Chiral ferromagnetism beyond Lifshitz invariants

TL;DR

This paper investigates a chiral contribution to micromagnetic energy in 2D Rashba ferromagnets, revealing new terms beyond traditional Lifshitz invariants caused by spin-orbit coupling effects.

Contribution

It demonstrates that certain chiral energy terms cannot be expressed through Lifshitz invariants beyond linear SOC, highlighting the role of spin rotation symmetry breaking.

Findings

Identification of non-Lifshitz invariant terms in micromagnetic energy

Analysis of effects on magnetic phase diagrams and spin-wave dispersion

Classification of symmetry-allowed terms for crystallographic groups

Abstract

We consider a contribution $w_{\text{ch}}$ to the micromagnetic energy density that is linear with respect to the first spatial derivatives of the local magnetization direction. For a generalized 2D Rashba ferromagnet, we present a microscopic analysis of this contribution and, in particular, demonstrate that it cannot be expressed through Lifshitz invariants beyond the linear order in the spin-orbit coupling (SOC) strength. Terms in $w_{\text{ch}}$ beyond Lifshitz invariants emerge as a result of spin rotation symmetry breaking caused by SOC. Effects of these terms on the phase diagram of magnetic states and spin-wave dispersion are discussed. Finally, we present a classification of terms in $w_{\text{ch}}$, allowed by symmetry, for each crystallographic point group.

Tables

Table 1. Table 1: Classification of LI-type and non-LI-type terms in  w ch subscript 𝑤 ch w_{\text{ch}} allowed by point group symmetries. Here, 𝒜 i ​ j ( k ) = Θ i ​ j ( k ) , Φ i ​ j ( k ) subscript superscript 𝒜 𝑘 𝑖 𝑗 subscript superscript Θ 𝑘 𝑖 𝑗 subscript superscript Φ 𝑘 𝑖 𝑗 \mathcal{A}^{(k)}_{ij}=\Theta^{(k)}_{ij},\Phi^{(k)}_{ij} and we use the notations 𝒬 = 𝒜 x ​ x ( x ) − 𝒜 y ​ y ( x ) − 2 ​ 𝒜 x ​ y ( y ) 𝒬 subscript superscript 𝒜 𝑥 𝑥 𝑥 subscript superscript 𝒜 𝑥 𝑦 𝑦 2 subscript superscript 𝒜 𝑦 𝑥 𝑦 \mathcal{Q}=\mathcal{A}^{(x)}_{xx}-\mathcal{A}^{(x)}_{yy}-2\mathcal{A}^{(y)}_{xy} , 𝒮 = 𝒜 y ​ y ( y ) − 𝒜 x ​ x ( y ) − 2 ​ 𝒜 x ​ y ( x ) 𝒮 subscript superscript 𝒜 𝑦 𝑦 𝑦 subscript superscript 𝒜 𝑦 𝑥 𝑥 2 subscript superscript 𝒜 𝑥 𝑥 𝑦 \mathcal{S}=\mathcal{A}^{(y)}_{yy}-\mathcal{A}^{(y)}_{xx}-2\mathcal{A}^{(x)}_{xy} . For the classes C 2 ​ h subscript 𝐶 2 ℎ C_{2h} ( D 1 ​ d subscript 𝐷 1 𝑑 D_{1d} ), D 2 ​ h subscript 𝐷 2 ℎ D_{2h} , D 3 ​ d subscript 𝐷 3 𝑑 D_{3d} , S 2 subscript 𝑆 2 S_{2} , S 6 subscript 𝑆 6 S_{6} , T h subscript 𝑇 ℎ T_{h} , O h subscript 𝑂 ℎ O_{h} , and C n ​ h subscript 𝐶 𝑛 ℎ C_{nh} , D n ​ h subscript 𝐷 𝑛 ℎ D_{nh} with n > 3 𝑛 3 n>3 , the chiral energy density vanishes identically. Let us give an example of how to use this table. Consider a 2D system of the class C ∞ ​ v subscript 𝐶 𝑣 C_{\infty v} . According to the row 11, terms with the symmetry of LIs enter w ch subscript 𝑤 ch w_{\text{ch}} as a combination ℒ z ​ x ( x ) − ℒ y ​ z ( y ) subscript superscript ℒ 𝑥 𝑧 𝑥 subscript superscript ℒ 𝑦 𝑦 𝑧 \mathcal{L}^{(x)}_{zx}-\mathcal{L}^{(y)}_{yz} . This corresponds to the first term on the right hand side of Eq. ( 16 ). The combination 𝒜 z ​ x ( x ) + 𝒜 y ​ z ( y ) subscript superscript 𝒜 𝑥 𝑧 𝑥 subscript superscript 𝒜 𝑦 𝑦 𝑧 \mathcal{A}^{(x)}_{zx}+\mathcal{A}^{(y)}_{yz} with 𝒜 = Θ 𝒜 Θ \mathcal{A}=\Theta corresponds to the second term there, while 𝒜 z ​ x ( x ) + 𝒜 y ​ z ( y ) subscript superscript 𝒜 𝑥 𝑧 𝑥 subscript superscript 𝒜 𝑦 𝑦 𝑧 \mathcal{A}^{(x)}_{zx}+\mathcal{A}^{(y)}_{yz} with 𝒜 = Φ 𝒜 Φ \mathcal{A}=\Phi should be disregarded due to ∂ / ∂ ϕ ≡ 0 italic-ϕ 0 \partial/\partial\phi\equiv 0 for  C ∞ ​ v subscript 𝐶 𝑣 C_{\infty v} . In 2D, 𝒜 x ​ x ( z ) + 𝒜 y ​ y ( z ) subscript superscript 𝒜 𝑧 𝑥 𝑥 subscript superscript 𝒜 𝑧 𝑦 𝑦 \mathcal{A}^{(z)}_{xx}+\mathcal{A}^{(z)}_{yy} with 𝒜 = Θ , Φ 𝒜 Θ Φ \mathcal{A}=\Theta,\Phi vanish since ∇ z ≡ 0 subscript ∇ 𝑧 0 \nabla_{z}\equiv 0 .

