diff --git a/_thesis/0.1_Intro.html b/_thesis/0.1_Intro.html index 3868950..c434053 100644 --- a/_thesis/0.1_Intro.html +++ b/_thesis/0.1_Intro.html @@ -250,6 +250,15 @@ image: 
 
  • localisation
    • 
 
  • lengthscales
    • 
 
+
    Condsened Matter Systems
    
+
    Spin-Orbit Coupling
    
+
    Electronic wavefunctions can be understood as quantum extensions
+of
    
+
    This can be loosely understood as a consequence of that fact that
+electrons are ‘orbiting’ their host nucleus and in doing so they are
+moving with respect to an electric field generated by the positive
+charge of the nucleus. The electric field looks like a magnetic field in
+the rest frame of the electron and this magnetic field couples to the
+magnetic spin moment of the electron.
    
+
    This analogy is wrong on many levels but it suffices to understand
+that there should be such an effect.
    
+
    Going one level deeper we can estimate the scale of the effect by
+combining the non-relativistic quantum theory of a spin in a magnetic
+field with the classical relativistic electromagnetism prediction for
+how the electric field turns into a magnetic field in the rest frame of
+the electron. This gets us within a factor to two of the correct answer
+but it fails to account for an extra relativistic effect called Thomas
+Precession cite.
    
+
    The next level would be to compute this effect within relativistic QM
+using the Dirac equation. And finally, we could do the full calculation
+within Quantum Electrodynamics where we would find tiny corrections that
+come about from virtual processes involving particle-antiparticle pairs
+that spring form from the vacuum.
    
+
    Electronic
+correlations: The Hubbard Model
    
+
    
+
+
+
    
+
    These are easiest to understand within the context of the Hubbard
+model, if we take spin \(1/2\) fermions
+hopping on the lattice with hopping parameter \(t\) and interaction strength \(U\) \[ H = -t
+\sum_{\langle i,j \rangle \alpha} c^\dagger_{i\alpha} c_{j\alpha} +
+\sum_i c^\dagger_{i\uparrow} c_{i\downarrow}\]
    
+
    where \(c^\dagger_{i\alpha}\)
+creates a spin \(\alpha\) electron at
+site \(i\). Pauli exclusion prevents
+two electrons with the same spin being at the same site so which is why
+the interaction term only couples opposite spin electrons. The only
+physically relevant parameter here is \(U/t\) which compared the interaction
+strength \(U\) to the importance of
+kinetic energy \(t\).
    
+
    In the free fermion limit \(U/t =
+0\), we can just find the single particle eigenstates and fill
+them up to the fermi level. The many body ground state has no particular
+electron-electron correlations.
    
+
    In the interacting limit, \(t/U =
+0\), there’s no hopping so electrons just site wherever we put
+them. We can fill the system up until there is one electron per site
+without any energy penalty at all. The maximum we can fill the system up
+to
    
+
    
+
+
+
    
 
    __ Connection between 
    
+class="sourceCode python">
    diff --git a/_thesis/1.1_FK_Intro.html b/_thesis/1.1_FK_Intro.html index 62c4149..781661a 100644 --- a/_thesis/1.1_FK_Intro.html +++ b/_thesis/1.1_FK_Intro.html @@ -336,13 +336,35 @@ distribution} Trick +
  • Introduction
    • +
  • The Long-Ranged +Falikov-Kimball Model
    • +
  • The Phase +Diagram
    • +
  • Markov Chain +Monte Carlo and Emergent Disorder
    • +
  • Localisation Properties
    • +
  • Discussion & Conclusion
    • +
  • Acknowledgments
    • +
  • DETAILED BALANCE
    • +
  • UNCORRELATED DISORDER MODEL

    • Contributions

    This material is this chapter expands on work presented in

    -

    citekey? 1 One-dimensional long-range Falikov-Kimball model: Thermal phase transition and disorder-free localization, Hodson, T. and Willsher, J. and Knolle, @@ -351,42 +373,6 @@ J., Phys. Rev. B, 104, 4, 2021,

    stablise order in a one dimension Falikov-Kimball model. Josef developed a proof of concept during a summer project at Imperial. The three of us brought the project to fruition.

    -
    [WARNING] Citeproc: citation abanin_recent_2017 not found abaninRecentProgressManybody2017
-[WARNING] Citeproc: citation anderson_absence_1958-1 not found andersonAbsenceDiffusionCertain1958
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-[WARNING] Citeproc: citation croy_anderson_2011 not found croyAndersonLocalization1D2011
-[WARNING] Citeproc: citation dalessio_quantum_2016 not found dalessioQuantumChaosEigenstate2016
-[WARNING] Citeproc: citation dyson_existence_1969 not found dysonExistencePhasetransitionOnedimensional1969
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-[WARNING] Citeproc: citation goldshtein_pure_1977 not found goldshteinPurePointSpectrum1977
-[WARNING] Citeproc: citation huang_accelerated_2017 not found huangAcceleratedMonteCarlo2017
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-[WARNING] Citeproc: citation srednicki_chaos_1994 not found
-[WARNING] Citeproc: citation thouless_long-range_1969 not found
-[WARNING] Citeproc: citation yao_quasi-many-body_2016 not found

    Introduction

    Localisation

    The discovery of localisation in quantum systems surprising at the @@ -394,22 +380,21 @@ time given the seeming ubiquity of extended Bloch states. Later, when thermalisation in quantum systems gained interest, localisation phenomena again stood out as counterexamples to the eigenstate thermalisation hypothesisabanin_recent_2017?,srednicki_chaos_1994?, -allowing quantum systems to avoid to retain memory of their initial -conditions in the face of thermal noise.

    +data-cites="abaninRecentProgressManybody2017 srednickiChaosQuantumThermalization1994">2,3, allowing quantum systems to +avoid to retain memory of their initial conditions in the face of +thermal noise.

    The simplest and first discovered kind is Anderson localisation, first studied in 1958anderson_absence_1958-1? -in the context of non-interacting fermions subject to a static or -quenched disorder potential \(V_j\) -drawn uniformly from the interval \([-W,W]\)

    +data-cites="andersonAbsenceDiffusionCertain1958">4 in the context of +non-interacting fermions subject to a static or quenched disorder +potential \(V_j\) drawn uniformly from +the interval \([-W,W]\)

    \[ H = -t\sum_{\langle jk \rangle} c^\daggerger_j c_k + \sum_j V_j c_j^\daggerger c_j @@ -420,13 +405,11 @@ cannot contribute to transport processes. Initially it was thought that in one dimensional disordered models, all states would be localised, however it was later shown that in the presence of correlated disorder, bands of extended states can existizrailev_localization_1999?,croy_anderson_2011?,izrailev_anomalous_2012?.

