Understanding interactions in quantum many-body systems remains one of the most profound and difficult challenges in condensed matter physics. While free particle systems can be solved through single-particle techniques, the introduction of interactions turns the problem essentially many-body and quickly intractable. The Hilbert space grows exponentially, and traditional approaches become useless. To deal with this, one typically turns to numerical methods by discretizing the system on a lattice. This approach generally works well for many 1D problems, including those described by Schrödinger equation. However, when it comes to massless Dirac fermions with protected chirality, the situation becomes more complicated. Any naive discretization inevitably runs into the fermion doubling problem, formalized by the Nielsen–Ninomiya theorem, which forbids a straightforward lattice realization of a single chiral mode without either breaking symmetry or introducing unphysical degrees of freedom. My thesis is motivated by this problem: the need to describe 1D interacting systems, and the difficulty of doing so numerically due to these fundamental obstacles. The main focus of this work is to develop and explore lattice-based numerical methods for strongly correlated chiral fermions in one dimension, with the Luttinger liquid as a central case study.
Understanding interactions in quantum many-body systems remains one of the most profound and difficult challenges in condensed matter physics. While free particle systems can be solved through single-particle techniques, the introduction of interactions turns the problem essentially many-body and quickly intractable. The Hilbert space grows exponentially, and traditional approaches become useless. To deal with this, one typically turns to numerical methods by discretizing the system on a lattice. This approach generally works well for many 1D problems, including those described by Schrödinger equation. However, when it comes to massless Dirac fermions with protected chirality, the situation becomes more complicated. Any naive discretization inevitably runs into the fermion doubling problem, formalized by the Nielsen–Ninomiya theorem, which forbids a straightforward lattice realization of a single chiral mode without either breaking symmetry or introducing unphysical degrees of freedom. My thesis is motivated by this problem: the need to describe 1D interacting systems, and the difficulty of doing so numerically due to these fundamental obstacles. The main focus of this work is to develop and explore lattice-based numerical methods for strongly correlated chiral fermions in one dimension, with the Luttinger liquid as a central case study.
Understanding the effect of correlations in interacting many-body systems is one of the main challenges in quantum mechanics. While the general problem can only be addressed by approximate methods and numerical simulations, in some limiting cases, it is amenable to exact solutions. I will present a family of exact solutions for strongly interacting 1D quantum gases in arbitrary external potential, valid for bosons, multicomponent fermions and mixtures. It allows us to obtain experimental observables such as the density profiles and momentum distribution at all momentum scales, beyond the Luttinger liquid approach. It also predicts the exact quantum dynamics at all the times, including the small oscillation regime yielding the collective modes of the system and the large quench regime where the system parameters are changed considerably. The solution can be extended to describe finite-temperature conditions, spin, and magnetization effects. In particular I will present large-size calculations of the frequency- and momentum-resolved spectral function of strongly repulsive lattice bosons and a symmetry-breaking exact solution for a Bose-Bose mixture. Our approach directly describes the experiments with trapped ultracold atoms where the strongly correlated regime in one dimension has been achieved.
