Classification of spin Hall effect in two-dimensional systems

Abstract

Physical properties such as the conductivity are usually classified according to the symmetry of the underlying system using Neumann’s principle, which gives an upper bound for the number of independent components of the corresponding property tensor. However, for a given Hamiltonian, this global approach usually can not give a definite answer on whether a physical effect such as spin Hall effect (SHE) exists or not. It is found that the parity and types of spin-orbit interactions (SOIs) are good indicators that can further reduce the number of independent components of the spin Hall conductivity for a specific system. In terms of the parity as well as various Rashba-like and Dresselhaus-like SOIs, we propose a local approach to classify SHE in two-dimensional (2D) two-band models, where sufficient conditions for identifying the existence or absence of SHE in all 2D magnetic point groups are presented.

Graphical abstract

Keywords

spin Hall effect / symmetry / two-dimensional system

1 Introduction

Berry curvature related band geometric quantities are widely adopted to describe various Hall effects [1, 2]. For instance, nonzero Berry phase accounts for both quantum Hall effect [3] and quantum anomalous Hall effect [4], which are intrinsic responses and involve the breaking of time-reversal symmetry. In time-reversal invariant systems, extrinsic Hall effect can exist in the nonlinear response regime, such as the second-order Hall effect induced by Berry curvature dipole [5–7] and the third-order Hall effect induced by Berry-connection polarizability tensor [8], which can also be studied in multiterminal systems using the scattering matrix theory [9, 10]. Higher-order nonlinear anomalous Hall effects induced by Berry curvature multipoles such as quadrupole and hexapole have been discussed in certain materials with magnetic point group symmetry [11]. In particular, intrinsic second-order anomalous Hall effect has been discovered in $\mathcal{P}\mathcal{T}$-symmetric antiferromagnets [12–14] and intrinsic third-order anomalous Hall effect was discussed [15]. In addition to Hall currents, it was found that Berry curvature dipole and Berry curvature fluctuation can give rise to linear and nonlinear thermal Hall noises [16]. The existence of these nonlinear Hall effects are classified by symmetry and relevant constraints [5, 6, 11–13, 15], since band geometry is strongly affected by symmetry.

In fact, symmetry plays essential roles in classifying a variety of physical properties [17]. For example, the famous universal conductance fluctuation (UCF) [18–22] in mesoscopic transport depends on symmetry and dimensionality of the system. To describe UCF, the system Hamiltonian is categorized into three ensembles, i.e., Gaussian orthogonal/unirary/sympletic ensemble, based on the presence or absence of time-reversal and spin-rotation symmetries. When the particle-hole and chiral symmetries are further included, it has been extended to the ten-fold way [23, 24], which is widely used in classifying topological insulators (TI) and topological superconductors. Spin photovoltaic effect in antiferromagnetic materials can be classified in terms of the $\mathcal{P}\mathcal{T}$ symmetry and spacial symmetries [25]. However, there are also exceptions using symmetry analysis. The remarkable example is the phase transition between TI and band insulator, which occurs without symmetry breaking but with the changing of global (topological) invariants [26, 27]. In this work, we find another example: for a system with a given symmetry, spin Hall effect (SHE) can be present or absent depending on the parity and spin-orbit interaction (SOI) of the Hamiltonian.

SHE is a relativistic phenomenon, where charge current drives transverse spin current in spin-orbit coupled systems [28]. Similar to charge Hall effects, SHE can also be intrinsic or extrinsic. Intrinsic SHE [29, 30] is not influenced by transport process, which has been proposed in Weyl semimetals such as TaAs [31]. The orientation of spin polarization in SHE can be either in-plane or out-of-plane. In different materials, the dominant SOI could be Dresselhaus-like or Rashba-like, or the combination of them [32–34]. The existence of SHE and inverse SHE [35] have been solidly confirmed by a series of experiments [36, 37], and theoretical description of SHE usually involves spin Berry curvature [31]. However, less attention has focused on classifying SHE, which could be carried out using Neumann’s principle. Neumann’s principle has been applied to analyze the conductivity tensor for nonlinear Hall effects [11–13], which can be stated as: if a crystal is invariant with respect to certain symmetry, any of its physical properties must also be invariant with respect to the same symmetry. As a consequence of this principle, symmetry imposes constraint on physical quantities, including the spin Hall conductivity.

In this work, we demonstrate that the symmetry alone is insufficient to determine the existence of SHE in 2D systems, since SHE can be switched on or off by varying the SOI while maintaining the symmetry of the system. Compared with Neumann’s principle, the parity and types of SOIs are better indicators for identifying SHE. Based on the parity as well as different orders of in-plane or out-of-plane, Dresselhaus-like or Rashba-like SOIs, we propose a local approach to classify SHE, which gives sufficient conditions for the existence of SHE in 2D two-band models. Complete analysis on all 2D magnetic point groups (MPGs) are carried out and possible materials for experimental verification are discussed.

The paper is organized as follows. In Section 2, the Kubo formula for the spin Hall conductivity of a 2D two-band model is introduced. In Section 3, classification of 2D SHE according to Neumann’s principle is discussed. For a Hamiltonian with certain symmetry, this general classification can only give an indefinite answer on the presence or absence of SHE. In Section 4, a different symmetry analysis based on parity and types of SOIs constituting the 2D Hamiltonian is given, which allows identifying the existence of SHE. Finally, discussion and conclusion are given in Section 5.

