Existence and Computation of Generalized Wannier Functions for Non-Periodic Systems in Two Dimensions and Higher

Abstract

Exponentially-localized Wannier functions (ELWFs) are an orthonormal basis of the Fermi projection of a material consisting of functions which decay exponentially fast away from their maxima. When the material is insulating and crystalline, conditions which guarantee existence of ELWFs in dimensions one, two, and three are well-known, and methods for constructing ELWFs numerically are well-developed. We consider the case where the material is insulating but not necessarily crystalline, where much less is known. In one spatial dimension, Kivelson and Nenciu-Nenciu have proved ELWFs can be constructed as the eigenfunctions of a self-adjoint operator acting on the Fermi projection. In this work, we identify an assumption under which we can generalize the Kivelson–Nenciu–Nenciu result to two dimensions and higher. Under this assumption, we prove that ELWFs can be constructed as the eigenfunctions of a sequence of self-adjoint operators acting on the Fermi projection.

Appendices

A Proof of Properties of $P_j$ (Proposition 4.2)

It turns out that both of the estimates in Proposition 4.2 will easily follow by showing that the resolvent of PXP is exponentially localized in the following technical sense:

Proposition A.1

Suppose that P is an exponentially localized orthogonal projector and PXP has uniform spectral gaps with decomposition $\{ \sigma _j \}_{j \in {\mathcal {J}}}$ and corresponding contours $\{ {\mathcal {C}}_j \}_{j \in {\mathcal {J}}}$ (see Definition 4.2). There exists finite, positive constants $(C, \gamma ^*)$ so that for all $0 \le \gamma \le \gamma ^*$

$$\begin{aligned} \sup _{j \in {\mathcal {J}}} \sup _{\lambda \in {\mathcal {C}}_j} \Vert B_{\gamma } (\lambda - PXP)^{-1} B_{\gamma }^{-1} \Vert = \sup _{j \in {\mathcal {J}}} \sup _{\lambda \in {\mathcal {C}}_j} \Vert (\lambda - P_{\gamma } X P_{\gamma })^{-1} \Vert \le C. \end{aligned}$$

We will prove Proposition A.1 in Section A.1. We will then use Proposition A.1 to show that $P_j$ admits an exponentially localized kernel in Section A.2 and that $P_j$ is localized along lines where $X = \eta _j$ in Section A.3. The analysis given in Sections A.1, A.2, and A.3 is quite general and can be generalized to choices of position operator other than X and Y. We show how our analysis can be extended to a wide class of self-adjoint position operators in Section A.4.

1.1 Proof that the Resolvent of $P_j$ is Exponentially Localized (Proposition A.1)

One key part of Proposition A.1 is that the bound on $\Vert (\lambda - P_{\gamma } X P_{\gamma })^{-1} \Vert$ is uniform in the choice of $j \in {\mathcal {J}}$ as well as the element of the contour $\lambda \in {\mathcal {C}}_j$. By applying a naive argument, typically one finds that many of the estimates depend on $\vert {}\lambda \vert {}$. Since $\vert {}\lambda \vert {}$ can be arbitrarily large, this naive argument does not allow us to prove Proposition A.1 with uniform constants. To correct this issue we introduce a technical result, the shifting lemma (Lemma A.1), which allows one to effectively shift the contour ${\mathcal {C}}_j$ so that instead of getting a dependence of $\vert {}\lambda \vert {}$ we have a dependence of $\vert {} \lambda - \eta \vert {}$ where $\eta$ is an arbitrary element from $\sigma _j$. If $\lambda \in {\mathcal {C}}_j$ and $\eta \in \sigma _j$ then the difference $\vert {} \lambda - \eta \vert {}$ is bounded by the diameter of the contour ${\mathcal {C}}_j$ and therefore bounded by a constant uniform in $j \in {\mathcal {J}}$.

The core idea underlying the shifting lemma is the following simple calculation. First, recall that if we are given a projector P, we can define the complementary projector $Q := I - P$. Since P and Q act on orthogonal subspaces we should expect that, for any $\eta \ne \lambda$,

$$\begin{aligned} (\lambda - PXP)^{-1} P&= \Big ((\lambda - \eta + \eta ) - P(X - \eta + \eta )P\Big )^{-1} P \\&= \Big ((\lambda - \eta ) - P(X - \eta )P + \eta (I - P)\Big )^{-1} P \\&= \Big ((\lambda - \eta ) - P(X - \eta )P + \eta Q\Big )^{-1} P \\&= \Big ((\lambda - \eta ) - P(X - \eta )P\Big )^{-1} P \end{aligned}$$

By using this calculation, we are able to replace $\lambda$ with $\lambda - \eta$ which leads to uniform bounds as discussed above. Importantly, the above calculation does not require periodicity of the underlying system. By using some variations on this calculation, one can show the following lemma:

Lemma A.1

(Shifting Lemma) Suppose P admits an exponentially localized kernel and suppose that PXP has uniform spectral gaps with decomposition $\{ \sigma _j \}_{j \in {\mathcal {J}}}$ and corresponding contours $\{ {\mathcal {C}}_j \}_{j \in {\mathcal {J}}}$. Then there exists a $\gamma ^*$ so that the following are equivalent for all $0 \le \gamma \le \gamma ^*$:

• 1. There exists a $C > 0$, independent of j, such that

  $$\begin{aligned} \sup _{\lambda \in {\mathcal {C}}_j} \Vert (\lambda - P_{\gamma } X P_{\gamma })^{-1}\Vert \le C. \end{aligned}$$

• 2. There exists a $C' > 0$, independent of j, such that for each $j \in {\mathcal {J}}$:

  $$\begin{aligned} \sup _{\lambda \in {\mathcal {C}}_j} \sup _{\eta _j \in \sigma _j} \Vert (\lambda _{\eta _j} - P_{\gamma } X_{\eta _j} P_{\gamma })^{-1}\Vert \le C' \end{aligned}$$

Furthermore, for any $0 \le \gamma \le \gamma ^*$, if either $\Vert (\lambda - P_{\gamma } X P_{\gamma })^{-1}\Vert$ or $\Vert (\lambda _{\eta _j} - P_{\gamma } X_{\eta _j} P_{\gamma })^{-1}\Vert$ are bounded, we have for any $j \in {\mathcal {J}}$, $\lambda \in {\mathcal {C}}_j$ and any $\eta _j \in \sigma _j$ that:

$$\begin{aligned} (\lambda - P_{\gamma } X P_{\gamma })^{-1} P_{\gamma } = (\lambda _{\eta _j} - P_{\gamma } X_{\eta _j} P_{\gamma })^{-1} P_{\gamma }. \end{aligned}$$

Proof

Given in Appendix B. $\square$

As a consequence of Lemma A.1, to prove Proposition A.1 it is enough to fix some $j \in {\mathcal {J}}$, $\lambda \in {\mathcal {C}}_j$, and $\eta _j \in \sigma _j$ and show that

$$\begin{aligned} \Vert (\lambda _{\eta _j} - P_{\gamma } X_{\eta _j} P_{\gamma })^{-1}\Vert \le C' \end{aligned}$$

(A.1)

where the constant $C'$ is independent of the choice of j, $\lambda$, and $\eta _j$. For the remainder of this section, we will fix a choice of $j \in {\mathcal {J}}$, $\lambda \in {\mathcal {C}}_j$, and $\eta _j \in \sigma _j$ and prove Equation (A.1).

The path to proving Equation (A.1) is to use the following chain of implications (where USG is an abbreviation for uniform spectral gaps):

$$\begin{aligned} \begin{aligned} P X&P \text { has USG} \Longrightarrow \Vert (\lambda _{\eta _j} - P X_{\eta _j} P )^{-1} \Vert< \infty \\&\Longrightarrow \Vert (\lambda _{\eta _j} - P X_{\eta _j} P_{\gamma } )^{-1} \Vert< \infty \Longrightarrow \Vert (\lambda _{\eta _j} - P_{\gamma } X_{\eta _j} P_{\gamma } )^{-1} \Vert < \infty \end{aligned} \end{aligned}$$

(A.2)

In words, since PXP has uniform spectral gaps, we know that $\Vert (\lambda _{\eta _j} - P X_{\eta _j} P )^{-1} \Vert < \infty$ for all $\lambda \in {\mathcal {C}}_j$. Using this fact, along with the fact that P is exponentially localized, we can conclude that $\Vert (\lambda _{\eta _j} - P X_{\eta _j} P_{\gamma } )^{-1} \Vert < \infty$ for all $\gamma$ sufficiently small. Finally, once we know that $(\lambda _{\eta _j} - P X_{\eta _j} P_{\gamma } )^{-1}$ is bounded, we can use that estimate to show that $(\lambda _{\eta _j} - P_{\gamma } X_{\eta _j} P_{\gamma })^{-1}$ is bounded for all $\gamma$ sufficiently small. This completes the proof of the proposition.

