Momentum Transport in Ferromagnetic–Plasmon Heterostructures Within the Keldysh Formalism

Abstract

We investigate momentum transport in ferromagnetic–plasmon heterostructures using Keldysh field theory and energy–momentum tensor formalism. A three-layer model reveals that plasmon frequency shifts generate a non-zero expectation value for the $xz$-component of the energy–momentum tensor $⟨T_{xz}⟩$ through magnon–plasmon coupling. The momentum transport exhibits linear velocity dependence, with temperature behavior transitioning from exponential suppression at low temperatures to linear growth at high temperatures, governed by the magnon energy gap. Spatial oscillations follow $sin(2n\pi z/h)$ patterns within the ferromagnetic layer. This framework provides fundamental insights into quantum momentum transport mechanisms in magnetic systems.

1. Introduction

Momentum transport in quantum magnetic systems has emerged as a fundamental process underlying various energy dissipation phenomena. At microscopic scales, quantum fluctuations govern the transfer of momentum between different degrees of freedom, leading to energy dissipation through the excitation of quasiparticles such as phonons, electrons, plasmons, and magnons [1,2,3,4]. The energy–momentum tensor provides a comprehensive framework for quantifying these transport processes, capturing both energy flow and momentum transfer in quantum materials, while phononic mechanisms dominate at elevated temperatures, quantum effects become increasingly prominent at lower temperatures, where electronic and collective excitations mediate momentum transport through more subtle quantum channels [5,6,7,8,9,10].

In ferromagnetic heterostructures, the energy–momentum tensor formalism is essential for describing momentum transport phenomena arising from the interplay between spin and charge degrees of freedom. Recent experimental observations of magnetic friction, governed by magnon-mediated momentum transfer, have established the foundation for understanding spin-based transport mechanisms [11]. Theoretical descriptions have evolved from classical spin models with dipolar and exchange interactions, including temperature dependencies and critical behavior [11,12,13], to quantum treatments employing bosonic mappings that connect spin friction to electronic momentum transport [14]. However, existing frameworks predominantly address idealized interfaces of identical materials, neglecting the complexity of heterogeneous systems where quantum spins exchange momentum with diverse internal excitations. Although Casimir friction between magnetic and dielectric media has been explored using simplified oscillator models [15], the roles of dispersive magnons and momentum dissipation through intermediate layers, rather than vacuum gaps, remain largely uncharted territory.

This work addresses the fundamental challenge of describing non-equilibrium momentum transport in driven quantum systems by employing the Keldysh field theory framework within the path-integral formulation [16], which enables the treatment of time-dependent perturbations and energy dissipation beyond conventional equilibrium approaches. Building on our previous studies [8,14,17,18,19], we incorporate magnon–plasmon coupling, include temperature effects, and eliminate the steady-state requirement, yielding a more realistic and experimentally accessible description of the system’s dynamics. We develop a three-layer ferromagnetic–plasmon heterostructure model that captures the essential physics of momentum transfer mediated by magnon–plasmon coupling, providing the first systematic analysis of how plasmon frequency shifts induce momentum transport through quantum fluctuations. The energy–momentum tensor formalism reveals novel transport characteristics, including linear velocity dependence, spatially oscillatory patterns, and distinct temperature regimes governed by quantum statistics. Our approach resolves longstanding limitations in describing heterogeneous material interfaces and establishes a comprehensive theoretical foundation for understanding and controlling momentum transport in quantum magnetic systems.

This paper is organized as follows. Section 2 presents the fundamental model, detailing the Hamiltonian and Lagrange formulation of the system. Section 3 establishes the Keldysh formalism for analyzing momentum transport. The energy–momentum tensor formulation is given in Section 4. Then Section 5 analyzes the low-velocity regime, deriving perturbative expressions for the momentum transport, while Section 6 examines the weak-coupling limit to elucidate fundamental scaling relationships. Temperature-dependent behavior across quantum regimes is investigated in Section 7. This study concludes with Section 8.

2. Model and System

2.1. General Hamitonian of Ferromagnetic Insulator and Plasmon

To construct our model, we first present the parts of the Hamiltonian of the system. Considering a ferromagnetic insulator with uniaxial anisotropy, the easy axis is aligned along the z-direction, and an external magnetic field $\mathbf{B}=B\widehat{z}$ is applied along this axis. The spin chain is characterized by an exchange coupling constant J, which corresponds to the exchange stiffness A in the continuum limit by $J=2A/M_s$, where $M_s$ is the saturation magnetization. The magnetic film is assumed to be infinite in the x-y plane and has a finite thickness h along the z-direction. In the second-quantized formalism, the Hamiltonian of magnons can be expressed as follows:

$$H=\sum _n\int {\displaystyle \frac{d^2{k}_{\Vert }}{{\left(2\pi \right)}^2}}\hslash {\omega }_n\left({\mathbf{k}}_{\Vert }\right)a_{n,{\mathbf{k}}_{\Vert }}^{†}{a}_{n,{\mathbf{k}}_{\Vert }},$$

(1)

where $a_{n,\mathbf{k}}^{†}$ and $a_{n,\mathbf{k}}$ are the creation and annihilation operators for a magnon with wave vector ${\mathbf{k}}_{\Vert }=(k_x,k_y)$ and mode index n, and ${\omega }_n\left({\mathbf{k}}_{\Vert }\right)$ is the magnon dispersion relation.

The dispersion relation ${\omega }_n\left({\mathbf{k}}_{\Vert }\right)$ for magnons can be obtained by solving the spin wave equation in the ferromagnet with free boundary conditions as follows:

$$\hslash {\omega }_n\left({\mathbf{k}}_{\Vert }\right)=J{\mathbf{k}}_{\Vert }^2+J{\left({\displaystyle \frac{n\pi }{h}}\right)}^2+K+B.$$

(2)

The quantity $J{\mathbf{k}}_{\Vert }^2$ represents the in-plane exchange interaction, while $J{(n\pi /h)}^2$ denotes the quantized exchange energy along the thickness direction, with $n=0,1,2,\dots$. The parameter K is the anisotropy constant, corresponding to the magnetocrystalline anisotropy energy. The term B represents the Zeeman energy associated with the external magnetic field, given by $B=g{\mu }_B{B}_0$, where $B_0$ is the applied magnetic field strength.

Substituting the dispersion relation Equation (2) into the Hamiltonian expression Equation (1) yields the following:

$$\mathcal{H}=\sum _{n=0}^{\infty }\int {\displaystyle \frac{d^2{\mathbf{k}}_{\Vert }}{{\left(2\pi \right)}^2}}\left[J{\mathbf{k}}_{\Vert }^2+J{\left({\displaystyle \frac{n\pi }{h}}\right)}^2+K+B\right]a_{n,{\mathbf{k}}_{\Vert }}^{†}{a}_{n,{\mathbf{k}}_{\Vert }}.$$

(3)

Due to the finite film thickness h, the magnon wave vector along the z-direction becomes quantized as $k_z=n\pi /h$, leading to a set of discrete modes. For ${\mathbf{k}}_{\Vert }=0$ and $n=0$, the magnon spectrum exhibits an energy gap $\Delta =K+B$, which is induced by both the magnetic anisotropy and the external field. This linearized model is valid at low temperatures and for small spin deviations from the ordered ground state, consistent with the assumptions of linear spin wave theory.