symmetry	LI-type terms	non-LI-type terms
$C_2$ ($D_1$)	${ℒ}_{zx}^{\left(x\right)}$;+ ${ℒ}_{yz}^{\left(y\right)}$;+ ${ℒ}_{yz}^{\left(x\right)}$;+ ${ℒ}_{zx}^{\left(y\right)}$;+ ${ℒ}_{xy}^{\left(z\right)}$	$𝒬$;+ $𝒮$;+ ${𝒜}_{zx}^{\left(x\right)}$;+ ${𝒜}_{yz}^{\left(y\right)}$;+ ${𝒜}_{yz}^{\left(x\right)}$;+ ${𝒜}_{zx}^{\left(y\right)}$;+ ${𝒜}_{xy}^{\left(z\right)}$;+ ${𝒜}_{xx}^{\left(z\right)}$;+ ${𝒜}_{yy}^{\left(z\right)}$
$C_{2v}$ ($D_{1h}$)	${ℒ}_{zx}^{\left(x\right)}$;+ ${ℒ}_{yz}^{\left(y\right)}$;+ ${ℒ}_{yz}^{\left(x\right)}$;+ ${ℒ}_{zx}^{\left(y\right)}$;+ ${ℒ}_{xy}^{\left(z\right)}$	$𝒬$;+ $𝒮$;+ ${𝒜}_{zx}^{\left(x\right)}$;+ ${𝒜}_{yz}^{\left(y\right)}$;+ ${𝒜}_{yz}^{\left(x\right)}$;+ ${𝒜}_{zx}^{\left(y\right)}$;+ ${𝒜}_{xy}^{\left(z\right)}$;+ ${𝒜}_{xx}^{\left(z\right)}$;+ ${𝒜}_{yy}^{\left(z\right)}$
$D_2$	${ℒ}_{zx}^{\left(x\right)}$;+ ${ℒ}_{yz}^{\left(y\right)}$;+ ${ℒ}_{yz}^{\left(x\right)}$;+ ${ℒ}_{zx}^{\left(y\right)}$;+ ${ℒ}_{xy}^{\left(z\right)}$	$𝒬$;+ $𝒮$;+ ${𝒜}_{zx}^{\left(x\right)}$;+ ${𝒜}_{yz}^{\left(y\right)}$;+ ${𝒜}_{yz}^{\left(x\right)}$;+ ${𝒜}_{zx}^{\left(y\right)}$;+ ${𝒜}_{xy}^{\left(z\right)}$;+ ${𝒜}_{xx}^{\left(z\right)}$;+ ${𝒜}_{yy}^{\left(z\right)}$
$D_{2d}$	${ℒ}_{zx}^{\left(x\right)}+{ℒ}_{yz}^{\left(y\right)}$;+ ${ℒ}_{yz}^{\left(x\right)}-{ℒ}_{zx}^{\left(y\right)}$+ ${ℒ}_{xy}^{\left(z\right)}$	$𝒬$;+ $𝒮$;+ ${𝒜}_{zx}^{\left(x\right)}$;+ ${𝒜}_{yz}^{\left(y\right)}$;+ ${𝒜}_{yz}^{\left(x\right)}+{𝒜}_{zx}^{\left(y\right)}$;+ ${𝒜}_{xy}^{\left(z\right)}$;+ ${𝒜}_{xx}^{\left(z\right)}$;+ ${𝒜}_{yy}^{\left(z\right)}$
$C_3$	${ℒ}_{zx}^{\left(x\right)}-{ℒ}_{yz}^{\left(y\right)}$;+ ${ℒ}_{yz}^{\left(x\right)}+{ℒ}_{zx}^{\left(y\right)}$;+ ${ℒ}_{xy}^{\left(z\right)}$	$𝒬$;+ $𝒮$;+ ${𝒜}_{zx}^{\left(x\right)}+{𝒜}_{yz}^{\left(y\right)}$;+ ${𝒜}_{yz}^{\left(x\right)}-{𝒜}_{zx}^{\left(y\right)}$;+ ${𝒜}_{xy}^{\left(z\right)}$;+ ${𝒜}_{xx}^{\left(z\right)}+{𝒜}_{yy}^{\left(z\right)}$
$C_{3v}$	${ℒ}_{zx}^{\left(x\right)}-{ℒ}_{yz}^{\left(y\right)}$;+ ${ℒ}_{yz}^{\left(x\right)}+{ℒ}_{zx}^{\left(y\right)}$;+ ${ℒ}_{xy}^{\left(z\right)}$	$𝒬$;+ $𝒮$;+ ${𝒜}_{zx}^{\left(x\right)}+{𝒜}_{yz}^{\left(y\right)}$;+ ${𝒜}_{yz}^{\left(x\right)}-{𝒜}_{zx}^{\left(y\right)}$;+ ${𝒜}_{xy}^{\left(z\right)}$;+ ${𝒜}_{xx}^{\left(z\right)}+{𝒜}_{yy}^{\left(z\right)}$
$C_{3h}$		$𝒬$;+ $𝒮$;+ ${𝒜}_{zx}^{\left(x\right)}+{𝒜}_{yz}^{\left(y\right)}$;+ ${𝒜}_{yz}^{\left(x\right)}-{𝒜}_{zx}^{\left(y\right)}$;+ ${𝒜}_{xy}^{\left(z\right)}$;+ ${𝒜}_{xx}^{\left(z\right)}+{𝒜}_{yy}^{\left(z\right)}$
$D_3$	${ℒ}_{zx}^{\left(x\right)}-{ℒ}_{yz}^{\left(y\right)}$;+ ${ℒ}_{yz}^{\left(x\right)}+{ℒ}_{zx}^{\left(y\right)}$;+ ${ℒ}_{xy}^{\left(z\right)}$	$𝒬$;+ $𝒮$;+ ${𝒜}_{zx}^{\left(x\right)}+{𝒜}_{yz}^{\left(y\right)}$;+ ${𝒜}_{yz}^{\left(x\right)}-{𝒜}_{zx}^{\left(y\right)}$;+ ${𝒜}_{xy}^{\left(z\right)}$;+ ${𝒜}_{xx}^{\left(z\right)}+{𝒜}_{yy}^{\left(z\right)}$
$D_{3h}$		$𝒬$;+ $𝒮$;+ ${𝒜}_{zx}^{\left(x\right)}+{𝒜}_{yz}^{\left(y\right)}$;+ ${𝒜}_{yz}^{\left(x\right)}-{𝒜}_{zx}^{\left(y\right)}$;+ ${𝒜}_{xy}^{\left(z\right)}$;+ ${𝒜}_{xx}^{\left(z\right)}+{𝒜}_{yy}^{\left(z\right)}$
$C_n$, $n>3$	${ℒ}_{zx}^{\left(x\right)}-{ℒ}_{yz}^{\left(y\right)}$;+ ${ℒ}_{yz}^{\left(x\right)}+{ℒ}_{zx}^{\left(y\right)}$;+ ${ℒ}_{xy}^{\left(z\right)}$	$𝒬$;+ $𝒮$;+ ${𝒜}_{zx}^{\left(x\right)}+{𝒜}_{yz}^{\left(y\right)}$;+ ${𝒜}_{yz}^{\left(x\right)}-{𝒜}_{zx}^{\left(y\right)}$;+ ${𝒜}_{xy}^{\left(z\right)}$;+ ${𝒜}_{xx}^{\left(z\right)}+{𝒜}_{yy}^{\left(z\right)}$
$C_{nv}$, $n>3$	${ℒ}_{zx}^{\left(x\right)}-{ℒ}_{yz}^{\left(y\right)}$;+ ${ℒ}_{yz}^{\left(x\right)}+{ℒ}_{zx}^{\left(y\right)}$;+ ${ℒ}_{xy}^{\left(z\right)}$	$𝒬$;+ $𝒮$;+ ${𝒜}_{zx}^{\left(x\right)}+{𝒜}_{yz}^{\left(y\right)}$;+ ${𝒜}_{yz}^{\left(x\right)}-{𝒜}_{zx}^{\left(y\right)}$;+ ${𝒜}_{xy}^{\left(z\right)}$;+ ${𝒜}_{xx}^{\left(z\right)}+{𝒜}_{yy}^{\left(z\right)}$
$D_n$, $n>3$	${ℒ}_{zx}^{\left(x\right)}-{ℒ}_{yz}^{\left(y\right)}$;+ ${ℒ}_{yz}^{\left(x\right)}+{ℒ}_{zx}^{\left(y\right)}$;+ ${ℒ}_{xy}^{\left(z\right)}$	$𝒬$;+ $𝒮$;+ ${𝒜}_{zx}^{\left(x\right)}+{𝒜}_{yz}^{\left(y\right)}$;+ ${𝒜}_{yz}^{\left(x\right)}-{𝒜}_{zx}^{\left(y\right)}$;+ ${𝒜}_{xy}^{\left(z\right)}$;+ ${𝒜}_{xx}^{\left(z\right)}+{𝒜}_{yy}^{\left(z\right)}$
$S_4$	${ℒ}_{zx}^{\left(x\right)}+{ℒ}_{yz}^{\left(y\right)}$;+ ${ℒ}_{yz}^{\left(x\right)}-{ℒ}_{zx}^{\left(y\right)}$+ ${ℒ}_{xy}^{\left(z\right)}$	$𝒬$;+ $𝒮$;+ ${𝒜}_{zx}^{\left(x\right)}-{𝒜}_{yz}^{\left(y\right)}$;+ ${𝒜}_{yz}^{\left(x\right)}+{𝒜}_{zx}^{\left(y\right)}$;+ ${𝒜}_{xy}^{\left(z\right)}$;+ ${𝒜}_{xx}^{\left(z\right)}-{𝒜}_{yy}^{\left(z\right)}$
$T$	${ℒ}_{zx}^{\left(x\right)}-{ℒ}_{yz}^{\left(y\right)}$;+ ${ℒ}_{yz}^{\left(x\right)}+{ℒ}_{zx}^{\left(y\right)}+{ℒ}_{xy}^{\left(z\right)}$	$𝒬$;+ $𝒮$;+ ${𝒜}_{zx}^{\left(x\right)}+{𝒜}_{yz}^{\left(y\right)}$;+ ${𝒜}_{yz}^{\left(x\right)}+{𝒜}_{zx}^{\left(y\right)}+{𝒜}_{xy}^{\left(z\right)}$;+ ${𝒜}_{xx}^{\left(z\right)}+{𝒜}_{yy}^{\left(z\right)}$
$T_d$		$𝒬$;+ $𝒮$;+ ${𝒜}_{zx}^{\left(x\right)}+{𝒜}_{yz}^{\left(y\right)}$;+ ${𝒜}_{yz}^{\left(x\right)}+{𝒜}_{zx}^{\left(y\right)}+{𝒜}_{xy}^{\left(z\right)}$;+ ${𝒜}_{xx}^{\left(z\right)}+{𝒜}_{yy}^{\left(z\right)}$
$O$	${ℒ}_{zx}^{\left(x\right)}-{ℒ}_{yz}^{\left(y\right)}$;+ ${ℒ}_{yz}^{\left(x\right)}+{ℒ}_{zx}^{\left(y\right)}+{ℒ}_{xy}^{\left(z\right)}$

