Berry's Geometric Phase

Derivation

Berry's Geometric Phase

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Statement

For a nondegenerate eigenstate \(|n(\mathbf{R})\rangle\) of a Hamiltonian \(\hat{H}(\mathbf{R})\) that depends on external parameters \(\mathbf{R}\), transported adiabatically around a closed loop \(C\) in parameter space, the state acquires — in addition to the dynamical phase — a geometric phase \(\gamma_n(C) = \oint_C \mathbf{A}_n(\mathbf{R})\cdot d\mathbf{R}\), where \(\mathbf{A}_n = i\langle n(\mathbf{R})|\nabla_{\mathbf{R}} n(\mathbf{R})\rangle\) is the Berry connection. By Stokes' theorem this equals the flux \(\gamma_n(C) = \iint_S \mathbf{F}_n\cdot d\mathbf{S}\) of the Berry curvature \(\mathbf{F}_n = \nabla_{\mathbf{R}}\times\mathbf{A}_n\) through any surface \(S\) bounded by \(C\).

Why it matters

The Berry phase shows that a quantum state carries a memory of the path its Hamiltonian traversed, not merely of the time elapsed. This geometric quantity is gauge-invariant, independent of how slowly the loop is traversed, and depends only on the geometry of the loop in parameter space — a striking departure from the intuition that adiabatic transport simply returns a system to itself up to a trivial energy-time phase.

It is the organising principle behind the modern theory of polarization, the quantum Hall conductance, the Aharonov–Bohm effect, molecular Born–Oppenheimer sign changes, and topological band theory. Wherever a physical response is quantised or robust against smooth deformation, a Berry curvature flux is usually the reason.

Assumptions

The instantaneous spectrum is nondegenerate along the loop.If level \(n\) touches another, the adiabatic theorem fails at the crossing and the single-state phase is replaced by a non-Abelian holonomy matrix (Wilczek–Zee); the scalar \(\gamma_n\) is undefined there.

Evolution is adiabatic: the loop is traversed slowly compared with \(\hbar/\Delta E\).If the traversal is too fast, transitions to other levels populate them, the state leaves the ray \(|n\rangle\), and no single geometric phase can be assigned.

The eigenstate \(|n(\mathbf{R})\rangle\) can be chosen smoothly (single-valued and differentiable) over the surface \(S\).If no global smooth gauge exists (as around a degeneracy enclosed by \(C\)), Stokes' theorem must be applied patchwise and the flux picks up a quantised monopole contribution; the naive line integral becomes gauge-ambiguous by \(2\pi\) times an integer.

The loop \(C\) is closed and \(\hat H(\mathbf{R})\) returns exactly to itself.If the parameters do not return, \(|n\rangle\) is compared to a different basis state and the accumulated phase is gauge-dependent, hence not physically measurable on its own.

Derivation

\[ \hat{H}(\mathbf{R}(t))\,|n(\mathbf{R}(t))\rangle = E_n(\mathbf{R}(t))\,|n(\mathbf{R}(t))\rangle \]

Define the instantaneous eigenbasis at each point of the path; the adiabatic theorem guarantees the system stays in level \(n\). A

\[ |\psi(t)\rangle = e^{i\gamma_n(t)}\,\exp\!\left[-\frac{i}{\hbar}\int_0^t E_n(\mathbf{R}(t'))\,dt'\right]|n(\mathbf{R}(t))\rangle \]

Adiabatic ansatz: the state stays parallel to \(|n\rangle\) up to the dynamical phase and an as-yet-unknown extra phase \(\gamma_n(t)\). B

\[ i\hbar\,\frac{d}{dt}|\psi(t)\rangle = \hat{H}(\mathbf{R}(t))\,|\psi(t)\rangle \]

Impose the time-dependent Schrödinger equation — the ansatz must actually solve the dynamics. A

\[ i\hbar\!\left(i\dot{\gamma}_n|n\rangle -\tfrac{i}{\hbar}E_n|n\rangle + |\dot{n}\rangle\right) = E_n|n\rangle \]

