From b1749e8607b8f7b216aeed8caa301f32f8fb5f82 Mon Sep 17 00:00:00 2001 From: Tom Hodson Date: Sat, 17 Sep 2022 19:24:04 +0100 Subject: [PATCH] lots of LRFK figures! --- ...22-the-long-range-falikov-kimball-model.md | 2 +- _thesis/1_Introduction/1_Intro.html | 6 +- _thesis/2_Background/2.1_FK_Model.html | 2 +- _thesis/2_Background/2.2_HKM_Model.html | 71 +- _thesis/2_Background/2.4_Disorder.html | 134 +- .../3.1_LRFK_Model.html | 6 +- .../3.2_LRFK_Methods.html | 21 +- .../3.3_LRFK_Results.html | 102 +- .../4.1_AMK_Model.html | 2 +- .../amk_chapter/intro/amk_zoom/amk_zoom.svg | 20507 ++++++++++++++++ assets/thesis/fk_chapter/DOS/DOS.svg | 7363 ++++++ assets/thesis/fk_chapter/DOS/IPR_scaling.svg | 2429 ++ .../binder_cumulants/binder_cumulants.svg | 1399 ++ .../fk_chapter/disorder_model/DM_DOS.svg | 2884 +++ .../disorder_model/DM_IPR_scaling.svg | 3690 +++ .../fk_chapter/gap_opening/gap_opening_U2.svg | 1053 + .../fk_chapter/gap_opening/gap_opening_U5.svg | 1053 + .../phase_diagram/phase_diagram.svg | 1456 ++ 18 files changed, 42008 insertions(+), 172 deletions(-) create mode 100644 assets/thesis/amk_chapter/intro/amk_zoom/amk_zoom.svg create mode 100644 assets/thesis/fk_chapter/DOS/DOS.svg create mode 100644 assets/thesis/fk_chapter/DOS/IPR_scaling.svg create mode 100644 assets/thesis/fk_chapter/binder_cumulants/binder_cumulants.svg create mode 100644 assets/thesis/fk_chapter/disorder_model/DM_DOS.svg create mode 100644 assets/thesis/fk_chapter/disorder_model/DM_IPR_scaling.svg create mode 100644 assets/thesis/fk_chapter/gap_opening/gap_opening_U2.svg create mode 100644 assets/thesis/fk_chapter/gap_opening/gap_opening_U5.svg create mode 100644 assets/thesis/fk_chapter/phase_diagram/phase_diagram.svg diff --git a/_publications/2021-03-22-the-long-range-falikov-kimball-model.md b/_publications/2021-03-22-the-long-range-falikov-kimball-model.md index cd7d4ee..0643049 100644 --- a/_publications/2021-03-22-the-long-range-falikov-kimball-model.md +++ b/_publications/2021-03-22-the-long-range-falikov-kimball-model.md @@ -1,7 +1,7 @@ --- title: "The one-dimensional Long-Range Falikov-Kimball Model: Thermal Phase Transition and Disorder-Free Localisation" collection: publications -permalink: /publication/2021-03-22-the-long-range-falikov-kimball-model +permalink: /publication/2021-03-22-the-long-range-falicov-kimball-model excerpt: 'Disorder or interactions can turn metals into insulators. One of the simplest settings to study this physics is given by the Falikov-Kimball model, which describes itinerant fermions interacting with a classical Ising background field.' date: 2021-03-22 venue: 'Physics Review B' diff --git a/_thesis/1_Introduction/1_Intro.html b/_thesis/1_Introduction/1_Intro.html index 6dc4b9c..2f56c53 100644 --- a/_thesis/1_Introduction/1_Intro.html +++ b/_thesis/1_Introduction/1_Intro.html @@ -99,7 +99,7 @@ image: H_{\mathrm{FK}} = & -\;t \sum_{\langle i,j \rangle} c^\dagger_{i}c_{j} + \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}). \\ \end{aligned}\]

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].