The first implication, that uniform spectral gaps implies $(\lambda _{\eta _j} - P X_{\eta _j} P )^{-1}$ is bounded is an immediate consequence of Lemma A.1 by choosing $\gamma = 0$. Therefore, we only need to show the last two implications. We will prove the second implication in Section A.1.1 and the final implication in Section A.1.2.

1.1.1 Proof that Shifted $(\lambda - P X P_{\gamma } )^{-1}$ is Bounded

By adding and subtracting $P X_{\eta _j} P$ in the shifted resolvent we have that, formally,

$$\begin{aligned} (\lambda _{\eta _j}&- P X_{\eta _j} P_{\gamma } )^{-1} \\&= (\lambda _{\eta _j} - P X_{\eta _j} P + P X_{\eta _j} P - P X_{\eta _j} P_{\gamma } )^{-1} \\&= \Big (\lambda _{\eta _j} - P X_{\eta _j} P - P X_{\eta _j} (P_{\gamma } - P) \Big )^{-1} \\&= \Big (I - (\lambda _{\eta _j} - P X_{\eta _j} P)^{-1} P X_{\eta _j} (P_{\gamma } - P) \Big )^{-1} (\lambda _{\eta _j} - P X_{\eta _j} P)^{-1}. \end{aligned}$$

Since P admits an exponentially localized kernel, it is easy to verify that exists a constant C so that for all $\gamma$ sufficiently small

$$\begin{aligned} \Vert P_{\gamma } - P \Vert \le C \gamma . \end{aligned}$$

Therefore, if we can show that $\Vert (\lambda _{\eta _j} - P X_{\eta _j} P)^{-1} P X_{\eta _j} \Vert$ is bounded by an absolute constant, we can choose $\gamma$ sufficiently small so that

$$\begin{aligned} \Vert (\lambda _{\eta _j} - P X_{\eta _j} P)^{-1} P X_{\eta _j} (P_{\gamma } - P) \Vert \le \frac{1}{2}. \end{aligned}$$

This implies that $(\lambda _{\eta _j} - P X_{\eta _j} P_{\gamma } )^{-1}$ is bounded since

$$\begin{aligned} \Vert (\lambda _{\eta _j}&- P X_{\eta _j} P_{\gamma } )^{-1} \Vert \\&\le \Vert \Big (I - (\lambda _{\eta _j} - P X_{\eta _j} P)^{-1} P X_{\eta _j} (P_{\gamma } - P) \Big )^{-1} \Vert \Vert (\lambda _{\eta _j} - P X_{\eta _j} P)^{-1} \Vert \\&\le 2 \Vert (\lambda _{\eta _j} - P X_{\eta _j} P)^{-1} \Vert . \end{aligned}$$

To show that $\Vert (\lambda _{\eta _j} - P X_{\eta _j} P)^{-1} P X_{\eta _j} \Vert$ is bounded, we recall that $I = P + Q$ and so

$$\begin{aligned} (\lambda _{\eta _j}&- P X_{\eta _j} P)^{-1} P X_{\eta _j} \\&= (\lambda _{\eta _j} - P X_{\eta _j} P)^{-1} \Big ( P X_{\eta _j} P + P X_{\eta _j} Q \Big ) \\&= (\lambda _{\eta _j} - P X_{\eta _j} P)^{-1} \Big ( P X_{\eta _j} P - \lambda _{\eta _j} + \lambda _{\eta _j} + P X_{\eta _j} Q \Big ) \\&= -I + (\lambda _{\eta _j} - P X_{\eta _j} P)^{-1} \Big ( \lambda _{\eta _j} + P X_{\eta _j} Q \Big ) \end{aligned}$$

Therefore, recalling $\lambda _{\eta _j} = \lambda - \eta _j$, and $X_{\eta _j} = X - \eta _j$ we have that

$$\begin{aligned} \Vert (\lambda _{\eta _j} - P X_{\eta _j} P)^{-1} P X_{\eta _j} \Vert \le 1 + \Vert (\lambda _{\eta _j} - P X_{\eta _j} P)^{-1} \Vert \Big ( \vert {}\lambda - \eta _j \vert {} + \Vert P (X - \eta _j) Q \Vert \Big ). \end{aligned}$$

As discussed previously, by the choice of $\eta _j$ and the uniform spectral gaps assumption, we know that $\vert {}\lambda - \eta _j\vert {}$ is bounded by a constant independent of j, $\lambda$, and $\eta _j$. To see the second term is bounded by a constant, recall that $PQ = 0$ so that

$$\begin{aligned} \Vert P (X - \eta _j) Q \Vert = \Vert P X Q \Vert = \Vert [P, X] Q \Vert \le \Vert [P, X ] \Vert \end{aligned}$$

Since P admits an exponentially localized kernel, it is easily verified that $\Vert [P, X ] \Vert$ is bounded by an absolute constant. Therefore, $\Vert (\lambda _{\eta _j} - P X_{\eta _j} P)^{-1} P X_{\eta _j} \Vert$ is bounded and so by choosing

$$\begin{aligned} \gamma \le (2 C \Vert (\lambda _{\eta _j} - P X_{\eta _j} P)^{-1} P X_{\eta _j} \Vert )^{-1} \end{aligned}$$

by the previous logic $(\lambda _{\eta _j} - P X_{\eta _j} P_{\gamma } )^{-1}$ is bounded, proving the first implication.

1.1.2 Proof that Shifted $(\lambda - P_{\gamma } X P_{\gamma } )^{-1}$ is Bounded

Similar to before, we begin by adding and subtracting $P X_{\eta _j} P_{\gamma }$ in the shifted resolvent:

$$\begin{aligned} (\lambda _{\eta _j}&- P_{\gamma } X_{\eta _j} P_{\gamma } )^{-1} \\&= (\lambda _{\eta _j} - P X_{\eta _j} P_{\gamma } + P X_{\eta _j} P_{\gamma } - P_{\gamma } X_{\eta _j} P_{\gamma } )^{-1} \\&= (\lambda _{\eta _j} - P X_{\eta _j} P_{\gamma } - (P_{\gamma } - P) X_{\eta _j} P_{\gamma } )^{-1} \\&= \Big (I - (P_{\gamma } - P) X_{\eta _j} P_{\gamma } (\lambda _{\eta _j} - P X_{\eta _j} P_{\gamma })^{-1} \Big )^{-1} (\lambda _{\eta _j} - P X_{\eta _j} P_{\gamma })^{-1} \end{aligned}$$

Similar to before, since $\Vert P_{\gamma } - P \Vert \le C \gamma$, if we can show that $\Vert X_{\eta _j} P_{\gamma } (\lambda _{\eta _j} - P X_{\eta _j} P_{\gamma })^{-1} \Vert$ is bounded, then we can pick $\gamma$ sufficiently small so that

$$\begin{aligned} \Vert (P_{\gamma } - P) X_{\eta _j} P_{\gamma } (\lambda _{\eta _j} - P X_{\eta _j} P_{\gamma })^{-1} \Vert \le \frac{1}{2} \end{aligned}$$

which implies that $(\lambda _{\eta _j} - P_{\gamma } X_{\eta _j} P_{\gamma } )^{-1}$ is bounded.

