Out-of-time-order correlation in the quantum Ising Floquet spin system and magnonic crystals

Rohit Kumar Shukla

REVIEW 3 major objections 4 minor 254 references

Out-of-time-order correlation in the quantum Ising Floquet spin system and magnonic crystals

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The long-time averaged longitudinal OTOC is an order parameter for the four phases of a kicked transverse-field Ising chain, and randomized OTOC equals operator entanglement entropy.

desk verdict Solid OTOC analysis in the Floquet Ising chapters, but the quantum information diode is built on an ad hoc suppression rate that does not survive contact with the model. read the letter →

arxiv 2505.07550 v1 pith:ZARVZM3O submitted 2025-05-12 quant-ph

Rohit Kumar Shukla This is my paper · ORCID

classification quant-ph

keywords out-of-time-ordercorrelatorFloquettransverseIsingmodelquantumphasediagramJordan-WignertransformationoperatorentanglemententropyinformationdiodemagnoniccrystalDzyaloshinskii-Moriyainteraction

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

The paper sets out to make the out-of-time-order correlator (OTOC) a working diagnostic for driven spin systems: it claims that the long-time averaged longitudinal-magnetization OTOC is an order parameter that separates the four phases of the periodically kicked transverse-field Ising chain, and it supplies an exact Jordan-Wigner formula for the transverse-magnetization OTOC. A second claim is that once the OTOC is averaged over random observables drawn from the Gaussian unitary ensemble, it is exactly the operator entanglement entropy of the Floquet propagator, so exponential saturation of the OTOC in nonintegrable chains is the growth of operator entanglement. A third claim is that left- and right-propagating OTOCs through a yttrium-iron-garnet magnonic crystal with Dzyaloshinskii–Moriya interaction act as a quantum information diode whose rectification is controlled by an applied electric field. If these hold, OTOC becomes a structurally reliable probe of Floquet phase structure, a direct bridge to operator entanglement, and a design principle for directional magnonic information transport.

What carries the argument

The object carrying the argument is the out-of-time-order correlator $C^{l,m}(n)=1-\Re F^{l,m}(n)$ with $F^{l,m}(n)=\langle \sigma^l(n)\sigma^m\sigma^l(n)\sigma^m\rangle$, evaluated on polarized product states. The Floquet operator $\hat U=e^{-i\tau_1 H_{xx}}e^{-i\tau_0 H_z}$ supplies discrete-time Heisenberg evolution; the Jordan-Wigner transformation maps the integrable case to free fermions and yields the exact TMOTOC formula. For phase detection the load-bearing quantity is the long-time average $\overline{F^{l,l}_x}$, whose zero/nonzero value tracks the paramagnetic/ferromagnetic distinction. For the entanglement bridge, the GUE average of the four-point correlator uses the swap identity $\overline{\hat W\otimes\hat W}=\hat S$ to obtain $\overline C(n)=d^2 E_l[\hat U(n)]$. For the diode, the Dzyaloshinskii–Moriya term makes the magnon dispersion direction-dependent, $\omega(\pm D,k)=\omega(k)\pm D\sin(ka)$, so Bragg-matched right movers are damped by gate magnons while left movers are not, producing asymmetric left/right OTOCs.

What would settle it

Measure the left and right OTOC time integrals on a grooved YIG film as the electric field is varied: if $R(D)=\int C_R dt / \int C_L dt$ does not fall exponentially with $D=E_y g_{ME}$, or if reversing the field does not swap $C_L$ and $C_R$, the diode claim is falsified; a direct numerical simulation of the full spin Hamiltonian of Eq. (5.2), with no suppression interpolation, would also settle it.

Watch

Extended reading notes

Core claim

In the paper's own terms, the central discovery is a phase-detection scheme: starting from a polarized product state, the time-averaged longitudinal OTOC $\overline{F^{l,l}_x}$ stays zero in the $0$-paramagnetic and $0\pi$-paramagnetic regions and takes a positive value in the $0$-ferromagnetic and $\pi$-ferromagnetic regions, with finite-size critical lines tending to the known diagonal phase boundaries; the transverse OTOC always oscillates around a positive value and cannot serve this role, although it is exactly solvable. The exact solution shows that the transverse commutator departs from unity after a number of kicks equal to the separation between observables, with revival time and light-cone speed read off analytically. With block observables, the OTOC grows as a power law in both integrable and nonintegrable chains, and saturation to the random-matrix value is exponential; with GUE-random observables the averaged OTOC is exactly the operator entanglement entropy of the propagator. For the diode, nonreciprocal magnons produced by the Dzyaloshinskii–Moriya interaction make the left and right OTOCs different, and the ratio of their time integrals defines a rectification coefficient $R(D)$ that decreases with the electric-field-controlled DMI strength.