Differentiate the ansatz by the product rule and use \(\hat H|n\rangle=E_n|n\rangle\) on the right; the phase prefactors are common and cancel. B

\[ -\hbar\,\dot{\gamma}_n\,|n\rangle + i\hbar\,|\dot{n}\rangle = 0 \quad\Longrightarrow\quad \dot{\gamma}_n = i\,\langle n|\dot{n}\rangle \]

The \(E_n|n\rangle\) terms cancel; project onto \(\langle n|\) using \(\langle n|n\rangle=1\) to isolate \(\dot\gamma_n\). B

\[ \tfrac{d}{dt}\langle n|n\rangle = 0 = \langle \dot n|n\rangle + \langle n|\dot n\rangle \;\Rightarrow\; \langle n|\dot n\rangle = -\langle n|\dot n\rangle^{*} \]

Normalisation forces \(\langle n|\dot n\rangle\) to be pure imaginary, so \(\dot\gamma_n = i\langle n|\dot n\rangle\) is real — the phase is genuine. C

\[ |\dot{n}\rangle = \big(\nabla_{\mathbf{R}}|n\rangle\big)\cdot\dot{\mathbf{R}} \quad\Longrightarrow\quad \dot{\gamma}_n = i\,\langle n|\nabla_{\mathbf{R}} n\rangle\cdot\dot{\mathbf{R}} \]

Chain rule: the only time dependence enters through \(\mathbf{R}(t)\); this converts the rate into a parameter-space quantity. B

\[ \gamma_n(C) = \int_0^T \dot\gamma_n\,dt = \oint_C i\,\langle n(\mathbf{R})|\nabla_{\mathbf{R}} n(\mathbf{R})\rangle\cdot d\mathbf{R} \equiv \oint_C \mathbf{A}_n\cdot d\mathbf{R} \]

Integrate over the loop; the explicit time drops out, leaving a purely geometric line integral. Define the Berry connection \(\mathbf{A}_n = i\langle n|\nabla_{\mathbf R} n\rangle\). A

\[ |n\rangle \to e^{i\chi(\mathbf{R})}|n\rangle \;\Rightarrow\; \mathbf{A}_n \to \mathbf{A}_n - \nabla_{\mathbf{R}}\chi \]

Under a smooth phase (gauge) choice the connection shifts like a vector potential; the loop integral of a gradient over a closed \(C\) with single-valued \(\chi\) vanishes, so \(\gamma_n(C)\) is gauge-invariant mod \(2\pi\). C

\[ \gamma_n(C) = \oint_C \mathbf{A}_n\cdot d\mathbf{R} = \iint_S (\nabla_{\mathbf{R}}\times\mathbf{A}_n)\cdot d\mathbf{S} \equiv \iint_S \mathbf{F}_n\cdot d\mathbf{S} \]

Apply Stokes' theorem on any surface \(S\) with \(\partial S = C\), converting the gauge-dependent line integral into the flux of the manifestly gauge-invariant Berry curvature \(\mathbf{F}_n = \nabla_{\mathbf R}\times\mathbf{A}_n\). A

\[ \mathbf{F}_n = i\,\langle \nabla_{\mathbf{R}} n | \times | \nabla_{\mathbf{R}} n\rangle = -\,\mathrm{Im}\sum_{m\neq n}\frac{\langle n|\nabla_{\mathbf{R}}\hat H|m\rangle \times \langle m|\nabla_{\mathbf{R}}\hat H|n\rangle}{(E_m-E_n)^2} \]

Insert a complete set \(\sum_m|m\rangle\langle m|\) and use \(\langle m|\nabla\hat H|n\rangle=(E_n-E_m)\langle m|\nabla n\rangle\) for \(m\neq n\); the \(m=n\) term drops. This form is manifestly gauge-invariant and exposes the \(1/(E_m-E_n)^2\) enhancement near degeneracies. C

Result

\[ \gamma_n(C) = \oint_C i\,\langle n|\nabla_{\mathbf{R}} n\rangle\cdot d\mathbf{R} = \iint_S \mathbf{F}_n\cdot d\mathbf{S} \]

Reading. The extra phase left on an adiabatically transported eigenstate after one closed loop equals the circulation of the Berry connection around the loop, equivalently the flux of the Berry curvature through any capping surface. It is geometric — set entirely by the shape of \(C\) in parameter space and the local structure of the eigenstates — not by the clock. The curvature \(\mathbf{F}_n\) acts as a "magnetic field" living in parameter space whose sources are the degeneracies of the spectrum.

