Berry Phase

Berry phase is the geometric phase acquired when an isolated eigenstate follows a slowly changing Hamiltonian around a closed parameter loop. After the dynamical phase is removed, the ray returns to itself but its transported representative can have a different phase. Interference with a reference makes that phase measurable.

Required background. The adiabatic theorem reminder explains when an eigenstate follows a moving eigenspace. Rays and global phase explains why the resulting phase needs a comparison with another amplitude.

Geometric phase from the Schrödinger equation

Consider a Hamiltonian depending smoothly on external parameters $R$:

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

If the parameters follow a closed path $C$ slowly enough that the state remains in the same nondegenerate instantaneous eigenspace, then after one cycle the state returns to the same ray:

$$\lvert\psi(T)\rangle = e^{i\alpha_n} \lvert n(R(0))\rangle.$$

The total phase separates into a dynamical part and a geometric part:

$$\alpha_n = -\frac{1}{\hbar}\int_0^T E_n(t)\,dt + \gamma_n[C].$$

The second term is the Berry phase.

To obtain it, choose a normalized smooth instantaneous eigenvector along the path and use the leading adiabatic state

$$|\psi_{\mathrm{ad}}(t)\rangle =e^{i\alpha_n(t)}|n(t)\rangle, \qquad \alpha_n(0)=0.$$

Project the Schrödinger equation onto $\langle n(t)|$. Differentiating both the phase and the moving ket gives

$$-\hbar\dot\alpha_n+i\hbar\langle n|\dot n\rangle=E_n, \qquad \dot\alpha_n=-\frac{E_n}{\hbar}+i\langle n|\dot n\rangle.$$

The first term survives even for a fixed eigenvector. The second is required because the basis itself changes. Normalization makes $\langle n|\dot n\rangle$ purely imaginary, so it contributes a real phase. This is the evolution argument in Berry (1984), §2.

Projection matters: a moving one-state ansatz is generally not an exact solution of the full Schrödinger equation. With $P_n=|n\rangle\langle n|$, the remaining residual is

$$(i\hbar\partial_t-H)|\psi_{\mathrm{ad}}\rangle =i\hbar e^{i\alpha_n}(I-P_n)|\dot n\rangle.$$

The adiabatic theorem controls the resulting leakage under its gap and smoothness hypotheses. Deriving the phase does not prove those hypotheses or turn this residual into a general error bound.

Integrating the projected equation gives the phase relative to the chosen moving ket. If $|n(T)\rangle=e^{i\eta}|n(0)\rangle$, comparison with the initial ket adds $\eta$. That is why the closed-loop Berry phase includes an endpoint term whenever the representative is not periodic.

Berry Connection

With the convention used in this volume, the Berry connection is

$$\mathbf A_n(R) = i\langle n(R)|\nabla_R n(R)\rangle.$$

For a closed loop $C$, choose a smooth eigenvector along the loop whose final representative equals its initial one. Then the Berry phase is

$$\gamma_n[C] = \oint_C \mathbf A_n(R)\cdot dR.$$

A convenient calculation may use a representative that is not periodic. In that case the general cyclic formula is

$$\gamma_n[C]=\arg\langle n(0)|n(T)\rangle +\int_0^T i\langle n(t)|\dot n(t)\rangle\,dt \pmod{2\pi}.$$

The endpoint and integral change by opposite amounts under rephasing. In a parallel-transport gauge the integral is zero and the endpoint carries the entire phase. Berry connection explains the local one-form; parallel transport explains this second way to calculate the same result.

Dynamical Versus Geometric Phase

The dynamical phase is

$$-\frac{1}{\hbar}\int_0^T E_n(t)\,dt.$$

It changes if the same path is traversed more slowly or if the energy scale changes. The Berry phase depends on the path through parameter space, not directly on the rate of traversal, as long as the adiabatic approximation remains valid.

This distinction is experimentally important because geometric phases can survive changes that alter the timing while preserving the loop.

Gauge Dependence and Gauge Invariance

The instantaneous eigenvector has an arbitrary phase:

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

Under this change,

$$\mathbf A_n(R) \mapsto \mathbf A_n(R)-\nabla_R\chi(R).$$

Thus the Berry connection is gauge dependent. For a closed loop, however,

$$\gamma_n[C] \mapsto \gamma_n[C]-\oint_C\nabla_R\chi\cdot dR.$$

If the phase convention is single-valued around the loop, the extra integral is an integer multiple of $2\pi$, so the observable phase factor $e^{i\gamma_n}$ is gauge invariant.

Berry Curvature Preview

For three Cartesian parameters, the Berry curvature can be written as a curl:

$$\mathbf B_n(R) = \nabla_R\times\mathbf A_n(R).$$

In any parameter dimension the curvature is the two-form $F_n=dA_n$. Curl notation is just its three-dimensional representation.

Stokes’ theorem relates the phase to curvature flux through an oriented spanning surface:

$$\gamma_n[C] = \int_\Sigma \mathbf B_n\cdot d\mathbf S$$

when a single smooth gauge is available on the chosen surface. A gap must stay open over that surface. A noncontractible loop may have no spanning surface in the allowed region, even though its holonomy is well defined. Quantized integrals over closed surfaces are Chern numbers.