2 Formalism for spin Hall effect

The spin Hall conductivity is [29, 31, 38] ($\hslash =e=1$)

(1)${\sigma }_{xy}^{\alpha }={\displaystyle {\int }_k{\displaystyle \sum _n{f}_{n,\mathit{k}}{\mathrm{\Omega }}_{n,xy}^{\alpha }(\mathit{k}),}}$

where ${\int }_k={\int }_{BZ}{\mathrm{d}}^{2}k/(2\pi {)}^2$ and $\alpha$ labels the direction of spin polarization of SHE. The “spin” Berry curvature is defined as

(2)${\mathrm{\Omega }}_{n,xy}^{\alpha }(\mathit{k})=-2\mathrm{I}\mathrm{m}{\displaystyle \sum _{n^{\prime }\ne n}{\displaystyle \frac{⟨n|J_x^{\alpha }|n^{\prime }⟩⟨n^{\prime }|v_y|n⟩}{({ϵ}_n-{ϵ}_{n^{\prime }}{)}^2}},}$

with spin current operator $J_i^{\alpha }=\frac{1}{2}\{v_i,s_{\alpha }\}$, where $s_{\alpha }$ is the spin operator and $v_i=\frac{\mathrm{\partial }H}{\mathrm{\partial }{k}_i}$ is the velocity operator. Note that the “spin” Berry curvature resembles usual Berry curvature only if the spin is a good quantum number. In the presence of spin−orbit interaction (SOI), it is very different from the usual Berry curvature. We focus on a two-band model defined as $H=d_0{\sigma }_0+\mathit{d}\cdot \mathit{\sigma }$, from which we have

(3)$v_i={\mathrm{\partial }}_i{d}_0+({\mathrm{\partial }}_i{d}_j){\sigma }_j,$

where summation over repeated indices is implied and

(4)$J_i^{\alpha }={\displaystyle \frac{\hslash }{2}}({\sigma }_{\alpha }{\mathrm{\partial }}_i{d}_0+{\mathrm{\partial }}_i{d}_{\alpha }).$

After some algebra, we find the spin Berry curvature for the lower band

(5)${\mathrm{\Omega }}_{xy}^{\alpha }(\mathit{k})={\displaystyle \frac{{\mathrm{\partial }}_x{d}_0({\mathrm{\partial }}_y\mathit{d}\times \mathit{d}{)}_{\alpha }}{4d^3}},$

where $d^2=\mathit{d}\cdot \mathit{d}$. It is easy to show that for the linear Rashba SOI, Eq. (5) reproduces the spin Hall conductance of $e/(8\pi )$, which was obtained by Sinova et al. [29]. For comparison, we show the expression of Berry curvature

(6)${\mathrm{\Omega }}_{\pm ,xy}(\mathit{k})=\pm {\displaystyle \frac{\hslash {\mathrm{\partial }}_x\mathit{d}\cdot ({\mathrm{\partial }}_y\mathit{d}\times \mathit{d})}{2d^3}}.$

Obviously, if $H_0=0$ there is no linear SHE. From now on, we will work on systems with broken particle−hole symmetry ($d_0=k^2$) and focus on ${\sigma }_{xy}^{\alpha }$. The analysis of ${\sigma }_{yx}^{\alpha }$ is similar.

In order to have spin Hall effect, SOI must be present which can be classified according to the symmetry as well as linear or nonlinear orders of momentum. For instance, two typical in-plane linear order SOIs (IP-SOI) in 2D systems are the Rashba SOI $k_y{\sigma }_x-k_x{\sigma }_y$ and Dresselhaus SOI $k_x{\sigma }_x-k_y{\sigma }_y$. For the classification reason, we define two types of SOI, Dresselhaus-like ($H_D$) and Rashba-like ($H_R$) SOI, as follows. For IP-SOI, we define $A=\sum _{n=1}^N({\alpha }_n{k}_{+}^{N-n}{k}_{-}^n{\sigma }_{+}+{\beta }_n{k}_{-}^{N-n}{k}_{+}^n{\sigma }_{-})$ with $k_{\pm }=k_x\pm \mathrm{i}{k}_y$, then $H_D=A+A^{†}$ and $H_R=-\mathrm{i}(A-A^{†})$. For the out-of-plane SOI (OP-SOI), we have $H_D=B+B^{†}$ and $H_R=-\mathrm{i}(B-B^{†})$ where $B=\sum _{n=1}^N{\gamma }_n{k}_{+}^{N-n}{k}_{-}^n{\sigma }_z$. In Tab.1, we list a few IP-SOIs and OP-SOIs which are basic building blocks of the SOI Hamiltonian. In particular, the SOI with $N=3$ is called cubic Dresselhaus and Rashba SOI, respectively in the literatures [32, 38, 46–48]. According to our notation, $k_y^3{\sigma }_x-k_x^3{\sigma }_y\sim \mathrm{I}\mathrm{m}[(k_{+}^3+3k_{+}{k}_{-}^2){\sigma }_{+}]$, $(k_x^2-k_y^2)(k_y{\sigma }_x+k_x{\sigma }_y)\sim \mathrm{I}\mathrm{m}[(k_{+}^2+k_{-}^2)k_{+}{\sigma }_{+}]$, $k_x^2{k}_y^2(k_y{\sigma }_x-k_x{\sigma }_y)\sim \mathrm{I}\mathrm{m}[(k_{+}^2-k_{-}^2{)}^2{k}_{+}{\sigma }_{-}]$, and $k_x{k}_y(k_x{\sigma }_x-k_y{\sigma }_y)\sim \mathrm{I}\mathrm{m}[(k_{+}^2-k_{-}^2)k_{+}{\sigma }_{-}]$ belong to Rashba-like SOI with $C_{4v}$ symmetry [49–51] while $k_x{k}_y(k_y{\sigma }_x-k_x{\sigma }_y)\sim \mathrm{R}\mathrm{e}[(k_{+}^2-k_{-}^2)k_{+}{\sigma }_{-}]$ and $k_x^3{\sigma }_x+k_y^3{\sigma }_y\sim \mathrm{I}\mathrm{m}[(k_{+}^3+3k_{+}{k}_{-}^2){\sigma }_{+}]$ belongs to Dresselhaus-like SOI with $C_4$ symmetry [42, 52].

