Merge pull request #82 from noahsong-sdg/feature/readme-fix

update readme examples to match docs

Author: phjmsycc

README.md

@@ -62,24 +62,32 @@ julia> Pkg.add("TopologicalNumbers")
 
 ## Examples
 
-### The Su-Schriffer-Heeger (SSH) model
+### The Su-Schrieffer-Heeger (SSH) model
 
 Here's a simple example of the SSH Hamiltonian:
 
 ```julia
 julia> using TopologicalNumbers
-julia> function H₀(k, p)
+julia> function H₀(k, p) # SSH
+            t₁ = 1
+            t₂ = p
             [
-                0 p[1]+p[2]*exp(-im * k)
-                p[1]+p[2]*exp(im * k) 0
+                0  t₁ + t₂*exp(-im * k)
+                t₁ + t₂*exp(im * k) 0
             ]
         end
 ```
 
+Or you can use our preset Hamiltonian function:
+
+```julia
+julia> H₀ = SSH
+```
+
 The band structure is computed as follows:
 
 ```julia
-julia> H(k) = H₀(k, (0.9, 1.0))
+julia> H(k) = H₀(k, 1.1)
 julia> showBand(H; value=false, disp=true)
 ```
 
@@ -134,30 +142,35 @@ julia> sol = calcPhaseDiagram(prob, param; plot=true)
 Hamiltonian of Haldane model is given by:
 
 ```julia
-julia> function H₀(k, p) # landau
-           k1, k2 = k
-           J = 1.0
-           K = 1.0
-           ϕ, M = p
-
-           h0 = 2K * cos(ϕ) * (cos(k1) + cos(k2) + cos(k1 + k2))
-           hx = J * (1 + cos(k1) + cos(k2))
-           hy = J * (-sin(k1) + sin(k2))
-           hz = M - 2K * sin(ϕ) * (sin(k1) + sin(k2) - sin(k1 + k2))
-
-           s0 = [1 0; 0 1]
-           sx = [0 1; 1 0]
-           sy = [0 -im; im 0]
-           sz = [1 0; 0 -1]
-
-           h0 .* s0 .+ hx .* sx .+ hy .* sy .+ hz .* sz
-       end
+julia> function H₀(k, p) # Haldane
+            k1, k2 = k
+            t₁ = 1
+            t₂, ϕ, m = p
+
+            h0 = 2t₂ * cos(ϕ) * (cos(k1) + cos(k2) + cos(k1 + k2))
+            hx = t₁ * (1 + cos(k1) + cos(k2))
+            hy = t₁ * (-sin(k1) + sin(k2))
+            hz = m - 2t₂ * sin(ϕ) * (sin(k1) + sin(k2) - sin(k1 + k2))
+
+            s0 = [1 0; 0 1]
+            sx = [0 1; 1 0]
+            sy = [0 -im; im 0]
+            sz = [1 0; 0 -1]
+
+            h0 .* s0 .+ hx .* sx .+ hy .* sy .+ hz .* sz
+        end
+```
+
+Or you can use our preset Hamiltonian function:
+
+```julia
+julia> H₀ = Haldane
 ```
 
 The band structure is computed as follows:
 
 ```julia
-julia> H(k) = H₀(k, (π/3, 0.5))
+julia> H(k) = H₀(k, (1, π/3, 0.5))
 julia> showBand(H; value=false, disp=true)
 ```
 