To show that $X_{\eta _j} P_{\gamma } (\lambda _{\eta _j} - P X_{\eta _j} P_{\gamma })^{-1}$ is bounded, let us adopt the shorthand $E := P_{\gamma } - P$. Since $Q_{\gamma } = I - P_{\gamma }$ and $Q = I - P$ we also have that

$$\begin{aligned} E = P_{\gamma } - P = Q - Q_{\gamma } \Rightarrow Q = Q_{\gamma } + E. \end{aligned}$$

Next, we calculate

$$\begin{aligned} X_{\eta _j} P_{\gamma } (\lambda _{\eta _j} - P X_{\eta _j} P_{\gamma })^{-1}&= (P + Q) X_{\eta _j} P_{\gamma } (\lambda _{\eta _j} - P X_{\eta _j} P_{\gamma })^{-1} \\&= (P + Q_{\gamma } + E) X_{\eta _j} P_{\gamma } (\lambda _{\eta _j} - P X_{\eta _j} P_{\gamma })^{-1} \end{aligned}$$

Moving the term multiplied by E to the left hand side then gives

$$\begin{aligned} (I - E) X_{\eta _j}&P_{\gamma } (\lambda _{\eta _j} - P X_{\eta _j} P_{\gamma })^{-1} = (P + Q_{\gamma }) X_{\eta _j} P_{\gamma } (\lambda _{\eta _j} - P X_{\eta _j} P_{\gamma })^{-1} \end{aligned}$$

Since $\Vert E \Vert = \Vert P_{\gamma } - P \Vert \le C \gamma$, we can choose $\gamma$ sufficiently small so that $I - E$ is invertible and hence we conclude that

$$\begin{aligned} \Vert X_{\eta _j}&P_{\gamma } (\lambda _{\eta _j} - P X_{\eta _j} P_{\gamma })^{-1} \Vert \\ {}&\le \Vert (I - E)^{-1} \Vert \Big ( \Vert P X_{\eta _j} P_{\gamma } (\lambda _{\eta _j} - P X_{\eta _j} P_{\gamma })^{-1} \Vert \\ {}&\qquad + \Vert Q_{\gamma } X_{\eta _j} P_{\gamma } \Vert \Vert (\lambda _{\eta _j} - P X_{\eta _j} P_{\gamma })^{-1} \Vert \Big ) \\ {}&\le \Vert (I - E)^{-1} \Vert \Big ( 1 + \vert {}\lambda - \eta _j \vert {} \Vert (\lambda _{\eta _j} - P X_{\eta _j} P_{\gamma })^{-1} \Vert \\ {}&\qquad + \Vert Q_{\gamma } X_{\eta _j} P_{\gamma } \Vert \Vert (\lambda _{\eta _j} - P X_{\eta _j} P_{\gamma })^{-1} \Vert \Big ) \end{aligned}$$

Since $\vert {}\lambda - \eta _j\vert {}$ is bounded by construction and $\Vert (\lambda _{\eta _j} - P X_{\eta _j} P_{\gamma })^{-1} \Vert$ is bounded by the proof in Section A.1.1, the only term to bound is $\Vert Q_{\gamma } X_{\eta _j} P_{\gamma } \Vert$. Using that $Q_{\gamma } P_{\gamma } = 0$ we have that

$$\begin{aligned} \Vert Q_{\gamma } X_{\eta _j} P_{\gamma } \Vert = \Vert Q_{\gamma } X P_{\gamma } \Vert \le \Vert Q_{\gamma } \Vert \Vert [X, P_{\gamma } ] \Vert . \end{aligned}$$

Since P admits an exponentially localized kernel, it is easy to verify that $\Vert [X, P_{\gamma } ] \Vert$ is bounded. Since additionally,

$$\begin{aligned} \Vert Q_{\gamma } \Vert = \Vert I - P_{\gamma } \Vert \le 1 + \Vert P_{\gamma } \Vert \end{aligned}$$

we conclude that $\Vert X_{\eta _j} P_{\gamma } (\lambda _{\eta _j} - P X_{\eta _j} P_{\gamma })^{-1} \Vert$ is bounded by an absolute constant for all $\gamma$ sufficiently small. Therefore, by the previously discussed reasoning, we conclude that $\Vert (\lambda _{\eta _j} - P_{\gamma } X_{\eta _j} P_{\gamma } )^{-1} \Vert$ is bounded by a constant, completing the proof of the proposition.

1.2 Proof that $P_j$ Admits an Exponentially Localized Kernel (Proposition 4.2(1))

Let us recall the definition of $P_j$ (Definition 4.2)

$$\begin{aligned} P_j = \frac{1}{2 \pi i} \int _{{\mathcal {C}}_j} (\lambda - PXP)^{-1} P \text {d}{\lambda } \end{aligned}$$

By multiplying on the left by $B_{\gamma }$ and on the right by $B_{\gamma }^{-1}$ we therefore have that

$$\begin{aligned} P_{j,\gamma }&= \frac{1}{2 \pi i} \int _{{\mathcal {C}}_j} B_{\gamma } (\lambda - PXP)^{-1} P B_{\gamma }^{-1} \text {d}{\lambda } \\&= \frac{1}{2 \pi i} \int _{{\mathcal {C}}_j} B_{\gamma } (\lambda - PXP)^{-1} \Big ( B_{\gamma }^{-1} B_{\gamma }\Big ) P B_{\gamma }^{-1} \text {d}{\lambda } \\&= \frac{1}{2 \pi i} \int _{{\mathcal {C}}_j} (\lambda - P_{\gamma } X P_{\gamma } )^{-1} P_{\gamma } \text {d}{\lambda } \end{aligned}$$

Therefore, we conclude that

$$\begin{aligned} \Vert P_{j,\gamma } \Vert \le \frac{\ell {({\mathcal {C}})}}{2 \pi } \Vert P_{\gamma } \Vert \Big ( \sup _{\lambda \in {\mathcal {C}}_j} \Vert (\lambda - P_{\gamma } X P_{\gamma } )^{-1} \Vert \Big ) \end{aligned}$$

In Section A.1 we showed that there exists an absolute constant C so that

$$\begin{aligned} \sup _{\lambda \in {\mathcal {C}}_j} \Vert (\lambda - P_{\gamma } X P_{\gamma } )^{-1} \Vert \le C. \end{aligned}$$

Therefore, we have that there exists a constant C so that

$$\begin{aligned} \Vert P_{j,\gamma } \Vert \le C. \end{aligned}$$

we can now use this estimate to show that $P_j$ admits an exponentially localized kernel.

Proposition A.2

Suppose that P is an orthogonal projector which admits an exponentially localized kernel with rate $\gamma _1$ (Definition 4.1). Suppose further that $P_j$ is an orthogonal projector which satisfies the properties:

• 1. $P_j P = P P_j = P_j$

• 2. There exist a constant C such that for all $0 \le \gamma \le \gamma _2$

  $$\begin{aligned} \Vert P_{j,\gamma } \Vert \le C. \end{aligned}$$

Then $P_j$ admits an integral kernel $P_j(\cdot , \cdot ) : \mathbb {R}^2 \times \mathbb {R}^2 \rightarrow \mathbb {C}$ such that for all $\gamma \le \frac{1}{2} \min \{ \gamma _1, \gamma _2 \}$

$$\begin{aligned} \vert {}P_j(\varvec{x}, \varvec{x}')\vert {} \le C' e^{-\gamma \vert {}\varvec{x} - \varvec{x}'\vert {}} \qquad a.e. \end{aligned}$$

where the constant $C'$ only depends on $\gamma _1$, $\gamma _2$, and $\Vert P_{j,\gamma }\Vert$.

Proof

For this proof we will first show that $P_{j}$ admits a measurable integral kernel and then show that $P_{j,\gamma }$ is a bounded operator from $L^1$ to $L^\infty$. Once we can show these properties, the fact that $P_j$ admits an exponentially localized kernel follows by application of the Lebesgue differentiation theorem.

Define $\gamma ^* := \min \{ \gamma _1, \gamma _2 \}$, we will first show that $P_{j,\gamma }$ is a bounded operator from $L^2$ to $L^\infty$ for all $0 \le \gamma < \gamma ^*$ (that is that $P_{j,\gamma }$ is a Carleman operator). First, let us fix a realization of a function $f \in L^2(\mathbb {R}^2)$. Using the fact that $P_{j,\gamma } = P_{\gamma } P_{j,\gamma }$, we have that for almost all $\varvec{x}$,

$$\begin{aligned} \vert {}(P_{j,\gamma } f)(\varvec{x})\vert {} = \vert {}(P_{\gamma } P_{j,\gamma } f)(\varvec{x})\vert {}&= \left| {}\int _{\mathbb {R}^2} P_{\gamma }(\varvec{x}, \varvec{x}') (P_{j,\gamma } f)(\varvec{x}') \text {d}{\varvec{x}'}\right| {} \end{aligned}$$

(A.3)

$$\begin{aligned}&\le \int _{\mathbb {R}^2} C e^{-(\gamma _1 - \gamma ) \vert {}\varvec{x} - \varvec{x}'\vert {}} (P_{j,\gamma } f)(\varvec{x}') \text {d}{\varvec{x}'} \end{aligned}$$

