One of the most intriguing areas of physics is the study of strongly correlated many-body systems. In these kinds of systems, a large number of particles interacts via interactions that are strong enough to play a major role in determining the properties of the system, which is particularly interesting when multiple interaction terms compete and cannot be satisfied simultaneously. Examples of phases they realize range from simple magnetism to puzzling phenomena such as superconductivity or topological order. Of special interest in condensed matter are fermionic models which have an extra layer of complexity added by the possibly intricate sign structure of the wave function. \nHowever, exactly solvable models that exhibit the sought-after phenomena are scarce and Hamiltonians that are supposed to model experiments often elude analytical solutions precisely due to the strong interactions. Numerical methods are thus essential to gain insight into these systems and provide an important link between theoretical models and real world experiments. There are many different methods that in principle allow solving a problem numerically exact, most prominently exact diagonalization, tensor network based methods such as DMRG and (quantum) Monte Carlo. Exact diagonalization is certainly the most straightforward technique which allows measuring any observable of interest but its application to interacting models is severely limited by the exponential growth of the Hilbert space with the system size which only allows studying rather small systems. Tensor network methods excel in the study of gapped, one-dimensional systems and while they have shown some impressive results for two-dimensional systems, they need a lot of fine tuning and careful analysis to be considered reliable in this very important domain. Quantum Monte Carlo, on the other hand, is in principle neither limited by the size of the Hilbert space nor by the strength of the interactions and has thus been a tremendously important tool for studying condensed matter systems which is why it is the focus of this thesis. \nIt is necessary to continuously advance existing and develop new techniques to keep up with theoretical as well as with experimental progress. In this thesis, new methods to identify and characterize conventional and novel phases of matter within quantum Monte Carlo are developed and tested on archetypical models of condensed matter. \nIn the first part of this thesis, a method to study entanglement properties of strongly interacting fermionic models in quantum Monte Carlo is presented. Measurements of this kind are needed because some characteristics of a system remain hidden from conventional approaches based on the calculation of correlation functions. Prime examples are systems that realize so-called topological order entailing long-range entanglement that can be positively diagnosed using entanglement techniques. Realizing such measurements in a concrete algorithm is not straightforward for fermions. A large section of this part is thus devoted to developing a solution to the problem of implementing entanglement measurements and benchmarking the results against known data. \nThe second part is concerned with a truly novel and recent approach to the many-body problem, namely the symbiosis of machine learning and quantum Monte Carlo techniques. Out of the many different possibilities to combine the two, this part is concerne with exploring ways in which machine learning can help to effortlessly explore phase diagrams of hitherto unknown Hamiltonians. \nDespite the power and versatility of quantum Monte Carlo techniques, it does suffer from one significant weakness called the fermion sign problem. At its core related to the peculiar exchange statistics of fermions, it is in principle merely a technical problem of the Monte Carlo procedure but one that actually prohibits the use of Monte Carlo methods for a large number of interesting problems. The sign problem has been known for a long time but remains unsolved to this day. Thus, it is time to approach the problem from a different perspective with the hope of learning something new that can either help to better understand the problem itself or the properties of the models affected by it. Both of the two major research areas, entanglement measures and machine learning, are capable of providing such new perspectives and the sign problem will be revisited in more detail in the context of each of them.