Spin-One-Half Example

A spin-$1/2$ in a slowly rotating magnetic field gives the standard example. If the Hamiltonian is

$$H = -\mu\,\mathbf B\cdot\boldsymbol\sigma,$$

and the direction of $\mathbf B$ traces a closed loop on the unit sphere, the Berry phase for a spin state aligned with the field is

$$\gamma = -\frac{\Omega}{2},$$

where $\Omega$ is the solid angle enclosed by the loop. Here aligned means the $+1$ eigenstate of $\hat{\mathbf B}\cdot\boldsymbol\sigma$, and $\Omega$ is oriented. If $\mu>0$, this is the lower-energy branch; if $\mu<0$, it is the upper branch. Its geometric sign does not change merely by renaming its energy branch.

Changing the loop shape changes its solid angle continuously, so this example is geometric but generally not topologically quantized. Berry phase for spin-1/2 derives both eigenstate signs.

Holonomy Viewpoint

The Berry phase is the holonomy of the eigenstate connection. The analogy with carrying a tangent vector around a curved surface is useful: local parallel transport can leave a global mismatch. Here, however, the transported object is a complex state line, and its base space consists of Hamiltonian parameters.

For physical applications, Born–Oppenheimer Berry phase treats molecular coordinates, Zak phase treats one-dimensional crystal momentum, and Berry phase in the Aharonov–Bohm effect explains when real-space electromagnetic holonomy can be realized as an adiabatic Berry phase.

Common Mistakes

• Calling every phase accumulated in time a Berry phase.
• Forgetting that the Berry connection depends on phase convention.
• Thinking gauge dependence of $\mathbf A_n$ makes the closed-loop phase unphysical.
• Ignoring the adiabatic and nondegeneracy assumptions in the basic formula.
• Treating the spin-$1/2$ solid-angle result as convention-free without specifying the Hamiltonian and eigenstate.

Exercises

1. A phase becomes a probability. Prepare two coherent path amplitudes, $|a\rangle$ and $|b\rangle$, with the same internal spin state. In arm $a$, adiabatically cycle $H=-\Delta\mathbf n\cdot\boldsymbol\sigma/2$, $\Delta>0$, once with $\phi$ increasing from $0$ to $2\pi$ at fixed $\theta=\pi/3$. In arm $b$, hold its initial Hamiltonian fixed for the same duration. Start in the aligned lower-energy eigenstate; assume equal path amplitudes, matched additional path phases, and negligible leakage. Project the recombined path state onto

$$|+_\beta\rangle=\frac{e^{i\beta}|a\rangle+|b\rangle}{\sqrt2}.$$

Find the probability for $\beta=0$ and $\beta=\pi/2$. How does reversing the loop affect these measurements?

Solution

The lower energy is $-\Delta/2$ in both arms throughout the protocol, so their dynamical phases cancel in the relative phase. The fixed reference eigenray has no Berry phase. The cycled arm acquires $\gamma=-\pi(1-\cos\theta)=-\pi/2$, leaving the path state

$$|\Psi_{\mathrm{path}}\rangle =\frac{e^{i\gamma}|a\rangle+|b\rangle}{\sqrt2}.$$

Taking the overlap gives

$$P_\beta=\frac14|e^{i(\gamma-\beta)}+1|^2 =\frac{1+\cos(\gamma-\beta)}2.$$

Thus $P_0=1/2$ and $P_{\pi/2}=0$. Reversing the loop gives $\gamma=+\pi/2$: $P_0$ remains $1/2$, but $P_{\pi/2}=1$. The second analyzer setting distinguishes the two signs. An unreferenced global phase would produce neither difference.

2. A timing mismatch. After completing the preceding loop, hold arm $a$ at its final Hamiltonian for an extra time $\delta T$ while the reference arm acquires no additional phase. Assume the paths still recombine coherently with equal amplitudes. What happens to $P_\beta$? Does the Berry phase of the loop change?

Solution

The extra stationary interval adds the dynamical phase $-E_n\delta T/\hbar=\Delta\delta T/(2\hbar)$ to arm $a$. Its eigenray does not move, so it adds no geometric phase. Consequently

$$P_\beta=\frac{1+\cos\!\left(\gamma+\frac{\Delta\delta T}{2\hbar}-\beta\right)}2.$$

The Berry phase is unchanged, but the measured fringe moves. Timing independence of the geometric term does not remove the need to match or measure the dynamical phase in an interference experiment.

References

• M. V. Berry, “Quantal phase factors accompanying adiabatic changes”, Proceedings of the Royal Society A 392, 45–57, 1984, §§2 and 4 — evolution of the geometric phase and spin example.
• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• A. Shapere and F. Wilczek, eds., Geometric Phases in Physics, World Scientific, 1989.
• B. Simon, “Holonomy, the quantum adiabatic theorem, and Berry’s phase,” Physical Review Letters 51, 2167–2170, 1983.