Two-Dimensional Time-Reversal-Invariant Topological Insulators via Fredholm Theory

Abstract

We study spinful non-interacting electrons moving in two-dimensional materials which exhibit a spectral gap about the Fermi energy as well as time-reversal invariance. Using Fredholm theory we revisit the (known) bulk topological invariant, define a new one for the edge, and show their equivalence (the bulk-edge correspondence) via homotopy.

Appendix

1.1 A.1 More Proofs

Proof of the first statement in Theorem 3.1

We note that the Fermi projection $P \equiv \chi _{(-\infty , 0)}(H)$ can be written as P = g(H) in the spectral gap regime. Then using a special case of the Fedosov formula, Proposition A.1, we may write the RHS of (3.1) as

$$ \text{ind}\mathbb{W}_{1}(P{\Lambda}_{2}P) = \text{i } \text{tr}\exp({-2\pi iP{\Lambda}_{2}P}) \partial_{1} \exp({2\pi iP{\Lambda}_{2}P}) . $$

(A.1)

To do so, we must demonstrate that $\partial _{1} \exp ({2\pi iP{\Lambda }_{2}P})$ is trace class. Since P is local, ∂₂P is LOC₂ and so by the ideal property,

$$ (P{\Lambda}_{2}P)^{2}-P{\Lambda}_{2}P=\mathrm{i} P{\Lambda}_{2}P^{\perp}\partial_{2} P $$

is LOC₂ too. Hence we may apply Proposition A.4 on PΛ₂P to get that $\partial _{1} \exp ({2\pi iP{\Lambda }_{2}P})$ is trace class.

Next, we show that the RHS of (A.1) and the RHS of (3.2) can be written as the same integral:

$$ I := {\int}_{0}^{2\pi} \text{tr}~ \partial_{\alpha} (\exp({-\text{i}\alpha P{\Lambda}_{2}P}) P{\Lambda}_{1}P \exp({\text{i}\alpha P{\Lambda}_{2}P})) \text{d}{\alpha} . $$

(A.2)

Note that this formula is well-defined since the operator within the trace is trace-class:

$$ \begin{array}{@{}rcl@{}} \partial_{\alpha} (\exp(-\text{i}\alpha P{\Lambda}_{2}P) P{\Lambda}_{1}P\exp(\text{i}\alpha P{\Lambda}_{2}P)) &= \text{i} \exp(-\text{i}\alpha P{\Lambda}_{2}P) [P{\Lambda}_{1}P, P{\Lambda}_{2}P] \exp(\text{i}\alpha P{\Lambda}_{2}P) \\ &= -\text{i} \exp(-\text{i}\alpha P{\Lambda}_{2}P) P[\partial_{1} P,\partial_{2} P] \exp(\text{i}\alpha P{\Lambda}_{2}P) . \end{array} $$

Then expanding the integrand of (A.2) and using cyclicity of the trace and then integrating we obtain I equals the RHS of (3.2).

Conversely, as the integral in (A.2) converges strongly and the integrand is trace class we may exchange the trace and integral and then use the fundamental theorem of calculus to obtain

$$ I = \text{tr} \exp({-2\pi i P{\Lambda}_{2}P}) [P{\Lambda}_{1}P, \exp({2\pi i P{\Lambda}_{2}P})] = \text{i }\text{tr} \exp({-2\pi i P{\Lambda}_{2}P}) \partial_{1} \exp({2\pi i P{\Lambda}_{2}P}) $$

(A.3)

where the last equality follows from Lemma A.2. Thus, we have shown that the RHS of (3.1) is equal to the RHS of (3.2). Now [50, eq-n (4.6)], which is $\sigma _{\text {Hall}}(H)=\frac {1}{2\pi }\text {ind} F$, implies the first claim of Theorem 3.1. □

Proposition A.1 (A Fedosov type formula)

