# Topological index formula in physical waves: spectral flow, Chern index and topological contacts

Léon Monnier, Frédéric Faure

► To cite this version:

Léon Monnier, Frédéric Faure. Topological index formula in physical waves: spectral flow, Chern index and topological contacts. New Journal of Physics, 2025, 10.48550/arXiv.2510.25317. [hal-05338177]

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# Topological index formula in physical waves: spectral flow, Chern index and topological contacts

Léon Monnier

Univ. Grenoble Alpes, CNRS, Institut Fourier, F-38000 Grenoble, France
leonmonnier1@gmail.com

Frédéric Faure

Univ. Grenoble Alpes, CNRS, Institut Fourier, F-38000 Grenoble, France
frederic.faure@univ-grenoble-alpes.fr

2025 july 11

## Abstract

We study a family of pseudodifferential operators (quantum Hamiltonians) on $L^2(\mathbb{R}^n; \mathbb{C}^d)$ whose spectrum exhibits two energy bands exchanging a finite number of eigenvalues. We show that this number coincides with the Chern index of a vector bundle associated to the principal symbol (the classical Hamiltonian). This result provides a simple yet illustrative instance of the Atiyah-Singer index formula, with applications in areas such as molecular physics, plasma physics or geophysics. We also discuss the phenomenon of topological contact without exchange between energy bands – a feature that cannot be detected by the Chern index or K-theory, but rather reflects subtle torsion effects in the homotopy groups of spheres.

1

<footnote n="1">¹2010 Mathematics Subject Classification:
35Q86 PDEs in connection with geophysics,
81V55 Molecular physics,
55R50 Stable classes of vector space bundles, K-theory,
47A53 (Semi-) Fredholm operators; index theories,
19K56 Index theory,
81Q20 Semiclassical techniques, including WKB and Maslov methods
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics</footnote>
<figure><img src="image_2.png" /><figcaption>Figure 2.1: Energy levels (in $cm^{-1}$) of the $CD_4$ molecule (carbon with four deuterium atoms) as a function of the total angular momentum $J \in \mathbb{N}$ (a conserved quantity corresponding to rotational energy). The fine structure of the spectrum reflects the slow rotational motion, while the broad structure corresponds to the faster vibrational dynamics. The spectrum exhibits clusters of energy levels, with some levels crossing or connecting different clusters [4, 5, 6, 8, 13].</figcaption></figure>

$(\hat{x}, \hat{p})$ represent the quantization of local coordinates on the sphere $S^2$, which describes the rotational motion of the molecule, while the $\mathbb{C}^2$ space encodes the quantum dynamics of the fast vibrations of the molecule, restricted to an effective two-level system.

## 2.1 Spectral flow and spectral index

The next proposition describes the spectrum of the operator defined in Equation (2.6) and exhibits a single eigenvalue that moves upward, see in Figure 2.2. This behavior defines a spectral index $N_E = +1$. Throughout this paper, this elementary model will be referred to as model E.

**Proposition 2.3.** «Spectrum of $\hat{E}_\mu$». For each parameter $\mu \in \mathbb{R}$, the operator $\hat{E}_\mu$, (2.6), has discrete spectrum in $L^2(\mathbb{R}_x) \otimes \mathbb{C}^2$ given by

$$ \hat{E}_{\mu} \phi_{n}^{\pm} = \omega_{n}^{\pm} \phi_{n}^{\pm}, \quad n \ge 1, \qquad (2.3) $$

with for any $n \in \mathbb{N} \setminus \{0\}$, eigenvalues and eigenvectors are

$$ \omega_n^{\pm} = \pm \sqrt{\mu^2 + 2n} \qquad (2.4) $$

$$ \phi_n^{\pm} = \begin{pmatrix} \frac{\sqrt{2n}}{\mu + \omega_n^{\pm}} \varphi_{n-1} \\ \varphi_n \end{pmatrix} $$

and additionally for $n = 0$,

$$ \hat{E}_{\mu} \phi_0 = \omega_0 \phi_0, $$

with

$$ \omega_0 = \mu, \quad \phi_0 = \begin{pmatrix} 0 \\ \varphi_0 \end{pmatrix}. $$

($\varphi_n$)$_{n \ge 0}$ are the Hermite functions in $L^2(\mathbb{R}_x)$ of the harmonic oscillator defined by

$$ \varphi_0(x) = \frac{1}{\pi^{1/4}} e^{-\frac{1}{2}x^2}, \quad \varphi_{n+1} = \frac{1}{\sqrt{n+1}} a^{\dagger} \varphi_n, \quad a \varphi_n = \sqrt{n} \varphi_{n-1}. \qquad (2.5) $$

*Proof.* By direct computation, see [3, Arxiv version] for details. □

See figure 2.2. We remark that for $|\mu| > 1$ the spectrum of $\hat{E}_\mu$ has no eigenvalues in the interval $[-1, 1]$, this is called a spectral gap. When $\mu$ goes from $-1$ to $1$, we observe that there is $N_E = 1$ eigenvalue passing through this gap. Below, in a more general model, we will see that $N_E \in \mathbb{Z}$ is a
<figure><img src="image_3.png" /><figcaption>Figure 2.2: Spectrum of (2.3).</figcaption></figure>

**spectral index.** To study the origin of this phenomenon, we remark that the operator $\hat{E}_{\mu}$ can be seen as the quantization<fnref n="2" /> of the following matrix valued function $E_{\mu}: \mathbb{R}^2 \to \text{Herm}(\mathbb{C}^2)$ on a phase space.

**Definition 2.4.** The matrix valued function on a phase space.

$$E_{\mu}: \mathbb{R}^2 \to E_{\mu}(x, p) := \begin{pmatrix} -\mu & x + ip \\ x - ip & +\mu \end{pmatrix} \in \text{Herm}(\mathbb{C}^2). \qquad (2.6)$$

is called the **symbol** of $\hat{E}_{\mu}$ (also called classical Hamiltonian function in physics).

## 2.2 Chern index

In the next proposition, for each parameter $(\mu, x, p) \in \mathbb{R}^3 \setminus \{0\}$, we will consider the lower eigenvector $\psi(\mu, x, p) \in \mathbb{C}^2$ of the $2 \times 2$ matrix $E_{\mu}(x, p)$ in (2.6). Recall that this eigenvector is defined only up to a scalar, hence only the (complex) one dimensional eigenspace $F_{-}(\mu, x, p) = \{\lambda\psi(\mu, x, p), \lambda \in \mathbb{C}\} \subset \mathbb{C}^2$ is well defined. We get a family of one dimensional vector spaces $F_{-} := \{F_{-}(\mu, x, p) \subset \mathbb{C}^2, (\mu, x, p) \in \mathbb{R}^3 \setminus \{0\}\}$. This family $F_{-}$ is called a **complex vector bundle of rank 1 over the space of parameters $\mathbb{R}^3 \setminus \{0\}$**. We are interested by the “topology” (or isomorphism class)<fnref n="3" /> of this vector bundle $F_{-}$.

This topology is characterized by an integer $\mathcal{C}(F_{-}) \in \mathbb{Z}$ called the **Chern index**. There are many ways to define and compute it. One of the simplest is the following.

**Definition 2.5.** [9] We consider the unit sphere $S^2 = \{(\mu, x, p) \in \mathbb{R}^3, \|(\mu, x, p)\| = 1\}$ that can be decomposed as the union of two hemispheres joined at the equator $S^1$. On each hemisphere separately $H_1$ (respect. $H_2$), one can choose continuously an eigenvector $s_1$ (respect. $s_2$) in $F_{-}$ (but not globally on $S^2$). Then one observe that on each point $\theta \in S^1$ of the equator, $s_1, s_2$ are related by a phase $s_2 = e^{i\varphi(\theta)}s_1$, giving a map $\varphi: \theta \in S_1 \mapsto \varphi(\theta) \in S^1$, called the **clutching function**. The **Chern index** is the **winding number** (or **degree**, see (3.8)) of this map:

$$\mathcal{C}(F_{-}) := \deg(\varphi) \in \mathbb{Z}.$$

The proof written below gives more details. See also figures 2.3 and 2.4.

<footnote n="2">²Here quantization means that the position/momentum variables $x, p$ are replaced by the operators $\hat{x}, \hat{p}$. We will give a more precise definition later.</footnote>

<footnote n="3">³meaning this family up to equivalence under continuous (or homotopic) bundle map.</footnote>
<figure><img src="image_4.png" /><figcaption>Figure 2.3: Chern index $C$ of a rank one bundle over $S^2$, computed by the winding number of the clutching function.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 2.4: Eigenvalues from (2.7). We have $\omega^{-}(\mu, x, p) \le -|\mu|$, $\omega^{+}(\mu, x, p) \ge |\mu|$. The red domain represents the possible values of $\omega^{-}(\mu, x, p)$ with $\mu$ fixed and $(x, p) \in \mathbb{R}^2$. Similarly, the blue domain represents $\omega^{-}(\mu, x, p)$. The degeneracy is at $(\mu, x, p) = (0, 0, 0)$.</figcaption></figure>
**Proposition 2.6.** «Topological aspects of the symbol $E_\mu$». The eigenvalues of the matrix $E_\mu(x,p) \in \text{Herm}(\mathbb{C}^2)$, Eq. (2.6), are

$$ \omega_{\pm}(\mu, x, p) = \pm \sqrt{\mu^2 + x^2 + p^2} \qquad (2.7) $$

There is therefore a degeneracy $\omega_+ = \omega_-$ (only) for $(\mu, x, p) = (0, 0, 0)$. To each point on the unit sphere $(\mu, x, p) \in S^2 = \{(\mu, x, p) \in \mathbb{R}^3, \|(\mu, x, p)\| = 1\}$, we can associate the eigenspace $F_-(\mu, x, p) \subset \mathbb{C}^2$ corresponding to the lower eigenvalue $\omega_-(\mu, x, p)$. The Chern index of this rank 1 vector bundle $F_-$ is:

$$ \mathcal{C}_E := \mathcal{C}(F_-) = +1. $$

*Proof.* See [3, Arxiv version] for other alternative proofs. We have

$$ q(\omega) := \det (\omega \text{Id} - E_{\mu}(x, p)) = \det \begin{pmatrix} \omega + \mu & -(x + ip) \\ -(x - ip) & \omega - \mu \end{pmatrix} = \omega^2 - (\mu^2 + x^2 + p^2) $$

hence $q(\omega) = 0$ gives eigenvalues $\omega_{\pm} = \pm r$ with $r := \sqrt{\mu^2 + x^2 + p^2}$, i.e. Eq. (2.7). The eigenvectors of $E_\mu$ are respectively$^4$

$$ U_+ = \begin{pmatrix} -\mu + r \\ x - ip \end{pmatrix}, \quad U_- = \begin{pmatrix} -\mu - r \\ x - ip \end{pmatrix}, \qquad (2.8) $$

i.e. $E_\mu(x,p)U_\pm = \omega_\pm U_\pm$. Write $F_\pm(\mu,x,p) := \text{Vect}(U_\pm) \subset \mathbb{C}^2$ the associated eigenspaces. The spectral projector $\pi_-$ on $F_-$ is

$$ \pi_- = \frac{1}{\|U_-\|^2} U_- \langle U_-|, \rangle : \mathbb{C}^2 \to F_- (\mu, x, p). \qquad (2.9) $$

Consider $S^2 = \{(\mu, x, p) \in \mathbb{R}^3, r = \|(\mu, x, p)\| = 1\}$ the unit sphere in the parameter space and the northern and southern hemispheres $H_1 := \{(\mu, x, p) \in S^2, \mu \ge 0\}$, $H_2 := \{(\mu, x, p) \in S^2, \mu \le 0\}$. The projection of the fixed vector $\begin{pmatrix} 1 \\ 0 \end{pmatrix} \in \mathbb{C}^2$ on $F_-$ gives the global section:

$$ s_1(\mu, x, p) := \pi_- \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \frac{(-\mu - 1)}{((\mu + 1)^2 + x^2 + p^2)} \begin{pmatrix} -\mu - 1 \\ x - ip \end{pmatrix}. \qquad (2.10) $$

We have $\|s_1\|^2 = \frac{(\mu+1)^2}{((\mu+1)^2+x^2+p^2)} = \frac{1+\mu}{2}$ hence $\|s_1\|^2 \ne 0$ on $H_1$. Hence $s_1$ is a trivialization of $F_- \to H_1$ (i.e. a non zero section). We consider also the following trivialization of $F_- \to H_2$:

$$ s_2(\mu, x, p) := \pi_- \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \frac{(x + ip)}{((\mu + 1)^2 + x^2 + p^2)} \begin{pmatrix} -\mu - 1 \\ x - ip \end{pmatrix}, \qquad (2.11) $$

We have $\|s_2\|^2 = \frac{(x^2+p^2)}{((\mu+1)^2+x^2+p^2)} = \frac{1-\mu}{2}$ hence $\|s_2\|^2 \ne 0$ on $H_2$. The clutching function on the equator $S^1 = \{\mu = 0, x + ip = e^{i\theta}, \theta \in [0, 2\pi[\}$ is defined by

$$ \begin{align*} s_2(\theta) &= f_{21}(\theta) s_1(\theta) \\ \Leftrightarrow (x + i\xi) \begin{pmatrix} -1 \\ x - ip \end{pmatrix} &= -f_{21}(\theta) \begin{pmatrix} -1 \\ x - ip \end{pmatrix} \\ \Leftrightarrow f_{21}(\theta) &= -e^{i\theta}. \end{align*} $$

The degree (or winding number) of the function $f_{21}: \theta \in S^1 \to f_{21}(\theta) = -e^{i\theta} \in U(1) \equiv S^1$ is $C = \deg(f_{21}) = +1$. $\square$

<footnote>$^4$In xcas online, write: E:=[[-mu,x+i*xi],[x-i*xi,mu]]; eigenvals(E); eigenvects(E);</footnote>
## 2.3 Conclusion

In this very simple normal form model $E$, we have observed that there is $N_E = 1$ eigenvalue in the spectral flow of the quantum model $(\hat{E}_\mu)_{\mu \in \mathbb{R}}$ and a Chern index $C_E := C(F_-) = 1$ obtained from the eigenspaces of the symbol functions $(E_\mu)_{\mu \in \mathbb{R}}$. We also remark that the following equality holds true

$$C_E = N_E = 1. \qquad (2.12)$$

One may wonder whether this equality is merely a coincidence. We will explain that there is a very general theory called “Index theory” [1] which accounts for this relation called “index formula” and holds in a much more general setting (arbitrary dimensions and nonlinear symbols in $(x, p)$). For this, we need microlocal analysis. We present this more general model below.

# 3 General model

Here, we generalize the model introduced in Section 2. The general setting is defined in terms of a symbol (a matrix-valued function on phase space) under specific hypotheses that ensure both the existence of a spectral index $N$ for the quantized operators and a topological Chern index $C$ associated with a vector bundle constructed from the symbol. We then prove the equality $C = N$.

## 3.1 Definitions

### 3.1.1 Hypothesis on the symbol

We first define the model by its symbol: a matrix valued function on phase space depending on a parameter $\mu$. Below, Herm($\mathbb{C}^d$) denotes the vector space of hermitian matrices of dimension $d \times d$. There are (important but usual) conditions at infinity on the symbol that we will ignore in this article [7].

**Definition 3.1.** [3, ass. 2.1] Our model $H$ is defined from a continuous family of symbols depending on a parameter $\mu \in \mathbb{R}$:

$$H_{\mu} : (x, p) \in \mathbb{R}^{2n} \rightarrow H_{\mu}(x, p) \in \text{Herm}(\mathbb{C}^{d}) \qquad (3.1)$$

such that, if we denote the real and sorted eigenvalues of the matrix $H_\mu(x, p)$ by

$$\omega_1(\mu, x, p) \le \dots \le \omega_d(\mu, x, p), \qquad (3.2)$$

we suppose that there exists an index $r \in \{1, \dots, d-1\}$ and $C > 0$ such that for every $(\mu, x, p) \in \mathbb{R}^{1+2n}$ with $\|(\mu, x, p)\| \ge 1$, and $\mu \in (-2, 2)$, we have the **gap assumption**

$$\omega_r(\mu, x, p) < -C \text{ and } \omega_{r+1}(\mu, x, p) > +C. \qquad (3.3)$$

The hypothesis (3.3) is a spectral gap assumption, see figure 3.1. Clearly, the model $E_\mu$ (2.6) verifies this property with a gap constant $C = 1$.

### 3.1.2 Quantum operator

The following definition defines a family of operators $\hat{H}_{\mu, \epsilon} = \text{Op}_\epsilon(H_\mu)$ depending on a parameter $\epsilon > 0$ from the symbol $H_\mu$ given in (3.1). This is called quantization$^5$.

<footnote>$^5$In quantum mechanics, **quantization** is the procedure that gives a quantum Hamiltonian operator $\text{Op}_\epsilon(H)$ from a classical Hamiltonian function $H$. The parameter $\epsilon$ is sometimes written $\epsilon = h$ as the Planck constant, and the semi classical limit is $h \to 0$, where one can get some spectral properties of $\text{Op}_\epsilon(H)$ from the classical dynamics generated by $H$</footnote>
<figure><img src="image_6.png" /><figcaption>Figure 3.1: Illustration of the assumption (3.3). On figure (a), for parameters $(\mu, x, p) \in \mathbb{R} \times \mathbb{R}^n \times \mathbb{R}^n$ in the green domain, we assume that the spectrum of the hermitian matrix $H_\mu(x, p)$, has $r$ eigenvalues smaller than $-C$ and that the others are greater than $C > 0$. Equivalently, on figure (b), the spectrum $\omega$ of $H_\mu(x, p)$ for any $(x, p)$ is contained in the red domain.</figcaption></figure>

**Definition 3.2.** [15] The Weyl quantization of $H_\mu$ is the operator $\hat{H}_{\mu, \epsilon} = \text{Op}_\epsilon(H_\mu)$ acting in $L^2(\mathbb{R}^n; \mathbb{C}^d)$ defined on a vector valued functions $\psi \in L^2(\mathbb{R}^n; \mathbb{C}^d)$ by

$$ (\text{Op}_\epsilon(H_\mu)\psi)(x) = \frac{1}{(2\pi\epsilon)^n} \int_{\mathbb{R}^n \times \mathbb{R}^n} H_\mu\left(\frac{x+y}{2}, \xi\right) e^{i\xi\cdot(x-y)/\epsilon} \psi(y) dyd\xi $$

*Remark 3.3.* One can check that in our previous model, one has $\text{Op}_{\epsilon=1}(E_\mu) = \hat{E}_\mu$ for the parameter $\epsilon = 1$.

## 3.2 Spectral index

In the previous subsection, we described a family of symbols $(H_\mu)_{\mu \in \mathbb{R}}$ and how to quantize them getting a family of operators $(\hat{H}_{\epsilon, \mu})_{\epsilon > 0, \mu \in \mathbb{R}}$. Now, we state a theorem that defines a spectral index for the corresponding operators.

**Theorem 3.4.** [3, thm 2.2] A consequence of the gap assumption (3.3) is that for every $\alpha > 0$ there exists $\epsilon_0 > 0$ such that for every $0 < \epsilon < \epsilon_0$,

* for any $\mu$ such that $1 + \alpha < |\mu| < 2$, the operator $\hat{H}_{\mu, \epsilon}$ has no spectrum in the interval $]-C + \alpha, +C - \alpha[$.

* for any $\mu$ such that $|\mu| \le 1 + \alpha$, the operator $\hat{H}_{\mu, \epsilon}$ has discrete spectrum in the interval $]-C + \alpha, C - \alpha[$ that depends continuously on $\mu, \epsilon$.

Consequently one can define the spectral index of the model $(H_\mu)_\mu$ by

$$ \mathcal{N}_H := n_{\text{in}} - n_{\text{out}} \in \mathbb{Z} \qquad (3.4) $$

where the discrete spectrum has been labeled in the positive order and $n_{\text{in}}$, (respect. $n_{\text{out}}$) is the label of the first eigenvalue below the spectral gap “in” (respect. “out”). It does not depend on $\epsilon > 0$.

See figure 3.2. In other words, for a sufficiently small $\epsilon$, the operator $\hat{H}_{\mu, \epsilon}$ presents a spectral gap when $|\mu| > 1$. Moreover, when $\mu$ goes from $-1$ to $1$, a finite number of eigenvalues of $\hat{H}_\mu$ continuously crosses the gap. The proof uses microlocal analysis, but the idea is that, for a sufficiently small $\epsilon$, the
<figure><img src="image_7.png" /><figcaption>Figure 3.2: For $\epsilon > 0$ fixed, this is a schematic picture of the spectrum of the operator $\hat{H}_{\mu, \epsilon}$. In this example, $\mathcal{N} = n_{\text{in}} - n_{\text{out}} = 0 - (-2) = +2$ corresponding to the fact that $\mathcal{N} = +2$ eigenvalues are moving upward as $\mu$ increases.</figcaption></figure>

spectrum of $\hat{H}_{\mu}$ is close to the range of $H_{\mu}$, so the red zone of figure 3.2. Moreover the compactness of the ball $\|(\mu, x, p)\| \le 1$ implies discrete spectrum for $\hat{H}_{\mu}$ in that region (this is related to Weyl law or uncertainty principle). There are two energy bands (lower and upper) and the spectral index counts the number of states exchanged by them.

**Remark 3.5.** The integer $\mathcal{N}_H$ is continuous hence invariant under continuous variations (homotopy) in the space of symbols given by definition 3.1. This means that $\mathcal{N}_H$ is "topological". Then if two symbols are homotopic (one can continuously deform one to the other) then they have the same spectral index. But the converse is not true: we will study this point later in section 4.

We now give an easy but important proposition (or remark) related to so called “K-theory”. One can change the size of the matrix $H_{\mu}(x, p)$ without changing the spectral index: it suffices to add some constant eigenvalues outside of the gap:

**Proposition 3.6.** Consider $r \in \mathbb{N} \setminus \{0\}$ and $\omega_0 \in \mathbb{R}$ with $|\omega_0| > C$ and $(H_{\mu})_{\mu}$ verifying the definition 3.1. Then $\tilde{H}_{\mu}$ defined by

$$ \tilde{H}_{\mu}(x, p) = H_{\mu}(x, p) \oplus \omega_0 \mathrm{Id}_{\mathbb{C}^r} = \begin{pmatrix} H_{\mu}(x, p) & 0 \\ 0 & \omega_0 \mathrm{Id}_{\mathbb{C}^r} \end{pmatrix} \quad (3.5) $$

has the same spectral index than $H_{\mu}$:

$$ \mathcal{N}_{\tilde{H}} = \mathcal{N}_{H \oplus \omega_0 \mathrm{Id}_{\mathbb{C}^r}} = \mathcal{N}_H. \quad (3.6) $$

More generally we have the **additive property** for two symbols

$$ \mathcal{N}_{H \oplus H'} = \mathcal{N}_H + \mathcal{N}_{H'}. \quad (3.7) $$

*Proof.* The symbol $\omega_0 \mathrm{Id}_{\mathbb{C}^r}$ gives a quantized operator $\mathrm{Op}(\omega_0 \mathrm{Id}_{\mathbb{C}^r}) = \omega_0 \mathrm{Id}_{L^2(\mathbb{R}^n \otimes \mathbb{C}^r)}$, whose spectrum is constant eigenvalues $\{\omega_0\}$, independent of $(\mu, x, p)$. Hence the spectral flow index is $\mathcal{N}_{\omega_0 \mathrm{Id}_{\mathbb{C}^r}} = 0$. Due to the diagonal form of (3.5) the spectrum of $\mathrm{Op}(\tilde{H}_{\mu})$ is the superposition of the spectrum of $\mathrm{Op}(H_{\mu})$ and $\mathrm{Op}(\omega_0 \mathrm{Id}_{\mathbb{C}^r})$, hence $\mathcal{N}_{\tilde{H}_{\mu}} = \mathcal{N}_H + \mathcal{N}_{\omega_0 \mathrm{Id}_{\mathbb{C}^r}} = \mathcal{N}_H$. $\square$

## 3.3 Chern index

In the previous section, we were interested in the operators $\mathrm{Op}_{\epsilon}(H_{\mu})$ in order to define the spectral index $\mathcal{N}_H$. Now, we study the symbols $(H_{\mu})_{\mu \in \mathbb{R}}$.
<table><caption>Table 1: Equivalence groups of complex vector bundles of rank $r$ over sphere $S^k$. $\mathrm{Vec}^r(S^k) = \pi_{k-1}(U(r))$.</caption><thead><tr><th>Vec<sup>r</sup>(S<sup>k</sup>)</th><th>S<sup>2</sup></th><th>S<sup>3</sup></th><th>S<sup>4</sup></th><th>S<sup>5</sup></th><th>S<sup>6</sup></th><th>S<sup>7</sup></th></tr></thead><tbody><tr><td>Vec<sup>1</sup></td><td>Z</td><td>0</td><td>0</td><td>0</td><td>0</td><td>0</td></tr><tr><td>Vec<sup>2</sup></td><td>Z</td><td>0</td><td>Z</td><td>Z<sub>2</sub></td><td>Z<sub>2</sub></td><td>Z<sub>12</sub></td></tr><tr><td>Vec<sup>3</sup></td><td>Z</td><td>0</td><td>Z</td><td>0</td><td>Z</td><td>Z<sub>6</sub></td></tr><tr><td>Vec<sup>4</sup></td><td>Z</td><td>0</td><td>Z</td><td>0</td><td>Z</td><td>0</td></tr><tr><td>Vec<sup>5</sup></td><td>Z</td><td>0</td><td>Z</td><td>0</td><td>Z</td><td>0</td></tr><tr><td></td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td><td>...</td></tr><tr><td>K&#771;(S<sup>k</sup>)</td><td>Z</td><td>0</td><td>Z</td><td>0</td><td>Z</td><td>0</td></tr></tbody></table>

**Proposition 3.7.** Let $(H_\mu)_{\mu \in \mathbb{R}}$ be a family of symbols verifying definition 3.1 and let

$$S^{2n} := \{(\mu, x, p) \in \mathbb{R}^{1+2n}, \| (\mu, x, p) \| = 1\},$$

be the unit sphere in the space of parameters. The gap assumption (3.3) guarantees that for every parameter $(\mu, x, p) \in S^{2n}$, we have a well defined eigenspace $F_H(\mu, x, p)$ of dimension $r$ associated to the first $r$ eigenvalues $\omega_1 \dots \omega_r$ (direct sum of the first $r$ eigenspaces). This family of vector spaces is called a **smooth complex vector bundle of rank $r$ over the sphere $S^{2n}$** and denoted $F_H$ or

$$F_H \to S^{2n}.$$

The isomorphism class of this bundle $F_H$ denoted $\mathrm{Vec}_\mathbb{C}^r(S^{2n})$ is characterized by the isomorphism class of the **clutching function**$^a$ on the equator $g: S^{2n-1} \to U(r)$, denoted $\pi_{2n-1}(U(r))$ (see figure 1), i.e.

$$\mathrm{Vec}_{\mathbb{C}}^{r}(S^{2n}) = \pi_{2n-1}(U(r)).$$

Table 1 shows some class of isomorphism of vector bundles. We say that the vector bundle $F_H$ is **trivial** if the clutching function is homotopic to the identity. Remark that for $r \ge 2n$, $\mathrm{Vec}_\mathbb{C}^r(S^{2n})$ is always $\mathbb{Z}$, and this is the subject of the following important theorem.

**Theorem 3.8.** [9]«Bott periodicity theorem (1959)» For all $n, r \in \mathbb{N} \setminus 0$ with $r \ge n$, we have

$$\mathrm{Vec}_{\mathbb{C}}^{r}(S^{2n}) = \mathbb{Z}, \quad \mathrm{Vec}_{\mathbb{C}}^{r}(S^{2n-1}) = 0.$$

When $r < n$, the structure of $\mathrm{Vec}_\mathbb{C}^r(S^{2n})$ is more complicated and give rise to non trivial effects, see this **table**. More details will be given in the third section.

The next proposition gives a definition of the Chern index $\mathcal{C}(F_H)$ from the clutching function $g$ of proposition 3.7. This case is significantly less straightforward than in definition 2.5, i.e. dimension $n = 1$, where the Chern index simply reduces to the winding number (or degree) of the clutching function. Recall that the **degree** of a map $f: S^m \to S^m$ is

$$\deg(f) := \sum_{x \in f^{-1}(y)} \mathrm{sign}(\det(D_x f)) \in \mathbb{Z}, \qquad (3.8)$$

which is independent of the choice of the generic point $y \in S^m$. In the case $f: S^1 \to S^1$, the degree $\deg(f)$ is also called **winding number** of $f$.

<footnote>$^a$similarly as in definition 2.5 or figure 2.3. Here the equator is a sphere $S^{2n-1}$ and the clutching is a unitary map in $\mathbb{C}^r$ hence an element of $U(r)$.</footnote>
**Proposition 3.9.** [1, Section III.1.B, p.271] Let $g: S^{2n-1} \to U(r)$ be the clutching function of proposition 3.7. Suppose that $r \ge n$. Then, we can continuously deform $g$ so that $\forall x \in S^{2n-1}$, $g_x(e_r) = e_r$, where $(e_1, \dots, e_r)$ is the canonical basis of $\mathbb{C}^r$ and $g$ restricted to $\mathbb{C}^{r-1} \subset \mathbb{C}^r$ gives a function $g: S^{2n-1} \to U(r-1)$. By iteration we get the case $r = n$ with a clutching function $g: S^{2n-1} \to U(n)$ and we define the function

$$f: \begin{cases} S^{2n-1} & \to S^{2n-1} \subset \mathbb{C}^n \\ x & \to g_x(e_1) \end{cases} \qquad (3.9)$$

The Chern index of $F_H$ is an integer defined by

$$C_H := C(F) := \frac{\deg(f)}{(n-1)!} \in \mathbb{Z}, \qquad (3.10)$$

where $\deg(f)$ is the degree of $f$. Adding some constant eigenvalue in the symbol (as in (3.5)) does not change the Chern index:

$$C_{H \oplus \text{Id}_{\mathbb{C}^r}} = C_H. \qquad (3.11)$$

More generally we have the **additive property** for two symbols

$$C_{H \oplus H'} = C_H + C_{H'}. \qquad (3.12)$$

Hence if initially $r < n$, then we add a constant vector space to each fiber $F(\mu, x, p)$ (as in (3.5)) to reach $r = n$ and define $C(F)$ using (3.10).

*Remark 3.10.* A priori, formula (3.10) gives a rational number but proposition 3.9 claims that it is an integer.

## 3.4 Index formula

So far, for any family of symbols $(H_\mu)_\mu$ of definition 3.1, we have defined on the one hand, a spectral index $N_H \in \mathbb{Z}$ and on the other hand, a topological index $C_H \in \mathbb{Z}$. In this section we will present and explain the following result.

**Theorem 3.11.** [3, thm 2.7] «Index formula» Let$(H_\mu)_\mu$ be a family of symbols that satisfies the gap assumption (3.3). Let $N_H \in \mathbb{Z}$ be the spectral index defined in (3.4) and let $C_H \in \mathbb{Z}$ be the topological index defined by (3.10). We have:

$$N_H = C_H. \qquad (3.13)$$

*Remark 3.12.* Concerning the first model, this formula (3.13) has been observed in (2.12).

## 3.5 Sketch of proof of the index formula (3.13)

As explained in [3], the index formula (3.13) relies on the index theorem on Euclidean space of Fedosov-Hörmander given in [10, thm 7.3 p. 422], [1, Thm 1, page 252]. In this section we provide an convincing explanation of formula (3.13) using normal forms. We proceed in two steps.

1. First, we present some elementary models denoted $E^{(n,C)}$, for any $n \ge 1$ and $C \in \mathbb{Z}$, called normal forms, such that the phase space is $(x, p) \in \mathbb{R}^{2n}$, the symbol is a $r \times r$ matrix with $r$ large and the model gives the Chern index $C \in \mathbb{Z}$. We then observe that the index formula (3.13), holds true:

$$\mathcal{N}_E = C_E \qquad (3.14)$$

*Remark 3.13.* The model $E^{(1,1)} = E_\mu$ has already been presented in section 2. We present these normal form models in detail in section 3.5.1 below.
2. Second, for any model defined by a symbol $H_\mu$ (from definition 3.1), we first increase the size $r$ of the matrix by adding constant levels (as in (3.5)). According to (3.6) and (3.11) this operation does not change the value of $C_H$, $N_H$. If $r$ is large enough, the symbol $H_\mu$ can be deformed continuously to a normal form model $E^{(n,C)}$ with $C = C_H$. For this we use proposition 3.7 and Bott periodicity theorem 3.8. By continuity, this guarantees that $N_H = N_E$. We then deduce that $N_H = N_E = C_E = C = C_H$ giving formula (3.13).

### 3.5.1 The index formula with normal forms

We consider step 1. We consider dimension $n \ge 1$, i.e. phase space $(x, p) \in \mathbb{R}^n \times \mathbb{R}^n$ and we set

$$z = (z_1, \dots, z_n) := x + ip \in \mathbb{C}^n.$$

**Proposition 3.14 (Normal form model $E^{(n,C)}$).** For dimension $n \ge 1$, and Chern index $C = \pm 1$, we define a matrix symbol of size $2^n$

$$E^{(n, \pm 1)} := \begin{pmatrix} \mp \mu \operatorname{Id}_{2^{n-1}} & g_n(z) \\ (g_n(z))^{\dagger} & \pm \mu \operatorname{Id}_{2^{n-1}} \end{pmatrix}, \qquad (3.15)$$

where the matrix $g_n(z)$ of size $2^{n-1}$ is defined recurrently by

$$g_n \left( z_1, \underbrace{z_2, \dots, z_n}_{z'} \right) = \begin{pmatrix} z_1 \operatorname{Id}_{2^{n-2}} & -(g_{n-1}(z'))^{\dagger} \\ g_{n-1}(z') & \overline{z_1} \operatorname{Id}_{2^{n-2}} \end{pmatrix}, \quad g_1(z) = z. \qquad (3.16)$$

Then for any $C \in \mathbb{Z}$, we define the model $E^{(1,C)}$ by direct sum of $|C|$ copies

$$\text{If } C = 0, E^{(n,C)} := \operatorname{Id},$$

$$\text{If } C > 0, E^{(n,C)} := E^{(n,1)} \oplus \dots \oplus E^{(n,1)},$$

$$\text{If } C < 0, E^{(n,C)} := E^{(n,-1)} \oplus \dots \oplus E^{(n,-1)}.$$

For this model $E^{(n,C)}$, we have

$$N_E = C_E = C. \qquad (3.17)$$

*Proof.* We proceed with different cases.

* The case $C = 0$ with a constant matrix, gives fixed eigenvectors hence $C_E = 0$ and no exchange of states hence $N_E = 0$, so (3.17) holds true.

* For dimension $n = 1$ and $C = +1$, we get a $2 \times 2$ matrix symbol $E^{(1,1)}$ depending on $\mu, z = x + ip \in \mathbb{C}$ that coincides with the first normal form model (2.6), indeed:

$$E^{(1,1)} \underset{(3.15)}{=} \begin{pmatrix} -\mu & g_1(z) \\ (g_1(z))^{\dagger} & \pm \mu \operatorname{Id}_{2^{n-1}} \end{pmatrix} \underset{(3.16)}{=} \begin{pmatrix} -\mu & x + ip \\ x - ip & \mu \end{pmatrix} \underset{(2.6)}{=} E_{\mu}.$$

For that model, (3.17) has already been proven in (2.12). Indeed we by specific computations, we obtained $C_{E^{(1,1)}} = +1$ and $N_{E^{(1,1)}} = +1$.

* For the case $n = 1$ and $C = -1$, i.e. the model $E^{(1,-1)}$, a similar computation shows that (3.17) holds true (this is only a change of orientation on $S^2$).

* More generally, for the case of any dimension $n \ge 1$, we can similarly compute that $C_{E^{(n,\pm 1)}} = \pm 1$ using definition 3.10. For this, we refer to [14, section 1.1] and [3, prop B.25]. We also compute $N_{E^{(n,\pm 1)}} = \pm 1$. For this we first compute the Fredholm index of the operator $\hat{g}_n := \operatorname{Op}_1(g_n)$ and obtain in [3, prop B.26] that

$$\operatorname{Ind}(\hat{g}_n) := \operatorname{dimKer}(\hat{g}_n) - \operatorname{dimKer}(\hat{g}_n^{\dagger}) = +1.$$
Then in [3, Lemma 2.11] we compute in a direct way that $N_{E(n, \pm 1)} = \pm \mathrm{Ind}(\hat{g}_n)$, from which we deduce $N_{E(n, \pm 1)} = \pm 1$. Hence (3.17) holds true for the models $E^{(n, \pm 1)}$.

• Finally, for any $\mathcal{C} > 0$ (respect. $\mathcal{C} < 0$) and any dimension $n$, we use additive properties of $\mathcal{N}$ and $\mathcal{C}$ in (3.7) and (3.12) to deduce that (3.17) holds true for the models $E^{(n, \mathcal{C})}$.

□

*Remark 3.15.* In higher dimensions $n \ge 1$, the iteration (3.16) gives more complicated normal form matrices of size $2^n \times 2^n$. For example for $n = 2$, we get the symbol

$$E^{(2,1)}(\mu, x, p) \underset{(3.15,3.16)}{=} \begin{pmatrix} -\mu & 0 & z_1 & -\overline{z_2} \\ 0 & -\mu & z_2 & \overline{z_1} \\ \overline{z_1} & \overline{z_2} & +\mu & 0 \\ -z_2 & z_1 & 0 & +\mu \end{pmatrix}, \quad \text{with } z_1 = x_1 + ip_1, z_2 = x_2 + ip_2.$$

# 4 Topological contact without exchange

In this last section we discuss some interesting physical phenomenon called “topological contact” that may happen in the models $(H_\mu)_{\mu \in \mathbb{R}}$ of definition 3.1, if dimension $n$ of phase space is large enough. Consider a model $(H_\mu)_{\mu \in \mathbb{R}}$ given by definition 3.1. There are two cases and subcases:

1. If the vector bundle $F_H$ defined in proposition 3.7 is trivial (i.e. the isomorphism class is $[F_H] = 0$ in $\mathrm{Vec}_C^r(S^{2n})$), then $\mathcal{C} = 0$ and $\mathcal{N} = 0$. We can perturb continuously the symbol $(H_\mu)_\mu$ toward a constant matrix $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ (or with higher dimensions) so that we have a (big) gap for every values of $\mu \in \mathbb{R}$, i.e. we can “open the gap”. Conversely an open gap implies that the bundle is trivial.

2. If the vector bundle $F$ is non trivial, (i.e. $[F_H] \ne 0$ in $\mathrm{Vec}_C^r(S^{2n})$), then it means that the two bands are “topologically coupled” with a “topological contact” and we can not “open the gap”, or remove the contact between the two bands. However there are subcases:

(a) If $\mathcal{C} \ne 0$, then $\mathcal{N} = \mathcal{C} \ne 0$ also, there are exchange states between the bands. This situations holds if $r \ge n$ from Bott theorem 3.8.

(b) If $\mathcal{C} = 0$, then $\mathcal{N} = \mathcal{C} = 0$ also, there are no exchange states between the bands. From Bott theorem 3.8, this situation can not happen if $r \ge n$ but it may happen from table 1, if $r < n$ and $n \ge 3$. The simplest example with $n = 3$ degrees of freedom with rank $r = 2$, because $\mathrm{Vect}^2(S^6) = \mathbb{Z}_2 = \{0, 1\}$. Suppose for example that $F_H \to S^6$ is non trivial and with topological class $[F] = 1 \in \mathrm{Vect}^2(S^6) = \mathbb{Z}_2$. It means that the two bands have a “topological contact”, i.e. that we can not open the gap. Nevertheless $\mathcal{N} = \mathcal{C} = 0$ (because of the morphism $[F_H] \in \mathrm{Vect}^2(S^6) \to \mathcal{C} \in \mathbb{Z}$). This implies that there is no exchange of states between the two bands but there is some small gap smaller than $\sqrt{\epsilon}$, i.e. that goes to zero in the semi classical limit $\epsilon \to 0$. We can call this a **topological contact without exchange**. See figure below.

<figure><img src="image_8.png" /></figure>
If one adds a second similar contact (at some other value of $\mu$), then since $1 + 1 = 0$ in $\mathbb{Z}_2$, one recovers a trivial bundle, the two contact annihilate themselves and one can finally “open the gap”. See figure below.

<figure><img src="image_9.png" /></figure>

More generally these kind of phenomena may occur with models $(H_\mu)_\mu$ such that the vector bundle $F_H \to S^{2n}$ has rank $r < n$, see this **table of homotopy groups** that exhibits very rich, unexpected and complicated patterns.

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