Non-Abelian Berry Phase Preview

The ordinary Berry phase applies when an adiabatically followed instantaneous eigenstate is nondegenerate. A closed loop in parameter space returns the state to the same ray with a phase.

The non-Abelian Berry phase, or Wilczek–Zee holonomy, applies when an entire degenerate eigenspace is transported adiabatically. The state remains in the degenerate subspace, but it can rotate into a different superposition inside that subspace:

$$\lvert\psi\rangle \longmapsto \sum_b \left[ \mathcal U_{\mathrm{WZ}}(C) \right]_{ba} \lvert u_b\rangle.$$

The geometric object is no longer a number $e^{i\gamma}$. It is a unitary matrix acting on the degenerate subspace. Matrix multiplication is noncommutative, so the order of motion along the path matters.

An Isolated Degenerate Eigenspace

Let a Hamiltonian depend smoothly on parameters $R$:

$$H(R).$$

Assume there is an $N$-dimensional degenerate eigenspace separated by a gap from the rest of the spectrum. Choose an orthonormal local frame

$$\lvert u_a(R)\rangle, \qquad a=1,\ldots,N,$$

with common energy

$$H(R)\lvert u_a(R)\rangle = E(R)\lvert u_a(R)\rangle.$$

The physical subspace is the projector

$$P(R) = \sum_{a=1}^{N} \lvert u_a(R)\rangle\langle u_a(R)\rvert.$$

The chosen basis inside the subspace is not unique. At each $R$, it may be changed by a unitary matrix $G(R)\in U(N)$:

$$\lvert u'_a\rangle = \sum_b \lvert u_b\rangle G_{ba}.$$

This is the non-Abelian gauge freedom.

Matrix Berry Connection

The Wilczek–Zee connection is the matrix-valued one-form

$$(A)_{ab} = i\langle u_a|d u_b\rangle.$$

In coordinates,

$$(A_i)_{ab} = i\langle u_a|\partial_i u_b\rangle.$$

For $N=1$, this reduces to the ordinary abelian Berry connection

$$A = i\langle u|du\rangle.$$

For $N>1$, each $A_i$ is an $N\times N$ Hermitian matrix; its entries are complex numbers. They describe how the chosen frame twists and rotates as $R$ changes.

Adiabatic Transport

Write an adiabatic state inside the degenerate subspace as

$$\lvert\psi(t)\rangle = e^{-\frac{i}{\hbar}\int_0^t E(t')\,dt'} \sum_a c_a(t) \lvert u_a(R(t))\rangle.$$

The common dynamical phase is scalar because all states in the subspace have the same energy $E(t)$. The coefficients obey, in the adiabatic limit,

$$\dot c = iA_t c,$$

where

$$A_t = \sum_i A_i(R(t))\dot R^i(t).$$

Therefore transport along a path $C$ gives

$$c(T) = \mathcal U_{\mathrm{WZ}}[C]\,c(0),$$

with

$$\mathcal U_{\mathrm{WZ}}[C] = \mathcal P \exp\left( i\int_C A \right).$$

The symbol $\mathcal P$ means path ordering, with later factors acting on the left. It is needed because matrices $A_t$ at different times need not commute. For a loop, this coefficient transport is the holonomy only in a frame that matches at the endpoints. In general, multiply on the left by the endpoint matrix $S_{ab}=\langle u_a(0)|u_b(T)\rangle$.

Gauge Transformation

Under a frame change

$$U'(R) = U(R)G(R),$$

where $U$ denotes the column of basis vectors $\lvert u_a\rangle$, the connection transforms as

$$A' = G^{-1}AG + iG^{-1}dG.$$

This is the non-Abelian version of the abelian gauge law

$$A\mapsto A-d\chi.$$

The holonomy around a closed loop based at $R_0$ transforms by conjugation:

$$\mathcal U_{\mathrm{WZ}}[C] \mapsto G(R_0)^{-1} \mathcal U_{\mathrm{WZ}}[C] G(R_0).$$

Thus the full matrix depends on the chosen basis at the base point, but conjugation-invariant data such as eigenvalues and traces of powers are gauge invariant.

