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arxiv: 2605.10602 · v1 · submitted 2026-05-11 · ❄️ cond-mat.mes-hall · cond-mat.str-el· hep-th

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Inherent Altermagnetism on regular hyperbolic lattices

Eric Petermann , Kristian M{\ae}land , Haye Hinrichsen , Bj\"orn Trauzettel
Authors on Pith no claims yet

Pith reviewed 2026-05-12 04:24 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.str-elhep-th
keywords altermagnetismhyperbolic latticestight-binding modelspin splittinghyperbolic band theoryPoincaré diskantiferromagnetismmomentum space
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The pith

Next-nearest-neighbor hopping induces altermagnetism in certain hyperbolic lattices.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper aims to establish that altermagnetism occurs inherently in some regular hyperbolic lattices. It extends Euclidean tight-binding models of altermagnets to lattices on a discretized Poincaré disk and shows that next-nearest-neighbor hopping produces momentum-dependent spin splitting without net magnetization in bipartite cases. Certain families and one special case display this splitting while others remain antiferromagnetic. The result matters because hyperbolic lattices can be engineered in artificial systems, offering new platforms for controlling electron spins. Hyperbolic band theory supplies a momentum space of at least four dimensions, allowing the spin-splitting harmonics to be classified with four-dimensional atomic orbitals.

Core claim

By including next-nearest neighbor hopping in tight-binding models on regular hyperbolic lattices defined on a discretized Poincaré disk, we show that spin splitting is induced in an entire family and a special case of bipartite lattices. Hyperbolic crystallography and band theory establish that altermagnetism is therefore inherent to certain hyperbolic lattices. The at least four-dimensional momentum space permits classification of the leading spin-splitting harmonics using four-dimensional atomic orbitals.

What carries the argument

Tight-binding model on regular hyperbolic lattices with next-nearest-neighbor hopping, analyzed through hyperbolic band theory

Load-bearing premise

The discretized Poincaré-disk tight-binding model with next-nearest-neighbor hopping accurately captures the low-energy electronic structure of real or realizable hyperbolic lattices.

What would settle it

Fabricating a hyperbolic lattice and measuring its band structure to find no momentum-dependent spin splitting when next-nearest-neighbor hopping is present would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.10602 by Bj\"orn Trauzettel, Eric Petermann, Haye Hinrichsen, Kristian M{\ae}land.

Figure 1
Figure 1. Figure 1: The resulting lattice can be seen as a graph whose vertices correspond to atomic sites and whose edges repre￾sent nearest-neighbor connections. As we are interested in the electronic structure of these lattices, we require a periodic de￾scription of the system, which allows us to probe momentum space via Fourier transformation. To this end, we work with a finite repeating patch that serves as a fundamental… view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Examples of antiferromagnetic bipartite hyperbolic lattices (black) and their Schwarz triangles (gray/white). Atoms in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: (a) Bipartite [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: , it is possible to identify the symmetries that yield cer￾tain nodal lines. This is shown in Table II. We find that the nodal lines in the (k1, k3) and (k2, k4) subspaces are not en￾forced by an underlying symmetry, but instead are due to these specific choices of cuts and thus accidental nodal lines [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: (a) Bipartite exceptional [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Example of a hyperbolic [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
read the original abstract

Altermagnets are a novel class of magnetic systems characterized by their momentum-dependent spin splitting without net magnetization. In this work, we extend established Euclidean tight-binding models of altermagnets to regular hyperbolic lattices in two spatial dimensions defined on a discretized Poincar\'e disk. Using hyperbolic crystallography and hyperbolic band theory, we show that the inclusion of next-nearest neighbor hopping is sufficient to induce spin splitting in bipartite hyperbolic lattices. While certain families and special cases of hyperbolic lattices remain antiferromagnetic, we identify an entire family and a special case that show spin splitting in this framework. Hence, altermagnetism is inherent to certain hyperbolic lattices. Since hyperbolic band theory yields a momentum space that is at least four-dimensional, we classify the leading spin-splitting harmonics using four-dimensional atomic orbitals.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript extends Euclidean tight-binding models of altermagnets to regular hyperbolic lattices defined on a discretized Poincaré disk. Using hyperbolic crystallography and band theory, it shows that next-nearest-neighbor hopping induces momentum-dependent spin splitting (with zero net magnetization) in bipartite hyperbolic lattices for an entire family and one special case, while other families remain antiferromagnetic. The four-dimensional momentum space native to hyperbolic geometry is then used to classify the leading spin-splitting harmonics via four-dimensional atomic orbitals. The central claim is that altermagnetism is inherent to certain hyperbolic lattices.

