Neutron-Constrained Bilayer Spin Model

• The topic is a magnetic model that uses neutron measurements to quantitatively constrain bilayer-specific parameters including exchange topology, anisotropy, and interfacial magnetization profiles.
• It employs techniques such as inelastic neutron scattering and polarized neutron reflectometry to resolve mode splitting, parity channels, and depth-dependent magnetic behavior.
• The model discriminates between competing Hamiltonians, enabling precise extraction of exchange constants and identifying non-collinear textures, with implications for superconducting/ferromagnetic bilayers and magnetic domain analysis.

In the cited literature, a neutron-constrained bilayer spin model is a bilayer magnetic description in which neutron data place quantitative constraints on the admissible exchange topology, anisotropy, bilayer symmetry channel, internal field distribution, domain statistics, or depth-dependent magnetization profile. The relevant observables are not uniform across systems: inelastic neutron scattering constrains magnon dispersions and odd/even bilayer modes, polarized neutron reflectometry constrains nuclear and magnetic depth profiles, and waveguide-enhanced spin-flip scattering constrains non-collinear interfacial textures and internal stray fields. The resulting models range from Heisenberg Hamiltonians for coupled layers to exponential interfacial magnetization profiles in superconducting/ferromagnetic bilayers (Liu et al., 2022, Khaydukov et al., 2013, Khaydukov et al., 2021).

1. Neutron observables and the structure of the constraint

The defining feature of these models is that the measured neutron response fixes bilayer-specific degrees of freedom that are not directly accessible from bulk magnetization alone. In polarized neutron reflectometry on a single V(40 nm)/Fe(1 nm) superconducting/ferromagnetic bilayer, the central observable is the spin asymmetry

$S(Q)=\frac{R^{+}(Q)-R^{-}(Q)}{R^{+}(Q)+R^{-}(Q)},$

which is sensitive to both spatial distribution and absolute magnetization in the layers. Above $T_c$, the spin asymmetry is described using only the magnetic Fe layer, whereas below $T_c$ a systematic shift of the oscillations towards higher $Q$ requires a magnetic contribution inside the superconducting V layer (Khaydukov et al., 2013).

In bilayer superconductors, the constraint is instead symmetry-resolved. For CaKFe$_4$As$_4$, neutron spin resonance intensities separate into odd and even channels with respect to the Fe-As bilayer:

$$I_\text{odd}(L)\propto |F(Q)|^2 \sin^2(z\pi L), \qquad I_\text{even}(L)\propto |F(Q)|^2 \cos^2(z\pi L),$$

with the lower-energy resonances in the odd channel and the higher-energy resonance in the even channel. Here the neutron data constrain the bilayer eigenvectors and the necessity of an interlayer term in the effective spin model (Xie et al., 2018).

Waveguide-enhanced polarized neutron reflectometry extends the same logic to non-collinear textures. In CoO(20 nm)/Co($d_F$) bilayers, the resonance-enhanced spin-flip channels $R^{+-}$ and $R^{-+}$ are directly sensitive to magnetization components perpendicular to the applied field. For the trained sample with $T_c$0 nm, the spin-flip intensity at resonance reaches up to 30% of the incoming intensity; with increasing $T_c$1, it decreases linearly, while channel asymmetry requires an internal field beyond the external Zeeman field for $T_c$2 nm (Khaydukov et al., 2021).

A compact way to summarize the neutron constraint across representative systems is given below.

System	Neutron signature	Constrained result
Ni$T_c$3InSbO$T_c$4	20 meV spin-wave spectrum	$T_c$5, with $T_c$6
V(40 nm)/Fe(1 nm)	Shift of $T_c$7 below $T_c$8	Magnetic V sublayer with $T_c$9 kGs and $T_c$0 nm
CaKFe$T_c$1As$T_c$2	Odd/even $T_c$3-modulated resonance	Two odd modes at 9.5 and 13 meV, one even mode at 18.3 meV
LuFe$T_c$4O$T_c$5	Six magnon bands and one gap	Polar AB-bilayer with six-spin magnetic unit cell
Rare-earth doped bilayer nickelates	Split 45 meV mode and 60 meV mode	Enhanced $T_c$6 in Pr- and Nd-doped samples

These cases show that “constraint” does not refer to a single fitting protocol. It refers to the extraction of bilayer-specific information from neutron observables that depend on parity, depth, or non-collinearity, rather than on integrated magnetic moment alone (Liu et al., 2022, Khaydukov et al., 2013, Xie et al., 2018, Gaw et al., 2014, Zhou et al., 21 Jan 2026).

