Coupling phase enabled level transitions in pseudo-Hermitian magnon-polariton systems

Huang Xin; Lin Shirong; Liu Jingyu

arxiv: 2508.04298 · v1 · submitted 2025-08-06 · 🪐 quant-ph

Coupling phase enabled level transitions in pseudo-Hermitian magnon-polariton systems

Huang Xin , Liu Jingyu , Lin Shirong This is my paper

Pith reviewed 2026-05-19 00:42 UTC · model grok-4.3

classification 🪐 quant-ph

keywords pseudo-Hermitian systemsmagnon-polaritonexceptional pointslevel attractionlevel repulsionnon-Hermitian physicsspintronicscavity magnonics

0 comments

The pith

In pseudo-Hermitian magnon-polariton systems, symmetry breaking is linked to transitions in coupling modes.

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

This paper proposes a model of two magnon modes and two cavity modes interacting through phase-dependent coupling under pseudo-Hermitian conditions. It maps out how the energy levels behave as non-Hermiticity and the coupling phase change, showing exceptional points at symmetry breaking. The work connects these breaks to switches between level attraction and repulsion regimes. A reader might care because it suggests ways to manage damping in quantum hybrid systems by using balanced gain and loss, potentially improving control in spintronic applications.

Core claim

We propose a pseudo-Hermitian model with two magnon and two cavity modes coupled via phase-dependent interaction. Linking the energy spectrum to phase transitions, we observe exceptional points when pseudo-Hermitian symmetry breaks. Level attraction corresponds to four phase transitions and appears as a double Z-shaped energy spectrum, while level repulsion corresponds to two phase transitions with a gap depending on the coupling phase. In the phase diagram defined by non-Hermiticity and coupling phase, pseudo-Hermitian symmetry breaking is intrinsically linked to coupling mode transitions.

What carries the argument

The phase diagram in non-Hermiticity and coupling phase space that reveals the intrinsic link between pseudo-Hermitian symmetry breaking and coupling mode transitions.

If this is right

Exceptional points emerge precisely when the pseudo-Hermitian symmetry breaks.
Level attraction manifests as a double Z-shaped spectrum tied to four phase transitions.
Level repulsion shows a phase-dependent gap across two transitions.
This correspondence enables new strategies for controlling hybrid quantum states in spintronic systems.

Where Pith is reading between the lines

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

Similar phase-dependent mechanisms could be explored in other non-Hermitian hybrid systems to achieve tunable level structures.
Experimental setups with adjustable gain in magnon-cavity systems might test and utilize these transitions for quantum sensing.
The model suggests potential for designing devices that switch between attractive and repulsive regimes via phase control.

Load-bearing premise

Tunable external gain can be introduced at the macroscopic scale to maintain gain-loss balance in the magnon and cavity modes while preserving the pseudo-Hermitian description.

What would settle it

An experiment that varies the coupling phase and non-Hermiticity parameter and finds no correspondence between symmetry breaking and the expected level transitions or exceptional points would disprove the claimed link.

Figures

Figures reproduced from arXiv: 2508.04298 by Huang Xin, Lin Shirong, Liu Jingyu.

**Figure 3.** Figure 3: Evolution of the eigenfrequency ω/g with the magnon frequency ωm/g. The parameters γ/g and θ are set as follows: (a) 0.5, 0◦ ; (b) P, 0◦ ; (c) 1.6, 0◦ . The panels (d), (e), and (f) show the S21 spectra corresponding to panels (a), (b), and (c), respectively. The normalized S21 as a function of ω/g for different values of detuning ∆ as shown in panels (g), (h), and (i). The values of ωc,1/g = 24, ωc,2/g = … view at source ↗

**Figure 4.** Figure 4: (a) The system phase diagram at θ = 45◦ . (b) The system phase diagram at θ = 90◦ . The strong resonance region refers to the area near ωm = ωc, with the resonance center being at ωm = ωc. We then analyze the corresponding energy spectrum. For [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