Non-Abelian Curvature

With the Hermitian connection convention above, the curvature is

$$F = dA-iA\wedge A.$$

In coordinates,

$$F_{ij} = \partial_iA_j - \partial_jA_i - i[A_i,A_j].$$

The commutator term is the new non-Abelian feature. It vanishes in the one-dimensional abelian case. Physically, it records the fact that rotations within the degenerate subspace can fail to commute.

The trace of the curvature appears in first-Chern-number formulas for an occupied subspace:

$$C = \frac{1}{2\pi} \int_M \operatorname{Tr}F.$$

This integral is an integer for a smooth vector bundle over a closed oriented surface $M$. The curvature itself transforms as $F\mapsto G^{-1}FG$, so individual matrix entries are gauge covariant, while its trace is invariant.

Why Path Ordering Matters

For an ordinary Berry phase,

$$e^{i\oint A}$$

is just a phase factor. The order in which infinitesimal phase factors are multiplied does not matter.

For a degenerate subspace, the infinitesimal factors are matrices. If

$$[A_t(t_1),A_t(t_2)]\ne0,$$

then

$$e^{iA_t(t_2)\Delta t} e^{iA_t(t_1)\Delta t} \ne e^{iA_t(t_1)\Delta t} e^{iA_t(t_2)\Delta t}.$$

The path-ordered exponential preserves the physical order of transport along the loop.

Physical Interpretation

The non-Abelian Berry phase is not an additional dynamical Hamiltonian inside the degenerate subspace. It is a geometric rotation caused by how the subspace is embedded in the full Hilbert space as parameters change.

The adiabatic theorem says the state does not leak out of the isolated degenerate subspace. It does not say that a vector inside that subspace returns to itself. The Wilczek–Zee holonomy is precisely the allowed internal rotation.

This is why the correct invariant object is the projector family $P(R)$ and its parallel transport, not one preferred basis of degenerate eigenvectors.

Where It Appears

Non-Abelian Berry geometry appears in several settings:

• degenerate molecular electronic subspaces;
• conical intersections and molecular gauge structures;
• Kramers pairs transported under time-reversal-preserving conditions;
• multiple occupied Bloch bands treated as one occupied subspace;
• topological phases with matrix-valued Berry connections;
• holonomic quantum computation, where gates are geometric holonomies;
• adiabatic dark-state manifolds in atomic and optical systems.

These examples share the same structure but require different physical assumptions. Degenerate perturbation theory, environmental splitting, nonadiabatic errors, and symmetry protection all matter in applications.

Abelian Limit

When $N=1$, the frame change is

$$G=e^{i\chi}.$$

The connection law becomes

$$A' = A-d\chi,$$

and the holonomy becomes

$$\mathcal U[C] = e^{i\oint_C A}.$$

Thus the ordinary Berry phase is the one-dimensional case of Wilczek–Zee holonomy.

Degenerate Dynamics and Occupied-Band Geometry

An occupied subspace of several bands has the same matrix connection even when its bands are not degenerate. That is a geometric construction from its projector. Its actual time evolution, however, includes the projected energy matrix $E_{ab}=\langle u_a|H|u_b\rangle$:

$$i\hbar\dot c=(E-\hbar A_t)c.$$

Only when $E=E(t)I$ can one remove a common scalar dynamical phase and retain the pure Wilczek–Zee transport equation. Approximate degeneracy requires the accumulated splittings over the protocol to be considered, not just the gap to outside states.

A rotating basis of a fixed subspace also need not produce physical holonomy. If $P$ is constant and $U(t)=U(0)G(t)$, then $A_t=iG^\dagger\dot G$ and coefficient transport is $G^\dagger(t)G(0)$. The endpoint frame change cancels it. This is a useful check that geometric motion of the subspace has not been confused with a basis choice.

