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arxiv: 2606.01404 · v1 · pith:INAV6LBEnew · submitted 2026-05-31 · ❄️ cond-mat.mes-hall · cond-mat.mtrl-sci

Quantized orbital and spin Hall transport: interplay between sp-hybridization, altermagnetism and spin-orbit coupling

Saikat Saha , Banasree Sadhukhan , Tanay Nag This is my paper

Pith reviewed 2026-06-28 16:09 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.mtrl-sci
keywords orbital Hall effectspin Hall effectaltermagnetismRashba spin-orbit couplingsp-hybridizationBerry curvaturequantized transportNernst effect
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The pith

Orbital Hall conductivity quantizes when altermagnetic order exceeds sp-hybridization, while spin Hall quantizes when Rashba SOC falls below it, in a 2D model.

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

The paper examines how orbital and spin Hall effects arise in a two-dimensional model that includes inter-orbit sp-hybridization, relativistic Rashba spin-orbit coupling, and non-relativistic altermagnetic order. It reports that the orbital Hall response becomes quantized over a window of Fermi energies once altermagnetic order strength surpasses the hybridization scale, while the spin Hall response quantizes when Rashba strength lies below that scale. These quantized regimes remain independent of the other interaction term, trace back to Fermi-surface contributions in the orbital and spin Berry curvatures, and shrink or degrade with stronger intra-orbital coupling or rising temperature. The same energy windows produce clear dip-peak features in the orbital and spin Nernst coefficients.

Core claim

In the two-dimensional model Hamiltonian, the orbital Hall conductivity is quantized inside a Fermi-energy window whenever altermagnetic order exceeds sp-hybridization, while the spin Hall conductivity is quantized when Rashba SOC is weaker than sp-hybridization; the orbital quantization is independent of Rashba SOC and the spin quantization is independent of altermagnetic order, both originating from uniform finite orbital moments and vanishingly small spin moments of bands near the Fermi level that activate orbital and spin Berry curvatures at the Fermi surface.

What carries the argument

Two-dimensional model Hamiltonian with sp-hybridization, Rashba SOC, and altermagnetic order, whose orbital and spin Berry curvatures are evaluated to obtain the Hall conductivities.

If this is right

  • The width of each quantized window is set by the intra-orbital coupling strength.
  • Rising temperature degrades the spin-Hall quantization more rapidly than the orbital-Hall quantization.
  • Dip-peak structures appear in both orbital and spin Nernst coefficients exactly where the Hall conductivities are quantized.
  • The orbital-Hall quantization survives changes in Rashba SOC while the spin-Hall quantization survives changes in altermagnetic order.

Where Pith is reading between the lines

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

  • Similar quantization windows may appear in three-dimensional layered materials once the same hierarchy of energy scales is realized.
  • The independence of the two quantizations suggests separate experimental knobs for orbital versus spin responses in altermagnetic devices.
  • The Nernst signatures could be used to map the quantized windows without requiring direct Hall-bar measurements at low temperature.

Load-bearing premise

The idealized two-dimensional Hamiltonian and its Berry-curvature integrals capture the quantization windows without extra contributions from disorder, lattice effects, or higher-order terms.

What would settle it

Measured orbital or spin Hall conductivity that fails to show flat quantized plateaus inside the predicted Fermi-energy windows when altermagnetic order or Rashba SOC is tuned across the hybridization scale.

Figures

Figures reproduced from arXiv: 2606.01404 by Banasree Sadhukhan, Saikat Saha, Tanay Nag.

Figure 1
Figure 1. Figure 1: FIG. 1. Band Structure (a) with the distribution of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Variation of orbital Hall (OH) conductivities with [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Variation of spin Hall (SH) conductivities with Fermi [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Variation of SH conductivity with [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
read the original abstract

We here explore the emergence of orbital and spin Hall effects, originating beyond the $L$-$S$ coupling, and investigate the interplay between inter-orbit hybridization, relativistic Rashba spin-orbit coupling (SOC), and non-relativistic SOC, namely altermagnetic (AM) order, in a two-dimensional model Hamiltonian. The orbital (spin) Hall responses are remarkably found to be quantized within a window of Fermi energy when the strength of AM order (Rashba SOC) exceeds (falls below) the scale set by $sp$-hybridization. Importantly, orbital and spin Hall quantizations are independent of Rashba SOC and AM order, respectively, while the uniform profiles of finite orbital and vanishingly small spin moments of bands around the Fermi energy. The microscopic origin of such quantization comes from the Fermi surface-activated orbital and spin Berry curvatures. The extent of the quantized regime is strongly controlled by the intra-orbital coupling strength. As the temperature increases, the quantization is significantly compromised in the spin Hall case. We extend our analysis to the orbital and spin Nernst coefficients where the pronounced dip-peak structures signal the existence of the quantization leading to experimental relevance.

