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

@@ -24,13 +24,7 @@ import numpy as np
 import matplotlib.pyplot as plt
 import numpy as np
 import matplotlib.pyplot as plt
-import pymf.model as model
-import pymf.solvers as solvers
-import pymf.mf as mf
-import pymf.tb as tb
-import pymf.observables as observables
-import pymf.kwant_helper.kwant_examples as kwant_examples
-import pymf.kwant_helper.utils as kwant_utils
+import pymf
 ```
 
 ##  Preparing the model
@@ -38,6 +32,9 @@ import pymf.kwant_helper.utils as kwant_utils
 We first translate this model from a Kwant system to a tight-binding dictionary. In the tight-binding dictionary the keys denote the hoppings while the values are the hopping amplitudes.
 
 ```{code-cell} ipython3
+import pymf.kwant_helper.kwant_examples as kwant_examples
+import pymf.kwant_helper.utils as kwant_utils
+
 # Create translationally-invariant `kwant.Builder`
 graphene_builder, int_builder = kwant_examples.graphene_extended_hubbard()
 h_0 = kwant_utils.builder_to_tb(graphene_builder)
@@ -54,15 +51,15 @@ U=1
 V=0.1
 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)
+_model = pymf.Model(h_0, h_int, filling=2)
 ```
 
 To start the meanfield calculation we also need a starting guess. We will use our random guess generator for this. It creates a random Hermitian hopping dictionary based on the hopping keys provided and the number of degrees of freedom specified. As we don't expect the mean-field solution to contain terms more than the hoppings from the interacting part, we can use the hopping keys from the interacting part. We will use the same number of degrees as freedom as both the non-interacting and interacting part, so that they match.
 
 ```{code-cell} ipython3
-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)
+guess = pymf.generate_guess(frozenset(h_int), len(list(h_0.values())[0]))
+mf_sol = pymf.solver(_model, guess, nk=18, optimizer_kwargs={'M':0})
+full_sol = pymf.add_tb(h_0, mf_sol)
 ```
 
 After we have defined the guess, we feed it together with the model into the meanfield solver. The meanfield solver will return a hopping dictionary with the meanfield approximation. We can then add this solution to the non-interacting part to get the full solution. In order to get the solution, we specified the number of k-points to be used in the calculation. This refers to the k-grid used in the Brillouin zone for the density matrix.
@@ -73,7 +70,7 @@ We can now create a phase diagram of the gap of the interacting solution. In ord
 
 ```{code-cell} ipython3
 def compute_gap(h, fermi_energy=0, nk=100):
-    kham = tb.transforms.tb_to_khamvector(h, nk, ks=None)
+    kham = pymf.tb_to_khamvector(h, nk, ks=None)
     vals = np.linalg.eigvalsh(kham)
 
     emax = np.max(vals[vals <= fermi_energy])
@@ -83,35 +80,33 @@ 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 = pymf.Model(h_0, h_int, filling=2)
+    guess = pymf.generate_guess(frozenset(h_int), len(list(h_0.values())[0]))
+    mf_sol = pymf.solver(_model, guess, nk=18, optimizer_kwargs={'M':0})
+    gap = compute_gap(pymf.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 +136,63 @@ 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)
-
-rho, _ = mf.construct_density_matrix(full_sol, filling=2, nk=40)
-expectation_value = observables.expectation_value(rho, cdw_order_parameter)
-print(expectation_value)
+cdw_list = []
+for mf_sol in mf_sols.flatten():
+    rho, _ = pymf.construct_density_matrix(pymf.add_tb(h_0, mf_sol), filling=2, nk=40)
+    expectation_value = pymf.expectation_value(rho, cdw_order_parameter)
+    cdw_list.append(expectation_value)
 ```
 
-We can also perform the same calculation over the complete phase diagram where we calculated the gap earlier:
+```{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
+
+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
-expectation_value_list = []
+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)
+```
+
+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
+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)
+    rho, _ = pymf.construct_density_matrix(pymf.add_tb(h_0, mf_sol), filling=2, nk=40)
+    expectation_values = []
+    for order_parameter in order_parameter_list:
+        expectation_value = pymf.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()
 ```