**Figure 2.** Figure 2: Next, we analyze the phase transition of the [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 5.** Figure 5: The energy spectrum of the system shows the evolution of the eigenfrequency [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

**Figure 6.** Figure 6: The distribution of phase transitions and [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗

**Figure 8.** Figure 8: (a) shows the pseudo-Hermitian phase diagram determined by the coupling phase and the nonHermiticity. The green region represents the real domain, where selecting any point within this area ensures that all eigenvalues are real. The boundary of this phase diagram is also marked by EPs. Next, we calculate the pseudo-Hermitian energy band structure at points (θ, γ/g) in [PITH_FULL_IMAGE:figures/full_fi… view at source ↗

read the original abstract

While cavity-magnon hybridization offers intriguing physics, its practical implementation is hindered by intrinsic damping in both cavity and magnon modes, leading to short coherence times and constrained applications. Recently, with the emergence of tunable external gain at the macroscopic scale, the research focus has shifted from purely lossy systems to gain-loss balanced non-Hermitian systems. Here, we propose a pseudo-Hermitian model with two magnon and two cavity modes coupled via phase-dependent interaction. We link the energy spectrum to phase transitions, observing exceptional points when pseudo-Hermitian symmetry breaks. We also observed level attraction and level repulsion. The former corresponds to four phase transitions and manifests as a double Z-shaped energy spectrum, the latter corresponds to two phase transitions, with the repulsive gap depending on the coupling phase. In the phase diagram defined by non-Hermiticity and coupling phase, we reveal a distinctive correspondence: pseudo-Hermitian symmetry breaking is intrinsically linked to coupling mode transitions, enabling new strategies for controlling hybrid quantum states in spintronic systems.

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.

Desk Editor's Note private letter to a colleague

This paper maps coupling phase to symmetry breaking and mode transitions in a four-mode pseudo-Hermitian magnon-polariton model, with the spectra and phase diagram following directly from the equations.

read the letter

This paper shows how introducing a phase into the coupling term of a four-mode pseudo-Hermitian magnon-polariton system creates a direct link between symmetry breaking and transitions in the coupling modes. The energy levels form a double-Z shape in the attraction case, and the phase diagram reveals the correspondence across different values of non-Hermiticity and phase. The authors set up the system with two magnon modes and two cavity modes. They include a phase-dependent interaction and work out the resulting spectra. Exceptional points mark the symmetry breaking, level attraction corresponds to four transitions with the double-Z feature, and level repulsion has a phase-dependent gap. The full text includes the Hamiltonian and the numerical phase diagram, which backs up the abstract claims. What the paper does well is to make the phase control explicit and to plot the transitions clearly. The math follows through consistently, and there are no contradictions in how the spectra are obtained from the equations. This gives a concrete way to think about controlling the hybrid states. On the softer side, the observed linkage is built into the model by the choice of the coupling term, so it is more an illustration of the consequences than an independent test. The assumption that external gain can be tuned at the macroscopic scale to keep the gain-loss balance is central, and while the internal logic holds, practical challenges in realizing that balance are not explored in depth. The work is for researchers focused on non-Hermitian effects in spintronic or magnonic systems. A reader looking for new theoretical tools for phase-based control in these platforms would get value from the phase diagram and the spectra. Given the explicit construction and the absence of load-bearing flaws, it deserves a serious referee. I would recommend sending this to peer review.

Referee Report

1 major / 2 minor

Summary. The manuscript proposes a pseudo-Hermitian model with two magnon modes and two cavity modes coupled by a phase-dependent interaction. The authors derive the energy spectra, locate exceptional points associated with the breaking of pseudo-Hermitian symmetry, and present a phase diagram in the plane of non-Hermiticity strength and coupling phase. They report level attraction as a double Z-shaped spectrum linked to four phase transitions and level repulsion whose gap depends on the coupling phase and corresponds to two transitions. The central result is an observed correspondence between pseudo-Hermitian symmetry breaking and coupling-mode transitions.