    +data-cites="izrailevLocalizationMobilityEdge1999 croyAndersonLocalization1D2011 izrailevAnomalousLocalizationLowDimensional2012">5–7.

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

    \[ @@ -437,9 +420,10 @@ c_j^\daggerger c_j + U\sum_{jk} n_j n_k c^\dagger_j c_j\). Here, in contrast to the Anderson model, localisation phenomena can be proven robust to weak perturbations of the Hamiltonian. This is called many-body localisation (MBL)imbrie_many-body_2016?.

    +class="citation" +data-cites="imbrieManyBodyLocalizationQuantum2016">8.

    Both MBL and Anderson localisation depend crucially on the presence of quenched disorder. This has led to ongoing interest in the possibility of disorder-free localisation, in which the disorder @@ -451,29 +435,28 @@ the initial state.

    context of Helium mixtures1 and then extended to heavy-light +role="doc-biblioref">9 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 modelsyao_quasi-many-body_2016?,schiulaz_dynamics_2015? -instead find that the models thermalise exponentially slowly in system -size, which Ref.yao_quasi-many-body_2016? -dubs Quasi-MBL.

    +data-cites="yaoQuasiManyBodyLocalizationTranslationInvariant2016 schiulazDynamicsManybodyLocalized2015">10,11 instead find that the models +thermalise exponentially slowly in system size, which Ref.10 dubs Quasi-MBL.

    True disorder-free localisation does occur in exactly solveable models with extensively many conserved quantitiessmith_disorder-free_2017?. -As conserved quantites have no time dynamics this can be thought of as -taking the separation of timescales to the infinite limit.

    +data-cites="smithDisorderFreeLocalization2017">12. As conserved quantites have no +time dynamics this can be thought of as taking the separation of +timescales to the infinite limit.

    The Falikov Kimball Model

    In the Falikov Kimball (FK) model spinless fermions \(c_{i\uparrow}\) are coupled via a repulsive @@ -498,11 +481,11 @@ between the classical particles has been added by us to stabilise the formation of an ordered phase in 1D. The classical variables commute with the Hamiltonian \([H, n_{i\downarrow}] = 0\) so like the lattice gauge model in Refsmith_disorder-free_2017?} -the FK model has extensively many conserved quantities which can act as -an effective disorder potential for the electronic sector.

    +data-cites="smithDisorderFreeLocalization2017">12} the FK model has extensively +many conserved quantities which can act as an effective disorder +potential for the electronic sector.

    Due to Pauli exclusion, the maximum filling occurs when one of each species occupies each lattice site such that \(\sum_i (n_{i\downarrow} + n_{i\uparrow} )/ N = @@ -537,20 +520,21 @@ class="math inline">\(\mu = 0\) to explain the Mott metal-insulator (MI) transition, however it has seen applications to high-temperature superconductivity and become target for cold-atom optical trap experiments.noauthor_hubbard_2013?, -greiner_quantum_2002, jordens_mott_2008}. While simple, only a few -analytic results exist, namely the Bethe ansatzlieb_absence_1968?} -which proves the absence of even a zero temperature phase transition in -the 1D model and Nagaoka’s theoremnagaoka_ferromagnetism_1966?} -which proves that the three dimensional model has a ferromagnetic ground -state in the vicinity of half filling.

    +data-cites="HubbardModelHalf2013">13, greiner_quantum_2002, +jordens_mott_2008}. While simple, only a few analytic results exist, +namely the Bethe ansatz14} which proves the absence of +even a zero temperature phase transition in the 1D model and Nagaoka’s +theorem15} which proves that the three +dimensional model has a ferromagnetic ground state in the vicinity of +half filling.

    Falikov-Kimball model

    The Falikov-Kimball model corresponds to the case \(t_{\downarrow} = 0\). It can be interpreted @@ -563,10 +547,9 @@ and f electrons or electrons and ions. The model was first introduced by Hubbard in 1963 as a model of interacting localised and de-localised electron bands and gained its name from Falikov and Kimball’s use of it to study the MI transition in rare-earth materialshubbard_j._electron_1963?, -falicov_simple_1969}.

    +data-cites="hubbardj.ElectronCorrelationsNarrow1963">16, falicov_simple_1969}.

    Here we will use refer to the light spinless species as electrons' with creation operator $c^\dagger_{i}$ and the heavy species asions’ with density operator \(n_i\). When the @@ -627,34 +610,33 @@ class="math inline">\(U\) dependent critical temperature \(T_c(U)\) to a low temperature charge density wave state in which the ions occupy one of the two sublattices A and Bmaska_thermodynamics_2006-1?}. -The order parameter is the square of the staggered magnetisation: \[ +data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006">17}. The order parameter is the +square of the staggered magnetisation: \[ M = \sum_{i \in A} n_i - \sum_{i \in B} n_i \] % In the disordered phase Ref.antipov_interaction-tuned_2016-1?} -identifies an interplay between Anderson localisation at weak -interaction and a Mott insulator phase in the strongly interacting -regime.

    +data-cites="andersonAbsenceDiffusionCertain1958">4} identifies an interplay between +Anderson localisation at weak interaction and a Mott insulator phase in +the strongly interacting regime.

    In the one dimensional FK model, however, Peierls’ argumentpeierls_isings_1936?, -kennedy_itinerant_1986} and the Bethe ansatzlieb_absence_1968?} -make it clear that there is no ordered CDW phase. Peierls’ argument is -that one should 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 the order -parameter. In the Ising model this would be having the spins pointing up -in one part of the model and down in the other, for a CDW phase it means -having the ions occupy the A sublattice in one part and the B sublattice -in the other.

    +class="citation" +data-cites="peierlsIsingModelFerromagnetism1936">18, +kennedyItinerantElectronModel1986} and the Bethe ansatz14} make it clear that there is no +ordered CDW phase. Peierls’ argument is that one should 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 the order parameter. In the Ising model this would be +having the spins pointing up in one part of the model and down in the +other, for a CDW phase it means having the ions occupy the A sublattice +in one part and the B sublattice in the other.

    Short range interactions will produce a constant energy penalty for such a domain wall that does not scale with system size while in 1D there are \(L\) such states so the @@ -694,15 +676,16 @@ behaviour of our modified FK model.