(A.4)

$$\begin{aligned}&\le C \left( \int _{\mathbb {R}^2} e^{-2(\gamma _1 - \gamma ) \vert {}\varvec{x} - \varvec{x}'\vert {}} \text {d}{\varvec{x}'} \right) \Vert P_{j,\gamma } f \Vert \end{aligned}$$

(A.5)

where in the last line we have applied the Cauchy-Schwartz inequality. Taking the essential supremum over $\varvec{x}$ on both sides then gives

$$\begin{aligned} \sup _{\varvec{x}}\vert {}(P_{j,\gamma } f)(\varvec{x})\vert {}&\le C (\gamma _1 - \gamma )^{-1} \Vert P_{j,\gamma } \Vert \Vert f \Vert . \end{aligned}$$

(A.6)

Therefore, $P_{j,\gamma }$ is a bounded operator $L^2 \rightarrow L^\infty$. Since $P_{j,\gamma }$ is a bounded operator from $L^2 \rightarrow L^2$ and $L^2 \rightarrow L^\infty$, a standard result in the study of integral operators (see [74, Corollary A.1.2]) gives us that $P_{j,\gamma }$ admits an integral kernel which satisfies the following estimate:

$$\begin{aligned} \sup _{\varvec{x}} \left[ \int \vert {}P_{j,\gamma }(\varvec{x}, \varvec{x}')\vert {}^2 \text {d}{\varvec{x}'} \right] ^{1/2} < \infty . \end{aligned}$$

(A.7)

By the definition of $P_{j,\gamma }$ this implies that for any $\varvec{a} \in \mathbb {R}^2$

$$\begin{aligned} \sup _{\varvec{x}} \left[ \int \vert {}e^{\gamma \vert {} \varvec{x} - \varvec{a}\vert {}} P_{j}(\varvec{x}, \varvec{x}') e^{-\gamma \vert {} \varvec{x}' - \varvec{a}\vert {}}\vert {}^2 \text {d}{\varvec{x}'} \right] ^{1/2} < \infty . \end{aligned}$$

(A.8)

For reasons which will shortly be clear, we make the following observation. Since for all $\gamma$

$$\begin{aligned} \Vert P_{j,\gamma } \Vert = \Vert P_{j,\gamma }^\dagger \Vert = \Vert (B_{\gamma } P_j B_{\gamma }^{-1})^\dagger \Vert = \Vert P_{j,-\gamma } \Vert \end{aligned}$$

we can repeat the above steps replacing $\gamma$ with $-\gamma$. This implies that

$$\begin{aligned} \begin{aligned} \sup _{\varvec{x}}&\left[ \int \vert {}e^{-\gamma \vert {} \varvec{x} - \varvec{a}\vert {}} P_{j}(\varvec{x}, \varvec{x}') e^{\gamma \vert {} \varvec{x}' - \varvec{a}\vert {}}\vert {}^2 \text {d}{\varvec{x}'} \right] ^{1/2}< \infty \\&\Longleftrightarrow \sup _{\varvec{x}} \left[ \int \vert {}e^{\gamma \vert {} \varvec{x}' - \varvec{a}\vert {}} P_{j}(\varvec{x}', \varvec{x}) e^{-\gamma \vert {} \varvec{x} - \varvec{a}\vert {}}\vert {}^2 \text {d}{\varvec{x}'} \right] ^{1/2}< \infty \\&\Longleftrightarrow \sup _{\varvec{x}} \left[ \int \vert {} P_{j,\gamma }(\varvec{x}', \varvec{x})\vert {}^2 \text {d}{\varvec{x}'} \right] ^{1/2} < \infty \end{aligned} \end{aligned}$$

(A.9)

where we have used the fact that $P_j$ is self-adjoint and hence $P_j(\varvec{x},\varvec{x}') = \overline{P_j(\varvec{x}',\varvec{x})}$. Note that the difference between Equation (A.7) and (A.9) is that we have exchanged the arguments in the kernel for $P_{j,\gamma }$.

Having established that $P_{j,\gamma }$ has an integral kernel satisfying Equation (A.9), we will now use the existence of this kernel to show that $P_{j,\gamma }$ is a bounded operator from $L^1 \rightarrow L^\infty$. Once we show this, the fact that the kernel for $P_j$ is exponentially localized will follow as a consequence of the Lebesgue differentiation theorem.

Repeating the calculations that lead to (A.5) we have that

$$\begin{aligned} \vert {}(P_{j,\gamma } f)(\varvec{x})\vert {}&\le C \int _{\mathbb {R}^2} e^{-(\gamma _1 - \gamma ) \vert {}\varvec{x} - \varvec{x}'\vert {}} \vert {}(P_{j,\gamma } f)(\varvec{x}')\vert {} \text {d}{\varvec{x}'} \\&\le C \int _{\mathbb {R}^2} e^{-(\gamma _1 - \gamma ) \vert {}\varvec{x} - \varvec{x}'\vert {}} \int _{\mathbb {R}^2} \vert {}P_{j,\gamma }(\varvec{x}', \varvec{y})\vert {} \vert {}f(\varvec{y})\vert {} \text {d}{\varvec{y}} \text {d}{\varvec{x}'} \\&= C \int _{\mathbb {R}^2} \vert {}f(\varvec{y})\vert {} \int _{\mathbb {R}^2} e^{-(\gamma _1 - \gamma ) \vert {}\varvec{x} - \varvec{x}'\vert {}} \vert {}P_{j,\gamma }(\varvec{x}', \varvec{y})\vert {} \text {d}{\varvec{x}'} \text {d}{\varvec{y}} \\&\le C \Vert f \Vert _{L^1} \left( \sup _{\varvec{y}} \int _{\mathbb {R}^2} e^{-(\gamma _1 - \gamma ) \vert {}\varvec{x} - \varvec{x}'\vert {}} \vert {}P_{j,\gamma }(\varvec{x}', \varvec{y})\vert {} \text {d}{\varvec{x}'} \right) \end{aligned}$$

Taking the essential supremum over $\varvec{x}$ on both sides then gives

$$\begin{aligned} \sup _{\varvec{x}} \vert {}(P_{j,\gamma } f)(\varvec{x})\vert {}&\le C \Vert f \Vert _{L^1} \left( \sup _{\varvec{x},\varvec{y}} \int _{\mathbb {R}^2} e^{-(\gamma _1 - \gamma ) \vert {}\varvec{x} - \varvec{x}'\vert {}} \vert {}P_{j,\gamma }(\varvec{x}', \varvec{y})\vert {} \text {d}{\varvec{x}'} \right) \\&\le C \Vert f \Vert _{L^1} \left( \sup _{\varvec{x}} \int _{\mathbb {R}^2} e^{-2 (\gamma _1 - \gamma ) \vert {}\varvec{x} - \varvec{x}'\vert {}} \text {d}{\varvec{x}'} \right) \\&\quad \times \left( \sup _{\varvec{y}} \int _{\mathbb {R}^2} \vert {}P_{j,\gamma }(\varvec{x}', \varvec{y})\vert {}^2 \text {d}{\varvec{x}'} \right) \end{aligned}$$

where in the last line we have used the Cauchy-Schwarz inequality. Finally, using Equation (A.9) we conclude that

$$\begin{aligned} \sup _{\varvec{x}} \vert {}(P_{j,\gamma } f)(\varvec{x})\vert {} \le C (\gamma _1 - \gamma )^{-1} \Vert f \Vert _{L^1} \end{aligned}$$

hence $P_{j,\gamma }$ is a bounded operator from $L^1 \rightarrow L^\infty$.

