When you take many objects and let them interact together, it is often simpler to describe the behaviour of the group differently from the way one would describe the individual objects. Consider a flock of starlings like that of 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 of the individual birds.
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The behaviours of the flock are an emergent phenomena. The starlings are only interacting with their immediate six or seven neighbours [1,2], what a physicist would call a local interaction. There is much philosophical debate about how exactly to define emergence [3,4] but for our purposes it enough to say that emergence is the fact that the aggregate behaviour of many interacting objects may necessitate a description very different from that of the individual objects.
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When you take many objects and let them interact together, it is often simpler to describe the behaviour of the group in a different way to how one would describe the individual objects. Consider a flock of starlings like that of 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 in terms of the flock, not the individual birds.
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The behaviours of the flock are an emergent phenomenon. The starlings are only interacting with their immediate six or seven neighbours [1,2], what a physicist would call a local interaction. There is much philosophical debate about how exactly to define emergence [3,4] but for our purposes it enough to say that emergence is the fact that the aggregate behaviour of many interacting objects may necessitate a description very different from that of the individual objects.
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+Figure 1: A murmuration of starlings. Dorset, UK. Credit Tanya Hart, “Studland Starlings”, 2017, CC BY-SA 3.0
To give an example closer to the topic at hand, our understanding of thermodynamics began with bulk properties like heat, energy, pressure and temperature [5]. It was only later that we gained an understanding of how these properties emerge from microscopic interactions between very large numbers of particles [6].
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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.
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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 gas [7] are good examples. Including interactions into many-body physics leads to the Ising model [8], Landau theory [9] and the classical theory of phase transitions [10]. In contrast, condensed matter theory got it state in quantum many-body theory. Bloch’s theorem [11] 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 antiferromagnetism [12].
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The development of Landau-Fermi Liquid theory explained why band theory works so well even in cases where an analysis of the relevant energies suggests that it should not [13]. Landau Fermi Liquid theory demonstrates that in many cases where electron-electron interactions are significant, the system can still be described in terms on generalised non-interacting quasiparticles.
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However there are systems where even Landau Fermi Liquid theory fails. An effective theoretical description of these systems must include electron-electron correlations and they are thus called Strongly Correlated Materials [14], Correlated Electron systems or Quantum Materials. The canonical examples are superconductivity [15], the fractional quantum hall effect [16] and the Mott insulators [17,18]. We’ll start by looking at the latter but shall see that there are many links between three topics.
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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 the presence interactions 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.
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Historically, we first made headway in the study of many-body systems, ignoring interactions and quantum properties. The ideal gas law and the Drude classical electron gas [7] are good examples. Including interactions too leads to the Ising model [8], Landau theory [9] and the classical theory of phase transitions [10]. In contrast, condensed matter theory got its start in quantum many-body theory where the only electron-electron interaction considered is the Pauli exclusion principle. Bloch’s theorem [11], the core result of band theory, predicted the properties of non-interacting electrons in crystal lattices, in particular that band insulators arise when the electrons bands are filled, leaving the fermi level in a bandgap [7]. In the same vein, advances were made in understanding the quantum origins of magnetism, including ferromagnetism and antiferromagnetism [12].
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The development of Landau-Fermi Liquid theory explained why band theory works so well even in cases where an analysis of the relevant energies suggests that it should not [13]. Landau Fermi Liquid theory demonstrates that in many cases where electron-electron interactions are significant, the system can still be described in terms of generalised non-interacting quasiparticles. This happens when the properties of the quasiparticles in the interacting system can be smoothly connected to the free fermions of the non-interacting system.
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However there are systems where even Landau Fermi Liquid theory fails. An effective theoretical description of these systems must include electron-electron correlations and they are thus called Strongly Correlated Materials [14]. The canonical examples are superconductivity [15], the fractional quantum hall effect [16] and the Mott insulators [17,18]. We’ll start by looking at the latter but shall see that there are many links between the three topics.
Mott Insulators
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Mott Insulators are remarkable because their electrical insulator properties come from electron-electron interactions. 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. Band insulators derive their character from the characteristics of the underlying lattice. 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.
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Mott Insulators are remarkable because their electrical insulator properties come not from having filled bands but from electron-electron interactions other than Pauli exclusion. 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. Band insulators derive their character from the characteristics of the underlying lattice. A third kind of insulator, the Anderson Insulators, have only localised electronic states near the fermi level and therefore fail the second criteria. We will discuss Anderson insulators and the disorder that drives them, in a later section.
Both band and Anderson insulators occur without electron-electron interactions. Mott insulators, by contrast, require a many body picture to understand and thus elude band theory and single-particle methods.
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-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.
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+Figure 2: Three key adjectives. Many Body refers to systems considered 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.
The theory of Mott insulators developed out of the observation that many transition metal oxides are erroneously predicted by band theory to be conductive [19] leading to the suggestion that electron-electron interactions were the cause of this effect [20]. Interest grew with the discovery of high temperature superconductivity in the cuprates in 1986 [21] which is believed to arise as the result of a doped Mott insulator state [22].
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The canonical toy model of the Mott insulator is the Hubbard model [23–25] of \(1/2\) fermions hopping on the lattice with hopping parameter \(t\) and electron-electron repulsion \(U\)
The canonical toy model of the Mott insulator is the Hubbard model [23–25] of spin-\(1/2\) fermions hopping on the lattice with hopping parameter \(t\) and electron-electron repulsion \(U\)
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 \(i\). The sum runs over lattice neighbours \(\langle i,j \rangle\) including both \(\langle i,j \rangle\) and \(\langle j,i \rangle\) so that the model is Hermition.
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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.