    J(\abs{i-j}) \tau_i \tau_j = J \sum_{i\neq j} |i - j|^{-\alpha} \tau_i \tau_j\] % Rigorous renormalisation group arguments show that the LRI model has an ordered phase in 1D for $1 < < 2 $dyson_existence_1969?}. -Peierls’ argument can be extendedthouless_long-range_1969?} -to provide intuition for why this is the case. Again considering the -energy difference between the ordered state 19}. Peierls’ argument can be +extended20} to provide intuition for why +this is the case. Again considering the energy difference between the +ordered state \(\ket{\ldots\uparrow\uparrow\uparrow\uparrow\ldots}\) and a domain wall state \(\ket{\ldots\uparrow\uparrow\downarrow\downarrow\ldots}\). @@ -716,23 +699,23 @@ proved rigorously for a very general class of 1D systems, that if \(\Delta E\) or its many-body generalisation converges in the thermodynamic limit then the free energy is analyticruelle_statistical_1968?}. -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 \(J(n)\)dyson_existence_1969?}.

    +data-cites="ruelleStatisticalMechanicsOnedimensional1968">21}. 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 \(J(n)\)19}.

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

    1. $ = 0$ For infinite range interactions the Ising model is exactly solveable and mean field theory is exactlipkin_validity_1965?}.
      2. +data-cites="lipkinValidityManybodyApproximation1965">22}.
    2. $ $ For slowly decaying interactions \(\sum_n J(n)\) does not converge so the Hamiltonian is non-extensive, a case which won’t be further considered @@ -741,9 +724,10 @@ here.
      3. temperature.
    3. $ = 2 $ The energy of domain walls diverges logarithmically, and this turns out to be a Kostelitz-Thouless transitionthouless_long-range_1969?}.
      4. +class="citation" +data-cites="thoulessLongRangeOrderOneDimensional1969">20}.
    4. $ 2 < $ For quickly decaying interactions, domain walls have a finite energy penalty, hence Peirels’ argument holds and there is no phase transition.
      5. @@ -752,47 +736,45 @@ phase transition.

      On bipartite lattices in dimensions 2 and above the FK model exhibits a finite temperature phase transition to an ordered charge density wave (CDW) phasemaska_thermodynamics_2006-1?. -In this phase, the ions are confined to one of the two sublattices, -breaking the \(\mathbb{Z}_2\) -symmetry.

      +data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006">17. In this phase, the ions are +confined to one of the two sublattices, breaking the \(\mathbb{Z}_2\) symmetry.

      In 1D, however, Periel’s argumentpeierls_isings_1936?,kennedy_itinerant_1986? -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. -This leads to a disordered system that is gaped by the CDW background -but with correlated fluctuations leading to a disorder-free correlation -induced mobility edge in one dimension.

      +data-cites="peierlsIsingModelFerromagnetism1936 kennedyItinerantElectronModel1986">18,23 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. This leads to a disordered +system that is gaped by the CDW background but with correlated +fluctuations leading to a disorder-free correlation induced mobility +edge in one dimension.

      Markov Chain Monte Carlo

      To evaluate thermodynamic averages we perform a classical Markov Chain Monte Carlo random walk over the space of ionic configurations, at each step diagonalising the effective electronic Hamiltonianmaska_thermodynamics_2006-1?}. -Using a binder-cumulant methodbinder_finite_1981?,musial_monte_2002?, -we demonstrate the model has a finite temperature phase transition when -the interaction is sufficiently long ranged. We then estimate the -density of states and the inverse participation ratio as a function of -energy to diagnose localisation properties. We show preliminary results -that the in-gap states induced at finite temperature are localised while -the states in the unperturbed bands remain extended, evidence for a -mobility edge.

      +class="citation" +data-cites="maskaThermodynamicsTwodimensionalFalicovKimball2006">17}. Using a binder-cumulant +method24,25, we demonstrate the model has a +finite temperature phase transition when the interaction is sufficiently +long ranged. We then estimate the density of states and the inverse +participation ratio as a function of energy to diagnose localisation +properties. We show preliminary results that the in-gap states induced +at finite temperature are localised while the states in the unperturbed +bands remain extended, evidence for a mobility edge.

      Following Ref.abanin_recent_2017?, -consider the time evolution of a local operator \(\hat{O}\) \[ -\expval{\hat{O}}{\psi(t)} = \sum_{\alpha \beta} C^*_\alpha C_\beta -e^{i(E_\alpha - E_\beta)} O_{\alpha \beta}\]

      +data-cites="abaninRecentProgressManybody2017">2, consider the time evolution of +a local operator \(\hat{O}\) \[ \expval{\hat{O}}{\psi(t)} = \sum_{\alpha \beta} +C^*_\alpha C_\beta e^{i(E_\alpha - E_\beta)} O_{\alpha +\beta}\]

      Where \(C_\alpha\) are determined by the initial state and \(O_{\alpha \beta} = \expval{\alpha | \hat{O} | \beta}\) are the matrix elements of \(\hat{O}\) with respect to the energy eigenstates. Srednickisrednicki_chaos_1994?} -introduced the ansatz that for local operators:

      +data-cites="srednickiChaosQuantumThermalization1994">3} introduced the ansatz that for +local operators:

      \[O_{\alpha \beta} = O(E)\delta_{\alpha\beta} + e^{-S(E)/2} f(E,\omega) R_{\alpha\beta}\]

      @@ -864,15 +847,15 @@ entropic term \(e^{-S(E)}\) and the rapidly varying phase factors \(e^{i(E_\alpha - E_\beta)}\). This statement of the ETH has verified for the quantum hard sphere modelsrednicki_chaos_1994? -and numerically for other modelskhatami_fluctuation-dissipation_2013?,dalessio_quantum_2016?.

      +data-cites="srednickiChaosQuantumThermalization1994">3 and numerically for other +models26,27.

      An alternate view on ETH is the statement that in thermalising systems individual eigenstates look thermal when viewed locally. Take a eigenstate \(|\alpha\rangle\) with @@ -892,13 +875,13 @@ condition is broken by systems with localised states so a lack of thermalisation is often used as a diagnostic tool for localisation.