To show that $P_{j}$ admits an exponentially localized kernel, let us fix some arbitrary points $\varvec{a}, \varvec{b} \in \mathbb {R}^2$ and define a function $g_{\delta }$ as follows

$$\begin{aligned} g_{\delta }(\varvec{x}) = {\left\{ \begin{array}{ll} 1 &{} \varvec{x} \in {\mathcal {B}}_{\delta }(\varvec{0}) \\ 0 &{} \text {otherwise,} \end{array}\right. } \end{aligned}$$

where ${\mathcal {B}}_{\delta }(\varvec{0})$ is the ball of radius $\delta$ centered at $\varvec{0}$. Observe that $\Vert g_{\delta } \Vert _{L^1} = \vert {}{\mathcal {B}}_{\delta }\vert {}$ where $\vert {}{\mathcal {B}}_{\delta }\vert {}$ is the volume of the ball of radius $\delta$. With this notation, using our previous bound for $P_{j,\gamma }$ we have that for all $\varvec{a} \in \mathbb {R}^2$ and almost all $\varvec{b} \in \mathbb {R}^2$:

$$\begin{aligned}&\left| {} \int P_{j,\gamma }(\varvec{b}, \varvec{x}') g_{\delta }(\varvec{x}' - \varvec{a}) \text {d}{\varvec{x}'} \right| {} \le C' (\gamma _1 - \gamma )^{-1} \Vert g_{\delta } \Vert _{L^1} \\&\quad \Longrightarrow \frac{1}{\vert {}{\mathcal {B}}_{\delta }\vert {}} \left| {} \int _{{\mathcal {B}}_{\delta }(\varvec{a})} P_{j,\gamma }(\varvec{b}, \varvec{x}') \text {d}{\varvec{x}'} \right| {} \le C' (\gamma _1 - \gamma )^{-1} \\&\quad \Longrightarrow \frac{1}{\vert {}{\mathcal {B}}_{\delta }\vert {}} \left| {} \int _{{\mathcal {B}}_{\delta }(\varvec{a})} e^{\gamma \vert {} \varvec{b} - \varvec{a}\vert {}} P_{j}(\varvec{b}, \varvec{x}') e^{-\gamma \vert {} \varvec{x}' - \varvec{a}\vert {}} \text {d}{\varvec{x}'} \right| {} \le C' (\gamma _1 - \gamma )^{-1} \end{aligned}$$

where in the last line, we have substituted in the definition for the kernel for $P_{j,\gamma } = B_{\gamma ,\varvec{a}} P_j B_{\gamma ,\varvec{a}}^{-1}$. Since for each fixed $\varvec{b}$, the function $\varvec{x}' \mapsto P_{j,\gamma }(\varvec{b}, \varvec{x}')$ is in $L^2(\mathbb {R}^2)$ (Equation (A.9)), in particular this kernel is also in $L^1_{{{\,\mathrm{loc}\,}}}(\mathbb {R}^2)$. Therefore, by the Lebesgue differentiation theorem [79, Chapter 3, Theorem 1.2], for almost every $\varvec{a} \in \mathbb {R}^2$:

$$\begin{aligned} \lim _{\delta \rightarrow 0} \frac{1}{\vert {}{\mathcal {B}}_{\delta }\vert {}} \left| {} \int _{{\mathcal {B}}_{\delta }(\varvec{a})} e^{\gamma \vert {} \varvec{b} - \varvec{a}\vert {}} P_{j}(\varvec{b}, \varvec{x}') e^{-\gamma \vert {} \varvec{x}' - \varvec{a}\vert {}} \text {d}{\varvec{x}'} \right| {} = e^{\gamma \vert {} \varvec{b} - \varvec{a}\vert {}} \vert {}P_{j}(\varvec{b}, \varvec{a}) \vert {}. \end{aligned}$$

Hence, for almost every $\varvec{a}, \varvec{b} \in \mathbb {R}^2$:

$$\begin{aligned} \vert {}P_{j}(\varvec{b}, \varvec{a})\vert {} \le C' (\gamma _1 - \gamma )^{-1} e^{-\gamma \vert {} \varvec{b} - \varvec{a}\vert {}} \end{aligned}$$

which completes the proof. $\square$

1.3 Proof that $P_j$ is Localized in X (Proposition 4.2(2))

For this section, let us fix some $\eta _j \in \sigma _j$, we will prove that $\Vert (X - \eta _j) P_{j,\gamma }\Vert$ is bounded. The fact that $\Vert P_{j,\gamma } (X - \eta _j)\Vert$ is bounded follows by essentially the same steps. Recalling we define $\lambda _{\eta _j} := \lambda - \eta _j$ and $X_{\eta _j} := X - \eta _j$ and using Lemma A.1 we have

$$\begin{aligned} (X - \eta _j) P_{j,\gamma }&= \frac{1}{2 \pi i} \int _{{\mathcal {C}}_j} (X - \eta _j) (\lambda - P_{\gamma } X P_{\gamma })^{-1}P_{\gamma } \text {d}{\lambda } \\&= \frac{1}{2 \pi i} \int _{{\mathcal {C}}_j} X_{\eta _j} (\lambda _{\eta _j} - P_{\gamma } X_{\eta _j} P_{\gamma })^{-1} P_{\gamma } \text {d}{\lambda } \\&= \frac{1}{2 \pi i} \int _{{\mathcal {C}}_j} X_{\eta _j} P_{\gamma } (\lambda _{\eta _j} - P_{\gamma } X_{\eta _j} P_{\gamma })^{-1} \text {d}{\lambda } \\&= \frac{1}{2 \pi i} \int _{{\mathcal {C}}_j} (P_{\gamma } + Q_{\gamma }) X_{\eta _j} P_{\gamma } (\lambda _{\eta _j} - P_{\gamma } X_{\eta _j} P_{\gamma })^{-1} \text {d}{\lambda }. \end{aligned}$$

where we have used that $P_{\gamma }$ commutes with $(\lambda _{\eta _j} - P_{\gamma } X_{\eta _j} P_{\gamma })^{-1}$. Therefore,

$$\begin{aligned} \Vert (X - \eta _j) P_{j,\gamma }\Vert&\le \frac{\ell ({\mathcal {C}}_j)}{2 \pi } \bigg ( \Vert P_{\gamma } X_{\eta _j} P_{\gamma } (\lambda _{\eta _j} - P_{\gamma } X_{\eta _j} P_{\gamma })^{-1}\Vert \\&\quad + \Vert Q_{\gamma } X_{\eta _j} P_{\gamma }\Vert \Vert (\lambda _{\eta _j} - P_{\gamma } X_{\eta _j} P_{\gamma })^{-1}\Vert \bigg ). \end{aligned}$$

Note that

$$\begin{aligned} \Vert P_{\gamma } X_{\eta _j} P_{\gamma } (\lambda _{\eta _j} - P_{\gamma } X_{\eta _j} P_{\gamma })^{-1}\Vert \le 1 + \vert {}\lambda _{\eta _j}\vert {} \Vert (\lambda _{\eta _j} - P_{\gamma } X_{\eta _j} P_{\gamma })^{-1}\Vert . \end{aligned}$$

Therefore, since $\Vert (\lambda _{\eta _j} - P_{\gamma } X_{\eta _j} P_{\gamma })^{-1}\Vert$ and $\Vert Q_{\gamma } X_{\eta _j} P_{\gamma }\Vert$ are both bounded, we can conclude that $\Vert (X - \eta _j) P_{j,\gamma }\Vert$ is bounded as we wanted to show.

1.4 Extension to Other Position Operators

In Sections A.1, A.2, and A.3 we show that if PXP has uniform spectral gaps then if $\{ P_j \}_{j \in {\mathcal {J}}}$ are the band projectors for PXP then each $P_j$ admits an exponentially localized kernel and each $P_j$ is localized along a line $X = \eta _j$. In the main text we used these properties of $P_j$ to show that exponentially localized Wannier functions can be constructed by diagonalizing $P_j Y P_j$ for each $j \in {\mathcal {J}}$. In this section we show that this argument does not rely on X being precisely the position operator defined in (3.1). Instead, we will show that for operators ${\widehat{X}}$ which are, in a certain sense, close to the operator X, if $P {\widehat{X}} P$ has uniform spectral gaps, then if we define projections $P_j$ with respect to the spectral bands of $P {\widehat{X}} P$, the rest of our analysis goes through without modification. Specifically, we can prove that the projections $P_j$ admit exponentially localized kernels, localize along lines $X = \eta _j$, and can be used to form generalized Wannier functions by diagonalizing each $P_j Y P_j$ for $j \in {\mathcal {J}}$. We anticipate that this generalization may be important for two reasons.

• Depending on the application, it may be more natural to measure position by operators other than the standard ones.

• It may be that PXP does not have uniform spectral gaps, but $P {\widehat{X}} P$, where ${\widehat{X}}$ is an alternative position operator, does. In this case, the fact that our proofs can be generalized is essential to prove existence of exponentially localized generalized Wannier functions.

We consider various cases where alternative position operators are useful in [41]. The problem of constructing an alternative position operator ${\widehat{X}}$ which ensures that $P {\widehat{X}} P$ has gaps has been considered in [34].

We will prove the following Lemma, which gives sufficient conditions so that the band projectors for $P {\widehat{X}} P$, where ${\widehat{X}}$ is an alternative position operator, satisfy the results of Proposition 4.2.

