Symmetry-based theory of Dirac fermions on two-dimensional hyperbolic crystals: Coupling to the spin connection
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866911050535796736 |
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| author | Djordjević, Ana Ćirić, Marija Dimitrijević Juričić, Vladimir |
| author_facet | Djordjević, Ana Ćirić, Marija Dimitrijević Juričić, Vladimir |
| contents | Discrete fermionic and bosonic models for hyperbolic lattices have attracted significant attention across a range of fields since the experimental realization of hyperbolic lattices in metamaterial platforms, sparking the development of hyperbolic crystallography. However, a fundamental and experimentally consequential aspect remains unaddressed: fermions propagating in curved space inherently couple to the underlying geometry via the spin connection, as required by general covariance - a feature not yet incorporated in studies of hyperbolic crystals. Here, we introduce a symmetry-based framework for Dirac fermions on two-dimensional hyperbolic lattices, explicitly incorporating spin-curvature coupling via a discrete spin connection. Starting from the continuous symmetries of the Poincaré disk, we classify the irreducible representations and construct a symmetry-adapted basis, establishing a direct correspondence to the continuum Dirac theory. We show that this continuum theory predicts a finite density of states at zero energy for any finite curvature in $D-$dimensional hyperbolic space with $2\leq D \leq 4$, suggesting enhanced susceptibility of Dirac fermions to interaction-driven instabilities at weak coupling. We then derive explicit forms of discrete translational and rotational symmetries for lattices characterized by Schläfli symbols $\{p,q\}$, and explicitly construct the discrete spin connection, represented as hopping phases, via parallel transport. Our results pave the way for experimental realization of spin-curvature effects in metamaterial platforms and systematic numerical studies of correlated Dirac phases in hyperbolic geometries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_08276 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Symmetry-based theory of Dirac fermions on two-dimensional hyperbolic crystals: Coupling to the spin connection Djordjević, Ana Ćirić, Marija Dimitrijević Juričić, Vladimir Mesoscale and Nanoscale Physics General Relativity and Quantum Cosmology High Energy Physics - Lattice High Energy Physics - Theory Discrete fermionic and bosonic models for hyperbolic lattices have attracted significant attention across a range of fields since the experimental realization of hyperbolic lattices in metamaterial platforms, sparking the development of hyperbolic crystallography. However, a fundamental and experimentally consequential aspect remains unaddressed: fermions propagating in curved space inherently couple to the underlying geometry via the spin connection, as required by general covariance - a feature not yet incorporated in studies of hyperbolic crystals. Here, we introduce a symmetry-based framework for Dirac fermions on two-dimensional hyperbolic lattices, explicitly incorporating spin-curvature coupling via a discrete spin connection. Starting from the continuous symmetries of the Poincaré disk, we classify the irreducible representations and construct a symmetry-adapted basis, establishing a direct correspondence to the continuum Dirac theory. We show that this continuum theory predicts a finite density of states at zero energy for any finite curvature in $D-$dimensional hyperbolic space with $2\leq D \leq 4$, suggesting enhanced susceptibility of Dirac fermions to interaction-driven instabilities at weak coupling. We then derive explicit forms of discrete translational and rotational symmetries for lattices characterized by Schläfli symbols $\{p,q\}$, and explicitly construct the discrete spin connection, represented as hopping phases, via parallel transport. Our results pave the way for experimental realization of spin-curvature effects in metamaterial platforms and systematic numerical studies of correlated Dirac phases in hyperbolic geometries. |
| title | Symmetry-based theory of Dirac fermions on two-dimensional hyperbolic crystals: Coupling to the spin connection |
| topic | Mesoscale and Nanoscale Physics General Relativity and Quantum Cosmology High Energy Physics - Lattice High Energy Physics - Theory |
| url | https://arxiv.org/abs/2507.08276 |