-

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.

+

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.

Quantum Spin Liquids

@@ -118,9 +118,9 @@ H_{\mathrm{FK}} = & -\;t \sum_{\langle i,j \rangle} c^\dagger_{i}c_{j} + \;U

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].

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.

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].

-

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 genuine quantum phases, i.e. to long-range entangled quantum spin liquids (QSL)  [91–94].

+

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].

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.

-

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 Falikov-Kimball Model in one dimension while chapter 4 focusses on the Amorphous Kitaev Model.

+

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 Falikov-Kimball Model in one dimension while chapter 4 focusses on the Amorphous Kitaev Model.

Next Chapter: 2 Background

diff --git a/_thesis/2_Background/2.1_FK_Model.html b/_thesis/2_Background/2.1_FK_Model.html index f98595a..a5ac2a3 100644 --- a/_thesis/2_Background/2.1_FK_Model.html +++ b/_thesis/2_Background/2.1_FK_Model.html @@ -99,7 +99,7 @@ H_{\mathrm{FK}} = & \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\

In the disordered region above \(T_c(U)\) there are two insulating phases. For weak interactions \(U << t\), thermal fluctuations in the spins act as an effective disorder potential for the fermions, causing them to localise and giving rise to an Anderson insulating state  [16] which we will discuss more in section 2.3. For strong interactions \(U >> t\), the spins are not ordered but nevertheless their interaction with the electrons opens a gap, leading a Mott insulator analogous to that of the Hubbard model  [17].

By contrast, in the one dimensional FK model there is no finite-temperature phase transition (FTPT) to an ordered CDW phase  [18]. Indeed dimensionality is crucial for the physics of both localisation and FTPTs. In one dimension, disorder generally dominates: even the weakest disorder exponentially localises all single particle eigenstates. Only longer-range correlations of the disorder potential can potentially induce localisation-delocalisation transitions in one dimension  [19–21]. Thermodynamically, short-range interactions cannot overcome thermal defects in one dimension which prevents ordered phases at non-zero temperature  [22–24].

However, the absence of an FTPT in the short ranged FK chain is far from obvious because the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction mediated by the fermions  [25–28] decays as \(r^{-1}\) in one dimension  [29]. This could in principle induce the necessary long-range interactions for the classical Ising background to order at low temperatures  [30,31]. However, Kennedy and Lieb established rigorously that at half-filling a CDW phase only exists at \(T = 0\) for the one dimensional FK model  [32].

-

Based on this primacy of dimensionality, we will go digging into the one dimensional case. In chapter 3 we will construct a generalised one-dimensional FK model with long-range interactions which induces the otherwise forbidden CDW phase at non-zero temperature. To do this we will draw on theory of the Long Range Ising Model which is the subject of the next section.

+

Based on this primacy of dimensionality, we will go digging into the one dimensional case. In chapter 3 we will construct a generalised one-dimensional FK model with long-range interactions which induces the otherwise forbidden CDW phase at non-zero temperature. To do this we will draw on theory of the Long Range Ising Model which is the subject of the next section.

Long Ranged Ising model

diff --git a/_thesis/2_Background/2.2_HKM_Model.html b/_thesis/2_Background/2.2_HKM_Model.html index 45df0b2..542f4f7 100644 --- a/_thesis/2_Background/2.2_HKM_Model.html +++ b/_thesis/2_Background/2.2_HKM_Model.html @@ -36,7 +36,6 @@ image:
  • An Emergent Gauge Field
  • Anyons, Topology and the Chern number
  • Ground State Phases
    • -
  • Glossary
  • Bibliography
    • @@ -60,7 +59,6 @@ image:
  • An Emergent Gauge Field
  • Anyons, Topology and the Chern number
  • Ground State Phases
    • -
  • Glossary
  • Bibliography
    • @@ -88,30 +86,30 @@ image:

    The Spin Model

    +
    +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.