On the other hand, for plasmons in a two-dimensional electron gas, the effective Lagrangian density can be written as follows:

$$\mathcal{L}={\displaystyle \frac{1}{2}}{\left({\partial }_t\varphi \right)}^2-{\displaystyle \frac{1}{2}}{c}^2{(\nabla \varphi )}^2-{\displaystyle \frac{1}{2}}{m}^2{\varphi }^2,$$

(4)

where $\varphi (t,{\mathbf{r}}_{\Vert })$ is a real scalar field describing plasmon excitations, c is the effective plasmon velocity, which in this model corresponds to the plasmon group velocity, and m is a parameter related to the plasmon energy gap.

To obtain the plasmon Hamiltonian, we performed a Fourier expansion of the field as follows:

$$\varphi (\mathbf{r},t)=\int {\displaystyle \frac{d^{2}k}{{\left(2\pi \right)}^2}}{\displaystyle \frac{1}{\sqrt{2{\omega }_k}}}\left(a_k{e}^{i(\mathbf{k}·\mathbf{r}-{\omega }_{k}t)}+a_k^{†}{e}^{-i(\mathbf{k}·\mathbf{r}-{\omega }_{k}t)}\right),$$

(5)

with the dispersion relation ${\omega }_k=\sqrt{c^2{k}^2+m^2}$. The canonical momentum density is

$$\pi (\mathbf{r},t)={\displaystyle \frac{\partial \mathcal{L}}{\partial \left({\partial }_t\varphi \right)}}={\partial }_t\varphi ,$$

(6)

which can be expanded in Fourier space as

$$\pi (\mathbf{r},t)=-i\int {\displaystyle \frac{d^{2}k}{{\left(2\pi \right)}^2}}\sqrt{{\displaystyle \frac{{\omega }_k}{2}}}\left(a_k{e}^{i(\mathbf{k}·\mathbf{r}-{\omega }_{k}t)}-a_k^{†}{e}^{-i(\mathbf{k}·\mathbf{r}-{\omega }_{k}t)}\right).$$

(7)

Imposing the equal-time commutation relation

$$[\varphi (\mathbf{r},t),\pi ({\mathbf{r}}^{\prime },t)]=i\hslash {\delta }^2(\mathbf{r}-{\mathbf{r}}^{\prime })$$

(8)

requires the creation and annihilation operators to satisfy the bosonic commutation relations

$$[a_k,a_{k^{\prime }}^{†}]={\left(2\pi \right)}^2{\delta }^2(\mathbf{k}-{\mathbf{k}}^{\prime }),[a_k,a_{k^{\prime }}]=[a_k^{†},a_{k^{\prime }}^{†}]=0.$$

(9)

The plasmon Hamiltonian in real space is then

$$H={\displaystyle \frac{1}{2}}\int d^{2}r\left[{\pi }^2+c^2{(\nabla \varphi )}^2+m^2{\varphi }^2\right],$$

(10)

which, after substituting the Fourier expansions, becomes

$$H=\int {\displaystyle \frac{d^{2}k}{{\left(2\pi \right)}^2}}\hslash {\omega }_k\left(a_k^{†}{a}_k+{\displaystyle \frac{1}{2}}\right),$$

(11)

where the $1/2$ term represents the zero-point energy.

For two-dimensional plasmons, the gap parameter is typically $m=0$, giving a linear dispersion relation ${\omega }_k=c\left|{\mathbf{k}}_{\Vert }\right|$. In a two-dimensional electron gas, c depends on parameters such as the electron density and effective mass. The zero-point energy term ${\displaystyle \frac{1}{2}}\int {\displaystyle \frac{d^{2}k}{{\left(2\pi \right)}^2}}\hslash {\omega }_k$ is usually omitted when calculating observable quantities.

2.2. The Lagrangian of System

We consider a three-layer system consisting of a ferromagnetic insulator sandwiched between two metallic plasmon layers. The top layer, located at $z>h$, supports two-dimensional plasmons described by the field ${\varphi }_1({\mathbf{r}}_{\Vert },t)$. The middle layer, spanning $0\le z\le h$, is a ferromagnetic insulator characterized by the magnon field $\psi ({\mathbf{r}}_{\Vert },z,t)$. The bottom layer, located at $z<0$, is another two-dimensional plasmon layer with field ${\varphi }_2({\mathbf{r}}_{\Vert },t)$. Here, ${\mathbf{r}}_{\Vert }=(x,y)$ denotes the in-plane coordinates. Then, the total Lagrangian density of the system consists of the following three parts:

$${\mathcal{L}}_{\mathrm{total}}={\mathcal{L}}_{\mathrm{plasmon}}+{\mathcal{L}}_{\mathrm{magnon}}+{\mathcal{L}}_{\mathrm{coupling}}.$$

(12)

From Equation (4), the Lagrangian density for the two-dimensional plasmon layers can be written as follows:

$${\mathcal{L}}_{\mathrm{plasmon}}=\sum _{i=1}^2\left[{\displaystyle \frac{1}{2}}{\left({\partial }_t{\varphi }_i\right)}^2-{\displaystyle \frac{1}{2}}{c}_i^2{\left({\nabla }_{\Vert }{\varphi }_i\right)}^2-{\displaystyle \frac{1}{2}}{m}_i^2{\varphi }_i^2\right]\delta (z-z_i),$$

(13)

where ${\varphi }_i({\mathbf{r}}_{\Vert },t)$ represents the corresponding plasmon field, and $i=1,2$ denotes the top and bottom plasmon layers, respectively. The layer positions are given by $z_1=h$ and $z_2=0$. The parameters $c_i$ and $m_i$ denote the effective plasmon velocity and mass term for the i-th layer, respectively. The operator ${\nabla }_{\Vert }=({\partial }_x,{\partial }_y)$ represents the in-plane gradient, and $\delta (z-z_i)$ enforces the confinement of plasmons to the two-dimensional planes.

The Lagrangian density for magnons in the ferromagnetic insulator is as follows:

$${\mathcal{L}}_{\mathrm{magnon}}={\displaystyle \frac{i}{2}}({\psi }^{†}{\partial }_t\psi -{\partial }_t{\psi }^{†}\psi )-{\displaystyle \frac{1}{2M}}{|\nabla \psi |}^2-{\displaystyle \frac{\Delta }{2}}{\left|\psi \right|}^2.$$

(14)

The parameter M represents the effective mass associated with the exchange stiffness, and $\Delta$ corresponds to the magnon energy gap, incorporating contributions from both the magnetic anisotropy and the external magnetic field. The operator $\nabla =({\partial }_x,{\partial }_y,{\partial }_z)$ is the three-dimensional gradient.

The interaction occurs between the plasmon layers and the ferromagnetic layer, a form of coupling that guarantees that the Lagrangian remains real. Thus, we use a linear coupling between plasmons and magnons to describe that interaction as follows:

$${\mathcal{L}}_{\mathrm{coupling}}=\sum _{i=1}^2{g}_i\left[{\varphi }_i({\mathbf{r}}_{\Vert },t){\psi }^{†}({\mathbf{r}}_{\Vert },z_i,t)+{\varphi }_i({\mathbf{r}}_{\Vert },t)\psi ({\mathbf{r}}_{\Vert },z_i,t)\right]\delta (z-z_i),$$

(15)

where, $g_i$ denotes the coupling constant at the i-th interface.