One of the most intriguing areas of physics is the study of strongly correlated many-body systems. In these kinds of systems, a large number of particles interacts via interactions that are strong enough to play a major role in determining the properties of the system, which is particularly interesting when multiple interaction terms compete and cannot be satisfied simultaneously. Examples of phases they realize range from simple magnetism to puzzling phenomena such as superconductivity or topological order. Of special interest in condensed matter are fermionic models which have an extra layer of complexity added by the possibly intricate sign structure of the wave function. \nHowever, exactly solvable models that exhibit the sought-after phenomena are scarce and Hamiltonians that are supposed to model experiments often elude analytical solutions precisely due to the strong interactions. Numerical methods are thus essential to gain insight into these systems and provide an important link between theoretical models and real world experiments. There are many different methods that in principle allow solving a problem numerically exact, most prominently exact diagonalization, tensor network based methods such as DMRG and (quantum) Monte Carlo. Exact diagonalization is certainly the most straightforward technique which allows measuring any observable of interest but its application to interacting models is severely limited by the exponential growth of the Hilbert space with the system size which only allows studying rather small systems. Tensor network methods excel in the study of gapped, one-dimensional systems and while they have shown some impressive results for two-dimensional systems, they need a lot of fine tuning and careful analysis to be considered reliable in this very important domain. Quantum Monte Carlo, on the other hand, is in principle neither limited by the size of the Hilbert space nor by the strength of the interactions and has thus been a tremendously important tool for studying condensed matter systems which is why it is the focus of this thesis. \nIt is necessary to continuously advance existing and develop new techniques to keep up with theoretical as well as with experimental progress. In this thesis, new methods to identify and characterize conventional and novel phases of matter within quantum Monte Carlo are developed and tested on archetypical models of condensed matter. \nIn the first part of this thesis, a method to study entanglement properties of strongly interacting fermionic models in quantum Monte Carlo is presented. Measurements of this kind are needed because some characteristics of a system remain hidden from conventional approaches based on the calculation of correlation functions. Prime examples are systems that realize so-called topological order entailing long-range entanglement that can be positively diagnosed using entanglement techniques. Realizing such measurements in a concrete algorithm is not straightforward for fermions. A large section of this part is thus devoted to developing a solution to the problem of implementing entanglement measurements and benchmarking the results against known data. \nThe second part is concerned with a truly novel and recent approach to the many-body problem, namely the symbiosis of machine learning and quantum Monte Carlo techniques. Out of the many different possibilities to combine the two, this part is concerne with exploring ways in which machine learning can help to effortlessly explore phase diagrams of hitherto unknown Hamiltonians. \nDespite the power and versatility of quantum Monte Carlo techniques, it does suffer from one significant weakness called the fermion sign problem. At its core related to the peculiar exchange statistics of fermions, it is in principle merely a technical problem of the Monte Carlo procedure but one that actually prohibits the use of Monte Carlo methods for a large number of interesting problems. The sign problem has been known for a long time but remains unsolved to this day. Thus, it is time to approach the problem from a different perspective with the hope of learning something new that can either help to better understand the problem itself or the properties of the models affected by it. Both of the two major research areas, entanglement measures and machine learning, are capable of providing such new perspectives and the sign problem will be revisited in more detail in the context of each of them.
Understanding strongly correlated quantum many-body systems requires dealing with a largeconfiguration space comprised of a set of all possible measurable configurations. The total number of configurations grows exponentially with the size of the system, which makes exact numerical calculations impossible for even moderately sized systems. Since the advent of high-temperature superconductivity and the subsequent interest in strongly correlated phases of matter, progress in the field has been marked by ingenuity of the field to overcome these computational limitations. The first techniques to emerge were inspired by quantum chemistry and based on exact diagonalization[1], which is limited to small clusters, and different variants of quantum Monte Carlo[2], that suffers from the sign problem in fermionic and frustrated systems. Circa 1992, the density matrix renormalization group (DMRG) [3, 4, 5, 6, 7] became one of the major breakthroughs of the past few decades. Despite the success of DMRG for one-dimensional(1D) and quasi-one-dimensional geometries, extensions to actual two-dimensional systems remain challenging and applications are constrained to long cylinders and strips. One promising alternative is to combine one the oldest methods, variational wave functions (VWFs), with neural networks, which have been used to recently with great success in machine learning. The focus of this dissertation is on developing novel variational Monte Carlo algorithms to make neural network variational wave functions applicable to many problems in quantum many-body physics. These algorithms draw from existing variational wave-function, tensor network, and machine learning methods. We calculate dynamical spectral functions using the vector correction and Chebyshev expansion method. We do finite temperature calculations with minimally entangled thermal states. Finally, we do ground state calculations for the frustrated J1−J2 Heisenberg model. We benchmark these results using restricted Boltzmann machines as the neural network for the the variational wave-functions.--Author's abstract