    In the spin hamiltonian eq. 1 we can already tease out a set of conserved fluxes that will be key to the model’s solution. These fluxes are the expectations of Wilson loop operators

    \[\hat{W}_p = \prod_{\langle i,j\rangle_\alpha\; \in\; p} \sigma_i^{\alpha}\sigma_j^{\alpha},\]

    -

    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. 2. 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.

    +

    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.

    -Figure 2: 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. - +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.

    As with the Falikov-Kimball model, the KH model has a extensive number of conserved quantities, the fluxes. As with the FK model it will make sense to work in the simultaneous eigenbasis of the fluxes and the Hamiltonian so that we can treat the fluxes like a classical degree of freedom. This is part of what makes the model tractable. We will find that the ground state of the model corresponds to some particular choice of fluxes. We will refer to local excitations away from the flux ground state as vortices. In order to fully solve the model however, we must first move to a Majorana picture.

    The Majorana Model

    -
    -Figure 3: A visual introduction to the Kitaev Model. - -

    Majorana fermions are something like ‘half of a complex fermion’ and are their own antiparticle. From a set of \(N\) fermionic creation \(f_i^\dagger\) and anhilation \(f_i\) operators we can construct \(2N\) Majorana operators \(c_m\). We can do this construction in multiple ways subject to only mild constraints required to keep the overall commutations relations correct  [1]. Majorana operators square to one but otherwise have standard fermionic commutation relations.

    \(N\) spins can be mapped to \(N\) fermions with the well known Jordan-Wigner transformation and indeed this approach can be used to solve the Kitaev model  [15]. Here I will introduce the method Kitaev used in the original paper as this forms the basis for the results that will be presented in this thesis. Rather than mapping to \(N\) fermions, Kitaev maps to \(4N\) Majoranas, effectively \(2N\) fermions. In contrast to the Jordan-Wigner approach which makes fermions out of strings of spin operators in order to correctly produce fermionic commutation relations, the Kitaev transformation maps each spin locally to four Majoranas. The downside is that this enlarges the Hilbert space from \(2^N\) to \(4^N\). We will have to employ a projector \(\hat{P}\) to come back down to the physical Hilbert space later. As everything is local, I will drop the site indices \(ijk\) in expressions that refer to only a single site.

    The mapping is defined in terms of four Majoranas per site \(b_i^x,\;b_i^y,\;b_i^z,\;c_i\) such that

    \[\tilde{\sigma}^x = i b^x c,\; \tilde{\sigma}^y = i b^y c,\; \tilde{\sigma}^z = i b^z c\qquad{(2)}\]

    The tildes on the spin operators \(\tilde{\sigma_i^\alpha}\) emphasis that they live in this new extended Hilbert space and are only equivalent to the original spin operators after applying a projector \(\hat{P}\). The form of the projection operator can be understood in a few ways. From a group-theoretic perspective, before projection, the operators \(\{\tilde{\sigma}^x, \tilde{\sigma}^y, \tilde{\sigma}^z\}\) form a representation of the gamma group \(G_{3,0}\). The gamma groups \(G_{p,q}\) have \(p\) generators that square to the identity and \(q\) that square (roughly) to \(-1\). The generators otherwise obey standard anticommutation relations. The well known gamma matrices \(\{\gamma^0, \gamma^1, \gamma^2, \gamma^3\}\) represent \(G_{1,3}\) the quaternions \(G_{0,3}\) and the Pauli matrices \(G_{3,0}\).

    The Pauli matrices, however, have the additional property that the chiral element \(\sigma^x \sigma^y \sigma^z = i\), this relation is not determined by the group properties of \(G_{3,0}\). Therefore to fully reproduce the algebra of the Pauli matrices we must project into the subspace where \(\tilde{\sigma}^x \tilde{\sigma}^y \tilde{\sigma}^z = i\). The chiral element of the gamma matrices for instance \(\gamma_5 = i\gamma^0 \gamma^1 \gamma^2 \gamma^3\) is of central importance in quantum field theory. See  [16] for more discussion of this group theoretic view.

