In Chapter 3 I will introduce a generalized FK model in one
-dimension. With the addition of long-range interactions in the
-background field, the model shows a similarly rich phase diagram. I use
-an exact Markov chain Monte Carlo method to map the phase diagram and
-compute the energy-resolved localization properties of the fermions. I
-then compare the behaviour of this transitionally invariant model to an
-Anderson model of uncorrelated binary disorder about a background charge
-density wave field which confirms that the fermionic sector only fully
-localizes for very large system sizes.
+
In Chapter 3 I will introduce a generalized Falikov-Kimball model in
+one dimension I call the Long-Range Falikov-Kimball model. With the
+addition of long-range interactions in the background field, the model
+shows a similarly rich phase diagram its higher dimensional cousins. I
+use an exact Markov chain Monte Carlo method to map the phase diagram
+and compute the energy-resolved localization properties of the fermions.
+I then compare the behaviour of this transitionally invariant model to
+an Anderson model of uncorrelated binary disorder about a background
+charge density wave field which confirms that the fermionic sector only
+fully localizes for very large system sizes.
Quantum Spin Liquids
-
To turn to the other key topic of this thesis, we have discussed the
-question of the magnetic ordering of local moments in the Mott
-insulating state. The local moments may form an AFM ground state.
-Alternatively they may fail to order even at zero temperature [28,29], giving rise to what is known as a
-quantum spin liquid (QSL) state.
-
Landau-Ginzburg-Wilson theory characterises phases of matter as
+
To turn to the other key topic of this thesis, we have already
+discussed the AFM ordering of local moments in the Mott insulating
+state. Landau-Ginzburg-Wilson theory characterises phases of matter as
inextricably linked to the emergence of long range order via a
-spontaneously broken symmetry. The fractional quantum Hall (FQH) state,
-discovered in the 1980s is an explicit example of an electronic system
-that falls outside of the Landau-Ginzburg-Wilson paradigm. FQH systems
-exhibit fractionalised excitations linked to their ground state having
-long range entanglement and non-trivial topological properties [42]. Quantum spin liquids are the
-analogous phase of matter for spin systems. Remarkably the existence of
-QSLs was first suggested by Anderson in 1973 [ [43].
+role="doc-biblioref">42] that if long range order is
+suppressed by some mechanism, it might lead to a liquid-like state even
+at zero temperature, the Quantum Spin Liquid (QSL).
+
This QSL state would exist at zero or very low temperatures, so we
+would expect quantum effects to be very strong, which will turn out to
+have far reaching consequences. It was the discovery of a different
+phase, however that really kickstarted interest in the topic. The
+fractional quantum Hall (FQH) state, discovered in the 1980s is an
+explicit example of an interacting electron system that falls outside of
+the Landau-Ginzburg-Wilson paradigm. It shares many phenomenological
+properties with the QSL state. They both exhibit fractionalised
+excitations, braiding statistics and non-trivial topological
+properties [43]. The many-body ground state of such
+systems acts as a complex and highly entangled vacuum. This vacuum can
+support quasiparticle excitations with properties unbound from that of
+the Dirac fermions of the standard model.
+
How do we actually make a QSL? Frustration is one mechanism that we
+can use to suppress magnetic order in spin models [44]. Frustration can be geometric,
+triangular lattices for instance cannot support AFM order. It can also
+come about as a result of spin-orbit coupling or other physics. There
+are also other routes to QSLs besides frustrated spin systems that we
+will not discuss here [45–47].
+
+
44].
-
The main route to QSLs, though there are others [45–47], is via frustration of spin models
-that would otherwise order have AFM order. This frustration can come
-geometrically, triangular lattices for instance cannot support AFM
-order. It can also come about as a result of spin-orbit coupling.