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The Mott insulating phase occurs at half filling \(\mu = \tfrac{U}{2}\) where there is one electron per lattice site [26]. Here the model can be rewritten in a symmetric form \[ H_{\mathrm{H}} = -t \sum_{\langle i,j \rangle \alpha} c^\dagger_{i\alpha} c_{j\alpha} + U \sum_i (n_{i\uparrow} - \tfrac{1}{2})(n_{i\downarrow} - \tfrac{1}{2})\]
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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. Depending on the lattice, the local moments may then order antiferromagnetically. Originally it was proposed that this antiferromagnetic order was the cause of the gap opening [27]. However, Mott insulators have been found [28,29] without magnetic order. Instead the local moments may form a highly entangled state known as a quantum spin liquid, which will be discussed shortly.
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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) [30] and dynamical mean-field theory [31]. None of these approaches are 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 simulations [32].
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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 look at quantum spin liquids and the Kitaev honeycomb model.
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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 ground state is a direct product of the local Hilbert spaces \(|0\rangle, |\uparrow\rangle, |\downarrow\rangle, |\uparrow\downarrow\rangle\). At half filling, one electron per site, each site becomes a local moment in the reduced Hilbert space \(|\uparrow\rangle, |\downarrow\rangle\) and thus acts like a spin-\(1/2\) [26].
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The Mott insulating phase occurs at half filling \(\mu = \tfrac{U}{2}\). Here the model can be rewritten in a symmetric form \[ H_{\mathrm{H}} = -t \sum_{\langle i,j \rangle \alpha} c^\dagger_{i\alpha} c_{j\alpha} + U \sum_i (n_{i\uparrow} - \tfrac{1}{2})(n_{i\downarrow} - \tfrac{1}{2}).\]
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The basic reason that the half filled state is insulating seems 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. Depending on the lattice, the local moments may then order antiferromagnetically. Originally it was proposed that this antiferromagnetic (AFM) order was actually the reason for the insulating behaviour. This would make sense since AFM order doubles the unit cell and can turn a system into a band insulator with an even number of electrons per unit cell [27]. However, Mott insulators have been found [28,29] without magnetic order. Instead the local moments may form a highly entangled state known as a quantum spin liquid, which will be discussed shortly.
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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) [30], dynamical mean-field theory [31], density matrix renormalisation group methods [32–34] and Markov chain Monte Carlo [35–37]. None of these approaches are 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 simulations [38].
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From here the discussion will branch in two directions. First, we will discuss a limit of the Hubbard model called the Falikov-Kimball Model. Second, we will look at quantum spin liquids and the Kitaev honeycomb model.
The Falikov-Kimball Model
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Though not the original reason for its introduction, the Falikov-Kimball (FK) model is the limit of the Hubbard model as the mass ratio of the spin up and spin down electron is taken to infinity. This gives a model with two fermion species, one itinerant and one entirely immobile. The number operators for the immobile fermions are therefore conserved quantities and can be be treated like classical degrees of freedom. For our purposes it will be useful to replace the immobile fermions with a classical Ising background field \(S_i = \pm1\).
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+Figure 3: The Falicov-Kimball Model can be viewed as a model of classical spins \(S_i\) coupled to spinless fermions \(\hat{c}_i\) where the fermions are mobile with hopping \(t\) and the fermions are coupled to the spins by an Ising type interaction with strength \(U\).
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Originally introduced to describe the metal-insulator transition in f-electron system [25,39], the Falikov-Kimball (FK) model is the limit of the Hubbard model as the mass of one of the spins states of the electron is taken to infinity. This gives a model with two fermion species, one itinerant and one entirely immobile. The number operators for the immobile fermions are therefore conserved quantities and can be be treated like classical degrees of freedom. For our purposes it will be useful to replace the immobile fermions with a classical Ising background field \(S_i = \pm1\).
Given that the physics of states near the metal-insulator (MI) transition is still poorly understood [33,34] the FK model provides a rich test bed to explore interaction driven MI transition physics. Despite its simplicity, the model has a rich phase diagram in \(D \geq 2\) dimensions. It shows an Mott insulator transition even at high temperature, similar to the corresponding Hubbard Model [35]. In 1D, the ground state phenomenology as a function of filling can be rich [36] but the system is disordered for all \(T > 0\) [37]. The model has also been a test-bed for many-body methods, interest took off when an exact dynamical mean-field theory solution in the infinite dimensional case was found [38–41].
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In chapter 3 I will introduce a generalized Falikov-Kimball model in one dimension I call the Long-Range Falikov-Kimball model. With the addition of long-range interactions in the background field, the model shows a similarly rich phase diagram its higher dimensional cousins. I use an exact Markov chain Monte Carlo method to map the phase diagram and compute the energy-resolved localization properties of the fermions. I then compare the behaviour of this transitionally invariant model to an Anderson model of uncorrelated binary disorder about a background charge density wave field which confirms that the fermionic sector only fully localizes for very large system sizes.
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Given that the physics of states near the metal-insulator (MI) transition is still poorly understood [40,41] the FK model provides a rich test bed to explore interaction driven MI transition physics. Despite its simplicity, the model has a rich phase diagram in \(D \geq 2\) dimensions. It shows an Mott insulator transition even at high temperature, similar to the corresponding Hubbard Model [42]. In 1D, the ground state phenomenology as a function of filling can be rich [43] but the system is disordered for all \(T > 0\) [44]. The model has also been a test-bed for many-body methods, interest took off when an exact dynamical mean-field theory solution in the infinite dimensional case was found [45–48].
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In chapter 3 I will introduce a generalized Falikov-Kimball model in one dimension I call the Long-Range Falikov-Kimball model. With the addition of long-range interactions in the background field, the model shows a similarly rich phase diagram as its higher dimensional cousins. Our goal is to understand the Mott transition in more detail, the the phase transition into a charge density wave (CDW) state and how the localisation properties of the fermionic sector behave in one dimension. We were particularly interested to see if correlations in the disorder potential are enough to bring about localisation effects such mobility edges that are normally only seen in higher dimensions. I use an exact Markov chain Monte Carlo method to map the phase diagram and compute the energy-resolved localization properties of the fermions. We observe what seems like a hint of coexisting localised and delocalised states. However after careful comparison to an Anderson model of uncorrelated binary disorder about a background charge density wave field, we confirm that the fermionic sector does fully localize at larger system sizes as expected for one dimensional systems.
