Topological Adiabatic Transport in Quantum Wires

(1)

Topological Adiabatic Transport in

Quantum Wires

by

Maurits Tepaske

Faculteit der Natuurwetenschappen, Wiskunde en Informatica Institute for Theoretical Physics Amsterdam

Report Bachelor Project Physics and Astronomy

Conducted between 03-04-2017 and 05-07-2017 15 EC

Student Number: 10780831

Supervisor: dr. J. van Wezel

(2)

Populair Wetenschappelijke Samenvatting

Al bestaat de wereld om ons heen voornamelijk uit maar drie deeltjes: protonen, neutronen en elektronen, ervaren wij als mens dagelijks dat grote hoeveelheden van deze deeltjes zich onderling toch heel verschillend kunnen gedragen. Vergelijk bijvoorbeeld hout en metaal: door het verschil in rangschikking van de deeltjes, wat cruciaal is voor het gedrag van deze deeltjes, laat hout nauwelijks elektronen door (isolator) maar metaal juist heel goed (geleider). De invloed van rangschikking op eigenschappen zoals stroomgeleiding komt nog veel extremer terug wanneer we subtielere scenarios beschouwen, die iets verder van de menselijke ervaring verwijderd zijn.

In deze thesis onderzoeken we golven van elektronen in eendimensionale isolator die bestaat uit een rij atomen, zoals weergegeven in het onderstaande figuur.

Hier is de elektronen-dichtheid als golf weergegeven; de elektronen bevinden zich in de rij (blauwe) atomen. De variatie boven en onder de gemiddelde dichtheid, aangegeven met de gestreepte lijn, kunnen als plus en min lading beschouwt worden. Deze rangschikking van elektronen als golf wordt de Charge Density Wave (CDW) genoemd.

Wanneer we de CDW vastpakken en vervolgens voor de afstand λ (in het figuur) slepen, wat gelijk is aan ´e´en golflengte, dan hebben we even veel negatieve als positieve lading verplaatst. Oftewel, er lijkt netto geen lading verplaatst te worden. Maar nu blijkt het tegendeel waar te zijn: er wordt na iedere verplaatsing λ altijd x aantal elektronen verplaatst, bijvoorbeeld x = 1, van het meest linker atoom 1 naar het meest rechter atoom 2 (zie het figuur).

Wanneer we de rij van atomen andere eigenschappen geven, waardoor het systeem zich her-rangschikt, ontstaat soms weer een CDW. Deze transporteert vaak weer precies x elektronen na λ afstand! Dit betekent dat er dus een mechanisme moet bestaan die al deze verschillende soorten atoom-rijen verbindt, en daarmee ervoor zorgt dat x constant blijft.

In deze thesis zoeken we een verklaring voor de constantheid van x, en leren we precies hoe de ’verboden’ elektronen van 1 naar 2 getransporteerd worden. Hiervoor duiken we een abstracte tak van de wiskunde in: de topologie, waarbij alleen groffe eigenschappen van een systeem tellen. Deze grofheid verklaart waarom verschillende CDW systemen gelijke x hebben, ondanks dat ze andere eigenschappen hebben.

(3)

Abstract

In this bachelorthesis we consider the effect of band topology on transport properties of various incarnations of the 1D adiabatic charge pump: regular electrically and kinetically induced charge density waves (CDW), spin density waves (SDW), and CDW on a triangular lattice ladder.

We derive the relation between a bulk band’s Chern number and quantized topological transport, taking place during adiabatic CDW sliding, in the presence of edges. Furthermore, we show that adiabatically slid CDW and longitudinal SDW (LSDW) are mathematically equivalent to the lattice integer quantum Hall effect (IQHE), and we apply Hatsugai’s transfer matrix analysis to both DW phases on an open geometry. This allows us to study the states living on the edges in detail, and in particular how they explain the anti-intuitive adiabatic transport, occurring during the sliding.

Since transverse SDW and CDW on a triangular ladder are merely mathematically similar to the IQHE, we only get to determine how the Chern number relates to transport, and we qualitatively discuss the edge states. Crucially, due to the robustness of the Chern number, we can extrapolate most of the obtained results to more complicated systems.

(4)

1 Introduction

In the last few decades, the broad new field of ’topological condensed matter physics’ has emerged, where the systems under investigation depend on certain topological properties in crucial and often unexpected ways [1–5]. The impact of the initial discoveries on physics has recently been acknowledged with the Nobel prize (2016). Most notably Thouless’ early contributions have shown the generality of such topology-dependent properties [3, 6, 7], paving the way for reinvestigation of a wide range of systems.

One of such systems is the charge density wave (CDW) in a one-dimensional (1D) lattice, which is a bulk insulating phase, and was well-known before the advent of topological phases [8]. Basically, the CDW forms out of a 1D metal because a periodic electronic charge density modulation is energetically favorable, the period of which depends on the electron filling factor of the dispersion. The topological properties of this system and some its extensions, and the various consequences, will be the subject of this thesis.

As we will see shortly, the topology concerns only a small part of CDW, but is nevertheless essential for a complete description. It will also lead to counter-intuitive predictions. In particular, we will show that under certain circumstances quantized electron transport occurs, in between the lattice edges, even though the CDW bulk is insulating. This hints at interesting phenomena, such as gapless states that live on the boundary of the 1D lattice. These will be found to mediate the topological transport. Edge states are actually the hallmark property of a lot of topological phases; for instance, they also arise in the integer quantum Hall effect (IQHE) and topological insulators [9].

The analogy between the IQHE and CDW goes even deeper, since these phases will turn out to be mathematically equivalent. Physically, the IQHE is also characterized by an insulating bulk, and also has gapless edge states and the possibility of edge-to-edge transport [10]. However, since the IQHE involves a magnetic field, and hence must obey gauge-invariance, the specifics of the transport remain obscure. By contrast, the CDW topological transport requires no electromagnetic fields, and can therefore be studied without problem.

We will start the thesis with a brief review of general CDW phenomenology, acquiring the tools to prove mathematical equivalence between the IQHE and CDW. This equivalence will allow us to study CDW using analytic tools developed for the IQHE. It will also allow us to use the gauge-less CDW to reason about the topological IQH transport. To prove it, we will apply mean field theory (MFT) to various CDW-forming Hamiltonians, which we will then compare to the lattice IQH Hamiltonian. In the MFT approximation the interaction Hamil-tonian is decoupled by replacing some operators with their average, implying the dynamical interactions are approximated by a static potential, therefore reducing to a solvable

(6)

single-particle Hamiltonian. By doing this we neglect the fluctuations that are normally present due to the interactions.

Afterwards, we will study the topology of the CDW energy bands, which will be shown to indicate whether edge-to-edge transport takes place, and thus if gapless edge states must be present. To this end, we will formulate a geometrical description of quantum adiabatic transport. To investigate the edge states, we will use an analysis due to Hatsugai [11], originally used to study IQH edge states. The found edge state characteristics will allow us to study the transport in detail. Finally, we will try to apply the mentioned analyses to various extensions of the 1D CDW, namely two variations of the 1D spin density wave, and the CDW on a triangular lattice ladder.

As will be emphasized throughout the text, the topological and consequently robust nature of the properties under investigation allows us to be very general. It permits us to focus mostly on a single CDW model, and extrapolate the results to a large topological equivalence class of CDW Hamiltonians.

(7)

2 CDW formation

To understand how topology enters the 1D CDW phase, we first have to understand why CDW form, and their basic properties. Because the focus of this thesis is CDW topology and its consequences, the scope of the review below is narrow. For a detailed account we refer to [12], for example.

