From a72ac1774bb133c6e0eb16c128fccba229a3b980 Mon Sep 17 00:00:00 2001 From: Tom Hodson Date: Wed, 21 Sep 2022 10:57:53 +0100 Subject: [PATCH] draft 21 Sept --- _thesis/1_Introduction/1_Intro.html | 12 +- _thesis/2_Background/2.1_FK_Model.html | 66 +- _thesis/2_Background/2.2_HKM_Model.html | 50 +- _thesis/2_Background/2.4_Disorder.html | 80 +- .../3.1_LRFK_Model.html | 120 + .../3.2_LRFK_Methods.html | 167 + .../3.3_LRFK_Results.html | 185 + .../3.1_LRFK_Model.html | 122 +- .../3.2_LRFK_Methods.html | 160 +- .../3.3_LRFK_Results.html | 27 +- .../A.2_Markov_Chain_Monte_Carlo.html | 255 +- _thesis/toc.html | 13 +- .../amk_chapter/intro/amk_zoom/amk_zoom.svg | 9988 ++++++++--------- .../background_chapter/fk_phase_diagram.svg | 5208 +++++---- .../phase_diagram/phase_diagram.svg | 300 +- .../kitaev_material_phase_diagram.svg | 3610 ++++++ 16 files changed, 12018 insertions(+), 8345 deletions(-) create mode 100644 _thesis/3_Long_Range_Falicov_Kimball/3.1_LRFK_Model.html create mode 100644 _thesis/3_Long_Range_Falicov_Kimball/3.2_LRFK_Methods.html create mode 100644 _thesis/3_Long_Range_Falicov_Kimball/3.3_LRFK_Results.html create mode 100644 assets/thesis/intro_chapter/kitaev_material_phase_diagram.svg diff --git a/_thesis/1_Introduction/1_Intro.html b/_thesis/1_Introduction/1_Intro.html index 2f56c53..c691d66 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

@@ -109,18 +109,18 @@ H_{\mathrm{FK}} = & -\;t \sum_{\langle i,j \rangle} c^\dagger_{i}c_{j} + \;U
-Figure 3: From  [44]. - +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]. +

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 \(t\).

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].

-

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.

+

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.

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 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.

+

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 Falicov-Kimball Model in one dimension. 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 a5ac2a3..7ffdf53 100644 --- a/_thesis/2_Background/2.1_FK_Model.html +++ b/_thesis/2_Background/2.1_FK_Model.html @@ -92,33 +92,34 @@ H_{\mathrm{FK}} = & \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\

Phase Diagrams

-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] +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.

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.

+

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. In the one dimensional Kitaev model this means the whole spectrum is localised at all finite temperatures, though at low temperatures the localisation length may be so large that the states appear extended in finite size systems. 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].

+

The one dimensional FK model has been studied numerically, as a perturbation in interaction strength \(U\) and in the continuum limit  [25] with the main results beings for attractive \(U > U_c\) the system forms electron spin bound state ‘atoms’ which repel on another  [26] and that the ground state phase diagram has a has a fractal structure as a function of electron filling  [27].

+

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  [28–31] decays as \(r^{-1}\) in one dimension  [32]. This could in principle induce the necessary long-range interactions for the classical Ising background to order at low temperatures  [33,34]. However, Kennedy and Lieb established rigorously that at half-filling a CDW phase only exists at \(T = 0\) for the one dimensional FK model  [35].

+

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

The suppression of phase transitions is a common phenomena in one dimensional systems and the Ising model serves as a great illustration of this. In terms of classical spins \(S_i = \pm \frac{1}{2}\) the standard Ising model reads

\[H_{\mathrm{I}} = \sum_{\langle ij \rangle} S_i S_j\]

-

Like the FK model, the Ising model shows an FTPT to an ordered state only in two dimensions and above. This can be understood via Peierls’ argument  [31,32] to be a consequence of the low energy penalty for domain walls in one dimensional systems.

+

Like the FK model, the Ising model shows an FTPT to an ordered state only in two dimensions and above. This can be understood via Peierls’ argument  [34,35] to be a consequence of the low energy penalty for domain walls in one dimensional systems.

Following Peierls’ argument, consider the difference in free energy \(\Delta F = \Delta E - T\Delta S\) between an ordered state and a state with single domain wall in a discrete order parameter. If this value is negative it implies that the ordered state is unstable with respect to domain wall defects, and they will thus proliferate, destroying the ordered phase. If we consider the scaling of the two terms with system size \(L\) we see that short range interactions produce a constant energy penalty \(\Delta E\) for a domain wall. In contrast, the number of such single domain wall states scales linearly with system size so the entropy is \(\propto \ln L\). Thus the entropic contribution dominates (eventually) in the thermodynamic limit and no finite temperature order is possible. In two dimensions and above, the energy penalty of a domain wall scales like \(L^{d-1}\) which is why they can support ordered phases. This argument does not quite apply to the FK model because of the aforementioned RKKY interaction. Instead this argument will give us insight into how to recover an ordered phase in the one dimensional FK model.

In contrast the long range Ising (LRI) model \(H_{\mathrm{LRI}}\) can have an FTPT in one dimension.

\[H_{\mathrm{LRI}} = \sum_{ij} J(|i-j|) S_i S_j = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j\]

-

Renormalisation group analyses show that the LRI model has an ordered phase in 1D for \(1 < \alpha < 2\)   [33]. Peierls’ argument can be extended  [30] to long range interactions to provide intuition for why this is the case. Again considering the energy difference between the ordered state \(|\ldots\uparrow\uparrow\uparrow\uparrow\ldots\rangle\) and a domain wall state \(|\ldots\uparrow\uparrow\downarrow\downarrow\ldots\rangle\). In the case of the LRI model, careful counting shows that this energy penalty is \[\Delta E \propto \sum_{n=1}^{\infty} n J(n)\]

-

because each interaction between spins separated across the domain by a bond length \(n\) can be drawn between \(n\) equivalent pairs of sites. The behaviour then depends crucially on the sum scales with system size. Ruelle proved rigorously for a very general class of 1D systems, that if \(\Delta E\) or its many-body generalisation converges to a constant in the thermodynamic limit then the free energy is analytic  [34]. This rules out a finite order phase transition, though not one of the Kosterlitz-Thouless type. Dyson also proves this though with a slightly different condition on \(J(n)\)  [33].

+

Renormalisation group analyses show that the LRI model has an ordered phase in 1D for \(1 < \alpha < 2\)   [36]. Peierls’ argument can be extended  [33] to long range interactions to provide intuition for why this is the case. Again considering the energy difference between the ordered state \(|\ldots\uparrow\uparrow\uparrow\uparrow\ldots\rangle\) and a domain wall state \(|\ldots\uparrow\uparrow\downarrow\downarrow\ldots\rangle\). In the case of the LRI model, careful counting shows that this energy penalty is \[\Delta E \propto \sum_{n=1}^{\infty} n J(n)\]

+

because each interaction between spins separated across the domain by a bond length \(n\) can be drawn between \(n\) equivalent pairs of sites. The behaviour then depends crucially on the sum scales with system size. Ruelle proved rigorously for a very general class of 1D systems, that if \(\Delta E\) or its many-body generalisation converges to a constant in the thermodynamic limit then the free energy is analytic  [37]. This rules out a finite order phase transition, though not one of the Kosterlitz-Thouless type. Dyson also proves this though with a slightly different condition on \(J(n)\)  [36].

