EDAD DENTAL

Having now taken a whistle-stop tour of phase transitions, disordered systems and the main theoretical tools which we will be using in this thesis, we are now equipped to move on to look at the original research I have conducted during my PhD. In this thesis, we will examine a number of di↵erent types of physical systems subject to a variety of types of disorder.

• Chapter 2 - We begin with an investigation into the behaviour of how disorder can couple to quantum fluctuations at finite temperatures. In this chapter, we examine fluctuation-induced phase reconstruction near an itinerant ferromagnetic quantum critical point. Using the fermionic quantum order-by-disorder mechanism, we recover the finding from previous work that fluctuations near the quantum critical point favour the formation of an incommensurate spiral phase in the absence of disorder. Upon the addition of quenched charge disorder and application of the replica trick, the incommensurate phase becomes stabilised over a slightly larger region of the phase diagram but it no longer displays long-range order, instead adopting short-range correlations and a strongly anisotropic correlation length. The disorder causes local changes in the pitch of the spiral magnetism, resulting in a novel phase we term a ‘helical glass’.

• Chapter 3- Having looked perturbatively at quantum fluctuations at finite temperatures, we now move on to a zero-temperature renormalisation group study of the Bose glass, a rare-region Griffiths phase found in the disordered Bose-Hubbard model. Restricting ourselves to random mass disorder only, we recap previous work done on this model in the context of a replica symmetric field theory and for the first time consider the e↵ects of replica symmetry breaking on the renormalisation group equations. Allowing for replica symmetry breaking in the most general Parisi form, we find the Mott insulator to Bose glass transition is governed by a one-step replica symmetry broken fixed point, signifying a breakdown of ergodicity, and derive Edwards-Anderson order parameters to quantify this.

• Chapter 4 - Motivated by the need to find experimental systems where replica symmetry breaking could be experimentally detected, we conduct a related renormalisation group study of a dimerised quantum antiferromag- net with random intra-dimer bond disorder, which maps onto a hard-core Bose-Hubbard model. This model contains magnetic analogues of the Mott insulating and superfluid phases of the conventional Bose-Hubbard model as well as fractionally-filled ‘checkerboard’ phases. When disorder is added, a magnetic analogue of the Bose glass phase intervenes between the Mott insulating and superfluid phases. Using the full replica symmetry breaking renormalisation group technique developed in Chapter 3, we show that without replica symmetry breaking the renormalisation group procedure does not correctly capture the physics of the Bose glass. In addition, we find a strong suppression of the compressibility near the tip of the Mott lobes which we identify with a Mott glass, an incompressible rare-region phase never before analytically predicted to exist in a three-dimensional bosonic system.

• Chapter 5 - Switching to a fully numerical study for the first time in this thesis, we examine the capabilities of current-generation experimental setups to probe the local properties of glassy phases. Specifically, we consider the case of the quantum gas microscope. Capable of single- atom resolved fluorescence imaging of both bosons and fermions, quantum

gas microscopes can directly image exotic glassy phases. Using a Bose- Hubbard model with mean-field numerics, we model the experimental system, reproduce existing experimental results in the clean case and show that the addition of disorder leads to changes that are well within current experimental resolution, even at finite experimental temperatures. We show for the first time that quantum gas microscopes can measure the thermally-averaged Edwards-Anderson order parameter, serving as a proof- of-principle that quantum gas microscopes are ideal for investigating the properties of disordered strongly correlated phases. With such local probes, many of the theoretical predictions in the rest of the thesis can now be experimentally tested and an exciting new experimental toolbox is opened up.

In all of the following, I work in units where all natural constants are equal to unity, e.g. ~= 1, kB = 1 andc= 1.

Chapter 2

Disordered Itinerant

Ferromagnetic Quantum Critical

Points

While the majority of this thesis is concerned with the e↵ects of disorder in bosonic systems, the earliest work I did as part of my PhD was a study of the e↵ects of quenched charge disorder on phase reconstruction near an itinerant ferromagnetic quantum critical point. This work was my introduction to disordered systems before I moved on to the bosonic work that forms the bulk of this thesis.

In this work, I look at how disorder can generate new quantum fluctuations in the vicinity of a ferromagnetic quantum critical point and show that this leads to a novel glassy phase known as a helical glass which persists up until a tricritical point at non-zero temperature.

The following work was published in “Helical glasses near ferromagnetic quantum criticality”, S. J. Thomson, F. Kr¨uger and A. G. Green,Physical Review B 87, 224203 (2013) [79].

2.1

Background

The conventional theory of itinerant ferromagnetic quantum criticality is the Hertz-Millis (or Hertz-Millis-Moriya) theory [80–82], which works by determining an e↵ective action for a conducting fermionic system in terms of dynamical fluctuations of a bosonic order parameter and from there allowing calculations of the free energy and other thermodynamic properties. Although Hertz- Millis theory is successful at describing many aspects of itinerant ferromagnetic quantum critical points, it fails to predict some of the more striking phenomena observed in experiments such as fluctuation-induced first order behaviour and the emergence of new phases near quantum critical points.

The fermionic order-by-disorder mechanism [83] is one of the proposed ways to extend and repair the conventional Hertz-Millis theory by including low- energy particle-hole fluctuations which couple to the order parameter and result in additional non-analytic corrections to the free energy. These fluctuation corrections reproduce the experimentally seen first-order behaviour in the vicinity of quantum critical points as well as the stabilisation of incommensurately ordered magnetic phases.

The name ‘order-by-disorder’ refers to the competition between internal energy and entropy rather than to any impurities or randomness. Previous work dealt purely with clean systems. My contribution to this mechanism, motivated by Ref. [84], was to consider the addition of quenched charge disorder into the system. We found that new fluctuation-corrections arise due to the presence of disorder in the system and these change the nature of the incommensurate phase from a long-range-ordered spiral ferromagnet to a new fluctuation-induced short-range ordered helical glass phase unlike anything predicted before in the literature.