Units check. \(|n(\mathbf{R})\rangle\) is dimensionless, so \(\nabla_{\mathbf{R}} n\) has units of \([\,\mathbf{R}\,]^{-1}\); the connection \(\mathbf{A}_n\) then carries \([\,\mathbf{R}\,]^{-1}\), and \(\mathbf{A}_n\cdot d\mathbf{R}\) is dimensionless. Likewise \(\mathbf{F}_n\) has units \([\,\mathbf{R}\,]^{-2}\) and \(\mathbf{F}_n\cdot d\mathbf{S}\) is dimensionless. Thus \(\gamma_n\) is a pure number (radians), exactly as a phase must be.

Limiting cases

Real Hamiltonian, no enclosed degeneracy: eigenstates can be chosen real, \(\mathbf{A}_n=0\) locally, and \(\gamma_n(C)=0\) — the ordinary case with no geometric phase.
Real Hamiltonian encircling a conical intersection: the eigenvector flips sign, giving \(\gamma_n(C)=\pi\) (the molecular Longuet-Higgins / Herzberg sign change), independent of loop size.
Spin-\(\tfrac12\) in a magnetic field, \(\mathbf{R}=\mathbf{B}\): \(\mathbf{F}_\pm = \mp\tfrac12\,\hat{\mathbf{B}}/B^2\) (a monopole at \(B=0\)) so \(\gamma_\pm(C)=\mp\tfrac12\Omega(C)\), half the solid angle the loop subtends at the origin.
Shrinking loop \(C\to\) point: \(\gamma_n\to \mathbf{F}_n\cdot\Delta\mathbf{S}\to 0\) linearly in the enclosed area — the phase is a genuine flux, not an intrinsic offset.
Fast (diabatic) traversal: the adiabatic premise fails; the result does not apply and level mixing dominates.

Breaks when

A degeneracy sits on the loop \(C\). The gap \(E_m-E_n\to 0\) makes \(\mathbf{F}_n\) diverge and the adiabatic theorem fails; the state cannot be tracked and \(\gamma_n\) is undefined there.
Levels are degenerate throughout the loop. A single scalar phase is inadequate; transport is governed by a non-Abelian \(U(N)\) holonomy (Wilczek–Zee), a matrix rather than a number.
Traversal is not adiabatic. If \(T \lesssim \hbar/\Delta E\), Landau–Zener transitions leak amplitude to other bands; the state is no longer \(\propto|n\rangle\) and no geometric phase can be extracted cleanly.
No global smooth gauge exists over \(S\). When \(C\) encircles a curvature source, \(|n(\mathbf{R})\rangle\) cannot be chosen single-valued everywhere; Stokes' theorem yields the flux only if applied patchwise, and the line integral is defined only modulo \(2\pi\).

Failure modes

Forgetting the dynamical phase. Students equate the total phase with \(\gamma_n\); the measurable phase is \(\gamma_n - \tfrac1\hbar\int E_n\,dt\), and only closed-loop, gauge-invariant \(\gamma_n\) is the geometric part.
Dropping the factor \(i\). Writing \(\mathbf{A}_n = \langle n|\nabla n\rangle\) (no \(i\)) gives a pure-imaginary "phase"; the \(i\) is exactly what makes \(\gamma_n\) real, by normalisation.
Believing \(\gamma_n\) depends on speed. The result is reparametrisation-invariant; going twice as slow changes the dynamical phase but not \(\gamma_n\).
Claiming \(\mathbf{A}_n=0\) proves \(\gamma_n=0\). A vanishing connection in one gauge patch says nothing about the enclosed flux if a singularity is inside; check the curvature, not the connection.
Treating an open path's phase as physical. For non-closed \(\mathbf{R}(t)\), \(\gamma_n\) is gauge-dependent and not observable without a reference (e.g. an interferometer).
Confusing solid angle with area. For spin-\(\tfrac12\) the phase is half the solid angle at the field-space origin, not the geometric area of \(C\).