With a power law form for \(J(n)\), there are a few cases to consider:

-

For \(\alpha = 0\) i.e infinite range interactions, the Ising model is exactly solvable and mean field theory is exact  [35]. This limit is the same as the infinite dimensional limit.

-

For \(\alpha \leq 1\) we have very slowly decaying interactions. \(\Delta E\) does not converge as a function of system size so the Hamiltonian is non-extensive, a topic not without some considerable controversy  [36–38] that we will not consider further here.

+

For \(\alpha = 0\) i.e infinite range interactions, the Ising model is exactly solvable and mean field theory is exact  [38]. This limit is the same as the infinite dimensional limit.

+

For \(\alpha \leq 1\) we have very slowly decaying interactions. \(\Delta E\) does not converge as a function of system size so the Hamiltonian is non-extensive, a topic not without some considerable controversy  [39–41] that we will not consider further here.

For \(1 < \alpha < 2\), we get a phase transition to an ordered state at a finite temperature, this is what we want!

-

For \(\alpha = 2\), the energy of domain walls diverges logarithmically, and this turns out to be a Kostelitz-Thouless transition  [30].

+

For \(\alpha = 2\), the energy of domain walls diverges logarithmically, and this turns out to be a Kostelitz-Thouless transition  [33].

Finally, for \(2 < \alpha\) we have very quickly decaying interactions and domain walls again have a finite energy penalty, hence Peirels’ argument holds and there is no phase transition.

-

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 \(\alpha\)   [39]. 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  [40].

+

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].

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. @@ -201,53 +202,62 @@ H_{\mathrm{FK}} = & \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\
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M. C. Angelini, G. Parisi, and F. Ricci-Tersenghi, Relations Between Short-Range and Long-Range Ising Models, Phys. Rev. E 89, 062120 (2014).
diff --git a/_thesis/2_Background/2.2_HKM_Model.html b/_thesis/2_Background/2.2_HKM_Model.html index 542f4f7..f0bfabd 100644 --- a/_thesis/2_Background/2.2_HKM_Model.html +++ b/_thesis/2_Background/2.2_HKM_Model.html @@ -34,7 +34,7 @@ image:
  • The Majorana Model
  • The Fermion Problem
  • An Emergent Gauge Field
    • -
  • Anyons, Topology and the Chern number
    • +
  • Anyons, Topology and the Chern number
  • Ground State Phases
  • Bibliography
    • @@ -57,7 +57,7 @@ image:
  • The Majorana Model
  • The Fermion Problem
  • An Emergent Gauge Field
    • -
  • Anyons, Topology and the Chern number
    • +
  • Anyons, Topology and the Chern number
  • Ground State Phases
  • Bibliography
    • @@ -81,8 +81,7 @@ image:

    This section introduces the Kitaev honeycomb (KH) model. The KH model is an exactly solvable model of interacting spin\(-1/2\) spins on the vertices of a honeycomb lattice. Each bond in the lattice is assigned a label \(\alpha \in \{ x, y, z\}\) and that bond couple two spins along the \(\alpha\) axis. See fig. 1 for a diagram of the setup.

    This gives us the Hamiltonian \[ H = - \sum_{\langle j,k\rangle_\alpha} J^{\alpha}\sigma_j^{\alpha}\sigma_k^{\alpha}, \qquad{(1)}\] where \(\sigma^\alpha_j\) is the \(\alpha\) component of a Pauli matrix acting on site \(j\) and \(\langle j,k\rangle_\alpha\) is a pair of nearest-neighbour indices connected by an \(\alpha\)-bond with exchange coupling \(J^\alpha\). Kitaev introduced this model in his seminal 2006 paper  [1].

    The KH can arise as the result of strong spin-orbit couplings in, for example, the transition metal based compounds  [2–6]. The model is highly frustrated: each spin would like to align along a different direction with each of its three neighbours. This cannot be achieved even classically  [7,8]. This frustration leads the the model to have a quantum spin liquid (QSL) ground state, a complex many body state with a high degree of entanglement but no long range magnetic order even at zero temperature. While the possibility of a QSL ground state was suggested much earlier  [9], the KH model was one of the first concrete models of the QSL state. The KH model has a rich ground state phase diagram with gapless and gapped phases, the latter supporting fractionalised quasiparticles with both Abelian and non-Abelian quasiparticle excitations. Anyons have been the subject of much attention because, among other reasons, they can be braided through spacetime to achieve noise tolerant quantum computations  [10]. At finite temperature the KH model undergoes a phase transition to a thermal metal state  [11]. The KH model can be solved exactly via a mapping to Majorana fermions. This mapping yields an extensive number of static \(\mathbb Z_2\) fluxes tied to an emergent gauge field with the remaining fermions are governed by a free fermion hamiltonian.

    -

    This section will go over the standard model in detail, first discussing the spin model, then detailing the transformation to a Majorana hamiltonian that allows a full solution while enlarging the Hamiltonian. We will discuss the properties of the emergent gauge fields and the projector. We will then discuss the ground state found via Lieb’s theorem as well as work on generalisations of the ground state to other lattices. Finally we will look at the phase diagram.

    -

    The next section will discuss anyons, topology and the Chern number, using the Kitaev model as an explicit example of these topics.

    +

    This section will go over the standard model in detail, first discussing the spin model, then detailing the transformation to a Majorana hamiltonian that allows a full solution while enlarging the Hamiltonian. We will discuss the properties of the emergent gauge fields and the projector. The next section will discuss anyons, topology and the Chern number, using the Kitaev model as an explicit example of these topics. Finally will then discuss the ground state found via Lieb’s theorem as well as work on generalisations of the ground state to other lattices. Finally we will look at the phase diagram.

    The Spin Model

    @@ -109,7 +108,7 @@ image:

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

    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 $^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 \(P = \prod_i P_i\).

    +

    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 \(\tilde \sigma^x \tilde \sigma^y \tilde\sigma^z = iD\) 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 \(D_i|\phi\rangle = |\phi\rangle\) 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.

    \[\begin{aligned} @@ -151,23 +150,36 @@ H &= i \sum_{\langle i,j\rangle_\alpha} J^{\alpha} u_{ij} \hat{c}_i \hat{c} \end{aligned}\qquad{(5)}\]

    Thus we can interpret the fluxes \(\phi_i\) as the exponential of magnetic fluxes of some fictitious gauge field \(\vec{A}\) and the bond operators as \(u_{ij} = \exp i \int_i^j \vec{A} \cdot d\vec{l}\). In this analogy to classical electromagnetism, the sets \(\{u_ij\}\) that correspond to the same \(\{\phi_i\}\) are all gauge equivalent. The fact that fluxes can be written as products of bond operators and composed is a consequence of (the exponential of) Stokes’ theorem. The additional phase factors of \(i^n\) can be incorporated as additional \(\tfrac{\pi}{2}\) phases but then make little difference when all the plaquettes are hexagons or have an even number of sides. However if the lattice contains odd plaquettes, as in the Yao-Kivelson model  [22], the complex fluxes that appear are a sign that chiral symmetry has been broken.

    Understanding \(u_{ij}\) as a gauge field provides another way to understand the action of the projector. The local projector \(P_i = \frac{1 + D_i}{2}\) applied to a state constructs a superposition of the original state and the gauge equivalent state linked to it by flipping the three \(u_{ij}\) around site \(i\). The overall projector \(P = \prod_i P_i\) can thus be understood as a symmetrisation over all gauge equivalent states, removing the gauge degeneracy introduced by the mapping from spins to Majoranas.

