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

+---
+jupytext:
+  text_representation:
+    extension: .md
+    format_name: myst
+    format_version: 0.13
+    jupytext_version: 1.14.4
+kernelspec:
+  display_name: Python 3 (ipykernel)
+  language: python
+  name: python3
+---
+
+# Graphene and extended Hubbard
+
+## The physics
+
+This tutorial serves as a simple example of using the meanfield algorithm in two dimensions in combination with using Kwant. We will consider a simple tight-binding model of graphene with a Hubbard interaction. The graphene system is first created using Kwant. For the basics of creating graphene with Kwant we refer to [this](https://kwant-project.org/doc/1/tutorial/graphene) tutorial.
+
+We begin with the basic imports
+
+```{code-cell} ipython3
+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
+```
+
+##  Preparing the model
+
+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
+# Create translationally-invariant `kwant.Builder`
+graphene_builder, int_builder = kwant_examples.graphene_extended_hubbard()
+h_0 = kwant_utils.builder_to_tb(graphene_builder)
+```
+
+We also use Kwant to create the Hubbard interaction. The interaction terms are  described by:
+
+$$ Hubbardd $$
+
+Once we have both the non-interacting and the interacting part, we can assign the parameters for the Hubbard interaction and then combine both, together with a filling, into the model.
+
+```{code-cell} ipython3
+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)
+```
+
+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)
+```
+
+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.
+
+## Creating a phase diagram of the gap
+
+We can now create a phase diagram of the gap of the interacting solution. In order to calculate the gap we first create a function which takes a hopping dictionary and a Fermi energy and returns the indirect gap. The gap is defined as the difference between the highest occupied and the lowest unoccupied energy level. We will use a dense k-grid to calculate the gap. In order to obtain the Hamiltonian on a dense k-grid, we use the `tb_to_khamvector` function from the `transforms` module.
+
+```{code-cell} ipython3
+def compute_gap(h, fermi_energy=0, nk=100):
+    kham = tb.transforms.tb_to_khamvector(h, nk, ks=None)
+    vals = np.linalg.eigvalsh(kham)
+
+    emax = np.max(vals[vals <= fermi_energy])
+    emin = np.min(vals[vals > fermi_energy])
+    return np.abs(emin - emax)
+```
+
+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):
+  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 then plot it
+
+```{code-cell} ipython3
+Us = np.linspace(0, 4, 5)
+Vs = np.linspace(0, 1.5, 5)
+gap, mf_sols = compute_phase_diagram(Us, Vs, int_builder, h_0)
+plt.imshow(gap.T, extent=(Us[0], Us[-1], Vs[0], Vs[-1]), origin='lower', aspect='auto')
+plt.colorbar()
+plt.xlabel('V')
+plt.ylabel('U')
+plt.title('Gap')
+plt.show()
+```
+
+This phase diagram has gap openings at the same places as shown in the [literature](https://arxiv.org/abs/1204.4531).
+
+## Order parameters
+
+We might also want to calculate order parameters of the mean field solution. From literature we know to expect a charge density wave (CDW) and a spin density wave (SDW) in the mean field solution in different regions of the phase diagram. Here we show how to create both the CDW and SDW order parameters and evaluate them for the mean field solution in the phase diagram.
+
+### Charge density wave
+
+We first start with the CDW order parameter. In CDW the spins on different sublattices have opposite signs. This means that our order parameter can be constructed as:
+
+```{code-cell} ipython3
+sz = np.array([[1, 0], [0, -1]])
+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 perform this calculation over the complete phase diagram where we calculated the gap earlier:
+
+```{code-cell} ipython3
+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
+
+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)
+```
+
+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_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
+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('Spin Density Wave Order Parameter')
+plt.show()
+```