Examples

Everything related to constructing examples that motivate and teach how to use our package. This includes:

• coming up with new interesting physical systems
• cleaning up the existing ones

and likely more.

docs/source/graphene_example.md

@@ -83,35 +83,33 @@ def compute_gap(h, fermi_energy=0, nk=100):
 
 Now that we can calculate the gap, we create a phase diagram of the gap as a function of the Hubbard interaction strength $U$ and the nearest neighbor interaction strength $V$. We vary the onsite Hubbard interactio $U$ strength from $0$ to $2$ and the nearest neighbor interaction strength $V$ from $0$ to $1.5$.
 
+```{code-cell} ipython3
+def gap_and_mf_sol(U, V, int_builder, h_0):
+    params = dict(U=U, V=V)
+    h_int = kwant_utils.builder_to_tb(int_builder, params)
+    _model = model.Model(h_0, h_int, filling=2)
+    guess = tb.utils.generate_guess(frozenset(h_int), len(list(h_0.values())[0]))
+    mf_sol = solvers.solver(_model, guess, nk=18, optimizer_kwargs={'M':0})
+    gap = compute_gap(tb.tb.add_tb(h_0, mf_sol), fermi_energy=0, nk=300)
+    return gap, mf_sol
+```
+
 ```{code-cell} ipython3
 def compute_phase_diagram(Us, Vs, int_builder, h_0):
-    gap = []
-    mf_sols = []
-    for U in Us:
-      gap_U = []
-      mf_sols_U = []
-      for V in Vs:
-        params = dict(U=U, V=V)
-        h_int = kwant_utils.builder_to_tb(int_builder, params)
-        _model = model.Model(h_0, h_int, filling=2)
-        converged=False
-        while not converged:
-          guess = tb.utils.generate_guess(frozenset(h_int), len(list(h_0.values())[0]))
-          try:
-            mf_sol = solvers.solver(_model, guess, nk=18, optimizer_kwargs={'M':0})
-            converged=True
-          except:
-            converged=False
-        gap_U.append(compute_gap(tb.tb.add_tb(h_0, mf_sol), fermi_energy=0, nk=300))
-        mf_sols_U.append(mf_sol)
-      guess = None
-      gap.append(gap_U)
-      mf_sols.append(mf_sols_U)
-    return np.asarray(gap, dtype=float), np.asarray(mf_sols)
+  gaps = []
+  mf_sols = []
+  for U in Us:
+    for V in Vs:
+      gap, mf_sol = gap_and_mf_sol(U, V, int_builder, h_0)
+      gaps.append(gap)
+      mf_sols.append(mf_sol)
+  gaps = np.asarray(gaps, dtype=float).reshape((len(Us), len(Vs)))
+  mf_sols = np.asarray(mf_sols).reshape((len(Us), len(Vs)))
+  return gaps, mf_sols
 ```
 We chose to initialize a new guess for each $U$ value, but not for each $V$ value. Instead, for consecutive $V$ values we use the previous mean-field solution as a guess. We do this because the mean-field solution is expected to be smooth in the interaction strength and therefore by using an inspired guess we can speed up the calculation.
 
-We can now compute the phase diagram and plot it.
+We can now compute the phase diagram and then plot it
 
 ```{code-cell} ipython3
 Us = np.linspace(0, 4, 5)
@@ -141,37 +139,63 @@ cdw_order_parameter = {}
 cdw_order_parameter[(0,0)] = np.kron(sz, np.eye(2))
 ```
 