The foundation of all calculation in the first few chapters will be the 1D nearest-neighbor (nn) hopping Hamiltonian H = N X j=1 −t(a†_j+1aj+ a†_jaj+1) + Hint, (1)

describing spinless electrons (i.e. fully polarized), denoted by the ladder operators aj and

a†_j, hopping from sites j to j ± 1 with amplitude t > 0 (on a N -site lattice with spacing a). We have omitted the operator hats, which will be maintained throughout the text. The Hint

term denotes a possible interaction term, which will soon be our focus; for now we neglect it. Employing periodic boundary conditions (PBC), and taking the Fourier transform (FT) of (1) with Hint= 0, using

aj = 1 √ N X BZ eikXj0_a k a†_j = 1 √ N X BZ0 e−ik0Xj0_a† k0, (2)

where X_j0 = ja are the atomic equilibrium positions, immediately yields the diagonal Hamil-tonian

H =X

BZ

ka†kck. (3)

Here k≡ −2t cos(ka) − µ defines the bare (i.e. interaction-less) dispersion, where we added

the chemical potential µ to fix the filling factor through Ef.

Before consider a Hint that caused CDW-formation, we will first qualitatively consider the

phenomenology of a simple CDW model.

2.1 A qualitative picture: CDW in a dimerized lattice

The original example of CDW formation: by dimerization of a 1D lattice with negligible inter-electronic interactions [8, 12], is shown in figure 1 for 1/2-filling of the bare band k.

The lattice distortion has obviously changed the electronic density period, since electrons are tightly bound to the atoms, thereby motivating the terminology ’CDW’.

=⇒

(8)

In this figure, the interaction between electrons and phonons, which is derived and analyzed in appendix A, has caused the unit cell a to double in size, therefore halving the Brillouin zone (BZ). In particular, BZ = [−π, π] has been rescaled to the reduced Brillouin zone (RBZ): [−π/2, π/2], implying that, upon dimerization, the bare dispersion k will be folded into the

RBZ.

In the next section and appendix A we will see that this rescaling is accompanied by the opening of gaps at the RBZ boundary, due to the interaction term, implying that the dimerized CDW dispersion is gapped. Therefore, upon dimerization, the 1/2-filling bare dispersion will change according to figure 2 (where Ef = 0).

→

Figure 2: The result of dimerization, as in figure 1, on the bare nn-dispersion [12], plotted for t = a = 1 and Ef = 0. The BZ in the right plot has been halved, and gaps have opened at the new BZ boundary.

The new upper band is shifted above the lower band.

From the right panel it becomes obvious that all states in the lowest band are occupied, and none in the upper band. Therefore, the gaps at the RBZ boundary lower the CDW energy relative to the metallic phase, implying that the CDW phase is stable.1 Currently, the gaps open due to the electron-phonon interaction, but in the next section we will see that electron-electron interactions also induce CDW.

In the story above we considered only 1/2-filling, for which the lattice period doubled due to formation of periodic charge order with k = 2kf ≡ q, where kf is the Fermi wavevector (π/2a

for half-filling). However, CDW formation is easily seen to occur for general m/n-filling, with m and n integer, characterized by a periodic electronic charge density with k = 2kf = 2π/na.

In the CDW model above this corresponds to the grouping of n atoms, with m electrons per group, since a filled bare dispersion corresponds to one electron per atom. Therefore, the BZ length gets divided by n, and hence the CDW dispersion now consists n separate bands, of which m are filled.

As mentioned, (static) periodic charge ordering occurs for interaction Hamiltonians other

1_{More precisely, the CDW phase is stable because the lowering of energy due to the gaps is larger than the} gain in energy due to the dimerization strain [12]

(9)

than the ’kinetic’ electron-phonon interaction discussed qualitatively above (and quantita-tively in appendix A). Next, we will consider CDW-formation in a system of nn Coulomb-interacting electrons, on a 1D lattice, which will be the main model used in the rest of the thesis. In this case the lattice is taken to be rigid, implying that a periodically modulated electron density arises without deformation of the lattice.

2.2 A quantitative picture: Coulomb interacting electrons

As stated, we continue by considering nn Coulomb-interacting spinless electrons on a 1D lattice, for which we have

Hint= U N

X

j=1

a†_jaj(a†_j+1aj+1+ a†_j−1aj−1), (4)

where U parametrizes the Coulomb coupling strength between neighboring electrons. This model will prove to be most useful in determining properties of the edge states that mediate the topological transport, as it will be seen that the MFT approximation of this Hamiltonian is mathematically equivalent to the IQH phase on a lattice (unlike the ’kinetic’ system from the previous section and appendix A). This is not the only model with mathematical IQH equivalence. For example, if Hint represents the interaction between the lattice polarization

and the electrons, we will also find equivalence [13, 14].2

Instead of a qualitative picture, we now opt for a quantitative description, in which MFT will be used to show that gaps indeed open at the RBZ boundary, and that in this approxi-mation mathematical equivalence with the IQHE holds.

Starting off, we identify the CDW order parameter as the electronic charge density, which we expect to become periodic with k = q in the CDW phase. Therefore, we approximate a†_j±1aj±1 in (4) as ha†_j±1aj±1i = ρ0cos(qa(j ± 1) + φ), where ρ0 signifies the amplitude of

charge density modulation. Crucially, we incorporated the ’sliding angle’ φ, which will turn out to be crucial for the topological aspect of the analysis. As a result, the CDW dispersion is extended to (k, φ)-space. Also, we have omitted the average charge density, which merely shifts the bare dispersion in energy.

φ signifies the possibility of (experimentally) shifting the charge density with respect to the lattice, allowing us to slide the CDW a full wavelength. In figure 3 we show how a strong potential spike U ’pins’ a valley of the periodic charge density modulation ∆ρ, which could either be the sliding mechanism or a positively charged impurity [12]. By slowly sliding the potential with velocity v, the CDW will shift along with it. We will talk more about experimental aspects of the sliding in a later chapter.

2

(10)

Figure 3: Pinning of the periodic charge modulation ∆ρ(x) by a strong potential spike U (t), which is time-dependent (via the velocity v) if it resembles the sliding mechanism, and time-independent if it is a charged impurity. The striped line denotes the average charge density and the signs signify charge hills and valleys.

Back to the Hamiltonian, we FT the MFT approximation of (4), yielding the quadratic Hamiltonian

Hint≈

X

BZ

ζeiφc†_k+qck+ h.c., (5)

where ζ ≡ ρ0U cos(qa)/N . At this point, the k-sum still runs over the BZ, while in writing

the order parameter we have assumed the system is in the CDW phase. Therefore, we must manually rescale the summation interval, while maintaining the Hamiltonian’s physical struc-ture. This is accomplished by adding extra k-shifted operator products, such as a†_k+2qak+q.

To make this clear, it is shown in figure 4 how this allows us to reach the original BZ from the RBZ, in the case of m/3-filling. From this figure we also see that ck+3q = ck.

Figure 4: Rescaling of the k-sum, by employing extra ladder operators to reach the original BZ from the RBZ, shown for m/3-filling. The different arrows, combined with the colored curve, show the domains of the different operators.

(11)

section, the period of the purely electronic CDW arising out of (4) is not strictly commensurate with the lattice (i.e. the order parameter does not require q = 2π/na). This is because the CDW period is no longer based on groups of atoms, since the lattice is relaxed by construction. Nevertheless, the rescaling procedure depicted in figure 4 fails when q 6= π/na, thereby excluding incommensurate CDW from our analysis. Hence, the rest of the theory developed in this thesis, built around the rescaling, will only concern commensurate CDW.

Back to the the m/3-filling CDW, we rewrite (5) as

Hint= ζ

X

RBZ

eiφ(c†_k+qck+ c_k+2q† ck+q + c†_kck+2q) + h.c., (6)

where we used ck+3q = ck and defined RBZ ≡ [−π/na, π/na] (of course quantized in ∆k =

2π/L). After substituting (6) in (1), the full MFT Hamiltonian in matrix form finally becomes

H = X RBZ h a†_k a†_k+q a†_k+2q i    

k ζe−iφ ζeiφ

ζe−iφ k+q ζe−iφ

ζeiφ ζe−iφ k+2q

        ak ak+q ak+2q     . (7)

In exactly analogous manner, we determine the half-filling MFT Hamiltonian as

H = X RBZ h a†_k a†_k+q i " k 2ζ cos(φ) 2ζ cos(φ) k+q # " ak ak+q # . (8)

It is easily seen that the rescaling procedure generalizes to arbitrary m/n-filling, allowing us to construct a n × n matrix similar to those above. However, we will not do this, because in this thesis we only consider 1/2- and m/3-filling CDW in detail.

