From 5a856d2182ca8741efe732d17f4ac3d3dfc82cf4 Mon Sep 17 00:00:00 2001 From: Tom Hodson Date: Tue, 20 Sep 2022 10:59:29 +0100 Subject: [PATCH] add localisation stuff --- _thesis/2_Background/2.4_Disorder.html | 64 +- .../3.2_LRFK_Methods.html | 42 +- .../3.3_LRFK_Results.html | 41 +- .../localisation_radius_vs_length.svg | 2109 +++++++++++++++++ 4 files changed, 2184 insertions(+), 72 deletions(-) create mode 100644 assets/thesis/background_chapter/localisation_radius_vs_length.svg diff --git a/_thesis/2_Background/2.4_Disorder.html b/_thesis/2_Background/2.4_Disorder.html index d0ed54d..31f7558 100644 --- a/_thesis/2_Background/2.4_Disorder.html +++ b/_thesis/2_Background/2.4_Disorder.html @@ -29,8 +29,7 @@ image: @@ -47,8 +46,7 @@ image: @@ -62,11 +60,6 @@ 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.

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 @@ -83,17 +76,31 @@ H = -t\sum_{\langle jk \rangle} c^\dagger_j c_k + \sum_j V_j c_j^\dagger c_j + U

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

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

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

-

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

-

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

- -
-

Disorder and Spin liquids

-
-
-

Amorphous Magnetism

+

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].

+
+

Diagnosing Localisation in practice

+
+Figure 1: A localised state \psi in an potential well that has formed from random fluctuations in the disorder potential V(x). The localisation length \lambda governs how quickly the state decays away from the well while the diameter R of the state is controlled by the size of the well. Reproduced from  [6]. + +
+

Looking at practical tools for diagnosing localisation, there are a few standard methods  [6].

+

The most direct method would be to fit a function of the form \(\psi(x) = f(x) e^{-|x-x_0|/\lambda}\) to each single particle wavefunction to extract the localisation length \(\lambda\). This method is little used in practice since it requires storing and processing full wavefunctions which quickly becomes expensive for large systems.

+

For low dimensional systems with quenched disorder, transmission matrix methods can be used to directly extract the localisation length. These work by turning the time independent Schrödinger equation \(\hat{H}|\psi\rangle = E|\psi\rangle\) into a matrix equation linking the amplitude of \(\psi\) on each \(d-1\) dimensional slice of the system to the next and looking at average properties of this transmission matrix. This method is less useful for systems like the FK model where the disorder as a whole must be sampled from the thermodynamic ensemble. It is also problematic for the Kitaev Model on an amorphous lattice as the slicing procedure is complex to define in the absence of a regular lattice.

+

A more versatile method is based on the inverse participation ratio. The inverse participation ratio is defined for a normalised wave function \(\psi_i = \psi(x_i), \sum_i |\psi_i|^2 = 1\) as its fourth moment  [6]:

+

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

+

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 \(\lambda\).

+

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].

+

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

@@ -181,8 +188,23 @@ H = -t\sum_{\langle jk \rangle} c^\dagger_j c_k + \sum_j V_j c_j^\dagger c_j + U
[27]
M. Schrauth, J. Portela, and F. Goth, Violation of the Harris-Barghathi-Vojta Criterion, Physical Review Letters 121, (2018).
+
+
[28]
J. Nasu, M. Udagawa, and Y. Motome, Thermal Fractionalization of Quantum Spins in a Kitaev Model: Temperature-Linear Specific Heat and Coherent Transport of Majorana Fermions, Phys. Rev. B 92, 115122 (2015).
+
+
+
[29]
J. Knolle, Dynamics of a Quantum Spin Liquid, Max Planck Institute for the Physics of Complex Systems, Dresden, 2016.
+
-
[28]
J.-J. Wen et al., Disordered Route to the Coulomb Quantum Spin Liquid: Random Transverse Fields on Spin Ice in ${\Mathrm{Pr}}_{2}{\mathrm{Zr}}_{2}{\mathrm{O}}_{7}$, Phys. Rev. Lett. 118, 107206 (2017).
+
[30]
J.-J. Wen et al., Disordered Route to the Coulomb Quantum Spin Liquid: Random Transverse Fields on Spin Ice in ${\Mathrm{Pr}}_{2}{\mathrm{Zr}}_{2}{\mathrm{O}}_{7}$, Phys. Rev. Lett. 118, 107206 (2017).
+
+
+
[31]
G.-Y. Zhu and M. Heyl, Subdiffusive Dynamics and Critical Quantum Correlations in a Disorder-Free Localized Kitaev Honeycomb Model Out of Equilibrium, Phys. Rev. Research 3, L032069 (2021).
+
+
+
[32]
B. L. Altshuler, D. Khmel’nitzkii, A. I. Larkin, and P. A. Lee, Magnetoresistance and Hall Effect in a Disordered Two-Dimensional Electron Gas, Phys. Rev. B 22, 5142 (1980).
+
+
+
[33]
S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge University Press, Cambridge, 1995).
diff --git a/_thesis/3_Long_Range_Falikov_Kimball/3.2_LRFK_Methods.html b/_thesis/3_Long_Range_Falikov_Kimball/3.2_LRFK_Methods.html index 58e65c4..fc53e24 100644 --- a/_thesis/3_Long_Range_Falikov_Kimball/3.2_LRFK_Methods.html +++ b/_thesis/3_Long_Range_Falikov_Kimball/3.2_LRFK_Methods.html @@ -35,10 +35,6 @@ image:
  • Two Step Trick
  • Scaling
  • Binder Cumulants
    • -
  • Diagnostics of Localisation -
  • Convergence Time
  • Auto-correlation Time
    • @@ -63,10 +59,6 @@ image:
  • Two Step Trick
  • Scaling
  • Binder Cumulants
    • -
  • Diagnostics of Localisation -
  • Convergence Time
  • Auto-correlation Time
    • @@ -136,27 +128,20 @@ H_c& = \sum_i U S_i c^\dagger_{i}c_{i} -t(c^\dagger_{i}c_{i+1} + c^\dagger_{

