Condensed Matter Applications

In a crystal, symmetry tells us which momenta label states, which terms can appear in a band Hamiltonian, and which degeneracies survive perturbations. Band geometry then describes how eigenvectors change with momentum. These are different kinds of information: two models can have similar energy dispersions but different eigenvectors, Berry curvature, and responses. This page develops the connection for periodic, effectively noninteracting band models; interacting phases require further tools in Quantum Matter.

Helpful background. Crystalline symmetry explains lattice translations, time reversal explains antiunitary constraints, and Berry curvature defines the geometry of an eigenstate family.

Crystal momentum and band labels

For a lattice-periodic Hamiltonian, translations by lattice vectors $\mathbf R$ commute with $H$. With the active convention

$$[T(\mathbf R)\psi](\mathbf r)=\psi(\mathbf r-\mathbf R),$$

a Bloch eigenstate satisfies

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

and has translation eigenvalue $e^{-i\mathbf k\cdot\mathbf R}$. The minus sign is fixed by the stated active convention. Since $e^{-i\mathbf G\cdot\mathbf R}=1$ for every reciprocal vector $\mathbf G$, $\mathbf k$ and $\mathbf k+\mathbf G$ label the same crystal momentum. Crystal momentum is thus a point of the Brillouin-zone torus; it is not generally the mechanical momentum of an electron.

At a fixed $\mathbf k$, a point-group operation is a symmetry within that momentum sector only when it returns $\mathbf k$ modulo a reciprocal vector. These operations form its little group. Its representations constrain degeneracies and mixing; an operation carrying $\mathbf k$ elsewhere relates different sectors instead. See Bloch’s theorem and crystalline symmetry.

Two-level Hamiltonians and spin–orbit terms

An effective two-level Bloch Hamiltonian can be written

$$H(\mathbf k)=d_0(\mathbf k)I+\mathbf d(\mathbf k)\cdot\boldsymbol\sigma, \qquad E_\pm=d_0\pm|\mathbf d|.$$

The Pauli matrices might act on physical spin, sublattice, or orbital labels. The spectrum alone does not specify their meaning, and the symmetry operators must be supplied before constraining $\mathbf d$.

For a genuine spin-$1/2$ pair with time reversal $\Theta=i\sigma_yK$, where $K$ is complex conjugation, $\Theta\boldsymbol\sigma\Theta^{-1}=-\boldsymbol\sigma$. Therefore

$$\Theta H(\mathbf k)\Theta^{-1}=H(-\mathbf k) \quad\Longrightarrow\quad d_0(-\mathbf k)=d_0(\mathbf k),\qquad \mathbf d(-\mathbf k)=-\mathbf d(\mathbf k).$$

This permits a two-dimensional Rashba term

$$H_R=\alpha(k_y\sigma_x-k_x\sigma_y),$$

where $\alpha$ has units of energy times length. It breaks ordinary inversion while preserving time reversal. Its two branches can split away from time-reversal-invariant momenta; their splitting does not itself imply broken time reversal. Kramers partners occur at opposite momenta, and must be degenerate at a momentum equivalent to its negative. This discussion assumes the effective Hamiltonian and its symmetry action remain valid in the region considered.

For a spinless orbital model with $\Theta=K$, the constraints differ: $d_x,d_z$ are even in momentum and $d_y$ is odd. Substituting a spinful time-reversal matrix into a sublattice model without justification gives incorrect conclusions. The angular origin of spin–orbit coupling is developed in spin–orbit coupling; crystal symmetry determines its allowed band terms.

Berry curvature and its symmetry constraints

For an isolated nondegenerate band, use normalized cell-periodic eigenvectors and define

$$\mathbf A_n=i\langle u_{n\mathbf k}|\nabla_{\mathbf k}u_{n\mathbf k}\rangle, \qquad \boldsymbol\Omega_n=\nabla_{\mathbf k}\times\mathbf A_n.$$

A phase change of the eigenvector changes the connection but leaves the curvature invariant. An isolated band invariant under time reversal has odd curvature, $\boldsymbol\Omega_n(\mathbf k)=-\boldsymbol\Omega_n(-\mathbf k)$; inversion gives even curvature. If a symmetry exchanges bands, the relation instead connects the corresponding bands. When both constraints apply to the same nondegenerate band, its curvature vanishes pointwise.

The qualification matters for spinful systems: Kramers degeneracies prevent a globally isolated single-band description of a time-reversal pair. One instead treats the occupied subspace, whose Berry connection is matrix-valued. Its traced curvature is odd under time reversal, and its total first Chern number vanishes. This does not rule out a nontrivial time-reversal-protected $\mathbb Z_2$ insulator.

In two dimensions, an isolated band has integer

$$C_n=\frac{1}{2\pi}\int_{\mathrm{BZ}}\Omega_{n,z}(\mathbf k)\,d^2k.$$

Local curvature can change continuously under a deformation while this integral remains fixed. Changing the Chern number requires loss of the isolated-band condition somewhere along the deformation. For a filled group of bands the appropriate quantity is the Chern number of the occupied projector; crossings within the occupied group need not change it if the gap to unoccupied states stays open. The review by Xiao, Chang, and Niu connects these geometric quantities to electronic response. See Chern numbers and quantum Hall geometry.

An SSH chain makes the allowed deformation explicit

For a one-dimensional chain with two sublattices per unit cell, real intracell hopping $t_1$, and real intercell hopping $t_2$, choose

$$H(k)=(t_1+t_2\cos ka)\sigma_x+t_2\sin ka\,\sigma_y.$$

Here the Pauli matrices act on sublattices, and $a$ is the unit-cell length. The operator $\Gamma=\sigma_z$ anticommutes with $H$: this chiral symmetry confines $\mathbf d(k)$ to a plane. As $k$ traverses the Brillouin zone, $d_x+id_y=t_1+t_2e^{ika}$ traces a circle. For positive hoppings it winds once around the origin if $t_2>t_1$ and zero times if $t_1>t_2$. At $t_1=t_2$ the circle reaches the origin and the band gap closes.

Now add $m\sigma_z$. It breaks this chiral symmetry and gives

$$E_\pm(k)=\pm\sqrt{t_1^2+t_2^2+2t_1t_2\cos ka+m^2}.$$

Keeping $m\ne0$ allows $t_1$ and $t_2$ to pass through equality without closing the gap. Thus the chiral winding distinguishes phases under deformations preserving the specified symmetry, not under every conceivable gapped deformation. The winding convention also uses a chosen unit cell; physical boundary predictions must use a compatible termination. The SSH model develops the edge-state calculation.

Exercises

1. A two-level model uses physical spin and $\Theta=i\sigma_yK$. Which of $m\sigma_z$ and $\alpha k_x\sigma_z$ preserves time reversal? Take real constants $m,\alpha$ and no transformation of external controls.

Solution

Spin reverses under $\Theta$. The constant term maps to $-m\sigma_z$ but has the same value at $-\mathbf k$, so it violates time reversal when $m\ne0$. The second term maps to $-\alpha k_x\sigma_z$, exactly its value at $-\mathbf k$, and preserves time reversal. This tests the symmetry algebra only; other crystal symmetries can still forbid the second term.

2. For positive $t_1,t_2$, find the minimum direct gap of the SSH model with the staggered term $m\sigma_z$. Explain why the two hopping regimes can be connected when chiral symmetry is relaxed.

Solution

The minimum occurs at $ka=\pi$, where $\cos ka=-1$, and the gap is

$$\Delta=2\sqrt{(t_1-t_2)^2+m^2}.$$

It closes only if both $t_1=t_2$ and $m=0$. One can turn on $m$, vary the hoppings through equality, and turn off $m$ after reaching the other regime, keeping the gap open throughout. This path breaks the specified chiral symmetry in its middle.

References

• N. W. Ashcroft and N. D. Mermin, Solid State Physics, Saunders, 1976. See the treatment of Bloch states and crystal symmetry.
• B. A. Bernevig and T. L. Hughes, Topological Insulators and Topological Superconductors, Princeton University Press, 2013.
• M. Z. Hasan and C. L. Kane, “Colloquium: Topological insulators,” Reviews of Modern Physics 82, 3045–3067 (2010). doi:10.1103/RevModPhys.82.3045.
• D. Xiao, M.-C. Chang, and Q. Niu, “Berry phase effects on electronic properties,” Reviews of Modern Physics 82, 1959–2007 (2010). doi:10.1103/RevModPhys.82.1959.