Combining all parts, the complete Lagrangian density of the total system is as follows:

$$\begin{array}{cc}\hfill {\mathcal{L}}_{\mathrm{total}}=& \sum _{i=1}^2\left[{\displaystyle \frac{1}{2}}{\left({\partial }_t{\varphi }_i\right)}^2-{\displaystyle \frac{1}{2}}{c}_i^2{\left({\nabla }_{\Vert }{\varphi }_i\right)}^2-{\displaystyle \frac{1}{2}}{m}_i^2{\varphi }_i^2\right]\delta (z-z_i)\hfill \\ \hfill & +{\displaystyle \frac{i}{2}}({\psi }^{†}{\partial }_t\psi -{\partial }_t{\psi }^{†}\psi )-{\displaystyle \frac{1}{2M}}{|\nabla \psi |}^2-{\displaystyle \frac{\Delta }{2}}{\left|\psi \right|}^2\hfill \\ \hfill & +\sum _{i=1}^2{g}_i\left[{\varphi }_i{\psi }^{†}+{\varphi }_i\psi \right]\delta (z-z_i).\hfill \end{array}$$

(16)

3. Keldysh Formalism

3.1. Basics of Keldysh Formalism

We evaluate the energy–momentum tensor in a non-equilibrium steady state by deriving the real-time Green’s functions for the coupled magnon–plasmon dynamics within the path-integral formulation of Keldysh field theory. In the Keldysh formalism, field variables are defined on a closed time contour, consisting of a forward branch (+) from $-\infty$ to $+\infty$ and a backward branch (−) from $+\infty$ to $-\infty$. For bosonic fields, we introduce classical components ${\psi }_{\mathrm{cl}},{\varphi }_{i,\mathrm{cl}}$ and quantum components ${\psi }_{\mathrm{q}},{\varphi }_{i,\mathrm{q}}$, defined as follows:

$$\begin{array}{cc}\hfill & {\psi }_{\mathrm{cl}}={\displaystyle \frac{1}{\sqrt{2}}}({\psi }_{+}+{\psi }_{-}),{\psi }_{\mathrm{q}}={\displaystyle \frac{1}{\sqrt{2}}}({\psi }_{+}-{\psi }_{-}),\hfill \\ \hfill & {\varphi }_{i,\mathrm{cl}}={\displaystyle \frac{1}{\sqrt{2}}}({\varphi }_{i+}+{\varphi }_{i-}),{\varphi }_{i,\mathrm{q}}={\displaystyle \frac{1}{\sqrt{2}}}({\varphi }_{i+}-{\varphi }_{i-}).\hfill \end{array}$$

(17)

Thus, the Keldysh action of the system is as follows:

$$S_{\mathrm{Keldysh}}=\int dt\int d^2{\mathbf{r}}_{\Vert }{\int }_0^{h}dz{\mathcal{L}}_{\mathrm{Keldysh}},$$

(18)

where the Keldysh Lagrangian density is as follows:

$${\mathcal{L}}_{\mathrm{Keldysh}}={\mathcal{L}}_{\mathrm{plasma}}+{\mathcal{L}}_{\mathrm{magnon}}+{\mathcal{L}}_{\mathrm{coupling}}.$$

(19)

Within the Keldysh formalism, transformation to frequency–momentum space plays a central role. This procedure converts complex spatiotemporal differential operations into algebraic ones, thereby facilitating the analysis of equations of motion and Dyson equations [20,21]. Moreover, it provides direct insight into the intrinsic physics of the system, revealing properties such as excitation spectra and mode couplings. By performing a Fourier transform, the field can be expressed as follows:

$$\psi ({\mathbf{r}}_{\Vert },z,t)=\int {\displaystyle \frac{d\omega }{2\pi }}{\displaystyle \frac{d^2{\mathbf{k}}_{\Vert }}{{\left(2\pi \right)}^2}}{e}^{i({\mathbf{k}}_{\Vert }·{\mathbf{r}}_{\Vert }-\omega t)}\psi ({\mathbf{k}}_{\Vert },z,\omega ),$$

(20)

and the action is then written as follows:

$$S_{\mathrm{Keldysh}}=\int {\displaystyle \frac{d\omega }{2\pi }}{\displaystyle \frac{d^2{\mathbf{k}}_{\Vert }}{{\left(2\pi \right)}^2}}{\int }_0^{h}dz{\mathcal{L}}_{\mathrm{Keldysh}}({\mathbf{k}}_{\Vert },z,\omega ).$$

(21)

3.2. The Green’s Function of Keldysh Space

In Keldysh space, the magnon Green’s function is represented as a $2\times 2$ matrix as follows:

$${\widehat{G}}_{\psi }({\mathbf{k}}_{\Vert },z,z^{\prime },\omega )=\left(\begin{array}{cc}{G}^R& G^K\\ 0& G^A\end{array}\right),$$

(22)

where $G^R$, $G^A$, and $G^K$ denote the retarded, advanced, and Keldysh Green’s functions, respectively. In thermal equilibrium, the fluctuation–dissipation theorem relates $G^K$ to $G^R$ and $G^A$ as follows:

$$G^K\left(\omega \right)=\left[G^R\left(\omega \right)-G^A\left(\omega \right)\right]coth\left({\displaystyle \frac{\omega }{2T}}\right).$$

(23)

The retarded Green’s function for free magnons can be obtained from the Lagrangian as follows:

$$\left[\omega -{\displaystyle \frac{k_{\Vert }^2}{2M}}-{\displaystyle \frac{1}{2M}}{\partial }_z^2-\Delta +i{0}^{+}\right]G_0^R({\mathbf{k}}_{\Vert },z,z^{\prime },\omega )=\delta (z-z^{\prime }),$$

(24)

subject to free boundary conditions ${\partial }_z{G}_0^R{{|}_{z=0}={\partial }_z{G}_0^R|}_{z=h}=0$. The solution admits the following mode expansion:

$$G_0^R({\mathbf{k}}_{\Vert },z,z^{\prime },\omega )=\sum _{n=0}^{\infty }{\displaystyle \frac{{\phi }_n\left(z\right){\phi }_n\left(z^{\prime }\right)}{\omega -{\epsilon }_n\left({\mathbf{k}}_{\Vert }\right)+i{0}^{+}}},$$

(25)

with mode functions

$${\phi }_n\left(z\right)=\sqrt{{\displaystyle \frac{2-{\delta }_{n0}}{h}}}cos\left({\displaystyle \frac{n\pi z}{h}}\right),$$

(26)

and magnon energies

$${\epsilon }_n\left({\mathbf{k}}_{\Vert }\right)={\displaystyle \frac{k_{\Vert }^2}{2M}}+{\displaystyle \frac{1}{2M}}{\left({\displaystyle \frac{n\pi }{h}}\right)}^2+\Delta .$$

(27)

Similarly, the retarded Green’s function for free plasmons is given by

$$D_{0,i}^R({\mathbf{k}}_{\Vert },\omega )={\displaystyle \frac{1}{{\omega }^2-c_i^2{k}_{\Vert }^2-m_i^2+i{0}^{+}}}.$$

(28)