Quantum Spin Liquids
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To turn to the other key topic of this thesis, we have already discussed the AFM ordering of local moments in the Mott insulating state. Landau-Ginzburg-Wilson theory characterises phases of matter as inextricably linked to the emergence of long range order via a spontaneously broken symmetry. So within this paradigm we would not expect any interesting phases of matter not associated with AFM or other long-range order. However, Anderson first proposed in 1973 [42] that if long range order is suppressed by some mechanism, it might lead to a liquid-like state even at zero temperature, the Quantum Spin Liquid (QSL).
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This QSL state would exist at zero or very low temperatures, so we would expect quantum effects to be very strong, which will turn out to have far reaching consequences. It was the discovery of a different phase, however that really kickstarted interest in the topic. The fractional quantum Hall (FQH) state, discovered in the 1980s is an explicit example of an interacting electron system that falls outside of the Landau-Ginzburg-Wilson paradigm. It shares many phenomenological properties with the QSL state. They both exhibit fractionalised excitations, braiding statistics and non-trivial topological properties [43]. The many-body ground state of such systems acts as a complex and highly entangled vacuum. This vacuum can support quasiparticle excitations with properties unbound from that of the Dirac fermions of the standard model.
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How do we actually make a QSL? Frustration is one mechanism that we can use to suppress magnetic order in spin models [44]. Frustration can be geometric, triangular lattices for instance cannot support AFM order. It can also come about as a result of spin-orbit coupling or other physics. There are also other routes to QSLs besides frustrated spin systems that we will not discuss here [45–47].
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To turn to the other key topic of this thesis, we have already discussed the AFM ordering of local moments in the Mott insulating state. Landau-Ginzburg-Wilson theory characterises phases of matter as inextricably linked to the emergence of long range order via a spontaneously broken symmetry. So within this paradigm we would not expect any interesting phases of matter not associated with AFM or other long-range order. However, Anderson first proposed in 1973 [49] that if long range order is suppressed by some mechanism, it might lead to a liquid-like state even at zero temperature, the Quantum Spin Liquid (QSL).
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This QSL state would exist at zero or very low temperatures, so we would expect quantum effects to be very strong, which will turn out to have far reaching consequences. It was the discovery of a different phase, however that really kickstarted interest in the topic. The fractional quantum Hall (FQH) state, discovered in the 1980s is an explicit example of an interacting electron system that falls outside of the Landau-Ginzburg-Wilson paradigm. It shares many phenomenological properties with the QSL state. They both exhibit fractionalised excitations, braiding statistics and non-trivial topological properties [50]. The many-body ground state of such systems acts as a complex and highly entangled vacuum. This vacuum can support quasiparticle excitations with properties unbound from that of the Dirac fermions of the standard model.
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How do we actually make a QSL? Frustration is one mechanism that we can use to suppress magnetic order in spin models [51]. Frustration can be geometric, triangular lattices for instance cannot support AFM order. It can also come about as a result of spin-orbit coupling or other physics. There are also other routes to QSLs besides frustrated spin systems that we will not discuss here [52–54].
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-Figure 3: How Kitaev materials fit into the picture of strongly correlated systems. Interactions are required to open a Mott gap and localise the electrons into local moments, while spin-orbit correlations are required to produce the strongly anisotropic spin-spin couplings of the Kitaev Model. Reproduced from [44].
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+Figure 4: How Kitaev materials fit into the picture of strongly correlated systems. Interactions are required to open a Mott gap and localise the electrons into local moments, while spin-orbit correlations are required to produce the strongly anisotropic spin-spin couplings of the Kitaev Model. Reproduced from [51].
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Spin-orbit coupling is a relativistic effect, that very roughly corresponds to the fact that in the frame of reference of a moving electron, the electric field of nearby nuclei look like magnetic fields to which the electron spin couples. This effectively couples the spatial and spin parts of the electron wavefunction, meaning that the lattice structure can influence the form of the spin-spin interactions leading to spatial anisotropy. This anisotropy will be how we frustrate the Mott insulators [48,49]. As we saw with the Hubbard model, interaction effects are only strong or weak in comparison to the bandwidth or hopping integral \(t\) so what we need to see strong frustration is a material with strong spin-orbit coupling \(\lambda\) relative to its bandwidth \(t\).
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In certain transition metal based compounds, such as those based on Iridium and Ruthenium, the lattice structure, strong spin-orbit coupling and narrow bandwidths lead to effective spin-\(\tfrac{1}{2}\) Mott insulating states with strongly anisotropic spin-spin couplings. These transition metal compounds, known Kitaev Materials, draw their name from the celebrated Kitaev Honeycomb Model which is expected to model their low temperature behaviour [44,50–53].
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At this point we can sketch out a phase diagram like that of fig. 3. When both electron-electron interactions \(U\) and spin-orbit couplings \(\lambda\) are small relative to the bandwidth \(t\) we recover standard band theory of band insulators and metals. In the upper left we have the simple Mott insulating state as described by the Hubbard model. In the lower right, strong spin-orbit coupling gives rise to Topological insulators (TIs) characterised by symmetry protected edge modes and non-zero Chern number. Kitaev materials occur in the region where strong electron-electron interaction and spin-orbit coupling interact. See [54] for a much more expansive version of this diagram.
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The Kitaev Honeycomb model [55] was the first concrete spin model with a QSL ground state. It is defined on the two dimensional honeycomb lattice and provides an exactly solvable model that can be reduced to a free fermion problem via a mapping to Majorana fermions. This yields an extensive number of static \(\mathbb Z_2\) fluxes tied to an emergent gauge field. The model is remarkable not only for its QSL ground state but also for its fractionalised excitations with non-trivial braiding statistics. It has a rich phase diagram hosting gapless, Abelian and non-Abelian phases [56] and a finite temperature phase transition to a thermal metal state [57]. It been proposed that its non-Abelian excitations could be used to support robust topological quantum computing [58–60].