Most interacting many-body systems in physics are not analytically solvable. Instead, numerical methods are needed for the study of these complex and high-dimensional problems. At present, there are many interesting problems in strongly correlated systems that remain unsolved with current methods. At the heart of this problem is finding an efficient representation that incorporates symmetries, correlations and general features. \n \nIn the context of computer science, machine learning techniques have had astonishing success at reducing the dimensionality of data. The leading method is through the use of artificial neural networks. These networks have been enormously successful at sifting through vast amounts of data to find patterns and regularities. In a sense, neural networks are themselves a statistical system whose properties are adjusted to mimic the features of the data. By finding an effective low-dimensional representation of the data, machine learning has greatly subdued the curse of dimensionality found in many real-world problems. \n \nIn this Thesis, we apply several machine learning techniques to the study of interacting many-body systems in classical and quantum statistical physics. We explore supervised classification of phases of matter with an emphasis on physical interpretation of the net- work. In doing so, we design a custom network architecture that possesses rotational symmetry as an inductive bias. We further investigate connections between the renormalization group and deep learning through applying a super-resolving neural network to the classical Ising model. Towards experimental efforts, we also repurpose generative machine learning to quantum state tomography for the calibration and testing of quantum devices. We conclude with a latent variable model inspired by near-term quantum algorithms. This maps to a variational Monte Carlo ansatz that produces samples efficiently for interacting quantum systems.
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.
The study of interacting quantum many-body systems is one of the central challenges in modern physics. Interestingly, by understanding the physical properties of such systems from a microscopic description, one can explain many complex quantum phenomena such as the high-temperature superconductivity, fractional quantum Hall effect, etc. However, due to high complexity of these systems, theoretical investigations such as analytical and perturbation theories are limited to studying only a small subset of problems. In the past few decades, numerical simulation has emerged as an indispensable method for studying many phenomena, and has led to immense success and progress in recent years. The main obstacle in addressing quantum many-body systems is that the number of parameters required to represent a state of the system scales exponentially with the system size. However, one hardly needs the full Hilbert space but only a small subspace to describe physical properties of the system. From the point of view in quantum information theory, quantum entanglement is one of the main factors that is responsible for the complexity of studying quantum many-body systems. In particular, the amount of quantum entanglement in such system quantifies the correlation between constituents and therefore it determines how big the subspace is required to represent the system. In this thesis, we are interested in using tensor network states, a new emerging numerical technique to study dynamical and ground state properties of the system. The key ingredient is that a tensor network state which consists of a fairly small number of parameters can represent very well the complete wave function of a system. In one spatial dimension, Matrix Product State (MPS) has become a standard tool, combined with related algorithms such as Time Evolving Block Decimation algorithm (TEBD) to study dynamical properties of a locally-perturbed quantum lattice system. In this thesis, we introduce a technique called ``infinite boundary conditions'' that allows us to use a finite-size MPS to represent the state of an infinite-size system. As a result, one can simulate dynamics of a system for a longer time with a significant computational cost reduction compared to traditional methods. In two spatial dimensions, the tensor network known as Projected Entangled Pair States (PEPS) is emerging as a useful approach. However algorithms for optimizing the matrix elements of these tensors are complex and not very efficient. In particular, the efficiency of the optimization depends critically on the quality of the so-called ``environment'', which is expensive to compute. We have introduced a new method to accelerate the convergence of calculating the PEPS ground state by recycling the environment tensors for many optimization steps, vastly reducing the number of times that the full environment needs to be calculated.