The self-energy due to the magnon–plasmon coupling reads as follows:

$${\mathrm{\Sigma }}^R({\mathbf{k}}_{\Vert },z,z^{\prime },\omega )=\sum _{i=1}^2{g}_i^2\delta (z-z_i)\delta (z^{\prime }-z_i)D_{0,i}^R({\mathbf{k}}_{\Vert },\omega ).$$

(29)

The exact retarded Green’s function for magnons satisfies the Dyson equation:

$$G^R=G_0^R+G_0^R\circ {\mathrm{\Sigma }}^R\circ G^R,$$

(30)

which in position space is equivalent to the integral equation:

$$G^R({\mathbf{k}}_{\Vert },z,z^{\prime },\omega )=G_0^R({\mathbf{k}}_{\Vert },z,z^{\prime },\omega )+\sum _{i=1}^2{g}_i^2{D}_{0,i}^R({\mathbf{k}}_{\Vert },\omega )G_0^R({\mathbf{k}}_{\Vert },z,z_i,\omega )G^R({\mathbf{k}}_{\Vert },z_i,z^{\prime },\omega ).$$

(31)

Restricting the magnon Green’s function to the interfaces, we define the boundary Green’s function:

$$G_{ij}^R({\mathbf{k}}_{\parallel },\omega )=G^R({\mathbf{k}}_{\parallel },z_i,z_j,\omega ),i,j=1,2.$$

(32)

The Dyson equation becomes a matrix equation as follows:

$$\left(\begin{array}{cc}{G}_{11}^R& G_{12}^R\\ G_{21}^R& G_{22}^R\end{array}\right)=\left(\begin{array}{cc}{G}_{0,11}^R& G_{0,12}^R\\ G_{0,21}^R& G_{0,22}^R\end{array}\right)+\left(\begin{array}{cc}{g}_1^2{D}_{0,1}^R{G}_{0,11}^R& g_1^2{D}_{0,2}^R{G}_{0,12}^R\\ g_2^2{D}_{0,1}^R{G}_{0,21}^R& g_2^2{D}_{0,2}^R{G}_{0,22}^R\end{array}\right)\left(\begin{array}{cc}{G}_{11}^R& G_{12}^R\\ G_{21}^R& G_{22}^R\end{array}\right).$$

(33)

The solution is

$$\left(\begin{array}{cc}{G}_{11}^R& G_{12}^R\\ G_{21}^R& G_{22}^R\end{array}\right)={\left[\mathbb{I}-\left(\begin{array}{cc}{g}_1^2{D}_{0,1}^R{G}_{0,11}^R& g_1^2{D}_{0,2}^R{G}_{0,12}^R\\ g_2^2{D}_{0,1}^R{G}_{0,21}^R& g_2^2{D}_{0,2}^R{G}_{0,22}^R\end{array}\right)\right]}^{-1}\left(\begin{array}{cc}{G}_{0,11}^R& G_{0,12}^R\\ G_{0,21}^R& G_{0,22}^R\end{array}\right).$$

(34)

3.3. Non-Equilibrium Green’s Function

By substituting the boundary Green’s functions, the retarded Green’s function can be expressed in iterative form as follows:

$$\begin{array}{cc}\hfill G^R({\mathbf{k}}_{\Vert },z,z^{\prime },\omega )=& G_0^R({\mathbf{k}}_{\Vert },z,z^{\prime },\omega )\hfill \\ \hfill & +\sum _{i,j=1}^2{g}_i{g}_j{D}_0^R({\mathbf{k}}_{\Vert },\omega )G_0^R({\mathbf{k}}_{\Vert },z,z_i,\omega )G_{ij}^R({\mathbf{k}}_{\Vert },\omega )G_0^R({\mathbf{k}}_{\Vert },z_j,z^{\prime },\omega ).\hfill \end{array}$$

(35)

The corresponding advanced Green’s function is given by

$$G^A({\mathbf{k}}_{\Vert },z,z^{\prime },\omega )={\left[G^R({\mathbf{k}}_{\Vert },z^{\prime },z,\omega )\right]}^{*},$$

(36)

while the Keldysh Green’s function in thermal equilibrium reads as follows:

$$G^K({\mathbf{k}}_{\Vert },z,z^{\prime },\omega )=\left[G^R({\mathbf{k}}_{\Vert },z,z^{\prime },\omega )-G^A({\mathbf{k}}_{\Vert },z,z^{\prime },\omega )\right]coth\left({\displaystyle \frac{\omega }{2T}}\right).$$

(37)

The coupling to plasmons introduces self-energy corrections ${\mathrm{\Sigma }}^R$ for magnons, modifying both their dispersion and lifetime. Boundary coupling gives rise to hybridized plasmon-magnon modes. The Keldysh component accounts for quantum and thermal fluctuations under equilibrium conditions, whereas the retarded and advanced Green’s functions describe energy transfer across the interfaces.

4. Energy-Momentum Tensor of Magnons

In our study, the relative motion of the top plasmon layer along the x-direction with velocity v induces a Galilean boost in its frequency spectrum. This boost shifts the frequency in the retarded Green’s function from $\omega$ to $\omega -v{k}_x$, yielding $D_{0,2}^R({\mathbf{k}}_{\Vert },\omega )$, while the free retarded Green’s function for the bottom plasmon remains unchanged as $D_{0,2}^R({\mathbf{k}}_{\Vert },\omega )$. Magnons are assumed to be in thermal equilibrium at temperature T at initial time, so the Keldysh Green’s function satisfies the fluctuation-dissipation theorem. The system structure is maintained with the top plasmon layer at $z=h$, the ferromagnetic insulator layer for $0\le z\le h$, and the bottom plasmon layer at $z=0$.

Definition of the Magnon Energy-Momentum Tensor

For the magnon field $\psi$, its energy–momentum tensor derives from the Lagrangian density ${\mathcal{L}}_{\mathrm{magnon}}$ of Equation (14), thus, the $xz$-component of the energy–momentum tensor can be written as follows:

$$\begin{array}{cc}\hfill T_{xz}& ={\displaystyle \frac{\partial \mathcal{L}}{\partial \left({\partial }_x\psi \right)}}{\partial }_z\psi +{\displaystyle \frac{\partial \mathcal{L}}{\partial \left({\partial }_x{\psi }^{†}\right)}}{\partial }_z{\psi }^{†}-{\delta }_{xz}\mathcal{L}\hfill \\ \hfill & =-{\displaystyle \frac{1}{2M}}\left({\partial }_x{\psi }^{†}{\partial }_z\psi +{\partial }_x\psi {\partial }_z{\psi }^{†}\right).\hfill \end{array}$$

(38)

In thermal equilibrium, the expectation value of the $xz$-component of the energy–momentum tensor then becomes the following:

$$\langle T_{xz}\rangle =-{\displaystyle \frac{1}{2M}}〈{\partial }_x{\psi }^{†}{\partial }_z\psi +{\partial }_x\psi {\partial }_z{\psi }^{†}〉.$$

(39)

Utilizing the properties of the equal-time limit of Green’s functions, this can be expressed as follows:

$$\langle {\partial }_x{\psi }^{†}(\mathbf{r},t){\partial }_z\psi (\mathbf{r},t)\rangle =\underset{t^{\prime }\to t}{lim}{\partial }_x{\partial }_z\langle {\psi }^{†}({\mathbf{r}}^{\prime },t^{\prime })\psi (\mathbf{r},t)\rangle .$$

(40)

In Fourier space, its corresponds to

$$\langle {\partial }_x{\psi }^{†}{\partial }_z\psi \rangle =\int {\displaystyle \frac{d\omega }{2\pi }}{\displaystyle \frac{d^2{\mathbf{k}}_{\Vert }}{{\left(2\pi \right)}^2}}(-i{k}_x){\partial }_z{G}^{<}({\mathbf{k}}_{\Vert },z,z^{\prime },\omega ){|}_{z^{\prime }=z},$$

(41)

where $G^{<}$ is the lesser Green’s function.