2. Hamiltonian forms and parameter extraction

For local-moment bilayers, the constrained object is commonly a Heisenberg Hamiltonian. In Ni$T_c$7InSbO$T_c$8, the spin-wave analysis uses

$T_c$9

with four dominant exchange interactions $Q$0 through $Q$1. Linear spin-wave theory, implemented in SpinW and powder-averaged to match the experiment, distinguishes a constrained model with $Q$2 from an unconstrained model in which $Q$3. The best fit is the latter, with $Q$4 meV, $Q$5 meV, $Q$6 meV, and $Q$7 meV, and with the simulated spectrum fitted as

$Q$8

The powder spectrum contains two prominent spin-wave features arising from the $Q$9 and $_4$0 points, and their relative sharpness and $_4$1 boundaries discriminate strongly between $_4$2 and $_4$3 (Liu et al., 2022).

In rare-earth doped bilayer nickelates, the effective constrained model is a nearest-neighbor bilayer Heisenberg model on stripe-type antiferromagnetic orders,

$_4$4

with experimentally relevant parameters reported as products $_4$5. INS on La$_4$6Ni$_4$7O$_4$8 shows a weak flat mode around 45 meV, while Pr and Nd doping split this feature into two modes and add a weaker mode near 60 meV. Fitting requires strong interlayer and weak intralayer couplings: $_4$9–56.5 meV in La$_4$0Ni$_4$1O$_4$2, $_4$3–69.5 meV in La$_4$4PrNi$_4$5O$_4$6, and $_4$7–73.5 meV in La$_4$8NdNi$_4$9O$$I_\text{odd}(L)\propto |F(Q)|^2 \sin^2(z\pi L), \qquad I_\text{even}(L)\propto |F(Q)|^2 \cos^2(z\pi L),$$0, with $$I_\text{odd}(L)\propto |F(Q)|^2 \sin^2(z\pi L), \qquad I_\text{even}(L)\propto |F(Q)|^2 \cos^2(z\pi L),$$1 much smaller (Zhou et al., 21 Jan 2026).

The common methodological point is that the neutron spectrum selects among Hamiltonians that may look comparably plausible at the level of crystal chemistry alone. In Ni$$I_\text{odd}(L)\propto |F(Q)|^2 \sin^2(z\pi L), \qquad I_\text{even}(L)\propto |F(Q)|^2 \cos^2(z\pi L),$$2InSbO$$I_\text{odd}(L)\propto |F(Q)|^2 \sin^2(z\pi L), \qquad I_\text{even}(L)\propto |F(Q)|^2 \cos^2(z\pi L),$$3, the data require a strongly anisotropic exchange hierarchy. In the bilayer nickelates, the data require a large $$I_\text{odd}(L)\propto |F(Q)|^2 \sin^2(z\pi L), \qquad I_\text{even}(L)\propto |F(Q)|^2 \cos^2(z\pi L),$$4 ratio and exclude models whose spectra remain wavy rather than flat or split (Liu et al., 2022, Zhou et al., 21 Jan 2026).

3. Depth-resolved bilayers and interfacial magnetization models

Not all neutron-constrained bilayer spin models are lattice Hamiltonians. In superconducting/ferromagnetic bilayers, the constrained object may instead be a depth profile for induced magnetization. For V(40 nm)/Fe(1 nm), theory predicts an inverse-proximity profile

$$I_\text{odd}(L)\propto |F(Q)|^2 \sin^2(z\pi L), \qquad I_\text{even}(L)\propto |F(Q)|^2 \cos^2(z\pi L),$$5

and the fit used below $$I_\text{odd}(L)\propto |F(Q)|^2 \sin^2(z\pi L), \qquad I_\text{even}(L)\propto |F(Q)|^2 \cos^2(z\pi L),$$6 is

$$I_\text{odd}(L)\propto |F(Q)|^2 \sin^2(z\pi L), \qquad I_\text{even}(L)\propto |F(Q)|^2 \cos^2(z\pi L),$$7

with $$I_\text{odd}(L)\propto |F(Q)|^2 \sin^2(z\pi L), \qquad I_\text{even}(L)\propto |F(Q)|^2 \cos^2(z\pi L),$$8 fixed from the nuclear reflectivity profile. The measured shift of spin asymmetry towards higher $$I_\text{odd}(L)\propto |F(Q)|^2 \sin^2(z\pi L), \qquad I_\text{even}(L)\propto |F(Q)|^2 \cos^2(z\pi L),$$9 below $d_F$0 cannot be reproduced by changing Fe magnetization alone; the fit requires $d_F$1 kGs and $d_F$2 nm within the superconducting V layer. The same fit reproduces a 40% increase in total magnetic moment below $d_F$3, consistent with SQUID and waveguide-enhanced PNR measurements (Khaydukov et al., 2013).