@@ -220,31 +233,39 @@ julia> sol = calcPhaseDiagram(prob, param1, param2; plot=true)
 As an example of a two-dimensional topological insulator, the BHZ model is presented here:
 
 ```julia
+julia> using LinearAlgebra
 julia> function H₀(k, p) # BHZ
-    k1, k2 = k
-    tₛₚ = 1
-    t₁ = ϵ₁ = 2
-    ϵ₂, t₂ = p
-
-    R0 = -t₁*(cos(k1) + cos(k2)) + ϵ₁/2
-    R3 = 2tₛₚ*sin(k2)
-    R4 = 2tₛₚ*sin(k1)
-    R5 = -t₂*(cos(k1) + cos(k2)) + ϵ₂/2
-
-    s0 = [1 0; 0 1]
-    sx = [0 1; 1 0]
-    sy = [0 -im; im 0]
-    sz = [1 0; 0 -1]
-
-    a0 = kron(s0, s0)
-    a1 = kron(sx, sx)
-    a2 = kron(sx, sy)
-    a3 = kron(sx, sz)
-    a4 = kron(sy, s0)
-    a5 = kron(sz, s0)
-
-    R0 .* a0 .+ R3 .* a3 .+ R4 .* a4 .+ R5 .* a5
-end
+            k1, k2 = k
+            tₛₚ = 1
+            t₁ = ϵ₁ = 1
+            ϵ₂, t₂ = p
+
+            ϵ = -t₁*(cos(k1) + cos(k2)) + ϵ₁/2
+            R1 = 0
+            R2 = 0
+            R3 = 2tₛₚ*sin(k2)
+            R4 = 2tₛₚ*sin(k1)
+            R0 = -t₂*(cos(k1) + cos(k2)) + ϵ₂/2
+
+            s0 = [1 0; 0 1]
+            sx = [0 1; 1 0]
+            sy = [0 -im; im 0]
+            sz = [1 0; 0 -1]
+
+            I0 = Matrix{Int64}(I, 4, 4)
+            a1 = kron(sz, sx)
+            a2 = kron(sz, sy)
+            a3 = kron(sz, sz)
+            a4 = kron(sy, s0)
+            a0 = kron(sx, s0)
+
+            ϵ .* I0 .+ R1 .* a1 .+ R2 .* a2 .+ R3 .* a3 .+ R4 .* a4 .+ R0 .* a0
+        end
+```
+Alternatively, you can use our preset Hamiltonian:
+
+```julia
+julia> H₀ = BHZ
 ```
 
 To calculate the dispersion, execute:
@@ -254,7 +275,7 @@ julia> H(k) = H₀(k, (2, 2))
 julia> showBand(H; value=false, disp=true)
 ```
 
-![Dispersion of BHZ model](https://github.com/KskAdch/TopologicalNumbers.jl/assets/139373570/de14907c-777f-4667-810b-54c10888dfa1)
+![Dispersion of BHZ model](./docs/src/2D/assets/Band_BHZ.png)
 
 
 
@@ -272,7 +293,7 @@ The output is:
 Z2Solution{Vector{Int64}, Nothing, Int64}([1, 1], nothing, 0)
 ```
 
-The first argument `TopologicalNumber` in the named tuple is an vector that stores the $\mathbb{Z}_2$ number for Energy bands below and above some filling condition that you selected in the options (the default is the half-filling). 
+The first argument `TopologicalNumber` in the named tuple is a vector that stores the $\mathbb{Z}_2$ number for Energy bands below and above some filling condition that you selected in the options (the default is the half-filling). 
 The vector is arranged in order of bands, starting from the one with the lowest energy.
 The second argument `Total` stores the total of the $\mathbb{Z}_2$ numbers for each pair of two energy bands.
 `Total` is a quantity that should always return zero.
@@ -290,7 +311,7 @@ julia> sol = calcPhaseDiagram(prob, param; plot=true)
 (param = -2.0:0.004004004004004004:2.0, nums = [0 0; 0 0; … ; 0 0; 0 0])
 ```
 
-![One-dimensional phase diagram of BHZ model](https://github.com/KskAdch/TopologicalNumbers.jl/assets/139373570/5d6d5364-68d0-4423-8ecf-49bf0538af63)
+![One-dimensional phase diagram of BHZ model](./docs/src/2D/assets/phase_diagram1D_BHZ.png)
 
 
 Also, two-dimensional phase diagram is given by:
@@ -304,7 +325,7 @@ julia> calcPhaseDiagram(prob, param1, param2; plot=true)
 ```
 