Tab.1 SOI Hamiltonians of various orders and symmetries.

In-plane Dresselhaus-like SOI (symmetry)		In-plane Rashba-like SOI (symmetry)
$H_2=\mathrm{\Re }(k_{+}{\sigma }_{+})$ (21')		$H_1=k_y{\sigma }_x-k_x{\sigma }_y\sim \mathrm{\Im }(k_{+}{\sigma }_{-})$ ($C_{\mathrm{\infty }}$, $\mathcal{T}$)
$H_{3x}=\mathrm{\Re }(k_{+}^2{\sigma }_{-})$ (2'mm') [33, 39], $H_{4x}=\mathrm{\Re }(k_{+}^2{\sigma }_{+})$ (6'm'm) [33, 39]		$H_{3y}=\mathrm{\Im }(k_{+}^2{\sigma }_{-})$ (2'm'm) [33], $H_{4y}=\mathrm{\Im }(k_{+}^2{\sigma }_{+})$ (6'm'm) [33]
$H_2^{\prime }=\mathrm{\Re }(k_{+}^3{\sigma }_{-})$ (21'), $H_5=\mathrm{\Re }(k_{+}^3{\sigma }_{+})$ (41')		$H_6=\mathrm{\Im }(k_{+}^3{\sigma }_{-})$ (2mm1') [33], $H_7=\mathrm{\Im }(k_{+}^3{\sigma }_{+})$ (4mm1') [33]
$H_5^{\prime }=\mathrm{\Re }(k_{+}^5{\sigma }_{-})$ (41'), $H_8=\mathrm{\Re }(k_{+}^5{\sigma }_{+})$ (61')		$H_7^{\prime }=\mathrm{\Im }(k_{+}^5{\sigma }_{-})$ (4mm1'), $H_9=\mathrm{\Im }(k_{+}^5{\sigma }_{+})$ (6mm1')
Out-of-plane Dresselhaus-like SOI		Out-of-plane Rashba-like SOI
$\mathrm{\Re }(k_{+}){\sigma }_z$ (m1') [40]		$\mathrm{\Im }(k_{+}){\sigma }_z$ (m1')
$\mathrm{\Re }(k_{+}^2){\sigma }_z$ (4'm'm) [11]		$\mathrm{\Im }(k_{+}^2){\sigma }_z$ (4'm'm)
$\mathrm{\Re }(k_{+}^3){\sigma }_z$ (3m) [32, 41–43]		$\mathrm{\Im }(k_{+}^3){\sigma }_z$ (3m) [32, 42, 44, 45]

3 Classification of SHE from Neumann’s principle

SHE can be classified using Neumann’s principle. Note that the spin current is obtained from spin (pseudo-vector) and velocity (vector) operators, it forms a second rank pseudo-tensor while the electric field is a vector. As a result, the spin conducticity tensor ${\sigma }_{\beta \gamma }^{\alpha }$ is a third rank pseudo-tensor ($\beta ,\gamma =x,y$) [53, 54]. Different from the Hall effect, generally speaking, spin conductance does not enjoy antisymmetric property with respective to $\beta$ and $\gamma$. According to Neumann’s principle, the spin conductivity tensor is expressed in terms of a rotation matrix $R$ as

(7)${\sigma }_{\beta \gamma }^{\alpha }=\eta \mathrm{d}\mathrm{e}\mathrm{t}(R)R_{\alpha {\alpha }^{\prime }}{R}_{\beta {\beta }^{\prime }}{R}_{\gamma {\gamma }^{\prime }}{\sigma }_{{\beta }^{\prime }{\gamma }^{\prime }}^{{\alpha }^{\prime }}.$

where $\eta =-1$ for prime operation and the presence of $\mathrm{d}\mathrm{e}\mathrm{t}(R)$ is needed for a pseudo-tensor. From Eq. (1), it can also be shown as

(8)${\sigma }_{\beta \gamma }^{\alpha }={\displaystyle {\int }_k{\mathrm{\Omega }}_{\beta \gamma }^{\alpha }(\mathit{k})=\eta \mathrm{d}\mathrm{e}\mathrm{t}(\mathrm{R}){\displaystyle {\int }_k{\mathrm{\Omega }}_{\beta \gamma }^{\alpha }(R\mathit{k}),}}$

Since 2D MPG is imbedded in 3D MPG, we can use Bilbao Crystallographic Server [55] to find nonzero components of ${\sigma }_{\beta \gamma }^{\alpha }$, which is similar to the discussion of higher-order anomalous Hall effects [11–13].