Significance. If the explicit band-structure calculations and symmetry analysis hold, the work provides a concrete realization of altermagnetism in non-Euclidean geometry. This leverages the higher-dimensional momentum space of hyperbolic lattices to produce spin-splitting harmonics that have no direct Euclidean analog, potentially enabling new platforms for momentum-space spintronics in artificial or metamaterial systems. The systematic classification of 4D harmonics is a useful technical contribution.

major comments (2)
  1. [§3] §3 (or equivalent model-construction section): the claim that next-nearest-neighbor hopping alone is sufficient to produce the reported spin splitting rests on the specific choice of hopping amplitude and the bipartite nature of the lattice; the manuscript should explicitly demonstrate that the splitting vanishes identically when this term is removed, to confirm it is not an artifact of the chosen parameter value.
  2. [§4] §4 (harmonic classification): the mapping of the leading spin-splitting terms onto four-dimensional atomic orbitals is presented as complete, but the manuscript does not show the explicit decomposition of the 4D harmonics or verify orthogonality with respect to the hyperbolic Brillouin zone; this step is load-bearing for the claim that the classification is exhaustive.
minor comments (2)
  1. Figure captions for the band-structure plots should state the numerical value of the next-nearest-neighbor hopping amplitude used and whether it is in units of the nearest-neighbor hopping.
  2. The abstract states that 'certain families and special cases remain antiferromagnetic'; the main text should tabulate which specific hyperbolic lattices (by Schläfli symbol or vertex figure) fall into each category.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive evaluation of our work and for the constructive comments, which we address point by point below. We have prepared a revised manuscript incorporating the suggested clarifications.

read point-by-point responses

  1. Referee: [§3] §3 (or equivalent model-construction section): the claim that next-nearest-neighbor hopping alone is sufficient to produce the reported spin splitting rests on the specific choice of hopping amplitude and the bipartite nature of the lattice; the manuscript should explicitly demonstrate that the splitting vanishes identically when this term is removed, to confirm it is not an artifact of the chosen parameter value.

    Authors: We agree that an explicit demonstration strengthens the claim. In the revised manuscript we will add a direct comparison of the spin-resolved band structures for the relevant lattices (both the identified family and the special case) computed with the next-nearest-neighbor hopping amplitude set identically to zero. These calculations confirm that the momentum-dependent spin splitting vanishes throughout the hyperbolic Brillouin zone, leaving only conventional antiferromagnetic degeneracy. The new panels will be placed in §3 immediately following the finite-hopping results, together with a brief statement that the splitting is induced solely by the next-nearest-neighbor term for any nonzero amplitude. revision: yes

  2. Referee: [§4] §4 (harmonic classification): the mapping of the leading spin-splitting terms onto four-dimensional atomic orbitals is presented as complete, but the manuscript does not show the explicit decomposition of the 4D harmonics or verify orthogonality with respect to the hyperbolic Brillouin zone; this step is load-bearing for the claim that the classification is exhaustive.

    Authors: We acknowledge that the explicit decomposition and orthogonality check were omitted for brevity. In the revised version we will expand §4 to include (i) the explicit linear combinations expressing the leading spin-splitting harmonics in the four-dimensional atomic-orbital basis and (ii) a verification that these harmonics are mutually orthogonal when integrated over the hyperbolic Brillouin zone (computed via the appropriate inner product on the discretized Poincaré disk). These additions will be presented as a short table and accompanying paragraph, confirming that the classification is exhaustive within the symmetry-allowed channels. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs a tight-binding Hamiltonian on regular hyperbolic lattices via hyperbolic crystallography and band theory, then demonstrates that next-nearest-neighbor hopping produces momentum-dependent spin splitting (zero net magnetization) for specific families. This follows directly from symmetry-allowed terms in the model and explicit diagonalization in the native higher-dimensional momentum space; no parameter is fitted to the target spin-splitting data and then re-labeled as a prediction, no self-definitional loop appears in the altermagnetism definition, and load-bearing steps rely on standard Euclidean altermagnet precedents plus independent hyperbolic geometry results rather than self-citation chains. The derivation remains self-contained and externally falsifiable.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the applicability of hyperbolic band theory to discretized Poincaré-disk lattices and on the assumption that a simple tight-binding extension captures the essential physics.

free parameters (1)
  • next-nearest-neighbor hopping amplitude
    Its inclusion is stated to be sufficient to induce splitting; no explicit value or fitting procedure is given in the abstract.
axioms (2)
  • domain assumption Hyperbolic band theory correctly describes the momentum-space structure of regular hyperbolic lattices on the discretized Poincaré disk
    Invoked to obtain at least four-dimensional momentum space and to classify spin-splitting harmonics.
  • domain assumption Bipartite hyperbolic lattices admit a well-defined tight-binding description with nearest- and next-nearest-neighbor terms
    Underlying the statement that NNN hopping induces spin splitting while preserving antiferromagnetic character in some families.

pith-pipeline@v0.9.0 · 5448 in / 1346 out tokens · 40273 ms · 2026-05-12T04:24:37.923958+00:00 · methodology

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Reference graph

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