Recall that if one confines a free quantum particle onto a circle \(S^1\) of radius \(R\), then the de Broglie wavelength of the \(n\)-th \(H=P^2/2m\)-eigenstate must be quantized in the obvious manner \(\lambda_n=2\pi R/n\), leading to the angular wavenumber \(k_n=n/R\), the momenta \(p_n=n\hbar/R\) and the energies \(E_n=n^2\hbar^2/2I\) with \(I=mR^2\) the classical moment of inertia. Thus, more broadly, because the spatial domain \(S^1\) of the quantum particle was compact, it follows that the conjugate momenta \(p_n\) were discrete. However, it should not be a surprise to hear that the converse holds too by virtue of Pontryagin duality. That is to say, if one instead begins by discretizing space \(S^1\mapsto\textbf Z/N\textbf Z\) into a lattice of \(N\) equally spaced atomic sites such as in a crystal, then one should now expect the momenta \(p\) to lie in some compact interval (which will turn out to be called the Brillouin zone). Thus, roughly speaking, compactness and discreteness are Pontryagin/Fourier duals of each other. In some sense, this is just another manifestation of the Heisenberg uncertainty principle.
One might think to just stick with the usual free Hamiltonian \(H=P^2/2m\), however this doesn’t work because \(P\) is undefined on \(\mathcal H=L^2(\textbf Z/N\textbf Z\to\textbf C)\). Instead, we employ a so-called tight-binding Hamiltonian \(H_{\text{tight-binding}}\) defined by:
\[H_{\text{tight-binding}}:=E_01-E_{\text{hop}}\sum_{n=1}^N|n+1\rangle\langle n|+|n\rangle\langle n+1|\]
where for \(n=1,2,…,N\), \(|n\rangle\) denotes a position \(X\)-eigenstate of the quantum particle localized on the \(n\)-th atomic lattice point, with probability \(E_{\text{hop}}^2/E_0^2\) of locally tunneling from \(|n\rangle\) to a neighbouring atom \(|n\pm 1\rangle\). Some remarks:
- These \(N\) position \(X\)-eigenstates span the finite \(N\)-dimensional state space \(\mathcal H=\text{span}_{1\leq n\leq N}|n\rangle\).
- The matrix elements of the tight-binding Hamiltonian \(H_{\text{tight-binding}}\) in the \(X\)-eigenbasis \(\{|n\rangle:n=1,…,N\}\) are given by \(\langle m|H_{\text{tight-binding}}|n\rangle=E_0\delta_{nm}-E_{\text{hop}}\delta_{n,m+1}-E_{\text{hop}}\delta_{n,m-1}\) or one can combine \(\delta_{n,m+1}+\delta_{n,m-1}=\delta_{n,m\pm 1}\) into a single Kronecker delta. Such an \(N\times N\) Hermitian matrix \([H_{\text{tight-binding}}]_{(|n\rangle:n=1,…,N)}^{(|n\rangle:n=1,…,N)}\in i\frak u\)\((N)\) has \(E_0\) on the diagonal and \(-E_{\text{hop}}\) on the subdiagonal and superdiagonal, in addition to a \(-E_{\text{hop}}\) at the bottom-left and top-right corners due to the periodicity \(|N+1\rangle\cong |1\rangle\) of \(\textbf Z/N\textbf Z\). This reinforces the local tunneling interpretation earlier.
- \(E_0\) is the “on-site energy” associated with being in a particular atomic wave while \(E_{\text{hop}}\) is like a “delocalization energy” for reasons that will become clear.
