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Interacting Quantum Many Body Systems

When you take many objects and let them interact together, it is often simpler to describe the behaviour of the group differently than one would describe the individual objects. Consider a flock (technically called a murmuration) of starlings like fig. 1. Watching the flock you’ll see that it has a distinct outline, that waves of density will sometimes propagate through the closely packed birds and that the flock seems to respond to predators as a distinct object. The natural description of this phenomena is couched in terms of the flock rather than the individual birds.

The behaviours of the flock are an emergent phenomena. The starlings are only interacting with their immediate six or seven neighbours1,2. This is what a physicist would call a local interaction. There is much philosophical debate about how exactly to define emergence3,4 but for our purposes it enough to say that emergence is the fact that the aggregate behaviour of many interacting objects may be very different from the individual behaviour of those objects.

Figure 1: A murmuration of starlings. Dorset, UK. Credit Tanya Hart, “Studland Starlings”, 2017, CC BY-SA 3.0

To give another example, our understanding of thermodynamics began with bulk properties like heat, energy, pressure and temperature5. It was only later that we gained an understanding of how these properties emerge from microscopic interactions between very large numbers of particles6.

Condensed Matter is, at its heart, the study of what behaviours emerge from large numbers of interacting quantum objects at low energy. When these three properties are present together: a large number of objects, those objects being quantum and there are interaction between the objects, we call it an interacting quantum many body system. From these three ingredients nature builds all manner of weird and wonderful materials.

Historically, we made initial headway in the study of many-body systems, ignoring interactions and quantum properties. The ideal gas law and the Drude classical electron gas7 are good examples. Including interactions into many-body physics leads to the Ising model8, Landau theory9 and the classical theory of phase transitions10. In contrast, condensed matter theory got it state in quantum many-body theory. Bloch’s theorem11 predicted the properties of non-interacting electrons in crystal lattices, leading to band theory. In the same vein, advances were made in understanding the quantum origins of magnetism, including ferromagnetism and antiferromagnetism12.

However, at some point we had to start on the interacting quantum many body systems. Some phenomena cannot be understood without a taking into account all three effects. The canonical examples are superconductivity13, the fractional quantum hall effect14 and the Mott insulators15,16. We will discuss the latter in more detail.

Electrical conductivity, the bulk movement of electrons, requires both that there are electronic states very close in energy to the ground state and that those states are delocalised so that they can contribute to macroscopic transport. Band insulators are systems whose Fermi level falls within a gap in the density of states and thus fail the first criteria. Anderson Insulators have only localised electronic states near the fermi level and therefore fail the second criteria. We will discuss Anderson insulators and disorder in a later section.

Both band and Anderson insulators occur without electron-electron interactions. Mott insulators, by contrast, are by these interactions and hence elude band theory and single-particle methods.

Figure 2: Three key adjectives. Many Body, the fact of describing systems in the limit of large numbers of particles. Quantum, objects whose behaviour requires quantum mechanics to describe accurately. Interacting, the constituent particles of the system affect one another via forces, either directly or indirectly. When taken together, these three properties can give rise to what are called strongly correlated materials.

Mott Insulators and The Hubbard Model

The theory of Mott insulators developed out of the observation that many transition metal oxides are erroneously predicted by band theory to be conductive17 leading to the suggestion that electron-electron interactions were the cause of this effect18. Interest grew with the discovery of high temperature superconductivity in the cuprates in 198619 which is believed to arise as the result of doping a Mott insulator state20.

The canonical toy model of the Mott insulator is the Hubbard model21–23 of \(1/2\) fermions hopping on the lattice with hopping parameter \(t\) and electron-electron repulsion \(U\)

\[ H = -t \sum_{\langle i,j \rangle \alpha} c^\dagger_{i\alpha} c_{j\alpha} + U \sum_i n_{i\uparrow} n_{i\downarrow} - \mu \sum_{i,\alpha} n_{i\alpha}\]

where \(c^\dagger_{i\alpha}\) creates a spin \(\alpha\) electron at site \(i\) and the number operator \(n_{i\alpha}\) measures the number of electrons with spin \(\alpha\) at site \(i\). In the non-interacting limit \(U << t\), the model reduces to free fermions and the many-body ground state is a separable product of Bloch waves filled up to the Fermi level. In the interacting limit \(U >> t\) on the other hand, the system breaks up into a product of local moments, each in one the four states \(|0\rangle, |\uparrow\rangle, |\downarrow\rangle, |\uparrow\downarrow\rangle\) depending on the filing.

The Mott insulating phase occurs at half filling \(\mu = \tfrac{U}{2}\) where there is one electron per lattice site24. Here the model can be rewritten in a symmetric form \[ H = -t \sum_{\langle i,j \rangle \alpha} c^\dagger_{i\alpha} c_{j\alpha} + U \sum_i (n_{i\uparrow} - \tfrac{1}{2})(n_{i\downarrow} - \tfrac{1}{2})\]

The basic reason that the half filled state is insulating seems is trivial. Any excitation must include states of double occupancy that cost energy \(U\), hence the system has a finite bandgap and is an interaction driven Mott insulator. Originally it was proposed that antiferromagnetic order was a necessary condition for the Mott insulator transition25 but later examples were found without magnetic order cite.

Various theoretical treatments of the Hubbard model have been made, including those based on Fermi liquid theory, mean field treatments, the local density approximation (LDA)26 and dynamical mean-field theory27. None of these approaches is perfect. Strong correlations are poorly described by the Fermi liquid theory and the LDA approaches while mean field approximations do poorly in low dimensional systems. This theoretical difficulty has made the Hubbard model a target for cold atom simulations28.

From here the discussion will branch two directions. First, we will discuss a limit of the Hubbard model called the Falikov Kimball Model. Second, we will go down the rabbit hole of strongly correlated systems without magnetic order. This will lead us to Quantum spin liquids and the Kitaev honeycomb model.

An exactly solvable model of the Mott Insulator - demonstrate mott insulator in hubbard model, briefly tease the falikov kimball model in order to lay the ground work to talk about the falikov kimball model later

An exactly solvable Quantum Spin Liquid - relationship between mott insulators and spin liquids: the electrons in a mott insulator form local moments that normally form an AFM ground state but if they don’t, due to frustration or other reason, the local moments may form a QSL at T=0 instead.29,30

Figure 3: From32.

kinds of mott insulators: Mott-Heisenberg (AFM order below Néel temperature) Mott-Hubbard (no long-range order of local magnetic moments) Mott-Anderson (disorder + correlations) Wigner Crystal

Outline

This thesis is composed of two main studies of separate but related physical models, The Falikov-Kimball Model and the Kitaev-Honeycomb Model. In this chapter I will discuss the overarching motivations for looking at these two physical models. I will then review the literature and methods that are common to both models.

In Chapter 2 I will look at the Falikov-Kimball model. I will review what it is and why we would want to study it. I’ll survey what is already known about it and identify the gap in the research that we aim to fill, namely the model’s behaviour in one dimension. I’ll then introduce the modified model that we came up with to close this gap. I will present our results on the thermodynamic phase diagram and localisation properties of the model

In Chapter 3 I’ll study the Kitaev Honeycomb Model, following the same structure as Chapter 2 I will motivate the study, survey the literature and identify a gap. I’ll introduce our Amorphous Kitaev Model designed to fill this gap and present the results.

Finally in chapter 4 I will summarise the results and discuss what implications they have for our understanding interacting many-body quantum systems.

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