The discrete quantum Hall system

Our model of interest will be the discrete or tight-binding quantum Hall system as considered in [MC96]. This model allows our constructions and computations to be as transparent as possible. In the case without boundary, where H = `2(Z2), we have magnetic translations Ub and Vb acting as unitary operators on `2(Z2). These operators commute with the unitaries U and V that generate the Hamiltonian H = U +U∗+V +V∗. We choose the Landau gauge so that

(U λ)(m, n) =b λ(m−1, n), (V λ)(m, n) =b e−2πiφmλ(m, n−1), (U λ)(m, n) =e−2πiφnλ(m−1, n), (V λ)(m, n) =λ(m, n−1),

whereφhas the interpretation as the magnetic flux through a unit cell andλ∈`2(Z2). We note that UbVb = e2πiφVbUb and U V = e−2πiφV U, so C∗(U ,b Vb) ∼= Aφ, the rotation algebra, and C∗(U, V) ∼= A−φ. We can also interpret A−φ ∼= Aop_φ , where Aop_φ is the opposite algebra with multiplication (ab)op = bopaop. To see this identification we compute, b UopVbop = b VUb op =e−2πiφUbVb op =e−2πiφVbopUbop.

Our choice of gauge also means that C∗(U ,b Vb) ∼= C∗(Ub)o_η Z, where Vb is imple- menting the crossed-product structure via the automorphism η(Ubm) =Vb∗UbmVb. Such a decomposition of the algebraC∗(U ,b Vb) will be useful for when we consider a bulk-edge system (see Section 4.2.2and Remark4.2.7).

Next, we impose a boundary on the system and consider the Hamiltonian acting on the half-plane `2₍

Z×N). The Hamiltonian now takes the formHhp=Uhp+Uhp∗ + Vhp+Vhp∗, where (Uhpλ)(m, n) =e−2πiφnλ(m−1, n), (Vhpλ)(m, n) =χ(n−1)λ(m, n−1), and χ(n) =    n, n≥0 0, n <0 .

Our gauge choice comes with the corresponding magnetic translations

(Ub_hpλ)(m, n) =λ(m−1, n), (Vb_hpλ)(m, n) =χ(n−1)e−2πiφmλ(m, n−1). Lemma 4.2.1. The operators Uhp andVhp commute with Ub_hp andVb_hp.

Proof. We shall consider the case [Uhp,Vb_hp]. One checks that (UhpVb_hpλ)(m, n) =e−2πiφn(Vb_hpλ)(m−1, n)

=χ(n−1)e−2πiφne−2πiφ(m−1)λ(m−1, n−1), (VhpUhpλ)(m, n) =b χ(n−1)e−2πiφm(Uhpλ)(m, n−1)

=χ(n−1)e−2πiφme−2πiφ(n−1)λ(m−1, n−1) and so [Uhp,Vb_hp] = 0. The other cases follow similar arguments.

We find that in the presence of a boundary there are still unitaries Uhp and Uhp,b but nowVhp and Vb_hp are partial isometries, where

V_hp∗Vhp =Vb_hp∗ Vb_hp = 1, V_hpV_hp∗ =Vb_hpVb_hp∗ = 1−Pn=0.

Remark 4.2.2 (A note on boundary conditions). Our choice of translations are encoding Dirichlet-style boundary conditions at n = 0. We note that changing the boundary conditions in the discrete/tight-binding picture will change the operators Vhp and Vhpb by a finite-rank operator. Hence the difference is a compact perturbation.

One of our reasons for choosing ‘Dirichlet translations’ is that such a choice has a natural link to the representation theory of semigroups (say Z×N or R×[0,∞)). Define Wk = Uk1

hpV k2

hp and cWk = Ub_hpk1Vb_hpk2 for k ∈ Z×N and σ(k, k0) = e2πiφk

0

1k2 _for

k, k0 ∈_Z×_N. It is a simple check that σ is a semigroup 2-cocycle forZ×N.

Lemma 4.2.3. The operator W generates a σ-representation of Z×N. The operator c

Proof. We first compute that (Wkλ)(m, n) = (Uk1 hpV k2 hpλ)(m, n) =e −2πiφk1n_(V hpλ)(m−k1, n) =χ(n−k2)e−2πiφk1n_{λ(m−k1, n−k2).}

We need to show that WkWk0 =σ(k, k0)Wk+k0. We compute, (WkWk0λ)(m, n) =χ(n−k2)e−2πiφk1n(Wk 0 λ)(m−k1, n−k2) =χ(n−k2)e−2πiφk1n_{χ(n−k2−k}0 2)e−2πiφk 0 1(n−k2)_{λ(m−k1−k}0 1, n−k2−k02) =e2πiφk10k2_{χ(n−k2−k}0 2)e −2πiφ(k1+k01)n_λ(m−_k1−_k0 1, n−k2−k 0 2) =σ(k, k0)(Wk+k0λ)(m, n),

where we have used that χ(n−k2)χ(n−k2−k0₂) =χ(n−k2−k0₂) for all k2, k₂0 ∈_N. This gives the result for Wk.

Next we find (Wckλ)(m, n) = (Ub_hpk1Vb_hpk2λ)(x) =χ(n−k2)e−2πiφ(m−k1)λ(m−k1, n−k2). Then, (cWkWck 0 λ)(m, n) =χ(n−k2)e−2πiφ(m−k1)χ(n−k2−k20)e−2πiφk 0 2(m−k1−k01) ×λ(m−k1−k01, n−k2−k20) =e−2πiφk2k10_χ(n−_k2−_k0 2)e−2πiφ(k2+k 0 2)(m−k1−k01)_λ(m−_k1−_k0 1, n−k2−k20) =σ(k, k0)(Wck+k 0 λ)(m, n)

as required. Finally [W,Wc] = 0 by Lemma4.2.1.

We may use analogous arguments from the proof of Lemma4.2.3to obtain a similar result for the adjoint operators W∗ and Wc∗. We omit the details for brevity.

Lemma 4.2.4. Let σ∗(k, k0) =e−2πiφk1k20 for k, k0 ∈_Z×_N. The operatorsW∗ (resp. c

W∗) generate a σ∗-representation (resp. σ∗-representation) ofZ×N. Furthermore, the

two representations commute.

Our notation of σ∗(k, k0) = e−2πiφk1k20 is reasonable as σ∗(k, k0) = σ(k0, k). To recapitulate, the map k 7→ Wk _{is a right σ-representation of the semigroup}

Z×N, with k 7→ Wck the corresponding left σ-representation that commutes with Wk. An analogous result holds fork7→(W∗)k with σ∗ and k7→(cW∗)k.

One can also consider the HamiltonianHhp =Uhp+U_hp∗ +Vhp+V_hp∗ +g(m, n), where g(m, n) is a bounded periodic potential, that isg∈`∞(Z×N) andg(m+k1, n+k2) = g(m, n) for any k ∈ _Z×_N. Such a Hamiltonian will still commute with Wck, which encodes the magnetic translations.

Considering algebras with shift operators on the half-plane has put us in the do- main of Toeplitz algebras and, in particular, Toeplitz extensions. It was observed in [SBKR00,SBKR02] that we can link a system without boundary with an edge sys- tem via such an extension.