    -

    A honeycomb lattice (in black) along with its dual (in red). (Left) The product of sets of D_j operators (Bold Vertices) can be used to construct arbitrary contractible loops that flip u_{ij} values. If we take the product of every D_j the boundary contracts to a point and disappears. This is a visual proof that \prod_i D_i \propto \mathbb{1}. This observation forms a key part of constructing an explicit expression for the projector, see appendix A.5. (Right) In black and red the edges and dual edges that must be flipped to add vortices at the sites highlighted in orange. Flipping all the plaquettes in the system is not equivalent to the identity. Not that the edges that must be flipped can always be chosen from a spanning tree since loops can always be removed by a gauge transformation. 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].

    + +
    +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.

    -
    +

    Anyons, Topology and the Chern number

    -Figure 5: Worldlines of particles in two dimensions can become tangled or braided with one another. - +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.

    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.

    -

    First we realise that in two dimensions, swapping identical particles twice is not topologically equivalent to the identity, see fig. 5. 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.

    +

    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.

    -
    -Figure 6: The different kinds of strings and loops that we can make by flipping bond variables or transporting vortices around. (a) Flipping a single bond u_{ij} makes a pair of vortices on either side. (b) Flipping a string of bonds separates the vortex pair spatially. The flipped bonds form a path (in red) on the dual lattice. (c) If we create a vortex-vortex pair, transport one of them around a loop and then annihilate them, we can change the bond sector without changing the vortex sector. This is a manifestation of the gauge symmetry of the bond sector. (d) If we transport a vortex around the major or minor axes of the torus, we create a non-contractable loop of bonds \hat{\mathcal{T}}_{x/y}. Unlike all the other dual loops, These operators cannot be constructed from the contractable loops created by D_j. operators and they flip the value of the topological fluxes. - -
    +

    In condensed matter systems, the existence of anyonic excitations automatically implies the system has topological ground state degeneracy on the torus  [25] and indeed anyons and topology are intimately linked  [26–28]. Originally a concept used to describe complex vector bundles in algebraic topology  [29], the Chern number has found use in physics as a powerful tool diagnostic tool for topological systems. Kitaev showed that vortices in the KH model are Abelian when the Chern number is even and non-Abelian when the Chern number is odd. In the odd case the non-Abelian statistics of the vortices arise due to unpaired Majorana modes that are bound to them.

    Recently, topological systems have gained interest because of proposals to use their ground state degeneracy to implement both passively fault tolerant and actively stabilised quantum computations  [30–32].

    @@ -182,16 +194,8 @@ 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.

    +

    To surmise, 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 dynamical spin-spin correlation functions are zero beyond nearest neighbour separation  [35]. 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]

    -

    Next Section: Disorder and Localisation

    diff --git a/_thesis/2_Background/2.4_Disorder.html b/_thesis/2_Background/2.4_Disorder.html index 31f7558..459bbe2 100644 --- a/_thesis/2_Background/2.4_Disorder.html +++ b/_thesis/2_Background/2.4_Disorder.html @@ -60,24 +60,25 @@ image:

    Disorder and Localisation

    -

    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.

    +

    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. In this model one would expect the electrical conductivity to be proportional to the mean free path  [1], decreasing smoothly as the number of defects increases. However, Anderson in 1958  [2] showed that in a simple model, there is some critical level of disorder at which 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].

    +

    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 \(W\). Later Mott showed that in other contexts extended Bloch states and localised states can 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].

    +\] Here, in contrast to the Anderson model, localisation phenomena are robust to weak perturbations of the Hamiltonian. This is called many-body localisation (MBL)  [13,14].

    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].

    +

    The concept of disorder-free localisation was first proposed in the context of Helium mixtures  [15] 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  [16,17] instead find that the models thermalise exponentially slowly in system size, which Ref.  [16] dubs Quasi-MBL.

    +

    True disorder-free localisation does occur in exactly solvable models with extensively many conserved quantities  [18]. 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  [19].

    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 anti-correlations 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 flux  [28] and bond  [29] disorder. In some instances it seems that disorder can even promote the formation of a QSL ground state  [30]. I will look at how adding lattice disorder to the mix affects the picture. It has also been shown that the KH model exhibits disorder free localisation after a quantum quench  [31].

    +

    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   [20–23]. 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. A standard method for generating such graphs with coordination number \(d+1\) is Voronoi tessellation  [24,25]. The Harris  [26] and the Imry-Mar  [27] 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 while the Imry-Ma criterion simply forbids the formation of long range ordered states in \(d \leq 2\) dimensions in the presence of disorder. Both these criteria are modified for the case of topological disorder where the Euler equation an vertex degree constraints lead to strong anti-correlations which mean that topological disorder is effectively weaker than standard disorder in two dimensions  [28,29]. This does not apply to the three dimensional Voronoi lattices where the Euler equation contains an extra volume term and so is effectively a weaker constraint.

    +

    Lastly it is worth exploring how quantum spin liquids and disorder interact. The KH model has been studied subject to both flux  [30] and bond  [31] disorder. In some instances it seems that disorder can even promote the formation of a QSL ground state  [32]. It has also been shown that the KH model exhibits disorder free localisation after a quantum quench  [33].

    +

    In chapter 4 we will put the Kitaev model onto two dimensional Voronoi lattices and show that much of the rich character of the model is preserved despite the lack of long range order.

    Diagnosing Localisation in practice

    @@ -91,17 +92,19 @@ H = -t\sum_{\langle jk \rangle} c^\dagger_j c_k + \sum_j V_j c_j^\dagger c_j + U

    \[ P^{-1} = \sum_i |\psi_i|^4 \]

    -

    The name derive from the fact that this operator acts as a measure of the volume where the wavefunction is significantly different from zero. They can alternatively be thougt of as providing a measure of the average diameter \(R\) from \(R = P^{1/d}\), see fig. 1 for the distinction between \(R\) and \(\lambda\).

    +

    The name derive from the fact that this operator acts as a measure of the volume where the wavefunction is significantly different from zero. They can alternatively be thought of as providing a measure of the average diameter \(R\) from \(R = P^{1/d}\). See fig. 1 for the distinction between \(R\) and \(\lambda\).

    For localised states, the inverse participation ratio \(P^{-1}\) is independent of system size while for plane wave states in \(d\) dimensions \(P^{-1} = L^{-d}\). States may also be intermediate between localised and extended, described by their fractal dimensionality \(d > d* > 0\):

    \[ P(L)^{-1} \sim L^{-d*} \]

    -

    For finite size systems, these relations only hold once the system size \(L\) is much greater than the localisation length. When the localisation length is comparable to the system size the states contribute to transport. This is called weak localisation  [32,33].

    +

    Such intermediate states tend to appear as critical phenomena near mobility edges  [34]. For finite size systems, these relations only hold once the system size \(L\) is much greater than the localisation length. When the localisation length is comparable to the system size the states contribute to transport. This is called weak localisation  [35,36].

    For extended states \(d* = 0\) while for localised ones \(d* = 0\). In both chapters I will use an energy resolved IPR \[ DOS(\omega) = \sum_n \delta(\omega - \epsilon_n)\\ IPR(\omega) = DOS(\omega)^{-1} \sum_{n,i} \delta(\omega - \epsilon_n) |\psi_{n,i}|^4 \] Where \(\psi_{n,i}\) is the wavefunction corresponding to the energy \(\epsilon_n\) at the ith site. In practice I bin the energies and IPRs into a fine energy grid and use the mean within each bin.

