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Copy pathSpin_Chern.m
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Spin_Chern.m
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function Spin_Chern(parameters,a1,a2,grid,M)
%This function allows to calculate the spin chern numbers for both spin-up
%and spin-down bands
% The formula used in this calulation is as follows:
% c_up,down = sum_{k in EBZ}(berry_up,down(kx,ky) )
%-------------------------------------------------------------
%Initial parameters
t = parameters(1); %Nearest Neighbours
V = parameters(2); %staggered potential
LSO = parameters(3); %Next Nearest Neigbours
LR = parameters(4); %Rashba term
%--------------------------------------------------------------
%Matrices
%Pauli matrices
sig_x = [0 1;1 0];
sig_y = [0 -i;i 0];
sig_z = [1 0;0 -1];
I = [1 0;0 1];
T = i*kron(sig_y,I); %Time-reversal operator
%Gamma matrices
G1 = kron(sig_x,I);
G2 = kron(sig_z,I);
G3 = kron(sig_y,sig_x);
G4 = kron(sig_y,sig_y);
G5 = kron(sig_y,sig_z);
G15 = 1/(2*i)*( G1*G5 - G5*G1 );
G23 = 1/(2*i)*( G2*G3 - G3*G2 );
G24 = 1/(2*i)*( G2*G4 - G4*G2 );
G34 = 1/(2*i)*( G3*G4 - G4*G3 );
G35 = 1/(2*i)*( G3*G5 - G5*G3 );
G45 = 1/(2*i)*( G4*G5 - G5*G4 );
%--------------------------------------------------------------
%Calculations
dkx = pi/norm(a1)/grid;
dky = pi/norm(a2)/grid;
k_x = -pi/norm(a1):dkx:2*pi/norm(a1); %kx grid
k_y = -pi/norm(a2):dky:2*pi/norm(a2); %ky grid
Gamma1 = 2*M(1,:);
Gamma2 = 2*M(2,:);
c_Vdown = 0; %Chern numbers for each band
c_Vup = 0;
c_Cdown = 0;
c_Cup = 0;
for jj = 1:length(k_y)
ky = k_y(jj);
for ii = 1:length(k_x)
kx = k_x(ii);
if((kx>=0) && (kx<=Gamma1(1))) %Conditions for being inside of EBZ
if( ky >= Gamma1(2)/Gamma1(1)*kx )
if(ky <=( (Gamma2(2)-Gamma1(2))/(Gamma2(1)-Gamma1(1))*kx + Gamma2(2)) )
%Eigenvectors first----------------
k = [kx, ky];
k1 = dot(k,a1); k2 = dot(k,a2);
%Double Haldane model for different spin
d45 = t*(1 + cos(k1) + cos(k2)); % Re( NN )
d35 = t*( sin(k1) + sin(k2) ); % Im( NN )
d15 = 2*LSO*( sin(k1) - sin(k2) - sin(k1-k2)); % spin-orbit coupling
d34 = V; %stagerred potential
H = d45*G45 + d35*G35 + d15*G15 + d34*G34;
%Rashba term
x = (k1 + k2)/2;
y = (k2 - k1)/2;
d3 = sqrt(3)*LR*sin(y)*cos(x);
d4 = -sqrt(3)*LR*sin(x)*sin(y);
d23 = -LR*sin(x)*cos(y);
d24 = LR*( 1 - cos(x)*cos(y) );
H_R = d3*G3 + d4*G4 + d23*G23 + d24*G24;
H = H + H_R;
H = H + H';
[Pk1,~] = eig(H);
Pk1(2,:) = Pk1(2,:)*exp(i*(k1+k2)/2);
Pk1(4,:) = Pk1(4,:)*exp(i*(k1+k2)/2);
%Properly reordering the bands
if(abs( Pk1(1,1) )==0)
temp = Pk1(:,1);
Pk1(:,1) = Pk1(:,2);
Pk1(:,2) = temp;
end
if(abs( Pk1(1,4) )==0)
temp = Pk1(:,3);
Pk1(:,3) = Pk1(:,4);
Pk1(:,4) = temp;
end
%-------------------
%-----------------------------------------
%Eigenvectors second----------------------
k = [kx+dkx, ky];
k1 = dot(k,a1); k2 = dot(k,a2);
%Double Haldane model for different spin
d45 = t*(1 + cos(k1) + cos(k2)); % Re( NN )
d35 = t*( sin(k1) + sin(k2) ); % Im( NN )
d15 = 2*LSO*( sin(k1) - sin(k2) - sin(k1-k2)); % spin-orbit coupling
d34 = V; %stagerred potential
H = d45*G45 + d35*G35 + d15*G15 + d34*G34;
%Rashba term
x = (k1 + k2)/2;
y = (k2 - k1)/2;
d3 = sqrt(3)*LR*sin(y)*cos(x);
d4 = -sqrt(3)*LR*sin(x)*sin(y);
d23 = -LR*sin(x)*cos(y);
d24 = LR*( 1 - cos(x)*cos(y) );
H_R = d3*G3 + d4*G4 + d23*G23 + d24*G24;
H = H + H_R;
H = H + H';
[Pk2,~] = eig(H);