      Anderson Localisation

      Localisation was first studied by Anderson in 1958anderson_absence_1958-1? -in the context of non-interacting fermions subject to a static or -quenched disorder potential \(V_j\) -drawn uniformly from the interval \([-W,W]\):

      +class="citation" +data-cites="andersonAbsenceDiffusionCertain1958">4 in the context of +non-interacting fermions subject to a static or quenched disorder +potential \(V_j\) drawn uniformly from +the interval \([-W,W]\):

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

      @@ -909,41 +892,41 @@ cannot contribute to diffusive transport processes. Except in 1D where any disorder strength is sufficient. Intuitively this happens because hopping processes between nearby sites become off-resonant, hindering the hybridisation that would normally lead to extended Bloch stateskramer_localization_1993?.

      +class="citation" +data-cites="kramerLocalizationTheoryExperiment1993">28.

      In one and two dimensions, all the states in the Anderson model are localised. In three dimensions there are mobility edges. Mobility edges are critical energies in the spectrum which separate delocalised states in a band from localised states which form a band tailabanin_recent_2017?}. -An argument due to Lifshitz shows that the density of state of the band -tail should decay exponentially and localised and extended stats cannot -co-exist at the same energy as they would hybridise into extended -stateskramer_localization_1993?}.

      +class="citation" data-cites="abaninRecentProgressManybody2017">2}. An argument due to Lifshitz +shows that the density of state of the band tail should decay +exponentially and localised and extended stats cannot co-exist at the +same energy as they would hybridise into extended states28}.

      It was thought that mobility edges could not exist in 1D because all the states localised in the presence of any amount of disorder. This is true for uncorrelated potentialsgoldshtein_pure_1977?}. -However, it was shown that if the disorder potential \(V_j\) contains spatial correlations -mobility edges do exist in 1Dizrailev_localization_1999?, -izrailev_anomalous_2012}. Ref.croy_anderson_2011?} -extends this work to look at power law decay of the correlations: \[ C(l) = \expval{V_i V_{i+l}} \propto l^{-\alpha} -\] % Figure 29}. However, it was shown that if +the disorder potential \(V_j\) contains +spatial correlations mobility edges do exist in 1D5, +izrailevAnomalousLocalizationLowDimensional2012}. Ref.6} extends this work to look at +power law decay of the correlations: \[ C(l) += \expval{V_i V_{i+l}} \propto l^{-\alpha} \] % Figure \(\ref{fig:anderson_dos}\) shows numerical calculations of the Localisation length (see later) and density of states for the power law correlated Anderson model. At the unperturbed @@ -962,17 +945,17 @@ U\sum_{jk} n_j n_k c_j\) Here, in contrast to the Anderson model, localisation phenomena can be proven robust to weak perturbations of the Hamiltonianimbrie_many-body_2016?}.

      +data-cites="imbrieManyBodyLocalizationQuantum2016">8}.

      MBL is defined by the emergence of an extensive number of quasi-local operators called local integrals of motions (LIOMs) or l-bits. Following -Ref.abanin_recent_2017?}, -using a spin system with variables \(\sigma^z_i\), any operator can be written -in the general form:

      +Ref.2}, using a spin system with +variables \(\sigma^z_i\), any operator +can be written in the general form:

      \[ \tau^z_i = \sigma^z_i + \sum_{\alpha\beta kl} f_{kl}^{\alpha\beta} \sigma^\alpha_{i+k} \sigma_z\beta_{i+k} + ...\] % what defines a MBL system is that @@ -998,9 +981,9 @@ distant l-bits can only become entangled on a timescale of:

      \frac{\hbar}{J_0} e^{r/\Bar{\xi}} \] % and hence quantum correlations and entanglement propagates logarithmically in MBL systemsimbrie_diagonalization_2016?}.

      +data-cites="imbrieDiagonalizationManyBodyLocalization2016">30}.

      Disorder Free localisation

      Both Anderson localisation and MBL depend on the presence of quenched disorder. Recently the idea of disorder-free localisation has gained @@ -1009,34 +992,33 @@ can be generated entirely from the dynamics of a model itself.

      The idea was first proposed in the context of Helium mixtures1} and then extended to +role="doc-biblioref">9} and then extended to heavy-light mixtures in which multiple species with large mass ratios interact, the idea being that the heavier particles act as an effective disorder potential for the lighter ones, inducing localisation. Two such modelsyao_quasi-many-body_2016?,schiulaz_dynamics_2015} +data-cites="yaoQuasiManyBodyLocalizationTranslationInvariant2016">10,schiulazDynamicsManybodyLocalized2015} instead find that the models thermalise exponentially slowly in system size, which Ref.yao_quasi-many-body_2016?} -dubs Quasi-MBL. A. Smith, J. Knolle et al instead looked at models -containing an extensive number of conserved quantities and demonstrated -true disorder free localisationsmith_disorder-free_2017?}.

      +data-cites="yaoQuasiManyBodyLocalizationTranslationInvariant2016">10} dubs Quasi-MBL. A. Smith, J. +Knolle et al instead looked at models containing an extensive number of +conserved quantities and demonstrated true disorder free +localisation12}.

      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 momentkramer_localization_1993?}: -\[ +data-cites="kramerLocalizationTheoryExperiment1993">28}: \[ P^{-1} = \sum_i \abs{\psi_i}^4 \] % It acts as a measure of the portion of space occupied by the wave function. For localised states it will be independent of system @@ -1048,10 +1030,10 @@ 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 IPRantipov_interaction-tuned_2016-1?: -\[ +class="citation" +data-cites="andersonAbsenceDiffusionCertain1958">4: \[ 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 @@ -1063,14 +1045,15 @@ smoothing if necessary.

      Transfer Matrix Approach

      The transfer matrix method (TMM) can be used to calculate the localisation length of the eigenstates of a system. Following Refs.kramer_localization_1993?, -smith_dynamical_2018}, for bi-linear, 1D Hamiltonians the method -represents the action of \(H\) on a -state \(\ket{\psi} = \sum_i \psi_i -\ket{i}\) with energy E by a matrix equation: \[ +class="citation" +data-cites="kramerLocalizationTheoryExperiment1993">28, +smithDynamicalLocalizationMathbbZ2018}, for bi-linear, 1D Hamiltonians +the method represents the action of \(H\) on a state \(\ket{\psi} = \sum_i \psi_i \ket{i}\) with +energy E by a matrix equation: \[ H &= - \sum_i (c^\dagger_i c_{i+1} + c^\dagger_{i+1} c_{i}) - \sum_i h_i c^\dagger_i c_i \\ E\ket{\psi} &= H \ket{\psi} \\ @@ -1112,13 +1095,12 @@ system size, finds the smallest eigenvalue q and estimates the localisation length by: \[ \lambda = \frac{L}{\ln{q}} \] % As noted bysmith_dynamical_2018? -this method can be numerically unstable because the matrix elements -diverge or vanish exponentially. To get around this, the authors break -the matrix multiplication into chunks and the logarithms of the -eigenvalues of each are stored separately.