Equations

$${\mathcal{L}}_{ij}^{(k)}=n_i{\nabla }_k{n}_j-n_j{\nabla }_k{n}_i,$$

$$w_{\text{ch}}=\sum _{\beta \gamma }{\Omega }_{\beta \gamma }^{\text{ch}}{\nabla }_{\beta }{n}_{\gamma }$$

$$\mathcal{H}=\xi (p)+{\alpha }_{\text{R}}\zeta (p)[\mathbf{p}\times \mathbf{\sigma }{]}_z+J_{\text{sd}}S\mathbf{n}(\mathbf{r})\cdot \mathbf{\sigma },$$

$$w_{\text{ch}}=-D\left({\mathcal{L}}_{zx}^{(x)}-{\mathcal{L}}_{yz}^{(y)}\right)=D\mathbf{n}\cdot [[{\mathbf{e}}_z\times \nabla ]\times \mathbf{n}],$$

$$\begin{gathered} w_{\text{ch}}=D_{\parallel }(n_z^2)\mathbf{n}\cdot [[{\mathbf{e}}_z\times \nabla ]\times {\mathbf{n}}_{\parallel }] \\ +D_{\perp }(n_z^2)\mathbf{n}\cdot [[{\mathbf{e}}_z\times \nabla ]\times {\mathbf{n}}_{\perp }], \end{gathered}$$

$$\mathbf{n}={\mathbf{n}}_{\parallel }+{\mathbf{n}}_{\perp },{\mathbf{n}}_{\perp }={\mathbf{e}}_z{n}_z={\mathbf{e}}_z\cos \theta .$$

$$\phantom{\Biggr{|}}\Omega^{\text{ch}}_{\beta\gamma}=T\frac{J_{\text{sd}}S}{2\pi\hbar}\operatorname{Re}\int d\varepsilon\,g(\varepsilon)\int\frac{d^{2}p}{(2\pi)^{2}}\\ \times\operatorname{Tr}{\Bigl{(}G^{R}\sigma_{\gamma}\,G^{R}\,v_{\beta}\,G^{R}-G^{R}\,v_{\beta}\,G^{R}\sigma_{\gamma}\,G^{R}\Bigr{)}},$$

$$G^R=\frac{\epsilon -\xi (p)+{\alpha }_{\text{R}}\zeta (p)[\mathbf{p}\times \mathbf{\sigma }{]}_z+J_{\text{sd}}S\mathbf{n}\cdot \mathbf{\sigma }}{(\epsilon -{\epsilon }_{+}(\mathbf{p})+i0)(\epsilon -{\epsilon }_{-}(\mathbf{p})+i0)},$$

$$\Delta (\mathbf{p})=\sqrt{{\Delta }_{\text{sd}}^2+[{\alpha }_{\text{R}}p\zeta (p){]}^2-2{\alpha }_{\text{R}}\varsigma {\Delta }_{\text{sd}}p\zeta (p)\sin \theta \sin \phi },$$

$${\Delta }_{\text{sd}}=|J_{\text{sd}}|S,\varsigma =\mathrm{sign}{J}_{\text{sd}}.$$

$$\begin{gathered} {\Omega }_{\beta \gamma }^{\text{ch}}=W{n}_{\gamma }{n}_{\beta }+D_{\parallel }(1-{\delta }_{\gamma z})\sum _{ij}{n}_i{ϵ}_{ij\gamma }{ϵ}_{jz\beta } \\ +D_{\perp }{\delta }_{\gamma z}\sum _{ij}{n}_i{ϵ}_{ij\gamma }{ϵ}_{jz\beta }, \end{gathered}$$

$$D_a=\frac{{\alpha }_{\text{R}}{\Delta }_{\text{sd}}^{2}T}{2\hslash }\int \frac{d^{2}p}{(2\pi {)}^2}{\mathcal{D}}_a(\mathbf{p})\left(\frac{g_{+}-g_{-}}{[\Delta (\mathbf{p}){]}^3}-\frac{g_{+}^{\prime }+g_{-}^{\prime }}{[\Delta (\mathbf{p}){]}^2}\right),$$

$$\displaystyle g_{\pm}=g(\varepsilon_{\pm}(\bm{p})),\quad g_{\pm}^{\prime}=\partial g/\partial\varepsilon\bigr{|}_{\varepsilon=\varepsilon_{\pm}(\bm{p})},$$

$${\mathcal{D}}_{\parallel }(\mathbf{p})=\zeta (p)+p{\zeta }^{\prime }(p){\sin }^2\phi ,$$

