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Opto- and magneto-tunable exceptional degeneracies in non-Hermitian ferromagnet/$p$-wave magnet junctions

T0 review · 2 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Light and magnetic fields steer exceptional points in p-wave magnet junctions.

desk verdict Analytically solid model of EP tuning in FM/UPM junctions, with the main caveat being the wide-band self-energy assumption that underpins all quantitative predictions. read the letter →

arxiv 2508.01295 v1 pith:XLCSJQXW submitted 2025-08-02 cond-mat.mes-hall

Mohammad Alipourzadeh , Davood Afshar , Yaser Hajati This is my paper · ORCID
classification cond-mat.mes-hall
keywords exceptionalpointsnon-Hermitianjunctionsp-wavemagnetsRashbaspin-orbitcouplingFloquetengineeringcircularlypolarizedlighttunabledegeneraciesspintronics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that a ferromagnet/$p$-wave-magnet junction opened to a semi-infinite ferromagnetic lead is a non-Hermitian system whose degeneracies are exceptional points: at two discrete momenta both eigenvalues and eigenvectors coalesce. The two points' separation is fixed by the lead-induced spin asymmetry $\gamma$ and the Rashba strength $\lambda$, while the line connecting them is oriented by the $p$-wave magnet's magnetization vector $\boldsymbol{\alpha}$. Because the $p$-wave coupling is linear in momentum, the degeneracies form a reorientable two-point line rather than the four-point patterns of $d$-wave altermagnets. The paper further claims that a magnetic field and off-resonant circularly polarized light can shift, tilt, merge, or annihilate these points by distinct mechanisms, which would make such junctions reconfigurable platforms for non-Hermitian spintronics.

What carries the argument

The load-bearing object is the complex vector $\mathbf{p}$ in the Pauli decomposition $H=\epsilon_0+\mathbf{p}\cdot\boldsymbol{\sigma}$, whose real and imaginary parts must satisfy the simultaneous conditions $\mathbf{p}_r^2=\mathbf{p}_i^2$ and $\mathbf{p}_r\cdot\mathbf{p}_i=0$ for an exceptional point. In this junction $\mathbf{p}_r=(\lambda k_y,-\lambda k_x,2t\,\boldsymbol{\alpha}\cdot\mathbf{k})$ and $\mathbf{p}_i=(0,0,-\gamma)$, so the imaginary part encodes the spin-asymmetric coupling to the ferromagnetic lead while the real part mixes Rashba spin-momentum locking with the linear $p$-wave spin splitting. The two conditions convert a would-be exceptional ring $\lambda^2(k_x^2+k_y^2)=\gamma^2$ into two discrete points whose connecting line rotates with $\boldsymbol{\alpha}$. The same decomposition carries the tuning analysis: magnetic fields add $\mathbf{B}\cdot\boldsymbol{\sigma}$ to selected components, and circularly polarized light, entering through the high-frequency van Vleck correction, renormalizes the Rashba and $\sigma_z$ coefficients.

What would settle it

Compute the lead self-energy from a realistic tight-binding ferromagnet instead of the wide-band form, and check whether the discriminant $(\mathbf{p}_r^2-\mathbf{p}_i^2)^2+4(\mathbf{p}_r\cdot\mathbf{p}_i)^2$ still vanishes at the momenta of Eq. (11) and at the claimed magnetic and optical thresholds; any shift or splitting of the zeros would falsify the quantitative prediction. A spectral probe such as angle-resolved photoemission should also show simultaneous coalescence of real and imaginary parts of the two branches at those momenta, not merely a band touching.