If Q is a projection and V is a unitary such that [Q, V ] is trace class then (using (2.2))

$$\text{ind}~\mathbb{Q}V=\text{tr}~ V[Q,V^{\ast}] .$$

Proof

This is a special case of [39, Prop. 2.4], where the difference of projections Q − V^∗QV is trace class. □

Lemma A.2

If V is unitary and Q, R are projections such that $V = \mathbb {R}V$ and [Q, V ] is trace class then

$$ \text{tr}~ V^{\ast}[Q,V] = \text{tr}~ V^{\ast}[RQR,V] . $$

Proof

Since $V = \mathbb {R}V$ we have [R, V ] = 0, so that (using cyclicity)

$$ \begin{array}{@{}rcl@{}} \text{tr} V^{\ast}[RQR,V] = \text{tr} R V^{\ast} R[Q,V] = \text{tr} (V^{\ast} - R^{\perp}) [Q,V] ,\end{array} $$

but using cyclicity again trR^⊥[Q, V ] = trR^⊥[Q, V ]R^⊥ and since R^⊥V = R^⊥ this term vanishes. □

1.2 A.2 Locality Properties

Proposition A.3

If $A\in \mathcal {F}({\mathscr{H}})$ and $\partial _{1} A\in \mathcal {K}({\mathscr{H}})$ (the ideal of compact operators) then

Proof

Recall Atkinson’s characterization of Fredholm operators: they are operators invertible modulo a compact operator [40]. Thus it suffices to construct an explicit parametrix for the operator of interest. With B a parametrix for A define and then

and similarly for . Since ∂₁A is compact and B is a parametrix for A we find that G is a parametrix for indeed. □

Proposition A.4

If A is local such that A² − A is LOC₂, then

is also LOC₂. It follows that for such A, $\partial _{1} \exp (-2\pi \text {i} A)$ is trace class.

Proof

For any $n\in \mathbb {Z}$ and x, we may re-write

$$ \begin{array}{@{}rcl@{}} &= \underbrace{\sum\limits_{l=2}^{\infty}\frac{1}{l!}\left( -2\pi\text{i} n\right)^{l}\sum\limits_{k=0}^{l-2}x^{k}}_{=:h(x)}\left( x^{2}-x\right). \end{array} $$

Now h is analytic, so by the holomorphic functional calculus and the Combes-Thomas estimate (A.4) we find that this term is local. Furthermore, A² − A is assumed to be LOC₂. Finally by the algebraic closure property of LOC₂ [32, Lemma 3.12] we conclude h(A)(A² − A) is LOC₂.

This implies (by the results of [32, Section 3.3]) that $\partial _{1} \exp (-2\pi \text {i} A) =$ is local and confined in both directions and hence trace-class. □

Corollary A.5

If $A(t):[0,1] \to {\mathscr{B}}({\mathscr{H}})$ is a continuous map such that A(0) = A₀ and A(1) = A₁ and A(t)² − A(t) is LOC₂ for all t, then there is a homotopy within $\mathcal {F}({\mathscr{H}})$ from

$$ \mathbb{W}_{1} A_{0} \longrightarrow \mathbb{W}_{1} A_{1} $$

and if [A(t),Θ] = 0 then it passes within $\mathcal {F}_{\Theta }({\mathscr{H}})$, so that both ind and ind₂ agree on these two operators.

Proof

Since by Proposition A.4 we have that $\partial _{1} \exp (-2\pi \text {i} A(t))$ is trace-class, it is compact, so we may apply Proposition A.4 to get that for any t, $\mathbb {W}_{1} A(t) \in \mathcal {F}({\mathscr{H}})$. □

1.3 A.3 The Smooth Functional Calculus

In this section we want to establish that both Λ₂(g(Λ₂HΛ₂) − g(H))Λ₂ and $g(\iota ^{\ast } H \iota )-g(\hat {H})$ are LOC₂, which is used in Lemma 3.4 and Proposition A.10 respectively.

