On the hyperbolic Bloch transform

Ákos Nagy

University of California, Santa Barbara

Joint with Steve Rayan (University of Saskatchewan)

This talk is based on 2208.02749

Pacific Rim Mathematical Association Congress 2022, Vancouver, Canada

Friday, December 9, 2022.

this presentation can be viewed at akosnagy.com/talks/HBT_Vancouver/hyperbolic_bloch_transform.html

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
    \begin{equation} \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). \end{equation}

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

  • 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
    \begin{equation} \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). \end{equation}

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



    \begin{equation} \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). \end{equation}
  • 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: $\mathcal{B} \left( \left( \Delta + V \right) \psi \right) = \left( D^* D + V \right) \mathcal{B} \left( \psi, \lambda \right)$.

  • Interpretation: $\mathcal{B} \left( \psi \circ \gamma^{- 1}, \lambda \right) = \lambda \left( \gamma \right) \mathcal{B} \left( \psi, \lambda \right) \: \Rightarrow \: \mathcal{B}$ 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
    \begin{equation} \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). \end{equation}

Hyperbolic Bloch transform through stable bundles, part 2.



    \begin{equation} \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). \end{equation}
  • 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: $\mathcal{B} \left( \left( \Delta + V \right) \psi \right) = \left( D^* D + V \right) \mathcal{B} \left( \psi, \varrho \right)$.

  • Interpretation remains: $\mathcal{B} \left( \psi \circ \gamma^{- 1}, \varrho \right) = \varrho \left( \gamma \right) \mathcal{B} \left( \psi, \varrho \right) \: \Rightarrow \: \mathcal{B}$ 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: \begin{equation} \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). \end{equation}

  • \begin{equation} \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). \end{equation}

  • 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.

Thank you for your attention!