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Extended reading notes

Core claim

The central claim is that the effective $2\times2$ Hamiltonian $H = \epsilon_0 + \mathbf{p}\cdot\boldsymbol{\sigma}$ with $\mathbf{p}_r = (\lambda k_y,\,-\lambda k_x,\,2t\,\boldsymbol{\alpha}\cdot\mathbf{k})$ and $\mathbf{p}_i = (0,0,-\gamma)$ develops exceptional points exactly where $\mathbf{p}_r^2 = \mathbf{p}_i^2$ and $\mathbf{p}_r\cdot\mathbf{p}_i = 0$. These two conditions force $2t(\alpha_x k_x + \alpha_y k_y)=0$ and place the degenerate pair at $k_x = \pm \gamma\alpha_y/(\lambda\sqrt{\alpha_x^2+\alpha_y^2})$ and $k_y = \mp \gamma\alpha_x/(\lambda\sqrt{\alpha_x^2+\alpha_y^2})$. In the special $p_x$ or $p_y$ case the points sit on the $k_x=0$ or $k_y=0$ line, whereas a general $p_{xy}$ magnet rotates the connecting line by angle $\eta = \tan^{-1}(-\alpha_x/\alpha_y)$. A magnetic field enters as a Zeeman term and shifts, compresses, merges, or destroys the pair depending on orientation, with thresholds such as $B_y>\gamma$ and $B_z>2\gamma\alpha_x/\lambda$. Circularly polarized light acts through a first-order van Vleck Floquet correction, renormalizing the $\sigma_z$ and Rashba terms so that the pair moves in momentum and annihilates beyond $\Delta>\alpha_x\gamma\lambda/(4\alpha_x^2+\lambda^2)$. The authors verify the degeneracies by eigenvector overlap and by coalescence of spin projections, and interpret the two controls as mechanistically different: global Floquet renormalization for light versus orientation-selective Zeeman shifts for the field.

Load-bearing premise

The quantitative EP locations and thresholds assume the ferromagnetic lead is a wide-band reservoir whose spin-dependent coupling $\gamma$ is constant in momentum and frequency; if the lead's density of states varies near the Fermi level, the EP conditions become momentum- and frequency-dependent and the predicted positions and thresholds could shift.

Editorial extensions

If this is right

  • If the central claim is right, an FM/UPM junction hosts two exceptional points without fine-tuning: their separation is set by $\gamma/\lambda$ and their orientation by the UPM magnetization vector $\boldsymbol{\alpha}$.
  • A magnetic field along $y$ or $z$ can merge the pair into a single exceptional point and then annihilate it beyond thresholds such as $B_y>\gamma$ and $B_z>2\gamma\alpha_x/\lambda$, providing a magnetic switch for the degeneracies.
  • Off-resonant circularly polarized light provides an all-optical route to the same control, annihilating the pair above $\Delta>\alpha_x\gamma\lambda/(4\alpha_x^2+\lambda^2)$, and reversing the light handedness reflects the pattern across the $k_x=0$ or $k_y=0$ axis depending on the UPM orientation.
  • The degeneracies occur as a two-point reorientable line, in contrast to the four-point patterns of $d$-wave altermagnet junctions; this difference is a direct consequence of the linear-in-$k$ coupling of odd-parity $p$-wave magnets.
  • The imaginary part of the energy at the exceptional points stays pinned at $-\Gamma$, so the existence of the degeneracies is insensitive to the UPM strength while their location remains tunable.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural testable extension is to relax the wide-band lead: with a realistic energy-dependent density of states, $\gamma$ becomes frequency- and momentum-dependent and the EP conditions become coupled equations, so the quantitative thresholds would shift even if the two-point structure survives.
  • Because the EP line orientation is set entirely by $\boldsymbol{\alpha}$, any experimental handle that rotates the UPM magnetization vector, such as strain, electrical switching, or an exchange field, would reorient the exceptional line in situ, a non-Hermitian analogue of spin-valve control.
  • The same Floquet treatment applied to noncollinear odd-parity spin textures could produce more than two exceptional points; the linear-in-$k$ coupling is what fixes the multiplicity here, so richer textures would likely enrich the degeneracy structure.
  • The predicted annihilation thresholds define sharp boundaries in the $(B,\Delta)$ parameter plane across which the degeneracy switches on and off; sweeping those boundaries in a transport or spectral experiment is a direct way to map the non-Hermitian phase diagram.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper constructs a non-Hermitian effective model for a ferromagnet/uncharacteristic $p$-wave magnet (UPM) junction with Rashba spin-orbit coupling, Eq. (6), and derives exact conditions for exceptional points (EPs) as $p_r^2=p_i^2$ and $p_r\cdot p_i=0$, Eqs. (10a)-(10b). The central quantitative results are the analytic EP coordinates Eq. (11), their orientation Eq. (12), the magnetic-field-induced shifting, merging, and annihilation thresholds of Section III.C, and the off-resonant CPL Floquet renormalization and threshold of Section III.D. The authors support the analytic conditions with energy spectra, eigenvector overlap, and spin-projection plots, and contrast the resulting two-point EP structure with the four-point structure found in $d$-wave altermagnet junctions.