In the next section, we will use (7) to compare the IQHE and CDW.

2.3 Mathematical CDW-IQH equivalence

With the Hamiltonians (7) and (8) we have all ingredients necessary to study the topology arising in 1/2- and m/3-filling CDW systems. However, before proceeding with this, we will first compare (7) and (8) to the corresponding lattice IQH k-space Hamiltonians, which will reveal the mathematical equivalence. As mentioned in the introduction, this will indicate whether we can Hatsugai’s IQHE analysis to study CDW edge states [11]. Moreover, it will allow us to draw analogies between the topological transport in both phases.

To fully grasp the connection between the IQHE and CDW, we start off with a qualitative overview of some basic IQH aspects; for a proper exposition we refer to [10]. For later convenience, let us consider a long conducting cylinder in a strong perpendicular magnetic field B, shown from top view in figure 5.

(12)

Figure 5: A top-down view of the IQH-setup. The long cylinder is a conductor, and the arrows represent a strong perpendicular magnetic field B, which causes the bulk to become insulating [10].

Semi-classically, we know that this magnetic field localizes the free bulk electrons, since these are now forced to follow small ’cyclotron’ orbits around a field line [15]. The different types of orbits constitute different energy levels, and it is well-known that for a rational amount ν = m/n of magnetic flux quanta per unit cell, the dispersion consists of n separate bands. This is analogous to the CDW theory discussed above.

The only exception to this localization are the electrons at the edges, located at the bases of the cylinder, which follow a cycloid along the edge, as they cannot complete a full orbit without ’hitting the wall’. Moreover, because all electrons orbit in the same direction, the edge electrons must propagate in opposite directions, resulting in currents of opposite chirality. Both types of motion are shown in figure 6.

Figure 6: The different types of motion that electrons in a periodic slab of 2D conductor execute, under the influence of a strong perpendicular magnetic field [10]. Cyclotron motion results in a localized state, whereas the cycloid skipping orbits result in extended chiral states. The figure is adapted from topocondmat.org.

We will later see, by extrapolation from the CDW scenario, that these edge states medi-ate the topological edge-to-edge transport. Moreover, these chiral currents turn out to be

(13)

quantized by the same mechanism as for the topological CDW transport, namely by being proportional to the Chern number of an energy band(s) [3, 7].

If we would consider nn-hopping electrons on a square 2D lattice in a strong perpendicular magnetic field, with lattice spacing a, and FT the corresponding Hamiltonian, we would find the following Hamiltonian k-components for ν = 1/2 and ν = m/3, respectively:

" ˜ k −2t cos(kxa) −2t cos(k_xa) ˜k+q #     ˜

k −te−ikxa −teikxa

−te−ikxa _˜

k+q −te−ikxa

−teikxa _−te−ikxa _˜

k+2q     , (9)

where ˜k≡ −2t cos(kya) and q ≡ 2πν/a [16]. Comparison with (7) and (8) shows

mathemat-ical equivalence, with the identifications kxa ↔ φ, ky ↔ k, and on the off-diagonal −t ↔ ζ.

For the CDW ν corresponds to the filling factor.

Crucially, instead of the usual dispersion that lives fully in k-space, such as for the IQH model considered above (since it is 2D), the CDW Hamiltonian has a dispersion in (k, φ)-space. The dynamical parameter φ entered Hintas the order parameter’s sliding angle, implying that

different φ (mod 2π) characterize different systems, with the electron density slid differently with respect to the lattice. Therefore, we cannot talk about filling the CDW bands up to Ef. Instead, we will have to talk about filling each separate system, labeled by φ, up to

Ef. Moreover, since the CDW MFT Hamiltonian conserves the electron number,3 all these

different systems must have an equal amount of electrons. This difference in interpretation of the dispersion will mainly have implications later on, when we consider the edge state dynamics.

Concluding, we have shown that the IQH and CDW phases have the same spectrum, saving for the difference in numerical constants and interpretation. Properties depending only on global aspects of the spectrum are consequently also equivalent for both phases, including the ’band topology’, which will play a key role in the remainder of this thesis. Importantly, the equivalence requires the possibility of CDW sliding, encapsulated by φ.

In the next chapter we will introduce and study the notion of band topology in detail, and see how it manifests in observable phenomena.

3

To see this, notice that all terms in e.g. (7) consist of an equal amount of creation and annihilation operators.

(14)

3 Topologically quantized adiabatic transport

At the end of last chapter we hinted at the equivalence of the IQH and CDW bands in some topological property, but why would we care about such an abstract attribute? Usually observables involve local quantities; think for example of the electron group velocity, which depends on the dispersion’s slope. By contrast, topology is inherently global.

3.1 Berry’s phase

Since the sliding of CDW is required for quantized transport to occur, we will first consider the conditions this sliding must obey, and its consequences. Starting off, remember that the CDW was found to be an insulating ground state. Therefore, to prevent the CDW from collapsing, φ will have to be varied ’adiabatically’, basically implying that the CDW has to be slid very slowly, so as not to provide enough energy to excite electrons across the bandgap [17].4 Hence, even though φ modifies the Hamiltonian, the CDW always remains in the instantaneous ground state, occupying for each value of φ only the lowest bands of the 1D dispersion. Consequently, the only remaining degree of freedom is its phase.

At first sight, this seems to indicate a dead end, since global U (1) phase can usually be gauged away. However, it turns out that under some conditions the adiabaticity requirement allows this phase to become physical [19]. To investigate this, we will consider in detail the dynamics of the CDW ground state as we move through the (k, φ)-manifold, e.g. the lowest 1/3-filling CDW band, never exiting across the gap.

We start from the Schr¨odinger equation (~ = 1)

i∂t|ψ(t)i = H( ~P (t)) |ψ(t)i , (10)

where |ψ(t)i denotes the CDW state living in the (k, φ)-manifold, ~P (t) = (k(t), φ(t)) is a time-dependent vector in the manifold, and H( ~P (t)) is in our case defined by (7). The time dependence enters solely through k and φ, which are the parameters we vary adiabatically. At this point, k seems out of place, as it does not label the CDW as a whole. However, we will only care about its integral, so for now we act as if it is a genuine control parameter. Now, suppose |ψi starts out in the state |ψ0(0)i, the ground state of H(k(0), φ(0)). Since we

vary (k, φ) adiabatically, |ψi at a later time must equal

|ψ(t)i = eiθ(t)|ψ0(t)i = e−i Rt

0E0(t 0_)dt0

eiγ(t)|ψ0(t)i . (11)

Here we separated the usual dynamical phase, depending on the energy eigenvalue E0(t) of

|ψ₀(t)i, from possibly another contribution γ(t). By using µ to set E(t) = 0, we may drop

4

More precisely, if electrons are exited across the bandgap, they screen the gap-inducing interaction, and hence close the gaps after some threshold number of excitations [18]. This destroys the CDW phase.

(15)

the dynamical phase. Crucially, the system must always remain in the ground state of its current Hamiltonian |ψ0(t)i, not the initial |ψ0(0)i.

To determine γ for the case of adiabatic variation, we simply substitute (11) in (10), yielding

i∂t|ψ0(t)i = (∂tγ(t)) |ψ0(t)i . (12)

To obtain this, we divided out the exponential and made use of the energy eigenvalue equation. Importantly, this step requires that for all t the ground state is non-degenerate, i.e. the bands never cross. By taking the left inner product with hψ0(t)|, we find

∂tγ(t) = i hψ0(t)| ∂t|ψ0(t)i . (13)

Using that ∂t= ∇_{P (t)}~ · ∂tP (t), the last equation becomes~

∂tγ(t) = i hψ0(t)| ∇_P~|ψ0(t)i · ∂tP (t) ≡ A( ~~ P (t)) · ∂tP (t),~ (14)

which is easily seen to be solved by

γ(t) = Z

P

A( ~P (t0)) · d ~P . (15)

In (14) we defined the ’Berry connection’ 1-form A, and we integrate over some path P in the manifold. Notice that time dependence enters solely through the initial and final points of P.