$${\mathcal{D}}_{\perp }(\mathbf{p})={\mathcal{D}}_{\parallel }(\mathbf{p})+\frac{p{\zeta }^{\prime }(p)\cos 2\phi }{{\sin }^2\theta }-\frac{{\alpha }_{\text{R}}p{\zeta }^2(p)\sin \phi }{\varsigma {\Delta }_{\text{sd}}\sin \theta },$$

$$w_{\text{ch}}=D_{\perp }({\mathbf{n}}_{\parallel }\cdot \nabla )n_z-D_{\parallel }{n}_z(\nabla \cdot {\mathbf{n}}_{\parallel }).$$

$$\begin{gathered} w_{\text{ch}}=-D_{\perp }{n}_z(\nabla \cdot {\mathbf{n}}_{\parallel })+D_{\parallel }({\mathbf{n}}_{\parallel }\cdot \nabla )n_z \\ -n_z\frac{\partial D_{\perp }}{\partial \theta }({\mathbf{n}}_{\parallel }\cdot \nabla )\theta +n_z\frac{\partial D_{\parallel }}{\partial \theta }({\mathbf{n}}_{\parallel }\cdot \nabla )\theta , \end{gathered}$$

$$w_{\text{ch}}=-D_{\text{as}}\left({\mathcal{L}}_{zx}^{(x)}-{\mathcal{L}}_{yz}^{(y)}\right)+D_{\text{diff}}{n}_z({\mathbf{n}}_{\parallel }\cdot \nabla )\theta ,$$

$$D_{\text{as}}=\frac{D_{\parallel }+D_{\perp }}{2},D_{\text{diff}}=\frac{\partial }{\partial \theta }\frac{D_{\parallel }-D_{\perp }}{2},$$

$$D_{\text{as}}$$

$$D_{\text{diff}}$$

$$\theta\bigr{|}_{x\to-\infty}=\pi,\quad\theta\bigr{|}_{x\to+\infty}=0,\quad\phi\equiv\phi_{0}$$

$$\mathcal{W}^{\text{DW}}_{\text{ch}}=-L\cos{\phi_{0}}\int_{\pi}^{0}{d\theta\,\Bigl{(}D_{\text{as}}-D_{\text{diff}}\sin{\theta}\cos{\theta}\Bigr{)}}\\ =\pi L\cos{\phi_{0}}\left(D^{(0)}_{\text{as}}-\frac{1}{4}D^{(2)}_{\text{diff}}\right),$$

$$w_{\text{ch}}=\sum _{ijk}{D}_{ijk}{n}_i{\nabla }_k{n}_j,$$

$$w_{\text{ch}}=\frac{1}{2}\sum _{ijk}\left[D_{ijk}^{\text{as}}{\mathcal{L}}_{ij}^{(k)}+D_{ijk}^{\text{sym}}{\nabla }_k\left(n_i{n}_j\right)\right],$$

$$w_{\text{ch}}=\frac{1}{2}\sum _{ijk}\left[D_{ijk}^{\text{as}}{\mathcal{L}}_{ij}^{(k)}-\frac{\partial D_{ijk}^{\text{sym}}}{\partial \theta }{\Theta }_{ij}^{(k)}-\frac{\partial D_{ijk}^{\text{sym}}}{\partial \varphi }{\Phi }_{ij}^{(k)}\right],$$

$${\Theta }_{ij}^{(k)}=n_i{n}_j{\nabla }_k\theta ,{\Phi }_{ij}^{(k)}=n_i{n}_j{\nabla }_k\varphi ,$$

$$D_{\text{diff}}^{(2)}$$

$$D_{\text{diff}}^{(2)}=-\frac{14m{\alpha }_{\text{R}}{\Delta }_{\text{sd}}}{3\pi \hslash }{\left(m{\alpha }_{\text{R}}^2\right)}^2{\Delta }_{\text{sd}}^3(\kappa -\lambda {)}^4(4\kappa -9\lambda ),$$

$$D_{\text{LLG}}=\frac{1}{2}\left(D_{\parallel }+D_{\perp }+n_z\frac{\partial D_{\parallel }}{\partial n_z}\right),$$

$$\mathbf{u}=\nabla n_z-{\mathbf{e}}_z(\nabla \cdot {\mathbf{n}}_{\parallel }).$$

$$\Delta f=\frac{2\gamma D_{\text{LLG}}}{\pi M_s}{\left[\mathbf{n}\times \mathbf{k}\right]}_z,$$

$$D_{\parallel }=-\frac{{\Delta }_{\text{sd}}^2\mathrm{sign}{\alpha }_{\text{R}}}{8\pi \hslash {\alpha }_{\text{R}}^2}T{\int }_0^{\infty }pdp\frac{p{\zeta }^{\prime }(p)+2\zeta (p)}{[p\zeta (p){]}^3}\left({\tilde{g}}_{-}-{\tilde{g}}_{+}\right)-\frac{{\Delta }_{\text{sd}}^2}{8\pi \hslash {\alpha }_{\text{R}}}T{\int }_0^{\infty }pdp\frac{p{\zeta }^{\prime }(p)+2\zeta (p)}{[p\zeta (p){]}^2}\left({\tilde{g}}_{-}^{\prime }+{\tilde{g}}_{+}^{\prime }\right),$$

Full text

The Dzyaloshinskii-Moriya interaction (DMI) Dzyaloshinsky (1958); Moriya (1960) is usually regarded as a key ingredient for the existence of chiral magnetism Bak and Jensen (1980); Bogdanov and Yablonskii (1989); Rößler et al. (2006); Bode et al. (2007); Tretiakov and Abanov (2010); Chen et al. (2013); Emori et al. (2013). In ferromagnets (FMs), DMI is described, in the continuum limit, by so-called Lifshitz invariants (LIs), antisymmetric combinations of the form

[TABLE]

where $\bm{n}$ is a unit vector of the local magnetization direction Bak and Jensen (1980); Bogdanov and Yablonskii (1989); Rößler et al. (2006); Tretiakov and Abanov (2010); Thiaville et al. (2012); Meynell et al. (2014); Freimuth et al. (2017). In a broader sense, one can consider a “general chiral contribution”

[TABLE]

to the micromagnetic energy density that is linear with respect to the first spatial derivatives of $\bm{n}$, but is not necessarily expressed only in terms of LIs. Below, we refer to $w_{\text{ch}}$ as the chiral energy density.

Time-reversal symmetry dictates that elements of the tensor $\Omega^{\text{ch}}$ should be odd with respect to a transformation $\bm{n}\to-\bm{n}$ Landau et al. (1984). Usually, it is simply assumed that $\Omega^{\text{ch}}_{\beta\gamma}(\bm{n})$ are linear functions of the components $n_{i}$. In this case, $w_{\text{ch}}$ reduces to a linear combination of LIs and the corresponding symmetric terms $\nabla_{k}(n_{i}n_{j})$ Hals and Everschor-Sitte (2017, 2019). The latter describe only the effects of boundaries Dzyaloshinskii (1964).