    In order to reduce the effects of the boundary conditions and the finite size of the system we redefine and normalise the coupling matrix to have 0 derivative at its furthest extent rather than cutting off abruptly.

    \[ \begin{aligned} -J'(x) &= \abs{\frac{L}{\pi}\sin \frac{\pi x}{L}}^{-\alpha} \\ +J'(x) &= \frac{L}{\pi}\left|\;\sin \frac{\pi x}{L}\;\right|^{-\alpha} \\ J(x) &= \frac{J_0 J'(x)}{\sum_y J'(y)} -\end{aligned}\] % The scaling ensures that, in the ordered phase, the overall potential felt by each site due to the rest of the system is independent of system size.

    +\end{aligned} +\]

    +

    The scaling ensures that, in the ordered phase, the overall potential felt by each site due to the rest of the system is independent of system size.

    Binder Cumulants

    -

    The Binder cumulant is defined as: \[U_B = 1 - \frac{\tex{\mu_4}}{3\tex{\mu_2}^2}\] % where \[\mu_n = \tex{(m - \tex{m})^n}\] % are the central moments of the order parameter m: \[m = \sum_i (-1)^i (2n_i - 1) / N\] % The Binder cumulant evaluated against temperature can be used as 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  [9,10].

    -
    -
    -

    Diagnostics of Localisation

    -
    -

    Inverse Participation Ratio

    -

    The inverse participation ratio is defined for a normalised wave function \(\psi_i = \psi(x_i), \sum_i \abs{\psi_i}^2 = 1\) as its fourth moment  [11]: \[ -P^{-1} = \sum_i \abs{\psi_i}^4 -\] % It acts as a measure of the portion of space occupied by the wave function. For localised states it will be independent of system size while for plane wave states in d dimensions $P = L^d $. States may also be intermediate between localised and extended, described by their fractal dimensionality \(d > d* > 0\): \[ -P(L) \goeslike L^{d*} -\] % For extended states \(d* = 0\) while for localised ones \(d* = 0\). In this work we take use an energy resolved IPR  [12]: \[ -DOS(\omega) = \sum_n \delta(\omega - \epsilon_n) -IPR(\omega) = DOS(\omega)^{-1} \sum_{n,i} \delta(\omega - \epsilon_n) \abs{\psi_{n,i}}^4 -\] Where \(\psi_{n,i}\) is the wavefunction corresponding to the energy \(\epsilon_n\) at the ith site. In practice we bin the energies and IPRs into a fine energy grid and use Lorentzian smoothing if necessary.

    - +

    The Binder cumulant is defined as:

    +

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

    +

    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  [9,10].

    +

    (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? . 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.

    MCMC sidesteps these issues by defining a random walk that focuses on the states with the greatest Boltzmann weight. At low temperatures this means we need only visit a few low energy states to make good estimates while at high temperatures the weights become uniform so a small number of samples distributed across the state space suffice. However we will see that the method is not without difficulties of its own.

