Symmetry dictionary on charge and spin nonlinear responses for all magnetic point groups with nontrivial topological nature

ABSTRACT Recently, charge or spin nonlinear transport with nontrivial topological properties in crystal materials has attracted much attention. In this paper, we perform a comprehensive symmetry analysis for all 122 magnetic point groups (MPGs) and provide a useful dictionary for charge and spin nonlinear transport from the Berry curvature dipole, Berry connection polarizability and Drude term with nontrivial topological nature. The results are obtained by conducting a full symmetry investigation of the matrix representations of six nonlinear response tensors. We further identify every MPG that can accommodate two or three of the nonlinear tensors. The present work gives a solid theoretical basis for an overall understanding of the second-order nonlinear responses in realistic materials.

The response charge current density is given by 3 , v a is the electric velocity in the a direction, and we omit the band index. Gao et al. [2] proposed a modification of Berry curvature under driving electric field, where A is the Berry curvature connection, the modification is A a = eG ac E c . G n ac (k) = 2Re m =n A nm a (k)A mn c (k) εn(k)−εm(k) is the Berry connection poloarizability (BCP). Here we can directly get the electric velocity in the a direction, v a = 1 ∂ε ∂k a + e abd E b Ω d + where abc is the Levi-Civita symbol and the last term is the modification. Thus, we can get the response charge current j a = Re{j 0 a + j 1 a + j 2 a }, where j 2 a refers to the second-order current in the direction of a in response to the driving electric field, which is the part of our work. In the low frequency limit, Therefore, the dependence of the coeffficient on τ is Here we figure out the second-order charge response coefficient is a rank-three tensor, contributing by the three parts, BCD [1] , BCP [3,4] and the Drude [5,6]. According to the Ref. [6], we can do some further calculations, and we have Further, if we take the electric spin into account, the second-order conductivity can be expressed as where the σ = ↑ or ↓ is spin index of electrons. Then the second-order charge current and the spin current can be expressed separately as j a = j a and j (s) a ) [7]. Hence, we can write the second-order charge and spin response tensors, where χ abc = χ ↑ abc + χ ↓ abc , χ (s) abc = χ ↑ abc − χ ↓ abc , and we use the upper index s to represent the spin response. These three terms are the major contributions of the second-order response. Take the charge response as example, for the BCD and BCP terms they all have a ↔ b anti-symmetries, which have the same 9 independent components, χ BCD/BCP However the Drude term have the interchangeably symmetric for any two indices, and have 10 independent components, We can get similar results for the second-order spin response tensors.

SII. THE INTRODUCTION OF 122 MAGNETIC POINT GROUPS
The magnetic point groups(MPGs) M, or we can call the Shubnikov groups can be divided into three classes, the number of which are 32, 32 and 58. In Table S1, we give the international symbols and Schoenfies symbols for the Class I, as well as the international symbols of the Class II and III. Next we figure out all the production matrices (generators) of the three classes of magnetic point groups, which are represented by the symbol Γ i , i = 0, 1, 2, 3, ..., 9 and Γ k = Γ k T , k = 0, 1, 2, 3, ..., 9, where the T is the time-reversal operation. In this way, we can obtain the general matrices representations of physical quantities. The general matrices of the Class I and Class III can be found in Table S2.

SIII. SYMMETRY ANALYSIS
Since there are published papers working on some groups and symmetry analysis, here we give a comparison between their derivation and this work. It is worth to mentioning that the second-order charge response tensor also can be represented as a rank-two pseudo-tensor Berry curvature dipole D ab = k f 0 (∂ a Ω b ). According to the Eq. S8, we have For the MPG 23 (T) and 432(O), we have Hence, the second-order charge response tensor can be written, Hence we can get the charge current Hence, there is no second-order response current for the MPGs T (23)and O(432).  Table S12 II(0/32) nonē  In order to distinction, we call these spin-dependent BCD, BCP and Drude terms. The lists of allowed MPGs are shown for spin-dependent BCD (Table S7), BCP (Table S8) and Drude terms (Table S9). According to the Table S4−S9, we can find the Class II omitted in Table S5, S6 and S7 but presented in S4, S8 and S9. This issue is due to the time reversal operator. Class II contains T as an element to satisfy the time reversal symmetry (TRS). If the tensor does not satisfy the TRS, the Class II MPGs must forbid the corresponding current. Under the time reversal operator, we have According to Eq. (S8), For the second charge current contributed by the BCD, Thus, the BCD term satisfies the time reversal symmetry. For the BCP and Drude terms, they are both zero for the time reversal symmetry. Therefore, the existence of time reversal symmetry enables the BCP and Drude vanishing but BCD existing, which leads to the Class II (T is an element for Class II MPGs) omitted in Table S5 (BCP) and S6 (Drude) while presented in S4 (BCD). For the spin current, the spin BCD, BCP and Drude terms are defined as By analogy with Eq. (S30) and Eq. (S31), we have And, For time reversal symmetry, the spin-dependent BCD is zero, while the spin-dependent BCP and Drude are allowed. So the Class II omitted in Table S7 (spin-dependent BCD) but presented in Table S8 (spin-dependent BCP) and S9 (spin-dependent Drude).
Another important message is that the third line in Table S4 (BCD for Class III) is identical to the third line in Table S8 (spin-BCP for class III), while the third line in Table S5 (BCP for Class III) is identical to the second line in Table S7 (spin-BCD for class III).
For the second charge conductivity, As the component of charge and spin response tensors is unchanged before and after the symmetry operation, we can find every nonzero component of the tensors. Based on Eq. (S35), the condition of non-zero charge current is For the second spin conductivity, And, Because the BCD and spin-dependent BCP have the same independent components, the first line in Eq. (S36) for BCD is the same as the second line in Eq. (S38) for the spin-dependent BCP term. Thus, the third line in Table S4  (BCD for Class III) is identical to the third line in table S8 (spin-BCP for class III).
Since the second line in Eq. (S36) for BCP is the same as the first in Eq. (S37) for spin-dependent BCD term, the third line in Table S5 (BCP for Class III) is identical to the second line in Table S7 (spin-BCD for class III).
However, the Drude term IS interchangeably symmetric for any two indices, and has 10 independent components, which causes the Drude term is special and have more allowed Class III MPGs (Table S6 and S9).
Next we give all corresponding matrix representations and candidate materials in Table S10 − S20.
In this table, the candidate materials are not repeated, which are the same as the TABLE S13.   Gd2Ti2O6 [86] 2221 ,42m1 AgNiO2 [28], GeCu2O4 [52] 2mm1 Ba6Co6ClO15.5 [87]      CsFeCl3 [71] 3m GaV4S8 [50], GaV4S8 [50] C. Two-dimensional matrix representations for all response tensors The second-order charge and spin response tensors can be expressed as j where the χ abc is a rank-three tensor and can be used a three-dimensional matrix to represent. Straightforwardly, we can also use a two-dimensional matrix to describe due to the commutative symmetry of the electric field components E b and E c . Hence, we blackefine a matrix to represent it [89], The current density is xyx . Thus, the three term of the second conductivity can be written as xxx xyy xzz 2xyz 2xxz 2xxy xxy yyy yzz 2yyz 2xyz 2xyy xxz yyz zzz 2yzz 2xzz 2xyz So, we give the two-dimensional matrix representations for the second-order charge and spin response tensor in the Table S21 and S23. This result is also completely consistent with the discussion about BCD term χ BCD , which is χ in in the Ref. [8] .