      +data-cites="smithDynamicalLocalizationMathbbZ2018">31 this method can be numerically +unstable because the matrix elements diverge or vanish exponentially. To +get around this, the authors break the matrix multiplication into chunks +and the logarithms of the eigenvalues of each are stored separately.

      Numerical Methods

      In this section we will define the Markov Chain Monte Carlo (MCMC) method in general then detail its application to the FK model. We will @@ -1132,11 +1114,11 @@ expectation values \(\expval{O}\) with respect to some physical system defined by a set of states \(\{x: x \in S\}\) and a free energy \(F(x)\)krauth_introduction_1996?}. -The thermal expectation value is defined via a Boltzmann weighted sum -over the entire states: \[ +data-cites="krauthIntroductionMonteCarlo1998">32}. The thermal expectation value +is defined via a Boltzmann weighted sum over the entire states: \[ \tex{O} &= \frac{1}{\Z} \sum_{x \in S} O(x) P(x) \\ P(x) &= \frac{1}{\Z} e^{-\beta F(x)} \\ \Z &= \sum_{x \in S} e^{-\beta F(x)} @@ -1203,9 +1185,9 @@ P(x) \T(x \rightarrow x') = P(x') \T(x' \rightarrow x) \] % In practice most algorithms are constructed to satisfy detailed balance though there are arguments that relaxing the condition can lead to faster algorithmskapfer_sampling_2013?}.

      +data-cites="kapferSamplingPolytopeHarddisk2013">33}.

      The goal of MCMC is then to choose \(\T\) so that it has the desired thermal distribution \(P(x)\) as its fixed @@ -1234,10 +1216,9 @@ x')\).

      original Metropolis algorithm that allows for non-symmetric proposal distributions $q(xx’) q(x’x) $. It can be derived starting from detailed balancekrauth_introduction_1996?}: -\[ +data-cites="krauthIntroductionMonteCarlo1998">32}: \[ P(x)\T(x \to x') &= P(x')\T(x' \to x) \\ P(x)q(x \to x')\A(x \to x') &= P(x')q(x' \to x)\A(x' \to x) \\ @@ -1335,11 +1316,12 @@ it is too high it implies the steps are too small, a problem because then the walk will take longer to explore the state space and the samples will be highly correlated. Ideal values for the acceptance rate can be calculated under certain assumptionsroberts_weak_1997?}. -Here we monitor the acceptance rate and if it is too high we re-run the -MCMC with a modified proposal distribution that has a chance to propose -moves that flip multiple sites at a time.

      +data-cites="robertsWeakConvergenceOptimal1997">34      }. Here we monitor the +acceptance rate and if it is too high we re-run the MCMC with a modified +proposal distribution that has a chance to propose moves that flip +multiple sites at a time.

      In addition we exploit the particle-hole symmetry of the problem by occasionally proposing a flip of the entire state. This works because near half-filling, flipping the occupations of all the sites will @@ -1348,10 +1330,9 @@ produce a state at or near the energy of the current one.

      of the process, a speed up can be obtained by modifying the proposal distribution to depend on the classical part of the energy, a trick gleaned from Ref.krauth_introduction_1996?}: -\[ +data-cites="krauthIntroductionMonteCarlo1998">32}: \[ q(k \to k') &= \min\left(1, e^{\beta (H^{k'} - H^k)}\right) \\ \A(k \to k') &= \min\left(1, e^{\beta(F^{k'}- F^k)}\right) @@ -1361,11 +1342,11 @@ method.

      An extension of this idea is to try to define a classical model with a similar free energy dependence on the classical state as the full quantum, Ref.huang_accelerated_2017?} -does this with restricted Boltzmann machines whose form is very similar -to a classical spin model.

      +data-cites="huangAcceleratedMonteCarlo2017">35      } does this with restricted +Boltzmann machines whose form is very similar to a classical spin +model.

      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 @@ -1384,10 +1365,10 @@ central moments of the order parameter m: \[m 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 pointbinder_finite_1981?, -musial_monte_2002}.

      +class="citation" data-cites="binderFiniteSizeScaling1981">24, +musialMonteCarloSimulations2002}.

      Markov Chain Monte-Carlo in Practice}

      Quick Intro to MCMC}

      @@ -1583,54 +1564,1197 @@ energy term is the sum of an easy to compute classical energy and a more expensive quantum free energy, we can split the acceptance function into two in such as way as to avoid having to compute the full exact diagonalisation some of the time:

      -
      
-current_state = initial_state
-
-for i in range(N_steps):
-    new_state = proposal(current_state)
-
-    df_classical = classical_free_energy_change(current_state, new_state, parameters)
-    if exp(-beta * df_classical) < uniform(0,1):
-        f_quantum = quantum_free_energy(current_state, new_state, parameters)
-    
-        if exp(- beta * df_quantum) < uniform(0,1):
-          current_state = new_state
-    
-        states[i] = current_state
-
      +
      
+current_state = initial_state
+
+for i in range(N_steps):
+    new_state = proposal(current_state)
+
+    df_classical = classical_free_energy_change(current_state, new_state, parameters)
+    if exp(-beta * df_classical) < uniform(0,1):
+        f_quantum = quantum_free_energy(current_state, new_state, parameters)
+    
+        if exp(- beta * df_quantum) < uniform(0,1):
+          current_state = new_state
+    
+        states[i] = current_state
+

      lets cite Figure1

      lets cite to person2. and then multple36. and then multple2,36,3. what is we surround it by +role="doc-biblioref">37. what is we surround it by spaces?2

      +role="doc-biblioref">36

      +
      +
      + + +
      +
      +

      Introduction

      +

      The FK model is one of the simplest +models of the correlated electron problem. It captures the essence of +the interaction between itinerant and localized electrons, equivalent to +a model of hopping fermions coupled to a classical Ising field. It was +originally introduced to explain the metal-insulator transition in +f-electron systems but in its long history it has been interpreted +variously as a model of electrons and ions, binary alloys or of crystal +formation 16,38–40. Despite its simplicity, the +FK model has a rich phase +diagram in \(D \geq 2\) dimensions. For +example, it shows an interaction-induced gap opening even at high +temperatures, similar to the corresponding Hubbard Model 41. In 1D, the ground state +phenomenology as a function of filling can be rich 42 but the system is disordered +for all \(T > 0\) 23. Moreover, the model has been a +test-bed for many-body methods, interest took off when an exact DMFT +solution in the infinite dimensional case was found 43–46.

