Hyperbolic band theory

• URL: http://arxiv.org/abs/2008.05489v2
• Date: Wed, 8 Sep 2021 18:00:03 GMT
• Title: Hyperbolic band theory
• Authors: Joseph Maciejko and Steven Rayan
• Abstract summary: We construct the first 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.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 the first 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.

Related papers

• Radiative transport in a periodic structure with band crossings [47.82887393172228]
  We derive the semi-classical model for the Schr"odinger equation in arbitrary spatial dimensions. We consider both deterministic and random scenarios. As a specific application, we deduce the effective dynamics of a wave packet in graphene with randomness.
  arXiv  Detail & Related papers  (2024-02-09T23:34:32Z)
• Hyperbolic lattices and two-dimensional Yang-Mills theory [0.0]
  Hyperbolic lattices are a new type of synthetic quantum matter emulated in circuit quantum electrodynamics and electric-circuit networks. We show that moments of the density of states of hyperbolic tight-binding models correspond to expectation values of Wilson loops in the quantum gauge theory.
  arXiv  Detail & Related papers  (2023-09-07T17:15:54Z)
• Non-Abelian hyperbolic band theory from supercells [0.0]
  hyperbolic lattices support non-Abelian Bloch states that have so far analytical treatments. By adapting the solid-state-physics notions of supercells and zone folding, we devise a method for the systematic construction of non-Abelian Bloch states.
  arXiv  Detail & Related papers  (2023-05-08T18:00:01Z)
• Measurement phase transitions in the no-click limit as quantum phase transitions of a non-hermitean vacuum [77.34726150561087]
  We study phase transitions occurring in the stationary state of the dynamics of integrable many-body non-Hermitian Hamiltonians. We observe that the entanglement phase transitions occurring in the stationary state have the same nature as that occurring in the vacuum of the non-hermitian Hamiltonian.
  arXiv  Detail & Related papers  (2023-01-18T09:26:02Z)
• Hyperbolic band theory through Higgs bundles [0.0]
  Hyperbolic lattices underlie a new form of quantum matter with potential applications to quantum computing and simulation. A corresponding hyperbolic band theory has emerged, extending 2-dimensional Euclidean band theory in a natural way to higher-genus configuration spaces. Higgs bundles enjoy natural interpretations in the context of band theory.
  arXiv  Detail & Related papers  (2022-01-30T00:25:38Z)
• Spectrum of localized states in fermionic chains with defect and adiabatic charge pumping [68.8204255655161]
  We study the localized states of a generic quadratic fermionic chain with finite-range couplings. We analyze the robustness of the connection between bands against perturbations of the Hamiltonian.
  arXiv  Detail & Related papers  (2021-07-20T18:44:06Z)
• Crystallography of Hyperbolic Lattices [0.0]
  We derive, for the first time, a list of example hyperbolic $p,q$ lattices and their hyperbolic Bravais lattices. This dramatically simplifies the computation of energy spectra of tight-binding Hamiltonians on hyperbolic lattices.
  arXiv  Detail & Related papers  (2021-05-03T18:00:07Z)
• Quantum particle across Grushin singularity [77.34726150561087]
  We study the phenomenon of transmission across the singularity that separates the two half-cylinders. All the local realisations of the free (Laplace-Beltrami) quantum Hamiltonian are examined as non-equivalent protocols of transmission/reflection. This allows to comprehend the distinguished status of the so-called bridging' transmission protocol previously identified in the literature.
  arXiv  Detail & Related papers  (2020-11-27T12:53:23Z)
• Pseudo-time-reversal-symmetry-protected topological Bogoliubov excitations of Bose-Einstein condensates in optical lattices [2.1532925122722744]
  Bogoliubov excitations of Bose-Einstein condensates in optical lattices may possess band topology in analogous to topological insulators in class AII of fermions. This work is a universal formalism to study all kinds of symmetry-protected topological bosonic Bogoliubov bands.
  arXiv  Detail & Related papers  (2020-10-09T03:15:28Z)
• Dynamical solitons and boson fractionalization in cold-atom topological insulators [110.83289076967895]
  We study the $mathbbZ$ Bose-Hubbard model at incommensurate densities. We show how defects in the $mathbbZ$ field can appear in the ground state, connecting different sectors. Using a pumping argument, we show that it survives also for finite interactions.
  arXiv  Detail & Related papers  (2020-03-24T17:31:34Z)
• From stochastic spin chains to quantum Kardar-Parisi-Zhang dynamics [68.8204255655161]
  We introduce the asymmetric extension of the Quantum Symmetric Simple Exclusion Process. We show that the time-integrated current of fermions defines a height field which exhibits a quantum non-linear dynamics.
  arXiv  Detail & Related papers  (2020-01-13T14:30:36Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.