Significance. If the derived correspondence holds, the work supplies a concrete theoretical link between symmetry breaking and mode transitions that could be exploited to control hybrid magnon-polariton states by tuning the coupling phase. The explicit construction of the metric operator and the numerical phase diagram constitute reproducible elements that strengthen the internal consistency of the mapping.

major comments (1)

[§4] §4, phase-diagram construction: the reported one-to-one linkage between symmetry-breaking points and mode-transition lines follows directly from the eigenvalue analysis of the non-Hermitian Hamiltonian; the manuscript should state explicitly whether this correspondence is an independent physical prediction or a direct consequence of the chosen phase-dependent coupling term.

minor comments (2)

[Figure 3] Figure 3: the color bar for the phase diagram lacks explicit labels for the symmetry-breaking and mode-transition regions, complicating direct visual comparison with the text description.
[Eq. (8)] Eq. (8): the definition of the metric operator is given without a brief derivation step; adding one sentence would improve traceability from the Hamiltonian to the pseudo-Hermitian condition.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and for the constructive comment regarding the phase-diagram construction. We address the point raised below and have made revisions to improve clarity.

read point-by-point responses

Referee: §4, phase-diagram construction: the reported one-to-one linkage between symmetry-breaking points and mode-transition lines follows directly from the eigenvalue analysis of the non-Hermitian Hamiltonian; the manuscript should state explicitly whether this correspondence is an independent physical prediction or a direct consequence of the chosen phase-dependent coupling term.

Authors: We agree that the reported one-to-one correspondence between pseudo-Hermitian symmetry-breaking points and coupling-mode transition lines follows directly from the eigenvalue analysis of the non-Hermitian Hamiltonian. This linkage is not presented as an independent physical prediction but arises as a direct consequence of the phase-dependent coupling term in our four-mode model. The phase dependence specifically enables the double Z-shaped spectrum associated with four transitions (level attraction) and the phase-tunable gap with two transitions (level repulsion), both tied to symmetry breaking. In the revised Section 4, we have added explicit language clarifying that the correspondence is a feature of the chosen interaction form and the resulting spectral structure, while noting its utility for controlling hybrid magnon-polariton states. This revision does not change the central results but addresses the requested distinction. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper defines a pseudo-Hermitian Hamiltonian with explicit phase-dependent coupling between magnon and cavity modes, then computes the energy spectra, exceptional points, level attraction/repulsion features, and the phase diagram directly from the eigenvalue problem of that Hamiltonian. The reported correspondence between pseudo-Hermitian symmetry breaking and coupling-mode transitions is an observed outcome of varying the non-Hermiticity strength and coupling phase within the model; it does not reduce to a fitted parameter renamed as prediction, a self-definitional loop, or a load-bearing self-citation. The derivation remains independent of external benchmarks and contains no quoted step that equates an output to its input by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that a phase-dependent interaction term can be engineered while maintaining pseudo-Hermiticity; no new particles or forces are postulated, but the external gain mechanism is treated as an available resource.

free parameters (1)

coupling phase
The phase angle in the interaction Hamiltonian is introduced as a tunable parameter that directly controls the observed attraction/repulsion behavior.

axioms (1)

domain assumption The four-mode system admits a pseudo-Hermitian description once external gain balances intrinsic losses.
Invoked to justify the existence of exceptional points and the phase diagram.

pith-pipeline@v0.9.0 · 5707 in / 1302 out tokens · 53656 ms · 2026-05-19T00:42:08.235138+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

50 extracted references · 50 canonical work pages

[1]

Huebl H, Zollitsch C W, Lotze J, Hocke F, Greifenstein M, Marx A, Gross R and Goennenwein S T B 2013 Phys. Rev. Lett. 111 127003 7

work page 2013
[2]

Zhang X, Zou C L, Jiang L and Tang H X 2014 Phys. Rev. Lett. 113 156401

work page 2014
[3]

Zhang X, Zou C L, Zhu N, Marquardt F, Jiang L and Tang H X 2015 Nat. Commun. 6 8914

work page 2015
[4]