-
Electron spin naturally couples to magnetic fields. Spin-orbit
-coupling is a relativistic effect, that very roughly corresponds to the
-fact that in the frame of reference of a moving electron, the electric
-field of nearby nuclei look like magnetic field to which the electron
-spin couples. In certain transition metal based compounds, such as those
-based on Iridium and Rutheniun, crystal field effects, strong spin-orbit
-coupling and narrow bandwidths lead to effective spin-Spin-orbit coupling is a relativistic effect, that very roughly
+corresponds to the fact that in the frame of reference of a moving
+electron, the electric field of nearby nuclei look like magnetic fields
+to which the electron spin couples. This effectively couples the spatial
+and spin parts of the electron wavefunction, meaning that the lattice
+structure can influence the form of the spin-spin interactions leading
+to spatial anisotropy. This anisotropy will be how we frustrate the Mott
+insulators [48,49]. As we saw with the Hubbard model,
+interaction effects are only strong or weak in comparison to the
+bandwidth or hopping integral \(t\) so
+what we need to see strong frustration is a material with strong
+spin-orbit coupling \(\lambda\)
+relative to its bandwidth \(t\).
+
In certain transition metal based compounds, such as those based on
+Iridium and Ruthenium, the lattice structure, strong spin-orbit coupling
+and narrow bandwidths lead to effective spin-\(\tfrac{1}{2}\) Mott insulating states with
-strongly anisotropic spin-spin couplings known as Kitaev Materials [44,48–51]. Kitaev
-materials draw their name from the celebrated Kitaev Honeycomb Model as
-it is believed they will realise the QSL state via the mechanisms of the
-Kitaev Model.
At this point we can sketch out a phase diagram like that of fig. 3. When both electron-electron
+interactions \(U\) and spin-orbit
+couplings \(\lambda\) are small
+relative to the bandwidth \(t\) we
+recover standard band theory of band insulators and metals. In the upper
+left we have the simple Mott insulating state as described by the
+Hubbard model. In the lower right, strong spin-orbit coupling gives rise
+to Topological insulators (TIs) characterised by symmetry protected edge
+modes and non-zero Chern number. Kitaev materials occur in the region
+where strong electron-electron interaction and spin-orbit coupling
+interact. See [54] for a much more expansive version of
+this diagram.
The Kitaev Honeycomb model [52] was the first concrete model with a
-QSL ground state. It is defined on the honeycomb lattice and provides an
-exactly solvable model whose ground state is a QSL characterized by a
-static \(\mathbb Z_2\) gauge field and
-Majorana fermion excitations. It can be reduced to a free fermion
-problem via a mapping to Majorana fermions which yields an extensive
-number of static \(\mathbb Z_2\) fluxes
-tied to an emergent gauge field. The model is remarkable not only for
-its QSL ground state, it supports a rich phase diagram hosting gapless,
-Abelian and non-Abelian phases 55] was the first concrete spin model
+with a QSL ground state. It is defined on the two dimensional honeycomb
+lattice and provides an exactly solvable model that can be reduced to a
+free fermion problem via a mapping to Majorana fermions. This yields an
+extensive number of static \(\mathbb
+Z_2\) fluxes tied to an emergent gauge field. The model is
+remarkable not only for its QSL ground state but also for its
+fractionalised excitations with non-trivial braiding statistics. It has
+a rich phase diagram hosting gapless, Abelian and non-Abelian
+phases [53] and a finite temperature phase
+role="doc-biblioref">56] and a finite temperature phase
transition to a thermal metal state [54]. It been proposed that its
+role="doc-biblioref">57]. It been proposed that its
non-Abelian excitations could be used to support robust topological
-quantum computing [ [55]; [56];
-nayakNonAbelianAnyonsTopological2008].
-
It is by now understood that the Kitaev model on any tri-coordinated
-\(z=3\) graph has conserved plaquette
-operators and local symmetries [57,58] which allow a mapping onto effective
-free Majorana fermion problems in a background of static \(\mathbb Z_2\) fluxes [59–62].
-However, depending on lattice symmetries, finding the ground state flux
-sector and understanding the QSL properties can still be
+quantum computing [58–60].
+
As Kitaev points out in his original paper, the model remains
+solvable on any tri-coordinated \(z=3\)
+graph which can be 3-edge-coloured. Indeed many generalisations of the
+model to [61–65].
+Notably, the Yao-Kivelson model [66]
+introduces triangular plaquettes to the honeycomb lattice leading to
+spontaneous chiral symmetry breaking. These extensions all retain
+translation symmetry, likely because edge-colouring and finding the
+ground state become much harder without it. Finding the ground state
+flux sector and understanding the QSL properties can still be
challenging [63,64].
