Laughlin arguments in arbitrary dimensions

arXiv:1705.04162v1 [math-ph] 11 May 2017

Laughlin arguments in arbitrary dimensions Alan L. Carey1 and Hermann Schulz-Baldes2 1

Mathematical Sciences Institute, Australian National University, Canberra, Australia

and the School of Mathematics and Applied Statistics, University of Wollongong, Australia 2 Department

Mathematik, Friedrich-Alexander-Universit¨ at Erlangen-N¨ urnberg, Germany

and Instituto de Matem´aticas, UNAM, Unidad Cuernavaca, Mexico

Abstract The standard Laughlin argument shows that inserting a magnetic flux into a twodimensional Hamiltonian leads to a spectral flow through a given gap which is equal to the Chern number of the associated Fermi projection. For higher even dimension, the insertion of a non-abelian Wu-Young monopole is shown to lead to a spectral flow which is again equal to the strong invariant given by a higher even Chern number. For odd dimensions, an associated chirality flow allows to calculate the strong invariant. This follows from an index theorem for the spectral flow between two unitaries which are conjugates of each other by a selfadjoint unitary. Keywords: monopole, spectral flow, index pairings MSC numbers: 58J30, 37B30

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Overview and summary of results

The classical form of Laughlin’s argument [11, 2] considers a Landau Hamiltonian describing a two-dimensional electron in a constant magnetic field in which a magnetic flux tube is inserted at some point. This produces supplementary discrete spectrum between Landau levels which flows through a given gap while pushing the flux through. The outcome is that the spectral flow is equal to the Chern number of the Fermi projection below the given gap. While the analysis in [11, 2] uses the particular form of the Landau operator, the equality of the spectral flow resulting from a flux insertion and the Chern number is a structural fact which can also be referred to as two-dimensional topological charge pump. In particular, no constant magnetic field is needed for a non-trivial spectral flow, merely a non-vanishing Chern number. For example, a flux inserted into a disordered, but gapped Haldane model leads to a unit spectral flow. These results were established in [6] for a gapped tight-binding Hamiltonian h on ℓ2 (Z2 )

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(based on ideas from [13]). Let us recall the crucial facts from Sections 2 and 3 of [6], which will also follow from the more general analysis given in this paper. Let the magnetic flux α ∈ R added to one specified cell of the lattice Z2 be realized by the Aharanov-Bohm gauge (obtained by integrating the continuous Aharanov-Bohm gauge, see Section 3 below). This results in a family α ∈ R 7→ hα of bounded Hamiltonians on ℓ2 (Z2 ) with h0 = h for which the following holds: (i) hα − h0 is compact, so that hα and h0 have the same essential spectrum. (ii) hα+1 = V ∗ hα V where V =

X1 +ıX2 |X1 +ıX2 |

encodes the phase of the (dual) Dirac operator.