Load-bearing premise

The rectification curve of the proposed diode is built on the assumption that suppression of right-moving magnons by gate magnons follows $\zeta(D)\approx e^{-D/5}$ and that the Holstein-Primakoff Hamiltonian truncated at quadratic order describes the YIG device at low magnon density; if either fails, the electric-field dependence of the rectification coefficient changes.

Editorial extensions

If this is right

The time-averaged longitudinal OTOC gives an experimentally accessible dynamical order parameter for the four Floquet phases, requiring only polarized-state initialization and local spin measurements rather than full spectroscopy.
The exact TMOTOC formula reduces the computational cost of the transverse correlator to $O(N^3)$, so light-cone speeds, revival times, and phase boundaries can be studied at system sizes beyond exact diagonalization.
Exponential saturation of block-operator OTOC to random-matrix values provides a dynamical signature of quantum chaos in driven spin chains even when early-time growth is only power law.
Because the GUE-averaged OTOC equals operator entanglement entropy, a measurement of randomized-observable OTOC is a measurement of the propagator's bipartite operator entanglement.
An electrically controlled YIG magnonic crystal should rectify quantum information flow, with rectification coefficient $R(D)$ tunable through the magnetoelectric coupling.

Reading between the lines

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

If the LMOTOC time-average is a genuine order parameter, the same construction should identify drive-induced transitions in other Floquet spin chains, including interacting or disordered ones, where the four-fold phase structure is not known in advance.
The equality of GUE-averaged OTOC and operator entanglement entropy is a kinematic identity for any bipartite unitary; the paper's numerical saturation curves suggest that randomized OTOC protocols could be used to measure operator entanglement growth in quantum simulators.
The diode's rectification curve rests on an interpolated suppression $\zeta(D)\approx e^{-D/5}$; a microscopic four-magnon scattering calculation would replace that interpolation and would predict the optimal groove spacing, gate density, and field range for maximal rectification.
The asymmetry in left/right OTOC growth times implies a directional butterfly velocity $v_g^+-v_g^-\approx 2D$; measuring that velocity difference in spin-wave experiments would connect the quantum-information diode to the nonreciprocal magnon literature.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You asked for a candid read of this one. The thesis is a mixed bag. The genuinely solid parts are Chapter 2 and Chapter 4. The exact Jordan-Wigner formula for the transverse magnetization OTOC is cross-checked against exact diagonalization, and using the long-time averaged LMOTOC to carve out the four Floquet phases is a reasonable extension of known results. The identity in Chapter 4, that the GUE-averaged OTOC equals the operator entanglement entropy, is a clean derivation and the most citable thing here. Credit where it is due: those two chapters show real ability.

The soft spots are concentrated in Chapter 5. The quantum information diode is the headline, but its quantitative output, the rectification curve R(D), is not derived from the spin Hamiltonian. The Holstein-Primakoff truncation at quadratic order gives noninteracting magnons, yet the mechanism requires four-magnon scattering of source magnons by gate magnons. The suppression rate ζ(D) is inserted by hand as e^{-D/5} to mimic that scattering. Worse, the interpolation is internally inconsistent: if ζ is a suppression rate, ζ(0)=1 means full suppression at zero DMI, which contradicts the paper's own claim that D=0 gives R=1. The D-dependence of the diode's rectification is therefore not trustworthy.

There are also smaller issues. The phase diagram in Chapter 2 uses system sizes only up to N=10, with the thermodynamic limit reached by extrapolation; the reported critical exponent 1/ν = 0.83 ± 0.11 comes from a small number of points. No code or data files are supplied, so the numerics are not independently checkable.

Who is this for? A reader working on OTOCs in Floquet spin chains will get value from Chapters 2 and 4. The diode chapter is speculative and should be treated as a proposal, not a result. I would not cite the thesis as a whole, but the OPEE identity is worth citing if it appears in a journal.