-
paragraph about amorphous lattices
-
In Chapter 4 I will introduce a soluble chiral amorphous quantum spin
-liquid by extending the Kitaev honeycomb model to random lattices with
-fixed coordination number three. The model retains its exact solubility
-but the presence of plaquettes with an odd number of sides leads to a
-spontaneous breaking of time reversal symmetry. I unearth a rich phase
-diagram displaying Abelian as well as a non-Abelian quantum spin liquid
-phases with a remarkably simple ground state flux pattern. Furthermore,
-I show that the system undergoes a finite-temperature phase transition
-to a conducting thermal metal state and discuss possible experimental
-realisations.
-
-
-
Outline
+href="#ref-eschmann2019thermodynamics" role="doc-biblioref">67,68]. Undeterred,
+this gap lead us to wonder what might happen if we remove translation
+symmetry from the Kitaev Model. This might would be a model of a
+tri-coordinated, highly bond anisotropic but otherwise amorphous
+material.
+
Amorphous materials do no have long-range lattice regularities but
+covalent compounds can induce short-range regularities in the lattice
+structure such as fixed coordination number \(z\). The best examples being amorphous
+Silicon and Germanium with \(z=4\)
+which are used to make thin-film solar cells [69,70].
+Recently is has been shown that topological insulating (TI) phases can
+exist in amorphous systems. Amorphous TIs are characterized by similar
+protected edge states to their translation invariant cousins and
+generalised topological bulk invariants [71–77]. However, research on amorphous
+electronic systems has been mostly focused on non-interacting systems
+with a few exceptions, for example, to account for the observation of
+superconductivity [78–82] in amorphous materials or very
+recently to understand the effect of strong electron repulsion in
+TIs [83].
+
Amorphous magnetic systems has been investigated since the
+1960s, mostly through the adaptation of theoretical tools developed for
+disordered systems [84–87] and with
+numerical methods [88,89].
+Research on classical Heisenberg and Ising models has been shown to
+account for observed behaviour of ferromagnetism, disordered
+antiferromagnetism and widely observed spin glass behaviour [90].
+However, the role of spin-anisotropic interactions and quantum effects
+in amorphous magnets has not been addressed. It is an open question
+whether frustrated magnetic interactions on amorphous lattices can give
+rise genuine quantum phases, i.e. to long-range entangled quantum spin
+liquids (QSL) [91–94].
+
In Chapter 4 I will introduce the Amorphous Kitaev model, a
+generalisation of the Kitaev honeycomb model to random lattices with
+fixed coordination number three. We will show that this model is a
+soluble chiral amorphous quantum spin liquid. The model retains its
+exact solubility but, as with the Yao-Kivelson model [66],
+the presence of plaquettes with an odd number of sides leads to a
+spontaneous breaking of time reversal symmetry. We will confirm prior
+observations that the form of the ground state can be written in terms
+of the number of sides of elementary plaquettes of the model [64,95]. We unearth a rich phase diagram
+displaying Abelian as well as a non-Abelian chiral spin liquid phases.
+Furthermore, I show that the system undergoes a finite-temperature phase
+transition to a conducting thermal metal state and discuss possible
+experimental realisations.
The next chapter, Chapter 2, will introduce some necessary background
to the Falikov-Kimball Model, the Kitaev Honeycomb Model, disorder and
-localisation.
-
In Chapter 3 I introduce the Long Range Falikov-Kimball Model in
-greater detail. I will present results that. Chapter 4 focusses on the
+localisation. Then Chapter 3 introduces and studies the Long Range
+Falikov-Kimball Model in one dimension while Chapter 4 focusses on the
Amorphous Kitaev Model.
@@ -820,22 +916,22 @@ href="https://doi.org/10.1103/PhysRevB.94.245114">Nonequilibrium
Dynamical Cluster Approximation Study of the Falicov-Kimball
Model, Phys. Rev. B 94, 245114 (2016).
-
-
[42]
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-Broholm, R. J. Cava, S. A. Kivelson, D. G. Nocera, M. R. Norman, and T.