Quite recently, such boundary effects came into the focus of phenomenological studies in systems with the $C_{\infty v}$ point group symmetry. The authors of Refs. Hals and Everschor-Sitte (2017); Mulkers et al. (2018) demonstrated that the terms $\nabla_{k}(n_{i}n_{j})$ in $w_{\text{ch}}$ may become important in thin film systems. In particular, it was suggested that such terms can lead to the formation of magnetic twist states Hals and Everschor-Sitte (2017) and contribute to the stability of skyrmions Mulkers et al. (2018).

In this paper, the chiral energy density is addressed beyond the assumption of the linear dependence of $\Omega^{\text{ch}}_{\beta\gamma}(\bm{n})$ on $n_{i}$. Both microscopically and phenomenologically, we demonstrate that LIs can be insufficient for describing the chirality of a ferromagnet in the continuum limit, even in the absence of boundary effects (e.g., when the system is effectively infinite).

Let us start with a microscopic analysis of $w_{\text{ch}}$ for a particular 2D model system with the $C_{\infty v}$ symmetry. We consider a FM layer coupled to a 2DEG with spin-orbit coupling (SOC) of Rashba type and assume that the 2DEG is described by the Hamiltonian

[TABLE]

where the term $\xi(p)$ parametrizes the nonrelativistic electron dispersion, while the function $\zeta(p)$ quantifies the momentum-dependent Rashba SOC of strength $\alpha_{\text{\tiny R}}$. In the last term of Eq. (3), $\bm{\sigma}$ stands for the vector of Pauli matrices, while $J_{\textrm{sd}}$ represents the strength of the $s$-$d$-type exchange interaction between the 2DEG and localized FM spins of the absolute value $S$.

Using the model of Eq. (3), $w_{\text{ch}}$ has been computed recently Ado et al. (2018) in the lowest (linear) order with respect to $\alpha_{\text{\tiny R}}$, with the result

[TABLE]

where $\bm{\nabla}=(\nabla_{x},\nabla_{y})$ and $D$ is a DMI constant proportional to $\alpha_{\text{\tiny R}}$. We are about to show that, beyond the linear order in the SOC strength, Eq. (4) transforms into

[TABLE]

where $D_{\parallel}$ differs from $D_{\perp}$, and $\bm{n}_{\parallel/\perp}$ denotes the in-plane/perpendicular-to-the-plane component of $\bm{n}$,

[TABLE]

Note that the right hand side of Eq. (5) is no longer expressed in terms of LIs, as one can deduce from a direct expansion of the vector products.

In order to derive Eq. (5), we use a general expression for the tensor $\Omega^{\text{ch}}$ Ado et al. (2018):

[TABLE]

where $\bm{v}=\partial\mathcal{H}/\partial\bm{p}$ is the velocity operator, the retarded Green’s function $G^{R}$ describes a system with homogeneous magnetization, and $\operatorname{Tr}$ stands for the matrix trace operation. In Eq. (7), we also use the notation $g(\varepsilon)=\mathop{\mathrm{ln}}\nolimits{\left(1+\exp{\left[(\mu-\varepsilon)/T\right]}\right)}$, where $\mu$ and $T$ are the chemical potential and temperature, respectively. The Green’s function $G^{R}$, in the momentum representation, takes the form

[TABLE]

where the spectral branches $\varepsilon_{\pm}(\bm{p})=\xi(p)\pm\Delta(\bm{p})$ are parameterized by

[TABLE]

Here, $\theta$ stands for the polar angle of $\bm{n}$ with respect to the $z$ axis, while $\varphi$ is the angle between the momentum $\bm{p}$ and the in-plane component $\bm{n}_{\parallel}$ of the vector $\bm{n}$.

Substitution of Eq. (8) into Eq. (7) followed by the matrix trace calculation and integration over $\varepsilon$ produces the following outcome:

[TABLE]

where $\epsilon_{q_{1}q_{2}q_{3}}$ denotes the three-dimensional Levi-Civita symbol, and $\delta_{q_{1}q_{2}}$ is the Kronecker delta. The functions $D_{\parallel}$ and $D_{\perp}$ can be expressed as

[TABLE]

where $a=\,\parallel,\perp$ and we use the notations

[TABLE]

with $\zeta^{\prime}(p)=\partial\zeta/\partial p$.

To translate Eq. (9) into the expression for $w_{\text{ch}}$, we first note that the value of $W$ is totally irrelevant for the final result. Indeed, upon substitution of Eq. (9) into Eq. (2), the first term on the right hand side of Eq. (9) produces a contribution that is equal to $(W/2)(\bm{n}_{\parallel}\cdot\bm{\nabla})\bm{n}^{2}$. Due to the constraint $\bm{n}^{2}\equiv 1$, it vanishes. The remaining two terms in Eq. (9) correspond to the double vector products in Eq. (5). Noting that the dependence of $D_{\parallel}$ and $D_{\perp}$ on the vector $\bm{n}$, in the highly symmetric model of Eq. (3), can be expressed as $D_{a}=D_{a}(n_{z}^{2})$ sup , we therefore conclude the microscopic derivation of Eq. (5).

The fact that $D_{\parallel}$ and $D_{\perp}$ both turn out to be functions of $\bm{n}$ has quite a few important consequences. Ignoring, for a moment, microscopic details, let us rewrite Eq. (5) as

[TABLE]

Integration over space defines the total micromagnetic chiral energy $\mathcal{W}_{\text{ch}}=\int{dx\,dy\,w_{\text{ch}}}$. Performing integration by parts and disregarding contributions from the boundaries, we obtain a different representation of the density,

[TABLE]

where we have taken into account that the spatial dependence of $D_{a}=D_{a}(\cos^{2}{\theta})$ originates solely from the spatial dependence of the polar angle $\theta=\theta(\bm{r})$. By taking the half sum of Eqs. (14) and (15), we arrive at the result

[TABLE]

that demonstrates an essential separation of $w_{\text{ch}}$ into LI-type contributions and contributions of a different symmetry.

The first term on the right hand side of Eq. (16) has the structure of the DMI energy density for a system of the $C_{\infty v}$ class, Eq. (4). The second term, however, displays a non-LI-type symmetry and therefore does not originate from DMI. Importantly, it cannot be “integrated out” by means of a partial integration as opposed to the “boundary terms” $\nabla_{k}(n_{i}n_{j})$. We explicitly note that $D_{\text{diff}}\neq 0$ requires at least one of the functions, $D_{\parallel}$ and $D_{\perp}$, to depend on $\bm{n}$, which is possible due to broken spin rotation symmetry. Similar orientational anisotropy of SOC-related phenomena has been observed recently  Buchner et al. (2016); Chen et al. (2018); Li et al. (2019).

Remarkably, the phase diagram of magnetic states is affected by $D_{\text{diff}}$ as well as by $D_{\text{as}}$. Both functions incorporate an infinite amount of Fourier harmonics,

[TABLE]

which, obviously, complicates the minimization of the micromagnetic energy functional. Nevertheless, the role of the term $D_{\text{diff}}\,n_{z}(\bm{n}_{\parallel}\cdot\bm{\nabla})\theta$ can be illustrated by using a simple example. Let us consider a domain wall (DW)

[TABLE]

with the fixed azimuthal angle $\phi$ of the vector $\bm{n}=\bm{n}(x)$. Assuming the DW size in the $y$ direction to be equal to $L$, we can compute the corresponding total chiral energy $\mathcal{W}^{\text{DW}}_{\text{ch}}=L\int{dx\,w_{\text{ch}}}$ from Eqs. (16), (18), and (19). Making use of the relation $(\nabla_{x}\theta)dx=d\theta$ to reduce the integration over $x$ to the integration over $\theta$, we find

[TABLE]

where the orthogonality of the sine functions has been taken into account. Note that this result is independent of the particular shape of the DW profile $\theta(x)$.