Therefore, we have found that for adiabatic sliding we have an extra phase γ(t). To uncover possible physical meaning, we will consider the gauge transformation |ψ₀0(t)i → exp[iG( ~P (t))] |ψ0(t)i, using the global U (1) freedom at each ~P (t). This transforms A

ac-cording to

A( ~P (t)) → A( ~P (t)) − ∇_P~G. (16)

Therefore, A is gauge-dependent, which is fine since we only care about γ(t). Substituting (16) in (15), we find that γ transforms as

γ(t) → γ(t) − [G(P (t)) − G(P (0))], (17)

where t denotes the final time. It follows that γ(t) is gauge-dependent for general t, and thus unphysical for general adiabatic processes.

However, something special occurs when P describes a closed path. Since |ψ₀0(t)i must be single-valued, and hence exp[iG(P (t))] also, it follows from P (t) = P (0) that G(P (t)) and G(P (0)) can only differ by 2π times an integer. It immediately follows from (17) that the ’Berry phase’ exp[γ] is now gauge-independent.5.

(16)

We conclude that adiabatic, periodic variation of H( ~P (t)) results in the physical Berry phase γ; see [20] for various incarnations. To derive an alternative expression for γ, which will be used in the remainder of the thesis, we recognize that a closed line integral on the (k, φ)-manifold can be converted to a surface integral via Stokes’ theorem. Therefore, we may rewrite (15) as γ = 1 2 Z S FµνdPµ∧ dPν. (18)

Here S is the surface bounded by P, and we introduced the ’Berry curvature’ 2-form F , defined in coordinates as Fµν= ∂A ν ∂Pµ − ∂A µ ∂Pν = i ∂ψ0(t) ∂Pµ ∂ψ0(t) ∂Pν + h.c. (19)

Note that the labels ’connection’ and ’curvature’ have actual geometrical meaning, since from (14) the Berry connection can be identified with an element of the Lie algebra u(1), making the Berry phase an abelian holonomy [10, 21]. Furthermore, (19) is simply the definition of an abelian curvature in terms of the corresponding connection.

These objects define a principal U (1)-bundle over the (k, φ)-manifold [22], which contains the global phase freedom at each state in the band (i.e. its fiber is C2( ~P )). Since this phase is the only remaining degree of freedom, we recognize this vector bundle to be the object that fully specifies the dynamics of adiabatically slid CDW.

3.2 Intermezzo: vector bundles

For those not acquainted with basic aspects of the U (1) vector bundle, we here give a heuristic explanation in terms of familiar physics. For a more in-depth and quantitative coverage, see for example [20–23].

Basically, this ’bundle’ is an extra layer of geometrical structure, assigning to each point in the manifold under consideration a vector space, called a ’fiber’. For example, the manifold could be the usual space-time, and the fiber at some space-time point could be the space of all vectors tangent to this point. All these extra vector spaces, one for each space-time point, together form the ’tangent bundle’. In this thesis the manifold is an energy band in (k, φ)-space, which we dubbed the (k, φ)-manifold,6 and instead of considering tangent vectors, we identified the global phase at (k, φ) as the relevant extra level of structure. Together, all these global phases form a local gauge phase, connected via the Berry phase. Therefore, we identify the complex plane as the appropriate fiber.

In the easiest case, the bundle E with fiber V over some manifold M is ’trivial’, implying that it can be written as M × V , in which case the bundle is globally defined. However, when

6

Since both dimensions are periodic, and all bands are non-degenerate, we easily identify this manifold as a torus.

(17)

E is ’non-trivial’, we can maximally write it as a set of Cartesian products {Ei× Mi}, defined

on different patches of manifold Mi (i.e. on different ’trivializations’) that together cover M .

The coordinates of these Mi are connected by ’transition functions’, and to compare vectors

at different points in M , in different Mi, we require a ’connection’ C.

For the tangent bundle C simply tells us how to parallel transport a vector. For the bundle over the (k, φ)-manifold C equaled A, and was found to encode the evolution of a state’s phase as we move adiabatically through the (k, φ)-manifold (see (15)). In particular, we can move a state through the whole manifold by multiplying an initial state with certain complex phases that are specific to the system under consideration, determined by the Berry curvature through (18). These phases are elements of the group U (1), consisting of 1 × 1 unitary matrices. Since these multiplications connect all the fibers C2 over the manifold, fully specifying the dynamics of adiabatically slid CDW, we call the corresponding bundle a ’U (1)-bundle’

Furthermore, there is a unique Berry connection to each adiabatically evolved system, since A is the single ”natural” connection on a U (1)-bundle [21]. Also, C directly defines the ’curvature’ of the bundle (see (19)), which importantly does not represent the actual bending of the bands.

Notice that by switching out the (k, φ)-manifold for Euclidean space, keeping the U (1)-bundle, we have the scenario where a state merely acquires phase as it moves through space. This is simply the ’Aharonov-Bohm effect’ for charged particles [17], which also becomes apparent from (15), (18) and (19), when we identify the vector potential with A and the field strength with F .

This concludes the intermezzo. In the next section, we will use these concepts in a topo-logical analysis of Berry’s phase.

3.3 From Berry’s phase to topological transport

It is easily seen how topology enters the adiabatic CDW, having identified the relevant geo-metrical structure. Namely, for an abelian bundle F /2π defines the first Chern class, known to result in the integer-valued first Chern number C1 when integrated over a closed manifold

(such as the (k, φ)-torus) [22, 24]. At first sight this seems to be a trivial statement, since if we consider some closed manifold M, and apply Stokes’ theorem, we find that

C1 ∝ Z M F =! Z M dA = Z ∂M A = 0, (20)

where we used that a closed M has an empty boundary ∂M. This seemingly implies that C1 ≡ 0.

(18)

However, in the second step of (20) we used F = dA, which is only valid if the interior of the (k, φ)-manifold is simply connected [22]. Therefore, whenever the interior is occupied by a F -monopole, i.e. whenever it contains a phase singularity, (20) does not hold, and C1 instead equals the ’winding number’ [24]. Consequently, it is impossible to construct a

single eigenvector that is well-defined on the whole manifold. This integer basically tells us how many times the manifold wraps around the singularity, and therefore increases when the amount of singularities grows. Thus, the magnitude of C1 indicates how topologically

non-trivial the band is, which will become clear when we compare the C1-calculations of the lowest

and middle m/3-filling bands. Again, the story above has a counterpart in electromagnetism, where integrating a magnetic field over a closed surface always gives zero, except when it originates from a magnetic monopole in the interior.

In the language of differential topology, a non-zero C1 indicates that the U (1)-bundle is

non-trivial, implying that we require transition functions to define the phase on the whole (k, φ)-manifold [24]. This is often stated as the fact that C1 6= 0 characterizes a twisted ’band

topology’. Of course, this viewpoint is equivalent to that involving the F -monopole, which is instead based in homotopy [22]. Because this non-triviality requires us to work in at least two different gauges, integration over the full manifold becomes less straightforward, which will become apparent when we compute C1 for the 1/3-filling CDW. Before doing this, let us

briefly sketch a general outline of this calculation.

For concreteness, let us consider a 1/3-filling CDW, in which case the eigenstate in (19) lives in the lowest band. Then, if we solve (7) for this eigenstate, subsequently use (19) to compute F for the lowest band eigenstate, and finally integrate it over the closed (k, φ)-manifold, we will find an integer that topologically characterizes the lowest CDW band. Since F is gauge-invariant, this integer is also gauge-invariant and therefore of physical relevance.

The obvious next question is: what physical property does C1 correspond to? Since the

CDW is a 1D band insulator, and φ is varied adiabatically and periodically by some external potential, it is easily seen to be another instance of the iconic ’adiabatic charge pump’ [3, 6]. By considering the first order adiabatic correction to the electron group velocity of a CDW band, which would reveal that this correction equals exactly the Berry curvature [25], we would have found

cn= − Z 2π 0 jn(φ)dφ = − Z 2π 0 dφ Z BZ F_n(k, φ) 2π dk, (21)

where jn is the adiabatically induced current at sliding angle φ, due the nth band, and cn is

the amount of transported electrons for this band after completing one φ-period.

