2.10. Discretizing continuous Hamiltonians

Introduction

In “Discretization of a Schrödinger Hamiltonian” we have learnt that Kwant works with tight-binding Hamiltonians. Often, however, one will start with a continuum model and will subsequently need to discretize it to arrive at a tight-binding model. Although discretizing a Hamiltonian is usually a simple process, it is tedious and repetitive. The situation is further exacerbated when one introduces additional on-site degrees of freedom, and tracking all the necessary terms becomes a chore. The continuum sub-package aims to be a solution to this problem. It is a collection of tools for working with continuum models and for discretizing them into tight-binding models.

See also

The complete source code of this tutorial can be found in discretize.py

Discretizing by hand

As an example, let us consider the following continuum Schrödinger equation for a semiconducting heterostructure (using the effective mass approximation):

\[\left( k_x \frac{\hbar^2}{2 m(x)} k_x \right) \psi(x) = E \, \psi(x).\]

Replacing the momenta by their corresponding differential operators

\[k_\alpha = -i \partial_\alpha,\]

for \(\alpha = x, y\) or \(z\), and discretizing on a regular lattice of points with spacing \(a\), we obtain the tight-binding model

\[H = - \frac{1}{a^2} \sum_i A\left(x+\frac{a}{2}\right) \big(\ket{i}\bra{i+1} + h.c.\big) + \frac{1}{a^2} \sum_i \left( A\left(x+\frac{a}{2}\right) + A\left(x-\frac{a}{2}\right)\right) \ket{i} \bra{i},\]

with \(A(x) = \frac{\hbar^2}{2 m(x)}\).

Using discretize to obtain a template

First we must explicitly import the kwant.continuum package:

import kwant.continuum

The function kwant.continuum.discretize takes a symbolic Hamiltonian and turns it into a Builder instance with appropriate spatial symmetry that serves as a template. (We will see how to use the template to build systems with a particular shape later).

template = kwant.continuum.discretize('k_x * A(x) * k_x')
print(template)
# Discrete coordinates: x

# Onsite element:
def onsite(site, A):
    (x, ) = site.pos
    _const_0 = (A(-0.5 + x))
    _const_1 = (A(0.5 + x))
    return (1.0*_const_0 + 1.0*_const_1)

# Hopping from (1,):
def hopping_1(site1, site2, A):
    (x, ) = site1.pos
    _const_0 = (A(0.5 + x))
    return (-1.0*_const_0)

It is worth noting that discretize treats k_x and x as non-commuting operators, and so their order is preserved during the discretization process.

Printing the Builder produced by discretize shows the source code of its onsite and hopping functions (this is a special feature of builders returned by discretize).

Technical details
  • kwant.continuum uses sympy internally to handle symbolic expressions. Strings are converted using kwant.continuum.sympify, which essentially applies some Kwant-specific rules (such as treating k_x and x as non-commutative) before calling sympy.sympify

  • The builder returned by discretize will have an N-D translational symmetry, where N is the number of dimensions that were discretized. This is the case, even if there are expressions in the input (e.g. V(x, y)) which in principle may not have this symmetry. When using the returned builder directly, or when using it as a template to construct systems with different/lower symmetry, it is important to ensure that any functional parameters passed to the system respect the symmetry of the system. Kwant provides no consistency check for this.

  • The discretization process consists of taking input \(H(k_x, k_y, k_z)\), multiplying it from the right by \(\psi(x, y, z)\) and iteratively applying a second-order accurate central derivative approximation for every \(k_\alpha=-i\partial_\alpha\):

    \[\partial_\alpha \psi(\alpha) = \frac{1}{a} \left( \psi\left(\alpha + \frac{a}{2}\right) -\psi\left(\alpha - \frac{a}{2}\right)\right).\]

    This process is done separately for every summand in Hamiltonian. Once all symbols denoting operators are applied internal algorithm is calculating gcd for hoppings coming from each summand in order to find best possible approximation. Please see source code for details.

  • Instead of using discretize one can use discretize_symbolic to obtain symbolic output. When working interactively in Jupyter notebooks it can be useful to use this to see a symbolic representation of the discretized Hamiltonian. This works best when combined with sympy Pretty Printing.

  • The symbolic result of discretization obtained with discretize_symbolic can be converted into a builder using build_discretized. This can be useful if one wants to alter the tight-binding Hamiltonian before building the system.

Building a Kwant system from the template

Let us now use the output of discretize as a template to build a system and plot some of its energy eigenstate. For this example the Hamiltonian will be

\[H = k_x^2 + k_y^2 + V(x, y),\]

where \(V(x, y)\) is some arbitrary potential.