In exchange-biased CoO/Co bilayers, the constrained variables are angular dispersion and internal field. For a single domain,

$d_F$4

while domain fluctuations are quantified by

$d_F$5

Experimentally, $d_F$6 for the thinnest Co layer, indicating isotropic-domain-like fluctuations, and the resonance asymmetry for thicker Co cannot be explained by the applied field alone. The model therefore requires an additional internal magnetic field attributed to stray fields from chiral Bloch domain walls, with the chirality explained by Dzyaloshinskii-Moriya interaction at the CoO/Co interface (Khaydukov et al., 2021).

These depth-resolved models differ from bulk Heisenberg fits in form, but not in logic. The neutron signal does not merely detect a moment; it fixes where that moment is located, whether it is collinear, and whether additional internal fields must be introduced to reproduce the observed spin dependence (Khaydukov et al., 2013, Khaydukov et al., 2021).

4. Bilayer parity channels, mode splitting, and spin-space anisotropy

A central achievement of neutron-constrained bilayer models is the resolution of bilayer parity channels. In CaKFe$d_F$7As$d_F$8, three neutron spin resonance modes appear below $d_F$9 K at $R^{+-}$0 meV, $R^{+-}$1 meV, and $R^{+-}$2 meV. The two lower-energy modes are odd under interchange of the Fe-As layers within the bilayer, while the 18.3 meV mode is even. The pronounced $R^{+-}$3-modulation shows that the Fe layers must be treated as a strongly coupled magnetic bilayer rather than as effectively independent planes (Xie et al., 2018).

Polarized INS further resolves the spin-space structure of the same resonance manifold. The high-energy even mode near 18 meV is isotropic in spin space, whereas the low-energy odd channel consists of a $R^{+-}$4-axis polarized mode around 9 meV together with another partially overlapped in-plane mode around 12 meV. The reported interpretation is that this anisotropy is induced by spin-orbit coupling in spin-vortex-type fluctuations, and that the anisotropy is weak below 6 meV in the normal state but shifts to higher energy and is enhanced in the odd resonance channel below $R^{+-}$5 (Xie et al., 2020).

The minimal bilayer model consistent with these observations has the form

$R^{+-}$6

with the odd/even classification fixing the role of $R^{+-}$7 and the polarization analysis fixing the need for explicit anisotropy or spin-orbit terms (Xie et al., 2020).

The same parity logic reappears in a different setting in the later bilayer cuprate optomagnonics study. There, a neutron-constrained bilayer spin model for YBa$R^{+-}$8Cu$R^{+-}$9O$R^{-+}$0 yields a complete $R^{-+}$1-point spectrum containing an in-plane acoustic $R^{-+}$2 mode that is gapless and Zeeman-linear and an in-plane optical $R^{-+}$3 mode stabilized by weak anisotropy and tunable from the gigahertz to terahertz range. When coupled to a single-mode microwave cavity, the $R^{-+}$4-photon interaction is magnetically tunable and the $R^{-+}$5-photon interaction is nearly field-independent (Parvini, 21 Sep 2025). This suggests that once neutron data have fixed bilayer mode content and anisotropy, the same model can be transplanted into hybrid magnon-photon calculations.

5. Interlayer coupling as a model discriminator

In several bilayer materials, the decisive neutron constraint concerns not the absolute scale of exchange alone but the very identity of the bilayer configuration. LuFe$R^{-+}$6O$R^{-+}$7 is the clearest example. The measured magnon spectrum contains six magnetic modes and a single prominent gap near 8 meV at the zone center, which is compatible with a single bilayer magnetic unit cell containing six spins. Linear spin-wave theory then supports a polar AB-bilayer made of one Fe$R^{-+}$8-rich monolayer and one Fe$R^{-+}$9-rich monolayer. The alternative AA-BB stacking would produce twelve modes, or two distinct spin gaps, neither of which is observed. The best-fit model further yields refined exchange parameters $T_c$00 meV, $T_c$01 meV, $T_c$02 meV, $T_c$03 meV, $T_c$04 meV, together with strongly different single-ion anisotropies for Fe$T_c$05 and Fe$T_c$06 (Gaw et al., 2014).