      +

      The presence of the classical field makes the model amenable to an +exact numerical treatment at finite temperature via a sign problem free +MCMC algorithm 17,47–51. The MCMC +treatment motivates a view of the classical background field as a +disorder potential, which suggests an intimate link to localisation +physics. Indeed, thermal fluctuations of the classical sector act as +disorder potentials drawn from a thermal distribution and the emergence +of disorder in a translationally invariant Hamiltonian links the FK +model to recent interest in disorder-free localisation 12,31,52.

      +

      Dimensionality is crucial for the physics of both localisation and +FTPTs. In 1D, disorder generally +dominates, even the weakest disorder exponentially localises +all single particle eigenstates. Only longer-range correlations +of the disorder potential can potentially induce delocalization 53–55. Thermodynamically, short-range +interactions cannot overcome thermal defects in 1D which prevents +ordered phases at nonzero temperature 4,28,29,56. 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 57–60 decays as \(r^{-1}\) in 1D 61. This could in principle induce +the necessary long-range interactions for the classical Ising +background 18,20. However, Kennedy and Lieb +established rigorously that at half-filling a CDW +phase only exists at \(T = 0\) for the +1D FK model 23.

      +

      Here, we construct a generalised one-dimensional FK +model with long-range interactions which induces the otherwise forbidden +CDW phase at non-zero +temperature. We find a rich phase diagram with a CDW FTPT and +interaction-tuned Anderson versus Mott localized phases similar to the +2D FK model 49. 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.

      +

      The paper is organised as follows. First, we introduce the model and +present its phase diagram. Second, we present the methods used to solve +it numerically. Last, we investigate the model’s localisation properties +and conclude.

      +

      The Long-Ranged +Falikov-Kimball Model

      +

      We interpret the FK model as a model of +spinless fermions, \(c^\dag_{i}\), +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, \(J_{ij} = 4\kappa J +(-1)^{\abs{i-j}} \abs{i-j}^{-\alpha}\), between the spins. The +normalisation, \(\kappa^{-1} = \sum_{i=1}^{N} +i^{-\alpha}\), renders the 0th order mean field critical +temperature independent of system size. 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 39.   \[\begin{aligned} +H_{\mathrm{FK}} = & \;U \sum_{i} S_i\;(c^\dag_{i}c_{i} - +\tfrac{1}{2}) -\;t \sum_{i} (c^\dag_{i}c_{i+1} + \textit{h.c.)}\\ +& + \sum_{i, j}^{N} J_{ij} S_i S_j \nonumber +\label{eq:HFK}\end{aligned}\]

      +

      In two or more dimensions, the \(J\!=0\!\) FK model has a FTPT +to the CDW phase with non-zero +staggered magnetisation \(m = N^{-1} \sum_i +(-1)^i \; S_i\) and fermionic order parameter \(f = 2 N^{-1}\abs{\sum_i (-1)^i \; +\expval{c^\dag_{i}c_{i}}}\) 17,49. This only exists at zero +temperature in the short ranged 1D model 23. To study the CDW +phase at finite temperature in 1D, we add an additional coupling that is +both long-ranged and staggered by a factor \((-1)^{|i-j|}\). The additional coupling +stabilises the Antiferromagnetic (AFM) order of the Ising spins which +promotes the finite temperature CDW phase of the fermionic +sector.

      +

      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. We recall that Peierls’ +classic argument can be extended 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 \(\alpha \leq 3/2\) 20,21,62. In the following, we choose +\(\alpha = 5/4\) to avoid this +additional complexity.

      +

      To improve the scaling of finite size effects, we make the +replacement \(\abs{i - j}^{-\alpha} +\rightarrow \abs{f(i - j)}^{-\alpha}\), in both \(J_{ij}\) and \(\kappa\), where \(f(x) = \frac{N}{\pi}\sin \frac{\pi x}{N}\), +which is smooth across the circular boundary 63. We only consider even system +sizes given that odd system sizes are not commensurate with a CDW +state.

      +

      The Phase Diagram

      +

      Figs. [1a] and [1b] show the phase diagram for +constant \(U=5\) and constant \(J=5\), respectively. We determined the +transition temperatures from the crossings of the Binder cumulants \(B_4 = \tex{m^4}/\tex{m^2}^2\) 24. For a representative set of +parameters, Fig. [1c] shows the order parameter +\(\tex{m}^2\). Fig. [1d] shows 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 our long-range FK +model mimics that of the LRI model and is not +significantly altered by the presence of the fermions, which shows that +the long range tail expected from a basic fermion mediated RKKY +interaction between the Ising spins is absent.

      +

      Our main interest concerns the additional structure of the fermionic +sector in the high temperature phase. Following Ref. 49, we can distinguish between the +Mott and Anderson insulating phases. The former is characterised by a +gapped DOS in the absence of a CDW. +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. [3a]. The Anderson phase is gapless +but, as we explain below, shows localised fermionic eigenstates.

      +

      Markov Chain +Monte Carlo and Emergent Disorder

      +

      The results for the phase diagram were obtained with a classical +MCMC method which we discuss +in the following. It allows us to solve our long-range FK +model efficiently, yielding unbiased estimates of thermal expectation +values and linking it to disorder physics in a translationally invariant +setting.

