Numerical Studies of Superconductivity and Charge-Density-Waves: Progress on the 2D Holstein Model and a Superconductor-Metal Bilayer

The problem of superconductivity has been central in many areas of condensed matter physics for over 100 years. Despite this long history, there is still no theory capable of describing both conventional and unconventional superconductors. Recent experimental observations such as the dilute superconductivity in SrTiO3 and near room-temperature superconductivity in hydride compounds under extreme pressure have renewed interest in electron-phonon systems. Adding to this is evidence that electron-phonon coupling may play a supporting role in unconventional systems like the cuprates and monolayer FeSe on SrTiO3. One way to make sense of these observations is to construct simple models that capture the essential physics. Among the models with electron-phonon interactions, the simplest and most studied is the two-dimensional Holstein model. It describes a single band of electrons that hop between sites on a square lattice and interact with atomic oscillators by coupling linearly to their displacements. This model gives rise to superconductivity and charge-density-wave order spanning different regions of doping. Surprisingly, even this model is not entirely understood. First, we present a comprehensive study of the Holstein model phase diagram using self-consistent many-body perturbation theory. We then discuss one potential avenue for accelerating non-perturbative quantum Monte Carlo simulations of electron-phonon models using artificial neural networks. Following these topics, we wrap up the electron-phonon-related part by discussing the importance of nonlinear interaction terms and moving beyond the Holstein model. The last problem of this dissertation revisits a proposal by Steve Kivelson. He hypothesized and later showed that coupling a superconductor with a large pairing scale but low phase stiffness to a metal raises the transition temperature (Tc). Expanding on previous work, we studied a more general case with a 2D negative-U Hubbard model coupled with a metallic layer via single-particle tunneling. Here, we use the dynamical cluster approximation to estimate Tc, finding it is maximal for finite tunneling values, thereby confirming Kivelson’s hypothesis in the general case. Collectively, the results in this dissertation shed new light on superconductivity in conventional systems and demonstrate a need to incorporate more aspects of real materials into models

Quantum Monte Carlo studies of a metallic spin-density wave transition

Plenty experimental evidence indicates that quantum critical phenomena give rise to much of the rich physics observed in strongly correlated itinerant electron systems such as the high temperature superconductors. A quantum critical point of particular interest is found at the zero-temperature onset of spin-density wave order in two- dimensional metals. The appropriate low-energy theory poses an exceptionally hard problem to analytic theory, therefore the unbiased and controlled numerical approach pursued in this thesis provides important contributions on the road to comprehensive understanding. After discussing the phenomenology of quantum criticality, a sign- problem-free determinantal quantum Monte Carlo approach is introduced and an extensive toolbox of numerical methods is described in a self-contained way. By the means of large-scale computer simulations we have solved a lattice realization of the universal effective theory of interest. The finite-temperature phase diagram, showing both a quasi-long-range spin-density wave ordered phase and a d-wave superconducting dome, is discussed in its entirety. Close to the quantum phase transition we find evidence for unusual scaling of the order parameter correlations and for non-Fermi liquid behavior at isolated hot spots on the Fermi surface

Disentangling and machine learning the many-fermion problem

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. However, 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. It 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. In 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. The 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. Despite 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

Real-Time Simulation of Open Quantum Spin Chains with Inchworm Method

We study the real-time simulation of open quantum systems, where the system is modeled by a spin chain, with each spin associated with its own harmonic bath. Our method couples the inchworm method for the spin-boson model and the modular path integral methodology for spin systems. In particular, the introduction of the inchworm method can significantly suppress the numerical sign problem. Both methods are tweaked to make them work seamlessly with each other. We represent our approach in the language of diagrammatic methods, and analyze the asymptotic behavior of the computational cost. Extensive numerical experiments are done to validate our method

Theory and practice of modeling van der Waals interactions in electronic-structure calculations

The accurate description of long-range electron correlation, most prominently including van der Waals (vdW) dispersion interactions, represents a particularly challenging task in the modeling of molecules and materials. vdW forces arise from the interaction of quantum-mechanical fluctuations in the electronic charge density. Within (semi-)local density functional approximations or Hartree–Fock theory such interactions are neglected altogether. Non-covalent vdW interactions, however, are ubiquitous in nature and play a key role for the understanding and accurate description of the stability, dynamics, structure, and response properties in a plethora of systems. During the last decade, many promising methods have been developed for modeling vdW interactions in electronic-structure calculations. These methods include vdW-inclusive Density Functional Theory and correlated post-Hartree–Fock approaches. Here, we focus on the methods within the framework of Density Functional Theory, including non-local van der Waals density functionals, interatomic dispersion models within many-body and pairwise formulation, and random phase approximation-based approaches. This review aims to guide the reader through the theoretical foundations of these methods in a tutorial-style manner and, in particular, highlight practical aspects such as the applicability and the advantages and shortcomings of current vdW-inclusive approaches. In addition, we give an overview of complementary experimental approaches, and discuss tools for the qualitative understanding of non-covalent interactions as well as energy decomposition techniques. Besides representing a reference for the current state-of-the-art, this work is thus also designed as a concise and detailed introduction to vdW-inclusive electronic structure calculations for a general and broad audience

Automated Algebra for the Quantum Virial Expansion of Strongly-coupled Matter

The thermodynamics of generic quantum many-body systems is a technically challenging area of research relevant to different fields covering a vast energy range, from condensed matter physics at the low-energy end, atomic systems in the middle, and nuclear and QCD at higher energies. An open question at the forefront of research concerns, for instance, the thermodynamics of neutron and nuclear matter at finite temperature, as it is directly relevant to the behavior of dense matter in neutron star mergers. With the advent of multimessenger astronomy, these astrophysical environments where nuclear many-body physics is essential have attracted considerable attention in recent years. This dissertation focuses on one of the most widely-applied methods in the calculation of the thermodynamics of quantum many-body systems, namely the Quantum Virial Expansion (QVE), which is an expansion of the grand canonical partition function in powers of the fugacity. While applications of the QVE have seen successes in systems as different as ultracold atoms and dilute neutron matter, most investigations have been limited to the lowest order, due to the increasing complexity of the quantum few-body problem. The main development of this work is the construction of a new class of semi-analytic methods which we will generally call "Automated-Algebra methods" to tackle the QVE, offering higher and more accurate analytic estimations for its coefficients than ever before. Due to general interest from the condensed matter, atomic physics, and nuclear astrophysics communities, we focus on a system of spin-1/2 fermions with and without external harmonic trapping. In both cases, we were able to push the calculation of the QVE to the unprecedented fifth order. The analytic nature of our method makes our results an analytic function of parameters such as dimension, coupling strength, and external trapping frequency, which greatly extends the range of prior results on lower-order coefficients. We applied our high-order QVE to examine properties, such as density, Tan's contact compressibility and spin susceptibility, and finding good agreement with existing experimental results. Thanks to the access to such high-order coefficients, we were able to apply series resummation methods, which greatly extended the applicability of the QVE to regimes at lower temperature, where the truncated finite-order expansion usually fails.Doctor of Philosoph