The contribution to $\mathcal{W}^{\text{DW}}_{\text{ch}}$ that originates from the “antisymmetric part” -D_{\text{as}}\bigl{(}\mathcal{L}_{zx}^{(x)}-\mathcal{L}_{yz}^{(y)}\bigr{)} of the chiral energy density has been computed before (see, e. g., Eq. (19) in Ref. Wakatsuki et al. (2015)). The second contribution provided by the term $D_{\text{diff}}\,n_{z}(\bm{n}_{\parallel}\cdot\bm{\nabla})\theta$ is the novel result of this paper. As can be seen from Eq. (21), the DW chiral energy depends equally on the $D^{(0)}_{\text{as}}$ and $D^{(2)}_{\text{diff}}$ Fourier harmonics. Thus, indeed, the chirality of a ferromagnet in general cannot be properly analysed (in the continuum limit) without consideration of the non-LI-type contributions to $w_{\text{ch}}$. Despite being simplified, the ansatz of Eq. (20) serves as a good illustration of the importance of such contributions. We certainly expect them to be relevant for more complex structures Tretiakov and Abanov (2010) as well.

To perform an analogous to Eq. (16) separation of $w_{\text{ch}}$ into LI-type and non-LI-type terms for an arbitrary FM, we assume that

[TABLE]

where $D_{ijk}$ are even functions of $\bm{n}$. Below, we refer to the tensor with the components $D_{ijk}$ as the chiral tensor. Symmetrization of Eq. (22) gives Hals and Everschor-Sitte (2017, 2019)

[TABLE]

with $D_{ijk}^{\text{as(sym)}}=(D_{ijk}\mp D_{jik})/2$. Applying integration by parts to the second term inside the brackets and disregarding contributions from the boundaries, we obtain

[TABLE]

where $\theta$ and $\phi$ are, as before, the polar and azimuthal angles of $\bm{n}$, respectively.

Using a standard symmetry analysis Authier (2003); Hals and Everschor-Sitte (2017), one can identify the coefficients $\mathcal{L}^{(k)}_{ij}$, $\Theta^{(k)}_{ij}$, $\Phi^{(k)}_{ij}$ that are allowed in $w_{\text{ch}}$ by a point group symmetry of a particular system. The corresponding results for all crystallographic point groups (except $C_{1}$, $C_{1v}$, and $C_{1h}$ that we address in the Supplemental Material sup ) are collected in Table 1. Remarkably, for the classes $C_{3h}$, $D_{3h}$, and $T_{d}$, DMI does not contribute to the chiral energy density. Non-LI-type terms are the only source of chirality in FMs described by these three groups.

Let us now use the results of Eqs. (10), (11), (12), (13) and return to the microscopic analysis of the functions $D_{\parallel}$ and $D_{\perp}$ for the generalized Rashba model of Eq. (3). First, it is easy to observe that, in the leading (linear) order with respect to small $\alpha_{\text{\tiny R}}$, the angle integration in Eq. (10) can be performed straightforwardly. This leads to the result $D_{\parallel}=D_{\perp}=D_{\text{as}}=D$, where $D$ is the DMI constant given by Eq. (5) of Ref. Ado et al. (2018).

In the other limit, $\Delta_{\text{sd}}\to 0$, the coefficients of the chiral tensor do not coincide: one generally finds $D_{\parallel}\neq D_{\perp}$, in the leading (second) order with respect to small $\Delta_{\text{sd}}$. Nevertheless, in this case, the quantity $D_{\parallel}-D_{\perp}$ turns out to be independent of $\theta$ and hence $D_{\text{diff}}=0$ sup . Therefore, for either weak SOC or weak $s$-$d$ exchange, the chiral energy density can be described by LIs alone, at least as long as the boundary effects are disregarded. Any possible effect of $D_{\text{diff}}$ is absent in these two limits.

A further asymptotic analysis sup shows that the function $D_{\text{diff}}$ does not contain contributions in the order $\alpha_{\text{\tiny R}}^{3}$. Indeed, in this order, $D_{\parallel}\neq D_{\perp}$ – yet, again, the difference between $D_{\parallel}$ and $D_{\perp}$ does not depend on $\theta$. In general, the expansions of $D_{\text{diff}}$ in small $\alpha_{\text{\tiny R}}$ and small $\Delta_{\text{sd}}$ start with the contributions of the order $\alpha_{\text{\tiny R}}^{5}$ and $\Delta_{\text{sd}}^{4}$, respectively. Moreover, for the leading-order asymptotics, only the first Fourier harmonic is nonvanishing, so that $D_{\text{diff}}=D^{(2)}_{\text{diff}}\sin{2\theta}$.

For the particular Bychkov-Rashba model Bychkov and Rashba (1984) characterized by the choice $\xi(p)=p^{2}/2m$ and $\zeta(p)\equiv 1$ in Eq. (3), we find at $T=0$ for the first nonzero terms of the expansions (in $\alpha_{\text{\tiny R}}$ and $\Delta_{\text{sd}}$, respectively):

[TABLE]

where we have introduced $Q(x)=35x^{4}-30x^{2}+3$ and $R(x)=(35x^{2}+40x+12)/(1+2x)^{5/2}$, with $R(x)\equiv 0$ for $x<-1/2$. In principle, it is clear that, in this simple model, $D_{\text{diff}}$ is determined by three independent energy scales: $m\alpha_{\text{\tiny R}}^{2}$, $\Delta_{\text{sd}}$, and $\mu$. Intuitively, one would expect $|D_{\text{diff}}|$ to be maximal when $m\alpha_{\text{\tiny R}}^{2}$ and $\Delta_{\text{sd}}$ are of a comparable magnitude. Our perturbative analysis agrees with this conjecture. We also illustrate the latter in Fig. 1, by plotting the ratio $D^{(2)}_{\text{diff}}/D^{(0)}_{\text{as}}$ as a function of the SOC strength. The absolute value of this ratio, and even its sign, are sensitive to variation of the chemical potential. One might recognize this as a possibility to gain additional means of magnetic order tuning by means of gate voltage control.

Notably, in the Bychkov-Rashba model, the leading-order asymptotics of $D_{\text{diff}}$ vanish at zero temperature when both spin subbands are partly occupied. This is not accidental. In fact, chiral terms in the micromagnetic energy density (including those originating from DMI) are totally absent in this case, $D_{\parallel}\equiv 0$ and $D_{\perp}\equiv 0$, regardless of the values of $\alpha_{\text{\tiny R}}$ and $\Delta_{\text{sd}}$ sup . Such a peculiarity, however, is a property of the specific model and does not characterize the symmetry class ($C_{\infty v}$) to which the latter corresponds. Indeed, one may consider a slightly more general example, with $\xi(p)=(p^{2}/2m)/\left(1+\kappa\,p^{2}/2m\right)$ and $\zeta(p)=1/\left(1+\lambda\,p^{2}/2m\right)$, where the positive parameters $\kappa$ and $\lambda$ represent deviations from parabolic band dispersion Kane (1957); Nelson et al. (1987); Hu et al. (1999) and nonlinear dependence of the Rashba SOC on momentum Yang and Chang (2006); Gong et al. (2013); Liu et al. (2013), respectively. In this model, finite $w_{\text{ch}}$ for two partly occupied subbands is restored. In particular, for $\mu>\Delta_{\text{sd}}$, we find a surprisingly compact result in the leading $\alpha_{\text{\tiny R}}^{5}$ order,

[TABLE]

where the temperature is set to zero and $\kappa\approx\lambda$ are both considered small in comparison with $\mu^{-1}$ and $\Delta_{\text{sd}}^{-1}$.

