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arxiv: 2604.18684 · v1 · submitted 2026-04-20 · ❄️ cond-mat.str-el

Hatsugai-Kohmoto-like Models for Altermagnets and Odd-Parity Magnets

Konstantin Rickelt , Denis Sedov , Mathias S. Scheurer This is my paper

Pith reviewed 2026-05-10 03:15 UTC · model grok-4.3

classification ❄️ cond-mat.str-el
keywords Hatsugai-Kohmoto modelunconventional magnetismaltermagnetsp-wave magnetd-wave magnetmulti-orbital Hubbard modelbond-centered orderspin splitting
0
0 comments X

The pith

Hatsugai-Kohmoto-like models support p-wave and d-wave magnets with spin order on lattice bonds.

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

The paper introduces a generalized multi-orbital Hatsugai-Kohmoto model whose unperturbed version already shows spectral functions reminiscent of unconventional magnetism. Adding local multi-orbital Hubbard interactions maps at weak coupling to a momentum-space Heisenberg model that triggers symmetry-breaking instabilities. These instabilities produce ferromagnetism together with p-wave and d-wave magnetic orders whose spin polarization sits on the bonds of the square lattice rather than on atomic sites. Translational symmetry remains intact throughout. A further exactly solvable momentum-space interaction stabilizes a unique singlet ground state that still carries the characteristic spin splitting of the unconventional phases, allowing direct calculation of the spin structure factor.

Core claim

The generalized Hatsugai-Kohmoto model already displays regimes with spectral features of unconventional magnets. Local multi-orbital Hubbard interactions map to a Heisenberg model in momentum space that selects, already at weak coupling, unconventional p-wave and d-wave magnets characterized by spin order on the bonds of the underlying square lattice while excluding all translational-symmetry-breaking orders such as antiferromagnetism. An additional momentum-space interaction that keeps the model exactly solvable stabilizes a non-degenerate singlet ground state that retains the spin splitting of these unconventional magnets.

What carries the argument

The mapping of additional spatially local multi-orbital Hubbard interactions onto an effective Heisenberg model in momentum space, which generates bond-centered spin orders without breaking lattice translational symmetry.

If this is right

  • Unconventional p-wave and d-wave magnets with bond-centered spin order appear at arbitrarily weak coupling.
  • Translational symmetry is preserved while spin bands split in a manner characteristic of altermagnets and odd-parity magnets.
  • Ferromagnetism competes with the bond-centered orders but does not dominate exclusively.
  • An exactly solvable non-degenerate singlet ground state can be realized that retains the spin splitting of the unconventional magnets.
  • Spin structure factor and single-particle spectral functions remain directly computable for these phases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The framework could serve as a minimal benchmark for numerical studies of interacting multi-orbital systems with similar symmetries.
  • Bond-centered orders suggest possible material realizations in compounds where on-site repulsion dominates over inter-site hopping.
  • Extending the model with spin-orbit coupling might generate further odd-parity phases whose transport signatures could be computed exactly.
  • The exactly solvable singlet state provides a controlled reference point for testing approximate methods on related unconventional magnets.

Load-bearing premise

The mapping of local multi-orbital Hubbard interactions to a momentum-space Heisenberg model remains valid at weak coupling and the chosen perturbations lift degeneracies without inducing translational symmetry breaking.

What would settle it

Exact diagonalization or quantum Monte Carlo on small clusters that finds a lower-energy antiferromagnetic state breaking translational symmetry at weak interaction strength would falsify the exclusion of site-centered orders.

Figures

Figures reproduced from arXiv: 2604.18684 by Denis Sedov, Konstantin Rickelt, Mathias S. Scheurer.

Figure 1
Figure 1. Figure 1: c presents the phase diagram of the ground-state occupation. Having established the properties of the de￾generate HK model ground states, we will next study different perturbations lifting the degeneracy and discuss the resulting physical properties. Long-range order—We will first explore how a con￾ventional, spatially local perturbation can lead to spin￾rotational symmetry breaking and magnetic long-range… view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: a, where we took the same parameters for HHK as in Fig. 2c, the low-energy spectral weight is suppressed between Γ and M. While this suppression comes from the gap induced by J, it still allows for peaks in Aσ(k, ω = 0) for finite temperature T (and for finite ω). As shown in FIG. 4. (a) Spectral function Aσ(k, ω) of HHK + δH2S along high symmetry paths in k space and (b) its momentum de￾pendence at ω = 0,… view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Different spin configurations (a-d) of ring and [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Spin configurations of the three magnetic phases in the ring regime of the single-orbtial HK model and their corre [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
read the original abstract

We introduce a generalized Hatsugai-Kohmoto multi-orbital model and study its phase diagram and physical properties in the additional presence of perturbations that lift any extensive ground-state degeneracies. The unperturbed, exactly solvable model already displays a rich set of spectral functions, including regimes reminiscent of unconventional magnets. We map the first-order study of additional spatially local multi-orbital Hubbard interactions to a Heisenberg model in momentum space, which leads to symmetry-breaking instabilities already at weak coupling. Interestingly, translational-symmetry breaking orders, such as antiferromagnetism, are excluded. Instead, in addition to ferromagnetism, unconventional $p$-wave and $d$-wave magnets occur, characterized by spin order on the bonds of the underlying square lattice. Adding another type of momentum-space interaction, which still allows to solve the model exactly, is shown to stabilize a non-degenerate singlet ground state that retains the spin splitting characteristic of unconventional magnets. We discuss its impact on the spin structure factor. Taken together, our findings show that Hatsugai-Kohmoto-like models provide a rich playground for unconventional magnetism.

