(tutorial16)=

# Tutorial 16

## Spin wave spectrum of the honeycomb lattice Na2IrO3

we reproduce the spin wave calculation of the paper S. K. Choi, et al. PRL, 108(12), 127204 (2012), [[http://link.aps.org/doi/10.1103/PhysRevLett.108.127204]].

### Crystal structure

We define the crystal structure using the space group C2/m, and taking the crystallographic parameters at 300 K (parameters are only slightly different at 5 K) and we add not only the magnetic Ir4+ ions with effective spin quantum number of 1/2 but also the non-magnetic atoms for plotting the structure.

```matlab
nairo = spinw;
nairo.genlattice('lat_const',[5.427 9.395 5.614],'angled',[90 109.037 90],'spgr','C 2/m')
nairo.addatom('label','MIr4','r',[1/2; 0.167; 0],'S',1/2,'color','DarkCyan');
nairo.addatom('r',[0 1/2 1/2;0 0 0.340; 0 1/2 1/2],'S',[0 0 0],'label',{'Na1 Na1+' 'Na2 Na1+' 'Na3 Na1+'},'color',{'lightGray' 'lightGray' 'lightGray'});
nairo.addatom('r',[0.748 0.711; 0.178 0; 0.789 0.204],'S',[0 0],'label',{'O1 O2-', 'O2 O2-'},'color',{'r' 'r'});
plot(nairo,'baseShift',[-2;0;0])
swplot.zoom(1.3)

% We generate all bonds up to 8 Angstrom length.
nairo.gencoupling('maxDistance',8);
```

```{image} /_static/img/tutorials/tutorial16/tutorial16_02.png
:alt: 
``` 

### Magnetic Hamiltonian

We define the three anisotropic exchange interactions and Heisenberg exchange J1, J2 and J3 and assign them according to the paper. We assign a constant 1 meV for all exchange matrix, that we will change later for every model. J1 is plotted with dashed line to see both J1 and the kitaev coupling on the plot ('-' at the end of the matrix label will be plotted with dashed line). The Jxx and Jyy Kitaev terms are assigned to symmetry equivalent bonds, but they don't transform according to the space group symmetry. Thus we use a second option 'subIdx' to subselect from the symmetry equivalent bonds. In this case SpinW won't apply the symmetry transformation on exchange matrices on these bonds, it can be also forced setting the 'sym' option false in the spinw.addcoupling() function.

```matlab
% Kitaev term
nairo.addmatrix('label','Jxx','value',1,'color','r');
nairo.addmatrix('label','Jyy','value',1,'color','g');
nairo.addmatrix('label','Jzz','value',1,'color','b');
% Heisenberg terms
nairo.addmatrix('label','J1-','value',1,'color','gray');
nairo.addmatrix('label','J2','value',1,'color','orange');
nairo.addmatrix('label','J3','value',1,'color','cyan');

% add J1, J2 and J3 and JK couplings
nairo.addcoupling('mat','J1-','bond',[1 2]);
nairo.addcoupling('mat','J2','bond',[3 4]);
nairo.addcoupling('mat','J3','bond',[7 8]);
% Plot all couplings.
plot(nairo,'range',[2 2 0.5],'atomMode','mag','cellMode','inside',...
    'atomLegend',false,'cellcolor','gray','bondMode','line','bondLinewidth0',2)
swplot.zoom(1.4)
```

```{image} /_static/img/tutorials/tutorial16/tutorial16_03.png
:alt: 
``` 

```matlab
snapnow

% add JJxx, Jyy and Jzz couplings
nairo.addcoupling('mat','Jxx','bond',1,'subidx',[3 4]);
nairo.addcoupling('mat','Jyy','bond',1,'subidx',[1 2]);
nairo.addcoupling('mat','Jzz','bond',2);
% Plot Kitaev couplings only.
plot(nairo,'range',[2 2 0.5],'atomMode','mag','cellMode','inside',...
    'atomLegend',false,'cellcolor','gray','bondMode','line','bondLinewidth0',2)
swplot.zoom(1.4)
```

```text
Warning: By subselecting equivalent bonds, the symmetry of the corresponding
bond(s) are reduced to P1! 
Warning: By subselecting equivalent bonds, the symmetry of the corresponding
bond(s) are reduced to P1!
```

```{image} /_static/img/tutorials/tutorial16/tutorial16_04.png
:alt: 
``` 