Finally, we have shown that topologically quantized electron transport occurs in the adi-abatically slid CDW. More precisely, the amount of electrons transported due to a band,

(19)

after a full sliding period, depends solely on the topology of the U (1)-bundle over this band. The strangeness of this transport becomes apparent upon the realization that by sliding the CDW for a single wavelength, we have pictorially transported both a hill and valley of charge density modulation, seemingly implying that the nett transport should equal zero (see figure 3). Hence, the topological transport is anti-intuitive.

It is also important to realize that this transport only occurs in the presence of lattice boundaries, as will become explicit in the next chapter. Namely, we determined the bulk (PBC) spectrum of the CDW to consist solely of bulk bands, either fully occupied or unoc-cupied, implying that for the pure bulk CDW there are simply no states that could carry the adiabatic current. This poses no conceptual problem, as (21) is derived by simply recognizing the first order adiabatic correction as the Berry curvature [25], i.e. without physical initiative. In the next chapter, we will incorporate boundaries into the lattice, which will result in the appearance of ’edge states’ that live on the edges and occupy the bandgap. These will be seen to mediate the adiabatic transport. Also, since nonzero cn dictates the presence of edge

states, even though cn is derived purely from the bulk Hamiltonian (7), there appears to be

a correspondence between edge and bulk.

Due to the global nature of topological properties, C1, and hence cn, is robust to local

perturbations of the Hamiltonian. In particular, since C1 is integer-valued, we cannot

con-tinuously deform it. Pictorially, this means that C1 does not change when we slightly deform

the corresponding band. Instead, changing C1 requires altercation of the band topology,

in-duced by closing the bandgap, which violates the prerequisites of F (see derivation of (12)). Therefore, due to the global nature of cn, it labels an equivalence class of gapped

Hamiltoni-ans, related to each other by adding or subtracting local perturbations, rather than a single Hamiltonian.

Importantly, even though the CDW model (7) is not necessarily realistic, we can make use of the robustness to add all sorts of higher order terms that make the Hamiltonian more realistic. Provided these terms do not close the gap, cn does not change, implying that the

cn’s of (7) will generalize to more realistic Hamiltonians. Finally, we stress the rarity of global

observables, making topological transport a truly remarkable phenomenon.7

3.4 Calculation of the 1/3-filling Chern number

Having identified cn ∝ C1 as the incarnation of topology in the adiabatically slid CDW, we

can now explicitly calculate cnfor the 1/3-filling CDW Hamiltonian (7). Since only the lowest

band is occupied, we use the energy eigenstate |ψi in the lowest band to compute F , and

7

Of course, since the IQH and CDW were shown to be mathematically equivalent, we could have easily guessed how topology entered, as the Hall conductance is quantized for the same reason as cn.

(20)

subsequently integrate over the (k, φ)-torus to determine the total transport ctot = c1. To find

|ψi, we need to diagonalize the matrix in (7). Ideally, we would do this exactly, and hence obtain a state defined on the whole (k, φ)-manifold. However, setting up the characteristic polynomial quickly shows that this is unrealistic, as we will find a third order polynomial with (k, φ)-dependent coefficients.

Therefore, we will resort to approximation, which we can justify by invoking the robustness of cn, as this implies that we can deform the Hamiltonian into one that is easier to solve,

without changing cn. To understand which deformations are allowed, we note that F is

con-centrated around the near-degeneracies. More precisely, since degenerate points are sources of Berry curvature [17], it follows that after formation of the CDW gaps (see figure 2), F is largest near the minimum bandgap energy. This indicates that we only need to maintain the structure of (7) around these near-degeneracies, since far from the source points F is small, implying that in this region we can deform the Hamiltonian to one easily solved. Similarly, when we consider H in the vicinity of a near-degeneracy, we only retain the degeneracy-lifting, off-diagonal elements most relevant to this point. This corresponds to the lowest order of per-turbation theory, which is again justified by the robustness of cn. Using the approximate

Hamiltonians at the near-degeneracies, we can determine the corresponding eigenvectors, and finally compute cn. If this happens to be nonzero, the band’s topology has been complicated

by one or multiple F -singularities in the (k, φ)-manifold’s bulk, which is a remnant of one or multiple bare degeneracies.

Let us now execute this procedure for (7). Firstly, the near-degeneracies of the lowest band obviously always occur at the RBZ boundary: ±kf = ±π/3a. Furthermore, using q = 2kf,

we find that the k− k+2q degeneracy occurs at k = π/3a, and the k− k+q degeneracy

at k = −π/3a. The degeneracy of k+q and k+2q is currently uninteresting, since 1/3-filling

involves only the lowest band. Starting at k = −π/3a, we approximate (7) as

H ≈     t ζe−iφ 0 ζeiφ t 0 0 0 −2t     , (22)

which is block diagonal and hence easily solved for the relevant eigenvalues

E±= t ± ζ, (23)

from which the gap at k = −π/3a becomes apparent. The corresponding normalized eigen-vectors are easily seen to be

ψ± −π 3a = √1 2     e−iφ ±1 0     , (24)

(21)

where |ψ−i lives in the lowest band, and |ψ+i in the middle band. Because in the used

approximation we consider the eigenstates at single k-points, k-dependence is lost, which is needed to compute F in (19). Therefore, we need to somehow implement this manually.

Turning to the other near-degeneracy point k = π/3a, we employ the exact same procedure as for k = −π/3a, yielding the eigenstates

ψ± π 3a E = √1 2     eiφ 0 ±1     , (25)

again corresponding to E± = t ± ζ. Crucially, the RBZ periodicity dictates that k = ±π/3a

label the same physical state. However, since (24) and (25) are not connectible via a global U (1) phase, these eigenstates seem to be different. This is simply a manifestation of the U (1)-bundle’s non-triviality, which is linked with the existence of a F -monopole in the interior of the lowest band. More precisely, both states live in different and incompatible trivializations, implying that a global definition of |ψ−i’s phase requires the use of transition functions. We

will circumvent this by cleverly integrating F . Remember that we noted a nonzero cn to

characterize this non-triviality, implying that this difficulty in globally defining the state’s phase is the fundamental ingredient for nonzero topological transport.8

To deal with this non-triviality, we divide the (k, φ)-manifold in two k-patches, trivialized such that (24) and (25) hold. For the first patch M1 we take [−π/3a, 0] × [0, 2π], and for the

second M2 we take [0, π/3a] × [0, 2π]. Hence, the patches’ boundaries, both of different

orien-tation (since we endow Mi with the orientation of M), coincide along k = 0 and k = ±π/3a.

The purpose of this division is that Stokes’ theorem now applies to both patches separately, as they are trivial by construction. Therefore, the k = 0 and k = ±π/3a boundaries contribute nothing to the sum of integrals over the patches, leaving an integral over the full (k, φ)-torus, which equals C1 if the integrand is F . Notice that the orientation of the (k, φ)-manifold is

fixed by (21), as is required, since transport has a well-defined direction (defined physically by the CDW sliding direction).

To integrate over M1 and M2, we first introduce k-dependence by defining the general

states |Θ(k, φ)i =     Θ1(k, φ)e−iφ Θ2(k, φ) Θ3(k, φ)     |∆(k, φ)i =     ∆1(k, φ)eiφ ∆2(k, φ) ∆3(k, φ)     , (26)

where |Θ(k, φ)i is defined on M1 and |∆(k, φ)i on M2. The φ-dependency is chosen to match

8

(22)

(24) and (25). Using these states, we find the real integrals cn= Z (k,φ) F 2π = Z M1 FΘ 2π + Z M2 F∆ 2π = 2 Z 0 −π/3a Θ1∂kΘ1dk − 2 Z π/3a 0 ∆1∂k∆1dk, (27)

where we used (19) to determine

F_∆01(k, φ) = −2 Re{∆1∂k∆1} FΘ01(k, φ) = 2 Re{Θ1∂kΘ1}, (28)

and also used the antisymmetry of F .

