[Paper Review] Algebra of Hyperbolic Band Theory under Magnetic Field

Kazuki Ikeda, Yoshiyuki Matsuki+1 more|arXiv (Cornell University)|Jul 22, 2021

Geometric and Algebraic Topology14 references4 citations

TL;DR

This paper introduces the magnetic Fuchsian group and hyperbolic magnetic Bloch states on a hyperbolic lattice under a uniform magnetic field, generalizing Euclidean band theory to negatively curved spaces. It establishes a correspondence between these states and automorphic forms of weight $|2B|$, with the magnetic flux through the fundamental domain quantized as $\phi = 4(g-1)\pi B$, enabling a new algebraic framework for hyperbolic band theory in curved space with magnetic fields.

ABSTRACT

We explore algebras associated with the hyperbolic band theory under a magnetic field for the first time. We define the magnetic Fuchsian group associated with a higher genus Riemann surface. By imposing the magnetic boundary conditions for the hyperbolic Bloch states, we construct the hyperbolic magnetic Bloch states and investigate their energy spectrum. We give a connection between such magnetic Bloch states and automorphic forms. Our theory is a general extension of the conventional algebra associated with the band theory defined on a Euclidean lattice/space into that of the band theory on a general hyperbolic lattice/Riemann surface.

Motivation & Objective

• To extend conventional band theory from Euclidean lattices to hyperbolic lattices on Riemann surfaces of genus $g \geq 2$ under a magnetic field.
• To develop a mathematical framework for hyperbolic band theory that incorporates magnetic translation symmetries and topological invariants.
• To establish a connection between hyperbolic magnetic Bloch states and automorphic forms of non-zero weight on the Poincaré upper half-plane.
• To generalize the algebraic structure of the $PSL_2(\mathbb{R})$ group to a magnetic version $PSL_2(\mathbb{R})[B]$ that encodes the magnetic flux and boundary conditions.
• To provide a foundation for testing hyperbolic band theory in quantum platforms such as cQED and quantum simulators.

Proposed method

• Define the magnetic $PSL_2(\mathbb{R})[B]$ Lie group generated by vector fields $S$, $T$, and $U$ that include magnetic corrections via the gauge potential $A = \frac{B}{y}dx$.
• Construct the magnetic Fuchsian group $\hat{\gamma}_j^B$ as exponentials of combinations of magnetic generators $\hat{S}_B$, $\hat{T}_B$, and $\hat{U}_B$, encoding magnetic translation symmetry.
• Impose magnetic boundary conditions on hyperbolic Bloch states using the magnetic Fuchsian group, ensuring quasi-periodicity with phase factors $e^{ik_j}$.
• Derive the magnetic flux condition $\hat{\gamma}_{2g}^B \cdots \hat{\gamma}_1^B (\hat{\gamma}_{2g}^B)^{-1} \cdots (\hat{\gamma}_1^B)^{-1} = e^{i4(g-1)\pi B}$, which generalizes the Gauss-Bonnet theorem.
• Construct hyperbolic magnetic Bloch states $\psi_{0,k}$ as products of automorphic forms $f$ of weight $|2B|$ and a phase-modulating function $u(z)$, ensuring invariance under the magnetic Fuchsian group.
• Use the transformation rule $\check{g}f(z) = j(g,z)^{2B}f(gz)$ for $g \in SL_2(\mathbb{R})$, showing that automorphic forms of weight $-2B$ transform consistently under the magnetic group action.

Experimental results

Research questions

• RQ1 How can the algebraic structure of band theory be generalized from Euclidean to hyperbolic lattices under a magnetic field?
• RQ2 What is the correct generalization of the magnetic translation group in hyperbolic space, and how does it relate to the fundamental group of a Riemann surface of genus $g \geq 2$?
• RQ3 How do hyperbolic magnetic Bloch states transform under the magnetic Fuchsian group, and what is their topological characterization?
• RQ4 What is the role of automorphic forms in describing the energy spectrum of electrons on a hyperbolic lattice with a magnetic field?
• RQ5 How does the magnetic flux through the fundamental domain of a $\{4g,4g\}$ tiling relate to the quantum numbers of the Bloch states?

Key findings

• The magnetic Fuchsian group $\hat{\gamma}_j^B$ is defined as a non-trivial extension of the conventional Fuchsian group, incorporating magnetic translation symmetry via the gauge field $A = \frac{B}{y}dx$.
• The magnetic flux through the fundamental domain of a genus-$g$ hyperbolic surface is quantized as $\phi = 4(g-1)\pi B$, and this flux appears in the commutation relation of the magnetic Fuchsian generators.
• When $B = 1/q$, the group $PSL_2(\mathbb{R})$ acts as a $q$-fold covering of the magnetic $PSL_2(\mathbb{R})[B]$ group, generalizing the magnetic translation structure from the torus to hyperbolic surfaces.
• Hyperbolic magnetic Bloch states $\psi_{0,k}$ are constructed as $\psi_{0,k}(z) = \psi_k(z) u(z)$, where $u(z)$ is a phase-modulating function derived from an automorphic form $f$ of weight $|2B|$, ensuring invariance under the magnetic Fuchsian group.
• The magnetic Hamiltonian $H'$ commutes with the generators $\check{S}, \check{T}, \check{U}$, and its spectrum is determined by the automorphic form $f$ of weight $|2B|$, linking quantum mechanics to modular forms.
• For $2B \in \mathbb{Z}$, the function $u(z)$ is well-defined and $\hat{\gamma}_j^B u(z) = u(z)$, ensuring the Bloch state $\psi_{0,k}$ satisfies the magnetic Bloch condition with phase $e^{ik_j}$.