    -

    Next Chapter: 3 The Long Range Falikov-Kimball Model

    +

    Chapter Summary

    +

    In this chapter we have covered the Falicov-Kimball model, the Kitaev Honeycomb model and the theory of disorder and localisation. We saw that the FK model is one of immobile species (spins) interacting with an itinerant quantum species (electrons). While the KH model is specified in terms of spins on a honeycomb lattice interacting via a highly anisotropic Ising coupling, it can be transformed into one of Majorana fermions interacting with a classical gauge field that supports immobile flux excitations. In each case it is the immobile species that makes each model exactly solvable. Both models have rich ground state and thermodynamic phase diagrams. The last part of this chapter dealt with disorder and how it almost inevitably leads to localisation. Both the FK and KH models are effectively disordered at finite temperatures by their immobile species. In the next chapter we will look at a version of the FK model in one dimension augmented with long range interactions in order to retain its ordered phase. The model is translation invariant but we will see that it exhibits disorder free localisation. After that we will look at the KH model defined on an amorphous lattice with vertex degree \(z=3\).

    +

    Next Chapter: 3 The Long Range Falicov-Kimball Model

    @@ -146,65 +149,74 @@ IPR(\omega) = DOS(\omega)^{-1} \sum_{n,i} \delta(\omega - \epsilon_n) |\psi_{n,i
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    diff --git a/_thesis/3_Long_Range_Falicov_Kimball/3.1_LRFK_Model.html b/_thesis/3_Long_Range_Falicov_Kimball/3.1_LRFK_Model.html new file mode 100644 index 0000000..65b0413 --- /dev/null +++ b/_thesis/3_Long_Range_Falicov_Kimball/3.1_LRFK_Model.html @@ -0,0 +1,120 @@ +--- +title: The Long Range Falikov-Kimball Model - The Model +excerpt: +layout: none +image: + +--- + + + + + + + The Long Range Falikov-Kimball Model - The Model + + + + + + + + + + + +{% capture tableOfContents %} +
    + +{% endcapture %} + + +{% include header.html extra=tableOfContents %} + +
    + + + + + +
    +

    3 The Long Range Falicov-Kimball Model

    +
    +
    +
    +

    3 The Long Range Falicov-Kimball Model

    +

    Contributions

    +

    This chapter expands on work presented in

    +

      [1] One-dimensional long-range Falikov-Kimball model: Thermal phase transition and disorder-free localization, Hodson, T. and Willsher, J. and Knolle, J., Phys. Rev. B, 104, 4, 2021,

    +

    The code is available online  [2].

    +

    Johannes had the initial idea to use a long range Ising term to stabilise order in a one dimension Falikov-Kimball model. Josef developed a proof of concept during a summer project at Imperial along with Alexander Belcik. I wrote the simulation code and performed all the analysis presented here.

    +

    Chapter Summary

    +

    The paper is organised as follows. First, I will introduce the long range Falicov-Kimball (LRFK) model and motivate its definition. Second, I will present the methods used to solve it numerically, including Markov chain Monte Carlo and finite size scaling. I will then present and interpret the results obtained.

    +
    +
    +

    The Model

    +

    Dimensionality is crucial for the physics of both localisation and phase transitions. We have already seen that the one dimensional standard FK model cannot support an ordered phase at finite temperatures and therefore has no finite temperature phase transition (FTPT).

    +

    On bipartite lattices in dimensions 2 and above the FK model exhibits a finite temperature phase transition to an ordered charge density wave (CDW) phase  [3]. In this phase, the spins order anti-ferromagnetically, breaking the \(\mathbb{Z}_2\) symmetry. In 1D, however, Periel’s argument  [4,5] states that domain walls only introduce a constant energy penalty into the free energy while bringing a entropic contribution logarithmic in system size. Hence the 1D model does not have a finite temperature phase transition. However 1D systems are much easier to study numerically and admit simpler realisations experimentally. We therefore introduce a long-range coupling between the ions in order to stabilise a CDW phase in 1D.

    +

    We interpret the FK model as a model of spinless fermions, \(c^\dagger_{i}\), hopping on a 1D lattice against a classical Ising spin background, \(S_i \in {\pm \frac{1}{2}}\). The fermions couple to the spins via an onsite interaction with strength \(U\) which we supplement by a long-range interaction, \[ +J_{ij} = 4\kappa J\; (-1)^{|i-j|} |i-j|^{-\alpha}, +\]

    +

    between the spins. The additional coupling is very similar to that of the long range Ising model, it stabilises the Antiferromagnetic (AFM) order of the Ising spins which promotes the finite temperature CDW phase of the fermionic sector.

    +

    The hopping strength of the electrons, \(t = 1\), sets the overall energy scale and we concentrate throughout on the particle-hole symmetric point at zero chemical potential and half filling  [6].

    +

    \[\begin{aligned} +H_{\mathrm{FK}} = & \;U \sum_{i} S_i\;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) -\;t \sum_{i} (c^\dagger_{i}c_{i+1} + \textit{h.c.)}\\ +& + \sum_{i, j}^{N} J_{ij} S_i S_j +\label{eq:HFK}\end{aligned}\]

    +

    Without proper normalisation, the long range coupling would render the critical temperature strongly system size dependent for small system sizes. Within a mean field approximation the critical temperature scales with the effective coupling to all the neighbours of each site, which for a system with \(N\) sites is \(\sum_{i=1}^{N} i^{-\alpha}\). Hence, the normalisation \(\kappa^{-1} = \sum_{i=1}^{N} i^{-\alpha}\), renders the critical temperature independent of system size in the mean field approximation. This greatly improves the finite size behaviour of the model.

    +

    Taking the limit \(U = 0\) decouples the spins from the fermions, which gives a spin sector governed by a classical LRI model. Note, the transformation of the spins \(S_i \to (-1)^{i} S_i\) maps the AFM model to the FM one. As discussed in the background section, Peierls’ classic argument can be extended to long range couplings to show that, for the 1D LRI model, a power law decay of \(\alpha < 2\) is required for a FTPT as the energy of defect domain then scales with the system size and can overcome the entropic contribution. A renormalisation group analysis supports this finding and shows that the critical exponents are only universal for \(\alpha \leq 3/2\)  [7–9]. In the following, we choose \(\alpha = 5/4\) to avoid the additional complexity of non-universal critical points.