It follows from (27) that the Hamiltonian enters the C1 calculation solely through the

limiting eigenvectors at k = ±π/3a and k = 0. Thus, to connect with the Hamiltonian (7), we simply require |∆(−π/3a, φ)i = |ψ−(−π/3a)i and |Θ(π/3a, φ)i = |ψ−(π/3a)i. Because

k = 0 is far from both near-degeneracies, we are allowed to use bare eigenvectors as the k = 0 integration limits. We choose these as

|γi =     eiφ 0 0     γ0 =     e−iφ 0 0     , (29)

for M1and M2, respectively. Finally, (27) yields c1= 1, i.e. the 1/3-filling CDW has nonzero

adiabatic topological transport.

Note the passive role of φ throughout this story: the non-triviality stems solely from the k-dimension, as can be seen from (26), where we could explicitly write the φ-dependence of the general states. Nevertheless, it is an essential ingredient, since only integration of F (k, φ) over the full (k, φ)-manifold is required to return an integer. Furthermore, the C1

-calculation depended solely on the eigenvectors (24) and (25), which are independent of the system-specific constants, reflecting the global nature of C1.

Next, we will calculate the Chern numbers of the remaining bands.

3.5 Calculation of C1 for other fillings

We continue our study of CDW, covering a large topological equivalence class of Hamiltonians related to (4), by considering a few other filling factors. We start with 2/3-filling, which also corresponds to (7), only now with the two lowest bands filled. Here, the topological transport is governed by the Chern numbers of both bands, which simply add since the bundles over these bands are independent. Luckily, the only relevant difference between these bands is the amount of near-degeneracies. In particular, the second band has three such points, instead of two; from figure 4 we see that the extra k+q− k+2q degeneracy is located at k = 0.

(23)

Since the calculation is almost exactly analogous to the 1/3-filling case, we will only state the main results, omitting various steps. For |ψ±(0)i, where + now corresponds to the third

band and − to the second, we find

|ψ±(0)i = 1 √ 2     0 1 ±eiφ     , (30)

and the k = ±π/3a eigenvectors are again (24). Of course, we now use the ’+’ states in (24). These three states are all in different trivializations, implying that the calculation of C1 for the

second band requires integration over four different patches. In other words, the U (1)-bundle over the second band is even less trivial. In this case, we place bare boundary eigenvectors at k = ±π/6, and define four general states to implement k-dependence. Determining F on all patches, and subsequently integrating, yields c2 = −2. As expected, we have found that

more non-triviality results in a larger |C1|. To specify ctot for the 2/3-filling CDW, we simply

add c1 and c2, yielding ctot = −1. We conclude that topological transport also occurs for

2/3-filling, with the same magnitude as for 1/3-filling, but in the different direction. For the 3/3-filling CDW there are never empty states, implying that ctot= 0 must hold, and therefore

that the third band must have c3 = 1.

The procedure above is easily seen to generalize to larger n and m, since figure 4 indicates that for any n the highest filled band always has either two or three near-degeneracies, at k = 0 and k = ±π/na.9 An exception to this rule is 1/2-filling, for which we derived (8). Namely, the off-diagonal components vanish at φ = π/2 and φ = 3π/2, implying that the two-band structure has degeneracies. Therefore, both bands form a single manifold, and integration of F over the lowest band, up to the degeneracies, is no longer required to return an integer. Physically, this occurs because at these φ the CDW is no longer insulating, therefore violating the assumption of adiabaticity and consequently a prerequisite for Berry’s phase. Summarizing, except for 1/2- and m/m-filling, all m/n-filling CDW arising out of (4), and more generally the corresponding equivalence classes, support topological transport.

Having used purely the bulk Hamiltonian to predict transport that must involve edge states, emphasizing the correspondence between the bulk and the edges, we will spend the next chapter analyzing CDW with actual lattice boundaries incorporated into the model.

9

However, as n grows the RBZ shrinks, until for some n F becomes significant everywhere. At this point the approximation is no longer valid.

(24)

4 Edge states

We concluded at the end of last chapter that adiabatic sliding of e.g. 1/3-filling CDW results in topologically quantized adiabatic transport. However, since we have been working with PBC (purely in the bulk), we could not investigate the implications. More precisely, ctot has

so far been a meaningless quantity, because the bulk dispersion was seen to contain no states capable of mediating adiabatic transport.

In this chapter, we consider CDW on an open geometry (OBC), in which case ctot

signi-fies actual transport, and use Hatsugai’s transfer matrix analysis to uncover the specifics of this transport [11]. At first sight, probing properties of CDW always seems to require the attachment of measuring devices to the sample, implying that any testable 1D CDW theory requires edges. We will get back to this important point in a later section.

The implications of finite ctot seem baffling at first sight, since the bulk described by (4) is

insulating for large U , seemingly leaving only the edges as mediators of the transport. This in turn seems to imply instantaneous tunneling between the well-separated edges. Luckily, the bulk spectrum will be seen to contain some delocalized states, which form conduction channels between the edges when occupied, causing charge to accumulate at one edge and deplete at the other. Since the transport takes place under adiabatic conditions, with filled bands and Ef in the gap, the edges must contain gapless states, ready to accept electrons.

We will find that these edge states, when occupied, mediate the topological transport. In particular, they will turn out to be highly dynamical, with different localization properties on different φ-intervals. By using Hatsugai’s analysis [11], these ideas will be made quantitative, showing that the CDW dispersion for OBC indeed encodes such dynamics.

Another argument for the existence of gapless edge states derives from the mentioned robustness of the Chern number. This implies that a boundary, separating a topologically non-trivial CDW from the trivial vacuum, must be accompanied by a closing of the bandgap, as only deformation this radical can change the Chern number.

We again emphasize that the results of this chapter apply to the whole equivalence class, consisting of a wide range of CDW Hamiltonians (and the IQHE). This is possible because we only care about global properties of the edge states, exemplified by the fact that we have not even solved a single MFT self-consistency equation. Because we expect such properties, e.g. the amount of edge states per gap, to depend on the bands’ Chern numbers, we also expect them to be conserved under local perturbation of the Hamiltonian. In physical terms, all these CDW systems have similar characteristics, indicating that the global edge state properties should not be much different either.

(25)

interacting Hamiltonian in the presence of edges.

4.1 CDW on an open geometry

Due to the presence of edges, the system’s wavefunction can no longer be translation-invariant, preventing us from FT and subsequently diagonalizing in k-space. Instead, we have to deter-mine the dispersion working in real space. Once again considering the Coulomb interacting CDW, we start from the real space MFT Hamiltonian

H ≈

N

X

j=1

−t(a†_jaj+1+ a†j+1aj) + ρ0Ucos qXj+10 + φ + cos qXj−10 + φ a †

jaj. (31)

To determine the spectrum of H, we proceed by substituting the general single particle state |ψi = P

j0ψ_j0(φ)a†

j0|0i into H |ψi = E |ψi,10 and require equality of the basis states’

coefficients on both sides of this vector equation, yielding

ψj+1+ ψj−1= −E t + ξj ψj −→ " ψj+1 ψj # = " −E/t + ξj −1 1 0 # " ψj ψj−1 # . (32)

Here ξj ≡ (2ρ0U/t) cos(qja + φ) cos(qa), |0i denotes the vacuum, and ψj is the probability

amplitude at the jth site. We recognize the left equation as the IQH Harper equation [11], with the usual identifications, as expected from the mathematical equivalence in k-space. The right equation is the ’transfer matrix’ representation of Harper’s equation, with the matrix Tj, which forms the core of Hatsugai’s analysis.

Since (32) describes the wavefunction by coupling neighboring sites, we can implement edges by forcing the wavefunction to vanish at some lattice sites. Ideally, we would like to consider a true lattice strip, where the wavefunction vanishes past the edge sites. However, as we will shortly see, this setup is incompatible with Hatsugai’s method.