-We choose a point in the phase diagram where we expect there to be a CDW phase and calculate the expectation value with the CDW order parameter. In order to do this we first construct the density matrix from the mean field solution.
+We choose a point in the phase diagram where we expect there to be a CDW phase and calculate the expectation value with the CDW order parameter. In order to do this we first construct the density matrix from the mean field solution. We perform this calculation over the complete phase diagram where we calculated the gap earlier:
 
 ```{code-cell} ipython3
-params = dict(U=0, V=2)
-h_int = kwant_utils.builder_to_tb(int_builder, params)
-_model = model.Model(h_0, h_int, filling=2)
-guess = tb.utils.generate_guess(frozenset(h_int), len(list(h_0.values())[0]))
-mf_sol = solvers.solver(_model, guess, nk=18, optimizer_kwargs={'M':0})
-full_sol = tb.tb.add_tb(h_0, mf_sol)
+cdw_list = []
+for mf_sol in mf_sols.flatten():
+    rho, _ = mf.construct_density_matrix(tb.tb.add_tb(h_0, mf_sol), filling=2, nk=40)
+    expectation_value = observables.expectation_value(rho, cdw_order_parameter)
+    cdw_list.append(expectation_value)
+```
+
+```{code-cell} ipython3
+cdw_list = np.asarray(cdw_list).reshape(mf_sols.shape)
+plt.imshow(np.abs(cdw_list.T.real), extent=(Us[0], Us[-1], Vs[0], Vs[-1]), origin='lower', aspect='auto')
+plt.colorbar()
+plt.xlabel('V')
+plt.ylabel('U')
+plt.title('Charge Density Wave Order Parameter')
+plt.show()
+```
+
+### Spin density wave
 
-rho, _ = mf.construct_density_matrix(full_sol, filling=2, nk=40)
-expectation_value = observables.expectation_value(rho, cdw_order_parameter)
-print(expectation_value)
+To check the other phase we expect in the graphene phase diagram, we construct a spin density wave order parameter. In our chosen graphene system the spin density wave has $SU(2)$ symmetry. This means that we need to sum over the pauli matrices when constructing this order parameter. We can construct the order parameter as:
+
+```{code-cell} ipython3
+sx = np.array([[0, 1], [1, 0]])
+sy = np.array([[0, -1j], [1j, 0]])
+
+s_list = [sx, sy, sz]
+
+order_parameter_list = []
+for s in s_list:
+    order_parameter = {}
+    order_parameter[(0,0)] =  np.kron(sz, s)
+    order_parameter_list.append(order_parameter)
 ```
 
-We can also perform the same calculation over the complete phase diagram where we calculated the gap earlier:
+Then, similar to what we did in the CDW phase, we calculate the expectation value of the order parameter with the density matrix of the mean field solution over the complete phase diagram. The main subtlety here is that we need to sum over expectation value of all SDW direction order parameters defined in order to get the total spin density wave order parameter.
 
 ```{code-cell} ipython3
-expectation_value_list = []
+sdw_list = []
 for mf_sol in mf_sols.flatten():
     rho, _ = mf.construct_density_matrix(tb.tb.add_tb(h_0, mf_sol), filling=2, nk=40)
-    expectation_value = observables.expectation_value(rho, cdw_order_parameter)
-    expectation_value_list.append(expectation_value)
+    expectation_values = []
+    for order_parameter in order_parameter_list:
+        expectation_value = observables.expectation_value(rho, order_parameter)
+        expectation_values.append(expectation_value)
+
+    sdw_list.append(np.sum(np.array(expectation_values)**2))
 ```
 
 ```{code-cell} ipython3
-expectation_value_list = np.asarray(expectation_value_list).reshape(mf_sols.shape)
-plt.imshow(np.abs(expectation_value_list.T.real), extent=(Us[0], Us[-1], Vs[0], Vs[-1]), origin='lower', aspect='auto')
+sdw_list = np.asarray(sdw_list).reshape(mf_sols.shape)
+plt.imshow(np.abs(sdw_list.T.real), extent=(Us[0], Us[-1], Vs[0], Vs[-1]), origin='lower', aspect='auto')
 plt.colorbar()
 plt.xlabel('V')
 plt.ylabel('U')
-plt.title('Charge Density Wave Order Parameter')
+plt.title('Spin Density Wave Order Parameter')
 plt.show()
 ```