Chapter 32 Exercises: The Adiabatic Theorem and Berry Phase

Part A: Conceptual Questions (*)

These questions test your understanding of the core ideas. No calculations required.

A.1 Explain in your own words why the Berry phase depends on the path through parameter space but not on the speed at which the path is traversed. What condition must be satisfied for this statement to be valid?

A.2 A friend claims: "The Berry phase is just an artifact of choosing a particular phase convention for the eigenstates — it's not physical." Is your friend correct? If so, under what circumstances? If not, what makes the Berry phase physically real? Distinguish between open and closed paths in parameter space.

A.3 The adiabatic condition requires $|\langle m|\dot{n}\rangle|/(|E_m - E_n|/\hbar) \ll 1$ for all $m \neq n$. Explain physically what each of the three quantities (numerator, denominator, and their ratio) represents. What happens at a degeneracy point where $E_m = E_n$?

A.4 In the spin-1/2 Berry phase example, the geometric phase for the spin-up state is $\gamma_+ = -\frac{1}{2}\Omega_C$, where $\Omega_C$ is the solid angle. Why does the factor of $1/2$ appear? What would the corresponding formula be for a spin-$j$ particle?

A.5 The Berry curvature for a spin-1/2 particle on the Bloch sphere has the form of a magnetic monopole at the origin. But real magnetic monopoles have never been observed. How can the Berry curvature have a monopole form without contradicting Maxwell's equations?

A.6 In the Born-Oppenheimer approximation, what physical role does the Berry phase play? Why was it missed for nearly sixty years after Born and Oppenheimer's original paper?

A.7 Explain the difference between the dynamical phase and the geometric phase using the analogy of a pendulum on a rotating Earth. What is the "dynamical" aspect and what is the "geometric" aspect?

A.8 The Aharonov-Bohm effect demonstrates that a charged particle can be affected by a vector potential $\mathbf{A}$ even in a region where $\mathbf{B} = 0$. How is this related to the Berry phase? What is the parameter space, and what is the Berry connection?

Part B: Applied Problems (**)

These problems require direct application of the chapter's key equations.

B.1: Berry Phase for a General Cone

A spin-1/2 particle is in a magnetic field $\mathbf{B}(t) = B_0(\sin\alpha\cos\omega t, \sin\alpha\sin\omega t, \cos\alpha)$.

(a) Verify that the instantaneous eigenstates are:

$$|+;\alpha,\phi\rangle = \cos\frac{\alpha}{2}|{\uparrow}\rangle + \sin\frac{\alpha}{2}e^{i\phi}|{\downarrow}\rangle$$

by showing that $\hat{\mathbf{n}}\cdot\hat{\boldsymbol{\sigma}}|+;\alpha,\phi\rangle = +|+;\alpha,\phi\rangle$, where $\hat{\mathbf{n}} = (\sin\alpha\cos\phi, \sin\alpha\sin\phi, \cos\alpha)$.

(b) Compute the Berry connection $\mathcal{A}_\phi^+ = i\langle +|\partial_\phi|+\rangle$ directly.

(c) Compute the Berry connection $\mathcal{A}_\alpha^+ = i\langle +|\partial_\alpha|+\rangle$. Is it zero?

(d) Evaluate the Berry phase $\gamma_+$ for a circular path at fixed $\alpha$, $\phi \in [0, 2\pi]$.

(e) Verify your result using Stokes' theorem and the Berry curvature $\Omega_{\alpha\phi} = \partial_\alpha \mathcal{A}_\phi - \partial_\phi \mathcal{A}_\alpha$.

B.2: Berry Phase for the Spin-Down State

(a) Starting from $|-;\alpha,\phi\rangle = -\sin\frac{\alpha}{2}e^{-i\phi}|{\uparrow}\rangle + \cos\frac{\alpha}{2}|{\downarrow}\rangle$, compute $\gamma_-$ for the same circular path as in B.1.

(b) Show that $\gamma_- = -\gamma_+$ (the Berry phases of the two states are equal and opposite).

(c) Compute the Berry curvature for the spin-down state and show it corresponds to a monopole of opposite charge.

(d) What is the physical significance of $\gamma_+ - \gamma_- = -\Omega_C$?

B.3: Adiabatic Condition for NMR

A proton (gyromagnetic ratio $\gamma_p = 2.675 \times 10^8$ rad/(s$\cdot$T)) is in a magnetic field of magnitude $B_0 = 2.0$ T that rotates at angular frequency $\omega$.