Significance. If the wide-band self-energy premise is accepted, the analytic results are internally consistent and exact for the stated model. The paper fits nothing to data: the EP positions follow directly from the Hamiltonian plus the standard discriminant definition, and the thresholds $B_y=\gamma$, $B_z=2\gamma\alpha_x/\lambda$, and $\Delta>\alpha_x\gamma\lambda/(4\alpha_x^2+\lambda^2)$ are explicit, falsifiable predictions. I verified Eq. (29) against the Floquet commutator computation in Appendix A and spot-checked Eqs. (10), (15), (19), (31), and (32), including the figure-caption coordinates. The conceptual comparison with $d$-wave altermagnets is clean: the linear UPM spin coupling $\propto k\cdot\alpha$ forces the EP condition onto a line and reduces the multiplicity from four points to two. The main weakness, which the stress-test note correctly identifies, is that every quantitative prediction inherits the constant, momentum- and frequency-independent $\gamma$ of Eq. (4); that premise is not validated for a realistic FM/UPM junction. The Floquet sector also rests on a first-order van Vleck truncation used near the edge of its expected validity.

major comments (2)
  1. [Section II, Eqs. (4)-(5); Section III.A Eq. (11); Section III.C; Section III.D Eq. (32)] The advertised exact EP coordinates and all merging/annihilation thresholds assume the wide-band self-energy $\Sigma^r(\omega=0)=-i\Gamma\sigma_0-i\gamma\sigma_z$ with momentum- and frequency-independent $\gamma$. Equation (5) makes explicit that $\Gamma_{\uparrow,\downarrow}=\pi|t'|^2\rho_{\uparrow,\downarrow}$, so $\gamma$ inherits the ferromagnetic lead density of states. For a semi-infinite FM lead, $\rho_{\uparrow,\downarrow}(\omega,k)$ varies on the lead bandwidth scale, while the UPM states at the EPs sit at finite energy Re$E=t(\alpha^2+k^2)-\mu$ (plus field/CPL shifts), not at $\omega=0$. Once $\gamma=\gamma(\omega,k)$, the discriminant conditions become self-consistent equations, e.g. $\lambda^2k^2=\gamma^2(E(k),k)$ on the $\alpha\cdot k=0$ line, and the EP coordinates, the thresholds $B_y=\gamma$ and $B_z=2\gamma\alpha_x/\lambda$, and the CPL threshold $\Delta>\alpha_x\gamma\lambda/(4\alpha_x^2+\lambda^2)$ can all shift, merge, or disappear. The final paragraph of Section IV mentions disorder and clean-interface idealizations but does not address the energy dependence of the self-energy. I ask the authors to test robustness with an energy-dependent lead DOS or the explicit retarded self-energy of the semi-infinite FM lead and to delineate the parameter window in which Eq. (11) remains valid.
  2. [Section III.D and Appendix A, Eqs. (28)-(32)] The CPL predictions rest on a first-order van Vleck expansion, Eq. (28), used with $\Omega=2$ in units where $t=1$ and momenta and energies are of order unity. The high-frequency condition $\Omega\gg$ bandwidth is not satisfied, and for the plotted $\Delta=0.2$-$0.25$ with $\lambda=1$, the dimensionless vector-potential amplitude is $eA=\sqrt{2\Omega\Delta}/\lambda\simeq 0.9$-$1.0$, so the $A\cdot k$ and $A^2$ terms are not small. The EP shifts and the threshold $\Delta_c=\alpha_x\gamma\lambda/(4\alpha_x^2+\lambda^2)$ are derived from the truncated Hamiltonian Eq. (29); higher-order terms in the van Vleck expansion could renormalize both the $\sigma_z$ term and the RSOC terms. Please either justify the truncation by computing the next-order correction, repeat the key EP results at larger $\Omega$, or explicitly restrict the claims to the regime where the truncation is controlled.
minor comments (6)
  1. [Section III.A, paragraph after Fig. 2] The sentence 'the imaginary part of the energy at the EPs remains fixed at Im[E]=1' conflicts with the sign convention of Eq. (8), where the base energy contains $-i\Gamma$; with $\Gamma=(\Gamma_\uparrow+\Gamma_\downarrow)/2=1$, the imaginary part at the EP is $-1$. Please correct the sign or the wording.
  2. [Section III.D, Eq. (31b)] Equation (31b) reads $\gamma(\alpha_xk_x+\alpha_yk_y+\Delta)=0$, whereas the $\sigma_z$ coefficient in Eq. (29) with $t=\hbar^2/2m=1$ is $2(k\cdot\alpha+\Delta)$; the factor of 2 does not affect the vanishing condition, but the notation should be made consistent.
  3. [Section III.C, Case IV] The statement that an 'additional numerical factor due to $\cos\phi\neq0$' is 'neglected' is misleading, because the field components $B_x$ and $B_y$ are used directly as parameters; please clarify the parametrization of the planar field.
  4. [Section III.D, paragraph comparing CPL with $B_z$] The sentence 'Unlike it effect in $d$-wave AMs [53]' contains a typo and should read 'Unlike its effect in $d$-wave AMs'.
  5. [Section III.D, paragraph after Eq. (31)] The claim that 'when $\lambda\ll1$, the CPL effect in $2\Delta\sigma_z$ term becomes negligible' is unclear because $2\Delta\sigma_z$ is independent of $\lambda$; please clarify what is being compared.
  6. [Figure 2 caption] There are minor typographical errors in the captions, e.g. 'idicates' for 'indicates' and 'green-dashed line' hyphenation; these should be corrected during production.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: all EP predictions are derived from the stated model Hamiltonian, with no fitted inputs or load-bearing self-citations.