-Senthil, Quantum
-Spin Liquids, Science 367, eaay0668
-(2020).
C.
+Broholm, R. J. Cava, S. A. Kivelson, D. G. Nocera, M. R. Norman, and T.
+Senthil, Quantum
+Spin Liquids, Science 367, eaay0668
+(2020).
+
[44]
S.
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Kitaev Magnetism, Journal of Physics: Condensed Matter
29, 493002 (2017).
-
[51]
H.
+
[53]
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Takagi, T. Takayama, G. Jackeli, G. Khaliullin, and S. E. Nagler,
Concept and Realization of Kitaev Quantum Spin Liquids, Nature
Reviews Physics 1, 264 (2019).
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-Hermanns, K. O’Brien, and S. Trebst, Weyl Spin Liquids,
-Physical Review Letters 114, 157202 (2015).
-
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[63]
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Eschmann, P. A. Mishchenko, T. A. Bojesen, Y. Kato, M. Hermanns, Y.
Motome, and S. Trebst, Thermodynamics of a Gauge-Frustrated Kitaev
Spin Liquid, Physical Review Research 1, 032011(R)
(2019).
M.
+Costa, G. R. Schleder, M. Buongiorno Nardelli, C. Lewenkopf, and A.
+Fazzio, Toward Realistic Amorphous Topological Insulators, Nano
+Letters 19, 8941 (2019).
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+Agarwala, V. Juričić, and B. Roy, Higher-Order Topological
+Insulators in Amorphous Solids, Physical Review Research
+2, 012067 (2020).
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[76]
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+Spring, A. Akhmerov, and D. Varjas, Amorphous Topological Phases
+Protected by Continuous Rotation Symmetry, SciPost Physics
+11, 022 (2021).
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[77]
P.
+Corbae et al., Evidence for Topological Surface States in Amorphous
+Bi _ {2} Se _ {3}, arXiv Preprint arXiv:1910.13412 (2019).
+
+
+
[78]
W.
+Buckel and R. Hilsch, Einfluß Der Kondensation Bei Tiefen
+Temperaturen Auf Den Elektrischen Widerstand Und Die Supraleitung Für
+Verschiedene Metalle, Zeitschrift Für Physik 138,
+109 (1954).
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[79]
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+McMillan and J. Mochel, Electron Tunneling Experiments on Amorphous
+Ge 1- x Au x, Physical Review Letters 46, 556
+(1981).
T.
+Kaneyoshi, Introduction to Amorphous Magnets (World Scientific
+Publishing Company, 1992).
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+
[87]
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+Kaneyoshi, editor, Amorphous Magnetism (CRC Press, Boca Raton,
+2018).
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[88]
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+Fähnle, Monte Carlo Study of Phase Transitions in Bond-and
+Site-Disordered Ising and Classical Heisenberg Ferromagnets,
+Journal of Magnetism and Magnetic Materials 45, 279
+(1984).
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[89]
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+Plascak, L. E. Zamora, and G. P. Alcazar, Ising Model for Disordered
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[90]
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+Coey, Amorphous Magnetic Order, Journal of Applied Physics
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+W. Anderson, Resonating Valence Bonds: A New Kind of
+Insulator?, Mater. Res. Bull. 8, 153 (1973).
Like other exactly solvable models Like other exactly solvable models [6] and the Kitaev Model, the FK model
@@ -108,15 +110,17 @@ model exactly solvable, in contrast to the Hubbard model.
Due to Pauli exclusion, maximum filling occurs when each lattice site
is fully occupied, \(\langle n_c + n_d \rangle
= 2\). Here we will focus on the half filled case \(\langle n_c + n_d \rangle = 1\). Doping the
-model away from the half-filled point leads to rich physics including
-superconductivity [\(\langle n_c + n_d \rangle = 1\). The ground
+state phenomenology as the model is doped away from the half-filled
+state can be rich [7].
+role="doc-biblioref">7,8] but from this point we will only
+consider the half-filled point.