      +

      Since the spin configurations are classical, the Hamiltonian can be +split into a classical spin part \(H_s\) and an operator valued part \(H_c\). \[\begin{aligned} +H_s& = - \frac{U}{2}S_i + \sum_{i, j}^{N} J_{ij} S_i S_j \\ +H_c& = \sum_i U S_i c^\dag_{i}c_{i} -t(c^\dag_{i}c_{i+1} + +c^\dag_{i+1}c_{i}) \end{aligned}\] The partition function can +then be written as a sum over spin configurations, \(\vec{S} = (S_0, S_1...S_{N-1})\): \[\begin{aligned} +\Z = \Tr e^{-\beta H}= \sum_{\vec{S}} e^{-\beta H_s} \Tr_c e^{-\beta +H_c} .\end{aligned}\] 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. \[\begin{aligned} +\Z = \sum_{\vec{S}} e^{-\beta H_S[\vec{S}] - \beta F_c[\vec{S}]} = +\sum_{\vec{S}} e^{-\beta E[\vec{S}]}\end{aligned}\]

      +

      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 \(p(\vec{S}) = \Z^{-1} e^{-\beta E}\) 64–66. Hence, any observable can be +estimated as a mean over the states visited by the walk. \[\begin{aligned} +\label{eq:thermal_expectation} +\tex{O}& = \sum_{\vec{S}} p(\vec{S}) \tex{O}_{\vec{S}} = \sum_{i = +0}^{M} \tex{O}_{\vec{S}_i} + \mathcal{O}(\tfrac{1}{\sqrt{M}})\\ +\tex{O}_{\vec{S}}& = \sum_{\nu} n_F(\epsilon_{\nu}) +\expval{O}{\nu}\end{aligned}\] 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 67. Here, we incorporate a +modification to the standard Metropolis-Hastings algorithm 68 gleaned from Krauth 32. Let us first recall the +standard algorithm which decomposes the transition probability into +\(\T(a \to b) = \p(a \to b)\A(a \to +b)\). Here, \(\p\) is the +proposal distribution that we can directly sample from while \(\A\) is the acceptance probability. The +standard Metropolis-Hastings choice is \[\A(a +\to b) = \min\left(1, \frac{\p(b\to a)}{\p(a\to b)} e^{-\beta \Delta +E}\right)\;,\] with \(\Delta E = E_b - +E_a\). The walk then proceeds by sampling a state \(b\) from \(\p\) and moving to \(b\) with probability \(\A(a \to b)\). The latter operation is +typically implemented by performing a transition if a uniform random +sample from the unit interval is less than \(\A(a \to b)\) 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 \(\A\). Here, we flip a small number of sites +in \(b\) at random to generate +proposals, which is indeed symmetric.

      +

      In our computations 69 we employ a modification of the +algorithm which 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 speedup 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 and has the acceptance +function \[\A(a \to b) = \min\left(1, +e^{-\beta \Delta H_s}\right)\min\left(1, e^{-\beta \Delta +F_c}\right)\;.\]

      +

      See Appendix [app:balance] for a proof that this +satisfies the detailed balance condition.

      +

      For the model parameters used in Fig. 2, 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 35. 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.

      +

      Localisation Properties

      +

      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 56. An Anderson localised state +centered around \(r_0\) has magnitude +that drops exponentially over some localisation length \(\xi\) i.e \(|\psi(r)|^2 \sim \exp{-\abs{r - +r_0}/\xi}\). 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 \(\vec{S}\).

      +

      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 28,70. 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?

      +
      +
      + + +
      +
      +
      +
      + + +
      +
      +
      +
      + + +
      +
      +

      Fig. 2 shows the DOS +and \(\tau\), the scaling exponent of +the IPR with system size, for a representative set of parameters +covering all three phases. 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.

      +

      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 70. 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 6,29. 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. As a result, the IPR does not scale correctly until the system +size has grown much larger than \(\xi\). Fig. [4] 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. 3. 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 71 and infinite dimensional 72 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 71.

      +

      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, see Appendix [app:disorder_model]. Fig. [4] compares 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. 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. Overall, we see that the interplay of +interactions, here manifest as a peculiar binary potential, and +localization can be very intricate and the added advantage of our 1D +model is that we can explore very large system sizes for a complete +understanding.

      +

      Discussion & 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 1D by adding a novel long-ranged coupling designed to stabilise the +CDW phase present in dimension +two and above. Our hybrid MCMC approach elucidates a +disorder-free localization mechanism within our translationally +invariant system. Further, we demonstrate a significant speedup over the +naive method. We show that our long-range FK in 1D 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 nonzero 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 73. One could also investigate +whether the rich ground state phenomenology of the FK model as a +function of filling 42 such as the devil’s +staircase 74 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.

      +

      Acknowledgments

      +

      We wish to acknowledge the support of Alexander Belcik who was +involved with the initial stages of the project. We thank Angus +MacKinnon for helpful discussions, Sophie Nadel for input when preparing +the figures and acknowledge support from the Imperial-TUM flagship +partnership. This work was supported in part by the Engineering and +Physical Sciences Research Council (EPSRC) Project +No. 2120140.

      +

      DETAILED BALANCE

      +

      Given a MCMC algorithm with target +distribution \(\pi(a)\) and transition +function \(\T\) the detailed balance +condition is sufficient (along with some technical constraints66) to guarantee that in the long +time limit the algorithm produces samples from \(\pi\). \[\pi(a)\T(a \to b) = \pi(b)\T(b \to +a)\]

      +

      In pseudo-code, our two step method corresponds to two nested +comparisons with the majority of the work only occurring if the first +test passes:

      +
      current_state = initial_state
+
+for i in range(N_steps):
+  new_state = proposal(current_state)
+
+  c_dE = classical_energy_change(
+                               current_state,
+                               new_state)
+  if uniform(0,1) < exp(-beta * c_dE):
+    q_dF = quantum_free_energy_change(
+                                current_state,
+                                new_state)
+    if uniform(0,1) < exp(- beta * q_dF):
+      current_state = new_state
+
+    states[i] = current_state
      +

      Defining \(r_c = e^{-\beta H_c}\) +and \(r_q = e^{-\beta F_q}\) our target +distribution is \(\pi(a) = r_c r_q\). +This method has \(\T(a\to b) = q(a\to b)\A(a +\to b)\) with symmetric \(p(a \to b) = +\p(b \to a)\) and \(\A = \min\left(1, +r_c\right) \min\left(1, r_q\right)\)

      +

      Substituting this into the detailed balance equation gives: \[\T(a \to b)/\T(b \to a) = \pi(b)/\pi(a) = r_c +r_q\]

      +

      Taking the LHS and substituting in our transition function: \[\begin{aligned} +\T(a \to b)/\T(b \to a) = \frac{\min\left(1, r_c\right) \min\left(1, +r_q\right)}{ \min\left(1, 1/r_c\right) \min\left(1, +1/r_q\right)}\end{aligned}\]

      +

      which simplifies to \(r_c r_q\) as +\(\min(1,r)/\min(1,1/r) = r\) for \(r > 0\).