    -

    The projector must project onto the subspace where \(\tilde{\sigma}^x \tilde{\sigma}^y \tilde{\sigma}^z = i\). If we work this through we find that in general $xy^z = iD $ where \(D = b^x b^y b^z c\) must be the identity for every site. In other words, we can only work with physical states \(|\phi\rangle\) that satisfy $ D_i|= |$ for all sites \(i\). From this we construct an on-site projector \(P_i = \frac{1 + D_i}{2}\) and the overall projector is simply \(P = \prod_i P_i\).

    +

    The projector must project onto the subspace where \(\tilde \sigma^x \tilde \sigma^y \tilde \sigma^z = i\). If we work this through we find that in general $^x ^y ^z = iD $ where \(D = b^x b^y b^z c\) must be the identity for every site. In other words, we can only work with physical states \(|\phi\rangle\) that satisfy $ D_i|= |$ for all sites \(i\). From this we construct an on-site projector \(P_i = \frac{1 + D_i}{2}\) and the overall projector is simply \(P = \prod_i P_i\).

    Another way to see what this is doing physically is to explicitly construct the two intermediate fermionic operators \(f\) and \(g\) that give rise to these four Majoranas. Working through the algebra we see that the \(D\) operator corresponds to the fermion parity \(D = -(2n_f - 1)(2n_g - 1)\) where \(n_f,\; n_g\) are the number operators. Expanding the product \(\prod_i P_i\) out, we find that the projector corresponds to a symmetrisation over \(\{u_{ij}\}\) states within a flux sector and and overall fermion parity \(\prod_i D_i\). This tells us that any arbitrary state can be made to have non-zero overlap with the physical subspace via the addition or removal of a single fermion. This implies that in the thermodynamic limit the projection step is not generally necessary to extract physical results, see  [17] or appendix A.5 for more details.

    We can now rewrite the spin hamiltonian in Majorana form with caveat that they are only strictly equivalent after projection. The Ising interactions \(\sigma_j^{\alpha}\sigma_k^{\alpha}\) decouple into the form \(-i (i b^\alpha_i b^\alpha_j) c_i c_j\). We factor out the bond operators \(\hat{u}_{ij} = i b^\alpha_i b^\alpha_j\) which are Hermitian and, remarkably, commute with the Hamiltonian and each other.

    \[\begin{aligned} @@ -184,6 +182,8 @@ H &= i \sum_{\langle i,j\rangle_\alpha} J^{\alpha} u_{ij} \hat{c}_i \hat{c} \sum_{(i,j,k)} \sigma_i^{\alpha} \sigma_j^{\beta} \sigma_k^{\gamma} \] where the sum \((i,j,k)\) runs over consecutive indices around plaquettes. The addition of this to the spin model leads to two bond terms in the corresponding Majorana model. The effect of breaking chiral symmetry is to open a gap in the B phase. The vortices of the gapped B phase are non-Abelian anyons.

    At finite temperatures, recent work has shown that the KH model undergoes a transition to a thermal metal phase.

    +

    Summary

    +

    We have seen that…

    Summary The Kitaev Honeycomb model is remarkable because it combines three key properties. First, the form of the Hamiltonian plausibly be realised by a real material. Candidate materials, such as \(\alpha\mathrm{-RuCl}_3\), are known to have sufficiently strong spin-orbit coupling and the correct lattice structure to behave according to the Kitaev Honeycomb model with small corrections  [5,34]. Second, its ground state is the canonical example of the long sought after quantum spin liquid state. Its excitations are anyons, particles that can only exist in two dimensions that break the normal fermion/boson dichotomy.

