Topological Quantum Numbers

A topological quantum number is a global label that is stable under continuous deformations of the Hamiltonian, state family, or background fields, provided the assumptions defining the label remain valid. Unlike an ordinary quantum number such as $m$ or parity, it usually does not arise as the eigenvalue of a simple local observable. It labels a deformation class.

The phrase is a physics shorthand rather than a single mathematical definition. In this chapter it means labels such as:

• winding numbers of phases or maps;
• Berry phases defined modulo $2\pi$, especially when symmetry quantizes them;
• flux periods such as the Aharonov–Bohm flux quantum;
• Chern numbers of isolated states or bands.

The mathematical canonical entry is Topological Invariants. This page explains how those invariants function as quantum labels.

Ordinary and Topological Labels

An ordinary good quantum number usually comes from a compatible conserved observable:

$$[H,Q]=0, \qquad Q\lvert \psi\rangle=q\lvert \psi\rangle.$$

The eigenvalue $q$ is useful because time evolution preserves the eigenspaces of $Q$. The canonical page is Simultaneous Eigenstates and Good Quantum Numbers.

A topological quantum number is different. It is usually attached to a family of states, a bundle of eigenvectors, a map into a phase space, or a background gauge configuration. The typical statement is not

$$Q\lvert \psi\rangle=q\lvert \psi\rangle,$$

but rather:

$$\text{smooth deformation} \quad\Longrightarrow\quad \text{same integer or same phase class}.$$

For example, the Chern number of a filled band is not measured by diagonalizing a single-particle operator at one momentum. It is computed from the Berry curvature over the entire Brillouin-zone torus and remains fixed as long as the occupied subspace stays separated from the unoccupied subspace.

What Must Stay Fixed

Topological protection is conditional. Before calling a label topological, state the data that are being deformed and the constraints that are not allowed to fail.

Common constraints include:

• a spectral gap between the states being labeled and the states being excluded;
• a closed parameter space such as $S^1$, $S^2$, or $T^2$;
• smoothness of the family away from specified singularities;
• a symmetry such as inversion, chiral symmetry, or time reversal, when the label relies on that symmetry;
• fixed boundary conditions or fixed flux periodicity.

An integer invariant can change only when the problem leaves its defining class. In a band problem, that usually means a gap closing. In a winding problem, it may mean that the map becomes undefined somewhere. In a Berry-phase problem, it may mean that the loop crosses a degeneracy or that the symmetry quantizing the phase is broken.

Winding Number

The simplest topological quantum number is the winding number of a phase around a circle. Let

$$u(\lambda)=e^{i\theta(\lambda)}, \qquad \lambda\in S^1.$$

The winding number is

$$w = \frac{1}{2\pi} \oint_{S^1} d\theta = \frac{1}{2\pi i} \oint_{S^1}u^{-1}du \in\mathbb Z.$$

It counts how many times the phase wraps around $U(1)$ as the parameter goes around the loop. The local value of $\theta$ depends on a branch choice, but the total winding is invariant under smooth deformations that keep $u(\lambda)$ nonzero and single-valued.

This is the prototype for many one-dimensional topological labels. In the Zak phase preview, a Bloch eigenstate is transported around the Brillouin-zone circle. In the SSH model, a winding description appears after choosing a symmetry-compatible representation of the two-band Hamiltonian.

Berry Phase Modulo 2π

For a closed loop $C$ in parameter space, the Berry phase is the holonomy

$$e^{i\gamma[C]} = \exp\left(i\oint_C A\right), \qquad \gamma[C]\sim \gamma[C]+2\pi.$$

The phase representative $\gamma[C]$ is not an ordinary real-valued observable. The gauge-invariant object is the phase factor $e^{i\gamma[C]}$, or equivalently $\gamma$ modulo $2\pi$.

Not every Berry phase is a topological quantum number. In a generic smooth family, $\gamma[C]$ can vary continuously when the loop changes shape. It becomes topological in stronger situations, for example when:

• the loop cannot be shrunk without crossing a degeneracy or excluded region;
• a symmetry restricts the Berry phase to discrete values such as $0$ or $\pi$ modulo $2\pi$;
• the Berry phase is the boundary expression of an integer invariant over a closed surface.

This distinction is important. The Berry phase is geometric in general; it is topological only after extra global or symmetry assumptions are added.

Flux Quantum and Aharonov–Bohm Periodicity

For a particle of charge $q$ moving around a thin solenoid, the Aharonov–Bohm phase is

$$\Delta\varphi = \frac{q}{\hbar}\Phi,$$

where $\Phi$ is the enclosed magnetic flux. Fluxes differing by

$$\Phi_0 = \frac{h}{q}$$

give the same phase factor for that charge:

$$\exp\left(i\frac{q}{\hbar}(\Phi+\Phi_0)\right) = \exp\left(i\frac{q}{\hbar}\Phi\right).$$

Thus the physically relevant flux label is periodic. In superconducting contexts the relevant charge is usually $2e$, giving the familiar superconducting flux quantum $h/(2e)$. The exact charge and boundary conditions must always be specified.

The topological aspect is not that the magnetic field is large or small. It is that a loop around an excluded flux tube cannot be contracted without crossing the excluded region. The detailed vector potential can be changed by gauge transformations, but the holonomy around the noncontractible loop remains observable. See Berry Phase in the Aharonov–Bohm Effect and Aharonov–Bohm Effect.

Chern Number

A Chern number is a topological quantum number for a $U(1)$ eigenstate line bundle, or more generally for an isolated vector bundle of occupied states. For a nondegenerate state over a closed oriented two-dimensional parameter space $M$,

$$C = \frac{1}{2\pi} \int_M F \in\mathbb Z,$$

where $F$ is the Berry-curvature two-form.

The integer character is global. Locally one can write $F=dA$, and the curvature density can move around under deformations of the Hamiltonian. The total integral over $M$ cannot change continuously. It changes only when the eigenstate bundle stops being defined as a smooth isolated bundle, typically because a degeneracy or band closing occurs.

In a two-dimensional band insulator, the occupied Chern number controls the integer Hall conductance in the simplest clean noninteracting setting:

$$\sigma_{xy} = \frac{e^2}{h}C_{\mathrm{occ}}.$$

The detailed physics is developed in Chern Numbers and Quantum Hall Geometry Preview.

Good Quantum Number or Topological Quantum Number?

The distinction is practical:

Label	Usual source	Stability criterion
$m$ in a central potential	eigenvalue of $L_z$	rotational or axial symmetry preserving $[H,L_z]=0$
parity	eigenvalue of a unitary symmetry	Hamiltonian invariant under inversion
spin representation $j$	irreducible representation label	symmetry algebra and representation content
winding number $w$	homotopy class of a map	map remains continuous and nonzero where required
Berry phase modulo $2\pi$	holonomy around a loop	loop, gap, and gauge bundle remain well defined
Chern number $C$	curvature flux over a closed space	isolated bundle remains smooth over the closed space

The first three are symmetry or operator labels. The last three are global labels. Both kinds can coexist. A topological band may also have spin, crystal momentum, parity, or angular-momentum labels in high-symmetry settings.

Robust Does Not Mean Immune to Everything

Topological labels are robust against perturbations inside the allowed class. That phrase carries several caveats.

First, the invariant may require a gap. If a perturbation closes the relevant gap, the topological label can change. In band language, this is the standard route through a topological phase transition.

Second, the invariant may require a symmetry. A one-dimensional Berry phase quantized to $0$ or $\pi$ by inversion or chiral symmetry need not remain quantized after that symmetry is broken.

Third, robustness is not a statement about every observable. Edge spectra, finite-size splittings, local density profiles, and response functions can change substantially while the topological label stays fixed.

Fourth, continuous geometric quantities are not automatically topological. Berry curvature $\Omega(\mathbf k)$ can be redistributed over the Brillouin zone even when its integral remains fixed.

Common Mistakes

• Calling any Berry phase topological. A generic Berry phase is geometric and path-dependent.
• Forgetting the modulo $2\pi$ nature of phase labels.
• Treating a Chern number as a local property at one point in parameter space.
• Saying an invariant is protected without stating the gap, symmetry, or boundary-condition assumptions.
• Confusing an approximate plateau or near-integer numerical value with a mathematically quantized invariant.
• Assuming topology fixes microscopic wavefunctions. It fixes global deformation classes, not all local details.

Exercises

1. A family of phases is given by $u(\lambda)=e^{in\lambda}$ for $0\le\lambda<2\pi$, with integer $n$. Compute its winding number.

Solution

Here $\theta(\lambda)=n\lambda$, so

$$w = \frac{1}{2\pi} \int_0^{2\pi} n\,d\lambda = n.$$

The integer cannot be changed by a smooth deformation of $u:S^1\to U(1)$ unless the map stops being well-defined or single-valued.

2. For a charged particle encircling a thin solenoid, show that adding one flux quantum $\Phi_0=h/q$ does not change the Aharonov–Bohm phase factor.

Solution

The phase factor is

$$\exp\left(i\frac{q\Phi}{\hbar}\right).$$

After $\Phi\mapsto\Phi+h/q$,

$$\exp\left(i\frac{q}{\hbar}\left(\Phi+\frac{h}{q}\right)\right) = \exp\left(i\frac{q\Phi}{\hbar}\right) \exp\left(i\frac{h}{\hbar}\right) = \exp\left(i\frac{q\Phi}{\hbar}\right) e^{i2\pi}.$$

Since $e^{i2\pi}=1$, the phase factor is unchanged.

3. Explain why a Chern number of an isolated band cannot change under a small smooth perturbation that keeps the band gap open.

Solution

The perturbation smoothly deforms the eigenstate bundle over the same closed parameter space. The Chern number is an integer-valued continuous invariant of that bundle. A continuous deformation cannot change an integer unless the assumptions defining the bundle fail. In a band problem, the failure is usually a gap closing that prevents the occupied band or occupied subspace from being isolated everywhere.

4. Give an example of a label that is a good quantum number but not topological, and an example of a label that is topological but not an eigenvalue of a local observable.

Solution

In a central potential, the magnetic quantum number $m$ is a good quantum number because $[H,L_z]=0$, but it is not a topological invariant. It follows from rotational symmetry and operator compatibility.

A Chern number of an isolated Bloch band is topological but is not the eigenvalue of a local operator at one momentum. It is obtained by integrating Berry curvature over the full Brillouin-zone torus.

References

• M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proceedings of the Royal Society A 392, 45-57, 1984.
• B. Simon, “Holonomy, the quantum adiabatic theorem, and Berry’s phase,” Physical Review Letters 51, 2167-2170, 1983.
• D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Physical Review Letters 49, 405-408, 1982.
• D. Xiao, M.-C. Chang, and Q. Niu, “Berry phase effects on electronic properties,” Reviews of Modern Physics 82, 1959-2007, 2010.
• M. Nakahara, Geometry, Topology and Physics, 2nd ed., Institute of Physics Publishing, 2003.
• J. K. Asbóth, L. Oroszlány, and A. Pályi, A Short Course on Topological Insulators, Springer, 2016.