Bai L, Harder M, Hyde P, Zhang Z, Hu C M, Chen Y P and Xiao J Q 2017 Phys. Rev. Lett. 118 217201

work page 2017
[5]

Zhang D, Luo X Q, Wang Y P, Li T F and You J Q 2017 Nat. Commun. 8 1368

work page 2017
[6]

Wang Y P, Zhang G Q, Zhang D, Li T F, Hu C M and You J Q 2018 Phys. Rev. Lett. 120 057202

work page 2018
[7]

Lin S, Nie Z, Yan W, Liang Y, Lin H, Zhao Q and Jia B 2019 Nanophotonics 8 2177-2188

work page 2019
[8]

2024 Adv

Guo Z, Wang J, Malinowski G, Zhang B, Zhang W, Wang H, Lyu C, Peng Y, Vallobra P, Xu Y, et al. 2024 Adv. Mater. 36 2311643

work page 2024
[9]

Zhou C-L, Wu X-H, Zhang Y, Yi H-L, Antezza M 2021 Phys. Rev. B 103 155404

work page 2021
[10]

Wu H, Huang L, Fang C, Yang B S, Wan C H, Yu G Q, Feng J F, Wei H X, Han X F 2018 Phys. Rev. Lett. 120 097205

work page 2018
[11]

Han X, Wu H, Zhang T 2024 Appl. Phys. Lett. 125 2

work page 2024
[12]

Wu H, Wan C H, Zhang X, Yuan Z H, Zhang Q T, Qin J Y, Wei H X, Han X F and Zhang S 2016 Phys. Rev. B 93 060403

work page 2016
[13]

Guo Y, Zhang X, Huang Z, Chen J, Luo Z, Zhang J, Li J, Zhang Z, Zhao J, Han X and others 2024 npj Spintronics 2 36

work page 2024
[14]

Jiang Z, Lim J, Li Y, Pfaff W, Lo T H, Qian J, Schleife A, Zuo J M, Novosad V and Hoffmann A 2023 Appl. Phys. Lett. 123 13

work page 2023
[15]

Agarwal G S 1984 Phys. Rev. Lett. 53 1732

work page 1984
[16]

10 2786-2793

Liu Z X and Li Y Q 2022 Photonics Res. 10 2786-2793

work page 2022
[17]

Xie H, He L W, Shang X, Lin G W and Lin X M 2022 Phys. Rev. A 106 053707

work page 2022
[18]

Bhoi B, Cliff T, Maksymov I S, Kostylev M, Aiyar R P R C, Venkataramani N, Prasad S V G V A and Stamps R L 2014 J. Appl. Phys. 116 24

work page 2014
[19]

Tabuchi Y, Ishino S, Ishikawa T, Yamazaki R, Usami K and Nakamura Y 2014 Phys. Rev. Lett. 113 083603

work page 2014
[20]

Zhang D, Wang X M, Li T F, Luo X Q, Wu W, Nori F and You J Q 2015 NPJ Quantum Inf. 1 1-6

work page 2015
[21]

Yan P, Wang X S and Wang X R 2011 Phys. Rev. Lett. 107 177207

work page 2011
[22]

Nair J M P, Zhang Z, Scully M O and Agarwal G S 2020 Phys. Rev. B 102 104415

work page 2020
[23]

Yao B, Gui Y S, Rao J W, Kaur S, Chen X S, Lu W, Xiao Y, Guo H, Marzlin K-P and Hu C-M 2017 Nat. Commun. 8 1437

work page 2017
[24]

Lachance-Quirion D, Tabuchi Y, Ishino S, Noguchi A, Ishikawa T, Yamazaki R, Nakamura Y 2017 Sci. Adv. 3 e1603150

work page 2017
[25]

Bai L, Harder M, Chen Y P, Fan X, Xiao J Q and Hu C-M 2015 Phys. Rev. Lett. 114 227201

work page 2015
[26]

Grigoryan V L, Shen K and Xia K 2018 Phys. Rev. B 98 024406

work page 2018
[27]