Lemma A.2

Let ${\widehat{X}}$ be a symmetric operator satisfying $\Vert {\widehat{X}} - X \Vert \le C$. Then ${\widehat{X}}$ is self-adjoint ${\mathcal {D}}(X) \rightarrow L^2(\mathbb {R}^2)$ and $P {\widehat{X}} P$ is self-adjoint $\{ {\mathcal {D}}(X) \cap {{\,\mathrm{range}\,}}(P) \} \cup {{\,\mathrm{range}\,}}(P)^\perp \rightarrow L^2(\mathbb {R}^2)$. Suppose further that

• 1. $\Vert {\widehat{X}}_{\gamma } - {\widehat{X}} \Vert \le C' \gamma$

• 2. $P {\widehat{X}} P$ has uniform spectral gaps.

Then, if $\{ P_j \}_{j \in {\mathcal {J}}}$ are the band projectors of $P {\widehat{X}} P$, then $\{ P_j \}_{j \in {\mathcal {J}}}$ satisfies the results of Proposition 4.2.

Following the proof in the main text, an immediate corollary of Lemma A.2 is the following.

Corollary A.3

If it is possible to construct a symmetric operator ${\widehat{X}}$ satisfying the assumptions of Lemma A.2, then ${{\,\mathrm{range}\,}}{(P)}$ admits a basis of exponentially localized generalized Wannier functions.

Proof of Lemma A.2

The first claim of the lemma follows easily from the Kato-Rellich theorem since ${\widehat{X}} - X$ is a bounded perturbation of X. Assuming (2), we can define the corresponding band projectors $P_j$ via the Riesz formula

$$\begin{aligned} P_j := \frac{1}{2 \pi i} \int _{{\mathcal {C}}_j} (\lambda - P {\widehat{X}} P)^{-1} P \text {d}{\lambda }. \end{aligned}$$

Therefore, we can write $P_{j,\gamma }$ as follows

$$\begin{aligned} P_{j,\gamma } := \frac{1}{2 \pi i} \int _{{\mathcal {C}}_j} (\lambda - P_{\gamma } {\widehat{X}}_{\gamma } P_{\gamma })^{-1} P_{\gamma } \text {d}{\lambda }. \end{aligned}$$

Our goal is now to establish Proposition A.1 for these band projectors. Similar to before, to get bounds which are independent of the choice of $j \in {\mathcal {J}}$, appealing to the shifting lemma (Lemma A.1), to prove an analog of Proposition A.1, it suffices to show that for all $j \in {\mathcal {J}}$

$$\begin{aligned} \sup _{\lambda \in {\mathcal {C}}_j} \sup _{\eta _j \in {\mathcal {C}}_j} \Vert (\lambda _{\eta _j} - P_{\gamma } {\widehat{X}}_{\eta _j,\gamma } P_{\gamma })^{-1} \Vert \le C' \end{aligned}$$

(A.10)

where $\lambda _{\eta _j} = \lambda - \eta _j$ and ${\widehat{X}}_{\eta _j,\gamma } = {\widehat{X}}_{\gamma } - \eta _j$.

Equation (A.10) can be proved by the following series of implications:

$$\begin{aligned} \begin{aligned} P {\widehat{X}}&P \text { has USG} \Longrightarrow \Vert (\lambda _{\eta _j} - P {\widehat{X}}_{\eta _j} P )^{-1} \Vert< \infty \\&\Longrightarrow \Vert (\lambda _{\eta _j} - P {\widehat{X}}_{\eta _j} P_{\gamma } )^{-1} \Vert< \infty \\&\Longrightarrow \Vert (\lambda _{\eta _j} - P_{\gamma } {\widehat{X}}_{\eta _j} P_{\gamma } )^{-1} \Vert< \infty \\&\Longrightarrow \Vert (\lambda _{\eta _j} - P_{\gamma } {\widehat{X}}_{\eta _j,\gamma } P_{\gamma } )^{-1} \Vert < \infty . \end{aligned} \end{aligned}$$

(A.11)

Using $\Vert {\widehat{X}} - X \Vert \le C$, one can check that the proof in Section A.1 proves all of the implications except for the last one.

For the last of these implications, we observe that ${\widehat{X}}_{\eta _j, \gamma } - {\widehat{X}}_{\eta _j} = {\widehat{X}}_{\gamma } - {\widehat{X}}$ and recall that by assumption $\Vert {\widehat{X}}_{\gamma } - {\widehat{X}} \Vert \le C' \gamma$. Now observe that

$$\begin{aligned} (\lambda _{\eta _j}&- P_{\gamma } {\widehat{X}}_{\eta _j,\gamma } P_{\gamma } )^{-1} \\&= (\lambda _{\eta _j} - P_{\gamma } {\widehat{X}}_{\eta _j} P_{\gamma } )^{-1} \Big ( I - P_{\gamma } ({\widehat{X}}_{\gamma } - {\widehat{X}}) P_{\gamma } (\lambda _{\eta _j} - P_{\gamma } {\widehat{X}}_{\eta _j,\gamma } P_{\gamma } )^{-1} \Big )^{-1}. \end{aligned}$$

Since $(\lambda _{\eta _j} - P_{\gamma } {\widehat{X}}_{\eta _j} P_{\gamma } )^{-1}$ is bounded, we can choose $\gamma$ sufficiently small so that

$$\begin{aligned} \Vert P_{\gamma } ({\widehat{X}}_{\gamma } - {\widehat{X}}) P_{\gamma } (\lambda _{\eta _j} - P_{\gamma } {\widehat{X}}_{\eta _j,\gamma } P_{\gamma } )^{-1} \Vert \le \frac{1}{2}. \end{aligned}$$

which implies that $(\lambda _{\eta _j} - P_{\gamma } {\widehat{X}}_{\eta _j,\gamma } P_{\gamma } )^{-1}$ completing the proof of Proposition A.1 for the band projectors of $P {\widehat{X}} P$. Hence following the arguments in Sections A.2 and A.3, we can conclude the band projectors for $P {\widehat{X}} P$ satisfy the results of Proposition 4.2 proving the lemma. $\square$

B Proof of the Shifting Lemma (Lemma A.1)

The basic steps to prove Lemma A.1 are the following:

$$\begin{aligned}&(\lambda _{\eta } - P_{\gamma } X_{\eta } P_{\gamma })^{-1} \nonumber \\&\quad = (\lambda - \eta - P_{\gamma } (X - \eta ) P_{\gamma })^{-1} \nonumber \\&\quad = (\lambda - \eta - P_{\gamma } X P_{\gamma } + \eta P_{\gamma })^{-1} \nonumber \\&\quad = (\lambda - P_{\gamma } X P_{\gamma } - \eta Q_{\gamma })^{-1}. \end{aligned}$$

(B.1)

Since $P_{\gamma } + Q_{\gamma } = I$, because of this calculation we know that

$$\begin{aligned} \Vert (\lambda _{\eta } - P_{\gamma } X_{\eta } P_{\gamma })^{-1}\Vert&= \Vert (\lambda - P_{\gamma } X P_{\gamma } - \eta Q_{\gamma })^{-1}\Vert \\&\le \Vert (\lambda - P_{\gamma } X P_{\gamma } - \eta Q_{\gamma })^{-1} P_{\gamma }\Vert + \Vert (\lambda - P_{\gamma } X P_{\gamma } - \eta Q_{\gamma })^{-1} Q_{\gamma }\Vert . \end{aligned}$$

Since $P_{\gamma } Q_{\gamma } = Q_{\gamma } P_{\gamma } = 0$, we should expect that shifting by $\eta Q_{\gamma }$ should not change what happens on ${{\,\mathrm{range}\,}}{(P_{\gamma })}$. Similarly, the action of $P_{\gamma } X P_{\gamma }$ should not change what happens on ${{\,\mathrm{range}\,}}{(Q_{\gamma })}$. This observation leads us to expect that:

$$\begin{aligned}&(\lambda - P_{\gamma } X P_{\gamma } - \eta Q_{\gamma })^{-1} P_{\gamma } = (\lambda - P_{\gamma } X P_{\gamma })^{-1} P_{\gamma } \end{aligned}$$

(B.2)

$$\begin{aligned}&(\lambda - P_{\gamma } X P_{\gamma } - \eta Q_{\gamma })^{-1} Q_{\gamma } = (\lambda - \eta Q_{\gamma })^{-1} Q_{\gamma }. \end{aligned}$$

(B.3)

By similar reasoning:

$$\begin{aligned} (\lambda - \eta Q_{\gamma })^{-1} Q_{\gamma } = (\lambda - \eta + \eta P_{\gamma })^{-1} Q_{\gamma } = (\lambda - \eta )^{-1} Q_{\gamma }. \end{aligned}$$