In the Keldysh formalism, the relationship between the lesser Green’s function and the Keldysh Green’s function is

$$G^{<}={\displaystyle \frac{1}{2}}(G^K-G^R+G^A).$$

(42)

By the fluctuation–dissipation theorem of Equation (37), its can be written as follows:

$$\begin{array}{cc}\hfill G^{<}({\mathbf{k}}_{\Vert },z,z^{\prime },\omega )& ={\displaystyle \frac{1}{2}}\left[(G^R-G^A)coth\left({\displaystyle \frac{\omega }{2T}}\right)-G^R+G^A\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}(G^R-G^A)\left[coth\left({\displaystyle \frac{\omega }{2T}}\right)-1\right].\hfill \end{array}$$

(43)

Substituting the above results into Equation (39), the specific expression for $\langle T_{xz}\rangle$ is

$$\begin{array}{cc}\hfill \langle T_{xz}\rangle & ={\displaystyle \frac{i}{2M}}\int {\displaystyle \frac{d\omega }{2\pi }}{\displaystyle \frac{d^2{\mathbf{k}}_{\Vert }}{{\left(2\pi \right)}^2}}{k}_x\hfill \\ \hfill & \times {\left[{\partial }_z{G}^{<}({\mathbf{k}}_{\Vert },z,z^{\prime },\omega )+{\partial }_z^{\prime }{G}^{<}({\mathbf{k}}_{\Vert },z,z^{\prime },\omega )\right]}_{z^{\prime }=z}\hfill \\ \hfill & ={\displaystyle \frac{i}{4M}}\int {\displaystyle \frac{d\omega }{2\pi }}{\displaystyle \frac{d^2{\mathbf{k}}_{\Vert }}{{\left(2\pi \right)}^2}}{k}_x\left[coth\left({\displaystyle \frac{\omega }{2T}}\right)-1\right]\hfill \\ \hfill & \times {\left[{\partial }_z(G^R-G^A)+{\partial }_z^{\prime }(G^R-G^A)\right]}_{z^{\prime }=z}.\hfill \end{array}$$

(44)

Here, the derivatives ${\partial }_z$ and ${\partial }_z^{\prime }$ account for spatial variations within the finite-thickness ferromagnetic layer.

Considering the frequency shift of the plasmons, the plasmon Green’s function is modified by a Galilean boost, replacing $\omega$ with $\omega -v{k}_x$. This momentum-dependent shift alters the energy spectrum and induces anisotropic self-energy corrections for magnons, leading to a non-zero value of $\langle T_{xz}\rangle$. Consequently, the magnon Green’s function is modified by these self-energy corrections as follows:

$$\begin{array}{cc}\hfill G^R({\mathbf{k}}_{\Vert },z,z^{\prime },\omega )& =G_0^R({\mathbf{k}}_{\Vert },z,z^{\prime },\omega )\hfill \\ \hfill & +\sum _{i,j=1}^2{g}_i{g}_j{D}_{0,i}^R({\mathbf{k}}_{\Vert },{\tilde{\omega }}_i)G_0^R({\mathbf{k}}_{\Vert },z,z_i,\omega )G_{ij}^R({\mathbf{k}}_{\Vert },\omega )G_0^R({\mathbf{k}}_{\Vert },z_j,z^{\prime },\omega ).\hfill \end{array}$$

(45)

Here, ${\tilde{\omega }}_1=\omega -v{k}_x$ represents the frequency-shifted top plasmon, while ${\tilde{\omega }}_2=\omega$ corresponds to the bottom plasmon without a frequency shift. The boundary coupling coefficients $g_1$ and $g_2$ affect the energy–momentum tensor distribution through the Green’s function $G_{ij}^R$. The term $coth\left({\displaystyle \frac{\omega }{2T}}\right)-1$ captures the contribution of thermal fluctuations. In the zero-temperature limit $T\to 0$, it reduces to $\mathrm{sgn}\left(\omega \right)-1$, indicating that only negative frequencies contribute. By using the identity $coth\left({\displaystyle \frac{\omega }{2T}}\right)-1={\displaystyle \frac{2}{e^{\omega /T}-1}}$, the average energy–momentum tensor can be expressed as follows:

$$\langle T_{xz}\rangle ={\displaystyle \frac{i}{2M}}\int {\displaystyle \frac{d\omega }{2\pi }}{\displaystyle \frac{d^2{\mathbf{k}}_{\Vert }}{{\left(2\pi \right)}^2}}{\displaystyle \frac{k_x}{e^{\omega /T}-1}}{\left[{\partial }_z(G^R-G^A)+{\partial }_z^{\prime }(G^R-G^A)\right]}_{z^{\prime }=z}.$$

(46)

The quantity $\langle T_{xz}\rangle$ characterizes the transport of momentum along the x-direction through the z-direction, driven by the frequency shift parameter v. These results establish a framework for analyzing momentum transport, energy transfer, and anisotropic transport phenomena in heterostructures, particularly under non-equilibrium conditions induced by the driving field associated with the frequency shift $v{k}_x$.

From Equation (46) we know that the function $1/(e^{\omega /T}-1)$ has poles at $\omega =2\pi inT$ where $n=0,\pm 1,\pm 2,\dots$ with residues T, while $(G^R-G^A)$ has poles at the system’s eigenfrequencies ${\omega }_n\left({\mathbf{k}}_{\Vert }\right)$. Applying the residue theorem, the integral along the real axis can be expressed as a sum over the residues at the poles of the Bose distribution function and the Green’s function. Thus, by performing the frequency integral, Equation (46) can be written as follows:

$$\begin{array}{cc}\hfill & {\int }_{-\infty }^{\infty }{\displaystyle \frac{d\omega }{2\pi }}{\displaystyle \frac{1}{e^{\omega /T}-1}}{\left[{\partial }_z(G^R-G^A)+{\partial }_z^{\prime }(G^R-G^A)\right]}_{z^{\prime }=z}\hfill \\ \hfill & =-2\pi i\sum _{{\omega }_n}\mathrm{Res}\left[{\displaystyle \frac{1}{e^{\omega /T}-1}}{\left({\partial }_z{G}^R+{\partial }_z^{\prime }{G}^R\right)}_{z^{\prime }=z},\omega ={\omega }_n\right]\hfill \\ \hfill & +2\pi i\sum _{n\ne 0}\mathrm{Res}\left[{\displaystyle \frac{1}{e^{\omega /T}-1}}{\left({\partial }_z{G}^R+{\partial }_z^{\prime }{G}^R\right)}_{z^{\prime }=z},\omega =2\pi inT\right].\hfill \end{array}$$