Common Mistakes

• Treating a degenerate eigenspace as if each basis vector had its own independent abelian Berry phase.
• Forgetting that the basis inside the degenerate subspace is gauge freedom.
• Omitting path ordering when the Berry connection matrices do not commute.
• Confusing a gauge-dependent holonomy matrix with its gauge-invariant conjugacy data.
• Applying the Wilczek–Zee formula when the degenerate subspace is not separated from the rest of the spectrum.
• Ignoring dynamical splitting inside a subspace that is only approximately degenerate.
• Treating non-Abelian Berry phase as the same thing as non-Abelian gauge-field dynamics.

Exercises

1. Recover the abelian gauge law.

Set $N=1$ and $G=e^{i\chi}$. Show that

$$A' = G^{-1}AG+iG^{-1}dG$$

becomes $A'=A-d\chi$.

Solution

For $N=1$, $G^{-1}AG=A$ because all factors commute. Also,

$$G^{-1}dG = e^{-i\chi}d(e^{i\chi}) = i\,d\chi.$$

Therefore

$$iG^{-1}dG = i(i\,d\chi) = -d\chi.$$

Hence

$$A'=A-d\chi.$$

2. Explain why path ordering is unnecessary in the abelian case.

Solution

In the abelian case, $A_t(t)$ is a number rather than a matrix. Numbers commute at different times:

$$[A_t(t_1),A_t(t_2)]=0.$$

Therefore infinitesimal factors can be rearranged without changing the product, and the path-ordered exponential reduces to the ordinary exponential

$$\exp\left( i\int_C A \right).$$

3. Show that the holonomy transforms by conjugation.

Assume an open-path transport matrix transforms as

$$U(t) \mapsto G(t)^{-1}U(t)G(0).$$

What happens for a closed loop with $G(T)=G(0)$?

Solution

For a closed loop based at $R_0$,

$$G(T)=G(0)=G(R_0).$$

Thus

$$U(C) \mapsto G(R_0)^{-1}U(C)G(R_0).$$

This is conjugation. Eigenvalues, trace, and determinant are invariant under this transformation.

4. Compute the non-Abelian curvature when all matrices commute.

Suppose $[A_i,A_j]=0$ for all $i,j$. What does

$$F_{ij} = \partial_iA_j-\partial_jA_i-i[A_i,A_j]$$

become?

Solution

If the commutator vanishes, the last term is zero. Therefore

$$F_{ij} = \partial_iA_j-\partial_jA_i.$$

This is the abelian-looking curvature formula, although the connection may still be matrix-valued in a basis where all relevant components commute.

5. Two path segments have connection integrals $\alpha\sigma_x$ and $\beta\sigma_z$. Write the transport in their chronological order and show that exchanging them can matter.

Solution

The first segment acts first on the state, so $U=e^{i\beta\sigma_z}e^{i\alpha\sigma_x}$. The reversed path ordering gives $U_{\mathrm{rev}}=e^{i\alpha\sigma_x}e^{i\beta\sigma_z}$. Using $\sigma_z\sigma_x=i\sigma_y$ and $\sigma_x\sigma_z=-i\sigma_y$,

$$U-U_{\mathrm{rev}}=-2i\sin\alpha\sin\beta\,\sigma_y.$$

It is generally nonzero. Only special angles or commuting generators allow the ordering to be discarded.

References

• M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proceedings of the Royal Society A 392, 45–57, 1984.
• M. Nakahara, Geometry, Topology and Physics, 2nd ed., CRC Press, 2003.
• 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.
• F. Wilczek and A. Zee, “Appearance of gauge structure in simple dynamical systems,” Physical Review Letters 52, 2111–2114, 1984.
• P. Zanardi and M. Rasetti, “Holonomic quantum computation,” Physics Letters A 264, 94–99, 1999.