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As Kitaev points out in his original paper, the model remains solvable on any tri-coordinated \(z=3\) graph which can be 3-edge-coloured. Indeed many generalisations of the model to [61–65]. Notably, the Yao-Kivelson model [66] introduces triangular plaquettes to the honeycomb lattice leading to spontaneous chiral symmetry breaking. These extensions all retain translation symmetry, likely because edge-colouring and finding the ground state become much harder without it. Finding the ground state flux sector and understanding the QSL properties can still be challenging [67,68]. Undeterred, this gap lead us to wonder what might happen if we remove translation symmetry from the Kitaev Model. This might would be a model of a tri-coordinated, highly bond anisotropic but otherwise amorphous material.
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Amorphous materials do no have long-range lattice regularities but covalent compounds can induce short-range regularities in the lattice structure such as fixed coordination number \(z\). The best examples being amorphous Silicon and Germanium with \(z=4\) which are used to make thin-film solar cells [69,70]. Recently is has been shown that topological insulating (TI) phases can exist in amorphous systems. Amorphous TIs are characterized by similar protected edge states to their translation invariant cousins and generalised topological bulk invariants [71–77]. However, research on amorphous electronic systems has been mostly focused on non-interacting systems with a few exceptions, for example, to account for the observation of superconductivity [78–82] in amorphous materials or very recently to understand the effect of strong electron repulsion in TIs [83].
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Amorphous magnetic systems has been investigated since the 1960s, mostly through the adaptation of theoretical tools developed for disordered systems [84–87] and with numerical methods [88,89]. Research on classical Heisenberg and Ising models has been shown to account for observed behaviour of ferromagnetism, disordered antiferromagnetism and widely observed spin glass behaviour [90]. However, the role of spin-anisotropic interactions and quantum effects in amorphous magnets has not been addressed. It is an open question whether frustrated magnetic interactions on amorphous lattices can give rise to genuine quantum phases, i.e. to long-range entangled quantum spin liquids (QSL) [91–94].
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In chapter 4 I will introduce the Amorphous Kitaev model, a generalisation of the Kitaev honeycomb model to random lattices with fixed coordination number three. We will show that this model is a soluble chiral amorphous quantum spin liquid. The model retains its exact solubility but, as with the Yao-Kivelson model [66], the presence of plaquettes with an odd number of sides leads to a spontaneous breaking of time reversal symmetry. We will confirm prior observations that the form of the ground state can be written in terms of the number of sides of elementary plaquettes of the model [64,95]. We unearth a rich phase diagram displaying Abelian as well as a non-Abelian chiral spin liquid phases. Furthermore, I show that the system undergoes a finite-temperature phase transition to a conducting thermal metal state and discuss possible experimental realisations.
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Spin-orbit coupling is a relativistic effect that, very roughly, corresponds to the fact that in the frame of reference of a moving electron, the electric field of nearby nuclei looks like a magnetic field to which the electron spin couples. This effectively couples the spatial and spin parts of the electron wavefunction, meaning that the lattice structure can influence the form of the spin-spin interactions leading to spatial anisotropy in the effective interactions. This spatial anisotropy can frustrate the Mott insulators [55,56] leading to more exotic ground states than the AFM order we have seen so far. As we saw with the Hubbard model, interaction effects are only strong or weak in comparison to the bandwidth or hopping integral \(t\). Hence, we will see strong frustration in materials with strong spin-orbit coupling \(\lambda\) relative to their bandwidth \(t\).
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In certain transition metal based compounds, such as those based on Iridium and Ruthenium, the lattice structure, strong spin-orbit coupling and narrow bandwidths lead to effective spin-\(\tfrac{1}{2}\) Mott insulating states with strongly anisotropic spin-spin couplings. These transition metal compounds, known as Kitaev Materials, draw their name from the celebrated Kitaev Honeycomb Model which is expected to model their low temperature behaviour [51,57–60].
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At this point we can sketch out a phase diagram like that of fig. 4. When both electron-electron interactions \(U\) and spin-orbit couplings \(\lambda\) are small relative to the bandwidth \(t\) we recover standard band theory of band insulators and metals. In the upper left we have the simple Mott insulating state as described by the Hubbard model. In the lower right, strong spin-orbit coupling gives rise to Topological insulators (TIs) characterised by symmetry protected edge modes and non-zero Chern number. Kitaev materials occur in the region where strong electron-electron interaction and spin-orbit coupling interact. See [61] for a much more expansive version of this diagram.
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The Kitaev Honeycomb model [62] was the first exactly solvable spin model with a QSL ground state. It is defined on the two dimensional honeycomb lattice and provides an exactly solvable model that can be reduced to a free fermion problem via a mapping to Majorana fermions. This yields an extensive number of static \(\mathbb Z_2\) fluxes tied to an emergent gauge field. The model is remarkable not only for its QSL ground state but also for its fractionalised excitations with non-trivial braiding statistics. It has a rich phase diagram hosting gapless, Abelian and non-Abelian phases [63] and a finite temperature phase transition to a thermal metal state [64]. It been proposed that its non-Abelian excitations could be used to support robust topological quantum computing [65–67].
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The Kitaev model shares are lot with the FK model, they are both effectively models of spinless fermions coupled to a classical Ising background field. This is what makes them exactly solvable. At finite temperatures, fluctuations in the background field provide an effective disorder potential for the fermionic sector, so both models can be studied at finite temperature with Markov chain Monte Carlo methods [64,68].