First we recall [53, Theorem 10.5] the basic estimate:

Theorem A.6 (The Combes-Thomas Estimate)

If $A\in {\mathscr{B}}({\mathscr{H}})$ is exponentially local and self-adjoint then there are constants $C<\infty ,\mu >0$ such that

Next, we need a result about the smooth functional calculus [54] (and see references therein):

Theorem A.7 (The Helffer-Sjöstrand formula)

Let $f:\mathbb {R}\to \mathbb {C}$ be smooth and of compact support, and let $\tilde {f}:\mathbb {C}\to \mathbb {C}$ be a quasi-analytic extension of it (which is supported within some strip about the real axis). This implies that for all $N\in \mathbb {N}$, there is some $C_{N}<\infty$ such that

$$ \begin{array}{@{}rcl@{}} |(\partial_{\bar{z}} \tilde{f})(z)|\leq C_{N} |\Im\{z\}|^{N}\qquad(z\in\mathbb{C}) . \end{array} $$

(A.5)

Then for $A\in {\mathscr{B}}({{\mathscr{H}}})$ self-adjoint, we have

One combines these two results to obtain that the smooth functional calculus on exponentially local self-adjoint operators is polynomially local [22, Appendix A], a result we have been using freely.

Finally, we also get a similar statement to [22, Lemma A3], which says that the difference of the smooth functional calculus of operators whose difference is LOC₂ is itself LOC₂:

Proposition A.8

If A, B are two self-adjoint exponentially local operators such that A − B is exponentially LOC₂, and $f:\mathbb {R}\to \mathbb {C}$ is a smooth function then f(A) − f(B) is also LOC₂.

Proof

First note that since A, B are bounded, we may WLOG assume that f is of compact support, and define $K:=|\text {supp}(\partial _{\bar {z}} \tilde {f})|$. Hence we may use (A.5) to get

For any $N\in \mathbb {N}$, taking the $n,m\in \mathbb {Z}^{2}$ matrix elements, using the fact f has compact support and (A.5) as well as (A.4) we find:

$$ \begin{array}{@{}rcl@{}} \|(f(A)-f(B))_{n,m}\| \leqslant \frac{K}{2\pi}{\int}_{y\in\mathbb{R}} C_{N}|y|^{N}\sum\limits_{l,k} C |y|^{-1} \text{e}^{-\mu y \|n-l\|}\|(B-A)_{l,k}\|C |y|^{-1} \text{e}^{-\mu y \|k-m\|} \text{d}{y} . \end{array} $$

Now since A − B is assumed LOC₂ we have some $D<\infty ,\nu >0$ such that

$$ \|(B-A)_{l,k}\| \leqslant D \exp(-\nu(\|l-k\|+|l_{2}|+|k_{2}|))\qquad(l,k\in\mathbb{Z}^{2}) $$

and so all together

$$ \begin{array}{@{}rcl@{}} \|(f(A)-f(B))_{n,m}\| &\leqslant \frac{K C_{N} C^{2}D}{2\pi}{\int}_{y\in\mathbb{R}} |y|^{N-2}{\sum}_{l,k} \text{e}^{-\mu y \|n-l\|-\nu(\|l-k\|+|l_{2}|+|k_{2}|)-\mu y \|k-m\|} \text{d}{y} \\ &\leq C^{\prime}{\int}_{y\in\mathbb{R}} |y|^{N-2} \text{e}^{-\mu^{\prime} y (\|n-m\|+|n_{2}|+|m_{2}|)} \text{d}{y} . \end{array} $$

Multiple integrations by parts to get rid of the |y|^N− 2 factor in the integrand yield now polynomial decay at rate N − 1. □