    In implementation MCMC can be boiled down to choosing a transition function \(\mathcal{T}(\s_{t} \rightarrow \s_t+1)\) where \(\s\) are vectors representing classical spin configurations. We start in some initial state \(\s_0\) and then repeatedly jump to new states according to the probabilities given by \(\mathcal{T}\). This defines a set of random walks \(\{\s_0\ldots \s_i\ldots \s_N\}\). Fig. 2 shows this in practice: we have a (rather small) ensemble of \(M = 2\) walkers starting at the same point in state space and then spreading outwards by flipping spins along the way.

    @@ -166,7 +151,6 @@ IPR(\omega) = DOS(\omega)^{-1} \sum_{n,i} \delta(\omega - \epsilon_n) \abs{\psi_

    Where the sample_T function here produces a state with probability determined by the current_state and the transition function \(\mathcal{T}\).

    If we ran many such walkers in parallel we could then approximate the distribution \(p_t(\s; \s_0)\) which tells us where the walkers are likely to be after they’ve evolved for \(t\) steps from an initial state \(\s_0\). We need to carefully choose \(\mathcal{T}\) such that after a large number of steps \(k\) (the convergence time) the probability \(p_t(\s;\s_0)\) approaches the thermal distribution \(P(\s; \beta) = \mathcal{Z}^{-1} e^{-\beta F(\s)}\). This turns out to be quite easy to achieve using the Metropolis-Hasting algorithm.

    -

    Convergence Time

    Considering \(p(\s)\) as a vector \(\vec{p}\) whose jth entry is the probability of the jth state \(p_j = p(\s_j)\), and writing \(\mathcal{T}\) as the matrix with entries \(T_{ij} = \mathcal{T}(\s_j \rightarrow \s_i)\) we can write the update rule for the ensemble probability as: \[\vec{p}_{t+1} = \mathcal{T} \vec{p}_t \implies \vec{p}_{t} = \mathcal{T}^t \vec{p}_0\] where \(\vec{p}_0\) is vector which is one on the starting state and zero everywhere else. Since all states must transition to somewhere with probability one: \(\sum_i T_{ij} = 1\).

    @@ -214,12 +198,6 @@ IPR(\omega) = DOS(\omega)^{-1} \sum_{n,i} \delta(\omega - \epsilon_n) \abs{\psi_
    [10]
    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).
    -
    -
    [11]
    B. Kramer and A. MacKinnon, Localization: Theory and Experiment, Rep. Prog. Phys. 56, 1469 (1993).
    -
    -
    -
    [12]
    P. W. Anderson, Absence of Diffusion in Certain Random Lattices, Phys. Rev. 109, 1492 (1958).
    -
    diff --git a/_thesis/3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html b/_thesis/3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html index 607a70c..9bf1d31 100644 --- a/_thesis/3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html +++ b/_thesis/3_Long_Range_Falikov_Kimball/3.3_LRFK_Results.html @@ -67,23 +67,19 @@ image:

    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 \(T\).

    -
    -Figure 1: (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. 2 . 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. - -
    -

    Fig fig. 2 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 \(B_4 = \langle m^4 \rangle /\langle m^2 \rangle^2\)  [1]. For a representative set of parameters, fig. 1 shows the order parameter \(\langle m \rangle^2\) and the Binder cumulants, both as functions of system size and temperature. The crossings confirm that the system has a FTPT and that the ordered phase is not a finite size effect.

    +

    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 \(B_4 = \langle m^4 \rangle /\langle m^2 \rangle^2\)  [1]. For a representative set of parameters, fig. ¿fig:binder_cumulants? shows the order parameter \(\langle m \rangle^2\) and the Binder cumulants, both as functions of system size and temperature. The crossings confirm that the system has a FTPT and that the ordered phase is not a finite size effect.

    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  [2] 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.  [3], 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 \(T_c\), see also fig. ¿fig:gap_opening?. The Anderson phase is gapless but, as we explain below, shows localised fermionic eigenstates hence it also has insulating character.

    -Figure 2: 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. - +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. +

    Localisation Properties

    -Figure 3: 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. - +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 \[E_{\pm} = \pm\sqrt{\frac{1}{4}U^2 + 2t^2(1 + \cos ka)^2}\;.\]

    @@ -91,32 +87,39 @@ image: \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 4: 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. - +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. 3 shows the density of states as function of energy while fig. 4 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.

    +

    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 5: 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 - +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. ¿fig:DM_IPR_scaling? shows that the scaling of the IPR in the CDW phase does flatten out eventually.

    +

    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. ¿fig:gap_opening?. 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 6: 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 - +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. ¿fig:DM_DOS? and fig. ¿fig:DM_IPR_scaling? 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.

    +

    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.

    -

    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, 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. 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. 4 \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  [3], 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

    +
    +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  [3], 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 + +
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