At half-filling and on bipartite lattices, FK the model is
-particle-hole symmetric. That is, the Hamiltonian anticommutes with the
-particle hole operator \(\mathcal{P}H\mathcal{P}^{-1} = -H\). As a
consequence the energy spectrum is symmetric about \(E = 0\) and this is the Fermi energy. The
@@ -127,9 +131,11 @@ class="math inline">\(\epsilon_i = +1\) for the A sublattice and
\(-1\) for the even sublattice [8]. The absence of a hopping term for
+role="doc-biblioref">9]. The absence of a hopping term for
the heavy electrons means they do not need the factor of \(\epsilon_i\).
+class="math inline">\(\epsilon_i\). See appendix A.1
+for a full derivation of the PH symmetry.
We will later add a long range interaction between the localised
-electrons so we will replace the immobile fermions with a classical
-Ising field \(S_i = 1 - 2d^\dagger_id_i =
-\pm\tfrac{1}{2}\).
+electrons at which point we will replace the immobile fermions with a
+classical Ising field \(S_i = 1 -
+2d^\dagger_id_i = \pm\tfrac{1}{2}\) which I will refer to as the
+spins.
The FK model can be solved exaclty with dynamic mean field theory in
+
The FK model can be solved exactly with dynamic mean field theory in
the infinite dimensional [9–10–12].
-
-
displays disorder free localisation
-
+role="doc-biblioref">13].
Phase Diagrams
@@ -172,161 +176,245 @@ role="doc-biblioref">12].
+alt="Figure 2: Schematic Phase diagram of the Falikov-Kimball model in dimensions greater than two. At low temperature the classical fermions (spins) settle into an ordered charge density wave state (antiferromagnetic state). The schematic diagram for the Hubbard model is the same. Reproduced from [10,14]" />
Figure 2: Schematic Phase
-diagrams of the Fermi-Hubbard (left) and Falikov-Kimball model (right)
-showing temperature (T) and repulsive interaction strength (U). Hubbard
-model diagram adapted from [13], Falikov-Kimball model from [ [10,14,15]
+role="doc-biblioref">14]
-
-
rich phase diagram in 2d 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 [16].
-
-
At half filling and in dimensions greater than one, the FK model
-exhibits a phase transition at some \(U\) dependent critical temperature \(T_c(U)\) to a low temperature charge
-density wave state in which the spins order antiferromagnetically. This
-corresponds to the heavy electrons occupying one of the two sublattices
-A and B In dimensions greater than one, the FK model exhibits a phase
+transition at some \(U\) dependent
+critical temperature \(T_c(U)\) to a
+low temperature ordered phase [17]. In the disordered region above
-\(T_c(U)\) there is a transition
-between an Anderson insulator phase at weak interaction and a Mott
-insulator phase in the strongly interacting regime [15]. In terms of the immobile electrons
+this corresponds to them occupying only one of the two sublattices A and
+B this is known as a charge density wave (CDW) phase. In terms of spins
+this is an AFM phase.
+
In the disordered region above \(T_c(U)\) there are two insulating phases.
+For weak interactions \(U << t\),
+thermal fluctuations in the spins act as an effective disorder potential
+for the fermions, causing them to localise and giving rise to an
+Anderson insulating state [18].
-
-
superconductivity when doped
-
-
In 1D, the ground state phenomenology as the model is doped away from
-the half-filled state can be rich [19] but the system is disordered for all
-\(T > 0\) [20].
-
In the one dimensional FK model there is no ordered CDW phase 16] which we will discuss more in
+section 2.3.
+For strong interactions \(U >>
+t\), the spins are not ordered but nevertheless their interaction
+with the electrons opens a gap, leading a Mott insulator analogous to
+that of the Hubbard model [17].
+
By contrast, in the one dimensional FK model there is no
+finite-temperature phase transition (FTPT) to an ordered CDW phase [21]. The supression of phase transition
-is a common phenomena in one dimensional systems. It can be understood
-via Peierls’ argument [20,18]. Indeed dimensionality is crucial
+for the physics of both localisation and FTPTs. In one dimension,
+disorder generally dominates: even the weakest disorder exponentially
+localises all single particle eigenstates. Only longer-range
+correlations of the disorder potential can potentially induce
+localisation-delocalisation transitions in one dimension [19–21]. Thermodynamically, short-range
+interactions cannot overcome thermal defects in one dimension which
+prevents ordered phases at non-zero temperature [22–24].