Instead, we go back to the lattice chain (PBC), and imitate edges by forcing wavefunction zeros on two sites. To this end, we impose ψj = ψj+2nz, appropriate for a lattice totaling

2nz ≡ 2L/a sites, with nz integer. Here n is the amount of sites in an enlarged unit cell, and z denotes the total amount of such cells. By enforcing the constraint ψ0 = ψnz = 0, we

effectively divide the chain into two separate lattice strips. Note that this choice of OBC, which will turn out to be crucial to Hatsugai’s analytic scheme, is approximate on one side of the lattice. Namely, the zero at j = nz is placed just inside the final unit cell, implying we have one strip with nz−1 sites and one with nz+1. Therefore, both strips are incommensurate with the CDW period, which is commensurate with nz sites by construction. We will come

10

Remember, determining the spectrum of a single electron is enough, as the MFT Hamiltonian is non-interacting.

(26)

back to this issue later, ignoring it for now. Figure 7 contains a schematic of the setup, and in the rest of this chapter we only consider the nz − 1 site strip.

Figure 7: The lattice chain used in Hatsugai’s edge state analysis [11], where grey sites correspond to forced wavefunction zeros, which imitate boundaries.

In the next section, we will describe the general procedure that will be used to determine edge state dispersions.

4.2 Hatsugai’s transfer matrix method

In the last section, we have set the stage for application of Hatsugai’s transfer matrix method to study CDW edge states [11], which we expect to mediate adiabatic topological transport. In the following sections, we will fully develop this method, starting from the transfer matrix in (32).

Using (32), we construct the transfer matrix T that couples amplitudes at the ends of a unit cell: " ψn+1 ψn # = Tn...T2T1 " ψ1 ψ0 # ≡ T " ψ1 ψ0 # . (33)

Since the lattice strip consists of roughly z such cells, T can in turn be used to couple the boundary sites: " ψnz+1 ψnz # = (T )z " ψ1 ψ0 # , (34)

where ψnz+1 lies just outside a boundary and ψ1 just inside. Inserting ψ0 = ψnz = 0 results

in " ψnz+1 0 # = (T )z " ψ1 0 # −→ (Tz)21= 0, (35)

since ψ1 is arbitrary. This expression contains all information of our MFT Hamiltonian and

can thus be solved for the dispersion, consisting of both edge and bulk states. One solution is (T )21 = 0, because T is upper diagonal for this choice and a product of such matrices is

also upper diagonal [11]. Importantly, (T )21 = 0 holds for wavefunctions with the stricter

ψ0 = ψn = 0, implying that the full lattice problem is reduced to one with only n − 1 sites;

(27)

(T )21= 0 contains all edge state energies. This will also show how the bulk spectrum relates

to that of the boundary.

We start off by removing the forced zeros, therefore again considering solely the bulk. This restores translational symmetry, implying that Bloch’s theorem holds, and hence that the wavefunctions satisfy

ψj+n = ρψj = eiθψj, (36)

where θ depends on the energy ofP

jψj. Subduing (33) to this condition, we find

" ψn+1 ψn # = T " ψ1 ψ0 # = ρ " ψ1 ψ0 # , (37)

which is simply an eigenvalue equation, implying that ρ must satisfy

"

T11− ρ T12

T21 T22− ρ

#

= 0 −→ ρ2− ρ tr T + det T = 0. (38)

From (32) we see that det Tj = 1, allowing us to simplify and solve (38) by using det T =Qnj=1det Tj:

ρ2− ρ tr T + 1 = 0 −→ ρ±=

∆ ±√∆2_{− 4}

2 , (39)

where ∆ ≡ tr T . If ∆2 ≤ 4, it follows from√∆2_{− 4 = i}√_{4 − ∆}2 _{that |ρ}_±_{| = 1, implying that}

∆2 < 4 corresponds to bulk states. This also implies that ∆2 > 4 labels the non-bulk states, which are the edge states.

To show that T21= 0 contains only edge states, we first note that T21= 0 implies det T =

1 = T11T22, and consequently

∆ = T11+ T22= T11+

1 T11

.

By considering the first and second derivatives of this equation with respect to T11, we find

that T11= ±1 minimizes ∆, implying that |∆|2 > 4. We conclude that T21= 0 contains only

edge states. Since T21 is a (n − 1)th order polynomial, as is easily seen from (32) and (33),

there are n − 1 edge states in T21= 0. Furthermore, it turns out that each gap contains one

such state (when we have an approximate boundary) [11].11

To prove that these are the sole edge states, we first note from (32) and (33) that ∆ = tr T is a polynomial in E and therefore continuous. Hence, ∆2 _{= 4 defines smooth boundaries}

in φ-space, delimiting bulk from edge territory. In other words, these curves are the bulk band boundaries, where the bands are now understood as the ’thick’ bands that form by projecting the (k, φ)-band structure on the φ-axis. It follows from (32) that (T )11 is a nth

(28)

order polynomial, thus having n roots. Consequently, ∆2 = 4 has 2n roots, implying that there are 2n band boundaries, corresponding to n thick bands (as anticipated).

Finally, we show that (Tz)21 = 0 does not contain extra edge states. To this end, we

redefine the CDW wavelength as λ = nz, which is no problem because it is commensurate with λ = z. In this case, ∆ corresponds to Tz and is consequently of order nz. Therefore, ∆2 _{= 4 has 2nz roots, corresponding to nz bands. Since the only difference with the previous}

scenario is a redefinition, it follows that the band structure must be identical. Therefore, each group of z succeeding new bands, starting from the bottom, must coalesce, resulting in n bands. Since each gap contains a single edge state [11], it follows that the redefinition maximally introduces extra bulk states. It turns out that these are the states which glue the z new bands, for all n bands, implying we get z − 1 extra states per band. Adding these to the n − 1 edge states, we have a total of nz − 1 solutions. Because (Tz)21 = 0 has nz − 1

roots, it follows that we have found all solutions, and therefore that the only edge states are the n − 1 contained in T21= 0.

The argument above relied heavily on the fact that the amount of edge states depends on the amount of bulk bands, or vice versa. This again shows that the bulk is intertwined with the edges, which became apparent earlier on, when we predicted quantized transport (necessarily involving edge states) purely on basis of bulk topology. More precisely, since each gap contains a single edge state [11], cn fully specifies the edge dispersions’ global properties.

We will continue by determining the dispersions and localization properties of the m/3-filling edge states.

4.3 Edge states of m/3-filling CDW

Unfortunately, we cannot tackle T21= 0 without specifying a filling factor, as no closed form

has yet been found for the product of an arbitrary amount of Tj’s [9]. Here we will solve

(T )21= 0 for m/3-filling, for which the transfer matrix equals

T = T3T2T1 = " −E_t + ξ3 −E_t + ξ2 −E_t + ξ1 +E_t − ξ3+E_t − ξ1 ∗ −E t + ξ2 −E t + ξ1 − 1 E t − ξ2 # , (40)

where ∗ denotes an irrelevant expression. From (40) we immediately find the two edge state dispersions E±= t 2 ξ1+ ξ2± p (ξ1− ξ2)2+ 4 . (41)

Besides (41), the transfer matrix also encodes their ’behavior’. As mentioned, we expect these states to be localized at one edge for small φ, merge with the bulk for later φ, and finally localize at the other edge. To determine the specifics of this process, and to see if it coincides

(29)

with the predicted amount of transport ctot, we first use (T )21= 0 to write for z CDW unit cells [11]: " ψnz+1 0 # = " (T )z₁₁ ∗ 0 ∗0 # " ψ1 0 # , (42)

where both asterisks abbreviate irrelevant expressions. Crucially, the simplicity of the upper left entry is a result of (T )21= 0. Multiplying out (42) yields

ψnz+1

ψ1

= (T )z₁₁, (43)

which can be used to prove exponential localization of the edge states.