The results obtained from Bilbao Crystallographic Server, using Jahn Symbol ${\mathrm{e}\mathrm{V}}^3$, can be summarized as follows. For in-plane SHE (IP-SHE), we find: (i) for MPG m, m1', ${\sigma }_{xy}^y={\sigma }_{yx}^y=0$ and ${\sigma }_{xy}^x$ and ${\sigma }_{yx}^x$ are indefinite; (ii) for MPG 2', m', $\mathcal{T}$, both ${\sigma }_{xy}^{x/y}$ and ${\sigma }_{yx}^{x/y}$ are indefinite; (iii) for MPG 3, 31', 3m', 6', ${\sigma }_{xy}^{x/y}={\sigma }_{yx}^{x/y}$; (iv) for MPG 3m, 3m1', 6'mm', ${\sigma }_{xy}^x={\sigma }_{yx}^x=0$ and ${\sigma }_{xy}^y={\sigma }_{yx}^y$. For other 2D MPG elements, both ${\sigma }_{xy}^{x/y}$ and ${\sigma }_{yx}^{x/y}$ are zero. For out-of-plane SHE (OP-SHE), we have: (i) ${\sigma }_{xy}^z=-{\sigma }_{yx}^z$ for most of the high symmetry rotations: 4, 41', 4mm, 4mm1', 4'm'm, 4m'm', 3, 31', 3m, 3m1', 3m', 6, 61', 6', 6mm, 6mm1', 6'mm', 6m'm'; (ii) ${\sigma }_{xy}^z$ and ${\sigma }_{yx}^z$ are indefinite for the rest of the MPGs. The in-plane and out-of-plane SHEs are schematically shown in Fig.1.

Fig.1 Schematic plots of the in-plane (a) and out-of-plane (b) spin Hall effects. $J_s^{y/z}$ denotes the spin current, and $E$ is the driving electric field.

Full size|PPT slide

These results are summarized in Tab.2, from which we find that for many systems the existence of SHE can not be determined solely by the symmetry. Moreover, we observe that ${\sigma }_{xy}^{\alpha }$ can be switched on and off while maintaining the symmetry of the system. For instance, $H_{\mathrm{S}\mathrm{O}\mathrm{I}}=k_x^2{\sigma }_x+k_y^2{\sigma }_y$ has $C_2\mathcal{T}$ symmetry and it is easy to verify that ${\sigma }_{xy}^z$ is zero for $H_{\mathrm{S}\mathrm{O}\mathrm{I}}$ while ${\sigma }_{xy}^z\ne 0$ for $H_1+H_{\mathrm{S}\mathrm{O}\mathrm{I}}$ where $H_1$ is the linear Rashba SOI listed in Tab.1. This shows that for a given Hamiltonian, symmetry alone cannot characterize the OP-SHE. This conclusion is also valid for IP-SHE. Note that Eq. (7) relates different components of SHE and imposes constraint on them (global constraint). It gives an upper bound of the number of independent components solely from the symmetry of the third rank pseudo-tensor, regardless of the physical system. Once the expression of SHE is given, we can use Eq. (8) to find further constraint on a particular component of SHE (local constraint). It turns out that the parity and types ($H_D$ or $H_R$) of SOI are good indicators to classify SHE. We recognize that parity is also a kind of symmetry, but in this work the word “symmetry” refers solely to spatial symmetries. In the following, we will give the sufficient condition under which SHE may vanish.

Tab.2 Results from Neumann’s principle for IP-SHE. Here “N” represents no constraint. “S” and “A” refer to symmetric relation for ${\sigma }_{xy}^{x/y}$ and antisymmetric relation for ${\sigma }_{xy}^z$ with respective to $x$ and $y$, respectively. “0” stands for zero ${\sigma }_{xy}^{\alpha }$. The results with $m_x$ are listed. For MPGs containing $m_y$, ${\sigma }_{xy}^x$ and ${\sigma }_{xy}^y$ are interchanged. For example, for MPG $m_y$, $m_y$1', $m_y$', ${\sigma }_{xy}^x$ = N and ${\sigma }_{xy}^y$ = 0.