    Third, and perhaps most importantly, this model is a rare many body interacting quantum system that can be treated analytically. It is exactly solvable. We can explicitly write down its many body ground states in terms of single particle states  [1]. The solubility of the Kitaev Honeycomb Model, like the Falikov-Kimball model of chapter 1, comes about because the model has extensively many conserved degrees of freedom. These conserved quantities can be factored out as classical degrees of freedom, leaving behind a non-interacting quantum model that is easy to solve.

    “dynamical two spin correlation functions are identically zero beyond nearest neighbor separation in the Kitaev Model”  [35]

    @@ -192,57 +192,6 @@ H &= i \sum_{\langle i,j\rangle_\alpha} J^{\alpha} u_{ij} \hat{c}_i \hat{c}
  • really follows the Kitaev-Heisenberg model
  • experimental probes include inelastic neutron scattering, Raman scattering
    • -
    -
    -

    Glossary

    - -

    The Spin Hamiltonian

    - -

    The Majorana Model

    - -

    Flux Sectors

    - -

    Phases

    - -

    Vortices and their movements

    -

    See fig. 6 for a diagram of the next three paragraphs.

    -

    We started from the ground state of the model and flipped the sign of a single bond (fig. 6 (a)). In doing so, we will flip the sign of the two plaquettes adjacent to that bond. We will call these disturbed plaquettes vortices. We will refer to a particular choice values for the plaquette operators as a vortex sector.

    -

    If we chain multiple bond flips, we can create a pair of vortices at arbitrary locations (fig. 6 (b)). The chain of bonds that we must flip corresponds to a path on the dual of the lattice.

    -

    We can also create a pair of vortices, move one around a loop and finally annihilate it with its partner (fig. 6 (c)). This corresponds to a closed loop on the dual lattice. Applying such a bond flip leaves the vortex sector unchanged. We can also do the same thing but move the vortex around one the non-contractible loops of the lattice (fig. 6 (d)).

    -

    There is one kind of dual loop that we cannot build out of \(D_j\)s, the non-contractible loops.

    -
    -

    </i,j></i,j>

    Next Section: Disorder and Localisation

    diff --git a/_thesis/2_Background/2.4_Disorder.html b/_thesis/2_Background/2.4_Disorder.html index 03875a6..d0ed54d 100644 --- a/_thesis/2_Background/2.4_Disorder.html +++ b/_thesis/2_Background/2.4_Disorder.html @@ -29,10 +29,8 @@ image: @@ -49,10 +47,8 @@ image: @@ -66,73 +62,127 @@ image:

    Disorder and Localisation

    -
    -

    Localisation: Anderson, Many Body and Disorder-Free

    -
    + +

    Disorder is a fact of life for the condensed matter physicist. No sample will ever be completely free of contamination or of structural defects. The classical Drude theory of electron conductivity envisages electrons as scattering off impurities. Hence we would expect the electrical conductivity to be proportional to the mean free path  [1], decreasing smoothly as the number of defects increases. However, Anderson showed in 1958  [2] that at some critical level of disorder all single particle eigenstates localise. What would later be known as Anderson localisation is characterised by exponentially localised eigenfunctions \(\psi(x) \sim e^{-x/\lambda}\) which cannot contribute to transport processes. The localisation length \(\lambda\) is the typical scale of localised state and can be extracted with transmission matrix methods  [3]. Anderson localisation provided a different kind of insulator to that of the band insulator.

    +

    The Anderson model is about the simplest model of disorder one could imagine \[ +H = -t\sum_{\langle jk \rangle} c^\dagger_j c_k + \sum_j V_j c_j^\dagger c_j +\qquad{(1)}\]

    +

    It is one of non-interacting fermions subject to a disorder potential \(V_j\) drawn uniformly from the interval \([-W,W]\). The discovery of localisation in quantum systems was surprising at the time given the seeming ubiquity of extended Bloch states. Within the Anderson model, all the states localise at the same disorder strength \(W\) but later Mott showed that in other contexts extended Bloch states and localised states could coexist at the same disorder strength but different energies. The transition in energy between localised and extended states is known as a mobility edge  [4].

