Kitaev-Heisenberg model

This Hamiltonian has been implemented for Sato et al. (2024)Inácio et al. (2026). It defines a spin model on a honeycomb lattice as:

where $\langle \boldsymbol{i},\boldsymbol{j} \rangle_\gamma$ refers to nearest-neighbor bonds of type $\gamma=x,y,z$ with cyclic permutations $(\gamma,\alpha,\beta)$ of $(x,y,z)$, $\langle \boldsymbol{i},\boldsymbol{j} \rangle$ to all nearest-neighbor bonds, and $\langle\langle\langle \boldsymbol{i},\boldsymbol{j}\rangle\rangle\rangle$ to third-neighbor bonds. A magnetic field is included through a Zeeman coupling,

with an anisotropic $g$-tensor consistent with the crystallographic axes of $\alpha{\rm -RuCl}_3$. In $\alpha{\rm -RuCl}_3$, the orthonormal axes $\mathbf{e}_a$, and $\mathbf{e}_b$, and $\mathbf{e}_c$ are defined by the crystallographic directions, with $\mathbf{e}_c \parallel [111]$, $\mathbf{e}_a \parallel [11\bar{2}]$, and $\mathbf{e}_b \parallel [\bar{1}10]$. In this convention, $[xyz]$ denote the cubic spin-space axes, $\mathbf{e}_a$ and $\mathbf{e}_b$ lie in the honeycomb plane, and $\mathbf{e}_c$ is perpendicular to it. Within the honeycomb plane, $\mathbf{e}_b$ is parallel to one bond direction of the honeycomb lattice, while $\mathbf{e}_a$ is perpendicular to it. The magnetic field is applied in the $a$-$c$ plane, nearly parallel to the crystallographic $a$ axis, consistent with the experimental setup.

The following model-specific parameters are available:

To use the Hamiltonian, it has to be selected via its name “Kit_Heis”

&VAR_ham_name
ham_name = "Kit_Heis"  ! Name of Hamiltonian
/

The following values for VAR_lattice choose $4\times4 = 16$ honeycomb unit cells with 32 spins.

&VAR_lattice
L1 = 4  ! Length in direction a_1
L2 = 4  ! Length in direction a_2
/

The rest of the model-specific parameters are in VAR_Kit_Heis.

&VAR_Kit_Heis
Ham_J = -0.1d0  ! Nearest neighbor Heisenberg coupling
Ham_J3 = 0.1d0  ! Third nearest neighbor Heisenberg coupling
Ham_alphax = -1.0d0  ! Kitaev coupling on x-bonds
Ham_alphay = -1.0d0  ! Kitaev coupling on y-bonds
Ham_alphaz = -1.0d0  ! Kitaev coupling on z-bonds
Ham_Gx = 0.5d0  ! Gamma coupling on x-bonds
Ham_Gy = 0.5d0  ! Gamma coupling on y-bonds
Ham_Gz = 0.5d0  ! Gamma coupling on z-bonds
Ham_Gx_p = 0.0d0  ! Gamma prime coupling on x-bonds
Ham_Gy_p = 0.0d0  ! Gamma prime coupling on y-bonds
Ham_Gz_p = 0.0d0  ! Gamma prime coupling on z-bonds
Hab = 0.0579d0  ! mu_B * B / K ; Magnetic field strength
Htheta = 0.45d0  ! Magnetic field angle (in units of pi)
Ham_U = 7.14d0  ! Hubbard U, for freezing out charge fluctuations
Dtau = 0.1d0  ! Imaginary time step. Ltrot=Beta/dtau
Beta = 0.5d0  ! Inverse temperature
/

Usage example¶

A brief usage example using pyALF.

Use provided script to download ALF and link the Hamiltonian to it¶

Import necessary pyALF classes.¶

Create ALF source object and inspect available parameters¶

Create and run Simulation¶

Calculating magnetotropic susceptibilty¶

This repository also includes files for calculating the magnetotropic susceptibilty $k$ displayed in Sato et al. (2024)Inácio et al. (2026). The workflow will need simulations to be carried out with HDF5, legacy formats are not supported out of the box.

To this effect, we included the following files:

• calc_k.py: Main scripts for calculating the susceptibilty.

• generate_tau_replicas.py: Script for generating replica of time-displaced observables. This is used for error analysis with MaxEnt

• calc_k2_tau.F90: Custom program for preparing time-displaced observable that is used to calculate $k_2$ via MaxEnt.

The script clone_alf.sh already sets up calc_k2_tau.F90 to be compiled as part of the analysis. Furthermore, pyALF is needed. After that, calc_k.py is already ready to use.

Usage of calc_k.py:

usage: calc_k.py [-h] --alf-dir ALF_DIR [--sim-dir SIM_DIR] [--results-dir RESULTS_DIR] [--always]

Calculate the magnetotropic susceptibility k.

options:
  -h, --help            show this help message and exit
  --alf-dir ALF_DIR     Path to ALF directory
  --sim-dir SIM_DIR     Path to simulation directory
  --results-dir RESULTS_DIR
                        Path where analysis results should be written
  --always              Always calculate k, even if results exist