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//! Physics calculation methods for tight-binding models
use crate::Gauge;
use crate::Model;
use crate::error::{Result, TbError};
use crate::kpoints::gen_kmesh;
use crate::solve_ham::solve;
use ndarray::prelude::*;
use ndarray::*;
use ndarray_linalg::Inverse;
use num_complex::Complex;
use rayon::prelude::*;
use std::f64::consts::PI;
impl Model {
#[allow(non_snake_case)]
#[inline(always)]
#[cfg_attr(doc, katexit::katexit)]
///Performs Fourier transform, converting real-space Hamiltonian to reciprocal-space Hamiltonian.
///
///There are two gauge choices: lattice gauge and atomic gauge, corresponding to `Gauge::Lattice` and `Gauge::Atom`.
///
///For the atomic gauge, the transformation between real-space wavefunction $\ket{n\bm R}$ and reciprocal-space wavefunction $\ket{u_{\bm k,n}}$ is:
///
///$$\ket{u_{n\bm k}(\bm r)}=\sum_{\bm R} e^{i\bm k\cdot(\bm R+\bm\tau_n)}\ket{n\bm R}$$
///
///satisfying $\ket{u_{i\bm k}(\bm r+\bm R)}=\ket{u_{i\bm k}(\bm r)}$.
///
///For the Hamiltonian, we have:
///$$
///H_{mn,\bm k}=\bra{u_{m\bm k}}\hat H\ket{u_{n\bm k}}=\sum_{\bm R^\prime}\sum_{\bm R} \bra{m\bm R^\prime}\hat H\ket{n\bm R}e^{-i(\bm R'-\bm R+\bm\tau_m-\bm \tau_n)\cdot\bm k}.
///$$
///Due to translational symmetry, only $\bm R'-\bm R$ matters, thus:
///$$
///H_{mn,\bm k}=\sum_{\bm R} \bra{m\bm 0}\hat H\ket{n\bm R}e^{i(\bm R-\bm\tau_m+\bm \tau_n)\cdot\bm k}
///$$
///
///For the lattice gauge, we have $$\ket{\phi_{n\bm k}}=\sum_{\bm R} e^{i\bm k\cdot\bm R}\ket{n\bm R},$$ so:
///$$
///H_{mn,\bm k}=\sum_{\bm R} \bra{m\bm 0}\hat H\ket{n\bm R}e^{i(\bm R)\cdot\bm k}
///$$
///
///Here $\ket{\psi_{n\bm k}}$ is periodic in reciprocal space: $\ket{\phi_{n\bm k}(\bm r)}=\ket{\phi_{n\bm k+\bm G}(\bm r)}$.
pub fn gen_ham<S: Data<Elem = f64>>(
&self,
kvec: &ArrayBase<S, Ix1>,
gauge: Gauge,
) -> Array2<Complex<f64>> {
assert!(
kvec.len() == self.dim_r(),
"Wrong, the k-vector's length must equal to the dimension of model."
);
let Us = (self.hamR.mapv(|x| x as f64))
.dot(kvec)
.mapv(|x| Complex::<f64>::new(x, 0.0));
let Us = Us * Complex::new(0.0, 2.0 * PI);
let Us = Us.mapv(Complex::exp);
let hamk: Array2<Complex<f64>> = self
.ham
.outer_iter()
.zip(Us.iter())
.fold(Array2::zeros((self.nsta(), self.nsta())), |acc, (hm, u)| {
acc + &hm * *u
});
let hamk = match gauge {
Gauge::Lattice => hamk,
Gauge::Atom => {
let U0: Array1<f64> = self.orb.dot(kvec);
let U0: Array1<Complex<f64>> = U0.mapv(|x| Complex::<f64>::new(x, 0.0));
let U0 = U0 * Complex::new(0.0, 2.0 * PI);
let mut U0: Array1<Complex<f64>> = U0.mapv(Complex::exp);
let U0 = if self.spin {
let U0 = concatenate![Axis(0), U0, U0];
U0
} else {
U0
};
let U = Array2::from_diag(&U0);
let U_dag = Array2::from_diag(&U0.map(|x: &Complex<f64>| x.conj()));
// Next two steps fill in the phase due to orbital coordinates
let hamk: Array2<Complex<f64>> = U_dag.dot(&hamk);
let re_ham = hamk.dot(&U);
re_ham
}
};
hamk
}
#[allow(non_snake_case)]
pub fn dos(
&self,
k_mesh: &Array1<usize>,
E_min: f64,
E_max: f64,
E_n: usize,
sigma: f64,
) -> Result<(Array1<f64>, Array1<f64>)> {
//! 我这里用的算法是高斯算法, 其算法过程如下
//!
//! 首先, 根据 k_mesh 算出所有的能量 $\ve_n$, 然后, 按照定义
//! $$\rho(\ve)=\sum_N\int\dd\bm k \delta(\ve_n-\ve)$$
//! 我们将 $\delta(\ve_n-\ve)$ 做了替换, 换成了 $\f{1}{\sqrt{2\pi}\sigma}e^{-\f{(\ve_n-\ve)^2}{2\sigma^2}}$
//!
//! 然后, 计算方法是先算出所有的能量, 再将能量乘以高斯分布, 就能得到态密度.
//!
//! 态密度的光滑程度和k点密度以及高斯分布的展宽有关
let kvec: Array2<f64> = gen_kmesh(&k_mesh)?;
let nk = kvec.len_of(Axis(0));
let eigenvalues = self.solve_band_all_parallel(&kvec);
let E = Array1::linspace(E_min, E_max, E_n);
let dim: usize = k_mesh.len();
let centre = eigenvalues.into_raw_vec().into_par_iter();
let sigma0 = 1.0 / sigma;
let pi0 = 1.0 / (2.0 * PI).sqrt();
let dos = Array1::<f64>::zeros(E_n);
let dos = centre
.fold(
|| Array1::<f64>::zeros(E_n),
|acc, x| {
let A: Array1<f64> = (&E - x) * sigma0;
let f: Array1<f64> = (-&A * &A / 2.0).mapv(|x: f64| x.exp()) * sigma0 * pi0;
acc + &f
},
)
.reduce(|| Array1::<f64>::zeros(E_n), |acc, x| acc + x);
let dos = dos / (nk as f64);
Ok((E, dos))
}
}