In the final part of the paper, we briefly discuss how the chiral energy density with the symmetry of Eq. (5) affects spin-wave dispersion. The effective field arising due to $w_{\text{ch}}$ is proportional to the functional derivative $\delta\mathcal{W}_{\text{ch}}/\delta\bm{n}$. Taking advantage of the fact that both $D_{\parallel}$ and $D_{\perp}$ can be considered independent of $\bm{n}_{\parallel}=(n_{x},n_{y})$, we find $\delta\mathcal{W}_{\text{ch}}/\delta\bm{n}=2D_{\text{\tiny{LLG}}}\bm{u}$, where

[TABLE]

The corresponding contribution to the Landau-Lifshitz-Gilbert (LLG) equation shifts the frequency of a spin wave by a term linear in wave vector $\bm{k}$ Moon et al. (2013); Di et al. (2015); Belmeguenai et al. (2015). Importantly, the frequency difference $\Delta f$ between spin waves with wave vectors $\bm{k}$ and $-\bm{k}$ is experimentally measurable Zakeri et al. (2010); Di et al. (2015); Belmeguenai et al. (2015); Nembach et al. (2015). In the present case, such difference should equal

[TABLE]

where $M_{s}$ is a saturation magnetization and $\gamma$ denotes the gyromagnetic ratio.

Normally, for thin magnetic films and interfaces, it is assumed that $D_{\text{\tiny{LLG}}}$ is a DMI constant which is independent of $\bm{n}$ and defines the DMI energy density as $D_{\text{\tiny{LLG}}}\,\bm{n}\cdot[[\bm{e}_{z}\times\bm{\nabla}]\times\bm{n}]$ Di et al. (2015). According to Ref. Hals and Everschor-Sitte (2017), for a 2D system of the $C_{\infty v}$ class, $D_{\text{\tiny{LLG}}}$ in Eq. (28) should, in fact, coincide with $D_{\text{as}}$ given by Eq. (17). However, once the dependence of the functions $D_{\parallel}$ and $D_{\perp}$ on the vector $\bm{n}$ is taken into account, the equality $D_{\text{\tiny{LLG}}}=D_{\text{as}}$ should also be revised. As one can see from Eq. (26), the result for $D_{\text{\tiny{LLG}}}$ is different from $D_{\text{as}}$ by the term $n_{z}\left(\partial D_{\parallel}/\partial n_{z}\right)$. Interestingly, for the model of Eq. (3), its expansion in powers of $\alpha_{\text{\tiny R}}$ starts with $\alpha_{\text{\tiny R}}^{3}$ sup . Therefore, one could anticipate the effects of this term to be more pronounced than those of $D_{\text{diff}}$.

In Fig. 2, we plot $D_{\text{\tiny{LLG}}}$ as a function of the polar angle $\theta$ of magnetization direction, for the Bychkov-Rashba model. It is very clear that the Fourier harmonic $\cos{2\theta}$ is non-negligible, already for small values of $m\alpha_{\text{\tiny R}}^{2}/\Delta_{\text{sd}}$. Manifestly, in systems with strong SOC Ast et al. (2007); Ishizaka et al. (2011), components of the chiral tensor can depend on $\bm{n}$. It would be interesting to observe such dependence experimentally. Should this happen, the proper treatment of chiral ferromagnetism must extend beyond Lifshitz invariants.

.1 Formal proof of the relation $D_{a}=D_{a}(n_{z}^{2})$

The model of Eq. (3) describes a system with rotational invariance with respect to the $z$ axis. Hence, $D_{a}$ cannot depend on the azimuthal angle of $\bm{n}$. At the same time, a simultaneous change $\theta\to-\theta$ and $\varphi\to\varphi+\pi$ does not alter the result of integration in Eq. (7). Thus, $D_{\parallel}$ and $D_{\perp}$ are even functions of $\theta$ with a period equal to $\pi$. Since, for Fourier harmonics, $\cos{2n\theta}=F_{n}(\cos^{2}{\theta})$, we observe that, indeed, $D_{a}=D_{a}(n_{z}^{2})$.

.2 $D_{\parallel}$ and $D_{\perp}$: leading-order results for small $\Delta_{\text{sd}}$

In this section we assume $\zeta(p)>0$. Expansion of the integrands in Eq. (10) with respect to small $\Delta_{\text{sd}}$ results in the following general expressions for the corresponding leading-order asymptotics of the functions $D_{\parallel}$ and $D_{\perp}$:

[TABLE]

where $\tilde{g}_{\pm}^{(n)}=\partial^{n}\tilde{g}_{\pm}/\partial\xi^{n}$ and $\tilde{g}_{\pm}=g\left(\xi(p)\pm|\alpha_{\text{\tiny R}}|p\,\zeta(p)\right)$. Using the relation $\partial\tilde{g}_{\pm}/\partial\xi=\pm[p\,\zeta(p)]^{-1}\partial\tilde{g}_{\pm}/\partial|\alpha_{\text{\tiny R}}|$, one can also rewrite Eqs. (s1), (s2) in more compact forms

[TABLE]

It is instructive to apply the above general expressions comment_expansions_again to two paradigmatic models: the model of massive Dirac fermions (DF) and the Bychkov-Rashba model (BR). By setting $\xi(p)\equiv 0$ and $\zeta(p)\equiv 1$ in Eq. (3), we get for the former model at $T=0$:

[TABLE]

where we used that $Tg(\varepsilon)\to(\mu-\varepsilon)H\,(\mu-\varepsilon)$ and $T\partial g/\partial\varepsilon\to-\delta\,(\varepsilon-\mu)$ when temperature approaches zero (the notations $H$ and $\delta$ refer here to the Heaviside step function and the Dirac delta function, respectively). In fact, the result for $D_{\text{as}}=D_{\parallel}/2$ that follows from Eq. (s5) almost coincides with the non-perturbative one Tserkovnyak ; Koretsune ; Wakatsuki . The only difference is the absence of the band gap signature in Eq. (s5).

For the Bychkov-Rashba model, $\xi(p)=p^{2}/2m$ and $\zeta(p)\equiv 1$, we obtain

[TABLE]

where temperature is again set to zero. It is interesting to observe that application of a formal limit $m\to\infty$ in Eq. (s6) leads to the result

[TABLE]

which does not coincide with that of Eq. (s5).

Evidently, the two coefficients of the chiral tensor are not equal to each other, $D_{\parallel}\neq D_{\perp}$, in the leading order with respect to small $\Delta_{\text{sd}}$. At the same time, no dependence on $\bm{n}$ is present in this case (as we have stated in the main text of the paper). Hence, $D_{\text{diff}}=0$ and $D_{\text{\tiny{LLG}}}=D_{\text{as}}$, up to the order $\Delta_{\text{sd}}^{2}$.