This thesis focuses on the development and application of tensor networks methods for simulating one-dimensional quantum many-body systems. The general aim is to unveil numerical evidence for a new universality class of 1D interacting fermion systems referred to as quasi-Fermi liquids (qFLs). Realizing such systems beyond a purely analytical context and elucidating their peculiar characteristics is the principle value of this thesis.Chapter 1 begins by introducing the essential physics of one-dimensional systems, known as Luttinger liquids relative to their higher dimensional counterparts. Then it introduces non-linear Luttinger liquids juxtaposed against the defining quasi-Fermi liquid (qFL) field theory that motivates the following lattice studies. Chapter 2 provides an in-depth review of matrix product states (MPS), the main methodology used to explore quasi-Fermi liquids. The chapter serves as a pedagogical introduction to the field and details those novel algorithmic developments necessary for the study of fermionic systems. The central result of this chapter is the section on finite entanglement scaling techniques applied to dynamic correlation functions. Chapter 3 begins the study of lattice models that exhibit quasi-Fermi liquid characteristics. Here we explore the ground states exclusively by computing their static correlation functions. The central results include Luttinger parameter phase space diagrams which demonstrate the existence of the quasi-Fermi liquid across multiple lattice models. Chapter 4 studies the excitations of the qFL lattice models. We begin with results from non-MPS based methods which were essential for benchmarking our novel application of MPS to fermionic spectral functions. The central results of the chapter are the power-law exponents that demonstrate the low-energy universality of the qFL and our analysis of the dynamic structure factor. Chapter 5 concludes with a recapitulation of the main results and an outlook on future work. The Appendix contains those tensor diagrams necessary for a numerical implementation of those algorithms introduced in Chapter 2.--Author's abstract
Understanding the effect of correlations in interacting many-body systems is one of the main challenges in quantum mechanics. While the general problem can only be addressed by approximate methods and numerical simulations, in some limiting cases, it is amenable to exact solutions. I will present a family of exact solutions for strongly interacting 1D quantum gases in arbitrary external potential, valid for bosons, multicomponent fermions and mixtures, and in particular our new exact solution for the case of symmetric-breaking mixtures.The solutions allow one to obtain experimental observables such as the density profiles and momentum distribution at all momentum scales, beyond the Luttinger liquid approach, as well as to describe finite-temperature conditions, spin, and magnetization effects. I will present some uses of these solutions to obtain the exact quantum dynamics, both for the small oscillation regime and for the strongly out-of-equilibrium spin dynamics, where I will propose a protocol to observe symmetry oscillations in binary bosonic mixtures.
It is common in physics to treat 1D system as a toy model in order to test a physical theory because it is easier to work with low dimensional systems and find analytical solutions. One naively will say that one-dimensional models serve as toy models and they do not apply in the real world. But, one-dimensional systems, e.g quantum wires, exists in nature and have quite different physics from higher dimensional systems. It is remarkable that, in 1D, an interacting fermionic problem can be turned to a free bosonic theory. An interesting question to ask is how the system evolves after a change in the systems internal parameters. This is called a quantum quench. In this thesis we investigate quantum quenches in the Luttinger liquid model, which is a 1D model and serves as the analog of the Fermi liquid theory in one dimension. More specifically, we study the kinetic energy, when we periodically drive the internal parameters of the system, and the two point correlation function, after a periodic quench of finite duration.
With the discovery of the integer quantum Hall effect by von Klitzing and collaborators in 1980, the mathematical field of topology entered the world of condensed matter physics. Almost three decades later, this eventually led to the theoretical prediction and the experimental realization of many intriguing topological materials and topology-based devices. In this Ph.D. thesis, we will study the interplay between topology and another key topic in condensed matter physics, namely the study of inter-particle interactions in many-body systems. This interplay is analyzed from two different perspectives. Firstly, we studied how the presence of electron-electron interactions affects single-electron injection into a couple of counter-propagating one-dimensional edge channels. The latter appear at the edges of topologically non-trivial systems in the quantum spin Hall regime and they can also be engineered by exploiting the integer quantum Hall effect. Because of inter-channel interactions, the injected electron splits up into a couple of counter-propagating fractional excitations. Here, we carefully study and discuss their properties by means of an analytical approach based on the Luttinger liquid theory and the bosonization method. Our results are quite relevant in the context of the so-called electron quantum optics, a fast developing field which deeply exploits the topological protection of one-dimensional edge states to study the coherent propagation of electrons in solid-state devices. As an aside, we also showed that similar analytical techniques can also be used to study the time-resolved dynamics of a Luttinger liquid subject to a sudden change of the interaction strength, a protocol known as quantum quench which is gaining more and more attention, especially within the cold-atoms community. Secondly, we study how inter-particle interactions can enhance the topological properties of strictly one-dimensional fermionic systems. More precisely, the starting point is the seminal Kitaev chain, a free-fermionic lattice model which hosts exotic Majorana zero-energy modes at its ends. The latter are extremely relevant in the context of topological quantum computation because of their non-Abelian anyonic exchange statistics. Here we show that, by properly adding electron-electron interactions to the Kitaev chain, it is possible to obtain lattice models which feature zero-energy parafermionic modes, an even more intriguing generalization of Majoranas. To this end, we develop at first an exact mapping between Z4 parafermions and ordinary fermions on a lattice. We subsequently exploit this mapping to analytically obtain an exactly solvable fermionic model hosting zero-energy parafermions. We study their properties and numerically investigate their signatures and robustness even when parameters are tuned away from the exactly solvable point.
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.