+
This chapter expands on work presented in
- [1 1 ] One-dimensional
long-range Falikov-Kimball model: Thermal phase transition and
disorder-free localization , Hodson, T. and Willsher, J. and Knolle,
J., Phys. Rev. B, 104 , 4, 2021,
-the code for which is available is available atthe code for which is available is available at [2
+role="doc-biblioref">2].
Contributions
Johannes had the initial idea to use a long range Ising term to
stablise order in a one dimension Falikov-Kimball model. Josef developed
@@ -340,16 +401,16 @@ simulation code and performed all the analysis presented here.
Chapter Summary
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 Hamiltonian [ 3 3 ] [4 ,musialMonteCarloSimulations2002? ,
+role="doc-biblioref">musialMonteCarloSimulations2002? ],
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
@@ -361,25 +422,25 @@ mobility edge.
\(D \geq 2\) dimensions. For example,
it shows an interaction-induced gap opening even at high temperatures,
similar to the corresponding Hubbard Model [ 5 [ 6 \(T > 0\) 6 ]\(T > 0\) [7 . Moreover, the model has been a
+role="doc-biblioref">7]. 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 [ 8 –11 The Model
The Falikov-Kimball (FK) model \[\begin{aligned}
@@ -394,10 +455,10 @@ 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 [ 12 –15
+role="doc-biblioref">15].
One can look to the references above to see why one might write the
FK Model down directly. Here, in order to motivate the connection to
exactly solvable models, I will explain how the Falikov-Kimball model is
@@ -422,22 +483,22 @@ class="math inline">\(t_\alpha\)
time given the seeming ubiquity of extended Bloch states. Later, when
thermalisation in quantum systems gained interest, localisation
phenomena again stood out as counterexamples to the eigenstate
-thermalisation hypothesis [ 16 ,17 The simplest and first discovered kind is Anderson localisation,
-first studied in 1958 [ 18 \(V_j\) drawn uniformly from
-the interval \([-W,W]\)
+role="doc-biblioref">18] 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
@@ -447,12 +508,12 @@ class="math inline">\(\psi(x) = f(x) e^{-x/\lambda}\) which
cannot contribute to transport processes. Initially it was thought that
in one dimensional disordered models, all states would be localised,
however it was later shown that in the presence of correlated disorder,
-bands of extended states can exist [ 19 –21
+role="doc-biblioref">21].
Later localisation was found in interacting many-body systems with
quenched disorder:
\[
@@ -462,11 +523,10 @@ c_j^\daggerger c_j + U\sum_{jk} n_j n_k
where the number operators \(n_j =
c^\dagger_j c_j\) . Here, in contrast to the Anderson model,
localisation phenomena can be proven robust to weak perturbations of the
-Hamiltonian. This is called many-body localisation (MBL) [ 22
+role="doc-biblioref">22] .
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
@@ -475,31 +535,30 @@ dynamics of the model. This contracts with typical models of disordered
systems in which disorder is explicielty introduced into the Hamilton or
the initial state.
The concept of disorder-free localisation was first proposed in the
-context of Helium mixtures23 [ 23 ] [24 ,25 instead find that the models
-thermalise exponentially slowly in system size, which Ref.25] instead find that the models
+thermalise exponentially slowly in system size, which Ref. [ 24
+role="doc-biblioref">24] dubs Quasi-MBL.
True disorder-free localisation does occur in exactly solveable
-models with extensively many conserved quantities [ 26
+role="doc-biblioref">26]. 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
@@ -523,12 +582,12 @@ the “electronic sector”. The final term that induces interactions
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 Ref so like the lattice gauge model in Ref [26
+role="doc-biblioref">26] 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 =
@@ -562,23 +621,23 @@ redefinition of the chemical potential.
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. [ 27 ,greiner_quantum_2002? jordens_mott_2008? jordens_mott_2008? ].
While simple, only a few analytic results exist, namely the Bethe
-ansatz [ 28 28 ] [29 which proves that the three
+role="doc-biblioref">29] which proves that the three
dimensional model has a ferromagnetic ground state in the vicinity of
half filling.
Falikov-Kimball model
@@ -592,11 +651,12 @@ class="math inline">\(n_{i\downarrow}\). These are often called c
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 materials [ 12 ,falicov_simple_1969? 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 as
@@ -658,28 +718,28 @@ exhibits a phase transition at some \(U\) 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 B [ 3 \[
+role="doc-biblioref">3] . 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. % In the disordered phase Ref. [18
-In the one dimensional FK model, however, Peierls’ argumentIn the one dimensional FK model, however, Peierls’ argument [ 7 ,30 30 ] [28 make it clear that there is no
+role="doc-biblioref">28] 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
@@ -726,17 +786,17 @@ behaviour of our modified FK model.
\[H_{\mathrm{LRI}} = \sum_{ij}
J(\abs{i-j}) \tau_i \tau_j = J \sum_{i\neq j} |i - j|^{-\alpha} \tau_i
\tau_j\] % Rigorous renormalisation group arguments show that the
-LRI model has an ordered phase in 1D for $1 < < 2 $ [ 31 31 ] [32 to provide intuition for why
-this is the case. Again considering the energy difference between the
-ordered state 32] 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 \(\ket{\ldots\uparrow\uparrow\downarrow\downarrow\ldots}\) .