First, use discretize to obtain a builder that we will use as a template:

hamiltonian = "k_x**2 + k_y**2 + V(x, y)"
template = kwant.continuum.discretize(hamiltonian)
print(template)
# Discrete coordinates: x y

# Onsite element:
def onsite(site, V):
    (x, y, ) = site.pos
    _const_0 = (V(x, y))
    return (_const_0 + 4.0)

# Hopping from (1, 0):
(-1+0j)

# Hopping from (0, 1):
(-1+0j)

We now use this system with the fill method of Builder to construct the system we want to investigate:

def stadium(site):
    (x, y) = site.pos
    x = max(abs(x) - 20, 0)
    return x**2 + y**2 < 30**2

syst = kwant.Builder()
syst.fill(template, stadium, (0, 0));
syst = syst.finalized()

After finalizing this system, we can plot one of the system’s energy eigenstates:

def plot_eigenstate(syst, n=2, Vx=.0003, Vy=.0005):

    def potential(x, y):
        return Vx * x + Vy * y

    ham = syst.hamiltonian_submatrix(params=dict(V=potential), sparse=True)
    evecs = scipy.sparse.linalg.eigsh(ham, k=10, which='SM')[1]
    kwant.plotter.density(syst, abs(evecs[:, n])**2, show=False)
../_images/discretize_8_0.svg

Note in the above that we pass the spatially varying potential function to our system via a parameter called V, because the symbol \(V\) was used in the initial, symbolic, definition of the Hamiltonian.

In addition, the function passed as V expects two input parameters x and y, the same as in the initial continuum Hamiltonian.

Models with more structure: Bernevig-Hughes-Zhang

When working with multi-band systems, like the Bernevig-Hughes-Zhang (BHZ) model [1] [2], one can provide matrix input to discretize using identity and kron. For example, the definition of the BHZ model can be written succinctly as:

hamiltonian = """
   + C * identity(4) + M * kron(sigma_0, sigma_z)
   - B * (k_x**2 + k_y**2) * kron(sigma_0, sigma_z)
   - D * (k_x**2 + k_y**2) * kron(sigma_0, sigma_0)
   + A * k_x * kron(sigma_z, sigma_x)
   - A * k_y * kron(sigma_0, sigma_y)
"""

a = 20

template = kwant.continuum.discretize(hamiltonian, grid=a)

We can then make a ribbon out of this template system:

L, W = 2000, 1000

def shape(site):
    (x, y) = site.pos
    return (0 <= y < W and 0 <= x < L)

def lead_shape(site):
    (x, y) = site.pos
    return (0 <= y < W)

syst = kwant.Builder()
syst.fill(template, shape, (0, 0))

lead = kwant.Builder(kwant.TranslationalSymmetry([-a, 0]))
lead.fill(template, lead_shape, (0, 0))

syst.attach_lead(lead)
syst.attach_lead(lead.reversed())

syst = syst.finalized()

and plot its dispersion using kwant.plotter.bands:

params = dict(A=3.65, B=-68.6, D=-51.1, M=-0.01, C=0)

kwant.plotter.bands(syst.leads[0], params=params,
                    momenta=np.linspace(-0.3, 0.3, 201), show=False)

pyplot.grid()
pyplot.xlim(-.3, 0.3)
pyplot.ylim(-0.05, 0.05)
pyplot.xlabel('momentum [1/A]')
pyplot.ylabel('energy [eV]')
pyplot.show()
../_images/discretize_11_0.svg

In the above we see the edge states of the quantum spin Hall effect, which we can visualize using kwant.plotter.density:

# get scattering wave functions at E=0
wf = kwant.wave_function(syst, energy=0, params=params)

# prepare density operators
sigma_z = np.array([[1, 0], [0, -1]])
prob_density = kwant.operator.Density(syst, np.eye(4))
spin_density = kwant.operator.Density(syst, np.kron(sigma_z, np.eye(2)))

# calculate expectation values and plot them
wf_sqr = sum(prob_density(psi) for psi in wf(0))  # states from left lead
rho_sz = sum(spin_density(psi) for psi in wf(0))  # states from left lead

fig, (ax1, ax2) = pyplot.subplots(1, 2, sharey=True, figsize=(16, 4))
kwant.plotter.density(syst, wf_sqr, ax=ax1)
kwant.plotter.density(syst, rho_sz, ax=ax2)

ax = ax1
im = [obj for obj in ax.get_children()
      if isinstance(obj, mpl.image.AxesImage)][0]
fig.colorbar(im, ax=ax)

ax = ax2
im = [obj for obj in ax.get_children()
      if isinstance(obj, mpl.image.AxesImage)][0]
fig.colorbar(im, ax=ax)

ax1.set_title('Probability density')
ax2.set_title('Spin density')
pyplot.show()
../_images/discretize_12_0.svg

Limitations of discretization

It is important to remember that the discretization of a continuum model is an approximation that is only valid in the low-energy limit. For example, the quadratic continuum Hamiltonian

\[H_\textrm{continuous}(k_x) = \frac{\hbar^2}{2m}k_x^2\]

and its discretized approximation

\[H_\textrm{tight-binding}(k_x) = 2t \big(1 - \cos(k_x a)\big),\]

where \(t=\frac{\hbar^2}{2ma^2}\), are only valid in the limit \(E < t\). The grid spacing \(a\) must be chosen according to how high in energy you need your tight-binding model to be valid.

