Math Needed for Quantum Matter

Quantum matter uses a few mathematical ideas repeatedly: Fourier modes describe periodic structures, matrices describe bands and spin interactions, and geometry describes phases of parameter-dependent eigenstates. Choose a physical question first, then review the tools that make its calculation understandable.

Periodic structures and crystal momentum

For a single-particle Hamiltonian with lattice translation symmetry, Bloch states take the form

$$\psi_{n\mathbf k}(\mathbf r)=e^{i\mathbf k\cdot\mathbf r}u_{n\mathbf k}(\mathbf r), \qquad u_{n\mathbf k}(\mathbf r+\mathbf R)=u_{n\mathbf k}(\mathbf r),$$

where $\mathbf R$ is a lattice vector. The wavevector $\mathbf k$ labels translation eigenvalues and is defined modulo a reciprocal-lattice vector. It is not generally the mechanical momentum of an electron.

Begin with Fourier Series and Group Actions to understand periodic functions and translations. Then read Reciprocal Lattice and Bloch’s Theorem. Disorder, boundaries, and interactions require additional care; the single-particle Bloch formula is not a universal form for every many-body wavefunction.

Choose the tools for your topic

Physical question	Mathematical preparation	Next physical page
How does hopping produce a band?	Matrix Diagonalization and Fourier modes	Tight-Binding Models
How many states lie in an energy interval?	Delta Function and integration	Density of States
How do atoms vibrate collectively?	Normal-mode diagonalization and the harmonic oscillator	Phonons
How does matter respond to an applied field?	Fourier Transform, linear differential equations, and Green Functions	Transport and Response
How do interacting particles form collective states?	Tensor Products and Math Needed for Many-Body QM	Many-Particle Hamiltonians
What does band topology measure?	Berry Connection and Chern Numbers	Chern Numbers in Band Theory
How can a finite model be computed reliably?	Sparse Eigensolvers and Convergence Tests	Computational Quantum Matter

A small model shows the algebra

For a chain of spinless fermionic modes with real nearest-neighbor hopping $t$ and real onsite energies $\varepsilon_j$, an open-chain Hamiltonian is

$$H=-t\sum_{j=1}^{L-1}\left(c_j^\dagger c_{j+1}+c_{j+1}^\dagger c_j\right) +\sum_{j=1}^L\varepsilon_j c_j^\dagger c_j.$$

The two hopping terms are Hermitian conjugates. In the one-particle sector the Hamiltonian is an $L\times L$ tridiagonal matrix; in the full fermionic Fock space there are $2^L$ basis states. These are different spaces describing different questions, even though the operator notation looks identical. Occupation-Number Basis explains the distinction.

Use a two-site chain first. You can inspect the matrix, diagonalize it exactly, and check a numerical routine before increasing the size. For periodic boundaries, add the link between sites $L$ and $1$ with the same explicit Hermiticity convention.

When geometry becomes necessary

For a normalized local eigenvector of an isolated band, the Berry connection is

$$\mathcal A_i(\mathbf k)=i\langle u_{n\mathbf k}|\partial_{k_i}u_{n\mathbf k}\rangle.$$

In two dimensions, write the curvature component as $\Omega_{xy}=\partial_{k_x}\mathcal A_y-\partial_{k_y}\mathcal A_x$. With orientation $dk_x\wedge dk_y$, the Chern number is

$$C_n=\frac{1}{2\pi}\int_{\mathrm{BZ}}\Omega_{xy}(\mathbf k)\,dk_x\,dk_y.$$

A phase change of the eigenvector changes $\mathcal A$, while leaving the curvature invariant. A nonzero Chern number prevents choosing a single smooth periodic eigenvector across the entire Brillouin torus. If occupied bands cross each other, use the entire occupied projector provided that subspace remains separated from unoccupied bands. Do not assign an isolated-band formula to a band at a degeneracy.

You can postpone this geometry until you reach Topological Quantum Matter. Fourier analysis and ordinary matrix eigenvectors are enough for a first tight-binding band calculation.

Check your understanding

1. For two sites with $\varepsilon_1=\varepsilon_2=0$, write the one-particle hopping matrix and find its energies for $t>0$.

Solution

In the basis $|10\rangle,|01\rangle$,

$$H=\begin{pmatrix}0&-t\\-t&0\end{pmatrix}.$$

The symmetric state has energy $-t$ and the antisymmetric state has energy $+t$. Including only one directed hopping term would give a non-Hermitian matrix and would fail this basic check.

2. A computed Chern number changes as a parameter varies. Which possibilities should you check before concluding that a topological transition occurred?

Solution

Check whether the relevant band gap closed, whether the same band subspace and orientation were used, and whether the momentum mesh is sufficiently fine. A gauge change alone cannot alter the invariant of a fixed smooth isolated bundle. Numerical error or an inconsistent band choice can mimic a change.

References

• N. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt, Rinehart and Winston, 1976 — reciprocal space and electronic bands.
• B. A. Bernevig and T. L. Hughes, Topological Insulators and Topological Superconductors, Princeton University Press, 2013 — Berry geometry and topological bands.
• S. H. Simon, The Oxford Solid State Basics, Oxford University Press, 2013 — accessible introductions to lattices, bands, and collective modes.