@@ -749,24 +809,24 @@ class="math inline">\(n\) equivalent pairs of sites. Ruelle
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
-analytic [ 33 \(J(n)\) 33 ]\(J(n)\) [31 .
+role="doc-biblioref">31].
With a power law form for \(J(n)\) ,
there are three cases to consider:
$ = 0$ For infinite range interactions the Ising model is exactly
-solveable and mean field theory is exact [ 34
+role="doc-biblioref">34].
$ $ 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
@@ -774,11 +834,11 @@ here.
$ 1 < < 2 $ A phase transition to an ordered state at a finite
temperature.
$ = 2 $ The energy of domain walls diverges logarithmically, and
-this turns out to be a Kostelitz-Thouless transition [ 32
+role="doc-biblioref">32].
$ 2 < $ For quickly decaying interactions, domain walls have a
finite energy penalty, hence Peirels’ argument holds and there is no
phase transition.
@@ -786,18 +846,18 @@ phase transition.
Thermodynamics
On bipartite lattices in dimensions 2 and above the FK model exhibits
a finite temperature phase transition to an ordered charge density wave
-(CDW) phase [ 3 3] . In this phase, the ions are confined
+to one of the two sublattices, breaking the \(\mathbb{Z}_2\) symmetry.
-In 1D, however, Periel’s argumentIn 1D, however, Periel’s argument [7 ,30
hypothesis. It states that if a system thermalises, then we can isolate
small patches of a larger system, trace everyhing else out, and get a
thermal density matrix.
-Following Ref.Following Ref. [16 \(\hat{O}\) 16] , consider the time evolution of a
+local operator \(\hat{O}\) \[ \expval{\hat{O}}{\psi(t)} = \sum_{\alpha \beta}
C^*_\alpha C_\beta e^{i(E_\alpha - E_\beta)} O_{\alpha
\beta}\]
@@ -850,11 +910,11 @@ C^*_\alpha C_\beta e^{i(E_\alpha - E_\beta)} O_{\alpha
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. Srednicki [ 17 \[O_{\alpha \beta} =
O(E)\delta_{\alpha\beta} + e^{-S(E)/2} f(E,\omega)
R_{\alpha\beta}\]
@@ -879,16 +939,16 @@ equal to the thermal expectations with fluctuations suppressed by the
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 model [ 17 17 ] [35 ,36 .
+role="doc-biblioref">36].
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
@@ -907,14 +967,13 @@ must be sufficiently well coupled that energy transport occurs. This
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 1958Localisation was first studied by Anderson in 1958 [18 \(V_j\) drawn uniformly from
-the interval \([-W,W]\) :
+role="doc-biblioref">18] in the context of non-interacting
+fermions subject to a static or quenched disorder potential \(V_j\) drawn uniformly from the interval
+\([-W,W]\) :
\[
H = -t\sum_{\expval{jk}} c^\dagger_j c_k + \sum_j V_j c_j^\dagger c_j
\]
@@ -924,43 +983,43 @@ class="math inline">\(\psi(x) = f(x) e^{-x/\lambda}\) which
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 states [ 37 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 tail [ 16 16 ] [37 .
+role="doc-biblioref">37].
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 potentials [ 38 \(V_j\) contains
-spatial correlations mobility edges do exist in 1D38 ]\(V_j\) contains
+spatial correlations mobility edges do exist in 1D [19 ,21 . Ref.21 ] [20 extends this work to look at
-power law decay of the correlations: \[ C(l)
-= \expval{V_i V_{i+l}} \propto l^{-\alpha} \] % Figure
+role="doc-biblioref">20] extends this work to look at power
+law decay of the correlations: \[ C(l) =
+\expval{V_i V_{i+l}} \propto l^{-\alpha} \] % Figure
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 band edges \(\abs{E}
@@ -977,18 +1036,18 @@ U\sum_{jk} n_j n_k
\] % with \(n_j = c^\dagger_j
c_j\) Here, in contrast to the Anderson model, localisation
phenomena can be proven robust to weak perturbations of the
-Hamiltonian [ 22
+role="doc-biblioref">22].
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. [ 16 \(\sigma^z_i\) , any operator
-can be written in the general form:
+role="doc-biblioref">16], 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
@@ -1013,51 +1072,50 @@ distant l-bits can only become entangled on a timescale of:
\[ t_{\mathrm{ent}}(r) \propto
\frac{\hbar}{J_0} e^{r/\Bar{\xi}} \] % and hence quantum
correlations and entanglement propagates logarithmically in MBL
-systems [ 39
+role="doc-biblioref">39].