For peer review: this deserves a serious referee, but with a clear mandate. The OTOC material is solid enough for a focused paper, and the diode chapter needs either a real derivation of the suppression rate or a much softer claim. I would recommend engaging with the authors and asking them to split the thesis into separate papers: one on the OTOC phase structure and one on the operator entanglement connection, and a separate, heavily revised paper for the diode.

Referee Report

3 major / 4 minor

Summary. The manuscript studies out-of-time-order correlators (OTOCs) in periodically kicked Ising spin chains and in a square-lattice Heisenberg model with Dzyaloshinskii-Moriya interaction. Chapter 2 derives an exact analytical expression for the transverse-magnetization OTOC via the Jordan-Wigner transformation, compares it with exact diagonalization, and proposes the long-time averaged longitudinal-magnetization OTOC as an order parameter for the four Floquet phases. Chapter 3 characterizes the characteristic, dynamic, and near-saturation regimes of transverse and longitudinal magnetization OTOCs in integrable and nonintegrable Floquet systems, reporting power-law growth with separation-dependent exponents. Chapter 4 introduces block-spin and random block observables, shows power-law growth for the former and exponential saturation for the latter, and derives an identity between the GUE-averaged OTOC and the operator entanglement entropy. Chapter 5 proposes a quantum information diode based on YIG magnonic crystals, using left and right OTOCs to quantify asymmetric quantum information currents and an effective electric-field-controlled DMI term to tune the rectification coefficient R(D).

Significance. If the claims hold, the work would strengthen the case for OTOCs as a structurally reliable diagnostic of Floquet phase structure, provide a clean bridge between OTOCs and operator entanglement, and suggest a design principle for directional magnonic quantum information transport. Strengths of the manuscript include the exact analytical TMOTOC formula cross-checked against exact diagonalization in Fig. 2.2, the compact O(L^3) scaling of that formula, the explicit derivation of the GUE-averaged OTOC/operator-entanglement identity in Sec. 4.2.3, and the systematic numerical separation of characteristic, dynamic, and saturation regimes in Chapter 3. The QID proposal in Chapter 5 is experimentally motivated and clearly described. However, the quantitative rectification result in Chapter 5 is not derived from the microscopic Hamiltonian, and the phase-structure extrapolation in Chapter 2 rests on quite small system sizes; these issues need to be addressed before the central claims can be considered fully supported.

major comments (3)

[Sec. 5.2.4 and Fig. 5.3] The central quantitative result of the QID chapter, the rectification curve R(D), is not derived from the microscopic model. Section 5.2.2 diagonalizes Eq. (5.2) through a Holstein-Primakoff transformation truncated at quadratic order, yielding the noninteracting magnon Hamiltonian of Eq. (5.4). In such a quadratic model the four-magnon scattering process invoked in Sec. 5.2.1 is absent, yet the suppression rate is inserted by hand in Sec. 5.2.4 as ζ(D)≈e^{-D/5}. This means both the magnitude and the D-dependence of R(D) in Fig. 5.3 are properties of an assumed damping law rather than predictions of the model. In addition, the interpolation is internally inconsistent with the stated D=0 limit: Sec. 5.2.1 defines the suppression rate as ξ(D)=1−n_D^+/n_D^-, for which zero suppression corresponds to ξ=0, but Eq. (5.3) of Sec. 5.2.4 gives ζ(0)=1, i.e., maximal suppression at zero DMI. This contradicts the text immediately below, which states that D=0 gives no rectification, R=1. Please either derive ζ from a microscopic scattering calculation or explicitly reframe Sec. 5.2.4 as a phenomenological model and include a sensitivity analysis of R(D).
[Sec. 2.7, Figs. 2.7 and 2.8] The load-bearing claim that the long-time averaged LMOTOC can serve as an order parameter for the four Floquet phases is established numerically only for N=6, 8, and 10 in Fig. 2.7, with the finite-size scaling in Fig. 2.8 based on four system sizes and yielding 1/ν=0.8314±0.1122 from a log-log fit. At N=10 the critical lines are still visibly far from the diagonal thermodynamic-limit lines, and no data collapse is presented. The extrapolation to N→∞ is plausible but not quantitatively established. Please provide a data-collapse analysis, additional system sizes, or an independent check, including for the open-chain case where the approach to the diagonal is claimed to be slower.
[Sec. 3.4, Eqs. (3.16)-(3.21)] The power-law exponent formulas b(Δl) are presented as quantitative characterization of the dynamic regime, but they are fits to N=18 data with fitted constants κ, bmax, b0, and λ, without confidence intervals, fitting ranges, or collapse tests. In particular, the distinction between the triangular form of Eq. (3.16) and the quadratic form of Eq. (3.20) is based on a small number of data points in Fig. 3.6(d). Please report the fitting procedure, error bars, and a stability check with respect to the fitting interval, or soften the claims accordingly.

