Complex Structures on Riemann Surfaces

Kiremidjian, Garo K.

doi:10.2307/1996246

Cited by 3 publications

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“…[51][52][53], in which the stiffness and overlap matrices of the weak (variational) formulation of the hyperbolic Schrödinger equation ( 14) are first computed using FreeFEM++ with unconstrained boundary conditions, and the number of physical degrees of freedom is subsequently reduced using simple matrix operations before proceeding to numerical diagonalization. The generalized Bloch phases (25) are easily introduced at this second stage, as we now explain.…”

Section: Discussionmentioning

confidence: 99%

“…Note that the eigenvalues of J can be seen as the origin of the imaginary units ±i on each tangent space. Seeing that the Fuchsian quotient H/Γ always possesses an integrable complex structure J requires some careful mathematical analysis treated, for instance, in [25].…”

Section: Hyperbolic Particle-wave Duality: Complex Geometry and The A...mentioning

confidence: 99%

“…In the hyperbolic case, we wish to find the eigenvalues E and eigenfunctions ψ of the hyperbolic Laplacian −∆ on the hyperbolic octagon D with the boundary conditions (25). At the origin of the hyperbolic Brillouin zone, k = 0, those boundary conditions reduce to the condition that the solutions be strictly automorphic, ψ(γ(z)) = ψ(z), the case usually considered in mathematics [34].…”

Section: The Hyperbolic Bloch Problem On the Bolza Latticementioning

confidence: 99%

“…We use a freely available software package, FreeFEM++ [38], which was used successfully to study the spectrum of the Bolza surface with strictly automorphic boundary conditions [39]. Our implementation of the twisted boundary conditions (25) is discussed in Appendix B. As a check on our calculations, we first compute the spectrum {E n (0)}, i.e., with strictly automorphic boundary conditions [colored plots in Fig.…”

Section: The Hyperbolic Bloch Problem On the Bolza Latticementioning

confidence: 99%

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Hyperbolic band theory

Maciejko,

Rayan

2020

Preprint

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Motivated by the recent realization of hyperbolic lattices in circuit quantum electrodynamics, we exploit ideas from Riemann surface theory and algebraic geometry to construct the first generalization of Bloch band theory to hyperbolic lattices, which can be formulated despite the absence of commutative translation symmetries. For an arbitrary Hamiltonian with the symmetry of a {4g, 4g} tessellation of the hyperbolic plane, we produce a continuous family of eigenstates that acquire Bloch-like phase factors under the Fuchsian group of the tessellation, which is discrete but noncommutative. Quasi-periodic Bloch wavefunctions are then generalized to automorphic functions. Naturally associated with the Fuchsian group is a compact Riemann surface of genus g 2, arising from the pairwise identification of sides of a unit cell given by a hyperbolic 4g-gon. A hyperbolic analog of crystal momentum continuously parametrizes a discrete set of energy bands and lives in a 2g-dimensional torus, which is a maximal set of independent Aharonov-Bohm phases threading the 2g noncontractible cycles of the Riemann surface.This torus is known in algebraic geometry as the Jacobian of the Riemann surface. We propose the Abel-Jacobi map -which associates to each point of the Riemann surface a point in the Jacobian -as a hyperbolic analog of particle-wave duality. Point-group symmetries form a finite group of automorphisms or self-maps of the Riemann surface and act nontrivially on the Jacobian. The tight-binding approximation and Wannier functions are also suitably generalized. In genus 1, our theory reduces to known band theory for 2-dimensional Euclidean lattices. We demonstrate our theory by explicitly computing hyperbolic Bloch wavefunctions and bandstructures numerically for a regular {8, 8} tessellation associated with a particular Riemann surface of genus 2, the Bolza surface.

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Section: Discussionmentioning

confidence: 99%

Section: Hyperbolic Particle-wave Duality: Complex Geometry and The A...mentioning

confidence: 99%

Section: The Hyperbolic Bloch Problem On the Bolza Latticementioning

confidence: 99%

Section: The Hyperbolic Bloch Problem On the Bolza Latticementioning

confidence: 99%

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Hyperbolic band theory

Maciejko,

Rayan

2020

Preprint

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“…We note two things: 1) the I I k-norm defined in ? 3 is a natural extension of the one defined in [2], and 2) this norm, just as in [2], behaves as the usual Sobolev D-norm in the case of compact complex manifolds as far as crucial inequalities are concerned (Theorem 3. 4 and Theorem 3.…”

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confidence: 99%

Classification of Certain Complex Structures on the Polydisc

Kiremidjian¹

1973

American Journal of Mathematics

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Hyperbolic band theory

Maciejko

Rayan

2021

Sci. Adv.

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The notions of Bloch wave, crystal momentum, and energy bands are commonly regarded as unique features of crystalline materials with commutative translation symmetries. Motivated by the recent realization of hyperbolic lattices in circuit quantum electrodynamics, we exploit ideas from algebraic geometry to construct a hyperbolic generalization of Bloch theory, despite the absence of commutative translation symmetries. For a quantum particle propagating in a hyperbolic lattice potential, we construct a continuous family of eigenstates that acquire Bloch-like phase factors under a discrete but noncommutative group of hyperbolic translations, the Fuchsian group of the lattice. A hyperbolic analog of crystal momentum arises as the set of Aharonov-Bohm phases threading the cycles of a higher-genus Riemann surface associated with this group. This crystal momentum lives in a higher-dimensional Brillouin zone torus, the Jacobian of the Riemann surface, over which a discrete set of continuous energy bands can be computed.

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Complex Structures on Riemann Surfaces

Cited by 3 publications

References 0 publications

Hyperbolic band theory

Hyperbolic band theory

Classification of Certain Complex Structures on the Polydisc

Hyperbolic band theory

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