(47)

Since $G^R$ is analytic in the upper half-plane, the second term is typically zero. Therefore, the main contribution comes from the first term

$${\int }_{-\infty }^{\infty }{\displaystyle \frac{d\omega }{2\pi }}{\displaystyle \frac{1}{e^{\omega /T}-1}}{\left[{\partial }_z(G^R-G^A)+{\partial }_z^{\prime }(G^R-G^A)\right]}_{z^{\prime }=z}=-2\pi i\sum _n{\displaystyle \frac{1}{e^{{\omega }_n/T}-1}}{R}_n({\mathbf{k}}_{\Vert },z),$$

(48)

where $R_n({\mathbf{k}}_{\Vert },z)$ is the residue of ${[{\partial }_z{G}^R+{\partial }_z^{\prime }{G}^R]}_{z^{\prime }=z}$ at the pole $\omega ={\omega }_n\left({\mathbf{k}}_{\Vert }\right)$. And $G_n^R$ is the principal singular part of $G^R$ near the pole ${\omega }_n$. Substituting into the energy–momentum tensor expression yields the following:

$$\langle T_{xz}\rangle ={\displaystyle \frac{1}{2M}}\int {\displaystyle \frac{d^2{\mathbf{k}}_{\Vert }}{{\left(2\pi \right)}^2}}{k}_x\sum _n{\displaystyle \frac{R_n({\mathbf{k}}_{\Vert },z)}{e^{{\omega }_n\left({\mathbf{k}}_{\Vert }\right)/T}-1}}.$$

(49)

The summation is taken over all eigenmodes n, where ${\omega }_n\left({\mathbf{k}}_{\Vert }\right)$ denotes the n-th eigenfrequency of the coupled system. Since the retarded $G^R$ and advanced $G^A$ Green’s functions exhibit a pole structure for decay rate $\gamma >0$ as follows:

$$G^R\left(\omega \right)=\sum _n{\displaystyle \frac{R_n}{\omega -{\omega }_n+i\gamma }},G^A\left(\omega \right)=\sum _n{\displaystyle \frac{R_n^{*}}{\omega -{\omega }_n-i\gamma }},$$

(50)

the eigenfrequencies ${\omega }_n\left({\mathbf{k}}_{\Vert }\right)$ satisfy the Dyson equation for the coupled system as follows:

$$det\left[\mathbb{I}-G_0^R({\mathbf{k}}_{\Vert },\omega )·\mathrm{\Sigma }({\mathbf{k}}_{\Vert },\omega )\right]=0,$$

(51)

where $\mathrm{\Sigma }$ includes the effect of the plasmon frequency shift.

5. Low-Velocity Approximation for the Energy-Momentum Tensor

The low-velocity regime ($v\to 0$) is particularly significant as it permits a controlled perturbative expansion, revealing the intrinsic linear-response characteristics of the momentum transport. Based on the previous derivation of the residue form of the energy–momentum tensor, in the limit $v\to 0$, the eigenfrequencies can be expanded as follows:

$${\omega }_n\left({\mathbf{k}}_{\Vert }\right)={\omega }_n^{\left(0\right)}\left({\mathbf{k}}_{\Vert }\right)+v{k}_x{\alpha }_n\left({\mathbf{k}}_{\Vert }\right)+O\left(v^2\right),$$

(52)

where ${\omega }_n^{\left(0\right)}\left({\mathbf{k}}_{\Vert }\right)$ denotes the eigenfrequency for $v=0$, and ${\alpha }_n\left({\mathbf{k}}_{\Vert }\right)$ is the linear response coefficient describing the frequency shift. Similarly, the residue can be expanded as follows:

$$R_n({\mathbf{k}}_{\Vert },z)=R_n^{\left(0\right)}({\mathbf{k}}_{\Vert },z)+v{k}_x{\beta }_n({\mathbf{k}}_{\Vert },z)+O\left(v^2\right),$$

(53)

with ${\beta }_n({\mathbf{k}}_{\Vert },z)$ characterizing the change in the mode weight due to the frequency shift. The Bose distribution function expands to first order in v as follows:

$${\displaystyle \frac{1}{e^{{\omega }_n\left({\mathbf{k}}_{\Vert }\right)/T}-1}}={\displaystyle \frac{1}{e^{{\omega }_n^{\left(0\right)}/T}-1}}-v{k}_x{\alpha }_n{\displaystyle \frac{e^{{\omega }_n^{\left(0\right)}/T}}{T{(e^{{\omega }_n^{\left(0\right)}/T}-1)}^2}}+O\left(v^2\right),$$

(54)

where $n_B\left(\omega \right)=1/(e^{\omega /T}-1)$. Substituting these expansions into the expression for the energy–momentum tensor yields the following:

$$\begin{array}{cc}\hfill \langle T_{xz}\rangle & ={\displaystyle \frac{1}{2M}}\int {\displaystyle \frac{d^2{\mathbf{k}}_{\Vert }}{{\left(2\pi \right)}^2}}\sum _n{k}_x\left[R_n^{\left(0\right)}+v{k}_x{\beta }_n\right]\hfill \\ \hfill & \times \left[n_B\left({\omega }_n^{\left(0\right)}\right)-v{k}_x{\alpha }_n{\displaystyle \frac{e^{{\omega }_n^{\left(0\right)}/T}}{T}}{n}_B^2\left({\omega }_n^{\left(0\right)}\right)\right]+O\left(v^2\right).\hfill \end{array}$$

(55)

Expanding to first order in v gives

$$\langle T_{xz}\rangle ={\displaystyle \frac{1}{2M}}\int {\displaystyle \frac{d^2{\mathbf{k}}_{\Vert }}{{\left(2\pi \right)}^2}}\sum _n\left[k_x{R}_n^{\left(0\right)}{n}_B\left({\omega }_n^{\left(0\right)}\right)+v{k}_x^2{\beta }_n{n}_B\left({\omega }_n^{\left(0\right)}\right)-v{k}_x^2{\alpha }_n{R}_n^{\left(0\right)}{\displaystyle \frac{e^{{\omega }_n^{\left(0\right)}/T}}{T}}{n}_B^2\left({\omega }_n^{\left(0\right)}\right)\right].$$

(56)

Within the low-velocity regime $v\to 0$ of the perturbative expansion, and in this case, second-order effects have been neglected, which are much smaller than the first-order term.