minor comments (4)

[Sec. 5.2.1 vs. Sec. 5.2.4] The notation for the suppression rate changes from ξ in Sec. 5.2.1 to ζ in Sec. 5.2.4 and Fig. 5.2(c); please unify the notation and explicitly define the D=0 limits of both ξ and ζ.
[Fig. 2.4 caption] The caption labels both panels as "F^l,l_x(n)"; one of the panels is presumably F^l,l_z(n) based on the surrounding text.
[Sec. 5.2.3, text near Eq. (5.7)] The notation ωs(±D,k±_s) mixes a two-dimensional wave vector with the one-dimensional dispersion written in terms of kx, and the statement that m0 ranges from 1 to N is followed by the specific choice m0=N; please make the summation over m0 explicit.
[Throughout] The manuscript is formatted as a PhD thesis and contains institutional front matter and declarations; for journal review, the arXiv submission should be reformatted into a standard article structure with consolidated references.

Circularity Check

0 steps flagged · score 6.0 of 10

The QID rectification curve is built on a hand-inserted ζ(D)=e^{-D/5}, so the field-controlled diode rectification reduces by construction; the OTOC phase-structure and operator-entanglement claims are self-contained.

full rationale

The long-time-averaged LMOTOC order parameter in Chapter 2 is a numerical observable computed from the Floquet map Eq. (2.1) and compared with the known phase diagram; the TMOTOC formula is derived via Jordan-Wigner (Appendix A), so it is self-contained. Chapter 4's identity between the GUE-averaged OTOC and operator entanglement entropy is a mathematical derivation from the definitions of C2, C4 and the Schmidt decomposition (Eqs. (4.8)-(4.10)); it is not a renamed empirical pattern. Chapter 3's power-law exponents and saturation slopes are descriptive fits to the same numerical OTOC data they summarize, but the paper does not use them to predict independent quantities, so they do not constitute circular derivation. The circular step is localized to Chapter 5: the rectification coefficient R(D) is computed from an interpolated ζ(D)≈e^{-D/5}, and with D=E_y g_ME the field-dependence of the diode is exactly the assumed damping law. This makes the central QID claim partially circular. I also note the D=0 inconsistency (ζ(0)=1 versus the stated R=1) as a correctness concern, but the circularity verdict rests on the explicit free-parameter status of ζ. Score 6 reflects partial circularity in one of the thesis's three central claims; the OTOC phase-structure and operator-entanglement results remain independent.

Assumptions & free parameters 4 free parameters · 5 assumptions · 1 invented entities

The list separates the few derived inputs from the fitted and ad hoc inputs. The TMOTOC exact formula relies on the standard Jordan-Wigner mapping, and the OTOC equals OPEE identity is a mathematical derivation with stated random-matrix assumptions. The load-bearing fitted inputs are the power-law exponent constants, the saturation rates, and especially the suppression rate zeta(D), which sets the diode rectification. The low-magnon-density truncation of the spin-wave Hamiltonian is an unchecked experimental assumption that also supports the QID claim.

free parameters (4)

