FIGURE 4: linear spin-wave theory magnon bands and moment for the Heisenberg-Kitaev model with J2A and J2B.

Heisenberg-Kitaev angle \phi=5\pi/8 and
Heisenberg coupling J=cos\phi
Kitaev coupling K=sin\phi
J2A=0.2
J2B=-0.1


FILE: "Fig_4/HK_zigzag_phi_0.625pi_J2A_0.2_J2B_-0.1_magnon_bands.txt"

The magnon bands data are given in the text file "Fig_4/HK_zigzag_phi_0.625pi_J2A_0.2_J2B_-0.1_magnon_bands.txt"
Format:

t	wA/S	wB/S	wC/S	wD/S

where t goes from 0 to 4 and parametrizes the four different pieces forming the path in momentum space, see inset Fig. 4(b). The w[i]/S are the magnon bands divided by the spin magnitude S, as in Fig. 4(a).

0<=t<=1 ---> from X to \Gamma
1<=t<=2 ---> from \Gamma to Y
2<=t<=3 ---> from \Y to \Gamma'
3<=t<=4 ---> from \Gamma' to \Gamma


FILE: "Fig_5/HK_zigzag_phi_0.625pi_J2A_0.2_J2B_-0.1_moment_vs_size.txt"

For each considered size L, we calculated the respective corrected moment for each magnetic sublattice A,B,C,D. The data are provided in the text file "Fig_4/HK_zigzag_phi_0.625pi_J2A_0.2_J2B_-0.1_moment_vs_size.txt".
Format:

1/L	mA*ez	mB*ez	mC*ez	mD*ez

where m[i] are the corrected moments from LSWT and ez=[0,0,1].


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FIGURE 5: secondary order parameter magnitude in LSWT for several values of the Heisenber-Kitaev angle \phi for J2A=0.2, J2B=-0.1.

FILE: "Fig_5/HK_msec_vs_phi_J2A_0.2_J2B_-0.1.txt"

Format

\phi/\pi	|m_sec|

Note that only the ferromagnetic and antiferromagnetic data are given since the zigzag and stripz zero have zero secondarz order parameter.


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FIGURE 6: LSWT magnon bands, rescaled free energy density as function of external field for different temperatures and out-of-plane magnetization for a zigzag ground state of an Heisenberg-Kitaev-\Gamma model with J2A and J2B.

We have \phi=1.25\pi and \theta=0.0625\pi for
Heisenberg coupling J=sin\theta*cos\phi
Kitaev coupling K=sin\theta\sin\phi
\Gamma coupling \Gamma=\cos\theta
and
J2A=0.2
J2B=-0.1


FILE: "Fig_6/HKG_zigzag_phi_1.25pi_theta_0.0625pi_J2A_0.2_J2B_-0.1_magnon_bands.txt"

The magnon bands data are given in the text file "Fig_6/HKG_zigzag_phi_1.25pi_theta_0.0625pi_J2A_0.2_J2B_-0.1_magnon_bands.txt"
Format:

t       wA/S    wB/S    wC/S    wD/S

where t goes from 0 to 4 and parametrizes the four different pieces forming the path in momentum space, see inset Fig. 4(b). The w[i]/S are the magnon bands divided by the spin magnitude S, as in Fig. 6(a).

0<=t<=1 ---> from X to \Gamma
1<=t<=2 ---> from \Gamma to Y
2<=t<=3 ---> from \Y to \Gamma'
3<=t<=4 ---> from \Gamma' to \Gamma


FILE: "Fig_6/HKG_zigzag_phi_1.25pi_theta_0.0625pi_J2A_0.2_J2B_-0.1_f_vs_h_T_%.3f.txt"

Free energy density rescaled as (f-c)/h, see paragraph after Eq.(25), as a function of external magnetic field h for different temperatures T.
Format:

h	(f-c)/h

c is obtained with quadratic fit on the free energy density, see Eq.(24).


FILE: "Fig_6/HKG_zigzag_phi_1.25pi_theta_0.0625pi_J2A_0.2_J2B_-0.1_m_vs_T.txt"

Out-of-plane magnetization as fucntion of temperature, obtained from coefficient of linear term in h in the free energy density expansion in Eq.(24). The numerical data are small and dependent of the field range considered. In particular, they get smaller as h range is decreased.
Format:
T	m


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FIGURE 8: LSWT magnon bands, rescaled free energy density as function of external field for different temperatures and out-of-plane magnetization for a triple-q ground state of an Heisenberg-Kitaev-\Gamma model with J2A and J2B and local field.

We have phi=pi-atan(2), theta=pi/3
Heisenberg coupling J=sin\theta*cos\phi
Kitaev coupling K=sin\theta*sin\phi
\Gamma coupling \Gamma=cos\theta
and
J2A=0.2
J2B=-0.1
hloc=0.4


FILE: "Fig_8/HKG_tripleq_phi_pi-atan2_Gamma_0.5_J2A_0.2_J2B_-0.1_loc_0.4_magnon_bands.txt"

The magnon bands data are given in the text file "Fig_8/HKG_tripleq_phi_pi-atan2_Gamma_0.5_J2A_0.2_J2B_-0.1_loc_0.4_magnon_bands.txt"
Format:

t       wA/S    wB/S    wC/S    wD/S	wE/S	wF/S	wG/S	wH/S

where t goes from 0 to 4 and parametrizes the four different pieces forming the path in momentum space, see inset Fig. 4(b). The w[i]/S are the magnon bands divided by the spin magnitude S, as in Fig. 8(a).

0<=t<=1 ---> from X to \Gamma
1<=t<=2 ---> from \Gamma to Y
2<=t<=3 ---> from \Y to \Gamma'
3<=t<=4 ---> from \Gamma' to \Gamma


FILE: "Fig_8/HKG_tripleq_phi_pi-atan2_Gamma_0.5_J2A_0.2_J2B_-0.1_loc_0.4_f_vs_h_T_%.3f.txt"

Free energy density rescaled as (f-c)/h, see paragraph after Eq.(25), as a function of external magnetic field h for different temperatures T.
Format:

h	(f-c)/h

c is obtained with quadratic fit on the free energy density, see Eq.(24).


FILE: "Fig_8/HKG_tripleq_phi_pi-atan2_Gamma_0.5_J2A_0.2_J2B_-0.1_loc_0.4_m_vs_T.txt"

Out-of-plane magnetization as fucntion of temperature, obtained from coefficient of linear term in h in the free energy density expansion in Eq.(24).
Format:
T	m
