# Hyperbolic materials

• Recent development in condensed matter physics: realizations of periodic systems with effective theories on the hyperbolic plane.

• More precisely: $\{ p, q \}$ tessellations with nearest neighbor interactions.

• The symmetry group is then a Fuchsian group
$$\Gamma = \left\langle \alpha_1, \beta_1, \ldots, \alpha_g, \beta_g \middle| \left[ \alpha_1, \beta_1 \right] \cdots \left[ \alpha_g, \beta_g \right] = \mathbb{I} \right\rangle \quad (4g = p).$$

• Effective model: Hilbert space $\mathcal{H} \cong L^2 \left( \Gamma \right) \otimes \mathcal{H}_0$, with a Hamiltonian, $H$, that is $\Gamma$-periodic:
$$\Gamma \hookrightarrow \mathrm{Aut} \left( H \right) \cap \mathrm{U} \left( \mathcal{H} \right).$$

• In classical crystallography ($\Gamma \cong \mathbb{Z}^{\dim}$) one studies the system $\left( \mathcal{H}, H \right)$ via the Bloch transform.

### Warmup: The classical Bloch transform on the Euclidean plane, part 1.

• Let $\mathcal{H} = L^2 (\mathbb{R}^2)$ and $\Gamma = \mathbb{Z} \: v_1 \oplus \mathbb{Z} \: v_2 \cong \mathbb{Z}^2$.

• Let the Brillouin zone be $\mathrm{BZ} = \mathrm{Hom} \left( \Gamma, \mathrm{U} (1) \right) \cong$ torus.

• Each $\lambda \in \mathrm{BZ}$ gives a flat unitary connection, $\nabla^\lambda$, on $E = \mathbb{R}^2 / \Gamma \times \mathbb{C}$, unique up to gauge.

• Let $U_{\nabla^\lambda} (x)$ be the lift of the holonomy of $\nabla^\lambda$ from $(0,0)$ to $x \in \mathbb{R}^2$.

• If $\psi \in \mathbb{R}^2$ and $y \in \mathbb{R}^2 / \Gamma$, then let
$$\widetilde{\mathcal{B}} \left( \psi, \nabla^\lambda \right) \left( y \right) = \sum\limits_{x \in y} \psi (x) U_{\nabla^\lambda} (x) \in \mathrm{End} \left( E_y \right).$$

• Note: $\lambda \left( R \right) e^{2 \pi i k \cdot R}$, where $[k] \in \left( \mathbb{R}^2 \right)^* / \Gamma^*$.

### Warmup: The classical Bloch transform on the Euclidean plane, part 2.

$$\mathcal{B} \left( \psi, \lambda \right) = \left[ \lambda, y \mapsto \sum\limits_{x \in y} \psi (x) U_{\nabla^\lambda} (x) \right] \in \left( \mathrm{BZ} \times L^2 \left( \mathrm{End} \left( E \right) \right) \right) / \mathrm{U} (1) = : L^2 \left( \mathcal{E} \right).$$
• Now $\mathcal{B} \left( \psi \right)$ is an $L^2$ section of $\mathcal{E}$, a (trivial) $\mathrm{End} \left( E \right)$-bundle over $\mathrm{BZ}$, with some nice properties.

• Plancherel theorem: $\mathcal{B}$ is a unitary isomorphism.

• If $A : \mathcal{H} \rightarrow \mathcal{H}$ is a $\Gamma$-periodic operator, then $\widehat{A} = \mathcal{B} \circ A \circ \mathcal{B}^{- 1}$ is algebraic.

• In particular: $\mathcal{B} \left( \left( \Delta + V \right) \psi \right) = \left( \left( \nabla^\lambda \right)^* \nabla^\lambda + V \right) \mathcal{B} \left( \psi, \lambda \right)$.

• Interpretation: $\mathcal{B} \left( T_R \left( \psi \right), \lambda \right) = \lambda \left( R \right) \mathcal{B} \left( \psi, \lambda \right) \: \Rightarrow \: \mathcal{B}$ decomposes $\psi$ into quasi-periodic waves.

### Warmup: The classical Bloch transform on the Euclidean plane, part 3.

• (Non)commutative geometric point of view: $C_\Gamma =$ group-$C^*$-algebra of $\Gamma$ (a Neumann algebra).

• $\mathcal{H}$ is a projective $C_\Gamma$-module.

• $C_\Gamma \cong C^0 \left( \mathrm{torus} \right)$.

• $C^0$-projective modules are vector bundles, $C_\Gamma$ periodic operators (elements of another Neumann algebra) are endomorphism of that bundle, etc...

• This approach is more natural, but less geometric/analytic.

### Hyperbolic lattices

• Let $\mathcal{H} = L^2 \left( \mathbb{H} \right)$ and $\Gamma \subset \mathrm{Isom} \left( \mathbb{H} \right)$ be a Fuchsian group (of genus $g$).

• $\mathcal{H}$ is again a projective $C_\Gamma$-module.

• Problem: $C_\Gamma$ is a noncommutative space!

• Maciejko–Rayan: studied the rank-1 case of the Bloch transform.

• Rank-1 representation are insensitive to $\left[ \Gamma, \Gamma \right]$.

• In other words, any "geometrization" needs to use higher rank representations.

### Hyperbolic Bloch transform through stable bundles, part 1.

• One Brillouin zone for each rank $\: \mathrm{BZ}_\Gamma = \bigcup\limits_{n \in \mathbb{N}_+}^\infty \overbrace{\mathrm{Hom}_{\mathrm{irr}} \left( \Gamma, \mathrm{U} (n) \right) / \mathrm{U} (n)}^{\mathrm{BZ}_\Gamma^n}.$.

• Each $\varrho : \Gamma \rightarrow \mathrm{U} (n)$ irreducible representation determines an irreducible flat, Hermitian connection, $\nabla^\varrho$, on $E^n = \Sigma \times \mathbb{C}^n$, where $\Sigma = \mathbb{H} / \Gamma$, unique up to gauge. These are also the stable bundles of rank $n$ and degree $0$ (Narasimhan–Seshadri, Donaldson).

• Let $U_{\nabla^\varrho} : \mathbb{H} \rightarrow \mathrm{U} (n)$ as before.

• If $\psi \in \mathbb{H}$ and $y \in \Sigma$, then let
$$\widetilde{\mathcal{B}} \left( \psi, \nabla^\varrho \right) \left( y \right) = \sum\limits_{x \in y} \psi (x) U_{\nabla^\varrho} (x) \in \mathrm{End} \left( E_y^n \right).$$

### Hyperbolic Bloch transform through stable bundles, part 2.

$$\mathcal{B}_n \left( \psi, \varrho \right) = \left[ \varrho, y \mapsto \sum\limits_{x \in y} \psi (x) U_{\nabla^\varrho} (x) \right] \in \left( \mathrm{Hom}_{\mathrm{irr}} \left( \Gamma, \mathrm{U} (n) \right) \times L^2 \left( \mathrm{End} \left( E^n \right) \right) \right) / \mathrm{U} (n) = : L^2 \left( \mathcal{E}_\Gamma^n \right).$$
• Now $\mathcal{B} \left( \psi \right)$ is an $L^2$ section of $\mathcal{E}^n$, a (trivial) $\mathrm{End} \left( E^n \right)$-bundle over $\mathrm{BZ}_\Gamma^b$, with some nice properties.

• If $A : \mathcal{H} \rightarrow \mathcal{H}$ is a $\Gamma$-periodic operator, then $\widehat{A} = \mathcal{B} \circ A \circ \mathcal{B}^{- 1}$ is algebraic.

• In particular: $\mathcal{B} \left( \left( \Delta + V \right) \psi \right) = \left( \left( \nabla^\varrho \right)^* \nabla^\varrho + V \right) \mathcal{B} \left( \psi, \varrho \right)$.

• Interpretation remains: $\mathcal{B} \left( T_\gamma \left( \psi \right), \varrho \right) = \varrho \left( \gamma \right) \mathcal{B} \left( \psi, \varrho \right) \: \Rightarrow \: \mathcal{B}$ decomposes $\psi$ into quasi-periodic waves.

• Missing: Plancherel theorem!

### Main result

• Main Theorem (N–Rayan '22)
For each $\psi \in C_{\mathrm{cpt}}^0 \left( \mathbb{H} \right)$, the sequence $\left( \mathcal{B}_n^* \mathcal{B}_n \left( \psi \right) \right)_{n \in \mathbb{N}_+}$, converges to $\psi$ in the topology of $L^\infty \left( \mathbb{H} \right)$.

• Proof uses a recent result of Magee: $$\mathbb{E}_n \left( \gamma \right) = \int\limits_{\mathrm{BZ}_\Gamma^n} \mathrm{tr} \left( \varrho \left( \gamma \right) \right) \mathrm{d} \mu_n = \delta_{\mathbb{I}, \gamma} + O \left( \tfrac{1}{n} \right).$$

• $$\left\langle \mathcal{B}_n^* \mathcal{B}_n \left( \psi \right) \middle| \psi^\prime \right\rangle - \left\langle \psi \middle| \psi^\prime \right\rangle = \sum\limits_{\gamma_1, \gamma_2 \in \Gamma} \left( \mathbb{E}_n \left( \gamma_1^{- 1} \gamma_2 \right) - \delta_{\gamma_1^{- 1} \gamma_2, e} \right) \int\limits_C \overline{\psi \left( \gamma_1^{- 1} (x) \right)} \psi \left( \gamma_2^{- 1} (x) \right) dvol_{\mathbb{H}} (x).$$

• A stronger version of Magee's result can sharpen our theorem too.

### Future wish list

• Plancherel theorem for the hyperbolic Bloch transform via strengthening Magee's theorem.

• Study spectral properties of the Laplaians of stable bundles.

• Construction of topological phases.

• Understanding connections to the Hitchin moduli space.

• Understanding connections to the Helgason's hyperbolic Fourier transform.