    +

    Next Section: Methods

    +
    +
    +

    Bibliography

    +
    +
    +
    [1]
    T. Hodson, J. Willsher, and J. Knolle, One-Dimensional Long-Range Falikov-Kimball Model: Thermal Phase Transition and Disorder-Free Localization, Phys. Rev. B 104, 045116 (2021).
    +
    +
    +
    [2]
    T. Hodson, Markov Chain Monte Carlo for the Kitaev Model, (2021).
    +
    +
    +
    [3]
    M. M. Maśka and K. Czajka, Thermodynamics of the Two-Dimensional Falicov-Kimball Model: A Classical Monte Carlo Study, Phys. Rev. B 74, 035109 (2006).
    +
    +
    +
    [4]
    R. Peierls, On Ising’s Model of Ferromagnetism, Mathematical Proceedings of the Cambridge Philosophical Society 32, 477 (1936).
    +
    +
    +
    [5]
    T. Kennedy and E. H. Lieb, An Itinerant Electron Model with Crystalline or Magnetic Long Range Order, Physica A: Statistical Mechanics and Its Applications 138, 320 (1986).
    +
    +
    +
    [6]
    C. Gruber and N. Macris, The Falicov-Kimball Model: A Review of Exact Results and Extensions, Helvetica Physica Acta 69, (1996).
    +
    +
    +
    [7]
    D. Ruelle, Statistical Mechanics of a One-Dimensional Lattice Gas, Comm. Math. Phys. 9, 267 (1968).
    +
    +
    +
    [8]
    D. J. Thouless, Long-Range Order in One-Dimensional Ising Systems, Phys. Rev. 187, 732 (1969).
    +
    +
    +
    [9]
    M. C. Angelini, G. Parisi, and F. Ricci-Tersenghi, Relations Between Short-Range and Long-Range Ising Models, Phys. Rev. E 89, 062120 (2014).
    +
    +
    +
    + + +
    + + diff --git a/_thesis/3_Long_Range_Falicov_Kimball/3.2_LRFK_Methods.html b/_thesis/3_Long_Range_Falicov_Kimball/3.2_LRFK_Methods.html new file mode 100644 index 0000000..6f694db --- /dev/null +++ b/_thesis/3_Long_Range_Falicov_Kimball/3.2_LRFK_Methods.html @@ -0,0 +1,167 @@ +--- +title: The Long Range Falikov-Kimball Model - Methods +excerpt: +layout: none +image: + +--- + + + + + + + The Long Range Falikov-Kimball Model - Methods + + + + + + + + + + + +{% capture tableOfContents %} +
    + +{% endcapture %} + + +{% include header.html extra=tableOfContents %} + +
    + + + + + +
    +

    3 The Long Range Falicov-Kimball Model

    +
    +
    +
    +

    Methods

    +

    To evaluate thermodynamic averages I perform classical Markov Chain Monte Carlo random walks over the space of spin configurations of the LRFK model, at each step diagonalising the effective electronic Hamiltonian  [1]. Using a binder-cumulant method  [2,3], I demonstrate the model has a finite temperature phase transition when the interaction is sufficiently long ranged. In this section I will discuss the thermodynamics of the model and how they are amenable to an exact Markov Chain Monte Carlo method.

    +
    +

    Thermodynamics of the LRFK Model

    +
    +Figure 1: Two MCMC walks starting from the CDW state for a system with N = 100 sites and 10,000 MCMC steps but at a temperature close to but above the ordered state (left column) and much higher than it (right column). In this simulation only a single spin can be flipped per step according to the Metropolis-Hastings Algorithm. The staggered magnetisation m = N^{-1} \sum_i (-1)^i \; S_i order parameter is plotted below. At both temperatures the thermal average of m is zero, while the initial state has m = 1. The higher temperature allows the MCMC to converge more quickly and to fluctuate about the mean with a shorter autocorrelation time. t = 1, \alpha = 1.25, T = {2.5,5}, J = U = 5 + +
    +

    The classical Markov Chain Monte Carlo (MCMC) method which we discuss in the following allows us to solve our long-range FK model efficiently, yielding unbiased estimates of thermal expectation values.

    +

    Since the spin configurations are classical, the LRFK Hamiltonian can be split into a classical spin part \(H_s\) and an operator valued part \(H_c\).

    +

    \[\begin{aligned} +H_s& = - \frac{U}{2}S_i + \sum_{i, j}^{N} J_{ij} S_i S_j \\ +H_c& = \sum_i U S_i c^\dagger_{i}c_{i} -t(c^\dagger_{i}c_{i+1} + c^\dagger_{i+1}c_{i}) \end{aligned}\]

    +

    The partition function can then be written as a sum over spin configurations, \(\vec{S} = (S_0, S_1...S_{N-1})\):

    +

    \[\begin{aligned} +\mathcal{Z} = \mathrm{Tr} e^{-\beta H}= \sum_{\vec{S}} e^{-\beta H_s} \mathrm{Tr}_c e^{-\beta H_c} .\end{aligned}\]

    +

    The contribution of \(H_c\) to the grand canonical partition function can be obtained by performing the sum over eigenstate occupation numbers giving \(-\beta F_c[\vec{S}] = \sum_k \ln{(1 + e^{- \beta \epsilon_k})}\) where \({\epsilon_k[\vec{S}]}\) are the eigenvalues of the matrix representation of \(H_c\) determined through exact diagonalisation. This gives a partition function containing a classical energy which corresponds to the long-range interaction of the spins, and a free energy which corresponds to the quantum subsystem.

    +

    \[\begin{aligned} +\mathcal{Z} = \sum_{\vec{S}} e^{-\beta H_S[\vec{S}] - \beta F_c[\vec{S}]} = \sum_{\vec{S}} e^{-\beta E[\vec{S}]}\end{aligned}\]

    +
    +
    +

    Markov Chain Monte Carlo and Emergent Disorder

    +

    Classical MCMC defines a weighted random walk over the spin states \((\vec{S}_0, \vec{S}_1, \vec{S}_2, ...)\), such that the likelihood of visiting a particular state converges to its Boltzmann probability \(p(\vec{S}) = \mathcal{Z}^{-1} e^{-\beta E}\). Hence, any observable can be estimated as a mean over the states visited by the walk  [4–6],

    +

    \[\begin{aligned} +\label{eq:thermal_expectation} +\langle O \rangle & = \sum_{\vec{S}} p(\vec{S}) \langle O \rangle\\ + & = \sum_{i = 0}^{M} \langle O\rangle \pm \mathcal{O}(M^{-\tfrac{1}{2}}) +\end{aligned}\]

    +

    where the former sum runs over the entire state space while the later runs over all the state visited by a particular MCMC run.

    +

    \[\begin{aligned} +\langle O \rangle_{\vec{S}}& = \sum_{\nu} n_F(\epsilon_{\nu}) \langle O \rangle{\nu} +\end{aligned}\]

    +

    Where \(\nu\) runs over the eigenstates of \(H_c\) for a particular spin configuration and \(n_F(\epsilon) = \left(e^{-\beta\epsilon} + 1\right)^{-1}\) is the Fermi function.

    +

    The choice of the transition function for MCMC is under-determined as one only needs to satisfy a set of balance conditions for which there are many solutions  [7]. Here, we incorporate a modification to the standard Metropolis-Hastings algorithm  [8] gleaned from Krauth  [9].

    +

    The standard algorithm decomposes the transition probability into \(\mathcal{T}(a \to b) = p(a \to b)\mathcal{A}(a \to b)\). Here, \(p\) is the proposal distribution that we can directly sample from while \(\mathcal{A}\) is the acceptance probability. The standard Metropolis-Hastings choice is

    +

    \[\mathcal{A}(a \to b) = \min\left(1, \frac{p(b\to a)}{p(a\to b)} e^{-\beta \Delta E}\right)\;,\]

    +

    with \(\Delta E = E_b - E_a\). The walk then proceeds by sampling a state \(b\) from \(p\) and moving to \(b\) with probability \(\mathcal{A}(a \to b)\). The latter operation is typically implemented by performing a transition if a uniform random sample from the unit interval is less than \(\mathcal{A}(a \to b)\) and otherwise repeating the current state as the next step in the random walk. The proposal distribution is often symmetric so does not appear in \(\mathcal{A}\). Here, we flip a small number of sites in \(b\) at random to generate proposals, which is a symmetric proposal.