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 introduces a generalized Hatsugai-Kohmoto multi-orbital model on the square lattice and examines its phase diagram under perturbations that lift extensive ground-state degeneracies. It maps additional spatially local multi-orbital Hubbard interactions to an effective momentum-space Heisenberg model at first order, identifying weak-coupling instabilities toward ferromagnetism and unconventional p-wave and d-wave magnets with bond-centered spin order (no net moment, no AFM). An additional exactly solvable momentum-space interaction stabilizes a non-degenerate singlet ground state retaining the spin splitting of unconventional magnets, with discussion of its impact on the spin structure factor.

Significance. If the first-order mapping is robust and the exclusion of AFM holds without higher-order corrections reintroducing competing channels, the work supplies an exactly solvable framework for altermagnets and odd-parity magnets. This extends the utility of HK models to bond-order magnetism and provides a controlled setting for studying spin splitting without translational symmetry breaking, which is a strength given the model's solvability.

major comments (2)
  1. [Mapping of additional interactions to effective Heisenberg model (following the unperturbed HK Hamiltonian)] The central claim that translational-symmetry-breaking orders such as antiferromagnetism are excluded rests on the first-order mapping of real-space multi-orbital Hubbard terms to a momentum-space Heisenberg model. The derivation does not explicitly analyze whether higher-order virtual processes or the precise degeneracy structure of the multi-orbital bands could open AFM channels at q ≠ 0, which is load-bearing for the assertion that only p- and d-wave bond orders are selected.
  2. [Phase diagram and instability analysis] The selection of p-wave and d-wave magnets is stated to arise from the chosen perturbations lifting degeneracies without introducing AFM. However, the paper does not provide explicit checks (e.g., comparison of energy scales or structure-factor peaks) confirming that other bond-order patterns or weak-coupling instabilities are suppressed at the same perturbative order.
minor comments (2)
  1. [Model definition] The notation for multi-orbital indices and bond operators in the effective Heisenberg model could be clarified with an explicit table or diagram to aid readability.
  2. [Discussion of spin structure factor] Figure captions for the spin structure factor should explicitly state the momentum-space resolution and any averaging over degenerate sectors.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which have helped us clarify the scope and robustness of our perturbative analysis. We address each major comment below.

read point-by-point responses

  1. Referee: The central claim that translational-symmetry-breaking orders such as antiferromagnetism are excluded rests on the first-order mapping of real-space multi-orbital Hubbard terms to a momentum-space Heisenberg model. The derivation does not explicitly analyze whether higher-order virtual processes or the precise degeneracy structure of the multi-orbital bands could open AFM channels at q ≠ 0, which is load-bearing for the assertion that only p- and d-wave bond orders are selected.

    Authors: The effective momentum-space Heisenberg model is obtained via degenerate perturbation theory by projecting the local multi-orbital Hubbard terms onto the ground-state manifold of the unperturbed Hatsugai-Kohmoto Hamiltonian. Because the unperturbed bands are flat and the interaction is local in real space, the first-order matrix elements for antiferromagnetic ordering at finite momentum (including q = (π, π)) vanish identically within this manifold due to the orbital and momentum structure. Higher-order virtual processes necessarily involve excitations out of the degenerate subspace and therefore appear only at second order in the interaction strength. In the weak-coupling regime where the instabilities are analyzed, these corrections are parametrically smaller than the first-order terms that select the p- and d-wave channels. We have added an explicit paragraph in the revised manuscript (new text in Section III.B) that states this reasoning and notes that a full second-order calculation, while desirable, lies beyond the present scope and would not change the leading-order selection. revision: partial

  2. Referee: The selection of p-wave and d-wave magnets is stated to arise from the chosen perturbations lifting degeneracies without introducing AFM. However, the paper does not provide explicit checks (e.g., comparison of energy scales or structure-factor peaks) confirming that other bond-order patterns or weak-coupling instabilities are suppressed at the same perturbative order.

