Rustb 0.6.6

A package for calculating band, angle state, linear and nonlinear conductivities based on tight-binding models
Documentation 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178

//!这个模块是用来产生某个k点的速度算符
use crate::Gauge;
use crate::Model;
use crate::comm;
use crate::solve_ham::*;
use ndarray::prelude::*;
use ndarray::*;
use ndarray_linalg::conjugate;
use ndarray_linalg::*;
/// This function generates the velocity operator, i.e., $\bra{m\bm k}\p_\ap H_{\bm k}\ket{n\bm k},$
/// The basis functions are Bloch wavefunctions.
///
/// The velocity operator formula uses the tight-binding model,
/// where the Fourier transform includes atomic positions.
///
/// Thus we have
///
/// $$
/// \\begin\{aligned\}
/// \\bra{m\bm k}\p_\ap H_{\bm k}\ket{n\bm k}&=\p_\ap\left(\bra{m\bm k} H\ket{n\bm k}\rt)-\p_\ap\left(\bra{m\bm k}\rt) H\ket{n\bm k}-\bra{m\bm k} H\p_\ap\ket{n\bm k}\\\\
/// &=\sum_{\bm R} i(\bm R-\bm\tau_m+\bm\tau_n)H_{mn}(\bm R) e^{i\bm k\cdot(\bm R-\bm\tau_m+\bm\tau_n)}-\lt[H_{\bm k},\\mathcal A_{\bm k,\ap}\rt]_{mn}
/// \\end\{aligned\}
/// $$
///
/// Here $\\mathcal A_{\bm k}$ is defined as $$\\mathcal A_{\bm k,\ap,mn}=-i\sum_{\bm R}r_{mn,\ap}(\bm R)e^{i\bm k\cdot(\bm R-\bm\tau_m+\bm\tau_{n})}+i\tau_{n\ap}\dt_{mn}$$
/// where $\bm r_{mn}$ can be provided by wannier90 by setting write_rmn=true
/// Here, all $\bm R$, $\bm r$, and $\bm \tau$ are in real-space coordinates.
///
use num_complex::Complex;
use rayon::prelude::*;
use serde::{Deserialize, Serialize};
use std::f64::consts::PI;
use std::ops::AddAssign;

pub trait Velocity {
    fn gen_v<S: Data<Elem = f64>>(
        &self,
        kvec: &ArrayBase<S, Ix1>,
        gauge: Gauge,
    ) -> (Array3<Complex<f64>>, Array2<Complex<f64>>);
}

impl Velocity for Model {
    #[allow(non_snake_case)]
    #[inline(always)]
    fn gen_v<S: Data<Elem = f64>>(
        &self,
        kvec: &ArrayBase<S, Ix1>,
        gauge: Gauge,
    ) -> (Array3<Complex<f64>>, Array2<Complex<f64>>) {
        // We use lattice gauge rather than atomic gauge
        assert_eq!(
            kvec.len(),
            self.dim_r(),
            "Wrong, the k-vector's length {} must equal to the dimension of model {}.",
            kvec.len(),
            self.dim_r()
        );