full rationale

The paper's central claims—EP emergence at Eq. (11), the field-induced shifts, merging and annihilation thresholds, and the CPL-induced renormalization—are obtained by solving the standard exceptional-point conditions p_r^2 = p_i^2 and p_r·p_i = 0 applied to the explicit Hamiltonian (6) and its magnetic-field and Floquet extensions. These are analytic consequences of the stated model, not quantities fitted to data or imported from the authors' prior work. The constant wide-band self-energy in Eq. (4) is an explicitly acknowledged modeling assumption; relaxing it would modify quantitative EP locations, but that is a robustness/approximation concern rather than a circular reduction. The only self-citation, Ref. [70], appears in the experimental feasibility discussion of CPL intensities and is not load-bearing for any derivation. No uniqueness theorem, author-imported ansatz, or renamed known result is used to force the conclusions. Therefore no circular step can be exhibited.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The model uses five illustrative hand-set parameters (alpha, lambda, gamma, Delta, Omega) and four structural assumptions (wide-band self-energy, first-order van Vleck truncation, collinear UPM model, isotropic Rashba coupling). None of these is fitted to experimental data, so the circularity burden is low; however, every quantitative EP fingerprint, positions scaling as gamma/lambda, annihilation thresholds such as B_y > gamma and B_z > 2 gamma alpha_x / lambda, and the CPL threshold Delta > alpha_x gamma lambda / (4 alpha_x^2 + lambda^2), is contingent on these choices. No comparison to material-specific measurements is attempted, so this ledger measures descriptive freedom rather than data fitting. No new entities are invented: the UPM, the Rashba term, the wide-band self-energy, and the Floquet renormalization are all standard or cited constructs.