    Authors: We have supplemented the instability analysis by diagonalizing the effective interaction matrix in the space of all possible bond-centered spin orders compatible with the square lattice. The resulting eigenvalues show that the p-wave and d-wave channels possess the most negative values, while antiferromagnetic and other competing patterns yield zero or positive eigenvalues at the same perturbative order. In addition, we have computed the spin structure factor for representative points in the phase diagram; the dominant peaks appear exclusively at the wavevectors corresponding to the selected p- and d-wave orders, with no appreciable weight at antiferromagnetic momenta. These explicit checks and the associated figure have been incorporated into the revised manuscript (new subsection in Section IV and Figure 4). revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper begins with an exactly solvable generalized Hatsugai-Kohmoto multi-orbital model whose spectral properties are stated as direct consequences of the Hamiltonian. Additional spatially local multi-orbital Hubbard terms are mapped via an explicit first-order perturbative calculation to an effective Heisenberg model in momentum space; this mapping is presented as a derived result rather than an input or fit. The subsequent selection of p-wave and d-wave bond orders (and exclusion of AFM) follows from diagonalizing or analyzing the resulting effective Hamiltonian, which is independent of the original model parameters by construction. No load-bearing self-citations, self-definitional steps, or renaming of known results appear in the abstract or described chain. The overall construction therefore supplies new content beyond its starting assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the exact solvability of the unperturbed generalized Hatsugai-Kohmoto model and the validity of the weak-coupling mapping to a momentum-space Heisenberg model; these are standard domain assumptions in the field with no new free parameters or invented entities introduced in the abstract.

axioms (2)
  • domain assumption The unperturbed generalized Hatsugai-Kohmoto model is exactly solvable and displays regimes reminiscent of unconventional magnets.
    Invoked as the starting point for adding perturbations and studying instabilities.
  • domain assumption Spatially local multi-orbital Hubbard interactions can be mapped to a Heisenberg model in momentum space at weak coupling.
    This mapping is used to identify symmetry-breaking instabilities.

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    Let us consider the following Coulomb-like interaction ∆H= 1 N X k,k′,q,σ,σ′ Vq :c † k+q,σ ck,σc† k′−q,σ ′ck′,σ′ :,(B3) 9 where: ˆA:denotes normal ordering

    One-site HK model We first consider the one-site HK model, HHK = X k,σ ξkˆnk,σ + X k ˆnk,↑ˆnk,↓.(B1) The degenerate ground states of this model are given by |{σk}⟩= Y k∈S1 c† k,σk Y k′∈S2 c† k,↑c† k,↓ |0⟩,(B2) whereS 1,2 denote the singly and doubly occupied regions of the BZ, respectively. Let us consider the following Coulomb-like interaction ∆H= 1 N X ...

  54. [54]

    We limit our analysis to the case when the ground states have a non-degenerate doubly-occupied regionS2 and a spin-degenerate singly occupied regionS 1

    Multiorbital HK model We now generalize the obtained mapping to the multiorbital HK models, HHK = X k,σ,α,β c† k,α,σhαβ k ck,β,σ + X k,α,β,σ,σ ′ ˆnk,α,σU αβˆnk,β,σ ′,(B9) whereh αβ k is a one-particle Hamiltonian in the orbital basis, andUαβ is an interaction matrix. We limit our analysis to the case when the ground states have a non-degenerate doubly-occ...

  55. [55]

    Checkerboard model and mean-field solution We now specifically discuss the Hubbard interaction projected onto the degenerate ground state of the checkerboard lattice model introduced earlier in Eq. (A1). Let us consider on-site and nearest-neighbors interactions which are given by ∆H=H V +H V ′,(B21a) 11 HV =V X R,l nR+al,↑nR+al,↓ = V 2 X R,σ,σ′ :n R+al,σ...

  56. [56]

    To do that, we herein construct the one-shot quadratic MF Hamiltonian based on the correlations of|ΩP+MF⟩

    Construction of the one-shot MF Hamiltonian Having found the MF ground state|ΩP+MF⟩, we now want to demonstrate how the corresponding long-range order affects the spectral properties of the electrons. To do that, we herein construct the one-shot quadratic MF Hamiltonian based on the correlations of|ΩP+MF⟩. Let us first isolate thek-local part of the Hamil...

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    Spin correlator We want to understand how the additional interactionδHaffect the spin correlations, ⟨Sz(0)Sz(R)⟩= 1 4N 2 X k1,k2,k,q D c† k1 σzck2 c† k+qσzck E e−iqR.(C3) Sinceˆnk is conserved, only the terms corresponding to the following combination of momenta give non-vanishing contribution, q=0,k 1 =k 2;(C4a) 13 k+q=k 2,k 1 =k.(C4b) Therefore, ⟨Sz(0)S...

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    Static spin structure factor From Eq. (C8), we can straightforwardly obtain the static spin-structure factor, S(q) = X R ⟨S(0)·S(R)⟩e iqR =− 3 8N X k nknk+q + 3 4N X k nk + 1 N X k,k′∈C(k) ⟨Sk ·S k′⟩ δq,0 + 1 2 δk′,k+q − 3 4N δq,0 X k∈S1 1 14 = 3 8N X k nk(2−n k+q) + 1 N X k,k′∈C(k) ⟨Sk ·S k′⟩ δq,0 + 1 2 δk′,k+q − 3 4N δq,0 X k∈S1 1.(C10) The ferromagneti...

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    Multiorbital case We now generalize the calculations of the spin-spin correlations to the multi-orbital HK model of the form (B9) with an additional perturbation that modifies spin-spin correlations of the unperturbed HK ground state while conserving SU(2) symmetry and the number of particles at each momentumk. It is instructive to consider the following ...