(B.4)

Assuming Equations (B.2), (B.3), (B.4) are true, we conclude that:

$$\begin{aligned} \Vert (\lambda _{\eta } - P_{\gamma } X_{\eta } P_{\gamma })^{-1}\Vert \le \Vert (\lambda - P_{\gamma } X P_{\gamma })^{-1}\Vert \Vert P_{\gamma }\Vert + \vert {}\lambda - \eta \vert {}^{-1} \Vert Q_{\gamma }\Vert . \end{aligned}$$

(B.5)

Since P admits an exponentially localized kernel (Definition 4.1) we know that $\Vert P_{\gamma }\Vert$ and $\Vert Q_{\gamma }\Vert$ are bounded. Because of the uniform spectral gaps assumption on PXP, since we have chosen $\eta \in \sigma _j$ and $\lambda \in {\mathcal {C}}_j$ we also know that $\vert {} \lambda - \eta \vert {}^{-1}$ is bounded by a constant independent of j and $\eta$. Therefore, Equation (B.5) shows that

$$\begin{aligned} \Vert (\lambda - P_{\gamma } X P_{\gamma })^{-1}\Vert< \infty \Longrightarrow \Vert (\lambda _{\eta } - P_{\gamma } X_{\eta } P_{\gamma })^{-1}\Vert < \infty . \end{aligned}$$

We can prove the reverse implication by instead starting with the calculation

$$\begin{aligned} (\lambda - P_{\gamma } X P_{\gamma })^{-1}&= (\lambda - \eta + \eta - P_{\gamma } (X - \eta + \eta ) P_{\gamma })^{-1} \\&= (\lambda _{\eta } - P_{\gamma } X_{\eta } P_{\gamma } + \eta Q_{\gamma })^{-1}, \end{aligned}$$

and proceeding along similar steps.

What remains to finish the proof of Lemma A.1 is to prove that Equations (B.2), (B.3), (B.4) are all true. For this, we have the following technical lemma:

Lemma B.1

Let ${\tilde{P}}, {\tilde{Q}}$ be any pair of bounded operators such that ${\tilde{P}} {\tilde{Q}} = {\tilde{Q}} {\tilde{P}} = 0$. Next, let A, B be possibly unbounded operators densely defined on a common domain ${\mathcal {D}}$. Suppose further that $\Vert [{\tilde{P}},A]\Vert$ both $\Vert [{\tilde{Q}},B]\Vert$ are bounded.

If ${\tilde{\lambda }} \in \mathbb {C}$ is any scalar such that $\Vert ({\tilde{\lambda }} + {\tilde{P}}A{\tilde{P}})^{-1}\Vert$ is bounded then $\Vert ({\tilde{\lambda }} + {\tilde{P}}A{\tilde{P}} + {\tilde{Q}}B{\tilde{Q}})^{-1} {\tilde{P}}\Vert$ is also bounded and

$$\begin{aligned} ({\tilde{\lambda }} + {\tilde{P}}A{\tilde{P}})^{-1} {\tilde{P}} = ({\tilde{\lambda }} + {\tilde{P}}A{\tilde{P}} + {\tilde{Q}}B{\tilde{Q}})^{-1} {\tilde{P}}. \end{aligned}$$

Note that applying Lemma B.1 three times proves that Equations (B.2), (B.3), (B.4) are all true.

The assumption that $\Vert [{\tilde{P}}, A]\Vert$ and $\Vert [{\tilde{Q}}, B]\Vert$ are bounded is purely a technical assumption which ensures that ${\tilde{P}} A {\tilde{P}}$ and ${\tilde{Q}} B {\tilde{Q}}$ are well-defined operators on ${\mathcal {D}}$. To see why, observe that

$$\begin{aligned} {\tilde{P}} A {\tilde{P}} = {\tilde{P}} [A, {\tilde{P}}] + {\tilde{P}} {\tilde{P}} A \text { and } {\tilde{Q}} B {\tilde{Q}} = {\tilde{Q}} [B, {\tilde{Q}}] + {\tilde{Q}} {\tilde{Q}} B. \end{aligned}$$

For our purposes, the only unbounded operator we will need to be careful with is the operator X. Since P satisfies admits an exponentially localized kernel (Definition 4.1) we know that $\Vert [P_{\gamma }, X]\Vert = \Vert [Q_{\gamma }, X]\Vert < \infty$, therefore we may apply Lemma B.1 without worry.

Proof of Lemma B.1

First, note that ${\tilde{\lambda }} + {\tilde{P}} A {\tilde{P}} + {\tilde{Q}} B {\tilde{Q}}$ is injective on ${{\,\mathrm{range}\,}}({\tilde{P}})$ since for arbitrary non-zero $v \in {{\,\mathrm{range}\,}}({\tilde{P}}) \cap {\mathcal {D}}$,

$$\begin{aligned} \begin{aligned}&\left\| \left( {\tilde{\lambda }} + {\tilde{P}} A {\tilde{P}} + {\tilde{Q}} B {\tilde{Q}} \right) v \right\| = \left\| \left( {\tilde{\lambda }} + {\tilde{P}} A {\tilde{P}} + {\tilde{Q}} B {\tilde{Q}} \right) {\tilde{P}} v \right\| \\&\quad = \left\| \left( {\tilde{\lambda }} + {\tilde{P}} A {\tilde{P}} \right) v \right\| \ge \Vert ({\tilde{\lambda }} + PAP)^{-1} \Vert ^{-1} \Vert v \Vert . \end{aligned} \end{aligned}$$

Now observe that since ${\tilde{P}} {\tilde{Q}} = {\tilde{Q}} {\tilde{P}} = 0$

$$\begin{aligned} {[} ({\tilde{\lambda }} + {\tilde{P}}A{\tilde{P}} + {\tilde{Q}}B{\tilde{Q}}), ({\tilde{\lambda }} + {\tilde{P}}A{\tilde{P}})] = 0. \end{aligned}$$

Since $({\tilde{\lambda }} + {\tilde{P}}A{\tilde{P}})^{-1}$ is well defined, this implies that

$$\begin{aligned} {[} ({\tilde{\lambda }} + {\tilde{P}}A{\tilde{P}} + {\tilde{Q}}B{\tilde{Q}}), ({\tilde{\lambda }} + {\tilde{P}}A{\tilde{P}})^{-1} ] = 0. \end{aligned}$$

Since ${\tilde{Q}}{\tilde{P}} = 0$ we also have that

$$\begin{aligned}&({\tilde{\lambda }} + {\tilde{P}}A{\tilde{P}} + {\tilde{Q}}B{\tilde{Q}}) {\tilde{P}} = ({\tilde{\lambda }} + {\tilde{P}}A{\tilde{P}}) {\tilde{P}} \\&\quad ~~~ \Longleftrightarrow ({\tilde{\lambda }} + {\tilde{P}}A{\tilde{P}})^{-1} ({\tilde{\lambda }} + {\tilde{P}}A{\tilde{P}} + {\tilde{Q}}B{\tilde{Q}}) {\tilde{P}} = {\tilde{P}} \\&\quad ~~~ \Longleftrightarrow ({\tilde{\lambda }} + {\tilde{P}}A{\tilde{P}} + {\tilde{Q}}B{\tilde{Q}}) ({\tilde{\lambda }} + {\tilde{P}}A{\tilde{P}})^{-1} {\tilde{P}} = {\tilde{P}}. \end{aligned}$$

The final equality implies that ${{\,\mathrm{range}\,}}{({\tilde{P}})} \subseteq {{\,\mathrm{range}\,}}{({\tilde{\lambda }} + {\tilde{P}}A{\tilde{P}} + {\tilde{Q}}B{\tilde{Q}})}$. Since $({\tilde{\lambda }} - {\tilde{P}}A{\tilde{P}})^{-1}$ is bounded we can conclude that $({\tilde{\lambda }} + {\tilde{P}}A{\tilde{P}} + {\tilde{Q}}B{\tilde{Q}})$ is invertible on ${{\,\mathrm{range}\,}}{({\tilde{P}})}$ and moreover we have

$$\begin{aligned} ({\tilde{\lambda }} + {\tilde{P}}A{\tilde{P}})^{-1} {\tilde{P}} = ({\tilde{\lambda }} + {\tilde{P}}A{\tilde{P}} + {\tilde{Q}}B{\tilde{Q}})^{-1} {\tilde{P}}. \end{aligned}$$