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 two-dimensional tight-binding model incorporating sp-hybridization, altermagnetic (AM) order, and Rashba SOC to investigate orbital and spin Hall conductivities. It reports that the orbital Hall response is quantized over a Fermi-energy window when AM strength exceeds the sp-hybridization scale, while the spin Hall response is quantized when Rashba SOC falls below that scale; the two quantizations are independent of Rashba and AM, respectively. The origin is attributed to Fermi-surface Berry curvatures, with the window size controlled by intra-orbital coupling; temperature degrades the spin-Hall plateau more than the orbital one. The analysis is extended to orbital and spin Nernst coefficients, which exhibit dip-peak features signaling the quantized regimes.

Significance. If the reported quantization windows are robust within the model, the work identifies a parameter-controlled route to quantized orbital and spin transport that does not rely on conventional band topology or external fields, potentially relevant for altermagnetic heterostructures and orbitronics. The independence properties and Nernst signatures supply concrete, falsifiable predictions that could guide experiments once material parameters are mapped onto the model.

major comments (2)
  1. [Model Hamiltonian and Results] The central quantization claims rest on the clean 2D Hamiltonian plus Berry-curvature integrals alone. No analysis is supplied showing that the reported windows survive disorder broadening, next-nearest-neighbor hopping, or weak three-dimensional coupling, all of which are expected to be present in candidate materials and could close or smear the plateaus (Model Hamiltonian and Results sections).
  2. [Abstract and Discussion] The abstract states that quantization occurs 'when the strength of AM order (Rashba SOC) exceeds (falls below) the scale set by sp-hybridization,' yet the manuscript supplies no explicit mapping of these relative scales onto independently measurable quantities (e.g., ARPES gaps or transport coefficients), leaving the physical accessibility of the windows unverified.
minor comments (2)
  1. [Abstract] The final sentence of the abstract is grammatically incomplete ('while the uniform profiles of finite orbital and vanishingly small spin moments of bands around the Fermi energy').
  2. [Model Hamiltonian] Notation for the intra-orbital coupling strength is introduced without a clear symbol or equation reference in the early sections, complicating cross-referencing with the parameter scans.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We provide point-by-point responses to the major comments below.

read point-by-point responses

  1. Referee: [Model Hamiltonian and Results] The central quantization claims rest on the clean 2D Hamiltonian plus Berry-curvature integrals alone. No analysis is supplied showing that the reported windows survive disorder broadening, next-nearest-neighbor hopping, or weak three-dimensional coupling, all of which are expected to be present in candidate materials and could close or smear the plateaus (Model Hamiltonian and Results sections).

    Authors: We agree that the quantization results are obtained within the clean two-dimensional tight-binding model using Berry-curvature integrals over the Fermi surface. The manuscript does not include calculations incorporating disorder broadening, additional next-nearest-neighbor hoppings, or weak interlayer coupling. Such extensions would require substantial new numerical work outside the present scope. In the revised manuscript we will add an explicit limitations paragraph in the discussion section noting that the reported plateaus are demonstrated in the idealized limit and may be most accessible in high-quality 2D samples or heterostructures. revision: partial

  2. Referee: [Abstract and Discussion] The abstract states that quantization occurs 'when the strength of AM order (Rashba SOC) exceeds (falls below) the scale set by sp-hybridization,' yet the manuscript supplies no explicit mapping of these relative scales onto independently measurable quantities (e.g., ARPES gaps or transport coefficients), leaving the physical accessibility of the windows unverified.

    Authors: The conditions are expressed in terms of the effective model parameters (AM strength, Rashba SOC, and sp-hybridization) to isolate the interplay that produces the independent quantization windows. The manuscript does not provide a direct translation of these scales into specific experimental observables such as ARPES gap values. We will revise the abstract and add a short paragraph in the discussion clarifying that the parameters are effective couplings whose relative magnitudes can be estimated from material-specific band-structure calculations, thereby indicating how the windows might be accessed experimentally. revision: yes

Circularity Check

0 steps flagged

No circularity: results are direct model computations

full rationale

The paper constructs a 2D model Hamiltonian incorporating sp-hybridization, altermagnetic order, and Rashba SOC, then computes orbital and spin Hall conductivities via Berry curvature integrals over the Fermi surface. Quantization windows emerge as numerical outcomes when AM exceeds hybridization (or Rashba falls below it), with stated independence properties following from the band structure. No quoted step reduces a claimed prediction to a fitted parameter by construction, nor does any load-bearing premise rest on self-citation chains or smuggled ansatzes. The derivation is self-contained within the model's equations and parameter choices; external validity concerns (disorder, lattice effects) are separate from circularity.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of a 2D model Hamiltonian whose parameters (hybridization scale, intra-orbital coupling) set the quantization windows, plus the domain assumption that Berry curvature at the Fermi surface fully accounts for the Hall responses.

free parameters (2)
  • sp-hybridization strength
    Defines the energy scale that AM order and Rashba SOC are compared against to enter the quantized regime.
  • intra-orbital coupling strength
    Controls the width of the quantized Fermi-energy window.
axioms (1)
  • domain assumption Berry curvature evaluated at the Fermi surface determines the orbital and spin Hall conductivities
    Explicitly stated as the microscopic origin of the quantization.

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