Remark A.9

We note that the first expression we want to control is re-written as

$$ \begin{array}{@{}rcl@{}} {\Lambda}_{2}(g({\Lambda}_{2}H{\Lambda}_{2})-g(H)){\Lambda}_{2} &= {\Lambda}_{2}(g({\Lambda}_{2}H{\Lambda}_{2}+{\Lambda}_{2}^{\perp} H{\Lambda}_{2}^{\perp})-g(H)){\Lambda}_{2} \end{array} $$

and since

$$ \begin{array}{@{}rcl@{}}{\Lambda}_{2}H{\Lambda}_{2}+{\Lambda}_{2}^{\perp} H{\Lambda}_{2}^{\perp}-H = 2\Re\{{\Lambda}_{2} H{\Lambda}_{2}^{\perp}\}=2\Re\{\text{i}(\partial_{2} H){\Lambda}_{2}^{\perp}\} \end{array} $$

is LOC₂ by applying ∂₂ on a local operator H, we may apply Proposition A.8 on it to get that Λ₂(g(Λ₂HΛ₂) − g(H))Λ₂ is LOC₂ indeed.

The second expression admits a direct application of Proposition A.8 due to our hypothesis in Definition 2.9.

1.4 A.4 More General Boundary Conditions

We want to generalize (??) to any boundary conditions. This is achieved using the fact that

Proposition A.10

If $\hat {H}$ and H are compatible as in Definition 2.9, then there is a homotopy within $\mathcal {F}({\mathscr{H}})$ from

$$ \mathbb{W}_{1} g(\hat{H}) \longrightarrow \mathbb{W}_{1} g(\text{Ad}_{\iota^{\ast}}H) $$

and if $[\hat {H},{\Theta }]=0$ and [H,Θ] = 0 then it passes within $\mathcal {F}_{\Theta }({\mathscr{H}})$, so that both ind and ind₂ agree on these two operators.

Proof

Consider the continuous map $M:[0,1] \to {\mathscr{B}}({\mathscr{H}})$ given by $M(t) = \exp (-2\pi \text {i} A(t))$, with $A(t) = t g(\text {Ad}_{\iota ^{\ast }}H) + (1-t) g(\hat {H})$. After some algebra, we have

Now, $g(\text {Ad}_{\iota ^{\ast }}H) - g(\hat {H})$ is LOC₂ by the compatibility condition Definition 2.9 and application of Proposition A.8. So by the algebraic closure of LOC₂, the first line is LOC₂. Finally, we note that g² − g is only supported within the bulk spectral gap (since outside of it it takes either the value 1 or 0, in each case g² − g is zero). Hence $(g^{2}-g)(\hat {H})$ is LOC₂ as well by [22, Lemma A3 (iii)] (we apply it with G := g² − g).

So by (A.5) we conclude the continuous interpolation between both operators remains Fredholm. If $\hat {H}$ and H are TRI then in particular we have [A(t),Θ] = 0 hence the interpolation passes within $\mathcal {F}_{\Theta }({\mathscr{H}})$. □

1.5 A.5 Equivalence of $\mathbb {Z}_{2}$ Indices

In order to see that our definition of $\mathcal {N}$ is equivalent to the Fu-Kane-Mele [8, 55] invariant, to the Schulz-Baldes [33] invariant, to the Katsura-Koma [34] invariant, as well as to the Graf-Porta [26], as well as prove (??), we construct a model with fermionic time reversal symmetry and then show that our definition of $\mathcal {N}$ agrees with the the Schulz-Baldes [33] invariant (which has been related to the other invariants already) on both the trivial and non-trivial classes.