Discussion

The deepest lesson is that the phase of a quantum state is not a private bookkeeping convention: transported around a loop, its change becomes gauge-invariant and physical. The Berry connection \(\mathbf{A}_n\) plays the mathematical role of a \(U(1)\) gauge potential over parameter space, and \(\mathbf{F}_n\) is its field strength. This is not an analogy imposed by hand — the connection genuinely defines parallel transport of the eigenray, and \(\gamma_n(C)\) is the holonomy of that connection, the failure of the ray to return to itself after a loop.

Because \(\mathbf{F}_n = \nabla\times\mathbf{A}_n\), any curvature must be sourced by "charges" that are the spectral degeneracies. For a two-level system \(\hat H = \mathbf{R}\cdot\vec\sigma\), the degeneracy at \(\mathbf{R}=0\) is a magnetic monopole of strength \(\mp\tfrac12\) in parameter space, and Gauss's law forces \(\gamma_\pm = \mp\tfrac12\Omega\). The half-integer charge is why a real spin returns with a minus sign after a \(2\pi\) rotation — the same physics as the \(\pi\) phase around a molecular conical intersection.

When the enclosed flux is topologically protected — an integral of curvature over a closed manifold rather than a disc — it is quantised: \(\frac{1}{2\pi}\iint_{T^2}\mathbf{F}_n\cdot d\mathbf{S} = C_n \in \mathbb{Z}\), the Chern number. This integer is the TKNN invariant that pins the quantised Hall conductance \(\sigma_{xy} = C_n\,e^2/h\), and it cannot change under smooth deformation of the Hamiltonian without closing a gap. The Berry phase is thus the microscopic seed of topological band theory: local geometry (curvature) integrates to a global topological invariant (Chern number), and robustness of the invariant is robustness of physical response.

Common misconceptions. The Berry phase is not "just the Aharonov–Bohm phase renamed" — AB is one special case where \(\mathbf{R}\) is real-space position and \(\mathbf{F}_n\) is the ordinary magnetic field; the general construction lives in arbitrary parameter space. Nor does it require a magnetic field at all: any smoothly varying Hamiltonian with the right eigenstate geometry produces it. And it is genuinely observable — through interference of a split beam whose two arms traverse different loops — despite \(\mathbf{A}_n\) being gauge-dependent, because only the loop integral survives.

Worked examples

Spin-\(\tfrac12\) in a magnetic field swept around a cone of half-angle \(\theta_0\).

Take \(\hat H = \tfrac12 g\mu_B\,\mathbf{B}\cdot\vec\sigma\) with \(\mathbf{B}=B(\sin\theta_0\cos\phi,\sin\theta_0\sin\phi,\cos\theta_0)\), \(\phi:0\to2\pi\). The lower state is \(|-\rangle\). A

\[ \mathbf{F}_{-} = +\frac{1}{2}\frac{\hat{\mathbf{B}}}{B^2},\qquad \gamma_-(C)=\iint_S \mathbf{F}_-\cdot d\mathbf{S}=\frac{1}{2}\,\Omega(C) \]

The curvature is a monopole of charge \(+\tfrac12\) at \(\mathbf B=0\); its flux through the cap is \(\tfrac12\) times the solid angle \(\Omega\) subtended at the origin. B

\[ \Omega = 2\pi(1-\cos\theta_0),\qquad \gamma_- = \pi(1-\cos\theta_0) \]

Solid angle of a cone; substitute. Numbers: \(\theta_0 = 60^\circ\Rightarrow\cos\theta_0=0.5\). A

\[ \gamma_-(C)=\pi(1-\tfrac12)=\frac{\pi}{2}\approx 1.571\ \text{rad} \]

Reading. A spin dragged once around a \(60^\circ\) cone in field space returns with an extra phase of \(\pi/2\), independent of \(B\), of the sweep rate, and of the dynamical phase \(-\tfrac1\hbar\int E_-\,dt\). Units: solid angle is dimensionless steradians, so \(\gamma_-\) is in radians.

Two-level model near an avoided crossing (parameter loop enclosing the degeneracy).