 
-![Two-dimensional phase diagram of BHZ model](https://github.com/KskAdch/TopologicalNumbers.jl/assets/139373570/802eedbe-c893-44b4-8267-d80e1745415a)
+![Two-dimensional phase diagram of BHZ model](./docs/src/2D/assets/phase_diagram2D_BHZ.png)
 
 
 

docs/src/1D/Kitaev-Chain.md

@@ -42,7 +42,7 @@ The output is:
 BPSolution{Vector{Int64}, Int64}([1, 1], 0)
 ```
 
-The first argument `TopologicalNumber` in the named tuple is an vector that stores the winding number for each band. 
+The first argument `TopologicalNumber` in the named tuple is a vector that stores the winding number for each band. 
 The vector is arranged in order of bands, starting from the one with the lowest energy.
 The second argument `Total` stores the total of the winding numbers for each band (mod 2).
 `Total` is a quantity that should always return zero.

docs/src/1D/SSH.md

@@ -1,4 +1,4 @@
-# The Su-Schriffer-Heeger (SSH) model
+# The Su-Schrieffer-Heeger (SSH) model
 
 Here's a simple example of the SSH Hamiltonian:
 
@@ -17,7 +17,7 @@ julia> function H₀(k, p)
 You can also use our preset Hamiltonian function `SSH` to define the same Hamiltonian matrix as follows:
 
 ```julia
-julia> H₀(k, p) = SSH(k, p)
+julia> H₀ = SSH
 ```
 
 The band structure is computed as follows:

docs/src/2D/BHZ.md

@@ -7,7 +7,7 @@ julia> using LinearAlgebra
 julia> function H₀(k, p) # BHZ
     k1, k2 = k
     tₛₚ = 1
-    t₁ = ϵ₁ = 2
+    t₁ = ϵ₁ = 1
     ϵ₂, t₂ = p
 
     ϵ = -t₁*(cos(k1) + cos(k2)) + ϵ₁/2
@@ -35,7 +35,7 @@ end
 You can also use our preset Hamiltonian function `BHZ` to define the same Hamiltonian matrix as follows:
 
 ```julia
-julia> H₀(k, p) = BHZ(k, p)
+julia> H₀ = BHZ
 ```
 
 To calculate the dispersion, execute:
@@ -45,7 +45,7 @@ julia> H(k) = H₀(k, (2, 2))
 julia> showBand(H; value=false, disp=true)
 ```
 
-![Dispersion of BHZ model](https://github.com/KskAdch/TopologicalNumbers.jl/assets/139373570/de14907c-777f-4667-810b-54c10888dfa1)
+![Dispersion of BHZ model](./assets/Band_BHZ.png)
 
 
 Next, we can compute the $\mathbb{Z}_2$ numbers using `Z2Problem`:
@@ -62,7 +62,7 @@ The output is:
 Z2Solution{Vector{Int64}, Nothing, Int64}([1, 1], nothing, 0)
 ```
 
-The first argument `TopologicalNumber` in the named tuple is an vector that stores the $\mathbb{Z}_2$ number for Energy bands below and above some filling condition that you selected in the options (the default is the half-filling).
+The first argument `TopologicalNumber` in the named tuple is a vector that stores the $\mathbb{Z}_2$ number for Energy bands below and above some filling condition that you selected in the options (the default is the half-filling).
 The vector is arranged in order of bands, starting from the lower energy.
 The second argument `Total` stores the total of the $\mathbb{Z}_2$ numbers for Energy bands below and above some filling condition (mod2).
 `Total` is a quantity that should always return zero.
@@ -91,7 +91,7 @@ julia> sol = calcPhaseDiagram(prob, param; plot=true)
 (param = -2.0:0.004004004004004004:2.0, nums = [0 0; 0 0; … ; 0 0; 0 0])
 ```
 