An illustrative case is that of \(N=5\) atoms, with a single electron \(e^-\) tunneling around this regular pentagon lattice such as one might find in a cyclopentane ring:
In this case, the \(5\times 5\) matrix of the tight-binding Hamiltonian looks like:
\[[H_{\text{tight-binding}}]_{(|1\rangle, |2\rangle, |3\rangle, |4\rangle, |5\rangle)}^{(|1\rangle, |2\rangle, |3\rangle, |4\rangle, |5\rangle)}=\begin{pmatrix}E_0&-E_{\text{hop}}&0&0&-E_{\text{hop}}\\-E_{\text{hop}}&E_0&-E_{\text{hop}}&0&0\\0&-E_{\text{hop}}&E_0&-E_{\text{hop}}&0\\0&0&-E_{\text{hop}}&E_0&-E_{\text{hop}}\\-E_{\text{hop}}&0&0&-E_{\text{hop}}&E_0\end{pmatrix}\]
To find the allowed energies \(E\) of the quantum particle (in this case the electron \(e^-\)) tunneling around on this pentagonal lattice, one could play the standard game of finite-dimensional linear algebra which is to solve \(\det([H_{\text{tight-binding}}]_{(|1\rangle, |2\rangle, |3\rangle, |4\rangle, |5\rangle)}^{(|1\rangle, |2\rangle, |3\rangle, |4\rangle, |5\rangle)}-E1)=0\) for the roots \(E\) of the characteristic polynomial of \([H_{\text{tight-binding}}]_{(|1\rangle, |2\rangle, |3\rangle, |4\rangle, |5\rangle)}^{(|1\rangle, |2\rangle, |3\rangle, |4\rangle, |5\rangle)}\), and then go back and find the corresponding \(H_{\text{tight-binding}}\)-eigenstates. Indeed, this is a few lines of Python code using the SymPy library:
init_printing(use_unicode=True)
E_0, E_hop = symbols("E_0 E_hop")
H_site = E_0 * eye(5)
H_tunneling = -E_hop * Matrix([[0, 1, 0, 0, 1], [1, 0, 1, 0, 0], [0, 1, 0, 1, 0], [0, 0, 1, 0, 1], [1, 0, 0, 1, 0]])
H = H_site + H_tunneling
H.eigenvects()which says that there are \(3\) energies, a singlet ground state at \(E=E_0-2E_{\text{hop}}\), a degenerate doublet at \(E=E_0-E_{\text{hop}}/\varphi\) and a highest-energy degenerate doublet at \(E=E_0+\varphi E_{\text{hop}}\) where \(\varphi=(1+\sqrt{5})/2\) is the golden ratio:
However, to handle the more general case, it is helpful to exploit the translational symmetry of the lattice group \(\textbf Z/N\textbf Z\cong C_N\). The way this manifests is that the tight-binding Hamiltonian matrix \([H_{\text{tight-binding}}]_{(|1\rangle, |2\rangle, |3\rangle, |4\rangle, |5\rangle)}^{(|1\rangle, |2\rangle, |3\rangle, |4\rangle, |5\rangle)}\) is a particular kind of Toeplitz matrix known as a circulant matrix, i.e. the rows and columns look like successive cyclic shifts of the same “circulant vector” \(\begin{pmatrix} -E_{\text{hop}}\\E_0\\-E_{\text{hop}}\\0\\0\end{pmatrix}\).
There is a fundamental universality theorem that everyone ought to know about circulant matrices (which is really not all that surprising), namely that they are diagonalized by the discrete Fourier transform (DFT). This is because any \(N\times N\) circulant matrix \(C\) defined by a circulant vector \(\textbf c\in\textbf C^N\) (the first column of \(C\)) acting on a vector \(\textbf x\in \textbf C^N\) produces the vector \(C\textbf x\in\textbf C^N\) which is nothing more than the discrete circular convolution \(C\textbf x=\textbf c*\textbf x=\textbf x*\textbf c\) of the circulant vector \(\textbf c\) with the input vector \(\textbf x\). So by the convolution theorem, \(F_NC\textbf x=F_N\textbf c\odot F_N\textbf x\) where \(F_N\in U(N)\) is the \(N\times N\) unitary DFT matrix of \(N\)-th roots of unity with components \((F_N)_{nm}=\omega_{N}^{-nm}/\sqrt{N}\) for \(0\leq n,m\leq N-1\) where \(\omega_N:=e^{2\pi i/N}\) is the \(N\)-th primitive root of unity. In particular, one can write the convolution theorem in the more suggestive form \(F_NCF_N^{-1}\textbf x=F_N\textbf c\odot \textbf x\). In particular, one can check that this means \(F_N\) (independent of the circulant matrix \(C\)!) diagonalizes \(C\) with the resultant diagonal matrix of eigenvalues \(F_NCF_N^{-1}=\text{diag}(F_N\textbf c)\) dependent on \(C\). In other words, the \(N\) columns of \(F_N\) are the eigenvectors of any \(N\times N\) circulant matrix, and are called the \(N\)-th Fourier modes.
Thus, all this discussion goes against the usual finite-dimensional linear algebra routine since we’ve written down the eigenvectors first before even finding their eigenvalues, but of course the symmetry here warrants it. Note that there is nothing inherently quantum-mechanical about any of this; for instance, if one has \(N\) classical masses \(m\) connected by identical springs \(k\) all sliding around \(S^1\), then the equation of motion for the state of such a system is also expressible via circulant matrices because each mass interacts only with its nearest neighbors.
Back to the physics then, we can immediately write down the normalized \(H_{\text{tight-binding}}\)-eigenstates for a \(5\)-atom pentagonal lattice. For instance, one of them is simply a maximally delocalized LCAO of all \(5\) \(X\)-eigenstates \(\frac{1}{\sqrt{5}}(|1\rangle+|2\rangle+|3\rangle+|4\rangle+|5\rangle)\). The others are the remaining columns of the \(5\times 5\) DFT matrix \(F_5\):
import sympy as sp
import numpy as np
init_printing(use_unicode=True)
DFT_matrix = Matrix(np.empty((5, 5), dtype=complex))
for n in np.arange(0, 5):
for m in np.arange(0, 5):
DFT_matrix[n, m] = nsimplify(sp.expand_complex(sp.exp(-2j * pi * n * m / 5)/sp.sqrt(5)))
DFT_matrix
Although keep in mind that all of these complex numbers have modulus \(1/\sqrt{5}\), so in some sense all of these are all fully delocalized around the ring.
The more interesting quantity one would like to compute are the energies \(E\). These were already calculated above using SymPy but of course if one already knows the eigenvectors then the energies are trivial to obtain. For instance, acting \([H_{\text{tight-binding}}]_{(|1\rangle, |2\rangle, |3\rangle, |4\rangle, |5\rangle)}^{(|1\rangle, |2\rangle, |3\rangle, |4\rangle, |5\rangle)}\) on \((1,1,1,1,1)^T\) (doesn’t even have to be normalized!) gives \(E=E_0-2E_{\text{hop}}\) so that \(\frac{1}{\sqrt{5}}(|1\rangle+|2\rangle+|3\rangle+|4\rangle+|5\rangle)\) is the ground state of the system. Proceeding along the columns of the DFT matrix \(F_5\), the next energy lies at \(E=E_0-e^{-2\pi i/5}E_{\text{hop}}-e^{-8\pi i/5}E_{\text{hop}}=E_0-E_{\text{hop}}/\varphi\), the next two both have maximum energy \(E=E_0+\varphi E_{\text{hop}}\), and finally the last one also has energy \(E=E_0-E_{\text{hop}}/\varphi\).
So this is all fine for \(N=5\), and has hopefully provided intuition for what’s going on. Generalizing, one finds that the \(n\)-th Fourier mode eigenvector \(\boldsymbol{\psi}_n=(1, e^{2\pi i n/N},…,e^{2\pi i n(N-1)/N})^T\) for \(n=0,1,…,N-1\) of the tight-binding Hamiltonian matrix \([H_{\text{tight-binding}}]_{(|n\rangle:n=1,…,N)}^{(|n\rangle:n=1,…,N)}\) has energy \(E_n=E_0-(e^{2\pi i n/N}+e^{2\pi i n(N-1)/N})E_{\text{hop}}\). But \(e^{2\pi i n(N-1)/N}=e^{-2\pi i n/N}\) so overall we get a real energy as required by Hermiticity of \(H_{\text{tight-binding}}\):
\[E_n=E_0-2E_{\text{hop}}\cos\left(\frac{2\pi n}{N}\right)\]
The \(\lfloor N/2\rfloor +1\) degeneracy \(E_n=E_{-n}\) of the energies is thus manifest via the even nature of the cosine function.
Finally, define the angular wavenumber \(k_n\) of the \(n\)-th Fourier mode \(\boldsymbol{\psi}_n\) by requiring \(k_n a:=\frac{2\pi n}{N}\) where \(a\) is the separation between neighbouring atoms. Then of course one can also write \(E_n=E_0-2E_{\text{hop}}\cos(k_n a)\). If we think of \(n\) as going from \(n\approx -N/2,…,N/2\), then the wavenumber \(k_n\) lies in the compact interval \(k_n\in[-\pi/a,\pi/a]\), called the Brillouin zone of the system. The dispersion relation for the \(N\)-atom ring is then given by:
This mimics the usual free-space dispersion relation \(E(k)=\frac{\hbar^2k^2}{2m}\) near \(k=0\) where the cosine is approximately parabolic. Indeed, performing a Taylor expansion, then for a slow-moving electron \(k\to 0\) it looks like a free particle with effective mass \(m_{\text{eff}}=\hbar^2/2a^2E_{\text{hop}}\) which has nothing to do with the actual mass \(m\) of the electron, being inherited from properties \(E_{\text{hop}}\) and \(a\) of the lattice.