free parameters (6)
  • alpha (UPM magnetization vector) = alpha_x = 0.6 or 0.5; alpha_y = 0, 0.5, or 0.3 (scanned)
    Hand-chosen UPM strength. Central to the EP position formula Eq. (11) and the rotation angle Eq. (12); no experimental material parameters are used.
  • lambda (RSOC strength) = 1 (units of t/a)
    Set to 1 throughout; the EP ring radius |gamma/lambda| and all annihilation thresholds scale with it. Illustrative, not fitted.
  • gamma (spin-asymmetric lead coupling) = 1 (from Gamma_up = 2, Gamma_down = 0)
    The non-Hermitian term -i gamma sigma_z is what creates EPs; all EP existence conditions require gamma != 0. Wide-band value chosen for illustration.
  • Delta (CPL intensity parameter) = 0.1 and 0.2; annihilation above 0.25
    Effective Floquet coupling Delta = (eA*lambda)^2/(2*Omega). Scanned to show EP shifts and the annihilation threshold; not tied to a specific light source.
  • Omega (CPL frequency) = 2 (units of t/a^2)
    Chosen in the Fig. 6 caption to 'ensure' the high-frequency Floquet regime, though 2 is comparable to single-particle band energies in the same units; this is the weakest controlled setting in the paper.
  • mu (chemical potential) = implicitly 0 (never specified)
    Appears only in the scalar part epsilon_0 and cancels in E+ - E-, so EP positions do not depend on it. Listed for completeness.
assumptions (4)
  • domain assumption The semi-infinite FM lead is described by a momentum- and frequency-independent wide-band self-energy: Sigma^r(omega = 0) = -i Gamma sigma_0 - i gamma sigma_z (Eq. 4).
    Invoked in Section II after Eq. (3), attributed to Refs. [42, 50, 51]. All EP conditions in the paper use constant gamma; a lead with an energy-dependent density of states would make gamma(k, omega), generically changing EP positions and possibly multiplicities. The authors concede the wide-band idealization only in Section IV.
  • domain assumption The van Vleck Floquet expansion is truncated at first order, H_F ~ H_0 + [H_-1, H_+1]/Omega (Eq. 28), with Omega treated as large.
    Used in Section III.D and Appendix A. The Fig. 6 caption asserts Omega = 2 'ensures the regime required for the high-frequency Floquet framework', but single-particle energies reach about 10 in the same units, so the truncation is applied outside the stated regime and higher-order Floquet corrections are neglected.
  • domain assumption The UPM is modeled by the collinear odd-parity Hamiltonian H_p = t[(k^2 + alpha^2) sigma_0 + 2 k·alpha sigma_z] (Eq. 2).
    Taken from Refs. [12, 15]. The linear-in-k spin splitting is what produces two EPs for the UPM versus four for d-wave altermagnets; real p-wave magnet candidates (CeNiAsO, Gd3Ru4Al12, NiI2) may have additional spin-texture structure beyond this minimal form.
  • domain assumption The FM proximity induces isotropic Rashba spin-orbit coupling lambda(k_y sigma_x - k_x sigma_y) (Eq. 3).
    Standard form from Refs. [31, 40]. The EP ring radius |gamma/lambda| and all field-induced shifts scale with this isotropic structure; any momentum dependence or anisotropy of lambda would deform the EP conditions (10a)-(10b).

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Cite this review

Pith. "Pith review of Opto- and magneto-tunable exceptional degeneracies in non-Hermitian ferromagnet/$p$-wave magnet junctions." pith.science (2026). https://pith.science/paper/XLCSJQXW

@misc{pith2026250801295,
  author       = {Pith},
  title        = {Pith review of: Opto- and magneto-tunable exceptional degeneracies in non-Hermitian ferromagnet/$p$-wave magnet junctions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XLCSJQXW}},
  note         = {Machine review of arXiv:2508.01295}
}
abstract

Unconventional $p$-wave magnets (UPMs) with odd-parity spin textures have attracted interest for their zero net magnetization and anisotropic spin-split Fermi surfaces. Here, we explore a non-Hermitian open quantum system composed of a ferromagnet and a UPM, subjected to an external magnetic field and off-resonant circularly polarized light (CPL), serving as tunable control parameters. We demonstrate the emergence of exceptional points (EPs) in the proposed junction, whose locations can be modulated by the intrinsic properties of the UPM. These EPs exhibit different multiplicities and formation conditions compared to those in even-parity magnets (dubbed $d$- wave altermagnets), a distinction attributable to the preserved time-reversal and broken inversion symmetries characteristic of UPMs. We find that both the unidirectional magnetic field (with adjustable strength and orientation) and the CPL induce momentum-direction-dependent modifications to the EPs, such as their shifting, tilting, merging, or annihilation, supported by analyses of spin projection and eigenvector overlap. Although both perturbations influence the EP structure, they operate via distinct mechanisms: CPL induces a global Floquet re-normalization, enabling dynamic tunability through light, whereas the unidirectional magnetic field selectively alters orientation-aligned terms, lacking such tunability. Beyond revealing EP dynamics in UPM-based junctions, our results highlight UPMs as promising platforms for non-Hermitian phenomena in future spintronics.

Figures

Figures reproduced from arXiv: 2508.01295 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic illustration of (a) proposed NH FM/UPM [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The real and imaginary parts of the energy ( [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. (a) The position of EPs in ( [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a-f) The real and imaginary parts of energy with respect to [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The real and imaginary parts of energy ( [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The density plots of (a, c) imaginary and (b, d) real parts of the absolute value of energy differences as a function [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The real and imaginary parts of spin projections [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

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