MPG	${\sigma }_{xy}^x$	${\sigma }_{xy}^y$	${\sigma }_{xy}^z$
$m_x$, $m_x$1', $m_x$'	0	N	N
2'	N	0	N
$\mathcal{T}$	N	N	N
3, 31', 6'	S	S	A
3$m_x$, 3$m_x$1', 3$m_x$', 6'$m_y$'$m_x$	0	S	A
4mm, 4mm1', 4'm'm, 4m'm', 4, 41', 6mm1', 6mm, 6, 61', 6m'm'	0	0	A
2, 21', 2mm, 2mm1', 2'm'm, 2m'm', 4'	0	0	N

4 Classification of SHE based on the parity and types of SOIs

The order of SOI, $N$, is determined by the power of $k$ (regardless of $k_x$ and $k_y$). In general, the parities of $H_D$ and $H_R$ are discussed in Appendix A. We find that the parity $(-1{)}^N$ plays a critical role in determining the existence of IP-SHE (${\sigma }_{xy}^{x/y}$). Specifically, we demonstrate that, for a system with several SOI components, the IP-SHE can be switched on and off by tuning these SOIs while maintaining the symmetry of the system. We further show that SHE can be classified by the parity of SOI.

For IP-SHE, there are 12 MPGs in Tab.3. The results are summarized as: (i) for MPG m, m1', 3m1, we find ${\sigma }_{xy}^x=0$ and ${\sigma }_{xy}^y=1$ (“1” stands for nonzero) when $m=m_x$; (ii) for MPG m', 3m, 3m', we find ${\sigma }_{xy}^x=0$ while ${\sigma }_{xy}^y=0/1$ (“0/1” stands for zero or nonzero depending on the parity of individual SOI component in the Hamiltonian); (iii) for MPG $\mathcal{T}$, 3, 31', ${\sigma }_{xy}^{x/y}=0/1$; (iv) for MPG 2', 21', 2'mm', 2mm1', 41', 4mm1', 6', 6'm'm, 61', and 6mm1', no SOI Hamiltonians are available for 2D two-band models with nonzero $d_z$.

Tab.3 Results of direct calculation. Here “0” and “1” stand for zero and nonzero ${\sigma }_{xy}^{\alpha }$, respectively. “0/1” means SHE can be switched on and off and “NA” means that SOI Hamiltonian is not available for $d_z\ne 0$. The results with $m_x$ are listed. For MPGs containing $m_y$, ${\sigma }_{xy}^x$ and ${\sigma }_{xy}^y$ are interchanged.

MPG	${\sigma }_{xy}^x$	${\sigma }_{xy}^y$	${\sigma }_{xy}^z$
$m_x$	0	1	0/1
$m_x$1', 3$m_x$1'	0	1	1
$\mathcal{T}$, 3, 31'	0/1	0/1	1
$m_x$', 3$m_x$, 3$m_x$'	0	0/1	1
4mm, 4'm'm, 4m'm', 4, 6mm, 6, 6m'm', 2, 2mm, 2'm'm, 2m'm', 4'	0	0	1
2', 21', 2'mm', 2mm1', 41', 4mm1', 6', 6'm'm, 61', 6mm1'	NA	NA	1

For OP-SHE, ${\sigma }_{xy}^z$ is nonzero for most of MPGs. The condition for ${\sigma }_{xy}^z=0$ is that $d_x{d}_y$ is an odd function in momentum space. We find that ${\sigma }_{xy}^z$ is nonzero for all 2D MPGs except three (m, 1, 2') with the following Hamiltonians: (i) the Hamiltonian with MPG m (mirror symmetry), e.g., $H=k_y^2{\sigma }_x+k_x{\sigma }_y$. Other Hamiltonian with the same symmetry may not have vanishing OP-SHE. (ii) the Hamiltonian with no symmetry at all, e.g., $H=k_x{\sigma }_x+k_y^2{\sigma }_y$. (iii) the Hamiltonian with MPG 2', e.g., $H=k_y^2{\sigma }_x-k_x^2{\sigma }_y$. Another straightforward condition is that the system either has the chiral symmetry ${\sigma }_x$ [56, 57] or chiral symmetry ${\sigma }_y$ [58]. In the following, we give an example to demonstrate our findings and then verify it in a general account.

We consider the following Hamiltonian

(9)$\begin{array}{rl}H=& k^2{\sigma }_0+{\lambda }_1(k_y{\sigma }_x-k_x{\sigma }_y)+{\displaystyle \frac{{\lambda }_2}{2}}(k_{+}^2{\sigma }_{+}+k_{-}^2{\sigma }_{-})\\ & -\mathrm{i}{\displaystyle \frac{{\lambda }_3}{2}}(k_{+}^5{\sigma }_{+}-k_{-}^5{\sigma }_{-})+{\displaystyle \frac{{\lambda }_4}{2}}(k_{+}^3+k_{-}^3){\sigma }_z-{\displaystyle \frac{\mathrm{i}{\lambda }_5}{2}}(k_{+}^3\\ & -k_{-}^3){\sigma }_z+{\displaystyle \frac{{\lambda }_6}{2}}(k_{+}^6+k_{-}^6){\sigma }_z-{\displaystyle \frac{\mathrm{i}{\lambda }_7}{2}}(k_{+}^6-k_{-}^6){\sigma }_z.\end{array}$