The study of strongly correlated materials remains one of the most challenging and active frontiers in condensed matter physics. In these systems, the intricate interplay between electrons' kinetic energy and their mutual Coulomb repulsion gives rise to a plethora of exotic phenomena, from high-temperature superconductivity to giant magnetoresistance, which cannot be explained by conventional one-electron theories. This necessitates the development and application of advanced theoretical and computational methods capable of capturing the essential physics of strong electronic correlations. A diverse array of numerical and analytical techniques has been developed to tackle the many-body problem. However, no single method serves as a panacea; each comes with its own set of strengths and weaknesses, and their domains of applicability vary significantly. The choice of an appropriate approximation is therefore highly dependent on the specific physical system and the scientific question at hand. Different problems may benefit from different, and sometimes complementary, methodological perspectives. This thesis focuses primarily on the application and development of local or quasi-local approaches. The rationale for this focus is rooted in the physical reality of many key correlated materials, such as transition metal oxides, heavy-fermion systems or molecular solids like alkali-doped fullerenes, where the local on-site electronic interaction is the dominant energy scale. By prioritizing the accurate treatment of local physics, these methods provide a robust and computationally efficient approximation for understanding a wide range of material properties. Recent experimental and theoretical advancements—spanning the rich physics of multiorbital compounds, the precise control over interactions in ultracold atomic gases, and the complex interplay in electron-phonon coupled systems—have created an urgent need for theoretical frameworks that can deliver reliable and rapid insights. The methods explored in this work are meant to address these challenges, offering a balance between accuracy and computational feasibility. Given the variety of topics addressed, from methodological developments to their application in diverse physical contexts, this thesis will try to be as detailed and self-contained as necessary, providing a clear exposition of the underlying concepts and technical aspects.
Strongly correlated systems, i.e., quantum materials for which the interactions between its constituents are strong, are good candidates for the development of applications based on quantum-mechanical principles, such as quantum computers. Two paradigmatic models of strongly correlated systems are heavy-fermionic systems and one-dimensional spin-12 systems, with and without quenched disorder. In the past decade, improvement in computational methods and a vast enhancement in computational power has made it possible to study these systems in a a non-perturbative manner. In this thesis we present state-of-the-art numerical methods to investigate the properties of strongly correlated systems, and we apply these methods to solve a couple of selected problems in quantum condensed matter theory. We start by revisiting the phase diagram of the Falicov-Kimball model on the square lattice which can be considered as a heavy-fermionic systems. This model describes an interplay between conduction electrons and heavy electrons and reveals several distinct metal-insulator phase transitions. Using a lattice Monte-Carlo method, we study the transport properties of the model. Our analysis describes the role of temperature and interaction strength on the metal-insulator phase transitions in the Falicov-Kimball model. The second part of the thesis investigate the spatial structure of the entanglement in ground and thermal statesof the transverse-field Ising chain. We use the logarithmic negativity as a measure for the entanglement between two disjoint blocks. We investigate how logarithmic negativity depends on the spatial separation between two blocks, which can be viewed as the entanglement analog of a spatial correlation function. We find sharp entanglement thresholds at a critical distance beyond which the logarithmic negativity vanishes exactly and thus the two blocks become unentangled. Our results hold even in the presence of long-ranged quantum correlations, i.e., at the system’s quantum critical point. Using Time-Evolving Block Decimation (TEBD), we explore this feature as a function of temperature and size of the two blocks. We present a simple model to describe our numerical observations. In the last part of this thesis, we introduce an order parameter for a many-body localized spin-glass (MBL-SG) phase. We show that many-body localized spin-glass order can also be detected from two-site reduced density matrices, which we use to construct an eigenstate spin-glass order parameter. We find that this eigenstate spin-glass order parameter captures spin-glass phases in random Ising chains, both in many-body eigenstates as well as in the nonequilibrium dynamics, from a local in time measurement. We discuss how our results can be used to observe MBL-SG order within current experiments in Rydberg atoms and trapped ion systems.