# Research

Ke Liu, Dario Hügel, Tobias Pfeffer, Sebastian Schulteß, Jonas Greitemann, Janik Schönmeier-Kromer, and Lode Pollet

## Physics of Cold Atomic Systems

We are interested in the phenomenology of cold atomic systems in a wide variety of settings. First, we explore the possibilities a microscope offers to the experimental study of cold fermionic samples aiming at identifying novel types of magnetism and superfluidity in Hubbard-like models. Second, we study the behavior of quantum systems in the presence of inhomogeneous conditions, either in space or in time, when they are close to a phase transition. Third, we are interested in the hydrodynamic description of multi-component superfluids and superfluid drag. The main tools are the renormalisation group theory of critical phenomena, numerical simulations (primarily quantum Monte Carlo and exact diagonalization), and the hydrodynamic theory of superfluidity.

## Lattice Gauge Theory and Exotic Order

Spontaneously breaking a large symmetry to its subgroups can lead to phases characterized by orders with an internal symmetry. Very often these phases possess non-trivial thermodynamical and topological properties. They are ubiquitous in both classical and quantum systems such as nematic phases in assembly of molecules, fluids of correlated electrons, and systems of spinor atoms. However, they remain largely unexplored, especially for those involving non-Abelian symmetries. We are interested in condensed matter systems and theoretical models that support such phases, in particular their realization in quantum regimes. The methods we use are based on lattice gauge theories, which appear to be convenient and mighty tools when traditional approaches become cumbersome.

The figure shows the phase diagram of orientational phases breaking the rotational O(3) symmetry (reproduced from Phys. Rev. X 6, 041025, K Liu et al.). The internal symmetries are indicated by the horizontal axes.

## Bosonic Dynamical Cluster Approximation for Systems with Magnetic Fields

While being the method of choice for many bosonic problems, quantum Monte Carlo simulations suffer from an intractable sign problem in the presence of artificial gauge fields, frustration or real frequencies. This motivates the need for new approximate numerical methods such as bosonic dynamical mean-field theory, bosonic self-energy functional theory, and reciprocal cluster mean-field. In these methods an impurity (cluster) is solved numerically and consequently embedded into the original system. The interplay between topological band structures and strong interactions is currently hotly debated. In the figure we show the phase diagram of the Harper-Hofstadter-Mott model for hardcore bosons and magnetic flux π/2, indicating symmetry broken phases, fractional Chern insulators, and symmetry protected phases, among others.

## Disordered Systems and Many-Body Localization

Our group has many years of experience in dealing with the ground state of disordered bosonic systems, featuring an ergodic phase as well as a non-ergodic glassy phase. Such phase diagrams have been obtained by path integral Monte Carlo simulations. The current frontier in this field is whether a generalized transition can also occur at finite energy and non-zero interaction strength. This is an intriguing question implying that there may exist a phase of matter which violates the eigenstate thermalization hypothesis of statistical mechanics and shows other unusual properties: The system would retain knowledge of its initial conditions, have a zero dc conductivity, and feature an area law, among others. A compelling description for this many-body localized phase is in terms of an extensive set of emergent local integrals of motion (see figure). We attempt to get an accurate description of those, and use them consequently in order to study the properties of excited eigenstates. Those are identified as ground states from a related system found by shifting the original action by the found local integrals of motion.

## Diagrammatic Monte Carlo

### Advances in Monte Carlo Methods: Homotopy Analysis Method

Developing first-principles methods for strongly-interacting many-body systems remains an active field of research in theoretical physics. Quantum Monte Carlo algorithms based on the path integral formulation are often the method of choice because of their versatility. However, in the absence of a positive expansion scheme, Monte Carlo algorithms scale exponentially. This is the case for interacting Fermi systems without special symmetry and frustrated spin systems.

An alternative approach to study such systems are semi-analytical techniques such as resummed perturbation theory or other skeleton approaches. Most notably are Dyson-Schwinger equations and functional renormalization group equations. In our work we combine the Monte Carlo method with such semi-analytical approaches. The key concept used is this work is the expansion in rooted trees and stochastic summation of numerical root finding algorithms which can be used to solve integral equations or even more complicated functional integro-differential equations. The goal of this work is to overcome the main limiting factors of these skeleton approaches by using the Monte Carlo method.

The figure shows the leading terms in the Dyson-Schwinger expansion of the φ^{4} model.

### The Homogeneous Electron Gas

Many-body calculations for correlated-electron systems pose a long-standing challenge in condensed matter physics. The main difficulty in such calculations stems from the long-range nature of the Coulomb interaction. This, for instance, manifests in a perturbative treatment of the homogeneous electron gas (jellium model), where certain classes of diagrams diverge. From a physical point of view one wants to understand collective phenomena such as the effective screening of the electron-electron interaction. A textbook example of a simple approximation, which tries to capture this physics and partly resolves the divergencies, is the so-called random phase approximation. It turns out to quite accurately reproduce the ground state energy of the electron gas at high densities. However, an accurate solution for a wider range of densities is of crucial interest, as jellium, while itself being a basic model for electrons in a solid, is also the starting point for more elaborate electronic structure calculations within density functional theory.

Expanding upon this is the idea of a series of skeleton diagrams, where all divergent contributions are excluded from the series, but are implicitly incorporated via a self-consistent scheme. This self-consistency can be combined with the technique of Diagrammatic Monte Carlo to evaluate higher-order diagrams. We investigate the applicability for various densities and temperatures, from the ground state up to the region of warm dense matter, depending on the convergence (or asymptotic) properties of the underlying series at the given regime.

### Spectral functions of the Froehlich polaron

The interaction of electrons with the crystal lattice gives rise to a multitude of interesting phenomena. The Coulomb attraction between a single impurity and vibrations of the lattice can be described within the Froehlich model, resulting in an expansion in terms of Feynman diagrams where the electron is dressed by phononic excitations, giving rise to a quasiparticle, the polaron, composed of the electron and its surrounding phonon cloud.

Diagrammatic Monte Carlo is routinely used to sample this diagrammatic series stochastically in imaginary time where the expansion is free of sign problems. The ground-state energy of the polaron has been extracted from the asymptotic behavior of the polaron Green's function in imaginary time. However, to access the excitation spectrum one has to rely on analytic continuation which is numerically ill-posed and prone to spurious spectral features. We aim to bypass the analytic continuation by sampling the diagrammatic series directly in real time (see figure). In order to manage the resulting severe sign (phase) problem, non-crossing diagrams are summed up self-consistently and the resulting bold propagator is used to sample the remaining crossing diagrams more efficiently.