Here ${\sigma }_{\pm }={\sigma }_x\pm \mathrm{i}{\sigma }_y$. In this Hamiltonian, different parts of SOI have been used before. For instance, the ${\lambda }_2$ term was used in Ref. [39] to address 2D dual topological insulator of Na${}_3$Bi. By fitting the experimental data, the Dresselhaus-like SOI (${\lambda }_3$ term) can account for the strong out-of-plane spin component at the Fermi surface of 2D Au/Ge(111) surface [34]. The ${\lambda }_4$ term was proposed [41] to explain the experimentally observed warping effect of Fermi surface for topological insulator Bi${}_2$Te${}_3$ and the ${\lambda }_5$ term is crucial in determining the origin of experimental finding, the giant Zeeman-type spin polarization in WSe${}_2$ [59]. Note that the linear Rashba SOI (${\lambda }_1$ term) has the highest symmetry while the ${\lambda }_2$ term (6'm'm), ${\lambda }_3$ term (6mm1'), ${\lambda }_6$ term (6), and ${\lambda }_7$ term (6mm) are hexagonal SOI. The ${\lambda }_4$ and ${\lambda }_5$ terms are trigonal SOI having $m_x$ and $m_y$ symmetries, respectively, making Eq. (9) a good testing platform for symmetry analysis. For instance, when ${\lambda }_2$ and ${\lambda }_5$ are nonzero, the system has ($C_3$, $m_y\mathcal{T}$) symmetry (see Appendix B for more details). Turning on and off any of ${\lambda }_1$ [60], ${\lambda }_3$, or ${\lambda }_6$ does not affect the symmetry of the system. Fixing ${\lambda }_2$ and ${\lambda }_4$ to be nonzero while switching off ${\lambda }_5$ and ${\lambda }_6$ changes the system symmetry to ($C_3$, $m_x$). Therefore IP-SHE of this trigonal symmetry can be studied by tuning any of parameters of ${\lambda }_1$, ${\lambda }_3$, and ${\lambda }_7$. For nonzero $({\lambda }_2$, ${\lambda }_4$, ${\lambda }_5)$, the system has $C_3$ symmetry. Its IP-SHE can be studied by varying ${\lambda }_1$. In addition, IP-SHE of hexagonal MPG 6 can be examined by setting $({\lambda }_2$, ${\lambda }_4$, ${\lambda }_5)=0$ and$({\lambda }_3$, ${\lambda }_6$, ${\lambda }_7)\ne 0$ while varying ${\lambda }_1$. IP-SHE of MPG 6mm (6m'm') can be studied by switching on ${\lambda }_3$ and ${\lambda }_7$ (${\lambda }_6$) and turning on and off ${\lambda }_1$.

In the following, we focus on $c_{\alpha }\equiv {\mathrm{\partial }}_x{d}_0({\mathrm{\partial }}_y\mathbf{d}\times \mathbf{d}{)}_{\alpha }$ which differs from ${\mathrm{\Omega }}_{xy}^{\alpha }$ by $1/(4d^3)$. For convenience, we define $a_N=k_{+}^N+k_{-}^N$ and $b_N=\mathrm{i}(k_{+}^N-k_{-}^N)$ so that all SOI Hamiltonians in Tab.1 can be expressed in terms of $a_N$, $b_N$, and ${\sigma }_{\alpha }$. The parities of $a_N$ and $b_N$ with respect to $k_x$ and $k_y$ are found to be $a_N\sim k_x^N$ and $b_N\sim k_x{k}_y{a}_N$ (see Appendix C). For Eq. (9), in terms of $a_N$ and $b_N$, we have $d_x={\lambda }_1{k}_y+{\lambda }_2{a}_2+{\lambda }_3{b}_5$, $d_y=-{\lambda }_1{k}_x+{\lambda }_2{b}_2-{\lambda }_3{a}_5$, and $d_z={\lambda }_4{a}_3+{\lambda }_5{b}_3+{\lambda }_6{a}_6+{\lambda }_7{b}_6$. Using Eq. (5), the expression of $c_{\alpha }$ is shown in Appendix D.

Now we examine the behaviors of IP-SHE for various symmetries 6mm (6m'm'), 6, ($C_3$, $m_y\mathcal{T}$), ($C_3$, $m_x$) and $C_3$. 1) For MPG 6mm or 6m'm', we find ${\sigma }_{xy}^{x/y}=0$. The reason behind this can be understood by simple parity analysis. For a 2D system having two mirror symmetries or both $m_{x/y}\mathcal{T}$ symmetries, $d^2$ is an even function of both $k_x$ and $k_y$, and hence we can focus on the parity of $c_{\alpha }$. For ${\sigma }_{xy}^{\alpha }$ to be nonzero, $c_{\alpha }$ must be an even function of $k_0$ if we set $k_x=k_y=k_0$. In addition, the in-plane (out-of-plane) SOIs must be odd (even) functions of $k_0$. This is because ${\sigma }_{\pm }$ rotates in the same way as $k_{\pm }$ while ${\sigma }_z$ remains unchanged under rotation. Since $c_{x/y}$ scales like $d_{y/x}{d}_z{k}_x{k}_y$ according to Eq. (5), it is impossible for $c_{x/y}$ to be an even function of $k_0$. Thus, we find that ${\sigma }_{xy}^{x/y}=0$ for systems with two mirror symmetries, which include symmetry groups: 2mm, 4mm, 6mm, 2mm1', 4mm1', 6mm1', 2m'm', 4m'm', and 6m'm'.