+
However, the absence of an FTPT in the short ranged FK chain is far
+from obvious because the Ruderman-Kittel-Kasuya-Yosida (RKKY)
+interaction mediated by the fermions [25–28] decays as \(r^{-1}\) in one dimension [29]. This could in principle induce the
+necessary long-range interactions for the classical Ising background to
+order at low temperatures [30,22] to be a consequence of the low
+role="doc-biblioref">31]. However, Kennedy and Lieb
+established rigorously that at half-filling a CDW phase only exists at
+\(T = 0\) for the one dimensional FK
+model [32].
+
Based on this primacy of dimensionality, we will go digging into the
+one dimensional case. In chapter 3
+we will construct a generalised one-dimensional FK model with long-range
+interactions which induces the otherwise forbidden CDW phase at non-zero
+temperature. To do this we will draw on theory of the Long Range Ising
+Model which is the subject of the next section.
+
+
+
Long Ranged Ising model
+
The suppression of phase transitions is a common phenomena in one
+dimensional systems and the Ising model serves as a great illustration
+of this. In terms of classical spins \(S_i =
+\pm \frac{1}{2}\) the standard Ising model reads
+
\[H_{\mathrm{I}} = \sum_{\langle ij
+\rangle} S_i S_j\]
+
Like the FK model, the Ising model shows an FTPT to an ordered state
+only in two dimensions and above. This can be understood via Peierls’
+argument [31,32] to be a consequence of the low
energy penalty for domain walls in one dimensional systems.
Following Peierls’ argument, consider the difference in free energy
\(\Delta F = \Delta E - T\Delta S\)
between an ordered state and a state with single domain wall in a
-discrete order parameter. Short range interactions produce a constant
-energy penalty for such a domain wall that does not scale with system
-size. In contrast, the number of such single domain wall states scales
-linearly so the entropy is \(\propto \ln
-L\). Thus the entropic contribution dominates (eventually) in the
-thermodynamic limit and no finite temperature order is possible. In two
-dimensions and above, the energy penalty of a domain wall scales like
-\(L^{d-1}\) so they can support ordered
-phases.
-
-
-
Long Ranged Ising model
-
Our extension to the FK model could now be though of as spinless
-fermions coupled to a long range Ising (LRI) model. The LRI model has
-been extensively studied and its behaviour may be bear relation to the
-behaviour of our modified FK model.
+discrete order parameter. If this value is negative it implies that the
+ordered state is unstable with respect to domain wall defects, and they
+will thus proliferate, destroying the ordered phase. If we consider the
+scaling of the two terms with system size \(L\) we see that short range interactions
+produce a constant energy penalty \(\Delta
+E\) for a domain wall. In contrast, the number of such single
+domain wall states scales linearly with system size so the entropy is
+\(\propto \ln L\). Thus the entropic
+contribution dominates (eventually) in the thermodynamic limit and no
+finite temperature order is possible. In two dimensions and above, the
+energy penalty of a domain wall scales like \(L^{d-1}\) which is why they can support
+ordered phases. This argument does not quite apply to the FK model
+because of the aforementioned RKKY interaction. Instead this argument
+will give us insight into how to recover an ordered phase in the one
+dimensional FK model.
+
In contrast the long range Ising (LRI) model \(H_{\mathrm{LRI}}\) can have an FTPT in one
+dimension.
Renormalisation group analyses show that the LRI model has an ordered
-phase in 1D for $1 < < 2 $ \(1 < \alpha <
+2\) [23]. Peierls’ argument can be
+role="doc-biblioref">33]. Peierls’ argument can be
extended [24] to long range interactions to
+role="doc-biblioref">30] to long range interactions to
provide intuition for why this is the case. Again considering the energy
difference between the ordered state \(\ket{\ldots\uparrow\uparrow\uparrow\uparrow\ldots}\)
+class="math inline">\(|\ldots\uparrow\uparrow\uparrow\uparrow\ldots\rangle\)
and a domain wall state \(\ket{\ldots\uparrow\uparrow\downarrow\downarrow\ldots}\).