$\square$

C Proof that the generalized Wannier functions of Theorem 1.1 are Wannier functions when A and V are periodic (Theorem 1.2)

We start with some notation. Let $\Lambda$ denote a two-dimensional lattice generated by non-parallel lattice vectors $\varvec{v}_1, \varvec{v}_2 \in \mathbb {R}^2$, that is

$$\begin{aligned} \Lambda = \left\{ m_1 \varvec{v}_1 + m_2 \varvec{v}_2 : ( m_1, m_2 ) \in \mathbb {Z}^2 \right\} . \end{aligned}$$

(C.1)

We assume that the functions A and V are periodic with respect to $\Lambda$ in the sense that for all $\varvec{x} \in \mathbb {R}^2$

$$\begin{aligned} A(\varvec{x} + \varvec{v}) = A(\varvec{x}), \quad V(\varvec{x} + \varvec{v}) = V(\varvec{x}) \text { for all }\varvec{v} \in \Lambda . \end{aligned}$$

(C.2)

Define $L_j := \vert {}\varvec{v}_j\vert {}$, $j = 1,2$. We work with co-ordinates defined with respect to the lattice vectors $\varvec{v}_1$ and $\varvec{v}_2$ so that for $\varvec{x} \in \mathbb {R}^2$,

$$\begin{aligned} \varvec{x} = x_1 \varvec{v}_1 + x_2 \varvec{v}_2 \end{aligned}$$

(C.3)

for $\varvec{x} = (x_1, x_2) \in \mathbb {R}^2$. We require the following assumption on the position operators X and Y.

Assumption C.1

We assume the position operators X and Y are defined with respect to the co-ordinates (C.3) by

$$\begin{aligned} X f(\varvec{x}) = L_1 x_1 f(\varvec{x}), \quad Y f(\varvec{x}) = L_2 x_2 f(\varvec{x}). \end{aligned}$$

(C.4)

Let $T_{\varvec{v}_1}$ and $T_{\varvec{v}_2}$ denote translation operators associated to the Bravais lattice vectors $\varvec{v}_1, \varvec{v}_2$, so that

$$\begin{aligned} T_{\varvec{v}_j} f(\varvec{r}) = f(\varvec{r} - \varvec{v}_j), \quad j \in \{1,2\}. \end{aligned}$$

(C.5)

First, note that by the Riesz projection formula, $[T_{\varvec{v}_j},P] = 0$, $j = 1,2$. Since $[T_{\varvec{v}_1},X] = - L_1 T_{\varvec{v}_1}$ and $[T_{\varvec{v}_2},X] = 0$, we have that

$$\begin{aligned} {[}T_{\varvec{v}_1},PXP] = - L_1 P T_{\varvec{v}_1}, \quad [T_{\varvec{v}_2},PXP] = 0. \end{aligned}$$

(C.6)

The first of these identities implies that the spectrum of PXP restricted to ${{\,\mathrm{range}\,}}(P)$ is the union of shifted copies of the spectrum of PXP restricted to the interval $[0,L_1)$, that is

$$\begin{aligned} \sigma ( PXP|_{{{\,\mathrm{range}\,}}(P)} ) = \bigcup _{j \in \mathbb {Z}} \{ \sigma _{[0,L_1)} + L_1 j \}, \end{aligned}$$

(C.7)

where $\sigma _{[0,L_1)} := \sigma ( PXP|_{{{\,\mathrm{range}\,}}(P)} ) \cap [0,L_1)$. The $L^2$ spectrum of PXP is then $\{ 0 \} \cup \sigma ( PXP|_{{{\,\mathrm{range}\,}}(P)} )$.

Using the uniform spectral gaps assumption, we can assume that, perhaps after a constant shift of X, the interval $(-\epsilon ,\epsilon ) \notin \sigma ( PXP |_{{{\,\mathrm{range}\,}}(P)} )$. Because of the symmetry (C.7) of the spectrum, we have that $(- \epsilon ,\epsilon ) + {\mathbb {Z}} L_1 \notin \sigma ( PXP|_{{{\,\mathrm{range}\,}}(P)} )$, and hence the components $\{ \sigma _{[0,L_1)} + L_1 j \}$ of $\sigma ( PXP |_{{{\,\mathrm{range}\,}}(P)} )$ are separated by gaps. We now assume the following.

Assumption C.2

We define the band projectors $P_j$ as in Definition 4.2 with $\sigma _j := \{ \sigma _{[0,L_1)} + L_1 j \}$ for $j \in {\mathbb {Z}}$.

Note that other choices of $\sigma _j$ which are compatible with the uniform spectral gaps assumption are possible if $\sigma _{[0,L_1)}$ has separated components, but for simplicity we do not consider this here. The key consequence of Assumption C.2 is that

$$\begin{aligned} T_{\varvec{v}_1} P_j = P_{j + 1} T_{\varvec{v}_1}. \end{aligned}$$

(C.8)

Using the Riesz formula again, we have that $[T_{\varvec{v}_2},P_j] = 0$. Using the facts that $[T_{\varvec{v}_1},Y] = 0$ and $[T_{\varvec{v}_2},Y] = - L_2 T_{\varvec{v}_2}$ we have that

$$\begin{aligned} \sigma (P_j Y P_j |_{{{\,\mathrm{range}\,}}( P_j )}) = \bigcup _{m \in \mathbb {Z}} \{ \sigma _{[0,L_2)} + L_2 m \} \end{aligned}$$

(C.9)

where $\sigma _{[0,L_2)} := \sigma (P_j Y P_j |_{{{\,\mathrm{range}\,}}( P_j )}) \cap [0,L_2)$. We can therefore index the eigenfunctions of $P_j Y P_j$ by $m \in \mathbb {Z}$ for every $j \in \mathbb {Z}$.

Now let $\{ \psi _{j,m} \}_{(j,m) \in \mathbb {Z} \times \mathbb {Z}}$ denote the set of eigenfunctions of the operators $P_j Y P_j$. We claim that this set is closed under $T_{\varvec{v}_1}$ and $T_{\varvec{v}_2}$. First, since $T_{\varvec{v}_2}$ commutes with $P_j$ it is clear that if $\psi _{j,m}$ is an eigenfunction of $P_j Y P_j$ with eigenvalue $\mu _{j,m}$, then

$$\begin{aligned} P_j Y P_j T_{\varvec{v}_2} \psi _{j,m} = T_{\varvec{v}_2} P_j (Y + L_2) P_j \psi _{j,m} = ( \mu _{j,m} + L_2 ) T_{\varvec{v}_2} \psi _{j,m} \end{aligned}$$

(C.10)

and hence $T_{\varvec{v}_2} \psi _{j,m}$ is another eigenfunction of $P_j Y P_j$ with eigenvalue $\mu _{j,m} + L_2$, so the set is closed under $T_{\varvec{v}_2}$. To see the set is closed under $T_{\varvec{v}_1}$, note that applying $T_{\varvec{v}_1}$ to both sides of the eigenequation $P_j ( Y - \mu ) P_j \psi _{j,m} = 0$ yields $P_{j'} ( Y - \mu ) P_{j'} T_{\varvec{v}_1} \psi _{j,m} = 0$ for some $j'$ and hence $T_{\varvec{v}_1} \psi _{j,m} = \psi _{j',m}$ for some $j' \in \mathbb {Z}$.

We now consider the centers $\{ (a_{j,m},b_{j,m}) \}_{(j,m) \in \mathbb {Z} \times \mathbb {Z}}$. Working in the co-ordinates (C.3), closure under translation in the lattice $\Lambda$ is equivalent to closure of the set of centers under component-wise integer addition. Now recall that for each j, in the statement of Theorem 1.1, the $a_{j,m}$ can be chosen as any point in $\sigma _j$, where $\sigma _j$ is the spectrum associated to the spectral projection $P_j$, for all m. So, since $\sigma _{j+1} = \sigma _j + L_1$ as we have already noted, in the co-ordinates (C.3) we can choose the $a_{j,m}$ such that $a_{j+1,m} = a_{j,m} + 1$, and hence the set of centers is closed under integer addition in the first component. To see the same is true of the second component, note that (C.10) shows that the eigenvalues of $P_j Y P_j$ are closed under addition of $L_2$ for each j. Since the centers $b_{j,m}$ are defined as the associated eigenvalues of $P_j Y P_j$ of the $\psi _{j,m}$ we see that the set of centers is also closed under integer addition in their second component (again working in the co-ordinates (C.3)).