### Plot Kitaev term

```matlab
nairo.addmatrix('label','Jxx','value',diag([1 0 0]),'color','r')
nairo.addmatrix('label','Jyy','value',diag([0 1 0]),'color','g')
nairo.addmatrix('label','Jzz','value',diag([0 0 1]),'color','b')

nairo.addmatrix('label','J1-','value',0);
nairo.addmatrix('label','J2','value', 0);
nairo.addmatrix('label','J3','value', 0);

plot(nairo,'range',[1 1 1/2],'atomMode','mag','bondRadius1',0.15,'bondMode','line',...
    'bondLineWidth','lin','bondLinewidth0',4,'atomLegend',false)
```

```{image} /_static/img/tutorials/tutorial16/tutorial16_05.png
:alt: 
``` 

### Q scans

We define a list of Q points, linear scans will be claculated between them with 200 points per line.

```matlab
Qp{1} = [ -1;   0; 0];
Qp{2} = [  0;   0; 0];
Qp{3} = [  0;   1; 0];
Qp{4} = [  1;   1; 0];
Qp{5} = [1/2; 1/2; 0];
Qp{6} = [  0;   0; 0];
Qp{7} = 500;
```

### Spin wave spectrum

We simulate every setting in the supplementary paper and denote each case with the figure plot index on Fig. S3. We create a Jfun() inline function that creates all exchange matrices from the scalar values. This can then directly be stored in sw.matrix.mat field.

```matlab
Jfun = @(x)cat(3,diag([-x(4) 0 0]),diag([0 -x(4) 0]),diag([0 0 -x(4)]),...
    eye(3)*x(1),eye(3)*x(2),eye(3)*x(3));
```

### Fig. S3(d) stripy order

We create stripy order, where energy per spin is -0.2913 meV. The spins are pointing along the c-axis.

```matlab
J1 =  1; J2 =  0; J3 =  0; JK =  1.33;
nairo.matrix.mat = Jfun([J1 J2 J3 JK]);
nairo.genmagstr('mode','func','func',@gm_planar,'x0',[[1/2 3/2 1/2 3/2]*pi 0 0 0 pi/2 0]);

specD = nairo.spinwave(Qp);
specD = sw_neutron(specD);
specD = sw_egrid(specD,'component','Sxx','Evect',linspace(0,4,1001));

figure
sw_plotspec(specD,'mode','color','dE',0.1,'axLim',[0 1]);
sw_plotspec(specD,'mode','disp','dashed',true,'colorbar',false,'legend',...
    false,'title',false,'sortMode',true,'colormap',[255  20 147]);
title('e) stripy, KH, S^{xx}','fontsize',15)
```

```{image} /_static/img/tutorials/tutorial16/tutorial16_06.png
:alt: 
``` 

### Fig. S3(e-f) stripy order

Here we have only J1 and Kitaev exchange. The magnetic ground state is stripy and parallel to the c-axis. The energy per spin is -0.375 meV.

```matlab
J1 = 1; J2 = 0; J3 = 0; JK = 2;
nairo.matrix.mat = Jfun([J1 J2 J3 JK]);
nairo.genmagstr('mode','func','func',@gm_planar,'x0',[[1/2 3/2 1/2 3/2]*pi 0 0 0 pi/2 0]);
specEF = nairo.spinwave(Qp,'hermit',false);
specEF = sw_neutron(specEF);

specEF = sw_egrid(specEF,'component','Sxx','evect',linspace(0,4,501));
figure
sw_plotspec(specEF,'mode','color','dE',0.1,'axLim',[0 1]);
sw_plotspec(specEF,'mode','disp','dashed',true,'colorbar',false,'legend',...
    false,'title',false,'sortMode',true,'colormap',[255  20 147]);
title('e) stripy, KH, S^{xx}','fontsize',15)

specEF = sw_egrid(specEF,'component','Syy','evect',linspace(0,4,501));
figure
sw_plotspec(specEF,'mode','color','dE',0.1,'axLim',[0 1]);
sw_plotspec(specEF,'mode','disp','dashed',true,'colorbar',false,'legend',...
    false,'title',false,'sortMode',true,'colormap',[255  20 147]);
title('f) stripy, KH, S^{yy}','fontsize',15)
```

```{image} /_static/img/tutorials/tutorial16/tutorial16_07.png
:alt: 
```

```{image} /_static/img/tutorials/tutorial16/tutorial16_08.png
:alt: 
``` 