Pk2(2,:) = Pk2(2,:)*exp(i*(k1+k2)/2);
Pk2(4,:) = Pk2(4,:)*exp(i*(k1+k2)/2);
%Properly reordering the bands
if(abs( Pk2(1,1) )==0)
temp = Pk2(:,1);
Pk2(:,1) = Pk2(:,2);
Pk2(:,2) = temp;
end
if(abs( Pk2(1,4) )==0)
temp = Pk2(:,3);
Pk2(:,3) = Pk2(:,4);
Pk2(:,4) = temp;
end
%-------------------
%Eigenvectors third--------------------
k = [kx+dkx, ky+dky];
k1 = dot(k,a1); k2 = dot(k,a2);
%Double Haldane model for different spin
d45 = t*(1 + cos(k1) + cos(k2)); % Re( NN )
d35 = t*( sin(k1) + sin(k2) ); % Im( NN )
d15 = 2*LSO*( sin(k1) - sin(k2) - sin(k1-k2)); % spin-orbit coupling
d34 = V; %stagerred potential
H = d45*G45 + d35*G35 + d15*G15 + d34*G34;
%Rashba term
x = (k1 + k2)/2;
y = (k2 - k1)/2;
d3 = sqrt(3)*LR*sin(y)*cos(x);
d4 = -sqrt(3)*LR*sin(x)*sin(y);
d23 = -LR*sin(x)*cos(y);
d24 = LR*( 1 - cos(x)*cos(y) );
H_R = d3*G3 + d4*G4 + d23*G23 + d24*G24;
H = H + H_R;
H = H + H';
[Pk3,~] = eig(H);
Pk3(2,:) = Pk3(2,:)*exp(i*(k1+k2)/2);
Pk3(4,:) = Pk3(4,:)*exp(i*(k1+k2)/2);
%Properly reordering the bands
if(abs( Pk3(1,1) )==0)
temp = Pk3(:,1);
Pk3(:,1) = Pk3(:,2);
Pk3(:,2) = temp;
end
if(abs( Pk3(1,4) )==0)
temp = Pk3(:,3);
Pk3(:,3) = Pk3(:,4);
Pk3(:,4) = temp;
end
%-------------------
%Eigenvectors fourth--------------------
k = [kx, ky+dky];
k1 = dot(k,a1); k2 = dot(k,a2);
%Double Haldane model for different spin
d45 = t*(1 + cos(k1) + cos(k2)); % Re( NN )
d35 = t*( sin(k1) + sin(k2) ); % Im( NN )
d15 = 2*LSO*( sin(k1) - sin(k2) - sin(k1-k2)); % spin-orbit coupling
d34 = V; %stagerred potential
H = d45*G45 + d35*G35 + d15*G15 + d34*G34;
%Rashba term
x = (k1 + k2)/2;
y = (k2 - k1)/2;
d3 = sqrt(3)*LR*sin(y)*cos(x);
d4 = -sqrt(3)*LR*sin(x)*sin(y);
d23 = -LR*sin(x)*cos(y);
d24 = LR*( 1 - cos(x)*cos(y) );
H_R = d3*G3 + d4*G4 + d23*G23 + d24*G24;
H = H + H_R;
H = H + H';
[Pk4,~] = eig(H);
Pk4(2,:) = Pk4(2,:)*exp(i*(k1+k2)/2);
Pk4(4,:) = Pk4(4,:)*exp(i*(k1+k2)/2);
%Properly reordering the bands
if(abs( Pk4(1,1) )==0)
temp = Pk4(:,1);
Pk4(:,1) = Pk4(:,2);
Pk4(:,2) = temp;
end
if(abs( Pk4(1,4) )==0)
temp = Pk4(:,3);
Pk4(:,3) = Pk4(:,4);
Pk4(:,4) = temp;
end
%-------------------
%---------------------------------------
%Calculating chern numbers for each band
x12 = dot(Pk1(:,1),Pk2(:,1));
x23 = dot(Pk2(:,1),Pk3(:,1));
x34 = dot(Pk3(:,1),Pk4(:,1));
x41 = dot(Pk4(:,1),Pk1(:,1));
c_Vdown = c_Vdown + angle(x12/abs(x12)*x23/abs(x23)*x34/abs(x34)*x41/abs(x41));
x12 = dot(Pk1(:,2),Pk2(:,2));
x23 = dot(Pk2(:,2),Pk3(:,2));
x34 = dot(Pk3(:,2),Pk4(:,2));
x41 = dot(Pk4(:,2),Pk1(:,2));
c_Vup = c_Vup + angle(x12/abs(x12)*x23/abs(x23)*x34/abs(x34)*x41/abs(x41));
x12 = dot(Pk1(:,3),Pk2(:,3));
x23 = dot(Pk2(:,3),Pk3(:,3));
x34 = dot(Pk3(:,3),Pk4(:,3));
x41 = dot(Pk4(:,3),Pk1(:,3));
c_Cdown = c_Cdown + angle(x12/abs(x12)*x23/abs(x23)*x34/abs(x34)*x41/abs(x41));
x12 = dot(Pk1(:,4),Pk2(:,4));
x23 = dot(Pk2(:,4),Pk3(:,4));
x34 = dot(Pk3(:,4),Pk4(:,4));
x41 = dot(Pk4(:,4),Pk1(:,4));
c_Cup = c_Cup + angle(x12/abs(x12)*x23/abs(x23)*x34/abs(x34)*x41/abs(x41));
%-----------------------------
end %endif
end %endif
end %endif
end
end
% fprintf('[c_Vdown, c_Vup, c_Cdown, c_Cup]');
% X = [c_Vdown, c_Vup, c_Cdown, c_Cup]/(2*pi)
C_up = (c_Vup + c_Cup)/(2*pi)
C_down = (c_Vdown + c_Cdown)/(2*pi)
Cs = (C_up - C_down)/2
% Spin_Chern_valence = 0.5*( c_Vup - c_Vdown)/(2*pi)
% Spin_Chern_conducting = 0.5*( c_Cup - c_Cdown)/(2*pi)
end