Let \(\hat H(\mathbf R)=R_x\sigma_x+R_y\sigma_y+R_z\sigma_z\) with the loop a circle in the \((R_x,R_y)\) plane at fixed \(R_z=d\), radius \(\rho\). This is the previous geometry with \(\hat{\mathbf R}\cdot\hat z=\cos\theta_0=d/\sqrt{\rho^2+d^2}\). B

\[ \gamma_-(C)=\pi\!\left(1-\frac{d}{\sqrt{\rho^2+d^2}}\right) \]

Insert \(\cos\theta_0=d/\sqrt{\rho^2+d^2}\) into \(\gamma_-=\pi(1-\cos\theta_0)\). A

\[ d=1,\ \rho=1:\quad \frac{d}{\sqrt{\rho^2+d^2}}=\frac{1}{\sqrt{2}}\approx0.7071 \]

Plug numbers (parameters in the same arbitrary units of \(\mathbf R\)). A

\[ \gamma_-(C)=\pi(1-0.7071)=0.293\pi\approx 0.920\ \text{rad} \]

Reading. As the offset \(d\to 0\) (loop in the degeneracy plane) \(\gamma_-\to\pi\), recovering the conical-intersection sign change; as \(d\to\infty\) (degeneracy far below the loop) \(\gamma_-\to0\). The phase smoothly interpolates and is dimensionless (radians).

Problems

Show that \(\mathbf{A}_n=i\langle n|\nabla_{\mathbf R}n\rangle\) is real.
Solution
Normalisation \(\langle n|n\rangle=1\) gives \(\nabla\langle n|n\rangle = \langle\nabla n|n\rangle+\langle n|\nabla n\rangle=0\). Since \(\langle\nabla n|n\rangle=\langle n|\nabla n\rangle^{*}\), we get \(\langle n|\nabla n\rangle + \langle n|\nabla n\rangle^{*}=0\), so \(\langle n|\nabla n\rangle\) is pure imaginary. Multiplying by \(i\) makes \(\mathbf A_n\) real. \(\blacksquare\)
A spin is carried around a cone of half-angle \(\theta_0=90^\circ\) (a great circle). Find \(\gamma_-\) and interpret.
Solution
\(\Omega=2\pi(1-\cos90^\circ)=2\pi\), so \(\gamma_-=\tfrac12\Omega=\pi\). The state returns with a sign flip \(e^{i\pi}=-1\): the spinor double-cover manifested as a geometric \(\pi\) phase for a loop that halves the sphere.
Under the gauge change \(|n\rangle\to e^{i\chi(\mathbf R)}|n\rangle\), show \(\gamma_n(C)\) changes only by a multiple of \(2\pi\) for single-valued \(\chi\).
Solution
\(\mathbf A_n\to\mathbf A_n-\nabla\chi\), so \(\gamma_n\to\gamma_n-\oint_C\nabla\chi\cdot d\mathbf R=\gamma_n-\Delta\chi\). For single-valued \(\chi\), \(\Delta\chi\) around a closed loop is \(2\pi\times\)(integer). Hence \(e^{i\gamma_n}\) is invariant and \(\gamma_n\) is defined mod \(2\pi\). \(\blacksquare\)
For \(\hat H=\mathbf R\cdot\vec\sigma\), verify that the total curvature flux of the lower band over a sphere enclosing \(\mathbf R=0\) is \(2\pi\) (Chern number \(1\)).
Solution
\(\mathbf F_-=\tfrac12\hat{\mathbf R}/R^2\). Flux over a sphere of radius \(R\): \(\iint \mathbf F_-\cdot d\mathbf S = \tfrac12\cdot\dfrac{1}{R^2}\cdot 4\pi R^2 = 2\pi\). Then \(C_-=\tfrac1{2\pi}(2\pi)=1\). The half-charge monopole integrates to a unit Chern number because the sphere subtends full solid angle \(4\pi\).
A spin loop subtends solid angle \(\Omega=0.40\) sr. An interferometer compares this arm with a reference arm of zero geometric phase; predict the fringe shift as a fraction of a full fringe.
Solution
\(\gamma_-=\tfrac12\Omega=0.20\) rad. A full fringe corresponds to \(2\pi\) rad, so the shift is \(0.20/(2\pi)=0.0318\) of a fringe, about \(3.2\%\). (The dynamical phase is arranged equal in both arms so only \(\gamma_-\) shifts the pattern.)