    +

    In our computations  [10] the modification to this algorithm that we employ is based on the observation that the free energy of the FK system is composed of a classical part which is much quicker to compute than the quantum part. Hence, we can obtain a computational speed up by first considering the value of the classical energy difference \(\Delta H_s\) and rejecting the transition if the former is too high. We only compute the quantum energy difference \(\Delta F_c\) if the transition is accepted. We then perform a second rejection sampling step based upon it. This corresponds to two nested comparisons with the majority of the work only occurring if the first test passes. This modified scheme has the acceptance function \[\mathcal{A}(a \to b) = \min\left(1, e^{-\beta \Delta H_s}\right)\min\left(1, e^{-\beta \Delta F_c}\right)\;.\]

    +

    For the model parameters used, we find that with our new scheme the matrix diagonalisation is skipped around 30% of the time at \(T = 2.5\) and up to 80% at \(T = 1.5\). We observe that for \(N = 50\), the matrix diagonalisation, if it occurs, occupies around 60% of the total computation time for a single step. This rises to 90% at N = 300 and further increases for larger N. We therefore get the greatest speedup for large system sizes at low temperature where many prospective transitions are rejected at the classical stage and the matrix computation takes up the greatest fraction of the total computation time. The upshot is that we find a speedup of up to a factor of 10 at the cost of very little extra algorithmic complexity.

    +

    Our two-step method should be distinguished from the more common method for speeding up MCMC which is to add asymmetry to the proposal distribution to make it as similar as possible to \(\min\left(1, e^{-\beta \Delta E}\right)\). This reduces the number of rejected states, which brings the algorithm closer in efficiency to a direct sampling method. However it comes at the expense of requiring a way to directly sample from this complex distribution, a problem which MCMC was employed to solve in the first place. For example, recent work trains restricted Boltzmann machines (RBMs) to generate samples for the proposal distribution of the FK model  [11]. The RBMs are chosen as a parametrisation of the proposal distribution that can be efficiently sampled from while offering sufficient flexibility that they can be adjusted to match the target distribution. Our proposed method is considerably simpler and does not require training while still reaping some of the benefits of reduced computation.

    +
    +
    +

    Scaling

    +
    +Figure 2: (Left) The order parameters, \langle m^2 \rangle(solid) and 1 - f (dashed) describing the onset of the charge density wave phase of the long-range 1D Falicov model at low temperature with staggered magnetisation m = N^{-1} \sum_i (-1)^i S_i and fermionic order parameter f = 2 N^{-1}|\sum_i (-1)^i \; \langle c^\dagger_{i}c_{i}| \rangle . (Right) The crossing of the Binder cumulant, B = \langle m^4 \rangle / \langle m^2 \rangle^2, with system size provides a diagnostic that the phase transition is not a finite size effect, it’s used to estimate the critical lines shown in the phase diagram fig. ¿fig:phase-diagram-lrfk?. All plots use system sizes N = [10,20,30,50,70,110,160,250] and lines are coloured from N = 10 in dark blue to N = 250 in yellow. The parameter values U = 5,\;J = 5,\;\alpha = 1.25 except where explicitly mentioned. + +
    +

    To improve the scaling of finite size effects, we make the replacement \(|i - j|^{-\alpha} \rightarrow |f(i - j)|^{-\alpha}\), in both \(J_{ij}\) and \(\kappa\), where \(f(x) = \frac{N}{\pi}\sin \frac{\pi x}{N}\). \(f\) is smooth across the circular boundary and its effect effect diminished for larger systems  [12]. We only consider even system sizes given that odd system sizes are not commensurate with a CDW state.

    +

    To identify critical points we use the the Binder cumulant \(U_B\) defined by

    +

    \[ +U_B = 1 - \frac{\langle\mu_4\rangle}{3\langle\mu_2\rangle^2} +\]

    +

    where \(\mu_n = \langle(m - \langle m\rangle)^n\rangle\) are the central moments of the order parameter \(m = \sum_i (-1)^i (2n_i - 1) / N\). The Binder cumulant evaluated against temperature is a diagnostic for the existence of a phase transition. If multiple such curves are plotted for different system sizes, a crossing indicates the location of a critical point while the lines do not cross for systems that don’t have a phase transition in the thermodynamic limit  [2,3].

    +

    Next Section: Results

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    +
    +
    +

    Bibliography

    +
    +
    +
    [1]
    M. M. Maśka and K. Czajka, Thermodynamics of the Two-Dimensional Falicov-Kimball Model: A Classical Monte Carlo Study, Phys. Rev. B 74, 035109 (2006).
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    [2]
    K. Binder, Finite Size Scaling Analysis of Ising Model Block Distribution Functions, Z. Physik B - Condensed Matter 43, 119 (1981).
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    [3]
    G. Musiał, L. Dȩbski, and G. Kamieniarz, Monte Carlo Simulations of Ising-Like Phase Transitions in the Three-Dimensional Ashkin-Teller Model, Phys. Rev. B 66, 012407 (2002).
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    [4]
    K. Binder and D. W. Heermann, Guide to Practical Work with the Monte Carlo Method, in Monte Carlo Simulation in Statistical Physics: An Introduction, edited by K. Binder and D. W. Heermann (Springer Berlin Heidelberg, Berlin, Heidelberg, 1988), pp. 68–112.
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    [5]
    J. Kertesz and I. Kondor, editors, Advances in Computer Simulation: Lectures Held at the Eötvös Summer School in Budapest, Hungary, 16–20 July 1996 (Springer-Verlag, Berlin Heidelberg, 1998).
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    [6]
    U. Wolff, Monte Carlo Errors with Less Errors, Computer Physics Communications 156, 143 (2004).
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    [7]
    F. P. Kelly, Reversibility and Stochastic Networks / F.P. Kelly, SERBIULA (Sistema Librum 2.0) 76, (1981).
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    [8]
    W. K. Hastings, Monte Carlo Sampling Methods Using Markov Chains and Their Applications, Biometrika 57, 97 (1970).
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    [9]
    W. Krauth, Introduction To Monte Carlo Algorithms, in Advances in Computer Simulation: Lectures Held at the Eötvös Summer School in Budapest, Hungary, 16–20 July 1996 (Springer-Verlag, Berlin Heidelberg, 1998).
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    [10]
    T. Hodson, Markov Chain Monte Carlo for the Kitaev Model, (2021).
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    [11]
    L. Huang and L. Wang, Accelerated Monte Carlo Simulations with Restricted Boltzmann Machines, Phys. Rev. B 95, 035105 (2017).
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    [12]
    K. Fukui and S. Todo, Order-N Cluster Monte Carlo Method for Spin Systems with Long-Range Interactions, Journal of Computational Physics 228, 2629 (2009).
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    + + +
    + + diff --git a/_thesis/3_Long_Range_Falicov_Kimball/3.3_LRFK_Results.html b/_thesis/3_Long_Range_Falicov_Kimball/3.3_LRFK_Results.html new file mode 100644 index 0000000..b16efbf --- /dev/null +++ b/_thesis/3_Long_Range_Falicov_Kimball/3.3_LRFK_Results.html @@ -0,0 +1,185 @@ +--- +title: The Long Range Falicov-Kimball Model - Results +excerpt: +layout: none +image: + +--- + + + + + + + The Long Range Falicov-Kimball Model - Results + + + + + + + + + + + +{% capture tableOfContents %} +
    + +{% endcapture %} + + +{% include header.html extra=tableOfContents %} + +
    + + + + + +
    +

    3 The Long Range Falicov-Kimball Model

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    +
    +

    Results

    +

    Looking at the results of our MCMC simulations, we find a rich phase diagram with a CDW FTPT and interaction-tuned Anderson versus Mott localized phases similar to the 2D FK model  [1]. We explore the localization properties of the fermionic sector and find that the localisation lengths vary dramatically across the phases and for different energies. Although moderate system sizes indicate the coexistence of localized and delocalized states within the CDW phase, we find quantitatively similar behaviour in a model of uncorrelated binary disorder on a CDW background. For large system sizes, i.e. for our 1D disorder model we can treat linear sizes of several thousand sites, we find that all states are eventually localized with a localization length which diverges towards zero temperature.

