propose a scalable scheme for full Hamiltonian tomography. The required number of measurements increases only polynomially with the system size, in contrast to an exponential scaling in common methods. Finally, we work toward the goal of demonstrating quantum supremacy. A number of sampling tasks, such as the boson sampling problem, have been proposed to be classically intractable under mild assumptions. An intermediate quantum computer can efficiently solve the sampling problem, but the correct operation of the device is not known to be classically verifiable. Toward practical verification, we present an experimental friendly scheme to extract useful and robust information from the quantum boson samplers based on coarse-grained measurements. In a separate study, we introduce a new model built from translation-invariant Ising-interacting spins. This model possesses several advantageous properties, catalyzing the ultimate experimental demonstration of quantum supremacy.