+class="math inline">\(|\ldots\uparrow\uparrow\downarrow\downarrow\ldots\rangle\).
In the case of the LRI model, careful counting shows that this energy
-penalty is: \[\Delta E \propto
+penalty is \[\Delta E \propto
\sum_{n=1}^{\infty} n J(n)\]
because each interaction between spins separated across the domain by
a bond length \(n\) can be drawn
between \(n\) equivalent pairs of
-sites. 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 \(\Delta E\) or its
+many-body generalisation converges to a constant in the thermodynamic
+limit then the free energy is analytic [25]. This rules out a finite order phase
+role="doc-biblioref">34]. This rules out a finite order phase
transition, though not one of the Kosterlitz-Thouless type. Dyson also
proves this though with a slightly different condition on \(J(n)\) [23].
+role="doc-biblioref">33].
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
+
For \(\alpha = 0\) i.e infinite
+range interactions, the Ising model is exactly solvable and mean field
+theory is exact [26].
-
$ $ 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
-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 35]. This limit is the same as the
+infinite dimensional limit.
+
For \(\alpha \leq 1\) we have very
+slowly decaying interactions. \(\Delta
+E\) does not converge as a function of system size so the
+Hamiltonian is non-extensive, a topic not without some considerable
+controversy [36–38] that we will not consider further
+here.
+
For \(1 < \alpha < 2\), we get
+a phase transition to an ordered state at a finite temperature, this is
+what we want!
+
For \(\alpha = 2\), the energy of
+domain walls diverges logarithmically, and this turns out to be a
+Kostelitz-Thouless transition [24].
-
$ 2 < $ For quickly decaying interactions, domain walls have a
+role="doc-biblioref">30].
+
Finally, for \(2 < \alpha\) we
+have very quickly decaying interactions and domain walls again have a
finite energy penalty, hence Peirels’ argument holds and there is no
-phase transition.
-
+phase transition.
+
One final complexity is that for \(\tfrac{3}{2} < \alpha < 2\)
+renormalisation group methods show that the critical point has
+non-universal critical exponents that depend on \(\alpha\) [39]. To avoid this potential confounding
+factors we will park ourselves at \(\alpha =
+1.25\) when we apply these ideas to the FK model.
+
Were we to extend this to arbitrary dimension \(d\) we would find that thermodynamics
+properties generally both \(d\) and
+\(\alpha\), long range interactions can
+modify the ‘effective dimension’ of thermodynamic systems [40].
-Figure 3:
+style="width:100.0%"
+alt="Figure 3: The thermodynamic behaviour of the long range Ising model H_{\mathrm{LRI}} = J \sum_{i\neq j} |i - j|^{-\alpha} S_i S_j as the exponent of the interaction \alpha is varied." />
+Figure 3: The thermodynamic
+behaviour of the long range Ising model \(H_{\mathrm{LRI}} = J \sum_{i\neq j} |i -
+j|^{-\alpha} S_i S_j\) as the exponent of the interaction \(\alpha\) is varied.
-
@@ -388,16 +476,24 @@ href="http://adsabs.harvard.edu/abs/2001AcPPB..32.3243J">Falicov--Kimball
Models of Collective Phenomena in Solids (A Concise Guide),
Acta Physica Polonica B 32, 3243 (2001).
A.
@@ -445,26 +532,9 @@ href="https://doi.org/10.1103/PhysRevLett.117.146601">Interaction-Tuned
Anderson Versus Mott Localization, Phys. Rev. Lett.
117, 146601 (2016).
S.
+Aubry and G. André, Analyticity Breaking and Anderson Localization
+in Incommensurate Lattices, Proceedings, VIII International
+Colloquium on Group-Theoretical Methods in Physics 3,
+18 (1980).
44].
+alt="Figure 2: Schematic Phase diagram of the Falikov-Kimball model in dimensions greater than two. At low temperature the classical fermions (spins) settle into an ordered charge density wave state (antiferromagnetic state). The schematic diagram for the Hubbard model is the same. Reproduced from [10,14]" />
-