    +

    Localisation phenomena are strongly dimension dependent. In three dimensions the scaling theory of localisation  [5,6] shows that Anderson localisation is a critical phenomenon with critical exponents both for how the conductivity vanishes with energy when approaching the mobility edge and for how the localisation length increases below it. By contrast, in one dimension disorder generally dominates. Even the weakest disorder exponentially localises all single particle eigenstates in the one dimensional Anderson model. Only long-range spatial correlations of the disorder potential can induce delocalisation  [7–12].

    +

    Later localisation was found in disordered interacting many-body systems:

    +

    \[ +H = -t\sum_{\langle jk \rangle} c^\dagger_j c_k + \sum_j V_j c_j^\dagger c_j + U\sum_{jk} n_j n_k +\] Here, in contrast to the Anderson model, localisation phenomena are robust to weak perturbations of the Hamiltonian. This is called many-body localisation (MBL)  [13].

    +

    Both MBL and Anderson localisation depend crucially on the presence of quenched disorder. Quenched disorder takes the form a static background field drawn from an arbitrary probability distribution to which the model is coupled. Disorder may also be introduced into the initial state of the system rather than the Hamiltonian. This has led to ongoing interest in the possibility of disorder-free localisation where the disorder is instead annealed. In this scenario the disorder necessary to generate localisation is generated entirely from the thermal fluctuations of the model.

    +

    The concept of disorder-free localisation was first proposed in the context of Helium mixtures  [14] and then extended to heavy-light mixtures in which multiple species with large mass ratios interact. The idea is that the heavier particles act as an effective disorder potential for the lighter ones, inducing localisation. Two such models  [15,16] instead find that the models thermalise exponentially slowly in system size, which Ref.  [15] dubs Quasi-MBL.

    +

    True disorder-free localisation does occur in exactly solvable models with extensively many conserved quantities  [17]. As conserved quantities have no time dynamics this can be thought of as taking the separation of timescales to the infinite limit. The localisation phenomena present in the Falikov-Kimball model are instead the result of annealed disorder. A strong separation of timescales means that the heavy species is approximated as immobile with respect to the lighter itinerant species. At finite temperature the heavy species acts as a disorder potential for the lighter one. However, in contract to quenched disorder, the probability distribution of annealed disorder is entirely determined by the thermodynamics of the Hamiltonian. In the two dimensional FK model this leads to multiple phases where localisation effects are relevant. At low temperatures the heavy species orders leading to a traditional band gap insulator. At higher temperatures however thermal disorder causes the light species to localise. At weak coupling, the localisation length can be very large, so finite sized systems may still conduct, an effect known as weak localisation  [18].

    +

    In Chapter 3 we will consider a generalised FK model in one dimension and how the disorder generated near a one dimensional thermodynamic phase transition interacts with localisation physics.

    +

    So far we have considered disorder as a static or dynamic field coupled to a model defined on a translation invariant lattice. Another kind of disordered system that worthy of study are amorphous systems.

    +

    Amorphous systems have disordered bond connectivity, so called topological disorder. As discussed in the introduction these include amorphous semiconductors such as amorphous Germanium and Silicon   [19–22]. While materials do not have long range lattice structure they can enforce local constraints such as the approximate coordination number \(z = 4\) of silicon.