Disorder Free localisation
Both Anderson localisation and MBL depend on the presence of quenched
disorder. Recently the idea of disorder-free localisation has gained
traction, asking whether the disorder necessary to generate localisation
can be generated entirely from the dynamics of a model itself.
-The idea was first proposed in the context of Helium mixtures23 The idea was first proposed in the context of Helium mixtures [23 ]
+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 models [24 ,25 25] instead find that the models
+thermalise exponentially slowly in system size, which Ref. [ 24 [ 26
+role="doc-biblioref">26].
Introduction
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 [ 3 ,40 –43 ,debskiPossibilityDetectionFinite2016? debskiPossibilityDetectionFinite2016? ].
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
@@ -1065,26 +1123,26 @@ 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 [ 26 ,44 ,45
+role="doc-biblioref">45].
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 [ 46 –48 [ 18 ,37 ,38 ,49 [ 50 –53 53] decays as \(r^{-1}\) in 1D [ 54 [ 30 ,32 \(T = 0\) for the 1D FK model [ 7
+role="doc-biblioref">7].
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 [ 42 \(t = 1\) , sets the overall
energy scale and we concentrate throughout on the particle-hole
symmetric point at zero chemical potential and half filling [ 14 14] . \[\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.)}\\
@@ -1169,16 +1226,16 @@ class="math inline">\(m = N^{-1} \sum_i (-1)^i \; S_i\) and
fermionic order parameter \(f = 2
N^{-1}\abs{\sum_i (-1)^i \; \expval{c^\dag_{i}c_{i}}}\) [ 3 ,42 [ 7 \((-1)^{|i-j|}\) . The additional coupling
stabilises the Antiferromagnetic (AFM) order of the Ising spins which
@@ -1195,15 +1252,15 @@ overcome the entropic contribution. A renormalisation group analysis
supports this finding and shows that the critical exponents are only
universal for \(\alpha \leq 3/2\) [ 32 ,33 ,55 \(\alpha = 5/4\) to avoid this
-additional complexity.
+role="doc-biblioref">55]. 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 [ 56
+role="doc-biblioref">56]. We only consider even system sizes
+given that odd system sizes are not commensurate with a CDW state.
Introduction and Motivation
@@ -1249,15 +1305,15 @@ conduction band and to be conductors when the gap is less that the
thermal energy scale \(k_bT\) . In many
systems band theory is enough to make good predictions about electrical,
mechanical and thermal properties [ 57 Later, the development of Landau Fermi Liquid theory explained why
band theory works so well even in cases where an analysis of the
relevant energies suggests that it should not [ 58
@@ -1265,9 +1321,9 @@ non-interacting quasiparticles.
fails. An effective theoretical description of these systems must
include electron-electron correlations and they are thus called Strongly
Correlated Materials [ 59 The transition metal oxides have half filled valence bands and hence
@@ -1276,23 +1332,23 @@ Experimentally they are of course found to be insulators, leading Mott
and others in 1937 to suggest that their insulating character is caused
by coulomb repulsion between electrons opening an energy gap that
prevents conduction [ 60
+role="doc-biblioref">60].
However a firmer theoretical description of the Mott-transition had
to wait until 1963 when Martin Gutzwiller [ 61 61] , Junjiro Kanamori [ 62 62] and John Hubbard [ 12
+role="doc-biblioref">12] independently proposed what would
+become known as the Hubbard Model.
The Hubbard Model is about the simplest interacting tight-binding
hamiltonian that could be written down. It is a tight binding model of
spin half electrons with finite bandwidth
transition, the Hubbard Model has seen applications to high-temperature
superconductivity and become the target for cold-atom optical trap
experiments [ noauthor_hubbard_2013? greiner_quantum_2002? jordens_mott_2008? jordens_mott_2008? [lieb_absence_1968?
+role="doc-biblioref">lieb_absence_1968? ]
which proves the absence of even a zero temperature phase transition in
-the 1D model and Nagaoka’s theorem [ nagaoka_ferromagnetism_1966? nagaoka_ferromagnetism_1966? ]
which proves that the three dimensional model has a ferromagnetic ground
state in the vicinity of half filling.
A full theoretical treatment of the Hubbard Model remains elusive
@@ -1363,520 +1419,502 @@ electons that can move and those that can’t.
-
+
-
3.
Maśka, M. M. & Czajka, K.
[3]
+Two-Dimensional Falicov-Kimball Model: A Classical Monte Carlo
+Study, Phys. Rev. B
74 , 035109 (2006).
-
8.
Antipov, A. E., Gull, E. & Kirchner, S.
[8]
-
11.
Herrmann, A. J., Tsuji, N., Eckstein, M. &
-Werner, P.
[11]
+Dynamical Cluster Approximation Study of the Falicov-Kimball
+Model, Phys. Rev. B
94 , 245114 (2016).
-
13.
Falicov, L. M. & Kimball, J. C.
[13]
+Transition-Metal Oxides, Phys. Rev. Lett.
22 ,
+997 (1969).
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