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As Kitaev points out in his original paper, the model remains solvable on any tri-coordinated \(z=3\) graph which can be 3-edge-coloured. Indeed many generalisations of the model exist [69–73]. Notably, the Yao-Kivelson model [74] introduces triangular plaquettes to the honeycomb lattice leading to spontaneous chiral symmetry breaking. These extensions all retain translation symmetry, likely because edge-colouring, finding the ground state and understanding the QSL properties are much harder without it [75,76]. Undeterred, this gap lead us to wonder what might happen if we remove translation symmetry from the Kitaev Model. This might would be a model of a tri-coordinated, highly bond anisotropic but otherwise amorphous material.
+
Amorphous materials do not have long-range lattice regularities but covalent compounds can induce short-range regularities in the lattice structure such as fixed coordination number \(z\). The best examples being amorphous Silicon and Germanium with \(z=4\) which are used to make thin-film solar cells [77,78]. Recently it has been shown that topological insulating (TI) phases can exist in amorphous systems. Amorphous TIs are characterized by similar protected edge states to their translation invariant cousins and generalised topological bulk invariants [79–85]. However, research on amorphous electronic systems has been mostly focused on non-interacting systems with a few exceptions, for example, to account for the observation of superconductivity [86–90] in amorphous materials or very recently to understand the effect of strong electron repulsion in TIs [91].
+
Amorphous magnetic systems have been investigated since the 1960s, mostly through the adaptation of theoretical tools developed for disordered systems [92–95] and with numerical methods [96,97]. Research on classical Heisenberg and Ising models has been shown to account for observed behaviour of ferromagnetism, disordered antiferromagnetism and widely observed spin glass behaviour [98]. However, the role of spin-anisotropic interactions and quantum effects in amorphous magnets has not been addressed. In this thesis I will address in the question of whether frustrated magnetic interactions on amorphous lattices can give rise to genuine quantum phases, i.e. to long-range entangled quantum spin liquids (QSL) [99–102]. We will find that the answer is yes.
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In chapter 4 I will introduce the Amorphous Kitaev model, a generalisation of the Kitaev honeycomb model to random lattices with fixed coordination number three. We will show that this model is a soluble chiral amorphous quantum spin liquid. The model retains its exact solubility but, as with the Yao-Kivelson model [74], the presence of plaquettes with an odd number of sides leads to a spontaneous breaking of time reversal symmetry. We will confirm prior observations that the form of the ground state can be written in terms of the number of sides of elementary plaquettes of the model [72,103]. We unearth a rich phase diagram displaying Abelian as well as a non-Abelian chiral spin liquid phases. Furthermore, I show that the system undergoes a finite-temperature phase transition to a conducting thermal metal state and discuss possible experimental realisations.
The next chapter, Chapter 2, will introduce some necessary background to the Falikov-Kimball Model, the Kitaev Honeycomb Model, disorder and localisation. Then chapter 3 introduces and studies the Long Range Falicov-Kimball Model in one dimension. Chapter 4 focusses on the Amorphous Kitaev Model.
A. Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. Kanász-Nagy, R. Schmidt, F. Grusdt, E. Demler, D. Greif, and M. Greiner, A Cold-Atom Fermi–Hubbard Antiferromagnet, Nature 545, 462 (2017).
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[38]
A. Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. Kanász-Nagy, R. Schmidt, F. Grusdt, E. Demler, D. Greif, and M. Greiner, A Cold-Atom Fermi–Hubbard Antiferromagnet, Nature 545, 462 (2017).
S. M. Winter, A. A. Tsirlin, M. Daghofer, J. van den Brink, Y. Singh, P. Gegenwart, and R. Valentí, Models and Materials for Generalized Kitaev Magnetism, Journal of Physics: Condensed Matter 29, 493002 (2017).
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[59]
S. M. Winter, A. A. Tsirlin, M. Daghofer, J. van den Brink, Y. Singh, P. Gegenwart, and R. Valentí, Models and Materials for Generalized Kitaev Magnetism, Journal of Physics: Condensed Matter 29, 493002 (2017).
-
[53]
H. Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and S. E. Nagler, Concept and Realization of Kitaev Quantum Spin Liquids, Nature Reviews Physics 1, 264 (2019).
+
[60]
H. Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and S. E. Nagler, Concept and Realization of Kitaev Quantum Spin Liquids, Nature Reviews Physics 1, 264 (2019).
T. Eschmann, P. A. Mishchenko, T. A. Bojesen, Y. Kato, M. Hermanns, Y. Motome, and S. Trebst, Thermodynamics of a Gauge-Frustrated Kitaev Spin Liquid, Physical Review Research 1, 032011(R) (2019).
+
[75]
T. Eschmann, P. A. Mishchenko, T. A. Bojesen, Y. Kato, M. Hermanns, Y. Motome, and S. Trebst, Thermodynamics of a Gauge-Frustrated Kitaev Spin Liquid, Physical Review Research 1, 032011(R) (2019).
M. Costa, G. R. Schleder, M. Buongiorno Nardelli, C. Lewenkopf, and A. Fazzio, Toward Realistic Amorphous Topological Insulators, Nano Letters 19, 8941 (2019).
+
[82]
M. Costa, G. R. Schleder, M. Buongiorno Nardelli, C. Lewenkopf, and A. Fazzio, Toward Realistic Amorphous Topological Insulators, Nano Letters 19, 8941 (2019).
-
[75]
A. Agarwala, V. Juričić, and B. Roy, Higher-Order Topological Insulators in Amorphous Solids, Physical Review Research 2, 012067 (2020).
+
[83]
A. Agarwala, V. Juričić, and B. Roy, Higher-Order Topological Insulators in Amorphous Solids, Physical Review Research 2, 012067 (2020).
-
[76]
H. Spring, A. Akhmerov, and D. Varjas, Amorphous Topological Phases Protected by Continuous Rotation Symmetry, SciPost Physics 11, 022 (2021).