Due to the rotational symmetry of the system when $v=0$, the first term integrates to zero. Therefore, the leading contribution arises from the linear term in v, and the final expression for the energy–momentum tensor in polar coordinates can be written as follows:

$$\langle T_{xz}\rangle ={\displaystyle \frac{v}{4M}}{\int }_0^{\infty }{\displaystyle \frac{k^{3}dk}{2\pi }}\sum _n\left[{\beta }_n({\mathbf{k}}_{\Vert },z)n_B\left({\omega }_n^{\left(0\right)}\right)-{\alpha }_n\left({\mathbf{k}}_{\Vert }\right)R_n^{\left(0\right)}({\mathbf{k}}_{\Vert },z){\displaystyle \frac{e^{{\omega }_n^{\left(0\right)}/T}}{T}}{n}_B^2\left({\omega }_n^{\left(0\right)}\right)\right].$$

(57)

Due to rotational symmetry, after integrating over the angle, only the dependence on $k=|{\mathbf{k}}_{\Vert }|$ remains. Therefore, the integration variable is simplified from ${\mathbf{k}}_{\Vert }$ to k, and the constant factor from the angular integration is absorbed. The frequency response coefficient ${\alpha }_n\left({\mathbf{k}}_{\Vert }\right)$ and residue response coefficient ${\beta }_n({\mathbf{k}}_{\Vert },z)$ can be computed using perturbation theory as follows:

$${\alpha }_n\left({\mathbf{k}}_{\Vert }\right)={\left.{\displaystyle \frac{\partial {\omega }_n\left({\mathbf{k}}_{\Vert }\right)}{\partial \left(v{k}_x\right)}}\right|}_{v=0},{\beta }_n({\mathbf{k}}_{\Vert },z)={\left.{\displaystyle \frac{\partial R_n({\mathbf{k}}_{\Vert },z)}{\partial \left(v{k}_x\right)}}\right|}_{v=0}.$$

(58)

The contribution of each mode n is further determined by the coefficients ${\alpha }_n$ and ${\beta }_n$, which quantify the effects of the frequency shift on the eigenfrequency and mode weight, respectively.

The energy–momentum tensor exhibits a linear response with respect to the velocity parameter v, such that $\langle T_{xz}\rangle \propto v$. The temperature dependence is reflected in two contributions: the first term, proportional to $n_B\left({\omega }_n^{\left(0\right)}\right)$, accounts for the effects of thermal excitations, while the second term, proportional to $n_B^2\left({\omega }_n^{\left(0\right)}\right)$, represents corrections arising from thermal fluctuations. The momentum dependence is encoded in the integration kernel $k^3$, indicating that higher-momentum modes contribute more significantly to the energy–momentum tensor. In this section, we have given the low-velocity approximation of $v\to 0$, and the unperturbed case is given in Appendix A. In this regime, the energy–momentum tensor scales linearly with v, and no critical velocity emerges within our model, implying that the magnon–plasmon mediated momentum transfer remains governed by linear response for sufficiently small driving velocities.

6. Simplified Case: Weak Coupling Limit

In the weak-coupling limit $(g_1,g_2\ll 1)$, the unperturbed eigenfrequency can be approximated by the free-magnon energy, i.e., ${\omega }_n^{\left(0\right)}$(${\mathbf{k}}_{\Vert })\approx {\epsilon }_n({\mathbf{k}}_{\Vert }$), and the energy–momentum tensor can be approximated as follows:

$$\langle T_{xz}\rangle \approx {\displaystyle \frac{v}{4M}}{\int }_0^{\infty }{\displaystyle \frac{k^{3}dk}{2\pi }}\sum _{n=1}^{\infty }\left[{\beta }_n({\mathbf{k}}_{\Vert },z)n_B\left({\epsilon }_n\right)-{\alpha }_n\left({\mathbf{k}}_{\Vert }\right)R_n^{\left(0\right)}({\mathbf{k}}_{\Vert },z){\displaystyle \frac{e^{{\epsilon }_n/T}}{T}}{n}_B^2\left({\epsilon }_n\right)\right],$$

(59)

where

$$n_B\left({\epsilon }_n\right)={\displaystyle \frac{1}{e^{{\epsilon }_n/T}-1}}.$$

(60)

The coefficients in this approximation are

$$\begin{array}{cc}\hfill & R_n^{\left(0\right)}({\mathbf{k}}_{\Vert },z)\approx 2{\displaystyle \frac{d{\phi }_n\left(z\right)}{dz}}{\phi }_n\left(z\right)=-{\displaystyle \frac{(2-{\delta }_{n0})n\pi }{h^2}}sin\left({\displaystyle \frac{2n\pi z}{h}}\right),\hfill \end{array}$$

(61)

$$\begin{array}{cc}\hfill & {\alpha }_n\left({\mathbf{k}}_{\Vert }\right)\approx -{\displaystyle \frac{2g_1^2{\epsilon }_n{\left|{\phi }_n\left(h\right)\right|}^2}{{[{\epsilon }_n^2-c_1^2{k}^2-m_1^2]}^2}},\hfill \end{array}$$

(62)

$$\begin{array}{cc}\hfill & {\beta }_n({\mathbf{k}}_{\Vert },z)\approx {\alpha }_n\left({\mathbf{k}}_{\Vert }\right){\displaystyle \frac{d}{d{\epsilon }_n}}{R}_n^{\left(0\right)}({\mathbf{k}}_{\Vert },z)=0,\hfill \end{array}$$

(63)

and $R_n^{\left(0\right)}$ does not depend on ${\epsilon }_n$. After simplification, the energy–momentum tensor reduces to

$$\langle T_{xz}\rangle \approx -{\displaystyle \frac{v}{4M}}{\int }_0^{\infty }{\displaystyle \frac{k^{3}dk}{2\pi }}\sum _{n=1}^{\infty }{\alpha }_n\left({\mathbf{k}}_{\Vert }\right)R_n^{\left(0\right)}({\mathbf{k}}_{\Vert },z){\displaystyle \frac{e^{{\epsilon }_n/T}}{T}}{n}_B^2\left({\epsilon }_n\right).$$

(64)

Substituting the explicit forms of the coefficients, we obtain the following:

$$\langle T_{xz}\rangle \approx {\displaystyle \frac{v{g}_1^2}{2M{h}^2}}{\int }_0^{\infty }{\displaystyle \frac{k^{3}dk}{2\pi }}\sum _{n=1}^{\infty }{\displaystyle \frac{(2-{\delta }_{n0})n\pi {\epsilon }_{n}sin\left(\frac{2n\pi z}{h}\right)e^{{\epsilon }_n/T}}{{[{\epsilon }_n^2-c_1^2{k}^2-m_1^2]}^{2}T}}{n}_B^2\left({\epsilon }_n\right).$$

(65)

From the analysis of the system dynamics with weak-coupling limit, we can get several key features of the energy–momentum tensor. The frequency shift effect is characterized by ${\alpha }_n<0$, indicating that the plasmon frequency shift reduces the magnon eigenfrequency, with pronounced mode selectivity where coefficients depend on the boundary values of the mode function ${\phi }_n\left(z\right)$, making boundary modes particularly susceptible. Furthermore, all coefficients exhibit momentum dependence through ${\mathbf{k}}_{\Vert }$, reflecting the spatial dispersion characteristics of the system, while the response coefficients scale quadratically with the coupling constants $g_i^2$, demonstrating that strong coupling significantly enhances the velocity effect. These formal expressions collectively elucidate how the velocity parameter v modulates the dynamical properties via plasmon–magnon coupling, thereby establishing a comprehensive theoretical framework for analyzing the energy–momentum tensor.