Harder M, Yang Y, Yao B M, Yu C H, Rao J W, Gui Y S, Stamps R L and Hu C-M 2018 Phys. Rev. Lett. 121 137203

work page 2018
[28]

Yuan H Y, Liu Q, Xia K, Yuan Z and Wang X R 2019 Europhys. Lett. 126 67006

work page 2019
[29]

Wang Y-P, Rao J W, Yang Y, Xu P-C, Gui Y S, Yao B M, You J Q and Hu C-M 2019 Phys. Rev. Lett. 123 127202

work page 2019
[30]

Zhang C, Jia C L, Shi Y Z, Jiang C J, Xue D S, Ong C K and Chai G 2021 Phys. Rev. B 103 184427

work page 2021
[31]

Yuan H Y, Yan P, Zheng S, He Q Y, Xia K and Yung M-H 2020 Phys. Rev. Lett. 124 053602

work page 2020
[32]

Lin S, Liang Y, Zhang J C, Chen M K and Tsai D P 2023 Appl. Phys. Lett. 123 22

work page 2023
[33]

Zhang G-Q and You J Q 2019 Phys. Rev. B 99 054404

work page 2019
[34]

Mostafazadeh A 2002 J. Math. Phys. 43 205–214

work page 2002
[35]

Vogt K, Fradin F Y, Pearson J E, Sebastian T, Bader S D, Hillebrands B, Hoffmann A and Schultheiss H 2014 Nat. Commun. 5 3727

work page 2014
[36]

Dark Univ

Flower G, Bourhill J, Goryachev M and Tobar M E 2019 Phys. Dark Univ. 25 100306

work page 2019
[37]

2021 Appl

Baity P G, Bozhko D A, Macˆ edo R, Smith W, Holland R C, Danilin S, Seferai V, Barbosa J, Peroor R R, Goldman S et al. 2021 Appl. Phys. Lett. 119 3

work page 2021
[38]

Liu H, Sun D, Zhang C, Groesbeck M, McLaughlin R and Vardeny Z V 2019 Sci. Adv. 5 eaax9144

work page 2019
[39]

Wang C and Wang X R 2020 Phys. Rev. B 101 165114

work page 2020
[40]

Wang C and Wang X R 2023 Phys. Rev. B 107 024202

work page 2023
[41]

Wang C and Wang X R 2022 Phys. Rev. B 106 045142

work page 2022
[42]

Zhang X, Liu C, Zhu Z, Zhang Y 2024 Phys. Rev. B 109 184426

work page 2024
[43]

Holstein T and Primakoff H 1940 Phys. Rev. 58 1098

work page 1940
[44]

Cao Y and Yan P 2019 Phys. Rev. B 99 214415

work page 2019
[45]

Shen X, Wang F, Li Z and Wu Z 2019 Phys. Rev. A 100 062514

work page 2019
[46]

Soykal ¨O O, Flatt´ e M E 2010Phys. Rev. Lett. 104 077202

work page
[47]

Lin S, Wang L, Yuan L and Chen X 2022 Phys. Rev. Appl. 17 064029

work page 2022
[48]

Gardin A, Bourhill J, Vlaminck V, Person C, Fumeaux C, Castel V and Tettamanzi G C 2023 Phys. Rev. Appl. 19 054069

work page 2023
[49]

Gardin A, Bourcin G, Bourhill J, Vlaminck V, Person C, Fumeaux C, Tettamanzi G C and Castel V 2024 Phys. Rev. Appl. 21 064033

work page 2024
[50]

Gardiner C W and Collett M J 1985 Phys. Rev. A 31 3761 8

work page 1985

[1] [1]

Huebl H, Zollitsch C W, Lotze J, Hocke F, Greifenstein M, Marx A, Gross R and Goennenwein S T B 2013 Phys. Rev. Lett. 111 127003 7

work page 2013

[2] [2]

Zhang X, Zou C L, Jiang L and Tang H X 2014 Phys. Rev. Lett. 113 156401

work page 2014

[3] [3]

Zhang X, Zou C L, Zhu N, Marquardt F, Jiang L and Tang H X 2015 Nat. Commun. 6 8914

work page 2015

[4] [4]

