update example

Author: mtian8

docs/example_problem.md

@@ -1,7 +1,6 @@
 # Example: Calculate Chern numbers for the Haldane Model
 
 ## Main Problem and Dependencies
-
 **1. Generate an array of Chern numbers for the Haldane model on a hexagonal lattice by sweeping the following parameters: the on-site energy to next-nearest-neighbor coupling constant ratio ($m/t_2$ from -6 to 6 with $N$ samples) and the phase ($\phi$ from -$\pi$ to $\pi$ with $N$ samples) values. Given the lattice spacing $a$, the nearest-neighbor coupling constant $t_1$, the next-nearest-neighbor coupling constant $t_2$, the grid size $\delta$ for discretizing the Brillouin zone in the $k_x$ and $k_y$ directions (assuming the grid sizes are the same in both directions), and the number of sweeping grid points $N$ for $m/t_2$ and $\phi$.**
 
 ``` python
@@ -27,16 +26,13 @@ phi_values: array of length N
     The swept phase values.
 '''
 ```
-
 ```python
 # Package Dependencies
 import numpy as np
 import cmath
 from math import pi, sin, cos, sqrt
 ```
-
 ## Subproblems
-
 **1.1 Write a Haldane model Hamiltonian on a hexagonal lattice, given the following parameters: wavevector components $k_x$ and $k_y$ (momentum) in the x and y directions, lattice spacing $a$, nearest-neighbor coupling constant $t_1$, next-nearest-neighbor coupling constant $t_2$, phase $\phi$ for the next-nearest-neighbor hopping, and the on-site energy $m$.**
 
 **_Scientists Annotated Background:_**
@@ -89,6 +85,7 @@ $$
 $$
 {d_3} = m - 2{t_2}\sin \phi \sum\nolimits_i {\sin (\mathbf{k} \cdot {\mathbf{b}_i})}  = m - 2{t_2}\sin \phi \left[ {\sin \left( {\sqrt 3 {k_x}a} \right) + \sin \left( { - \sqrt 3 {k_x}a/2 + 3{k_y}a/2} \right) + \sin \left( { - \sqrt 3 {k_x}a/2 - 3{k_y}a/2} \right)} \right] \\
 $$
+
 where $\sigma_i$ are the Pauli matrices and $I$ is the identity matrix.
 ```python
 def calc_hamiltonian(kx, ky, a, t1, t2, phi, m):
@@ -157,15 +154,17 @@ Source: Fukui, Takahiro, Yasuhiro Hatsugai, and Hiroshi Suzuki. "Chern numbers i
 
 
 Here we can discretize the two-dimensional Brillouin zone into grids with step $\delta {k_x} = \delta {k_y} = \delta$. If we define the U(1) gauge field on the links of the lattice as $U_\mu (\mathbf{k}_l) := \frac{\left\langle n(\mathbf{k}_l)\middle|n(\mathbf{k}_l + \hat{\mu})\right\rangle}{\left|\left\langle n(\mathbf{k}_l)\middle|n(\mathbf{k}_l + \hat{\mu})\right\rangle\right|}$, where $\left|n(\mathbf{k}_l)\right\rangle$ is the eigenvector of Hamiltonian at $\mathbf{k}_l$, $\hat{\mu}$ is a small displacement vector in the direction $\mu$ with magnitude $\delta$, and $\mathbf{k}_l$ is one of the momentum space lattice points $l$. The corresponding curvature (flux) becomes
+
 $$
 F_{xy}(\mathbf{k}_l) := \ln \left[U_x(\mathbf{k}_l)U_y(\mathbf{k}_l+\hat{x})U_x^{-1}(\mathbf{k}_l+\hat{y})U_y^{-1}(\mathbf{k}_l)\right]
 $$
+
 and the Chern number of a band can be calculated as
+
 $$
 c = \frac{1}{2\pi i} \Sigma_l F_{xy}(\mathbf{k}_l),
 $$
 where the summation is over all the lattice points $l$. Note that the Brillouin zone of a hexagonal lattice with spacing $a$ can be chosen as a rectangle with $0 \le {k_x} \le k_{x0} = 2\sqrt 3 \pi /(3a),0 \le {k_y} \le k_{y0} = 4\pi /(3a)$.
-
 ```python
 def compute_chern_number(delta, a, t1, t2, phi, m):
     """
@@ -225,7 +224,6 @@ assert np.allclose(compute_chern_number(delta, a, t1, t2, phi, m), target)
 ```
 
 **1.3 Make a 2D array of Chern numbers by sweeping the parameters: the on-site energy to next-nearest-neighbor coupling ratio ($m/t_2$ from -6 to 6 with $N$ samples) and phase ($\phi$ from -$\pi$ to $\pi$ with $N$ samples) values. Given the grid size $\delta$ for discretizing the Brillouin zone in the $k_x$ and $k_y$ directions (assuming the grid sizes are the same in both directions), the lattice spacing $a$, the nearest-neighbor coupling constant $t_1$, and the next-nearest-neighbor coupling constant $t_2$.**
-
 ```python
 def compute_chern_number_grid(delta, a, t1, t2, N):
     """
@@ -254,7 +252,6 @@ def compute_chern_number_grid(delta, a, t1, t2, N):
 ```
 
 ## Domain Specific Test Cases
-
 **Both the $k$-space and sweeping grid sizes are set to very rough values to make the computation faster, feel free to increase them for higher accuracy.**
 
 **At zero on-site energy, the Chern number is 1 for $\phi > 0$, and the Chern number is -1 for $\phi < 0$.**
@@ -272,7 +269,6 @@ t1 = 4.0
 t2 = 1.0
 N = 40
 ```
-
 ![](figures/chern_number_1.png)
 
 ```python
@@ -283,7 +279,6 @@ t1 = 5.0
 t2 = 1.0
 N = 40
 ```
-
 ![](figures/chern_number_2.png)
 
 ```python
@@ -294,6 +289,5 @@ t1 = 1.0
 t2 = 0.2
 N = 40
 ```
-
 ![](figures/chern_number_3.png)