7. Low-Temperature and High-Temperature Approximations

To compute the low-temperature and high-temperature approximations, we evaluate the integral over $k_{\Vert }$ in the energy–momentum tensor expression. The integrand peaks at a characteristic momentum $k_0$, allowing the use of the saddle-point approximation or the method of steepest descent. To make the calculation clearer, we define the integral as follows:

$$I_n={\int }_0^{\infty }{\displaystyle \frac{k^3{\epsilon }_n\left(k\right)}{{\left[{\epsilon }_n^2\left(k\right)-c_1^2{k}^2-m_1^2\right]}^2}}{\displaystyle \frac{e^{{\epsilon }_n/T}}{T}}{n}_B^2\left({\epsilon }_n\right)dk.$$

(66)

7.1. Low-Temperature Approximation ($T\ll \Delta$)

At low temperatures, the dominant contribution arises from the lowest mode $n=1$ and small k, with

$${\epsilon }_1\left(k\right)\approx {\Delta }_1+{\displaystyle \frac{k^2}{2M}},{\Delta }_1={\displaystyle \frac{1}{2M}}{\left({\displaystyle \frac{\pi }{h}}\right)}^2+\Delta .$$

(67)

The denominator can be approximated as follows:

$${\epsilon }_1^2-c_1^2{k}^2-m_1^2\approx {\Delta }_1^2-m_1^2+\left({\displaystyle \frac{{\Delta }_1}{M}}-c_1^2\right)k^2.$$

(68)

Define

$$P={\Delta }_1^2-m_1^2,Q={\displaystyle \frac{{\Delta }_1}{M}}-c_1^2.$$

(69)

The Bose function at low temperatures simplifies to the following:

$$n_B\left({\epsilon }_1\right)\approx e^{-{\epsilon }_1/T},{\displaystyle \frac{e^{{\epsilon }_1/T}}{T}}{n}_B^2\left({\epsilon }_1\right)\approx {\displaystyle \frac{1}{T}}{e}^{-{\epsilon }_1/T}.$$

(70)

The integral Equation (66) then becomes the following:

$$I_1\approx {\displaystyle \frac{{\Delta }_1}{T}}{e}^{-{\Delta }_1/T}{\int }_0^{\infty }{\displaystyle \frac{k^3}{{(P+Q{k}^2)}^2}}{e}^{-k^2/\left(2MT\right)}dk.$$

(71)

For $Q{k}^2\ll P$, the denominator can be approximated as constant $P^2$, yielding the following:

$$I_1\approx {\displaystyle \frac{{\Delta }_1}{T{P}^2}}{e}^{-{\Delta }_1/T}{\int }_0^{\infty }{k}^3{e}^{-k^2/\left(2MT\right)}dk={\displaystyle \frac{{\Delta }_1}{T{P}^2}}{e}^{-{\Delta }_1/T}·{\displaystyle \frac{1}{2}}{\left(2MT\right)}^2.$$

(72)

Therefore, the low-temperature integral evaluates to

$$I_1\approx {\displaystyle \frac{2M^{2}T{\Delta }_1}{P^2}}{e}^{-{\Delta }_1/T}.$$

(73)

Thus, the low-temperature limit ($T\ll \Delta$) yields the following:

$$\langle T_{xz}\rangle \approx {\displaystyle \frac{v{g}_1^{2}M}{h^2{({\Delta }_1^2-m_1^2)}^2}}\sum _{n=1}^{\infty }(2-{\delta }_{n0})n\pi {\Delta }_{n}Tsin\left({\displaystyle \frac{2n\pi z}{h}}\right)e^{-{\Delta }_n/T},$$

(74)

where

$${\Delta }_n={\displaystyle \frac{1}{2M}}{\left({\displaystyle \frac{n\pi }{h}}\right)}^2+\Delta .$$

(75)

in this case, $\langle T_{xz}\rangle \propto T{e}^{-{\Delta }_n/T}$, indicating exponential suppression at low temperature.

7.2. High-Temperature Approximation ($T\gg \Delta$)

At high temperatures, the Bose function can be approximated as follows:

$$n_B\left({\epsilon }_n\right)\approx {\displaystyle \frac{T}{{\epsilon }_n}},{\displaystyle \frac{e^{{\epsilon }_n/T}}{T}}{n}_B^2\left({\epsilon }_n\right)\approx {\displaystyle \frac{T}{{\epsilon }_n^2}}.$$

(76)

The integral becomes

$$I_n\approx T{\int }_0^{\infty }{\displaystyle \frac{k^3}{{\epsilon }_n{[{\epsilon }_n^2-c_1^2{k}^2-m_1^2]}^2}}dk.$$

(77)

For large k, ${\epsilon }_n\approx k^2/\left(2M\right)$, we can provide

$$I_n\approx 2MT{\int }_0^{\infty }{\displaystyle \frac{k}{{[k^4/\left(4M^2\right)-c_1^2{k}^2-m_1^2]}^2}}dk.$$

(78)

The integral converges as $k\to \infty$, with the main contribution from $k\sim 2M{c}_1$. So the high-temperature limit ($T\gg \Delta$) is

$$\langle T_{xz}\rangle \approx {\displaystyle \frac{v{g}_1^{2}MT}{h^2}}\sum _{n=1}^{\infty }(2-{\delta }_{n0})n\pi sin\left({\displaystyle \frac{2n\pi z}{h}}\right)f_n^{\mathrm{high}},$$

(79)

where $f_n^{\mathrm{high}}$ is an integral constant in the high-temperature limit, and $\langle T_{xz}\rangle$ grows linearly with T.

Beyond its temperature dependence, the energy–momentum tensor $\langle T_{xz}\rangle$ also exhibits distinct velocity, spatial, mode, and coupling dependencies. The tensor depends linearly on the velocity parameter v, $\langle T_{xz}\rangle \propto v$, reflecting the direct influence of the driving field. Spatially, $\langle T_{xz}\rangle$ displays oscillations within the ferromagnet following $sin(2n\pi z/h)$, and the contribution from low-order modes ($n=1,2,\dots$) dominates the tensor. Moreover, the tensor is proportional to the square of the coupling constant, $\langle T_{xz}\rangle \propto g_1^2$. These results demonstrate the linear effect of the velocity parameter on the energy–momentum tensor and provide a theoretical framework for analyzing momentum transport in heterostructures.

8. Conclusions

In this work, we developed a comprehensive theoretical framework for analyzing momentum transport in ferromagnetic–plasmon heterostructures using Keldysh field theory and energy–momentum tensor formalism. Our central finding is that in a three-layer model, a relative motion-induced plasmon frequency shift can generate significant momentum transport, characterized by a non-zero expectation value of the $xz$-component of the energy–momentum tensor, $\langle T_{xz}\rangle$. This effect is mediated by the magnon–plasmon coupling at the interfaces.

The momentum transport exhibits a linear dependence on the relative velocity v and distinct spatial oscillations following $sin(2n\pi z/h)$ patterns within the ferromagnetic layer. Its temperature dependence reveals a crossover from exponential suppression at low temperatures ($T\ll \Delta$), due to the magnon energy gap, to linear growth at high temperatures ($T\gg \Delta$). The weak-coupling analysis further shows that the response scales quadratically with the interfacial coupling constants $g_i^2$, underscoring the critical role of interface engineering in controlling the transport efficiency.

Our theoretical framework provides fundamental insights into quantum momentum transport mechanisms mediated by magnon–plasmon interactions in magnetic structures. The established formalism provides testable predictions for experimental studies of energy dissipation, including the velocity-linear damping, spatial oscillations of the momentum current, its distinct temperature scaling, and the quadratic dependence on interface coupling. These predictions offer a direct guide for designing optimized heterostructures for spintronic applications. Future work could extend this approach to more complex systems and explore potential applications in quantum information processing and energy transduction.