(iii) For µ in a gap of h and pµ = χ(h ≤ µ), (where χ denotes the characteristic function of the interval in question) one has Sf(α ∈ [0, 1] 7→ hα − µ) = Ind(pµ V pµ ). (iv) Ind(pµ V pµ ) is equal to the strong invariant given by the non-commutative Chern number Ch2 (pµ ) if one deals with a covariant family of Hamiltonians (e.g. periodic or disordered). Items (i) and (ii) are linked to the Aharanov-Bohm gauge. For other choices of the gauge, one may not have compactness of hα − h0 even though hα and h0 still have the same spectrum and the spectral flow is the same. Item (iii) is the main result of [6] while the index theorem of (iv) is by now a classical fact, e.g. [19]. Let us add a few comments. First of all, the particular relation hα+1 = V ∗ hα V is irrelevant, crucial is merely the continuity of α → hα and that the initial point h = h0 and final point h1 are unitarily equivalent. This follows from structural facts reviewed in Section 2. Furthermore let us stress that hα is not equal to (V α )∗ h0 V α where V α is the α-th root. Such a unitary equivalence would imply that there is no spectral flow. The first main result of this paper (Theorem 3) states that items (i), (ii) and (iii) above also hold for matrix-valued Hamiltonians in higher even dimensions, provided that the flux insertion results from a non-abelian Wu-Yang monople [24, 22]. The construction is done by following the strategy in the two-dimensional case [1] and is described in Section 3. In Section 4 the form of the Hamiltonian and the effect of the monopole on it is then specified. Given this, Theorem 3 is a direct consequence of Phillips’ result [16] (Theorem 1 recalled in Section 2) on the spectral flow between unitarily conjugate selfadjoint Fredholm operators. Item (iv) then also holds by the index theorem proved in [17], see also [19]. The second main result of this paper (Theorem 4) shows how inserting a Wu-Yang monopole into an odd-dimensional chiral system allows one to detect the strong invariant. This invariant is again given by an index of a Fredholm operator which is linked to a generalized winding number [20], also called an odd Chern number, by an index theorem [18, 19]. Associated to the insertion of the Wu-Yang monopole into a chiral Hamiltonian, one still has a spectral flow (albeit of the Fermi unitary) which is why we also call it a chirality flow for reasons explained below. To describe this flow, let us first recall that a (local) chiral symmetry of an invertible Hamiltonian H on ℓ2 (Zd , C2N ) is of the form JHJ = −H where J = diag(1N , −1N ) is the chiral symmetry operator. Then one finds that the so-called flat band Hamiltonian H|H|−1 is   0 U∗ −1 , H|H| = U 0 2

where the so-called Fermi unitary U acts on ℓ2 (Zd , CN ). In Section 5, it is shown that the insertion of the monopole leads to a path α ∈ [0, 1] 7→ Hα of chiral Hamiltonians with H0 = H. Generically, this path is invertible so that there are Fermi unitaries α ∈ [0, 1] 7→ Uα with U0 = U. Typically, the spectrum of these unitaries fills the whole unit circle. The crucial facts, corresponding to those in the even dimensional case, are: (i)′ Uα − U0 is compact. (ii)′ U1 = F U0 F where the selfadjoint unitary F is the phase of the (dual) Dirac operator. (iii)′ Sf(α ∈ [0, 1] 7→ F Uα U0∗ ) = Ind(ΠUΠ) where Π = 21 (F + 1) is the Hardy projection of F . (iv)′ Ind(ΠUΠ) is equal to the strong invariant given by the non-commutative odd Chern number Chd (U) if one deals with a covariant family of chiral Hamiltonians. Items (i)′ and (ii)′ follow again from the construction of the monopole. For d ≥ 3, there is also a relation Uα = F U1−α F which corresponds to the relation in (ii), but this is of no importance for the definition of the spectral flow and the claim in (iii)′ . Indeed, the path α ∈ [0, 1] 7→ F Uα U0∗ of unitaries connects two selfadjoint unitaries F and U0 F U0∗ with spectrum {−1, 1} and, as also F Uα U0∗ − F is compact, the above spectral flow counts the eigenvalues moving between them. Theorem 2, a general corollary of Phillips’ Theorem, allows us to show that this spectral flow is equal to the index in (iii)′ . A generalized statement of (iii)′ is given in Theorem 4 in Section 5. It is shows that the index in (iii)’ is equal to the spectral flow from JF to HJF H −1 which justifies the terminology chiral flow. Item (iv)′ is proved in [18, 19] and these references also contain the definition of Chd (U). To give the reader some traction, let us illustrate the claims (i)′ to (iii)′ on a simple onedimensional toy model, the Su-Schrieffer-Heeger model with vanishing mass and no disorder. The Hamiltonian with inserted flux α in one cell of the 2-strip acts on ℓ2 (Z, C2 ). Using Dirac’s bra-ket notations, it is   X 0 Sα α , S = |nihn + 1| + eıπα |0ih1| , (1) Hα = (S α )∗ 0 n6=0

so that the Fermi unitary is Uα = S α . The Dirac phase F is the sign of the position operator, namely X X F = |nihn| − |nihn| . (2) n>0

′

n≤0

′

One readily checks that all items (i) to (iii) hold. In particular, X X F S α (S 0 )∗ = |nihn| − eıπα |0ih0| − |nihn| , n>0

n