Bai L, Harder M, Hyde P, Zhang Z, Hu C M, Chen Y P and Xiao J Q 2017 Phys. Rev. Lett. 118 217201

work page 2017

[5] [5]

Zhang D, Luo X Q, Wang Y P, Li T F and You J Q 2017 Nat. Commun. 8 1368

work page 2017

[6] [6]

Wang Y P, Zhang G Q, Zhang D, Li T F, Hu C M and You J Q 2018 Phys. Rev. Lett. 120 057202

work page 2018

[7] [7]

Lin S, Nie Z, Yan W, Liang Y, Lin H, Zhao Q and Jia B 2019 Nanophotonics 8 2177-2188

work page 2019

[8] [8]

2024 Adv

Guo Z, Wang J, Malinowski G, Zhang B, Zhang W, Wang H, Lyu C, Peng Y, Vallobra P, Xu Y, et al. 2024 Adv. Mater. 36 2311643

work page 2024

[9] [9]

Zhou C-L, Wu X-H, Zhang Y, Yi H-L, Antezza M 2021 Phys. Rev. B 103 155404

work page 2021

[10] [10]

Wu H, Huang L, Fang C, Yang B S, Wan C H, Yu G Q, Feng J F, Wei H X, Han X F 2018 Phys. Rev. Lett. 120 097205

work page 2018

[11] [11]

Han X, Wu H, Zhang T 2024 Appl. Phys. Lett. 125 2

work page 2024

[12] [12]

Wu H, Wan C H, Zhang X, Yuan Z H, Zhang Q T, Qin J Y, Wei H X, Han X F and Zhang S 2016 Phys. Rev. B 93 060403

work page 2016

[13] [13]

Guo Y, Zhang X, Huang Z, Chen J, Luo Z, Zhang J, Li J, Zhang Z, Zhao J, Han X and others 2024 npj Spintronics 2 36

work page 2024

[14] [14]

Jiang Z, Lim J, Li Y, Pfaff W, Lo T H, Qian J, Schleife A, Zuo J M, Novosad V and Hoffmann A 2023 Appl. Phys. Lett. 123 13

work page 2023

[15] [15]

Agarwal G S 1984 Phys. Rev. Lett. 53 1732

work page 1984

[16] [16]

10 2786-2793

Liu Z X and Li Y Q 2022 Photonics Res. 10 2786-2793

work page 2022

[17] [17]

Xie H, He L W, Shang X, Lin G W and Lin X M 2022 Phys. Rev. A 106 053707

work page 2022

[18] [18]

Bhoi B, Cliff T, Maksymov I S, Kostylev M, Aiyar R P R C, Venkataramani N, Prasad S V G V A and Stamps R L 2014 J. Appl. Phys. 116 24

work page 2014

[19] [19]

Tabuchi Y, Ishino S, Ishikawa T, Yamazaki R, Usami K and Nakamura Y 2014 Phys. Rev. Lett. 113 083603

work page 2014

[20] [20]

Zhang D, Wang X M, Li T F, Luo X Q, Wu W, Nori F and You J Q 2015 NPJ Quantum Inf. 1 1-6

work page 2015

[21] [21]

Yan P, Wang X S and Wang X R 2011 Phys. Rev. Lett. 107 177207

work page 2011

[22] [22]

Nair J M P, Zhang Z, Scully M O and Agarwal G S 2020 Phys. Rev. B 102 104415

work page 2020

[23] [23]

Yao B, Gui Y S, Rao J W, Kaur S, Chen X S, Lu W, Xiao Y, Guo H, Marzlin K-P and Hu C-M 2017 Nat. Commun. 8 1437

work page 2017

[24] [24]

Lachance-Quirion D, Tabuchi Y, Ishino S, Noguchi A, Ishikawa T, Yamazaki R, Nakamura Y 2017 Sci. Adv. 3 e1603150

work page 2017

[25] [25]

