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
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
//! Physics calculation methods for tight-binding models
use crate::Gauge;
use crate::Model;
use crate::RMatrixData;
use crate::error::{Result, TbError};
use crate::kpoints::gen_kmesh;
use crate::solve_ham::Solve;
use ndarray::prelude::*;
use ndarray::*;
use num_complex::Complex;
use rayon::prelude::*;
use std::f64::consts::PI;
impl<const SPIN: bool, const DIM: usize, R: RMatrixData> Model<SPIN, DIM, R> {
#[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 nsta = self.nsta();
let mut hamk = Array2::<Complex<f64>>::zeros((nsta, nsta));
// Precompute phase factors exp(i 2π k·R) for each R vector.
// Dimension-dispatched: compile-time constant loop bound for R·k dot.
let Us: Vec<Complex<f64>> = match DIM {
1 => self
.hamR
.outer_iter()
.map(|r| Complex::new(0.0, 2.0 * PI * r[0] as f64 * kvec[0]).exp())
.collect(),
2 => self
.hamR
.outer_iter()
.map(|r| {
Complex::new(
0.0,
2.0 * PI * (r[0] as f64 * kvec[0] + r[1] as f64 * kvec[1]),
)
.exp()
})
.collect(),
3 => self
.hamR
.outer_iter()
.map(|r| {
Complex::new(
0.0,
2.0 * PI
* (r[0] as f64 * kvec[0]
+ r[1] as f64 * kvec[1]
+ r[2] as f64 * kvec[2]),
)
.exp()
})
.collect(),
_ => unreachable!(),
};
let hamk_slice = hamk.as_slice_mut().unwrap();
for (iR, &u) in Us.iter().enumerate() {
let hm = self.ham.index_axis(Axis(0), iR);
crate::ndarray_lapack::zaxpy(u, hm.as_slice().unwrap(), hamk_slice);
}
match gauge {
Gauge::Lattice => hamk,
Gauge::Atom => {
// Dimension-dispatched τ·k phase factors
let orb_phase: Vec<Complex<f64>> = match DIM {
1 => self
.orb
.outer_iter()
.map(|tau| Complex::new(0.0, 2.0 * PI * tau[0] * kvec[0]).exp())
.collect(),
2 => self
.orb
.outer_iter()
.map(|tau| {
Complex::new(0.0, 2.0 * PI * (tau[0] * kvec[0] + tau[1] * kvec[1]))
.exp()
})
.collect(),
3 => self
.orb
.outer_iter()
.map(|tau| {
Complex::new(
0.0,
2.0 * PI * (tau[0] * kvec[0] + tau[1] * kvec[1] + tau[2] * kvec[2]),
)
.exp()
})
.collect(),
_ => unreachable!(),
};
let norb = self.norb();
let orb_phase = Array1::from_vec(orb_phase);
// Build gauge phase vector: for spinful, duplicate orbital phases
let mut U0 = Array1::<Complex<f64>>::zeros(if SPIN { 2 * norb } else { norb });
U0.slice_mut(s![..norb]).assign(&orb_phase);
if SPIN {
U0.slice_mut(s![norb..]).assign(&orb_phase);
}
// Gauge transform: H'[m,n] = conj(U0[m]) * H[m,n] * U0[n]
for m in 0..nsta {
let mut row = hamk.slice_mut(s![m, ..]);
let conj_pm = U0[m].conj();
Zip::from(&mut row)
.and(&U0)
.for_each(|h, &pn| *h *= conj_pm * pn);
}
hamk
}
}
}
/// Computes the density of states $\rho(E)$ using Gaussian smearing.
///
/// The DOS is defined as:
///
/// $$\rho(E) = \frac{1}{N_k} \sum_{n,\mathbf{k}} \delta(E - E_{n\mathbf{k}})$$
///
/// The delta function is approximated by a Gaussian of width $\sigma$:
///
/// $$\delta(x) \approx \frac{1}{\sqrt{2\pi}\,\sigma}\, e^{-x^2 / (2\sigma^2)}$$
///
/// # Algorithm
///
/// 1. Generate a uniform k-mesh from `k_mesh`
/// 2. Diagonalize $H(\mathbf{k})$ at every k-point in parallel
/// 3. Convolve eigenvalues with the Gaussian kernel and sum
///
/// The smoothness depends on both the k-point density and $\sigma$.
///
/// # Arguments
///
/// * `k_mesh` — k-points along each direction, e.g. `[51, 51]`
/// * `E_min`, `E_max` — Energy range
/// * `E_n` — Number of energy bins
/// * `sigma` — Gaussian smearing width (same units as energy)
///
/// # Returns
///
/// `(energies, dos)` — energy grid and corresponding DOS.
#[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>)> {
if E_min >= E_max {
return Err(TbError::InvalidEnergyRange {
min: E_min,
max: E_max,
});
}
if E_n == 0 {
return Err(TbError::InvalidDosParameter {
parameter: "E_n",
message: "number of energy bins must be at least 1".to_string(),
});
}
if !sigma.is_finite() || sigma <= 0.0 {
return Err(TbError::InvalidDosParameter {
parameter: "sigma",
message: "Gaussian smearing width must be finite and positive".to_string(),
});
}
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_and_offset().0.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))
}
}