      +

      UNCORRELATED DISORDER MODEL

      +

      The disorder model referred to in the main text 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.   \[\begin{aligned} +H_{\mathrm{DM}} = & \;U \sum_{i} (-1)^i \; d_i \;(c^\dag_{i}c_{i} - +\tfrac{1}{2}) \\ +& -\;t \sum_{i} c^\dag_{i}c_{i+1} + c^\dag_{i+1}c_{i} +\nonumber\end{aligned}\]

      -

      +
      +
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      diff --git a/_thesis/1.2_FK_Methods.html b/_thesis/1.2_FK_Methods.html new file mode 100644 index 0000000..f65db41 --- /dev/null +++ b/_thesis/1.2_FK_Methods.html @@ -0,0 +1,186 @@ +--- +title: 1.2_FK_Methods +excerpt: +layout: none +image: + +--- + + + + + + + 1.2_FK_Methods + + + + + + + + + + + + + + +{% include header.html %} + +
      + +
      + + diff --git a/_thesis/1.2_FK_Results.html b/_thesis/1.2_FK_Results.html new file mode 100644 index 0000000..1161ae9 --- /dev/null +++ b/_thesis/1.2_FK_Results.html @@ -0,0 +1,186 @@ +--- +title: 1.2_FK_Results +excerpt: +layout: none +image: + +--- + + + + + + + 1.2_FK_Results + + + + + + + + + + + + + + +{% include header.html %} + +
      + +
      + + diff --git a/_thesis/2.1_AMK_Intro.html b/_thesis/2.1_AMK_Intro.html index 77d51ff..f67ace1 100644 --- a/_thesis/2.1_AMK_Intro.html +++ b/_thesis/2.1_AMK_Intro.html @@ -244,7 +244,19 @@ id="toc-open-boundary-conditions">Open boundary conditions

      which was a joint project of the first three authors with advice and guidance from Willian and Johannes. The project grew out of an interest Gino, Peru and I had in studying amorphous systems, coupled with -Johannes’ expertise on the Kitaev model.

      +Johannes’ expertise on the Kitaev model. The idea to use voronoi +partitions came from1 and Gino did the implementation +of this. The idea and implementation of the edge colouring using SAT +solvers, the mapping from flux sector to bond sector using A* search +were both entirely my work. Peru came up with the ground state +conjecture and implemented the local markers. Gino and I did much of the +rest of the programming for Koala while pair programming and +’whiteboard’ing, this included the phase diagram, edge mode and finite +temperature analyses as well as the derivation of the projector in the +amorphous case.

      Introduction

      The Kitaev Honeycomb model is remarkable because it combines three key properties.

      @@ -256,8 +268,8 @@ class="math inline">\(\alpha\mathrm{-RuCl}_3\)1,2.

      +role="doc-biblioref">2,3.

      Second, this model is deeply interesting to modern condensed matter theory. Its ground state is almost the canonical example of the long sought after quantum spin liquid state. Its excitations are anyons, @@ -267,14 +279,14 @@ because, among other reasons, they can be braided through spacetime to achieve noise tolerant quantum computations3.

      +role="doc-biblioref">4.

      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 states4. Its solubility comes about +role="doc-biblioref">5. Its solubility comes about because the model has many conserved degrees of freedom that mediate the interactions between quantum degrees of freedom.

      Amorphous Systems

      @@ -288,7 +300,7 @@ transformation to a Majorana hamiltonian. This discussion shows that, for the the model to be solvable, it needs only be defined on a trivalent, tri-edge-colourable lattice5.

      +role="doc-biblioref">6.

      The methods section discusses how to generate such lattices and colour them. It also explain how to map back and forth between configurations of the gauge field and configurations of the gauge @@ -478,7 +490,7 @@ class="math inline">\(\alpha\)-bond with exchange coupling \(J^\alpha\)4. For notational brevity, it is +role="doc-biblioref">5. For notational brevity, it is useful to introduce the bond operators \(K_{ij} = \sigma_j^{\alpha}\sigma_k^{\alpha}\) where \(u_{ij}\). What follows is relatively standard theory for quadratic Majorana Hamiltonians6.

      +role="doc-biblioref">7.

      Because of the antisymmetry of the matrix with entries \(J^{\alpha} u_{ij}\), the eigenvalues of the Hamiltonian \(\tilde{H}_u\) come in @@ -851,9 +863,18 @@ class="math inline">\(b^\alpha\) operators could be performed. </i,j></i,j>

      -
      1.
      Marsal, Q., Varjas, D. & Grushin, A. G. Topological +Weaire models of amorphous matter. Proceedings of +the National Academy of Sciences 117, 30260–30265 +(2020).
      +
      +
      +
      2.
      Banerjee, A. et al. Proximate Kitaev Quantum Spin Liquid Behaviour in {\alpha}-RuCl$_3$. @@ -861,7 +882,7 @@ Spin Liquid Behaviour in {\alpha}-RuCl$_3$.
      -
      2.
      3.
      Trebst, S. & Hickey, C. Kitaev materials. Physics Reports 950, 1–37 @@ -869,7 +890,7 @@ materials. Physics Reports 950, 1–37
      -
      3.
      4.
      Freedman, M., Kitaev, A., Larsen, M. & Wang, Z. Topological quantum @@ -878,14 +899,14 @@ computation. Bull. Amer. Math. Soc. 40,
      -
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      5.
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      -
      5.
      6.
      Nussinov, Z. & Ortiz, G. Bond algebras and exact solvability of Hamiltonians: Spin @@ -895,7 +916,7 @@ systems. Physical Review B 79, 214440 (2009).
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      6.
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      Blaizot, J.-P. & Ripka, G. Quantum theory of finite systems. (The MIT Press, 1986).
      diff --git a/_thesis/toc.html b/_thesis/toc.html index 15c10b3..6671f3e 100644 --- a/_thesis/toc.html +++ b/_thesis/toc.html @@ -2,6 +2,7 @@
    5. Introduction
    6. Chapter 1: The Long Range Falikov-Kimball Model
      7. + +
    7. Introduction
      8. +
    8. The Long-Ranged Falikov-Kimball Model
      9. +
    9. The Phase Diagram
      10. +
    10. Markov Chain Monte Carlo and Emergent Disorder
      11. +
    11. Localisation Properties
      12. +
    12. Discussion & Conclusion
      13. +
    13. Acknowledgments
      14. +
    14. []{#app:balance label="app:balance"} DETAILED BALANCE
      15. +
    15. []{#app:disorder_model label="app:disorder_model"} UNCORRELATED DISORDER MODEL
      16. +
    16. Chapter 2: The Amorphous Kitaev Model