Let H be some Hamiltonian (not necessarily such that [H,Θ] = 0). On a double Hilbert space $\hat {{\mathscr{H}}}:={\mathscr{H}}\oplus {\mathscr{H}}$ define $\tilde {H} := H \oplus {\Theta } H {\Theta }^{*}$ and

$$ \begin{array}{@{}rcl@{}} \tilde{\Theta} : = \left[\begin{array}{ccccc} 0 & {\Theta} \\ {\Theta} & 0 \end{array}\right] \end{array} $$

A calculation shows that and $[\tilde {H}, \tilde {\Theta }] =0$. So we see that $\tilde {H}$, $\tilde {\Theta }$, ${\mathscr{H}}\oplus {\mathscr{H}}$ defines a model with fermionic time reversal symmetry. From $\tilde {H}$ we may naturally define the Fermi projection

$$ \begin{array}{@{}rcl@{}} \tilde{P} := \chi_{(-\infty, 0)}(\tilde{H}) = \left[\begin{array}{ccccc} \chi_{(-\infty, 0)}(H) & 0 \\ 0 & \chi_{(-\infty, 0)}({\Theta} H {\Theta}^{*}) \end{array}\right] \end{array} $$

so that Θ being anti-unitary yields, via and Stone’s formula for $\chi _{(-\infty , 0)}$,

$$ \begin{array}{@{}rcl@{}} \tilde{P} = \left[\begin{array}{ccccc} P & 0 \\ 0 & {\Theta} P {\Theta}^{*} \end{array}\right] . \end{array} $$

We note also that X_j on ${\mathscr{H}}$ extends naturally to ${\mathscr{H}}\oplus {\mathscr{H}}$ as $\tilde {X_{j}} = X_{j} \oplus X_{j}$ and that since [Θ,X_j] = 0 we have $[\tilde {\Theta }, \tilde {X_{j}}] = 0$. We have a similar extension of Λ_j on ${\mathscr{H}}$ to $\tilde {\Lambda }_{j}$ on ${\mathscr{H}}\oplus {\mathscr{H}}$ as well.

Having constructed the model with fermionic time reversal symmetry the equivalence of our invariant to the known invariants is shown by the following proposition.

Proposition A.11

For the model for fermionic TRI symmetry defined previously in this section we have

(A.7)

where $\tilde {U} = \exp (\text {i} \arg (\tilde {X}_{1}+\text {i} \tilde {X}_{2}))$ is the unitary operator implementing the gauge transformation associated with a flux insertion at the origin.

Proof

Using the anti-unitary property of Θ and commutation relations discussed previously after some algebra we have

We stress the different sign in second block. Then using the elementary fact that $\dim \ker (A\oplus B) = \dim \ker A+\dim \ker B$, the fact that Θ is a bijection (so it doesn’t change dimensions of kernels), and the fact that n + m mod 2 = n − m mod 2, we get

where one notes that the last line is the usual Fredholm index. Then by the first statement of Theorem 3.1 we may write

Again using the anti-unitary property of Θ, commutation relations, and the observation that U satisfies (2.3) we also have

$$ \begin{array}{@{}rcl@{}} \tilde{\mathbb{P}}\tilde{U} = \left[\begin{array}{ccccc} \mathbb{P}U & 0\\ 0 & {\Theta}\mathbb{P}U^{\ast}{\Theta}^{*} \end{array}\right] \end{array} $$

so that following the same procedure, we obtain

$$ \begin{array}{@{}rcl@{}} \text{ind}_{2}\tilde{\mathbb{P}}\tilde{U} &= \text{ind}\mathbb{P}U\mod2 \end{array} $$

and so (A.7). □

In conclusion, since all the pre-existing $\mathbb {Z}_{2}$ indices are known to also admit such a direct sum decomposition relating them to the Chern number, our index agrees with them.

1.6 A.6 Θ-Odd Fredholm Theory

For convenience of the reader, we include here a repetition of some of the $\mathbb {Z}_{2}$ Fredholm theory phrased via the Θ-odd constraint (??). These proofs first appeared in [37] under the guise of skew-adjoint Fredholm theory and then new proofs were presented in [33] for odd-symmetric operators. In what follows, ${\mathscr{B}}_{\Theta }({\mathscr{H}}),\mathcal {K}_{\Theta }({\mathscr{H}})$ are the bounded linear operators and respectively compact operators obeying (??).