+
[84]
H. Spring, A. Akhmerov, and D. Varjas, Amorphous Topological Phases Protected by Continuous Rotation Symmetry, SciPost Physics 11, 022 (2021).
-
[77]
P. Corbae et al., Evidence for Topological Surface States in Amorphous Bi _ {2} Se _ {3}, arXiv Preprint arXiv:1910.13412 (2019).
+
[85]
P. Corbae et al., Evidence for Topological Surface States in Amorphous Bi _ {2} Se _ {3}, arXiv Preprint arXiv:1910.13412 (2019).
-
[78]
W. Buckel and R. Hilsch, Einfluß Der Kondensation Bei Tiefen Temperaturen Auf Den Elektrischen Widerstand Und Die Supraleitung Für Verschiedene Metalle, Zeitschrift Für Physik 138, 109 (1954).
+
[86]
W. Buckel and R. Hilsch, Einfluß Der Kondensation Bei Tiefen Temperaturen Auf Den Elektrischen Widerstand Und Die Supraleitung Für Verschiedene Metalle, Zeitschrift Für Physik 138, 109 (1954).
-
[79]
W. McMillan and J. Mochel, Electron Tunneling Experiments on Amorphous Ge 1- x Au x, Physical Review Letters 46, 556 (1981).
+
[87]
W. McMillan and J. Mochel, Electron Tunneling Experiments on Amorphous Ge 1- x Au x, Physical Review Letters 46, 556 (1981).
T. Kaneyoshi, Introduction to Amorphous Magnets (World Scientific Publishing Company, 1992).
+
[94]
T. Kaneyoshi, Introduction to Amorphous Magnets (World Scientific Publishing Company, 1992).
-
[87]
T. Kaneyoshi, editor, Amorphous Magnetism (CRC Press, Boca Raton, 2018).
+
[95]
T. Kaneyoshi, editor, Amorphous Magnetism (CRC Press, Boca Raton, 2018).
-
[88]
M. Fähnle, Monte Carlo Study of Phase Transitions in Bond-and Site-Disordered Ising and Classical Heisenberg Ferromagnets, Journal of Magnetism and Magnetic Materials 45, 279 (1984).
+
[96]
M. Fähnle, Monte Carlo Study of Phase Transitions in Bond-and Site-Disordered Ising and Classical Heisenberg Ferromagnets, Journal of Magnetism and Magnetic Materials 45, 279 (1984).
-
[89]
J. Plascak, L. E. Zamora, and G. P. Alcazar, Ising Model for Disordered Ferromagnetic Fe- Al Alloys, Physical Review B 61, 3188 (2000).
+
[97]
J. Plascak, L. E. Zamora, and G. P. Alcazar, Ising Model for Disordered Ferromagnetic Fe- Al Alloys, Physical Review B 61, 3188 (2000).
-
[90]
J. Coey, Amorphous Magnetic Order, Journal of Applied Physics 49, 1646 (1978).
+
[98]
J. Coey, Amorphous Magnetic Order, Journal of Applied Physics 49, 1646 (1978).
-
[91]
P. W. Anderson, Resonating Valence Bonds: A New Kind of Insulator?, Mater. Res. Bull. 8, 153 (1973).
+
[99]
P. W. Anderson, Resonating Valence Bonds: A New Kind of Insulator?, Mater. Res. Bull. 8, 153 (1973).
Due to Pauli exclusion, maximum filling occurs when each lattice site is fully occupied, \(\langle n_c + n_d \rangle = 2\). Here we will focus on the half filled case \(\langle n_c + n_d \rangle = 1\). The ground state phenomenology as the model is doped away from the half-filled state can be rich [7,8] but from this point we will only consider the half-filled point.
At half-filling and on bipartite lattices, FK the model is particle-hole (PH) symmetric. That is, the Hamiltonian anticommutes with the particle hole operator \(\mathcal{P}H\mathcal{P}^{-1} = -H\). As a consequence the energy spectrum is symmetric about \(E = 0\) and this is the Fermi energy. The particle hole operator corresponds to the substitution \(c^\dagger_i \rightarrow \epsilon_i c_i, d^\dagger_i \rightarrow d_i\) where \(\epsilon_i = +1\) for the A sublattice and \(-1\) for the even sublattice [9]. The absence of a hopping term for the heavy electrons means they do not need the factor of \(\epsilon_i\). See appendix A.1 for a full derivation of the PH symmetry.
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+Figure 1: The dispersion (upper row) and density of states (lower row) obtained from a cubic lattice model \(H = \sum_{i} V_i c^\dagger_{i}c_{i} - t \sum_{\langle i,j\rangle} c^\dagger_{i}c_{j}\) in one dimension. (a) With not external potential. (b) With a static charge density wave background \(V_i = (-1)^i\) (c) A static charge density wave background with 2% binary disorder.
We will later add a long range interaction between the localised electrons at which point we will replace the immobile fermions with a classical Ising field \(S_i = 1 - 2d^\dagger_id_i = \pm\tfrac{1}{2}\) which I will refer to as the spins.
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+Figure 2: Schematic Phase diagram of the Falikov-Kimball model in dimensions greater than two. At low temperature the classical fermions (spins) settle into an ordered charge density wave state (antiferromagnetic state). The schematic diagram for the Hubbard model is the same. Reproduced from [10,14]
In dimensions greater than one, the FK model exhibits a phase transition at some \(U\) dependent critical temperature \(T_c(U)\) to a low temperature ordered phase [15]. In terms of the immobile electrons this corresponds to them occupying only one of the two sublattices A and B this is known as a charge density wave (CDW) phase. In terms of spins this is an AFM phase.
One final complexity is that for \(\tfrac{3}{2} < \alpha < 2\) renormalisation group methods show that the critical point has non-universal critical exponents that depend on \(\alpha\) [42]. To avoid this potential confounding factors we will park ourselves at \(\alpha = 1.25\) when we apply these ideas to the FK model.