2) For the system with $C_6$ symmetry, (${\lambda }_3$, ${\lambda }_6$, ${\lambda }_7$) is nonzero and varying ${\lambda }_1$ does not affect the symmetry. We find ${\sigma }_{xy}^{x/y}=0$ because $d^2$ has a mirror symmetry $M_{x+y}$ with or without ${\lambda }_1$ so that terms involving $c_{x/y}^{(2)}$ and $c_{x/y}^{(3)}$ do not contribute to ${\sigma }_{xy}^{x/y}$. Discussion on the symmetry of energy spectrum or $d^2$ is shown in Appendix E. This result agrees with that obtained from Neumann’s principle.

3) For the system with ($C_3$, $m_y\mathcal{T}$) symmetry, we study the following cases. (a) The case when ${\lambda }_2$ and ${\lambda }_5$ are the only two nonzero parameters. Since there is only one SOI in either IP-SOI or OP-SOI, $d^2$ has two mirror symmetries. From Appendix D, we keep only $c_x^{(1)}$ and $c_y^{(1)}$ and find ${\sigma }_{xy}^{x/y}=0$ (no IP-SHE). As will be discussed later, if there is one IP-SOI and one OP-SOI with different parities, there is no IP-SHE. Case (3a) is just a special case. (b) Now we turn on ${\lambda }_1$ which respects $m_y$ symmetry. Since the ${\lambda }_2$ term does have $m_y$ symmetry, hence $d_x^2+d_y^2$ has only $m_x$ symmetry which is the only mirror symmetry possessed by both IP-SOIs. Note that the symmetry of $d_z^2$ remains the same as in (3a) which makes $d^2$ asymmetric about $k_y$. Including both $c_x^{(1)}$ and $c_x^{(3)}$, we obtain $c_x=3{\lambda }_1{\lambda }_5{a}_2{k}_x^2+{\lambda }_2{\lambda }_5(2a_1{b}_3-3a_2{b}_2)k_x$ and $c_y=0$, which gives ${\sigma }_{xy}^x\ne 0$ and ${\sigma }_{xy}^y=0$ [shown in Appendix F(a)]. (c) If we replace the ${\lambda }_1$ term by the ${\lambda }_6$ term, $d_x^2+d_y^2$ has two mirror symmetries as explained above. Note that $d_z^2$ has $m_x$ symmetry. We have $c_x=-4{\lambda }_2{\lambda }_6(a_1{a}_6+b_2{b}_5)k_x+2{\lambda }_2{\lambda }_5(3a_2{b}_2-2a_1{b}_3)k_x$ and $c_y=0$, leading to nonvanishing ${\sigma }_{xy}^x$ [shown in Appendix F(b)]. We see that ${\sigma }_{xy}^x$ and ${\sigma }_{xy}^y$ can be switched on and off while maintaining the system symmetry.

4) For the system with ($C_3$, $m_x$) symmetry, we require ${\lambda }_2$ and ${\lambda }_4$ to be nonzero and set ${\lambda }_5={\lambda }_6=0$. (a) When ${\lambda }_1={\lambda }_7=0$, $d^2$ has two mirror symmetries and we find ${\sigma }_{xy}^{x/y}=0$ from the symmetry argument which is the same as case (3a). (b) When turning on ${\lambda }_1$, $d^2$ becomes symmetric about $k_x$ only. Hence neglecting terms odd in $k_x$ we find ${\sigma }_{xy}^x=0$ and $c_y={\lambda }_1{\lambda }_4(a_3-3b_2{k}_y)k_x+$${\lambda }_2{\lambda }_4(2a_3{b}_1-3a_2{b}_2)k_x$ making ${\sigma }_{xy}^y\ne 0$ [shown in Appendix F(c)]. (c) When replacing the ${\lambda }_1$ term by the ${\lambda }_7$ term, the symmetry of $d^2$ remains the same. We obtain ${\sigma }_{xy}^x=0$ and $c_y={\lambda }_2{\lambda }_4(2a_3{b}_1-3a_2{b}_2)k_x$ which has nonzero contribution. The result is the same as (3b).

5) When $({\lambda }_2$, ${\lambda }_4$, ${\lambda }_5)$ are nonzero, the system has $C_3$ symmetry. It is easy to show that $d^2$ has only mirror symmetry $m_{x+y}$ and therefore only terms involving $c_{x/y}^{(1)}$ and $c_{x/y}^{(4)}$ contribute. We find ${\sigma }_{xy}^{x/y}=0$. When ${\lambda }_1$ is present, the system is still maintained at $C_3$ while $m_{x+y}$ symmetry is broken for $d^2$. We find $c_x=-6{\lambda }_1{\lambda }_5{a}_2{k}_x^2+$$6{\lambda }_1{\lambda }_4{b}_2{k}_x^2-2{\lambda }_2{\lambda }_4(2a_1{a}_3+3b_2^2)k_x+2{\lambda }_2{\lambda }_5(3a_2{b}_2-2a_1{b}_3)$$k_x$ and $c_y=2{\lambda }_1{\lambda }_4(a_3-3b_2{k}_y)k_x+2{\lambda }_1{\lambda }_5(b_3+3a_2{k}_y)k_x+$$2{\lambda }_2{\lambda }_5(2b_1{b}_3+3a_2^2)k_x+2{\lambda }_2{\lambda }_4(2a_3{b}_1-3a_2{b}_2)k_x$. Hence both ${\sigma }_{xy}^{x/y}\ne 0$.

