Berry Phase Problems

These solved problems practice geometric phase calculations: line integrals of Berry connections, gauge changes, spin-$1/2$ solid angles, Aharonov–Bohm holonomy, Berry-curvature flux, and Chern-number normalization.

Use Berry Phase for the physical setup, Berry Connection for gauge conventions, Berry Curvature for flux language, and Berry Phase for Spin-1/2 for the spin convention used below. The Aharonov–Bohm examples use Aharonov–Bohm Effect.

Conventions

For a nondegenerate instantaneous eigenstate,

$$H(R)\lvert n(R)\rangle = E_n(R)\lvert n(R)\rangle,$$

the Berry connection is

$$A_n = i\langle n|dn\rangle.$$

For a closed loop $C$,

$$\gamma_n[C] = \oint_C A_n.$$

Under a phase convention change

$$\lvert n(R)\rangle \mapsto e^{i\chi(R)} \lvert n(R)\rangle,$$

the connection transforms as

$$A_n\mapsto A_n-d\chi.$$

The curvature is

$$F_n=dA_n.$$

For the spin-$1/2$ Hamiltonian

$$H(\hat{\mathbf n}) = -\frac{\Delta}{2} \hat{\mathbf n}\cdot\boldsymbol\sigma, \qquad \Delta>0,$$

the lower state has

$$\gamma_+[C] = -\frac{\Omega(C)}{2},$$

where $\Omega(C)$ is the oriented solid angle enclosed by the loop of $\hat{\mathbf n}$ on the unit sphere.

For an electromagnetic Aharonov–Bohm loop,

$$\Delta\varphi_{\mathrm{AB}} = \frac{q}{\hbar} \oint_C\mathbf A\cdot d\mathbf r = \frac{q\Phi_B}{\hbar} \quad \operatorname{mod}2\pi.$$

For a closed oriented two-dimensional parameter space,

$$C_n = \frac{1}{2\pi} \int_M F_n.$$

Skill Map

Problem group	Skills	Preparation
Gauge changes	connection transformation, closed-loop phase	Berry Connection
Spin-$1/2$ loops	solid angle, dynamical versus geometric phase	Berry Phase for Spin-1/2
Curvature flux	exterior derivative, Stokes’ theorem, monopole analogy	Berry Curvature
Aharonov–Bohm holonomy	winding number, flux phase, gauge invariance	Aharonov–Bohm Effect
Chern numbers	normalized curvature flux, gap protection	Chern Numbers

Problems

1. Gauge Shift Around a Circle

On a parameter circle with coordinate $\phi\in[0,2\pi)$, suppose a Berry connection is

$$A=a\,d\phi,$$

where $a$ is constant. Compute the Berry phase around the circle. Then apply the gauge change $\chi(\phi)=N\phi$ with integer $N$ and check what happens to the phase factor.

Solution

The Berry phase is

$$\gamma = \oint A = \int_0^{2\pi}a\,d\phi = 2\pi a.$$

Under

$$A\mapsto A-d\chi,$$

and

$$d\chi=N\,d\phi,$$

the connection becomes

$$A'=(a-N)d\phi.$$

The transformed loop integral is

$$\gamma' = \int_0^{2\pi}(a-N)\,d\phi = 2\pi a-2\pi N.$$

The phase factor is unchanged:

$$e^{i\gamma'} = e^{i(2\pi a-2\pi N)} = e^{i2\pi a}.$$

The integer condition matters because $e^{i\chi(\phi)}=e^{iN\phi}$ is single-valued on the circle only when $N$ is an integer.

2. Constant-Polar-Angle Spin Loop

For the spin convention in this volume, the lower state has Berry connection

$$A_+ = -\frac{1-\cos\theta}{2}\,d\phi.$$

Compute the Berry phase for a loop at fixed $\theta=\theta_0$ with $\phi:0\to2\pi$. Evaluate the result for the north pole, the equator, and reversed orientation.

Solution

At fixed $\theta_0$,

$$\gamma_+ = \int_0^{2\pi} -\frac{1-\cos\theta_0}{2}\,d\phi.$$

Therefore

$$\gamma_+ = -\pi(1-\cos\theta_0).$$

The solid angle of the cap is

$$\Omega = 2\pi(1-\cos\theta_0),$$

so

$$\gamma_+ = -\frac{\Omega}{2}.$$

At the north pole, $\theta_0=0$, so

$$\gamma_+=0.$$

At the equator, $\theta_0=\pi/2$, so

$$\gamma_+=-\pi.$$

If the loop orientation is reversed, the line integral changes sign:

$$\gamma_+\mapsto-\gamma_+.$$

Only the phase factor $e^{i\gamma_+}$ is physically invariant modulo $2\pi$.

3. Dynamical Phase Versus Berry Phase

For

$$H(\hat{\mathbf n}) = -\frac{\Delta}{2}\hat{\mathbf n}\cdot\boldsymbol\sigma,$$

the system remains in the lower instantaneous eigenstate while $\hat{\mathbf n}$ traces a closed loop of solid angle $\Omega$ in time $T$. Assume $\Delta$ is constant. Find the total phase in the adiabatic approximation.

Solution

The lower-state energy is

$$E_+ = -\frac{\Delta}{2}.$$

The dynamical phase is

$$-\frac{1}{\hbar} \int_0^T E_+\,dt = -\frac{1}{\hbar} \left( -\frac{\Delta}{2} \right)T = \frac{\Delta T}{2\hbar}.$$

The Berry phase for the chosen convention is

$$\gamma_+ = -\frac{\Omega}{2}.$$

Thus the total phase is

$$\alpha_+ = \frac{\Delta T}{2\hbar} - \frac{\Omega}{2} \quad \operatorname{mod}2\pi.$$

Changing the traversal time changes the first term. Keeping the same oriented loop in the adiabatic limit keeps the second term fixed.

4. Curvature from the Spin Connection

Starting from

$$A_+ = -\frac{1-\cos\theta}{2}\,d\phi,$$

compute $F_+=dA_+$. Then integrate it over a spherical cap $0\le\theta\le\theta_0$, $0\le\phi<2\pi$.

Solution

Differentiate the connection:

$$F_+ = dA_+ = -\frac12\,d(1-\cos\theta)\wedge d\phi.$$

Since

$$d(1-\cos\theta) = \sin\theta\,d\theta,$$

the curvature is

$$F_+ = -\frac12 \sin\theta\, d\theta\wedge d\phi.$$

Integrating over the cap gives

$$\begin{aligned} \int_{\Sigma}F_+ &= -\frac12 \int_0^{2\pi} \int_0^{\theta_0} \sin\theta\,d\theta\,d\phi \\ &= -\frac12 (2\pi) (1-\cos\theta_0) \\ &= -\pi(1-\cos\theta_0). \end{aligned}$$

This equals the line-integral result for the boundary circle. The cap solid angle is $\Omega=2\pi(1-\cos\theta_0)$, so the flux is $-\Omega/2$.

5. Aharonov–Bohm Phase and Flux Period

A charged particle with charge $q$ winds once counterclockwise around an inaccessible magnetic flux $\Phi_B$. Find the Aharonov–Bohm phase. Then specialize to an electron with $q=-e$ and identify the flux period.

Solution

For one winding with the chosen orientation,

$$\oint_C\mathbf A\cdot d\mathbf r = \Phi_B.$$

Thus

$$\Delta\varphi_{\mathrm{AB}} = \frac{q\Phi_B}{\hbar} \quad \operatorname{mod}2\pi.$$

For an electron, $q=-e$, so

$$\Delta\varphi_{\mathrm{AB}} = -\frac{e\Phi_B}{\hbar}.$$

Using

$$\Phi_0=\frac{h}{e} = \frac{2\pi\hbar}{e},$$

this becomes

$$\Delta\varphi_{\mathrm{AB}} = -2\pi \frac{\Phi_B}{\Phi_0} \quad \operatorname{mod}2\pi.$$

The phase factor is periodic under

$$\Phi_B\mapsto\Phi_B+\Phi_0.$$

The sign depends on charge and loop orientation. The periodicity does not.

6. Flat Locally, Nontrivial Globally

Outside an ideal thin flux tube, take

$$\mathbf A = \frac{\Phi_B}{2\pi r} \hat{\boldsymbol\theta}$$

on the punctured plane $r>0$. Show that the field is zero locally outside the tube but that a circle around the origin has nonzero holonomy.

Solution

For $r>0$, this vector potential has zero curl:

$$\nabla\times\mathbf A=\mathbf 0.$$

The magnetic field is confined to the excluded flux tube. The accessible region is not simply connected, so zero local curl does not force every loop integral to vanish.

For a counterclockwise circle of radius $r$,

$$d\mathbf r = r\,d\theta\,\hat{\boldsymbol\theta}.$$

Therefore

$$\begin{aligned} \oint_C \mathbf A\cdot d\mathbf r &= \int_0^{2\pi} \frac{\Phi_B}{2\pi r} \left( r\,d\theta \right) \\ &= \Phi_B. \end{aligned}$$

For winding number $w$,

$$\oint_C \mathbf A\cdot d\mathbf r = w\Phi_B.$$

The Aharonov–Bohm phase is therefore sensitive to winding around the excluded region even though the local magnetic field vanishes along the path.

7. Chern Number of the Spin Sphere

For the lower spin state in the convention above,

$$F_+ = -\frac12 \sin\theta\,d\theta\wedge d\phi.$$

Compute the Chern number over the full parameter sphere with the standard orientation.

Solution

Integrate over the full sphere:

$$\begin{aligned} \int_{S^2}F_+ &= -\frac12 \int_0^{2\pi} \int_0^\pi \sin\theta\,d\theta\,d\phi \\ &= -\frac12 (2\pi)(2) \\ &= -2\pi. \end{aligned}$$

The Chern number is

$$C_+ = \frac{1}{2\pi} \int_{S^2}F_+ = -1.$$

The opposite eigenstate has the opposite Chern number in the matching convention. Reversing the orientation of the sphere also reverses the sign.

8. Toy Monopole Charge

Suppose a line bundle over a parameter sphere has curvature

$$F_N = \frac{N}{2} \sin\theta\,d\theta\wedge d\phi,$$

where $N$ is an integer. Compute its Chern number. Why should an arbitrary real coefficient not be accepted as a globally valid quantum line bundle without further checks?

Solution

The curvature flux is

$$\begin{aligned} \int_{S^2}F_N &= \frac{N}{2} \int_0^{2\pi} \int_0^\pi \sin\theta\,d\theta\,d\phi \\ &= \frac{N}{2} (2\pi)(2) \\ &= 2\pi N. \end{aligned}$$

Therefore

$$C_N = \frac{1}{2\pi} \int_{S^2}F_N = N.$$

The integrality is not a decorative convention. For a globally well-defined $U(1)$ quantum line bundle over a closed surface, transition functions between patches must be single-valued, and their winding numbers are integers. A random real coefficient would give a noninteger normalized flux, signaling that the assumed curvature cannot be the curvature of an ordinary globally defined eigenstate line bundle over the closed sphere without changing the setup.

9. Can the Chern Number Change Smoothly?

A Hamiltonian family over a closed two-dimensional parameter space has an isolated nondegenerate band with Chern number $C=1$. The Hamiltonian is smoothly deformed. Under what condition must the Chern number remain fixed, and what must happen for it to change?

Solution

The Chern number is integer-valued and stable under smooth deformations as long as the relevant eigenstate or band remains isolated over the entire parameter space. In band language, the spectral gap separating the band or occupied subspace from the rest of the spectrum must stay open everywhere.

If the gap never closes, $C$ cannot drift continuously from $1$ to another integer. A change requires a singular event for the bundle being tracked, typically a degeneracy or gap closing:

$$E_n(R,\lambda) = E_m(R,\lambda)$$

at some parameter value. At that point the isolated-band Berry curvature for that band is not defined, and after the gap reopens curvature flux can be redistributed between bands.

Common Mistakes

• Calling the dynamical phase part of the Berry phase.
• Treating the Berry connection as gauge invariant instead of its closed-loop holonomy or curvature.
• Forgetting that spin-$1/2$ solid-angle signs depend on the Hamiltonian and eigenstate convention.
• Applying Stokes’ theorem across a surface where no smooth gauge exists without patching.
• Confusing Aharonov–Bohm holonomy in real configuration space with Berry phase in Hamiltonian parameter space.
• Calling any curvature integral a Chern number without checking that the parameter space is closed and the eigenstate remains isolated.
• Assuming a Chern number can change without a gap closing or other singular loss of the isolated bundle.

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.
• Y. Aharonov and D. Bohm, “Significance of electromagnetic potentials in quantum theory,” Physical Review 115, 485–491, 1959.
• A. Shapere and F. Wilczek, eds., Geometric Phases in Physics, World Scientific, 1989.
• D. Xiao, M.-C. Chang, and Q. Niu, “Berry phase effects on electronic properties,” Reviews of Modern Physics 82, 1959–2007, 2010.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.