Power-law exponent constants (kappa, bmax, b0, lambda) = kappa=3.2, bmax between 24 and 32.9, b0 between 1.7 and 1.9, lambda=2.8
Chapter 3 Eqs. (3.16) and (3.20) describe the exponent of the power-law growth using constants fit to N=18 numerical OTOC data; the same data are then summarized by these formulas.
Saturation decay slopes mu = mu=0.002, 10^-5, 0.14, 0.77, 0.85 in different sections
Late-time OTOC saturation is fitted to linear or exponential decay in Chapters 3 and 4, and the resulting slopes are quoted without uncertainties.
Suppression rate zeta(D) = zeta(D) approximately e^{-D/5}
Sec. 5.2.4 states 'we interpolate the suppression rate' as zeta(D) approximately e^{-D/5}; this functional form is assumed, not derived, and it directly controls the rectification coefficient R(D).
Finite-size critical exponent 1/nu = 0.8314 +/- 0.1122
Chapter 2 Sec. 2.7.1 obtains 1/nu by fitting a straight line through log-log finite-size data for system sizes N=6 to 12; the value is a numerical fit, not an analytic result.

assumptions (5)

standard math The kicked transverse-field Ising model at hx=0 maps via the Jordan-Wigner transformation to noninteracting fermions.
Used in Chapter 2 Eq. (2.6) and Chapter 3 Eq. (3.10) to derive the exact TMOTOC formula.
domain assumption Trace over the full Hilbert space can be replaced by expectation values in a small number of Haar-random states for finite N.
Invoked in Chapter 3 Sec. 3.3 and Chapter 4 Sec. 4.2.2; the convergence of the random-state average to the trace is asserted, not quantified.
domain assumption The 2D DMI magnet with electric-field-induced polarization is accurately described by a Holstein-Primakoff boson Hamiltonian truncated at quadratic order.
Chapter 5 Sec. 5.2.2 assumes low magnon density in YIG so that magnon-magnon interactions are negligible; this is an experimental assumption, not a theorem.
domain assumption Fully polarized initial states reproduce random-state or trace behavior of OTOCs in all three regimes: characteristic, dynamic, and saturation.
Chapter 3 Sec. 3.3 states that no remarkable differences were found for selected parameters, but the equivalence is not proven in general.
standard math Spectral statistics after unfolding distinguish integrable (Poisson) and chaotic (Wigner-Dyson) phases.
Used in Chapter 4 Sec. 4.2.4 to classify the Floquet systems as integrable or nonintegrable.

invented entities (1)

Quantum information diode (QID)
purpose: A proposed device that rectifies the flow of quantum information using nonreciprocal magnons in a YIG magnonic crystal with Dzyaloshinskii-Moriya interaction.
No device has been built, and the quantitative rectification curve depends on the ad hoc suppression rate zeta(D) approximately e^{-D/5}; the setup borrows from the existing YIG magnonic transistor Ref. [89] but adds no independently falsifiable prediction.

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Cite this review

Pith. "Pith review of Out-of-time-order correlation in the quantum Ising Floquet spin system and magnonic crystals." pith.science (2026). https://pith.science/paper/ZARVZM3O

@misc{pith2026250507550,
  author       = {Pith},
  title        = {Pith review of: Out-of-time-order correlation in the quantum Ising Floquet spin system and magnonic crystals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZARVZM3O}},
  note         = {Machine review of arXiv:2505.07550}
}

read the original abstract

In recent times out-of-time-order correlators (OTOC) have been established as a tool to understand butterfly effects, quantum information scrambling, and many-body localization. They can also be useful in determining different phases of quantum critical systems. OTOCs can identify the quantum chaos within a system undergoing time evolution; and therefore, they can distinguish between chaotic and regular dynamics. This motivates us to study OTOCs in integrable and nonintegrable periodically kicked quantum spin models. A periodically kicked quantum Ising spin system, known as the quantum Ising Floquet system, is a variant of the transverse Ising model. In place of constant transverse magnetic fields in the transverse Ising system, time-periodic fields are applied in the form of delta pulses in the quantum Ising Floquet spin system. It provides very interesting and peculiar dynamics separate from that of the transverse Ising system.

Figures

Figures reproduced from arXiv: 2505.07550 by the authors.