Were we to extend this to arbitrary dimension \(d\) we would find that thermodynamics properties generally both \(d\) and \(\alpha\), long range interactions can modify the ‘effective dimension’ of thermodynamic systems [43].
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+Figure 3: The thermodynamic behaviour of the long range Ising model \(H_{\mathrm{LRI}} = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j\) as the exponent of the interaction \(\alpha\) is varied.
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+Figure 1: (a) The Kitaev honeycomb model is defined on a honeycomb lattice. The special feature of the honeycomb lattice that makes the model solvable is that each vertex is joined by exactly three bonds, i.e. the lattice is trivalent. One of three labels is assigned to each bond (x,y,z) here represented by colour. (b). After transforming to the Majorana representation we get an emergent gauge degree of freedom \(u_{jk} = \pm 1\) that lives on each bond, the bond variables. These are antisymmetric, \(u_{jk} = -u_{kj}\), so we represent them graphically with arrows on each bond that point in the direction that \(u_{jk} = +1\)(c). The Majorana transformation can be visualised as breaking each spin into four Majoranas \(b_i^x,\;b_i^y,\;b_i^z,\;c_i\). The x, y and z Majoranas then pair along the bonds forming conserved \(\mathbb{Z}_2\) bond operators \(u_{jk} = \langle i b_i^\alpha b_j^\alpha \rangle\). The remaining \(c_i\) operators form an effective quadratic Hamiltonian \(H = \frac{i}{4} \sum_{\langle i,j\rangle_\alpha} 2J^{\alpha} u_{ij} \hat{c}_i \hat{c}_j\).
@@ -86,7 +87,7 @@ image:
The Spin Model
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+Figure 2: A visual introduction to the Kitaev Model.
As discussed in the introduction, spin hamiltonians like that of the Kitaev model arise in electronic systems as the result the balance of multiple effects [5]. For instance, in certain transition metal systems with \(d^5\) valence electrons, crystal field and spin-orbit couplings conspire to shift and split the \(d\) orbitals into moments with spin \(j = 1/2\) and \(j = 3/2\). Of these, the bandwidth \(t\) of the \(j= 1/2\) band is small, meaning that even relatively meagre electron correlations (such those induced by the \(U\) term in the Hubbard model) can lead to the opening of a Mott gap. From there we have a \(j = 1/2\) Mott insulator whose effective spin-spin interactions are again shaped by the lattice geometry and spin-orbit coupling leading some materials to have strong bond-directional Ising-type interactions [12,13]. In the Kitaev Model the bond directionality refers to the fact that the coupling axis \(\alpha\) in terms like \(\sigma_j^{\alpha}\sigma_k^{\alpha}\) is strongly bond dependent.
the products of bonds winding around a closed path \(p\) on the lattice. These operators commute with the Hamiltonian and so have no time dynamics. The winding direction does not matter so long as it is fixed. By convention we will always use clockwise. Each closed path on the lattice is associated with a flux. The number of conserved quantities grows linearly with system size and is thus extensive, this is a common property for exactly solvable systems and can be compared to the heavy electrons present in the Falikov-Kimball model. The square of two loop operators is one so any contractible loop can be expressed as a product of loops around plaquettes of the lattice, as in fig. 3. For the honeycomb lattice the plaquettes are the hexagons. The expectations of \(\hat{W}_p\) through each plaquette, the fluxes, are therefore enough to describe the whole flux sector. We will focus on these fluxes, denoting them by \(\phi_i\). Once we have made the mapping to the Majorana Hamiltonian I will explain how these fluxes can be connected to an emergent \(B\) field which makes their interpretation as fluxes clear.
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+Figure 3: In the Kitaev Honeycomb model, Wilson loop operators \(\hat{W}_p = \prod_{\langle i,j\rangle_\alpha\; \in\; p} \sigma_i^{\alpha}\sigma_j^{\alpha}\) can be composed via multiplication to produce arbitrary contractible loops. As a consequence we need only to keep track of the value of the flux through each plaquette \(\phi_i\). This relationship between the \(u_{ij}\) around a region and fluxes with one is evocative of Stokes’ theorem for classical electromagnetism. In fact it turns out to be the exponential of it as we shall make explicit later.
It is worth noting in passing that the effective Hamiltonian for many Kitaev materials incorporates a contribution from an isotropic Heisenberg term \(\sum_{i,j} \vec{\sigma}_i\cdot\vec{\sigma}_j\), this is referred to as the Heisenberg-Kitaev Model [14]. Materials for which the Kitaev term dominates are generally known as Kitaev Materials. See [5] for a full discussion of Kitaev Materials.
@@ -135,7 +136,7 @@ H &= i \sum_{\langle i,j\rangle_\alpha} J^{\alpha} u_{ij} \hat{c}_i \hat{c}
We have transformed the spin Hamiltonian into a Majorana hamiltonian \(H = i \sum_{\langle i,j\rangle_\alpha} J^{\alpha} u_{ij} \hat{c}_i \hat{c}_j\) describing the dynamics of a classical field \(u_{ij}\) and Majoranas \(c_i\). It is natural to ask how the classical field \(u_{ij}\) relates to the fluxes of the original spin model. We can evaluate the fluxes \(\phi_i\) in terms of the bond operators
\[\phi_i = \prod_{\langle j,k\rangle \in \mathcal{P}_i} i u_{jk}.\qquad{(4)}\]
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+Figure 4: A honeycomb lattice with edges in light grey, along with its dual, the triangle lattice in light blue. The vertices of the dual lattice are the faces of the original lattice and, hence, are the locations of the vortices. (Left) The action of the gauge operator \(D_j\) at a vertex is to flip the value of the three \(u_{jk}\) variables (black lines) surrounding site \(j\). The corresponding edges of the dual lattice (red lines) form a closed triangle. (Middle) Composing multiple adjacent \(D_j\) operators produces a large closed dual loop or multiple disconnected dual loops. Dual loops are not directed like Wilson loops. (Right) A non-contractable loop which cannot be produced by composing \(D_j\) operators. All three operators can be thought of as the action of a vortex-vortex pair that is created, one of them is transported around the loop, and then the two annihilate again. Note that every plaquette has an even number of \(u_{ij}\)s flipped on its edge. Therefore, all retain the same flux \(\phi_i\).