In particular, if we consider a large crystal (i.e. z → ∞), |(T )11| < 1 results in |ψnz+1/ψ1| →

0, implying localization at the left boundary.12 Another option: |(T )11| > 1, corresponds to

localization at the right boundary. The last option is |(T )11| = 1, in which case |ψnz+1/ψ1| = 1

holds independent of z, implying that the wavefunction is delocalized. When we take U → ∞ in (4), electrons in the bulk are obviously localized, analogous to the IQHE cyclotron orbits. However, we have just now shown that states living on the boundary of the thick bands are delocalized. These must be the channels allowing for topological edge-to-edge transport, as will be confirmed later.

At this point, we can no longer ignore the approximate right boundary, since localization j = nz + 1 means that nearly all weight of the wavefunction is located outside the lattice (see figure 7). This is obviously unphysical, while for the delocalized state it is an excellent approximation on a large lattice. Therefore, to obtain exact results, we will do all calculations twice, also for an exact right boundary. Afterwards, we will discard the parts where a state is localized at an approximate boundary.

Unfortunately, we cannot avoid this difficulty by simply replacing ψn+1 with ψn−1 in (35),

by leaving out Tn in (33), since this spoils the analytical structure of Hatsugai’s scheme. In

particular, applying ψ0= ψnz= 0 to this new transfer matrix equation results in (T )z11 = 0,

which does not support a simple solution such as (T )11= 0. In other words, the (nz + 1)-site

is intrinsic to Hatsugai’s analysis, and we are forced to calculate twice. It also prevents us from using the usual PBC: ψ1 = ψnz+1.

We start by constructing a transfer matrix that admits exact right boundary. From (33) and the inverse of Tj:

T_j−1 = " 0 1 −1 −E/t + ξ # , (44)

(30)

it is clear that we may write " ψ1 ψ0 # = T₁−1T₂−1T₃−1 " ψ4 ψ3 # −→ " ψ0 ψ1 # = T1T2T3 " ψ3 ψ4 # , (45)

which indeed suits ψnz+1 = 0. We define this new transfer matrix as T0 = T1T2T3. Using

(32), T₂₁0 = 0 yields the energies

E±0 = t 2 ξ2+ ξ3± p (ξ2− ξ3)2+ 4 . (46)

We now have four dispersions: (41) and (46), two for both choices of approximate BC. To determine which pieces we drop, we first have to plot T11and T110 for φ ∈ [0, 2π], to determine

for which φ the state is localized at the ’bad’ boundary. This is done for (41) and (46), i.e. m/3-filling CDW, in figure 8.

Figure 8: Left: (T0)11 plotted for E±0 from φ = 0 to 2π, corresponding to a single adiabatic sliding

period. The lower curve is E−, the upper curve is E+, and the yellow and orange lines correspond to

|(T )11| = 1, which can be regarded as a localization phase boundary. The left boundary is approximate.

Right: (T )11 plotted for E± from φ = 0 to 2π. Everything holds as for (T0)11, except now with an

approximate right boundary.

The yellow and orange lines in figure 8 correspond to |(T )11| = 1, implying that everywhere

within these lines a state is localized at the left boundary and outside these lines at the right boundary; a state on one of the lines corresponds to delocalization. The left plot is for an exact left boundary, so that we must discard (46) for φ where |T11| > 1. The right plot is for

an exact right boundary, and we must therefore drop (41) at φ where |T11| < 1. The exact

edge dispersions are now obtained by combining the leftover parts (keeping the delocalized states). Before analyzing the resulting dispersions, we first turn to the bulk bands.

To incorporate the bulk bands into a plot of the edge dispersions, we use the earlier result that ∆2 = 4 contains the band boundaries. To find the full projected band structure we simply fill in areas between the appropriate boundaries.13 It is easily seen that tr T = tr T0,

(31)

which means that the bands computed from both transfer matrices are equal. From (40) we find ∆2= −E t + ξ1 −E t + ξ2 −E t + ξ3 +E t − ξ1+ E t − ξ2+ E t − ξ3. (47) Plotting the solutions of ∆2= 4 with (47) alongside the edge dispersions (41) and (46), on the appropriate intervals as indicated by figure 8, results in figure 9 (where we chose t = ρ0U = 1).

Figure 9: The projected bulk bands together with the edge dispersions, plotted for a single sliding period. Strings of edge states connect the different bands.

By comparing figure 9 to the plots in figure 8, we can see that the sign of an edge dispersion’s slope depends on its location, in analogy with the IQHE [10]. In particular, at φ with negative slope the state exists on the left edge and for positive slope on the right edge. Zero slope corresponds to delocalization, which only occurs at points where the edge string merges with a band, in accordance with the earlier result that the band boundaries correspond to delocalized states. There is one exception to this rule, since slope changes close to a band are seen not to change localization. We will soon show that figure 9 is in accordance with m/3-filling ctot, as

calculated in chapter 3. Also, it agrees with the numerical plot for the ν = m/3 IQHE from [9].

Since we no longer deal with approximate BC, the statement that each gap contains a single edge state has become invalid. Namely, we have φ-shifted part of the edge dispersions in figure 9 by correcting this approximation, resulting in ’overlap’. Nevertheless, we still have n − 1 edge dispersions at all φ.

Next, we turn to dynamical aspects of the topological transport, by considering the evolu-tion of the dispersion’s occupaevolu-tion during a sliding period.

(32)

4.4 Laughlin’s gauge argument

Knowing the edge states’ location as a function of φ, we finally have all ingredients necessary for a dynamic description of the topological transport. We again emphasize that the dispersion shown in figure 9 does not correspond to a single system, as for the IQHE, but rather to an infinite set of different systems, labeled by φ. It is also important to remember that figure 9 is plotted for arbitrary constants, not fixed by some MFT self-consistency equation. This means that only global properties are realistic, such as the amount of crossings per bandgap, in contrast with e.g. the φ for which they occur.

However, before analyzing the CDW dispersion, we briefly consider ”Laughlin’s gauge ar-gument” [26]. This predicts quantized edge-to-edge transport for an IQH sample in a tunable flux Φ(t) (on top of the strong background field B), purely on basis of gauge invariance. This argument will show us why a similar analysis for CDW, performed in the next section, shines light on the dynamics of IQH quantized transport. We again use the cylinder in figure 5 as the IQH setup, this time also with a uniform Φ(t) parallel to the axis.

We will use figure 9 as the IQH cylinder’s dispersion, which is fine since the mathematical CDW-IQH equivalence implies that this figure also corresponds to some IQH dual of the m/3-filling CDW. In this light we identify φ → k, where k corresponds to the circumferential direction, making explicit that for the IQHE the projected dispersion corresponds to a single system. The edge states now correspond to chiral Hall currents on the cylinder bases [10].14

We choose Ef to cross the lowest edge state degeneracy, and we fill the whole k-dispersion

with particles. Because the electrons are spinless, as a result of full polarization by the strong B, they only couple to Φ(t) through their charge, via k → k + A(t) (e ≡ 1), where A = Φ(t)/L with L the cylinder circumference [27]. This implies that varying Φ from 0 to 2π shifts all wavevectors with exactly one k-space quantum 2π/L. Therefore, the Hamiltonians at 0 and 2π have an equivalent dispersion, up to a possible reoccupation of states. Hence, they are connectible via a gauge transform. More precisely, the phase acquired at Φ = 2π can be removed by a gauge transformation that leaves the wavefunction boundary conditions, and hence the Hilbert space, invariant [27]. For 0 < Φ < 2π this is not possible, implying that the corresponding Hamiltonians characterize physically different systems.

If we vary Φ adiabatically, then upon completion of a Φ-period all occupied states in figure 9 will simply shift to the right with ∆k = 2π/L. This effectively excites one electron past Ef,

since an electron on the right-localized string is pushed above Ef, and a hole arises on the

left-localized string below Ef. Therefore, after one flux period an electron has been transported

14

Instead of using a T11 plot, we can now simply read off an edge state’s location from the dispersion, by employing the definition of velocity ˙q = ∂pH. To this end, remember that the semi-classical argument from section 2.3 implied that chirality is related to location via the magnetic field’s direction.

Topological Adiabatic Transport in Quantum Wires