Bai L, Harder M, Chen Y P, Fan X, Xiao J Q and Hu C-M 2015 Phys. Rev. Lett. 114 227201

work page 2015

[26] [26]

Grigoryan V L, Shen K and Xia K 2018 Phys. Rev. B 98 024406

work page 2018

[27] [27]

Harder M, Yang Y, Yao B M, Yu C H, Rao J W, Gui Y S, Stamps R L and Hu C-M 2018 Phys. Rev. Lett. 121 137203

work page 2018

[28] [28]

Yuan H Y, Liu Q, Xia K, Yuan Z and Wang X R 2019 Europhys. Lett. 126 67006

work page 2019

[29] [29]

Wang Y-P, Rao J W, Yang Y, Xu P-C, Gui Y S, Yao B M, You J Q and Hu C-M 2019 Phys. Rev. Lett. 123 127202

work page 2019

[30] [30]

Zhang C, Jia C L, Shi Y Z, Jiang C J, Xue D S, Ong C K and Chai G 2021 Phys. Rev. B 103 184427

work page 2021

[31] [31]

Yuan H Y, Yan P, Zheng S, He Q Y, Xia K and Yung M-H 2020 Phys. Rev. Lett. 124 053602

work page 2020

[32] [32]

Lin S, Liang Y, Zhang J C, Chen M K and Tsai D P 2023 Appl. Phys. Lett. 123 22

work page 2023

[33] [33]

Zhang G-Q and You J Q 2019 Phys. Rev. B 99 054404

work page 2019

[34] [34]

Mostafazadeh A 2002 J. Math. Phys. 43 205–214

work page 2002

[35] [35]

Vogt K, Fradin F Y, Pearson J E, Sebastian T, Bader S D, Hillebrands B, Hoffmann A and Schultheiss H 2014 Nat. Commun. 5 3727

work page 2014

[36] [36]

Dark Univ

Flower G, Bourhill J, Goryachev M and Tobar M E 2019 Phys. Dark Univ. 25 100306

work page 2019

[37] [37]

2021 Appl

Baity P G, Bozhko D A, Macˆ edo R, Smith W, Holland R C, Danilin S, Seferai V, Barbosa J, Peroor R R, Goldman S et al. 2021 Appl. Phys. Lett. 119 3

work page 2021

[38] [38]

Liu H, Sun D, Zhang C, Groesbeck M, McLaughlin R and Vardeny Z V 2019 Sci. Adv. 5 eaax9144

work page 2019

[39] [39]

Wang C and Wang X R 2020 Phys. Rev. B 101 165114

work page 2020

[40] [40]

Wang C and Wang X R 2023 Phys. Rev. B 107 024202

work page 2023

[41] [41]

Wang C and Wang X R 2022 Phys. Rev. B 106 045142

work page 2022

[42] [42]

Zhang X, Liu C, Zhu Z, Zhang Y 2024 Phys. Rev. B 109 184426

work page 2024

[43] [43]

Holstein T and Primakoff H 1940 Phys. Rev. 58 1098

work page 1940

[44] [44]

Cao Y and Yan P 2019 Phys. Rev. B 99 214415

work page 2019

[45] [45]

Shen X, Wang F, Li Z and Wu Z 2019 Phys. Rev. A 100 062514

work page 2019

[46] [46]

Soykal ¨O O, Flatt´ e M E 2010Phys. Rev. Lett. 104 077202

work page

[47] [47]

Lin S, Wang L, Yuan L and Chen X 2022 Phys. Rev. Appl. 17 064029

work page 2022

[48] [48]

Gardin A, Bourhill J, Vlaminck V, Person C, Fumeaux C, Castel V and Tettamanzi G C 2023 Phys. Rev. Appl. 19 054069

work page 2023

[49] [49]

Gardin A, Bourcin G, Bourhill J, Vlaminck V, Person C, Fumeaux C, Tettamanzi G C and Castel V 2024 Phys. Rev. Appl. 21 064033

work page 2024

[50] [50]

Gardiner C W and Collett M J 1985 Phys. Rev. A 31 3761 8

work page 1985