(a) Calculate the Larmor frequency $\omega_L = \gamma_p B_0$.

(b) What is the maximum $\omega$ for which the adiabatic approximation is good to 1% (i.e., the transition probability to the other spin state is less than 1%)? Use the Landau-Zener formula if needed, or simply estimate $\omega/\omega_L < 0.1$.

(c) For $\omega = 10^3$ rad/s and $\alpha = 60°$, calculate both the dynamical phase and the Berry phase accumulated over one complete rotation.

(d) How would you experimentally separate the Berry phase from the dynamical phase?

B.4: Berry Phase on a Square Path

Consider a 2D parameter space $(R_1, R_2)$ with the Hamiltonian:

$$\hat{H}(R_1, R_2) = R_1 \hat{\sigma}_x + R_2 \hat{\sigma}_z.$$

(a) Find the instantaneous eigenvalues $E_\pm(R_1, R_2)$.

(b) Find the instantaneous eigenstates (choose a convenient gauge).

(c) Compute the Berry curvature $\Omega_{12} = -2\,\text{Im}\langle \partial_1 +|\partial_2 +\rangle$.

(d) A path traces a square in parameter space: $(1,0) \to (1,1) \to (0,1) \to (0,0) \to (1,0)$. But this path passes through the degeneracy at $(0,0)$, where the Berry curvature diverges. Modify the path to avoid the degeneracy and compute the Berry phase for a path that encloses the origin.

B.5: Aharonov-Bohm Phase

An electron beam splits into two paths around a solenoid of radius $R = 1\;\mu\text{m}$ carrying a magnetic field $B = 0.01$ T.

(a) Calculate the total magnetic flux $\Phi = B \cdot \pi R^2$ through the solenoid.

(b) Calculate the Aharonov-Bohm phase $\Delta\phi = e\Phi/\hbar$.

(c) For what value of $B$ would the Aharonov-Bohm phase be exactly $\pi$? What would happen to the interference pattern at this field strength?

(d) Show that the Aharonov-Bohm phase is periodic in $\Phi$ with period $\Phi_0 = h/e$.

Part C: Advanced Problems (***)

These problems require synthesis of multiple concepts and substantial derivation.

C.1: Proof that the Berry Curvature is Gauge-Invariant

(a) Under the gauge transformation $|n(\mathbf{R})\rangle \to e^{i\chi(\mathbf{R})}|n(\mathbf{R})\rangle$, show that the Berry connection transforms as $\mathcal{A}_n \to \mathcal{A}_n - \nabla\chi$.

(b) Show that the Berry curvature $\boldsymbol{\Omega}_n = \nabla \times \mathcal{A}_n$ is gauge-invariant.

(c) Show that the Berry phase for a closed loop is gauge-invariant: $\gamma_n \to \gamma_n$ (unchanged). Hint: $\chi(\mathbf{R})$ must return to its starting value (up to $2\pi n$) around a closed loop for the eigenstate to be single-valued.

(d) What happens to the Berry phase for an open path? Why is it unphysical?

C.2: Berry Phase of the Harmonic Oscillator

Consider a 1D harmonic oscillator whose equilibrium position $x_0$ is slowly moved:

$$\hat{H}(x_0) = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2(\hat{x} - x_0)^2.$$

(a) Show that the instantaneous eigenstates are $|n(x_0)\rangle = e^{-i\hat{p}x_0/\hbar}|n\rangle$, where $|n\rangle$ are the stationary oscillator states.

(b) Compute the Berry connection $\mathcal{A}_n = i\langle n(x_0)|\partial_{x_0}|n(x_0)\rangle$.

(c) For a closed loop in which $x_0$ returns to its initial value, what is the Berry phase? (Hint: the parameter space is 1D. Can you have a nontrivial loop in 1D?)

(d) Now generalize to a 2D oscillator whose equilibrium is moved in a circle: $\mathbf{r}_0(t) = (x_0\cos\omega t, x_0\sin\omega t)$. Is the Berry phase still zero? Compute it.

C.3: Conical Intersection and the Molecular Berry Phase

Consider a two-level system (model for a conical intersection in a molecule) with Hamiltonian:

$$\hat{H}(x, y) = \begin{pmatrix} x & y \\ y & -x \end{pmatrix}.$$

(a) Find the eigenvalues $E_\pm(x, y)$. Show they form a cone at $(x, y) = (0, 0)$.

(b) Find the eigenstates using the polar parametrization $x = r\cos\theta$, $y = r\sin\theta$.