In bilayer nickelates, the discriminant is instead the dominance of the interlayer bond. The flat 45 meV mode in La$T_c$07Ni$T_c$08O$T_c$09, its splitting into two modes in Pr- and Nd-doped samples, and the additional weak mode near 60 meV all select a model with strong $T_c$10 and weak $T_c$11. The authors explicitly state that AFM-A and AFM-G models cannot reproduce the flat or split-band character of the observed excitations. Nd doping gives the largest extracted $T_c$12, and the corresponding spin fluctuations are stronger than in the undoped and Pr-doped compounds (Zhou et al., 21 Jan 2026).

In Ni$T_c$13InSbO$T_c$14, neutron fitting reaches a similar conclusion from a different spectral geometry: the data require $T_c$15, so the system is essentially a stack of weakly coupled two-dimensional honeycomb antiferromagnets, with $T_c$16 and $T_c$17 too weak to generate frustration-driven incommensurability. The helical order is therefore attributed instead to Dzyaloshinskii-Moriya interactions permitted by the noncentrosymmetric lattice (Liu et al., 2022). Across these examples, interlayer coupling is not merely a perturbation; it is often the parameter that determines whether one bilayer scenario survives neutron scrutiny.

6. Related bilayer theories, interpretive limits, and unresolved points

The neutron-constrained literature sits within a wider landscape of bilayer spin models in which the allowed phases are known theoretically even before neutron fitting is attempted. The spin-$T_c$18 honeycomb-bilayer $T_c$19–$T_c$20–$T_c$21 Heisenberg antiferromagnet exhibits a Néel phase bounded in the $T_c$22 plane by $T_c$23, $T_c$24, and a reentrant window that closes at $T_c$25, where $T_c$26 (Bishop et al., 2016). The bilayer Kugel-Khomskii model adds orbital degrees of freedom and supports VB$T_c$27, PVB, ESO, EPVB, and PVB-AF phases, demonstrating that factorized spin-and-orbital mean field can fail qualitatively when spin-orbital entanglement is strong (Brzezicki et al., 2011). In Sr$T_c$28Cr$T_c$29O$T_c$30, an effective Kugel-Khomskii Hamiltonian derived from maximally localized Wannier functions yields the experimentally proposed combination of antiferromagnetic in-plane spin order, ferromagnetic interplane spin order, and vertical pseudospin singlets (Aligia et al., 2018).

These models provide the phase-space context into which neutron-constrained fits are inserted. They also highlight a limit of the neutron-constrained approach: neutron data may strongly restrict parameters without uniquely eliminating every alternative microscopic mechanism. In the V/Fe bilayer, the inverse-proximity interpretation is the preferred one, but orbital effects associated with vortex pinning at the S/F interface yield $T_c$31 kGs, close to the fitted $T_c$32 (Khaydukov et al., 2013). In LuFe$T_c$33O$T_c$34, weak subgap scattering suggests that a fraction of the bilayers may contain other combinations of charged monolayers not included in the minimal six-spin model (Gaw et al., 2014). In CoO/Co, the domain interpretation depends on the domain size relative to neutron coherence lengths, so sufficiently small domains would require modeling beyond specular reflectivity, including off-specular scattering (Khaydukov et al., 2021).

A further limit emerges from frustrated bilayers. In the classical kagome bilayer Heisenberg antiferromagnet, neutron-relevant observables include pinch points at $T_c$35, half-moon features at finite $T_c$36, diffusive low-energy dynamics, and disorder-induced orphan-spin tails in the autocorrelation function. The study concludes that site disorder alone does not induce glassiness in the classical model (Saha et al., 2019). This suggests that when neutron spectra in real bilayers deviate from the constrained classical benchmark, additional ingredients such as quantum fluctuations, emergent rugged energy landscapes, or extra anisotropies must be incorporated.

Taken together, these results define the scope of neutron-constrained bilayer spin modeling. It is not a single Hamiltonian class, but a methodology in which bilayer-specific symmetry, depth, and mode structure are fixed by neutron data and then propagated into microscopic interpretation. Its most robust outputs are exchange hierarchies, bilayer parity assignments, anisotropy channels, interfacial magnetization profiles, and evidence for internal fields or minority stacking motifs; its persistent uncertainties arise when neutron signatures admit more than one microscopic origin or when the minimal bilayer model omits defects, orbital sectors, or off-specular structure.