    +

    Topological disorder can be qualitatively different from other disordered systems. Disordered graphs are constrained by fixed coordination number and the Euler equation. The Harris  [23] and the Imry-Mar  [24] criteria are key results on the effect of disorder on thermodynamic phase transitions. The Harris criterion signals when disorder will affect the universal of a thermodynamic critical point. It states that for a critical point in a \(d\)-dimensional system with correlation length scaling exponent, disorder will be relevant if \(\nu\) if \(d\nu < 2\). The Imry-Ma criterion simply forbids the formation of long range ordered states in \(d \leq 2\) dimensions in the presence of disorder. The latter criteria is violated in the presence of correlated disorder  [25] and both are modified for topological disorder. In chapter 4 we will put the Kitaev model onto two dimensional Voronoi lattices. These lattices are have fixed coordination number \(z=3\) and must satisfy the Euler equation for the plane, this leads to strong anticorrelations which mean that topological disorder is effectively weaker than standard disorder here  [26,27]]. This does not apply to the three dimensional Voronoi lattices where the Euler equation is a weaker constraint.

    +

    Lastly it is worth exploring how quantum spin liquids and disorder interact. The KH model has been studied subject to both bond and site disorder cite. In some instances it seems that disorder can even promote the formation of a QSL ground state  [28].

    +

    Disorder and Spin liquids

    Amorphous Magnetism

    -
    -
    -
    -

    Localisation

    -

    The discovery of localisation in quantum systems surprising at the time given the seeming ubiquity of extended Bloch states. Later, when thermalisation in quantum systems gained interest, localisation phenomena again stood out as counterexamples to the eigenstate thermalisation hypothesis  [1,2], allowing quantum systems to avoid to retain memory of their initial conditions in the face of thermal noise.

    -

    The simplest and first discovered kind is Anderson localisation, first studied in 1958  [3] in the context of non-interacting fermions subject to a static or quenched disorder potential \(V_j\) drawn uniformly from the interval \([-W,W]\)

    -

    \[ -H = -t\sum_{\langle jk \rangle} c^\dagger_j c_k + \sum_j V_j c_j^\dagger c_j -\]

    -

    this model exhibits exponentially localised eigenfunctions \(\psi(x) = f(x) e^{-x/\lambda}\) which cannot contribute to transport processes. Initially it was thought that in one dimensional disordered models, all states would be localised, however it was later shown that in the presence of correlated disorder, bands of extended states can exist  [4–6].

    -

    Later localisation was found in interacting many-body systems with quenched disorder:

    -

    \[ -H = -t\sum_{\langle jk \rangle} c^\dagger_j c_k + \sum_j V_j c_j^\dagger c_j + U\sum_{jk} n_j n_k -\]

    -

    where the number operators \(n_j = c^\dagger_j c_j\). Here, in contrast to the Anderson model, localisation phenomena can be proven robust to weak perturbations of the Hamiltonian. This is called many-body localisation (MBL)  [7].

    -

    Both MBL and Anderson localisation depend crucially on the presence of quenched disorder. This has led to ongoing interest in the possibility of disorder-free localisation, in which the disorder necessary to generate localisation is generated entirely from the dynamics of the model. This contracts with typical models of disordered systems in which disorder is explicitly introduced into the Hamilton or the initial state.

    -

    The concept of disorder-free localisation was first proposed in the context of Helium mixtures  [8] and then extended to heavy-light mixtures in which multiple species with large mass ratios interact. The idea is that the heavier particles act as an effective disorder potential for the lighter ones, inducing localisation. Two such models  [9,10] instead find that the models thermalise exponentially slowly in system size, which Ref.  [9] dubs Quasi-MBL.

    -

    True disorder-free localisation does occur in exactly solvable models with extensively many conserved quantities  [11]. As conserved quantities have no time dynamics this can be thought of as taking the separation of timescales to the infinite limit.

    -

    -link to the FK model

    -

    -link to the Kitaev Model

    -

    -link to the physics of amorphous systems

    Next Chapter: 3 The Long Range Falikov-Kimball Model

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    diff --git a/_thesis/3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html b/_thesis/3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html index cd647fe..af8d5ac 100644 --- a/_thesis/3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html +++ b/_thesis/3_Long_Range_Falikov_Kimball/3.1_LRFK_Model.html @@ -27,7 +27,7 @@ image: