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-Universität Erlangen-Nürnberg, Germany
and Instituto de Matemáticas, 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
1
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 )
1
(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<0
so that Sf α ∈ [0, 1] 7→ F S α (S 0 )∗ = 1 = Ind(ΠS 0 Π). While this results from a direct
elementary calculation, the homotopy invariance of both quantities in the equality in (iii)′
3
allows one to extend it to a model with a random mass term, as discussed in [14, 19]. The
higher odd-dimensional situation is treated in Section 5.
Let us add a short comment on what is omitted in this paper, but will be dealt with elsewhere. As shown in [6] for the two-dimensional case and in [4] for a particular one-dimensional
model, one can implement real symmetries to the Hamiltonian h0 or H0 and then analyze the
resulting symmetry properties of the spectral flow when a flux is inserted. These real symmetries are typically given by a time-reversal or a particle-hole symmetry. Such a symmetry
analysis is also possible for higher dimensional models. On the index side of the equalities in
(iii) and (iii)′ , this was already carried out in [7]. When the monopole is inserted, the equations
hα+1 = V ∗ hα V and Hα = F H1−α F will then be relevant. In some situations, one can then
prove the existence of bound states for half-flux α = 21 , see the example in [6].
2
Spectral flows between unitary conjugates
We assume that the reader is familiar with the definition of the spectral flow of a path of
selfadjoint Fredholm operators on a Hilbert space H and its basic properties as given in [15],
or alternatively [16, 5]. Let us begin by recalling a well-known result of Phillips [16] connecting
the spectral flow between unitarily conjugate selfadjoint Fredholm operators and an associated
index pairing.
Theorem 1 Let α ∈ [0, 1] 7→ Tα be a path of selfadjoint Fredholm operators with invertible end
points T0 and T1 . Furthermore, suppose that there exists a unitary U such that
T1 = U ∗ T0 U .
If P = χ(T0 ≤ 0), then P UP is a Fredholm operator on Ran(P ) with index given by
Ind(P UP ) = Sf α ∈ [0, 1] 7→ Tα .
The proof in [16] is based on the index of Fredholm pairs of projections [3]. A proof by a
homotopy argument is given in [5]. In this section we prove a unitary equivalent of this result
(based on the formulas on pp. 60-61 in [5]). For this purpose, the notion of spectral flow is first
slightly extended to also allow a path α ∈ [0, 1] 7→ Tα of normal operators with selfadjoint and
invertible end points T0 = T0∗ and T1 = T1∗ such that Tα − T0 is compact. In this situation, the
definition of the spectral flow using spectral projections as given in [15] immediately transposes
(one just considers spectral projections associated to eigenvalues lying in vertical strips of the
complex plane). Alternatively, one can simply use the spectral flow of self-adjoint Fredholm
operators, namely
Sf α ∈ [0, 1] 7→ Tα = Sf α ∈ [0, 1] 7→ ℜe(Tα ) ,
(3)
where ℜe(Tα ) = 21 (Tα + Tα∗ ) is a selfadjoint Fredholm operator.
4
Theorem 2 Let α ∈ [0, 1] 7→ Uα be a path of unitaries such that Uα − U0 is compact. Suppose
that there is a selfadjoint unitary F such that
U1 = F U0 F .
If Π = χ(F ≤ 0), then ΠU0 Π is a Fredholm operator on Ran(Π) with index given by
Ind(ΠU0 Π) = Sf α ∈ [0, 1] 7→ Wα ,
(4)
where
Wα = F Uα U0∗ .
Proof. To show that the spectral flow is well-defined, we note that Uα U0∗ − 1 = (Uα − U0 )U0∗
is compact by hypothesis, so that Wα − F is compact and so is ℜe(Wα ) − F . Moreover, W0 =
F = ℜe(W0 ) and W1 = U0 F U0∗ = ℜe(W1 ) are both self-adjoint, and ℜe(W1 ) = U0 ℜe(W0 )U0∗ .
Hence Theorem 1 can be applied to the family α ∈ [0, 1] 7→ Tα = ℜe(Wα ) to conclude.
✷
Let us stress that the spectral flow of unitaries in (4) does not distinguish whether the
eigenvalue travels on the upper or lower half of the unit circle, in contradistinction to e.g.
[9, 21]. Let us also note that it only depends on the end points of the path α ∈ [0, 1] 7→ Uα
(just as in Theorem 1). For example, choosing U0 Uα∗ F instead of Wα is another natural choice
giving a different path connecting F and U0 F U0∗ . The choices F Uα∗ U0 and U0∗ Uα F reverse the
path and thus the sign of the spectral flow. A standard form of a path from F to U0 F U0∗ ,
expressed merely in terms of U0 and F , is
Uα = U0 exp ıπ2 (F − 1 + α U0∗ [F, U0 ]) .
This was already given in [5] where it is also shown that this path establishes an isomorphism
between two K1 -groups, one of a C∗ -algebra containing U and one of the associated mapping
cone. Theorem 2 evaluates this K-theoretic fact, just as Theorem 1 results from an isomorphism
of the K0 -group of the algebra to the K1 -group of an ideal in the mapping cone [5].
From this K-theoretic perspective, it is natural to also consider paths α ∈ [0, 1] 7→ Aα of
invertible operators with compact differences Aα − A0 and A1 = F A0 F where F is a selfadjoint
unitary as above. Then consider the path α ∈ [0, 1] 7→ Tα = F Aα A−1
0 of invertibles connecting
−1
the selfadjoint unitary T0 = F to the operator T1 = A0 F A0 which also has spectrum {−1, 1}.
For this path, one would like to define a spectral flow. If the operators are normal, the above
procedure works. For a more general (not necessarily normal) case, let us suppose that the
path α ∈ [0, 1] 7→ Tα is real analytic and (merely) assume
Tα − T0 compact with T0 , T1 invertible with real spectrum .
As the unbounded component of the resolvent set of T0 contains C \ R, it then follows from analytic Fredholm theory that the essential spectrum of Tα coincides with the essential spectrum
of T0 . Hence by analytic perturbation theory [8] the discrete eigenvalues of Tα vary analytically away from level-crossings, at which there may be root singularities (Puiseux expansion).
5
Therefore the spectral flow through the imaginary axis along the path α ∈ [0, 1] 7→ Tα is welldefined by counting the finite number of eigenvalues passing from the left half-plane to the right
half-plane, minus those from passing right to left. In general, this spectral flow is not given by
the r.h.s. of (3). Indeed, if say T1 is not normal, the spectra of T1 and of ℜe(T1 ) may have
little in common. The generalized spectral flow will still be denoted by Sf α ∈ [0, 1] 7→ Tα . It
is invariant under analytic homotopies of the path provided the above conditions are satisfied.
This allows us to define the spectral flow for an analytic path α ∈ [0, 1] 7→ Aα as above.
A special case is the set-up in Theorem 2 in which Aα is unitary. To put ourselves in this
situation, let us use the real analytic homotopy of paths
s ∈ [0, 1] 7→ Aα,s = Aα |Aα |−s .
Then Aα,1 is indeed unitary so that Theorem 2 applies. Furthermore, both quantities in (4) are
homotopy invariant. Moreover, 2[Π, A0 ] = [F, A0 ] = (F A0 F − A0 )F = (A1 − A0 )F is compact
so that ΠA0 Π is Fredholm on Ran(Π) because ΠA−1
0 Π is a pseudo inverse. Finally, by standard
arguments s 7→ ΠA0,s Π is a continuous path of Fredholm operators which hence have constant
index. In conclusion, for an analytic part of invertibles with end points that are conjugate by
a selfadjoint unitary F = (1 − Π) − Π,
Ind(ΠA0 Π) = Sf α ∈ [0, 1] 7→ F Aα A−1
.
(5)
0
Let us note that the spectral flow on the r.h.s. is well-defined if merely the initial point A0 is
invertible. This will be explored further towards the end of this section.
The next series of comments shows that by passing to 2 × 2 matrices one can rewrite
Theorem 2 as a version of Theorem 1 with a supplementary symmetry. Indeed, given the
situation of Theorem 2, let us set
1
1 0
1
− Uα∗
,
J =
.
(6)
Pα =
1
0 −1
2 − Uα
These are both operators on H ⊗ C2 which becomes a Krein space with fundamental form J
which plays the role of the chiral symmetry operator in the application below. Let us also
extend F by identifying it (by abuse of notation) with F ⊗ 12 = diag(F, F ). Then JF =
F J = diag(F, −F ) is also a selfadjoint unitary with spectrum {−1, 1}. The projection Pα is
J-Lagrangian, namely J vanishes on the range of Pα and Pα is half-dimensional in the sense
that its J-orthogonal complement is equal to the orthogonal complement of Pα :
J Pα J = 1 − Pα .
Let us recall from [9, 21] some standard facts on J-Lagrangian projections on a Krein space.
Clearly, the set of these J-Lagrangian projections can be identified with the set of closed JLagrangian subspaces, also called the Lagrangian Grassmannian. Moreover, the J-Lagrangian
projections are always of the form (6) and are thus in bijection with the set of unitary operators
from the positive onto the negative eigenspaces of J. In our particular situation, Pα − P0 is
6
compact and this implies that Pα and JP0 form a Fredholm pair. This means that Pα and JP0
are essentially transversal, in the sense that the intersection of the ranges and co-ranges are
finite dimensional and the essential angle spectrum between the two ranges does not contain
0. In this situation, it is natural to consider the Bott-Maslov index counting the weighted
number (by orientation) of intersections of Pα with the singular cycle JP0 . This is given by the
oriented spectral flow of the path α ∈ [0, 1] 7→ U0∗ Uα through −1 (counter clockwise passages
give positive contributions) [9, 21]. This path has end points 1 and U0∗ F U0 F . Thus the BottMaslov index is not invariant under homotopic deformations of either U0 or F . It is not linked
to spectral flow in Theorem 2. We rather consider the path
0 Uα∗
,
α ∈ [0, 1] 7→ Tα = (1 − Pα ) − Pα =
Uα 0
of selfadjoint unitary Fredholm operators. It satisfies T1 = F T0 F as in Theorem 1, but,
moreover, it has the chiral symmetry JTα J = −Tα . The spectral flow of α ∈ [0, 1] 7→ Tα
vanishes so that Theorem 1 is trivial in this situation, and one has to proceed as in Theorem 2
and extract topological information from the path α ∈ [0, 1] 7→ Uα . In the following proposition
we now consider a more general path of chiral selfadjoint Fredholm operators. This path need
not consist of unitaries and indeed need not even consist of invertibles.
Proposition 1 Let α ∈ [0, 1] 7→ Tα be an
analytic path of selfadjoint Fredholm operators on
the Krein space H ⊗ C2 , J = diag(1, −1) satisfying the chiral symmetry JTα J = −Tα , as well
as T1 = F T0 F with an invertible T0 and a selfajoint unitary F = F ⊗ 12 . Due to the chiral
symmetry, there is a path α ∈ [0, 1] 7→ Aα of operators such that
0 A∗α
,
Tα =
Aα 0
Then ΠA0 Π with Π = 21 (F + 1) is a Fredholm operator on Ran(Π) with
2 Ind(ΠA0 Π) = Sf α ∈ [0, 1] 7→ JF Tα T0−1 .
Proof. Set Bα = JF Tα T0−1 . First of all, the path α ∈ [0, 1] 7→ Bα connects JF with −T0 JF T0−1
which both have spectrum {−1, 1}. Moreover, Bα − JF = JF (Tα − T0 )T0−1 is compact so that,
as above, the spectral flow of the analytic path α ∈ [0, 1] 7→ Bα is well-defined. Now Aα − A0
is compact and A1 = F A0 F . One has
F Aα A−1
0
0
.
Bα =
∗
0
−(A−1
0 Aα F )
In the case that all Tα are invertible, so too are all Aα and then the spectral flow of Bα is the
direct sum of two contributions which are both equal to the spectral flow on the r.h.s. of (5).
This implies the result in this case. In the other case, the invertibility of Tα can only fail on a
finite number of points. In each such point, the kernel is even dimensional and the restriction
of J to this kernel has vanishing signature. A generic finite dimensional perturbation on the
7
kernel will render the operator Tα invertible. We will not carry this out in detail, but provide
an example below.
✷
Example. Let us consider the S 0 and S 1 on ℓ2 (Z) as given in (1). Hence S 1 = F S 0 F with F
as in (2). Instead of the path α ∈ [0, 1] 7→ Hα of chiral selfadjoint unitaries given in (1), let us
consider another path connecting H0 and H1 :
0
S 0 − 2α |0ih1|
α ∈ [0, 1] 7→ Tα =
.
(S 0 − 2α |0ih1|)∗
0
Clearly this path consists again of chiral selfadjoint Fredholm operators, but for α = 21 the
invertibility is not given. However, replacing α ∈ [0, 1] 7→ (1 − 2α) in the definition of Tα by
any path from 1 to −1 avoiding 0, leads to the invertibility of the whole path.
⋄
3
Non-abelian monopole translations
Let γ1 , . . . , γd be a faithful representation of the d generators of the complex d-dimensional
Clifford algebra Cd on C2N , namely γi γj = −γj γi for i 6= j and (γj )2 = 12N . This representation
d−1
d
can be chosen irreducible if 2N = 2 2 for odd d and 2N = 2 2 for even d. If d is even, then
there exists a grading operator Γ on C2N with Γ2 = 1 and Γ∗ = Γ such that Γγj = −γj Γ. Now
introduce the Dirac operator D and its unitary, selfadjoint phase F = F ∗
D =
d
X
F = D|D|−1 ,
γ j Xj ,
j=1
where X1 , . . . , Xd are the components of the (unbounded selfadjoint) position operator on the
lattice. If d is even,
ΓDΓ = −D .
(7)
In the spectral representation of Γ, this leads to, still only for d even,
0 V∗
1 0
,
,
F =
Γ =
V 0
0 −1
(8)
for a unitary V which can also be expressed in terms of a lower-dimensional representation of
Cd−1 . Here we first work in the continuum and view D as a selfadjoint operator on L2 (Rd , C2N ),
1
but later on the aim is to discretize. Let us also set R = (X12 + . . . + Xd2 ) 2 = |D| which are
all diagonal operators on L2 (Rd , C2N ). If ∂1 , . . . , ∂d are the partial derivatives (also diagonal,
8
namely ∂j is identified with ∂j ⊗ 12N ), then
F [ı∂k , F ] =
=
=
=
D
γk
− Xk 3
ıF
R
R
Dγk
D2
ı
− Xk 4
R2
R
Dγk − Xk
ı
R2
ı [D, γk ]
,
2 R2
where we used {D, γk } = Dγk + γk D = 2Xk . One also has
F
ı [D, γk ]
D ı [D, γk ] D
F =
2
2 R
R 2 R2 R
ı
D(Dγk − γk D)D
=
2R4
ı [D, γk ]
.
= −
2 R2
(9)
From now on, let us suppose that d ≥ 2 as the following formulas are not interesting in the
case d = 1 for which a separate treatment is given towards the end of this section. For d ≥ 2,
the Wu-Yang monopole potential with charge 1 is introduced by [22]
ı [D, γk ]
,
2 R2
Ak (x) =
k = 1, . . . , d ,
and the covariant derivative with charge α ∈ R by
∇αk = − ı ∂k − α Ak (x) ,
k = 1, . . . , d .
(10)
Here the argument x in Ak (x) stresses that this is a multiplication operator on L2 (Rd , C2N ).
Note that for even d, one has ΓAk Γ = Ak so that Ak is diagonal in the spectral representation
(8) of Γ and thus so is ∇αk . Let us slightly generalize these notations. For v = (v1 , . . . , vd ) ∈ Rd ,
we introduce another 2N × 2N matrix-valued function x ∈ Rd 7→ Av (x) by
Av (x) =
d
X
vk Ak (x) = hv|A(x)i ,
k=1
where here h . | . i denotes the euclidean scalar product and A(x) = (A1 (x), . . . , Ad (x)). Similarly, with the vectors ∂ = (∂1 , . . . , ∂d ), ∇α = (∇α1 , . . . , ∇αd ) and γ = (γ1 , . . . , γd ), let us set
∂v = hv|∂i ,
∇αv = hv|∇αi ,
γv = hv|γi .
Then D = hX|γi and
Av (x) =
ı [hx|γi, hv|γi]
ı [γx , γv ]
=
.
2
2
R
2 R2
9
Note that Av (x) is a bounded selfadjoint matrix potential which decays (in matrix operator
norm) at infinity as
C
,
(11)
kAv (x)k ≤
R
for some constant C. Therefore ∇αv is a selfadjoint operator. It is also the covariant derivative
of a SU(N)-gauge theory. Moreover, the above relation (9) shows
F ∇αv F = − ı ∂v − F [ı∂v , F ] − α F Av (x)F = ∇1−α
.
v
(12)
As ∇αv is self-adjoint, it can be exponentiated to a one-parameter family
α
α
t ∈ R 7→ eıt∇v = eı∇tv
(13)
of unitary operators on L2 (Rd , C2N ). Proposition 2 below shows that these operators are translations, modified by a local matrix multiplication. This is similar to the magnetic translations
where the multiplication is merely by an abelian phase factor (see e.g. [1]), and therefore we
α
also call eıt∇v , non-abelian monopole translations associated to the Wu-Yang monopole. Moreover, the proposition shows that the non-abelian monopole translations in different directions
are connected by a unitary transformation which invokes a representation of the orthogonal
group O(d) that we first need to describe in some detail.
Let us first note that the orthogonal group naturally acts on the Clifford algebra Cd via
(O, γv ) 7→ γOv .
This definition, is explicitly only defined for Cd ⊂ Cd , and is then extended to products of γv ’s
and thus the whole Clifford algebra. This action is implemented by the restriction of the group
action of Pin(d) on Cd by adjunction to the subgroup O(d) [12]. More precisely, recall that
every O ∈ O(d) can be decomposed into a series of d or d − 1 quasi-reflections O = Rv1 · · · Rvk
where vj are unit vectors and Rvj (λvj + wj⊥ ) = λvj − wj⊥ for λ ∈ R and wj⊥ orthogonal to vj .
Then set gO = γv1 · · · γvk , which is a lift of O into Pin(d). Now one has
γOw = gO γw i(gO ) ,
(14)
where i is the transposition in Cd (an anti-involution). Note the slight twist w.r.t. the usual
treatment [12] where the Clifford generators square to minus the identity and one uses the usual
reflections defined by λvj + wj⊥ 7→ −λvj + wj⊥ . The proof of a decomposition O = Rv1 · · · Rvk
with quasi-reflections is the same as the usual one, based on the spectral decomposition of
O and on the fact that every rotation is a product of two quasi-reflections. While all this is
independent of the representation of Cd , we here only wrote it out in one given representation
on C2N , so that O ∈ O(d) 7→ gO is a unitary representation of O(d) on C2N . Now we can state
the following:
Proposition 2 Let v ∈ Rd and O ∈ O(d). The non-abelian monopole translations are of the
form
α
ψ ∈ L2 (Rd , C2N ) ,
(15)
(eı∇v ψ)(x) = Mvα (x) ψ(x + v) ,
10
where Mvα (x) ∈ U(2N) is a unitary matrix satisfying the normalization at infinity
lim Mvα (x) = 12N .
(16)
|x|→∞
The non-abelian phase factors satisfy the covariance property
α
gO Mvα (O ∗ x) gO∗ = MOv
(x) .
(17)
Moreover,
1−α
α
F eı∇v F = eı∇v
(18)
.
For even d and the chirality operator Γ of D, one also has
Γ Mvα (x) Γ = Mvα (x) ,
α
α
Γ eı∇v Γ = eı∇v .
(19)
Proof. Let v ∈ Rd first be fixed. We first solve the following ordinary differential equation on
the line Lx,v through x in the direction v, with selfadjoint and space-dependent coefficients:
∂v Nvα (x′ ) = ı α Av (x′ ) Nvα (x′ ) ,
x′ ∈ Lx,v .
(20)
Hence the solutions are unitaries t ∈ R 7→ Nvα (x + tv). As the coefficients decrease at infinity
due to the bound (11), the solution converges to a fixed unitary matrix at t = ±∞. The choice
of initial condition will be irrelevant for the following, as it cancels out in the definition (21)
of Mvα (x) below. Viewing Nvα as a unitary multiplication operator on L2 (Rd , C2N ), one readily
checks
Nvα ∇0v (Nvα )∗ = ∇αv .
As ∇0v = −ı∂v is the generator of the shift in the direction v, one hence has
α
0
0
eı∇v = Nvα eı∇v (Nvα )∗ = Mvα eı∇v ,
where Mvα is the unitary multiplication operator with
Mvα (x) = Nvα (x) Nvα (x + v)∗ .
(21)
This shows (15). Also (16) follows because Nvα (x) converges as x → ∞. We now establish the
covariance (17). The orthogonal group O(d) naturally acts on Rd and also on L2 (Rd ) by
(O · ψ)(x) = ψ(O ∗ x) ,
O ∈ O(d) , ψ ∈ L2 (Rd ) .
Without the monopole α = 0 and one can drop the fiber C2N Then one readily checks on the
0
0
group Oeıt∇v O ∗ = eıt∇Ov so that
O ∇0v O ∗ = ∇0Ov .
(22)
Now let us extend the action of O(d) to L2 (Rd , C2N ) by tensoring with the unitary representation
O ∈ O(d) 7→ gO . This action is again simply denoted by O. As Av is a multiplication operator
on L2 (Rd ), one has
O Av (x) O ∗ = O
ı [γx , γv ] ∗
ı [gO γO∗ x gO∗ , gO γv gO∗ ]
ı [γx , γOv ]
O
=
=
= AOv (x) .
2
2
2 R
2
R
2
R2
11
One deduces upon combination with (22)
O ∇αv O ∗ = ∇αOv .
(23)
Substituting this into (15) directly leads to (17). Furthermore, (18) follows from (12), and the
last claim from (7).
✷
Let us now introduce a notation for the monopole translation in direction k on the lattice
Hilbert space ℓ2 (Zd , C2N ):
α
Skα = eı∇ek .
α
Note that as the shift by ek leaves the lattice Zd invariant, the operators eı∇ek on the continuous
physical space indeed restrict to the lattice Hilbert space. Furthermore, Skα is by construction a
unitary operator. Let us note that (15) translates in Skα being the left-shift, namely for w ∈ C2N
specifying a localized state w ⊗ |ni at site n,
Skα w ⊗ |ni = Meαk (n)w ⊗ |n − ek i .
By (18), the lattice monopole translations satisfy
F Skα F = Sk1−α ,
(24)
where here also F is a unitary multiplication operator on ℓ2 (Zd , C2N ). Furthermore, for even d
Γ Skα Γ = Skα .
(25)
Corollary 1 The operators Skα − Sk are compact on ℓ2 (Zd , C2N ).
Proof. This follows immediately from Skα − Sk = (Meαk − 1)Sk and the asymptotics (16).
✷
Example: Let us consider the case d = 2. Then γ1 = σ1 , γ2 = σ2 and Γ = σ3 are expressed in
terms of the Pauli matrices. The Dirac operator is
0
X1 − ı X2
,
D = X1 σ1 + X2 σ2 =
X1 + ı X2
0
and the vector potential is
ı [x1 σ1 + x2 σ2 , σ1 ]
1
A1 (x) =
= 2
2
2
R
R
x2
0
,
0 − x2
where R2 = (x1 )2 + (x2 )2 . Now integration of (20) leads to
!
x
ı α arctan( x1 )
2
0
e
x
,
N2 (x) =
N1 (x) =
−ı α arctan( x1 )
2
0
e
1
A2 (x) = 2
R
− x1 0
,
0
x1
x
−ı α arctan( x2 )
e
0
1
0
x
ı α arctan( x2 )
e
1
!
.
From this M1α (x) and M2α (x) can readily be deduced. The first component of these formulas
and all the associated claims of Proposition 2 coincide (up to a sign change) with those in
12
[6] which were deduced by more complicated algebraic manipulations. Let us note that the
second component is obtained by replacing α by −α. This doubling will be further discussed
in Section 4.
⋄
Up to now, we only inserted the monopole in translations without constant magnetic field,
but it is also possible to add the monopole to the magnetic translations as defined, e.g., in [19].
These magnetic translations result from a U(1)-gauge potential which does not interfere with
the constructions above. Details are left to the interested reader.
4
Spectral flow of Hamiltonians with inserted monopole
We consider a Hamiltonian on ℓ2 (Zd , C2N ) of the form
H = ∆(S1 , . . . , Sd ) + W ,
(26)
where ∆ is a non-commutative polynomial with matrix coefficients specifying the kinetic part
and W = W ∗ is a matrix-valued potential. It is possible that both the coefficients of ∆ and W
are space dependent. However, we will assume throughout that, for any vector v ∈ Rd ,
F ∆(v)F = ∆(v) ,
FWF = W .
(27)
Let us note that there is a lower bound on N in terms of the dimension d because we assume
the representation of Cd on C2N to be faithful. Hence for d = 1, 2, it is possible to choose
N = 1, but for d = 3, 4 one needs N ≥ 2, and so on. The condition (27) is always satisfied
if ∆ and W are scalar, but may be stringent in the matrix case. However, one way to always
assure (27) is to start out with a given Hamiltonian H on ℓ2 (Zd , C2n ) for some n ≥ 0, and then
′
tensor a representation space C2n for the Clifford algebra and then set N = n + n′ . If the
′
Hamiltonian is extended as H ⊗ 12n′ , then (27) holds because F only acts on the fiber C2n .
This is the procedure applied in the proof of the index theorems in [17, 18, 19]. The second
main hypothesis is a gap condition at the Fermi level µ ∈ R, namely
µ 6∈ σ(H) .
(28)
Now a monopole is inserted into H by replacing Sj by Sjα :
Hα = ∆(S1α , . . . , Sdα ) + W .
It follows from (27) and (24) that
F Hα F = H1−α .
(29)
Moreover, Corollary 1 implies that α ∈ [0, 1] 7→ Hα − H0 is a compact selfadjoint operator.
Hence there is an associated spectral flow through µ. However, this spectral flow is not very
interesting:
Proposition 3 One has
Sf α ∈ [0, 1] 7→ Hα by µ
13
= 0.
Proof. The unitary equivalence (29) implies for the spectra
σ(Hα ) = σ(H1−α ) .
A moment of thought shows that there is no spectral flow due to this spectral symmetry.
✷
Nevertheless, the path α ∈ [0, 1] 7→ Hα can have interesting topology if there is some
supplementary symmetry. In this section we consider the case of even d. Then the Dirac
operator satisfies (7), but there is no relation between Γ and H = H0 . It is natural to either
impose ΓHΓ = H or ΓHΓ = −H. The latter leads to two off-diagonal operators in the grading
of Γ (namely, H and D are both off-diagonal), while the first relation leads to one diagonal and
one off-diagonal operator (H and D respectively). This choice turns out to be more interesting.
In general, ΓHΓ = H leads to H = diag(h, h′ ) in the grading of Γ. Moreover, the kinetic parts
of h and h′ are equal due to the form (26) of the Hamiltonian, and furthermore F W F = W
also implies that the potential parts are equal. Hence h = h′ . The insertion of the monopole
into H = H0 then leads to
hα 0
Hα =
.
(30)
0 h−α
Now (29) is indeed equivalent to the relation hα+1 = V ∗ hα V appearing in item (ii) in the
introduction. It is also obvious in this case that the spectral flow in Proposition 3 decomposes
into a direct sum of two spectral flows which cancel each other out. However, the remarkable
point is that each of these spectral flows is defined by itself.
Theorem 3 Let d be even, and suppose H = diag(h, h) with h ∈ ℓ2 (Zd , CN ) is of the form
(26) satisfying (27). For µ 6∈ σ(H), one then has
Sf α ∈ [0, 1] 7→ hα by µ = Ind(pµ V pµ ) ,
where pµ = χ(h ≤ µ) is the spectral projection of h on states below µ and V is the unitary entry
in the Dirac phase in (8).
Proof. This immediately follows from Theorem 1.
✷
For d = 2, Theorem 3 reproduces the result of [6] described in the introduction. Let us
next provide an explicit instance in higher dimension where Theorem 3 applies and leads to a
non-trivial spectral flow.
Example We follow Section 2.2.4 of [19]. For even d, let ν1 , . . . , νd be an irreducible repre′
′
sentation of Cd on C2n with grading ν0 . Then consider the Hamiltonian on ℓ2 (Zd , C2n ) given
by
!
d
d
X
1 X
1
h =
(Sj − Sj∗ )νj + m +
(Sj + Sj∗ ) ν0 .
2ı j=1
2 j=1
Then 0 6∈ σ(h) for m 6∈ {−d, −d + 2, . . . , d}. At these values the central gap at the Fermi level
µ = 0 closes, and actually the d-th Chern number Chd (p) of the Fermi projection p = χ(h ≤ 0)
changes its value. The reader is referred to [19] for an explicit calculation of all the values
14
taken. Now the second Clifford representation γ1 , . . . , γd on C2n is tensorized to the Hilbert
space so that H = diag(h ⊗ 12n , h ⊗ 12n ) and the Dirac operator D act on the same Hilbert
space ℓ2 (Zd , C2N ) where N = n + n′ . In this situation, Chapter 6 of [19] shows that
Chd (p) = Ind(pµ V pµ )
Thus the index appearing in Theorem 3 takes non-trivial values. Consequently, the spectral
flow of α ∈ [0, 1] 7→ hα past 0 is non-trivial as well.
⋄
5
Chirality flow in odd dimensions
In this section, d is odd and the Hilbert space is H = ℓ2 (Zd , C2N ) with N = n + n′ and both
n ≥ 1 and n′ ≥ 1. The Clifford representation and the non-abelian gauge potential of the
′
monopole (both described in Section 3) act on the fiber C2n and the translation invariant
chiral symmetry is
1n
0
J =
⊗ 12n′ .
0 − 1n
The Hamiltonian H on H is supposed to be of the form (26), satisfy (27) and furthermore have
the chiral symmetry
J H J = −H .
This implies that
H =
0 A
,
A∗ 0
′
′
with an operator A on ℓ2 (Zd , Cn ) ⊗ C2n acting trivially (as identity) on the fiber C2n . Again
A is the sum of a matrix-valued polynomial in the shift operators S1 , . . . , Sd and a matrix
potential. One can hence replace the shift operators Sj by the unitary monopole shifts Sjα to
obtain operators Aα and a family of chiral Hamiltonians Hα on H:
0 Aα
Hα =
.
A∗α 0
′
By Corollary 1 the differences Hα − H0 and Aα − A0 are compact (on H and ℓ2 (Zd , Cn+2n )
respectively). The Fermi level for chiral Hamiltonians is µ = 0 and the gap hypothesis (28)
states that H0 = H is invertible, and hence also A = A0 is invertible. Moreover, the path
α ∈ [0, 1] 7→ Hα is real analytic. Therefore Proposition 1 implies the following:
Theorem 4 Let d be odd and suppose that the chiral Hamiltonian H is of the form (26) and
satisfies (27) with µ = 0. Then
Sf(α ∈ [0, 1] 7→ JF Hα H0−1 ) = 2 Ind(ΠA0 Π) ,
where Π = 21 (F + 1) is the Hardy projection of F .
15
Let us note again that the spectral flow in the theorem results from a path connecting JF
to −H0 JF H0−1 that we also refer to as the chirality flow. If the whole path α ∈ [0, 1] 7→ Hα is
invertible, one can define the path of Fermi unitaries α ∈ [0, 1] 7→ Uα by
0 Uα∗
−1
,
Hα |Hα|
=
Uα 0
and Theorem 4 reduces to item (iii)′ in the introduction. This is the case in the SSH model
discussed in the introduction.
Example. Section 2.3.3 of [19] provides for odd d an example of a model with non-vanishing
strong invariant. As this is very similar to the even dimensional model presented in Section 4,
we do not provide further details.
⋄
Acknowledgements: The work of A. L. C. was supported by the Australian Research Council,
that of H. S.-B. partially by the DFG.
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17