### Fig. S3(g) stripy order

Stripy order again along c-axis.

```matlab
J1 =  1; J2 =  0.26; J3 = -0.2; JK = 0;
nairo.matrix.mat = Jfun([J1 J2 J3 JK]);
nairo.genmagstr('mode','func','func',@gm_planar,'x0',[[1/2 3/2 1/2 3/2]*pi 0 0 0 pi/2 0]);

specG = nairo.spinwave(Qp,'hermit',false);
specG = sw_neutron(specG);
specG = sw_egrid(specG,'component','Sxx','evect',linspace(0,2,501));

figure
sw_plotspec(specG,'mode','color','dE',0.1,'axLim',[0 1]);
sw_plotspec(specG,'mode','disp','dashed',true,'colorbar',false,'legend',...
    false,'title',false,'sortMode',true,'colormap',[255  20 147]);
title('g) stripy, J_{1,2,3}, S^{xx}(=S^{yy})','fontsize',15)
```

```{image} /_static/img/tutorials/tutorial16/tutorial16_09.png
:alt: 
``` 

### Fig. S3(h) zig-zag order

Zig-zag order along the b-axis.

```matlab
J1 =  1; J2 =  0.78; J3 =  0.9; JK =  0;
nairo.matrix.mat = Jfun([J1 J2 J3 JK]);
nairo.genmagstr('mode','func','func',@gm_planar,'x0',[[0 1 1 0]*pi 0 0 0 pi/2 0]);

specH = nairo.spinwave(Qp,'hermit',false);
specH = sw_neutron(specH,'pol',false);
specH = sw_egrid(specH,'component','Sxx','evect',linspace(0,4,501));

figure
sw_plotspec(specH,'mode','color','dE',0.1,'axLim',[0 1]);
sw_plotspec(specH,'mode','disp','dashed',true,'colorbar',false,'legend',...
    false,'title',false,'sortMode',true,'colormap',[255  20 147]);
title('h) zig-zag, J_{1,2,3}, S^{xx}(=S^{yy})','fontsize',15)
```

```text
Warning: Eigenvectors of defective eigenvalues cannot be orthogonalised at some
q-point!
```

```{image} /_static/img/tutorials/tutorial16/tutorial16_10.png
:alt: 
``` 

### Fig. S3(i-j) zig-zag order

Magnetic structure, where spins are pointing along the b-axis. In the paper the Sy'y' direction is the correlation function component perpendicular to the magnetic moment direction, that is Sxx for S||b moments.

```matlab
J1 =  1; J2 =  0.23; J3 =  0.51; JK =  1.33;
nairo.matrix.mat = Jfun([J1 J2 J3 JK]);
nairo.genmagstr('mode','func','func',@gm_planar,'x0',[[0 1 1 0]*pi 0 0 0 pi/2 0]);

specIJ = nairo.spinwave(Qp,'hermit',false);
specIJ = sw_neutron(specIJ);

figure
specIJ = sw_egrid(specIJ,'component','Szz','evect',linspace(0,3,501));
sw_plotspec(specIJ,'mode','color','dE',0.1,'axLim',[0 1]);
sw_plotspec(specIJ,'mode','disp','dashed',true,'colorbar',false,'legend',...
    false,'title',false,'sortMode',true,'colormap',[255  20 147]);
title('i) zig-zag, KH-J_2-J_3, S^{zz}','fontsize',15)

figure
specIJ = sw_egrid(specIJ,'component','Sxx','evect',linspace(0,3,501));
sw_plotspec(specIJ,'mode','color','dE',0.1,'axLim',[0 1]);
sw_plotspec(specIJ,'mode','disp','dashed',true,'colorbar',false,'legend',...
    false,'title',false,'sortMode',true,'colormap',[255  20 147]);
title('j) zig-zag, KH-J_2-J_3, S^{y''y''}=S^{xx}','fontsize',15)
```

```{image} /_static/img/tutorials/tutorial16/tutorial16_11.png
:alt: 
```

```{image} /_static/img/tutorials/tutorial16/tutorial16_12.png
:alt: 
``` 

```text
Written by
Sandor Toth
16-Jun-2014, 06-Feb-2017
```

Published with MATLAB® R2018b