**Figure 1.** Illustration of observable [PITH_FULL_IMAGE:figures/full_fig_p032_1.png] view at source ↗

**Figure 1.** (Left) We consider a state [PITH_FULL_IMAGE:figures/full_fig_p034_1.png] view at source ↗

Figures from the paper (32 more)

**Figure 1.** Figure 1: Schematics of single spin observables. One spin is considered as observable [PITH_FULL_IMAGE:figures/full_fig_p036_1.png]

**Figure 1.** Figure 1: Illustration of SBOs [PITH_FULL_IMAGE:figures/full_fig_p037_1.png]

**Figure 1.** Figure 1: Setup of Stern-Gerlach experiment [PITH_FULL_IMAGE:figures/full_fig_p042_1.png]

**Figure 1.** Figure 1: Arrangement of spins at lattices in one, two, and three dimensions. [PITH_FULL_IMAGE:figures/full_fig_p046_1.png]

**Figure 1.** Figure 1: One-dimensional lattice configuration with (Left) periodic boundary condition [PITH_FULL_IMAGE:figures/full_fig_p048_1.png]

**Figure 1.** Figure 1: Spin chain experience a periodic quench by non-commuting Hamiltonian functions [PITH_FULL_IMAGE:figures/full_fig_p049_1.png]

**Figure 1.** Figure 1: Local geometry to determine the DM vector’s orientation. [PITH_FULL_IMAGE:figures/full_fig_p052_1.png]

**Figure 1.** Figure 1: Pictorial representation of magnonic transistor in which YIG film with grooves is [PITH_FULL_IMAGE:figures/full_fig_p057_1.png]

**Figure 2.** Figure 2: Phase structure of the Floquet system with Floquet map given by Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p062_2.png]

**Figure 2.** Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p066_2.png]

**Figure 2.** Figure 2: Behaviour of (a) [PITH_FULL_IMAGE:figures/full_fig_p067_2.png]

**Figure 2.** Figure 2: (a) Variation of the real part of (a) [PITH_FULL_IMAGE:figures/full_fig_p068_2.png]

**Figure 2.** Figure 2: Variation of the real part of [PITH_FULL_IMAGE:figures/full_fig_p069_2.png]

**Figure 2.** Figure 2: Plot of [PITH_FULL_IMAGE:figures/full_fig_p071_2.png]

**Figure 2.** Figure 2: Regions with [PITH_FULL_IMAGE:figures/full_fig_p072_2.png]

**Figure 2.** Figure 2: Plot of the difference between finite size critical point and the infinite size critical [PITH_FULL_IMAGE:figures/full_fig_p074_2.png]

**Figure 2.** Figure 2: Heat map of the logarithmic dominant frequencies of [PITH_FULL_IMAGE:figures/full_fig_p075_2.png]

**Figure 3.** Figure 3: Schematic of the various regions of OTOC in a typical system. [PITH_FULL_IMAGE:figures/full_fig_p084_3.png]

**Figure 3.** Figure 3: Integrable transverse Ising Floquet system with [PITH_FULL_IMAGE:figures/full_fig_p086_3.png]

**Figure 3.** Figure 3: Integrable transverse Ising Floquet system with [PITH_FULL_IMAGE:figures/full_fig_p088_3.png]

**Figure 3.** Figure 3: Non-integrable closed chain transverse Ising Floquet system with [PITH_FULL_IMAGE:figures/full_fig_p089_3.png]

**Figure 3.** Figure 3: Integrable closed chain transverse Ising Floquet system with [PITH_FULL_IMAGE:figures/full_fig_p091_3.png]

**Figure 3.** Figure 3: Non-integrable closed chain transverse Ising Floquet system with [PITH_FULL_IMAGE:figures/full_fig_p094_3.png]

**Figure 4.** Figure 4: Schematics of SBOs defined in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p102_4.png]

**Figure 4.** Figure 4: Integrable [PITH_FULL_IMAGE:figures/full_fig_p109_4.png]

**Figure 4.** Figure 4: Nonitegrable [PITH_FULL_IMAGE:figures/full_fig_p111_4.png]

**Figure 4.** Figure 4: (a) [PITH_FULL_IMAGE:figures/full_fig_p113_4.png]

**Figure 4.** Figure 4: Integrable [PITH_FULL_IMAGE:figures/full_fig_p115_4.png]

**Figure 4.** Figure 4: (a) [PITH_FULL_IMAGE:figures/full_fig_p116_4.png]

**Figure 5.** Figure 5: Illustration of a quantum information diode: A plane of an YIG film with grooves [PITH_FULL_IMAGE:figures/full_fig_p121_5.png]

**Figure 5.** Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p126_5.png]

**Figure 5.** Figure 5: Rectification coefficient [PITH_FULL_IMAGE:figures/full_fig_p128_5.png]

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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