    +
    +Figure 1: Phase diagrams of the long-range 1D FK model. (Left) The TJ plane at U = 5: the CDW ordered phase is separated from a disordered Mott insulating phase by a critical temperature T_c, linear in J. (Right) The TU plane at J = 5: the disordered phase is split into two: at large/small U there’s a MI/Anderson phase characterised by the presence/absence of a gap at E=0 in the single particle energy spectrum. U_c is independent of temperature. At U = 0 the fermions are decoupled from the spins forming a simple Fermi gas. + +
    +
    +

    Phase Diagram

    +

    Using the MCMC methods described in the previous section I will now discuss the results of extensive MCMC simulations of the model, starting with the phase diagram in the fermion spin coupling \(U\), the strength of the long range spin-spin coupling \(J\) and the temperature \(T\).

    +

    Fig fig. 1 shows the phase diagram for constant \(U=5\) and constant \(J=5\), respectively. The transition temperatures were determined from the crossings of the Binder cumulants \(B_4 = \langle m^4 \rangle /\langle m^2 \rangle^2\)  [2].

    +

    The CDW transition temperature is largely independent from the strength of the interaction \(U\). This demonstrates that the phase transition is driven by the long-range term \(J\) with little effect from the coupling to the fermions \(U\). The physics of the spin sector in the long-range FK model mimics that of the long range Ising (LRI) model and is not significantly altered by the presence of the fermions. In two dimensions the transition to the CDW phase is mediated by an RKYY-like interaction  [3] but this is insufficient to stabilise long range order in one dimension. That the critical temperature \(T_c\) does not depend on \(U\) in our model further confirms this.

    +

    The main order parameters for this model is the staggered magnetisation \(m = N^{-1} \sum_i (-1)^i S_i\) that signals the onset of a charge density wave (CDW) phase at low temperature. However, my main interest concerns the additional structure of the fermionic sector in the high temperature phase. Following Ref.  [1], we can distinguish between the Mott and Anderson insulating phases. The Mott insulator is characterised by a gapped DOS in the absence of a CDW, instead the gap is driven entirely by interaction effect. Thus, the opening of a gap for large \(U\) is distinct from the gap-opening induced by the translational symmetry breaking in the CDW state below \(T_c\). The Anderson phase is gapless but, as we explain below, shows localised fermionic eigenstates hence it also has insulating character.

    +
    +
    +

    Localisation Properties

    +
    +Figure 2: Energy resolved DOS(\omega) against system size N in all three phases. The charge density wave phase is shown in both the high and low U regime for completeness. The top left panel shows the Anderson phase at U = 2 and high T = 2.5, this phase is gapless but does not conduct due to Anderson localisation. In the lower left pane at U = 2 and low T = 1.5, charge density wave order sets in, allowing the single particle eigenstates to become extended but opening a gap in their band structure. In the upper right panel at U = 5 and high T = 2.5 the states are localised by disorder and an interaction driven gap opens, a Mott insulator. Finally the charge density wave phase at U = 5 and T = 1.5 is qualitatively similar to the lower left panel except that the gap scales with U. For all the plots J = 5,\;\alpha = 1.25. + +
    +

    The MCMC formulation suggests viewing the spin configurations as a form of annealed binary disorder whose probability distribution is given by the Boltzmann weight \(e^{-\beta H_S[\vec{S}] - \beta F_c[\vec{S}]}\). This makes apparent the link to the study of disordered systems and Anderson localisation. While these systems are typically studied by defining the probability distribution for the quenched disorder potential externally, here we have a translation invariant system with disorder as a natural consequence of the Ising background field conserved under the dynamics.

    +

    In the limits of zero and infinite temperature, our model becomes a simple tight-binding model for the fermions. At zero temperature, the spin background is in one of the two translation invariant AFM ground states with two gapped fermionic CDW bands at energies \[E_{\pm} = \pm\sqrt{\frac{1}{4}U^2 + 2t^2(1 + \cos ka)^2}\;.\]

    +

    At infinite temperature, all the spin configurations become equally likely and the fermionic model reduces to one of binary uncorrelated disorder in which all eigenstates are Anderson localised  [4]. An Anderson localised state centered around \(r_0\) has magnitude that drops exponentially over some localisation length \(\xi\) i.e \(|\psi(r)|^2 \sim \exp{-|r - r_0|/\xi}\). Calculating \(\xi\) directly is numerically demanding. Therefore, we determine if a given state is localised via the energy-resolved IPR and the DOS defined as \[\begin{aligned} +\mathrm{DOS}(\vec{S}, \omega)& = N^{-1} \sum_{i} \delta(\epsilon_i - \omega)\\ +\mathrm{IPR}(\vec{S}, \omega)& = \; N^{-1} \mathrm{DOS}(\vec{S}, \omega)^{-1} \sum_{i,j} \delta(\epsilon_i - \omega)\;\psi^{4}_{i,j}\end{aligned}\] where \(\epsilon_i\) and \(\psi_{i,j}\) are the \(i\)th energy level and \(j\)th element of the corresponding eigenfunction, both dependent on the background spin configuration \(\vec{S}\).

    +
    +Figure 3: The IPR(\omega) scaling with N at fixed energy for each phase and for points both in the gap (\omega_0) and in a band (\omega_1). The slope of the line yields the scaling exponent \tau defined by \mathrm{IPR} \propto N^{-\tau}). \tau close to zero implies that the states at that energy are localised while \tau = -d corresponds to extended states where d is the system dimension. All but the bands of the charge density wave phase are approximately localised with \tau is very close to zero. The bands in the charge density wave phase are localised with long localisation lengths at finite temperatures that extend to infinity as the temperature approaches zero. For all the plots J = 5,\;\alpha = 1.25. The measured \tau_0,\tau_1 for each figure are: (a) (0.06\pm0.01, 0.02\pm0.01 (b) 0.04\pm0.02, 0.00\pm0.01 (c) 0.05\pm0.03, 0.30\pm0.03 (d) 0.06\pm0.04, 0.15\pm0.05 We show later that the apparent slight scaling of the IPR with system size in the localised cases can be explained by finite size effects due to the changing defect density with system size rather than due to delocalisation of the states. + +
    +

    The scaling of the IPR with system size

    +

    \[\mathrm{IPR} \propto N^{-\tau}\]

    +

    depends on the localisation properties of states at that energy. For delocalised states, e.g. Bloch waves, \(\tau\) is the physical dimension. For fully localised states \(\tau\) goes to zero in the thermodynamic limit. However, for special types of disorder such as binary disorder, the localisation lengths can be large comparable to the system size at hand, which can make it difficult to extract the correct scaling. An additional complication arises from the fact that the scaling exponent may display intermediate behaviours for correlated disorder and in the vicinity of a localisation-delocalisation transition  [5,6]. The thermal defects of the CDW phase lead to a binary disorder potential with a finite correlation length, which in principle could result in delocalized eigenstates.

    +

    The key question for our system is then: How is the \(T=0\) CDW phase with fully delocalized fermionic states connected to the fully localized phase at high temperatures?