In addition, the bond operators form a highly degenerate description of the system. The operators \(D_i = b^x_i b^y_i b^z_i c_i\) commute with \(H\) so form a set of local symmetries. The action of \(D_i\) on a state is to flip the values of the three \(u_{ij}\) bonds that connect to site \(i\). This changes the bond configuration \(\{u_{ij}\}\) but leaves the flux configuration \(\{\phi_i\}\) unchanged. Physically, we interpret \(u_{ij}\) as a gauge field with a high degree of degeneracy and \(\{D_i\}\) as the set of gauge symmetries. The Majorana bond operators \(u_{ij}\) are an emergent, classical, \(\mathbb{Z_2}\) gauge field! The flux configuration \(\{\phi_i\}\) is what encodes physical information about the system.
@@ -157,7 +158,7 @@ A honeycomb lattice (in black) along with its dual (in red). (Left) The product
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+Figure 5: Wilson loops that wind the major or minor diameters of the torus measure flux winding through the hole of the doughnut/torus or through the filling. If they made doughnuts with both had a jam filling and a hole, this analogy would be a lot easier to make [23].
A final but important point to mention is that is that the local fluxes \(\phi_i\) are not quite all there is. We’ve seen that products of \(\phi_i\) can be used to construct the flux associated with arbitrary contractible loops. On the plane contractible loops are all there are. However, on the torus we can construct two global fluxes \(\Phi_x\) and \(\Phi_y\) which correspond to paths tracing the major and minor axes. The four sectors spanned by the \(\pm1\) values of these fluxes are gapped away from one another but only by virtual tunnelling processes so the gap decays exponentially with system size [1]. Physically \(\Phi_x\) and \(\Phi_y\) could be thought of as measuring the flux that threads through the hole of the doughnut. In general, surfaces with genus \(g\) have \(g\) ‘handles’ and \(2g\) of these global fluxes. At first glance it may seem this would not have much relevance to physical realisations of the Kitaev model that will likely have a planar geometry with open boundary conditions. However these fluxes are closely linked to topology and the existence of anyonic quasiparticle excitations in the model, which we will discuss next.
@@ -165,11 +166,11 @@ A honeycomb lattice (in black) along with its dual (in red). (Left) The product
Anyons, Topology and the Chern number
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+Figure 6: Worldlines of particles in two dimensions can become tangled or braided with one another.
To discuss different ground state phases of the KH model we must first review the topic of anyons and topology. The standard argument for the existence of Fermions and Bosons goes like this: the quantum state of a system must pick up a factor of \(\pm1\) if two identical particles are swapped. Only \(\pm1\) are allowed since swapping twice must correspond to the identity. This argument works in three dimensions for states without topological degeneracy, which seems to be true of the real world, but condensed matter systems are subject to no such constraints.
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In gapped condensed matter systems, all equal time correlators decay exponentially with distance [24]. Put another way, the system supports quasiparticles with a definite location in space and a finite extent. As such it is meaningful to consider what would happen to the overall quantum state if we were to adiabatically carry out a series of swaps as described above. This is known as braiding.
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In gapped condensed matter systems, all equal time correlators decay exponentially with distance [24]. Put another way, gapped systems support quasiparticles with a definite location in space and finite extent. As such it is meaningful to consider what would happen to the overall quantum state if we were to adiabatically carry out a series of swaps as described above. This is known as braiding.
First we realise that in two dimensions, swapping identical particles twice is not topologically equivalent to the identity, see fig. 6. Instead it corresponds to encircling one particle around the other. This means we can in general pick up any complex phase \(e^{i\theta}\), hence the name any-ons upon exchange. These are known as Abelian anyons because complex multiplication commutes and hence the group of braiding operations forms and Abelian group.
The KH model has a topologically degenerate ground state with sectors labelled by the values of the topological fluxes \((\Phi_x\), \(\Phi_y)\). Consider the operation in which a quasiparticle pair is created from the ground state, transported around one of the non-contractible loops and then annihilated together, call them \(\mathcal{T}_{x}\) and \(\mathcal{T}_{y}\). These operations move us around within the ground state manifold and they need not commute. This leads to non-Abelian anyons. As Kitaev points out, these operations are not specific to the torus: the operation \(\mathcal{T}_{x}\mathcal{T}_{y}\mathcal{T}_{x}^{-1}\mathcal{T}_{y}^{-1}\) corresponds to an operation in which none of the particles crosses the torus, rather one simply winds around the other, hence these effects of relevant even for the planar case.
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+![Figure 3: How Kitaev materials fit into the picture of strongly correlated systems. Interactions are required to open a Mott gap and localise the electrons into local moments, while spin-orbit correlations are required to produce the strongly anisotropic spin-spin couplings of the Kitaev Model. Reproduced from [44].](/assets/thesis/intro_chapter/kitaev_material_phase_diagram.svg)
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+![Figure 2: Schematic Phase diagram of the Falikov-Kimball model in dimensions greater than two. At low temperature the classical fermions (spins) settle into an ordered charge density wave state (antiferromagnetic state). The schematic diagram for the Hubbard model is the same. Reproduced from [10,14]](/assets/thesis/background_chapter/fk_phase_diagram.svg)
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+![Figure 5: Wilson loops that wind the major or minor diameters of the torus measure flux winding through the hole of the doughnut/torus or through the filling. If they made doughnuts with both had a jam filling and a hole, this analogy would be a lot easier to make [23].](/assets/thesis/amk_chapter/topological_fluxes.png)
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