(c) Compute the Berry phase for the lower state when the parameters trace a circle of radius $R$ centered at the origin. Show that $\gamma_- = \pi$.

(d) Now consider a circle not enclosing the origin. What is the Berry phase? What is the physical implication for molecular dynamics?

(e) The sign change $e^{i\pi} = -1$ of the electronic wave function means the nuclear wave function must also change sign. What does this imply for the nuclear boundary conditions? How does this affect the nuclear energy spectrum?

C.4: Berry Phase in the Jaynes-Cummings Model (Theory)

The Jaynes-Cummings Hamiltonian describes a two-level atom coupled to a single cavity mode:

$$\hat{H} = \hbar\omega_c \hat{a}^\dagger\hat{a} + \frac{\hbar\omega_a}{2}\hat{\sigma}_z + \hbar g(\hat{a}^\dagger\hat{\sigma}_- + \hat{a}\hat{\sigma}_+).$$

For fixed photon number $n$, the Hamiltonian acts on a two-dimensional subspace spanned by $\{|n, e\rangle, |n+1, g\rangle\}$.

(a) Write the effective $2 \times 2$ Hamiltonian in this subspace as a function of the detuning $\Delta = \omega_a - \omega_c$.

(b) If the detuning is varied slowly around a loop in $(\Delta, g)$ space, compute the Berry phase for the lower dressed state.

(c) Under what conditions is the Berry phase close to $\pi$?

Part D: Numerical and Computational Problems

D.1 Using the berry.py code from this chapter, compute and plot the Berry phase $\gamma_+(\alpha)$ as a function of cone half-angle $\alpha$ for $\alpha \in [0, \pi]$. Verify that $\gamma_+(0) = 0$, $\gamma_+(\pi/2) = -\pi$, and $\gamma_+(\pi) = -2\pi$.

D.2 Modify the adiabatic evolution code to compute the Berry phase for a spin-1 particle ($S = 1$, 3-state system) in a rotating field. The Berry phase should be $\gamma_m = -m\Omega_C$ where $m = -1, 0, +1$. Verify this numerically.

D.3 Implement a numerical Berry curvature calculator: for a general parameterized $2 \times 2$ Hamiltonian $\hat{H}(R_1, R_2)$, compute $\Omega_{12}(R_1, R_2)$ on a grid and plot it as a heatmap. Apply it to the Hamiltonian from Problem B.4 and verify the monopole singularity at the origin.

D.4 Simulate the transition from adiabatic to diabatic evolution. Starting in the spin-up state, evolve a spin-1/2 in a field rotating at frequency $\omega$, and plot the probability of remaining in the instantaneous eigenstate as a function of $\omega/\omega_L$ (the adiabatic parameter). You should see a crossover from near-1 (adiabatic) to near-0.5 (diabatic) as $\omega/\omega_L$ increases from 0 to $\sim 1$.

Solutions to Selected Problems

A.1: The Berry phase is $\gamma_n = \oint_C \mathcal{A}_n \cdot d\mathbf{R}$, a line integral over the path $C$ in parameter space. Reparametrizing the path (e.g., traversing it at a different speed) does not change the integral, because $d\mathbf{R}$ tracks the geometric displacement, not the temporal one. The condition is that the adiabatic approximation must hold: the traversal must be slow enough that transitions to other levels are negligible. If the traversal is too fast, the system does not track the instantaneous eigenstate, and the concept of Berry phase for the $n$-th eigenstate breaks down.

B.1(b): $\mathcal{A}_\phi^+ = i \cdot i\sin^2(\alpha/2) = -\sin^2(\alpha/2)$.

B.1(d): $\gamma_+ = \int_0^{2\pi} \mathcal{A}_\phi^+ d\phi = -2\pi\sin^2(\alpha/2) = -\pi(1 - \cos\alpha)$.

B.5(a): $\Phi = 0.01 \times \pi \times (10^{-6})^2 = \pi \times 10^{-14}$ Wb.

B.5(b): $\Delta\phi = e\Phi/\hbar = (1.602 \times 10^{-19})(\pi \times 10^{-14})/(1.055 \times 10^{-34}) \approx 4.77 \times 10^{0} \approx 4.77$ rad.

C.3(c): The Berry phase for a loop enclosing the origin is $\gamma_- = \pi$, regardless of the radius of the circle. This is because the Berry curvature is concentrated at the degeneracy point (the conical intersection), and the total flux through any surface enclosing it is $\pi$.