Geometric Phase

Prerequisites: Canonical Quantum Mechanics, Parallel Transport. Full path (19 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

Intuitive

When a system moves once around a close loop, like for example a pendulum once around its suspension, we expect that it returns to exactly the same state that it started in.

However, this is not always the case. System can pick up a phase while moving once around a closed loop.

A famous example is a Foucault pendulum. Such a pendulum is expected to return to its original position after a full rotation of the earth (one sidereal day, about 23 hrs 56 min). However, it doesn’t. It picks up an angle, called Hannay’s angle.

• A nice laymen explanation of Berry phases is If You Walk in A Closed Loop, Do You End Up Where You Started? by Anshul Kogar
• Another great non-technical introduction is "The Geometric Phase" by Michael Berry.
• https://courses.cit.cornell.edu/ece5390/7_berry_phase.pdf

Concrete

Before Berry reinterpreted geometric phases it was already known that if the parameters of quantum system change slowly the system stays in the corresponding instantaneous eigenstate but picks up a phase. This is known as adiabatic theorem. However, people believed that this phase was of no physical significance since it can have any arbitrary value.

Berry suggested that the parameters should be varied in such a way that they end up at the values they started with. The phase that a system picks up after performing such a loop in parameter space is now called Berry's phase. This phase that the system picks up after a full cycle is non-arbitrary and has profound physical implications.

• For a nice introduction, see Quantum Phases And Angles by R. Jackiw
• A nice textbook that discusses Berry's phase is Griffith "Introduction to Quantum Mechanics", especially chapter 10.
• Geometric phases also exist in classical mechanics, like for example Hannay's angle. For a discussion of Hannay's angle, see Section 4.6.3 here and also Geometric phases in classical mechanics by Morten Ib Munk-Nielsen
• The standard reference is "Geometric Phases in Physics" edited by Alfred Shapere and Frank Wilczek
• A great discussion of the Hannay angle can be found in Spivak's Physics for Mathematicians.

Abstract

Let $H(R)$ be a family of Hamiltonians depending smoothly on parameters $R$ in a manifold $M$, and let $E_n(R)$ be a nondegenerate eigenvalue, separated from the rest of the spectrum for all $R\in M$, with normalized eigenvector $|n(R)\rangle$. The eigenlines $\{\lambda|n(R)\rangle\}$ form a complex line bundle $L_n\to M$, a subbundle of the trivial bundle $M\times\mathcal{H}$. Orthogonal projection of the trivial connection onto $L_n$ defines a connection, whose local connection 1-form in the gauge $|n(R)\rangle$ is the Berry connection $$\mathcal{A}_n=i\langle n|d\,n\rangle,\qquad \mathcal{A}_n\to\mathcal{A}_n-d\alpha\quad\text{under}\quad|n\rangle\to e^{i\alpha(R)}|n\rangle .$$ Normalization makes $\langle n|dn\rangle$ purely imaginary, so $\mathcal{A}_n$ is real. In the adiabatic limit a state that starts in $|n(R(0))\rangle$ and is transported around a closed loop $C$ in $M$ returns as $$e^{-\frac{i}{\hbar}\int_0^TE_n(R(t))\,dt}\;e^{i\gamma_n(C)}\;|n(R(0))\rangle,\qquad \gamma_n(C)=\oint_C\mathcal{A}_n .$$ The geometric phase $e^{i\gamma_n(C)}$ is the holonomy of the Berry connection around $C$ and is gauge invariant, while $\gamma_n$ itself is defined modulo $2\pi$. If $C=\partial S$, Stokes' theorem gives $\gamma_n(C)=\int_S\mathcal{F}_n$ with Berry curvature $\mathcal{F}_n=d\mathcal{A}_n=i\langle dn|\wedge|dn\rangle$. This interpretation of the adiabatic phase as holonomy in a line bundle is due to Simon (1983).

Universal description. The map $R\mapsto[|n(R)\rangle]$ sends $M$ into the projective Hilbert space $\mathbb{P}(\mathcal{H})$, and $L_n$ is the pullback of the tautological line bundle with its connection induced by the inner product. The curvature of that connection is proportional to the Fubini-Study Kähler form, so the Berry curvature is the pullback of a universal 2-form. Aharonov and Anandan used this to define a geometric phase for any cyclic evolution, adiabatic or not: it is the holonomy along the closed curve traced in $\mathbb{P}(\mathcal{H})$. For a degenerate level the eigenspaces form a vector bundle of rank $k$ and the holonomy is a $U(k)$ matrix (Wilczek and Zee).

Degeneracies and topology. The bundle $L_n$ is defined only away from points where $E_n$ meets another level. For a spin in a magnetic field, $H=-\mu\,\vec B\cdot\vec S$ with $M=\mathbb{R}^3\setminus\{0\}$, Berry's result is $\gamma_m(C)=-m\,\Omega(C)$ modulo $2\pi$ for the eigenstate with $\hat B\cdot\vec S=m\hbar$, where $\Omega(C)$ is the solid angle subtended by $C$ at $\vec B=0$. The Berry curvature is that of a monopole at the degeneracy, and its integral over a closed surface $\Sigma\subset M$ is $2\pi$ times an integer, the first Chern number of $L_n|_\Sigma$. When the parameter space is the Brillouin zone torus of a two-dimensional crystal, this Chern number of a filled band is the TKNN integer that gives the quantized Hall conductance.

The classical analogue is the Hannay angle: for an integrable system whose parameters are varied slowly around a loop, the action variables are adiabatic invariants and the angle variables acquire an extra shift given by a geometric formula analogous to $\oint\mathcal{A}$.

Why is it interesting?

Berry phases are, in some sense, the common origin of all anomaly effects.

This is stated, for example in Berry's phase, commutators, and the Dirac sea by Michael Stone and William E. Goff

Path to this page

1. Calculus
2. Imaginary Numbers
3. Linear Algebra
4. Functional
5. Hilbert Space
6. Legendre Transformation
7. Manifold
8. Newton's Second Law
9. Vector Calculus
10. Newtonian Mechanics
11. Parallel Transport
12. Variational Calculus
13. Configuration Space
14. Euler-Lagrange Equations
15. Lagrangian Formalism
16. Phase Space
17. Hamiltonian Formalism
18. Schrödinger Equation
19. Canonical Quantum Mechanics