-![One-dimensional phase diagram of BHZ model](https://github.com/KskAdch/TopologicalNumbers.jl/assets/139373570/5d6d5364-68d0-4423-8ecf-49bf0538af63)
+![One-dimensional phase diagram of BHZ model](./assets/phase_diagram1D_BHZ.png)
 
 
 Also, two-dimensional phase diagram is given by:
@@ -105,4 +105,4 @@ julia> calcPhaseDiagram(prob, param1, param2; plot=true)
 ```
 
 
-![Two-dimensional phase diagram of BHZ model](https://github.com/KskAdch/TopologicalNumbers.jl/assets/139373570/802eedbe-c893-44b4-8267-d80e1745415a)
+![Two-dimensional phase diagram of BHZ model](./assets/phase_diagram2D_BHZ.png)

docs/src/2D/Haldane.md

@@ -24,7 +24,7 @@ julia> function H₀(k, p) # Haldane
 You can also use our preset Hamiltonian function `Haldane` to define the same Hamiltonian matrix as follows:
 
 ```julia
-julia> H₀(k, p) = Haldane(k, p)
+julia> H₀ = Haldane
 ```
 
 The band structure is computed as follows:
@@ -51,7 +51,7 @@ The output is:
 FCSolution{Vector{Int64}, Int64}([1, -1], 0)
 ```
 
-The first argument `TopologicalNumber` in the named tuple is an vector that stores the first Chern number for each band. 
+The first argument `TopologicalNumber` in the named tuple is a vector that stores the first Chern number for each band. 
 The vector is arranged in order of bands, starting from the one with the lowest energy.
 The second argument `Total` stores the total of the first Chern numbers for each band.
 `Total` is a quantity that should always return zero.

docs/src/2D/Kane-Mele.md

@@ -57,7 +57,7 @@ The output is:
 Z2Solution{Vector{Int64}, Nothing, Int64}([1, 1], nothing, 0)
 ```
 
-The first argument `TopologicalNumber` in the named tuple is an vector that stores the $\mathbb{Z}_2$ number for Energy bands below and above some filling condition that you selected in the options (the default is the half-filling).
+The first argument `TopologicalNumber` in the named tuple is a vector that stores the $\mathbb{Z}_2$ number for Energy bands below and above some filling condition that you selected in the options (the default is the half-filling).
 The vector is arranged in order of bands, starting from the lower energy.
 The second argument `Total` stores the total of the $\mathbb{Z}_2$ numbers for Energy bands below and above some filling condition (mod2).
 `Total` is a quantity that should always return zero.

docs/src/2D/Kitaev-Honeycomb.md

@@ -49,7 +49,7 @@ The output is:
 FCSolution{Vector{Int64}, Int64}([-1, 1], 0)
 ```
 
-The first argument `TopologicalNumber` in the named tuple is an vector that stores the first Chern number for each band. 
+The first argument `TopologicalNumber` in the named tuple is a vector that stores the first Chern number for each band. 
 The vector is arranged in order of bands, starting from the one with the lowest energy.
 The second argument `Total` stores the total of the first Chern numbers for each band.
 `Total` is a quantity that should always return zero.

docs/src/2D/Thouless.md

@@ -56,7 +56,7 @@ The output is:
 Z2Solution{Vector{Int64}, Nothing, Int64}([1, 1], nothing, 0)
 ```
 
-The first argument `TopologicalNumber` in the named tuple is an vector that stores the $\mathbb{Z}_2$ number for Energy bands below and above some filling condition that you selected in the options (the default is the half-filling).
+The first argument `TopologicalNumber` in the named tuple is a vector that stores the $\mathbb{Z}_2$ number for Energy bands below and above some filling condition that you selected in the options (the default is the half-filling).
 The vector is arranged in order of bands, starting from the lower energy.
 The second argument `Total` stores the total of the $\mathbb{Z}_2$ numbers for Energy bands below and above some filling condition (mod2).
 `Total` is a quantity that should always return zero.