It is possible to set \(a\) through the grid parameter to discretize, as we will illustrate in the following example. Let us start from the continuum Hamiltonian

\[H(k) = k_x^2 \mathbb{1}_{2\times2} + α k_x \sigma_y.\]

We start by defining this model as a string and setting the value of the \(α\) parameter:

hamiltonian = "k_x**2 * identity(2) + alpha * k_x * sigma_y"
params = dict(alpha=.5)

Now we can use kwant.continuum.lambdify to obtain a function that computes \(H(k)\):

h_k = kwant.continuum.lambdify(hamiltonian, locals=params)
k_cont = np.linspace(-4, 4, 201)
e_cont = [scipy.linalg.eigvalsh(h_k(k_x=ki)) for ki in k_cont]

We can also construct a discretized approximation using kwant.continuum.discretize, in a similar manner to previous examples:

def plot(ax, a=1):
    template = kwant.continuum.discretize(hamiltonian, grid=a)
    syst = kwant.wraparound.wraparound(template).finalized()

    def h_k(k_x):
        p = dict(k_x=k_x, **params)
        return syst.hamiltonian_submatrix(params=p)

    k_tb = np.linspace(-np.pi/a, np.pi/a, 201)
    e_tb = [scipy.linalg.eigvalsh(h_k(k_x=a*ki)) for ki in k_tb]

    ax.plot(k_cont, e_cont, 'r-')
    ax.plot(k_tb, e_tb, 'k-')

    ax.plot([], [], 'r-', label='continuum')
    ax.plot([], [], 'k-', label='tight-binding')

    ax.set_xlim(-4, 4)
    ax.set_ylim(-1, 14)
    ax.set_title('a={}'.format(a))

    ax.set_xlabel('momentum [a.u.]')
    ax.set_ylabel('energy [a.u.]')
    ax.grid()
    ax.legend()

Below we can see the continuum and tight-binding dispersions for two different values of the discretization grid spacing \(a\):

../_images/discretize_16_0.svg

We clearly see that the smaller grid spacing is, the better we approximate the original continuous dispersion. It is also worth remembering that the Brillouin zone also scales with grid spacing: \([-\frac{\pi}{a}, \frac{\pi}{a}]\).

Advanced topics

The input to kwant.continuum.discretize and kwant.continuum.lambdify can be not only a string, as we saw above, but also a sympy expression or a sympy matrix. This functionality will probably be mostly useful to people who are already experienced with sympy.

It is possible to use identity (for identity matrix), kron (for Kronecker product), as well as Pauli matrices sigma_0, sigma_x, sigma_y, sigma_z in the input to lambdify and discretize, in order to simplify expressions involving matrices. Matrices can also be provided explicitly using square [] brackets. For example, all following expressions are equivalent:

sympify = kwant.continuum.sympify
subs = {'sx': [[0, 1], [1, 0]], 'sz': [[1, 0], [0, -1]]}

e = (
    sympify('[[k_x**2, alpha * k_x], [k_x * alpha, -k_x**2]]'),
    sympify('k_x**2 * sigma_z + alpha * k_x * sigma_x'),
    sympify('k_x**2 * sz + alpha * k_x * sx', locals=subs),
)

print(e[0] == e[1] == e[2])
True

We can use the locals keyword parameter to substitute expressions and numerical values:

subs = {'A': 'A(x) + B', 'V': 'V(x) + V_0', 'C': 5}
print(sympify('k_x * A * k_x + V + C', locals=subs))
V_0 + 5 + k_x*(B + A(x))*k_x + V(x)

Symbolic expressions obtained in this way can be directly passed to all discretizer functions.

Technical details
Because of the way that sympy handles commutation relations all symbols representing position and momentum operators are set to be non commutative. This means that the order of momentum and position operators in the input expression is preserved. Note that it is not possible to define individual commutation relations within sympy, even expressions such \(x k_y x\) will not be simplified, even though mathematically \([x, k_y] = 0\).

References

[1]Science, 314, 1757 (2006).
[2]Phys. Rev. B 82, 045122 (2010).