For OP-SHE, it is easy to see that: if OP-SHE vanishes for a particular $H_{SOI}$ then $H_1+H_{SOI}$ makes ${\sigma }_{xy}^z\ne 0$. Therefore, for a given symmetry, OP-SHE can also be tuned from zero to nonzero. One IP-SOI means a basic building block with a defnite parity and SOI type, such as the one in Tab.1.

Now we give the sufficient conditions for vanishing IP-SHE in 2D systems in the following, and present the proof in Appendix I. (i) If the SOI Hamiltonian has chiral symmetry ${\sigma }_z$, then ${\sigma }_{xy}^{x/y}=0$. (ii) If the SOI Hamiltonian contains only one in-plane and one out-of-plane components, there is no IP-SHE when the two componants have different parities [see the examples, cases (3a) and (4a)]. Supposing they have the same parity, we find that ${\sigma }_{xy}^y=0$ if both components are the same type of SOI while ${\sigma }_{xy}^x=0$ if they are different types of SOI. (iii) The case that $d^2$ has one mirror symmetry $m_x$. Supposing the Hamiltonian is given by $H=H_0+{\lambda }_1{R}_{IP}+{\lambda }_2{D}_{IP}+$${\lambda }_3{D}_{OP}+{\lambda }_4{R}_{OP}$ where $R_{IP}$ and $D_{OP}$ stand for in-plane Rashba-like SOI and out-of-plane Dresselhaus-like SOI, respectively. Each SOI $A_B$ with $A=R,D$ and $B=IP,OP$ may have several terms but must have the same parity. For instance, $D_{IP}=D_{IP}^a+D_{IP}^b$, IP-SHE vanishes only if $a$ and $b$ have the same parity. We find that the condition for ${\sigma }_{xy}^x=0$ is: the parity of $(D_{IP},R_{IP},D_{OP},R_{OP})=$$(\pm ,\mp ,\mp ,\pm )$ where $+$ stands for even parity while the condition for ${\sigma }_{xy}^y=0$ is: the parity of $(D_{IP},R_{IP},D_{OP},R_{OP})=(\pm ,\mp ,\pm ,\mp )$ [see the examples, cases (3b) and (4b-d)]. (iv) Assuming the mirror symmetry of $d^2$ is $m_y$, we find that the condition for ${\sigma }_{xy}^x=0$ is $H=R_{IP}^{-}+R_{IP}^{+}+D_{OP}^{-}+D_{OP}^{+}$ or $H=D_{IP}^{-}+D_{IP}^{+}+R_{OP}^{-}+R_{OP}^{+}$; while for ${\sigma }_{xy}^y=0$, we require $H=R_{IP}^{-}+R_{IP}^{+}+R_{OP}^{-}+R_{OP}^{+}$ or $H=D_{IP}^{-}+D_{IP}^{+}+D_{OP}^{-}+D_{OP}^{+}$. (v) If all IP-SOIs have one parity and all OP-SOIs have another parity, ${\sigma }_{xy}^{x/y}=0$.

We present examples of SOI Hamiltonians for all 2D MPGs in Table A2 of Appendix J. As shown in Appendix K, the vanishing of IP-SHE for all 2D MPGs with $d_z\ne 0$ in Tab.3 can be predicted using conditions (2)−(5) without involving Neumann’s principle.

5 Discussion and conclusion

In summary, we have demonstrated that the symmetry of the system along is not enough to characterize the existence of spin Hall effect, while the parity and symmetry types of constituent SOI play a crucial role. It is found that 2D in-plane SHE can be switched on and off by combining different SOI components, including in-plane and out-of-plane Rashba-like and Dresselhaus-like SOIs of different orders, while maintaining the system symmetry. Sufficient conditions for the existence of SHE are presented, accompanied by complete analysis on all 2D magnetic point groups.

The verification of our findings would be straightforward. SHE has been experimentally realized over fifteen years [36, 64]. In Ref. [65], $H=H_1+H_9+H_{13x}$ was used to model the surface states in the Bi${}_2$Te${}_3$ family of 3D TI using first-principles calculation, which has been experimentally verified [34]. However, we need a system with both types of SOI and different parities. In Ref. [66], a new class of atom-laser coupling schemes were introduced to describe the spin−orbit coupled Hamiltonian of ultracold neutral atoms, which has the form $H=H_1+H_{3x}+H_{3y}+H_5+H_6$ and could be adopted to the switching on and off of SHE.

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Declarations

The authors declare that they have no competing interests and there are no conflicts.

Acknowledgements

We acknowledge support from the National Natural Science Foundation of China (Grant Nos. 12034014, 12174262, and 12004442). L. Wang also thanks the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2021B1515130007) and the Shenzhen Natural Science Fund (the Stable Support Plan Program 20220810130956001).

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