    +

    For a representative set of parameters covering all three phases fig. 2 shows the density of states as function of energy while fig. 3 shows \(\tau\), the scaling exponent of the IPR with system size, The DOS is symmetric about \(0\) because of the particle hole symmetry of the model. At high temperatures, all of the eigenstates are localised in both the Mott and Anderson phases (with \(\tau \leq 0.07\) for our system sizes). We also checked that the states are localised by direct inspection. Note that there are in-gap states for instance at \(\omega_0\), below the upper band which are localized and smoothly connected across the phase transition.

    +
    +Figure 4: The DOS (a) and scaling exponent \tau (b) as a function of energy for the CDW to gapped Mott phase transition at U=5. Regions where the DOS is close to zero are shown in white. The scaling exponent \tau is obtained from fits to IPR(N) = A N^{-\lambda} for a range of system sizes. J = 5,\;\alpha = 1.25 + +
    +

    In the CDW phases at \(U=2\) and \(U=5\), we find for the states within the gapped CDW bands, e.g. at \(\omega_1\), scaling exponents \(\tau = 0.30\pm0.03\) and \(\tau = 0.15\pm0.05\), respectively. This surprising finding suggests that the CDW bands are partially delocalised with multi-fractal behaviour of the wavefunctions  [6]. This phenomenon would be unexpected in a 1D model as they generally do not support delocalisation in the presence of disorder except as the result of correlations in the emergent disorder potential  [7,8]. However, we later show by comparison to an uncorrelated Anderson model that these nonzero exponents are a finite size effect and the states are localised with a finite \(\xi\) similar to the system size, an example of weak localisation. As a result, the IPR does not scale correctly until the system size has grown much larger than \(\xi\). fig. 7 shows that the scaling of the IPR in the CDW phase does flatten out eventually.

    +

    Next, we use the DOS and the scaling exponent \(\tau\) to explore the localisation properties over the energy-temperature plane in fig. 5 and fig. 4. Gapped areas are shown in white, which highlights the distinction between the gapped Mott phase and the ungapped Anderson phase. In-gap states appear just below the critical point, smoothly filling the bandgap in the Anderson phase and forming islands in the Mott phase. As in the finite  [9] and infinite dimensional  [10] cases, the in-gap states merge and are pushed to lower energy for decreasing U as the \(T=0\) CDW gap closes. Intuitively, the presence of in-gap states can be understood as a result of domain wall fluctuations away from the AFM ordered background. These domain walls act as local potentials for impurity-like bound states  [9].

    +
    +Figure 5: The DOS (a) and scaling exponent \tau (b) as a function of energy for the CDW phase to the gapless Anderson insulating phase at U=2. Regions where the DOS is close to zero are shown in white. The scaling exponent \tau is obtained from fits to IPR(N) = A N^{-\lambda} for a range of system sizes. J = 5,\;\alpha = 1.25 + +
    +

    In order to understand the localization properties we can compare the behaviour of our model with that of a simpler Anderson disorder model (DM) in which the spins are replaced by a CDW background with uncorrelated binary defect potentials. This is defined by replacing the spin degree of freedom in the FK model \(S_i = \pm \tfrac{1}{2}\) with a disorder potential \(d_i = \pm \tfrac{1}{2}\) controlled by a defect density \(\rho\) such that \(d_i = -\tfrac{1}{2}\) with probability \(\rho/2\) and \(d_i = \tfrac{1}{2}\) otherwise. \(\rho/2\) is used rather than \(\rho\) so that the disorder potential takes on the zero temperature CDW ground state at \(\rho = 0\) and becomes a random choice over spin states at \(\rho = 1\) i.e the infinite temperature limit.

    +

    \[\begin{aligned} +H_{\mathrm{DM}} = & \;U \sum_{i} (-1)^i \; d_i \;(c^\dagger_{i}c_{i} - \tfrac{1}{2}) \\ +& -\;t \sum_{i} c^\dagger_{i}c_{i+1} + c^\dagger_{i+1}c_{i} +\end{aligned}\]

    +

    fig. 6 and fig. 7 compare the FK model to the disorder model at different system sizes, matching the defect densities of the disorder model to the FK model at \(N = 270\) above and below the CDW transition. We find very good, even quantitative, agreement between the FK and disorder models, which suggests that correlations in the spin sector do not play a significant role.

    +
    +Figure 6: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density 0 < \rho < 1 matched to the \rho for the largest corresponding FK model. As in fig. 2, the Energy resolved DOS(\omega) is shown. The DOSs match well implying that correlations in the CDW wave fluctuations are not relevant at these system parameters. + +
    +

    As we can sample directly from the disorder model, rather than through MCMC, the samples are uncorrelated. Hence we can evaluate much larger system sizes with the disorder model which enables us to pin down the correct localisation effects. In particular, what appear to be delocalized states for small system sizes eventually turn out to be states with large localization length. The localization length diverges towards the ordered zero temperature CDW state. The interplay of interactions, which here produce as peculiar binary potential, and localization can be very intricate and the added advantage of a 1D model is that we can explore very large system sizes.

    +
    +Figure 7: A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density 0 < \rho < 1 matched to the \rho for the largest corresponding FK model. As in fig. 3 \tau(\omega) the scaling of IPR(\omega) with system size, , is shown both in gap (\omega_0) and in the band (\omega_1). This data makes clear that the apparent scaling of IPR with system size at small sysis a finite size effect due to weak localisation  [1], hence all the states are indeed localised as one would expect in 1D. The disorder model \tau_0,\tau_1 for each figure are: (a) 0.01\pm0.05, -0.02\pm0.06 (b) 0.01\pm0.04, -0.01\pm0.04 (c) 0.05\pm0.06, 0.04\pm0.06 (d) -0.03\pm0.06, 0.01\pm0.06. The lines are fit on system sizes N > 400 + +
    +
    +
    +
    +

    Discussion and Conclusion

    +

    The FK model is one of the simplest non-trivial models of interacting fermions. We studied its thermodynamic and localisation properties brought down in dimensionality to one dimension by adding a novel long-ranged coupling designed to stabilise the CDW phase present in dimension two and above.

    +

    Our MCMC approach emphasises the presence of a disorder-free localization mechanism within our translationally invariant system. Further, it gives a significant speed up over the naive method. We show that our LRFK model retains much of the rich phase diagram of its higher dimensional cousins. Careful scaling analysis indicates that all the single particle eigenstates eventually localise at non-zero temperature albeit only for very large system sizes of several thousand.

    +

    Our work raises a number of interesting questions for future research. A straightforward but numerically challenging problem is to pin down the model’s behaviour closer to the critical point where correlations in the spin sector would become significant. Would this modify the localisation behaviour? Similar to other soluble models of disorder-free localisation, we expect intriguing out-of equilibrium physics, for example slow entanglement dynamics akin to more generic interacting systems  [11]. One could also investigate whether the rich ground state phenomenology of the FK model as a function of filling  [12] such as the devil’s staircase  [13] as well as superconductor like states  [14] could be stabilised at finite temperature.

    +

    In a broader context, we envisage that long-range interactions can also be used to gain a deeper understanding of the temperature evolution of topological phases. One example would be a long-ranged FK version of the celebrated Su-Schrieffer-Heeger model where one could explore the interplay of topological bound states and thermal domain wall defects. Finally, the rich physics of our model should be realizable in systems with long-range interactions, such as trapped ion quantum simulators, where one can also explore the fully interacting regime with a dynamical background field.

    +

    Next Chapter: 4 The Amorphous Kitaev Model

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    +
    +

    Bibliography

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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 af8d5ac..59629de 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: