Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2407.09601 · Physical Review Letters

Hyperbolic Spin Liquids

Patrick M. Lenggenhager, Santanu Dey, Tomáš Bzdušek, Joseph Maciejko

Published Jul 7, 2025• 13 citations
arXivHTMLPDF

Abstract

Hyperbolic lattices present a unique opportunity to venture beyond the conventional paradigm of crystalline many-body physics and explore correlated phenomena in negatively curved space. As a theoretical benchmark for such investigations, we extend Kitaev's spin-1/2 honeycomb model to hyperbolic lattices and exploit their non-Euclidean space-group symmetries to solve the model exactly. We elucidate the ground-state phase diagram on the {8,3} lattice and find a gapped Z_{2} spin liquid with Abelian anyons, a gapped chiral spin liquid with non-Abelian anyons and chiral edge states, and a Majorana metal whose finite low-energy density of states is dominated by non-Abelian Bloch states.

DOI
10.1103/s25y-s4fj

Related papers

Hyperbolic Spin Liquids

Patrick M. Lenggenhager, Santanu Dey, Tomáš Bzdušek +1 more

Hyperbolic lattices present a unique opportunity to venture beyond the conventional paradigm of crystalline many-body physics and explore correlated phenomena in negatively curved space. As a theoretical benchmark for such investigations, we extend Kitaev’s spin-1/2 honeycomb model to hyperbolic lattices and exploit their non-Euclidean space-group symmetries to solve the model exactly. We elucidate the ground-state phase diagram on the {8,3} lattice and find a gapped Z2 spin liquid with Abelian anyons, a gapped chiral spin liquid with non-Abelian anyons and chiral edge states, and a Majorana metal whose finite low-energy density of states is dominated by non-Abelian Bloch states.

•Aug 15, 2025•0 citations

Chiral Gapless Spin Liquid in Hyperbolic Space

Felix Dusel, Tobias Hofmann, Atanu Maity +5 more

We analyze the Kitaev model on the {9,3} hyperbolic lattice. The {9,3} is formed by a regular tricoordinated tiling of nonagons, where the three-color coding of bonds according to the inequivalent Kitaev Ising spin couplings yields the natural generalization of the original Kitaev model for Euclidean regular honeycomb tiling. Upon investigation of the bulk spectrum for large finite size droplets, we identify a gapless chiral Z_{2} spin liquid state featuring spontaneous time-reversal symmetry breaking. Because of its noncommutative translation group structure, such type of hyperbolic spin liquid is conjectured to feature chiral quasiparticles with a potentially non-Abelian Bloch profile.

arXiv:2407.15705•Physical Review Letters•Jun 26, 2025•12 citations

Exact chiral-spin-liquid state in a Kitaev-type spin model

Jianlong Fu

We study a frustrated two-dimensional lattice spin model with Kitaev-type interaction. The lattice is obtained from the honeycomb lattice by replacing half of its sites with triangles. Using the SO(3) Majorana representation of spin, the model is exactly mapped into a ${Z}_{2}$ lattice gauge theory of complex fermions with standard Gauss law constraint. We show that the ground state of the model is a chiral spin liquid, and it has gapless excitations. The Chern number of the ground state is $\ifmmode\pm\else\textpm\fi{}1$, implying the existence of a chiral edge mode. The vortex excitation of the model is bound with a Majorana zero mode, such mode behaves as non-Abelian anyons.

arXiv:1908.03123•Physical review. B./Physical review. B•Nov 19, 2019•12 citations

Lieb-Schultz-Mattis constraints for hyperbolic lattices

G. Shankar, Joseph Maciejko

The Lieb-Schultz-Mattis (LSM) theorem and its higher-dimensional extensions forbid the existence of a unique, symmetric, and gapped ground state at fractional fillings in quantum many-body systems with a conserved particle number (or spin angular momentum) and the conventional translation symmetry of Euclidean lattices. In this work, we propose a generalization of the LSM theorem to quantum many-body systems on hyperbolic lattices, i.e., regular tessellations of two-dimensional negatively curved space. By leveraging concepts from hyperbolic band theory in a many-body setting, we adapt Oshikawa's flux-threading argument to periodic hyperbolic lattices with a non-Euclidean (Fuchsian) translation symmetry and compute a lower-bound to the ground-state degeneracy as a function of filling and lattice geometry. We explore the consequences of LSM constraints for gapped phases of hyperbolic quantum matter and suggest frustrated spin models on hyperbolic analogs of the square and triangular lattices as promising platforms for realizing symmetric spin liquids in hyperbolic space.

cond-mat.str-elCross-listedarXiv:2605.15974•Physical review. B./Physical review. B•May 18, 2026•0 citations

Kitaev model on Hurwitz hyperbolic tilings: A non-Abelian gapped chiral spin liquid

Rémy Mosseri, Yasir Iqbal, Roger Vogeler +1 more

We study the Kitaev model on the trivalent heptagonal Hurwitz hyperbolic tiling. The presence of odd-length loops in the tiling is responsible for a spontaneous time-reversal symmetry breaking. Interestingly, at the isotropic point, the two degenerate ground states (Kramers pair) are shown to be gapped chiral spin liquids and the elementary excitations are non-Abelian Ising anyons.

arXiv:2411.03944•Physical review. B./Physical review. B•Feb 25, 2025•10 citations

Exact Chiral Spin Liquid with Non-Abelian Anyons

Hong Yao, Steven Allan Kivelson

We establish the existence of a chiral spin liquid (CSL) as the exact ground state of the Kitaev model on a decorated honeycomb lattice, which is obtained by replacing each site in the familiar honeycomb lattice with a triangle. This CSL state spontaneously breaks time reversal symmetry but preserves other symmetries. There are two topologically distinct CSL's separated by a quantum critical point. Interestingly, vortex excitations in the topologically nontrivial (Chern number +/-1) CSL obey non-Abelian statistics.

arXiv:0708.0040•Physical Review Letters•Dec 12, 2007•362 citations
Hyperbolic lattices present a unique opportunity to venture beyond the conventional paradigm of crystalline many-body physics and explore correlated phenomena in negatively curved space. As a theoretical benchmark for such investigations, we extend Kitaev's spin-1/2 honeycomb model to hyperbolic lattices and exploit their non-Euclidean space-group symmetries to solve the model exactly. We elucidate the ground-state phase diagram on the {8,3} lattice and find a gapped Z_{2} spin liquid with Abelian anyons, a gapped chiral spin liquid with non-Abelian anyons and chiral edge states, and a Majorana metal whose finite low-energy density of states is dominated by non-Abelian Bloch states.
Hyperbolic lattices present a unique opportunity to venture beyond the conventional paradigm of crystalline many-body physics and explore correlated phenomena in negatively curved space. As a theoretical benchmark for such investigations, we extend Kitaev’s spin-1/2 honeycomb model to hyperbolic lattices and exploit their non-Euclidean space-group symmetries to solve the model exactly. We elucidate the ground-state phase diagram on the {8,3} lattice and find a gapped Z2 spin liquid with Abelian anyons, a gapped chiral spin liquid with non-Abelian anyons and chiral edge states, and a Majorana metal whose finite low-energy density of states is dominated by non-Abelian Bloch states.
We analyze the Kitaev model on the {9,3} hyperbolic lattice. The {9,3} is formed by a regular tricoordinated tiling of nonagons, where the three-color coding of bonds according to the inequivalent Kitaev Ising spin couplings yields the natural generalization of the original Kitaev model for Euclidean regular honeycomb tiling. Upon investigation of the bulk spectrum for large finite size droplets, we identify a gapless chiral Z_{2} spin liquid state featuring spontaneous time-reversal symmetry breaking. Because of its noncommutative translation group structure, such type of hyperbolic spin liquid is conjectured to feature chiral quasiparticles with a potentially non-Abelian Bloch profile.
We study a frustrated two-dimensional lattice spin model with Kitaev-type interaction. The lattice is obtained from the honeycomb lattice by replacing half of its sites with triangles. Using the SO(3) Majorana representation of spin, the model is exactly mapped into a Z2{Z}_{2}Z2​ lattice gauge theory of complex fermions with standard Gauss law constraint. We show that the ground state of the model is a chiral spin liquid, and it has gapless excitations. The Chern number of the ground state is $\ifmmode\pm\else\textpm\fi{}1$, implying the existence of a chiral edge mode. The vortex excitation of the model is bound with a Majorana zero mode, such mode behaves as non-Abelian anyons.
The Lieb-Schultz-Mattis (LSM) theorem and its higher-dimensional extensions forbid the existence of a unique, symmetric, and gapped ground state at fractional fillings in quantum many-body systems with a conserved particle number (or spin angular momentum) and the conventional translation symmetry of Euclidean lattices. In this work, we propose a generalization of the LSM theorem to quantum many-body systems on hyperbolic lattices, i.e., regular tessellations of two-dimensional negatively curved space. By leveraging concepts from hyperbolic band theory in a many-body setting, we adapt Oshikawa's flux-threading argument to periodic hyperbolic lattices with a non-Euclidean (Fuchsian) translation symmetry and compute a lower-bound to the ground-state degeneracy as a function of filling and lattice geometry. We explore the consequences of LSM constraints for gapped phases of hyperbolic quantum matter and suggest frustrated spin models on hyperbolic analogs of the square and triangular lattices as promising platforms for realizing symmetric spin liquids in hyperbolic space.
We establish the existence of a chiral spin liquid (CSL) as the exact ground state of the Kitaev model on a decorated honeycomb lattice, which is obtained by replacing each site in the familiar honeycomb lattice with a triangle. This CSL state spontaneously breaks time reversal symmetry but preserves other symmetries. There are two topologically distinct CSL's separated by a quantum critical point. Interestingly, vortex excitations in the topologically nontrivial (Chern number +/-1) CSL obey non-Abelian statistics.