Theorem A.12

ind₂ is stable under norm continuous perturbations obeying (??).

Proof

Let $T\in \mathcal {F}_{\Theta }({\mathscr{H}})$. Then we may make the decompositions ${\mathscr{H}} = \ker (T)^{\perp }\oplus \ker T$ and ${\mathscr{H}} = \text {im} T\oplus \text {coker} T$ by $\text {im} T\in \text {Closed}({\mathscr{H}})$. With respect to this decomposition we may write

$$ \begin{array}{@{}rcl@{}} T = \left[\begin{array}{ccccc} T_{11} & 0 \\ 0 & 0 \end{array}\right] ,\quad {\Theta} = \left[\begin{array}{ccccc} {\Theta}_{11} & 0 \\ 0 & {\Theta}_{22} \end{array}\right] \end{array} $$

where $T_{11}:\ker (T)^{\perp } \to \text {im} T$ is an isomorphism. We note that Θ is diagonal in this decomposition due to T = −ΘT^∗Θ. Furthermore, Θ₁₁,Θ₂₂ are both anti-unitary operators such that due to the corresponding properties of Θ.

Now let $S\in {\mathscr{B}}_{\Theta }({\mathscr{H}})$ be given with S_ij for 1 ≤ i, j ≤ 2 denoting the blocks of S with respect to the stated decomposition of ${\mathscr{H}}$. For $\left \|S\right \|$ is sufficiently small we have T₁₁ + S₁₁ invertible. Define A := −S₂₁(T₁₁ + S₁₁)^− 1S₁₂ + S₂₂. Since A is a linear map from $\ker T \to \text {coker} T$ and T is Fredholm (so it has finite kernel), by the rank nullity theorem we have $\dim \ker T=\dim \text {im} A+\dim \ker A$. Performing an LDU-decomposition for T + S, we find

$$T+S = I_{1} ((T_{11}+S_{11})\oplus A) I_{2}$$

where I₁,I₂ are invertible. Since T₁₁ + S₁₁ is invertible, we conclude $\ker (T+S)=\ker A$ so that

$$ \begin{array}{@{}rcl@{}} \dim \ker(T+S) + \dim \text{im} A= \dim \ker T . \end{array} $$

Then as A is Θ₂₂-odd and it is a map between finite $\ker T\to \text {coker} T\cong \ker T$ (as indT = 0) we may use Lemma A.13 to conclude that $\dim \ker (T+S) = \dim \ker T \mod 2$, i.e., ind₂(T + S) = ind₂T. □

Lemma A.13

For $A\in {\mathscr{B}}_{\Theta }(\mathcal {V})$, if $\mathcal {V}$ is finite dimensional then $\dim \text {im} A$ is even.

Proof

Let ψ ∈imA. Consider AΘψ ∈imA. Noting A is Θ-odd and Θ is anti-unitary a calculation shows 〈AΘψ, ψ〉 = 0. As $\text {im} A \cong \ker (A)^{\perp }$ and Θ is invertible we conclude that each element of imA is paired with a distinct orthogonal element. Hence, $\dim \text {im} A$ is even. □

Theorem A.14

ind₂ is stable under perturbations in $\mathcal {K}_{\Theta }({\mathscr{H}})$.

Proof

The fact that $\mathcal {F}_{\Theta }({\mathscr{H}})+\mathcal {K}_{\Theta }({\mathscr{H}})\subseteq \mathcal {F}_{\Theta }({\mathscr{H}})$ follows from Atkinson’s theorem [40]. Then given $T\in \mathcal {F}_{\Theta }({\mathscr{H}})$ and $K\in \mathcal {K}_{\Theta }({\mathscr{H}})$ consider the norm continuous map [0, 1] ∋ t↦T + tK. This map passes within $\mathcal {F}_{\Theta }({\mathscr{H}})$ by the first statement. Hence, by Theorem A.12 the result follows. □