.3 Vanishing of $D_{\parallel}$, $D_{\perp}$ in the Bychkov-Rashba model when both spin subbands are partly occupied

For considerations of this section, it is useful to introduce the “magnetization” vector $\bm{M}=\varsigma\Delta_{\text{sd}}\bm{n}$ with the components $M_{\perp}=M_{z}=\varsigma\Delta_{\text{sd}}\cos{\theta}$ and $M_{\parallel}=\varsigma(M_{x}^{2}+M_{y}^{2})^{1/2}=\varsigma\Delta_{\text{sd}}\sin{\theta}$. With the help of the latter, under the condition $\zeta(p)\equiv 1$, we obtain from Eqs. (10), (12), (13) and the definition of $\Delta(\bm{p})$:

[TABLE]

where one can regard $g_{\pm}^{\prime}$ as $\partial g_{\pm}/\partial\xi$. Then, using the relations

[TABLE]

it is possible to bring the integral expressions of Eqs. (s8), (s9) to the concise forms

[TABLE]

Next, we differentiate the above results over the chemical potential to obtain

[TABLE]

where $f_{\pm}=T\partial g_{\pm}/\partial\mu$ is expressed in terms of the Fermi-Dirac distribution as $f_{\pm}=\left(1+\exp{\left[(\varepsilon_{\pm}(\bm{p})-\mu)/T\right]}\right)^{-1}$. Comparison of Eq. (s13) with Eq. (E15) of Ref. Ado shows that

[TABLE]

where $\delta S_{z}$ is the $z$-component of the total spin of conduction electrons in a unit cell of the area $A$. For the Bychkov-Rashba model, it was demonstrated Ado that, at zero temperature, $\delta S_{z}/M_{\perp}=-mA/(2\pi\hbar^{2})$, at least as long as $\mu>\Delta_{\text{sd}}$. From this, it immediately follows that $\partial D_{\parallel}/\partial\mu=\partial D_{\perp}/\partial\mu=0$, for such values of $\mu$. Therefore, for $T=0$, the functions $D_{\parallel}$ and $D_{\perp}$ do not depend on the chemical potential, once the latter exceeds $\Delta_{\text{sd}}$.

We now temporarily restrict the analysis to the case $\mu>\Delta_{\text{sd}}$ and return to the general situation later. We also assume that temperature is set to zero. In this case, $Tg_{\pm}$ in Eq. (s12) should be replaced with $(\mu-\varepsilon_{\pm}(\bm{p}))f_{\pm}$. Assuming $\xi(p)=p^{2}/2m$, we get

[TABLE]

The first integral inside the brackets is again proportional to $\delta S_{z}/M_{\perp}=-mA/(2\pi\hbar^{2})$ and vanishes after differentiation over $M_{\perp,\parallel}$. The third integral is the density of states, which, for $\mu>\Delta_{\text{sd}}$, is also independent of $M_{\perp,\parallel}$, as was shown in Ref. Kim . Thus, we are left with

[TABLE]

where $p_{\pm}$ are the angle dependent Fermi momenta $p_{\pm}$ corresponding to $\varepsilon_{\pm}(\bm{p})$ branches. Performing the integration over $p$ and utilizing the asymptotics

[TABLE]

we arrive at the relation

[TABLE]

which ultimately means that both coefficients of the chiral tensor vanish at $\mu=+\infty$. At the same time, as we have already learned, $D_{\parallel}$ and $D_{\perp}$ are independent of $\mu$ when $\mu>\Delta_{\text{sd}}$. Therefore, we conclude:

[TABLE]

It is easy to generalize the result (s19) and show that, in fact, $w_{\text{ch}}$ vanishes at zero temperature for all values of $\mu$ corresponding to the case of two partly occupied subbands, i. e., for $\mu>\min{\varepsilon_{+}(\bm{p})}$ (note that $\Delta_{\text{sd}}\geq\min{\varepsilon_{+}(\bm{p})}$). The proof presented so far is based on the results of Refs. Ado and Kim established under the assumption $\mu>\Delta_{\text{sd}}$. The role of this assumption was to ensure non-negativity of the discriminant $\Delta_{R}$ of the cubic function $R$ defined in Eq. (B15) of Ref. Kim . However, up to a positive prefactor, $\Delta_{R}$ coincides with $\Delta_{Q}(1)$, where $\Delta_{Q}\left(\sin^{2}{\phi}\right)$ is the discriminant of the quartic function $Q(p)=(\varepsilon_{+}(\bm{p})-\mu)(\varepsilon_{-}(\bm{p})-\mu)$ of the absolute value of momentum. And since $\mu>\min{\varepsilon_{+}(\bm{p})}$ is only possible if the equation $Q(p)=0$ has four real solutions for $p$ when $\sin^{2}{\phi}=1$, we deduce

[TABLE]

Hence, finally, for the Bychkov-Rashba model,

[TABLE]

.4 $D_{\text{diff}}$: leading-order results for weak SOC and for small $\Delta_{\text{sd}}$ (general formulas)

We expand the integrands in Eq. (10) with respect to small $\alpha_{\text{\tiny R}}$, up to the fifth order. Subsequent integration over $\varphi$ nullifies all linear and cubic contributions to $D_{\text{diff}}$, resulting in comment_expansions_again

[TABLE]

where $\hat{g}_{\pm}=g(\xi(p)\pm\Delta_{\text{sd}})$ and $\hat{g}_{\pm}^{(n)}=\partial^{n}\hat{g}_{\pm}/\partial\xi^{n}$.

Similarly, expansion up to $\Delta_{\text{sd}}^{4}$ leads to comment_expansions_again

[TABLE]

where $\tilde{g}_{\pm}^{(n)}=\partial^{n}\tilde{g}_{\pm}/\partial\xi^{n}$ and $\tilde{g}_{\pm}=g\left(\xi(p)\pm|\alpha_{\text{\tiny R}}|p\,\zeta(p)\right)$. In Eq. (s23), we also assume $\zeta(p)>0$.

.5 $D_{\parallel}$, $D_{\perp}$, and $D_{\text{\tiny{LLG}}}$ in the Bychkov-Rashba model: expansion up to $\alpha_{\text{\tiny R}}^{3}$

In the Bychkov-Rashba model, fixing $\mu<\Delta_{\text{sd}}$ and expanding the integrands in Eq. (10) up to $\alpha_{\text{\tiny R}}^{3}$, we obtain the asymptotic expressions

[TABLE]

According to the definition of Eq. (26), they determine the expansion

[TABLE]

which we use in Fig. 2 of the paper. Note that, in this model, $D_{\parallel}-D_{\perp}$ is independent of $\theta$, in the order $\alpha_{\text{\tiny R}}^{3}$, and $D_{\text{diff}}$, indeed, vanishes.

.6 Chiral energy density for the classes $C_{1}$, $C_{1v}$, and $C_{1h}$

In the chiral energy density, the classes $C_{1}$, $C_{1v}$, and $C_{1h}$ allow the following LIs:

[TABLE]

and non-LI-type terms with the following coefficients:

[TABLE]

where $\mathcal{A}^{(k)}_{ij}=\Theta^{(k)}_{ij},\Phi^{(k)}_{ij}$.