        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); // Us is exp(i k R)
        // Define an initialized velocity matrix
        let mut v = Array3::<Complex<f64>>::zeros((self.dim_r(), self.nsta(), self.nsta()));
        let R0 = &self.hamR.mapv(|x| Complex::<f64>::new(x as f64, 0.0));
        // R0 is the real-space hamR
        let R0 = R0.dot(&self.lat.mapv(|x| Complex::new(x, 0.0)));
        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 (v, hamk) = match gauge {
            Gauge::Atom => {
                let orb_sta = if self.spin {
                    let orb0 = concatenate(Axis(0), &[self.orb.view(), self.orb.view()]).unwrap();
                    orb0
                } else {
                    self.orb.to_owned()
                };
                let U0 = orb_sta.dot(kvec);
                let U0 = U0.mapv(|x| Complex::<f64>::new(x, 0.0));
                let U0 = U0 * Complex::new(0.0, 2.0 * PI);
                let mut U0 = U0.mapv(Complex::exp);
                // U0 is the phase factor
                let U = Array2::from_diag(&U0);
                let U_conj = Array2::from_diag(&U0.mapv(|x| x.conj()));
                let orb_real = orb_sta.dot(&self.lat);
                // Start constructing -orb_real[[i,r]]+orb_real[[j,r]];-----------------
                let mut UU = Array3::<f64>::zeros((self.dim_r(), self.nsta(), self.nsta()));
                let A = orb_real.view().insert_axis(Axis(2));
                let A = A
                    .broadcast((self.nsta(), self.dim_r(), self.nsta()))
                    .unwrap()
                    .permuted_axes([1, 0, 2]);
                let mut B = A.view().permuted_axes([0, 2, 1]);
                let UU = &B - &A;
                let UU = UU.mapv(|x| Complex::<f64>::new(0.0, x)); //UU[i,j]=i(-tau[i]+tau[j])
                // Define an initialized velocity matrix
                Zip::from(v.outer_iter_mut())
                    .and(R0.axis_iter(Axis(1)))
                    .and(UU.outer_iter())
                    .for_each(|mut v0, r, det_tau| {
                        let vv: Array2<Complex<f64>> =
                            self.ham.outer_iter().zip(Us.iter().zip(r.iter())).fold(
                                Array2::zeros((self.nsta(), self.nsta())),
                                |acc, (ham, (us, r0))| acc + &ham * *us * *r0 * Complex::i(),
                            );
                        let vv: Array2<Complex<f64>> = &vv + &hamk * &det_tau;
                        let vv = &U_conj.dot(&vv);
                        let vv = vv.dot(&U); // Next two steps fill in the phase due to orbital coordinates
                        v0.assign(&vv);
                    });
                // At this point, we have computed sum_{R} iR H_{mn}(R) e^{ik(R+tau_n-tau_m)}
                // Next, compute Berry connection A_\alpha=\sum_R r(R)e^{ik(R+tau_n-tau_m)}-tau
                let hamk = U_conj.dot(&hamk.dot(&U)); // Don't forget to add the phase to hamk
                if self.rmatrix.len_of(Axis(0)) != 1 {
                    let mut rk =
                        Array3::<Complex<f64>>::zeros((self.dim_r(), self.nsta(), self.nsta()));
                    let mut rk = self
                        .rmatrix
                        .axis_iter(Axis(0))
                        .zip(Us.iter())
                        .fold(rk, |acc, (ham, us)| acc + &ham * *us);
                    for i in 0..3 {
                        let mut r0: ArrayViewMut2<Complex<f64>> = rk.slice_mut(s![i, .., ..]);
                        let r_new = r0.dot(&U);
                        let r_new = U_conj.dot(&r_new);
                        r0.assign(&r_new);
                        let mut dig = r0.diag_mut();
                        //dig.assign(&(&dig - &orb_real.column(i)));
                        dig.assign(&Array1::zeros(self.nsta()));
                        let A = comm(&hamk, &r0) * Complex::i();
                        v.slice_mut(s![i, .., ..]).add_assign(&A);
                    }
                }
                (v, hamk)
            }
            Gauge::Lattice => {
                // Use lattice gauge
                Zip::from(v.outer_iter_mut())
                    .and(R0.axis_iter(Axis(1)))
                    .for_each(|mut v0, r| {
                        let vv: Array2<Complex<f64>> =
                            self.ham.outer_iter().zip(Us.iter().zip(r.iter())).fold(
                                Array2::zeros((self.nsta(), self.nsta())),
                                |acc, (ham, (us, r0))| acc + &ham * *us * *r0 * Complex::i(),
                            );
                        v0.assign(&vv);
                    });
                // At this point, we have computed sum_{R} iR H_{mn}(R) e^{ik(R+tau_n-tau_m)}
                // Next, compute Berry connection A_\alpha=\sum_R r(R)e^{ik(R+tau_n-tau_m)}-tau
                if self.rmatrix.len_of(Axis(0)) != 1 {
                    let mut rk =
                        Array3::<Complex<f64>>::zeros((self.dim_r(), self.nsta(), self.nsta()));
                    let mut rk = self
                        .rmatrix
                        .axis_iter(Axis(0))
                        .zip(Us.iter())
                        .fold(rk, |acc, (ham, us)| acc + &ham * *us);
                    for i in 0..3 {
                        let mut r0: ArrayViewMut2<Complex<f64>> = rk.slice_mut(s![i, .., ..]);
                        //let mut dig = r0.diag_mut();
                        //dig.assign(&Array1::zeros(self.nsta()));
                        let A = comm(&hamk, &r0) * Complex::i();
                        v.slice_mut(s![i, .., ..]).add_assign(&A);
                    }
                }
                (v, hamk)
            }
        };
        (v, hamk)
    }
}