Berry's geometric phase: a reviewhome | Group |
Any ''vector'' object which is parallel-trasported along a path back to the original place, may acquire an angle with respect to its initial direction prior to transport. This angle is a geometric property. Here, I illustrate (at a fairly elementary level) the basics and a few selected consequences of this almost ubiquitous phenomenon.
The reason for this rotation is purely geometrical-topological. In fact, it is connected to the intrinsic curvature of the sphere. No such phenomenon would appear if vectors are parallel-transported along a flat manifold, such as a plane or a cylinder. The rotation angle is in fact related to the integral of the curvature on the surface bounded by the loop.
Such rotation angles of geometrical origin are known as Berry phases. "Phase" is used meaning just "angle" for whatever possible argument of sin(·) or cos(·) or exp(i ·). Several specific cases had been recognized for many years, in particular the well-studied Aharonov-Bohm effect and Pancharatnam's work on the geometric phases in optics. M. Berry published in 1984 a very influential formal systematization of the closed-path geometric phase in quantum mechanical problems, some of which is summarized in the following paragraphs.
States in quantum mechanics are represented by vectors in a linear (complex) space. In many cases they can be visualized as wave functions. There is no reason that they should make exceptions to the general rule of acquiring phase angles after parallel transport along loops.
A standard, though slightly simplified, formulation goes as follows: consider the set of eigenstates of a (self-adjoint) Hamiltonian operator H, depending on several classical ''external'' parameters (which again we indicate collectively as q). Changing q from one point to another in its multi-dimensional space, the set of stationary states and the related eigenenergies of H change in a completely generic way, leading to totally new and unrelated eigensystem. Assuming however smooth dependence of H on the external parameters q, one expects that for infinitesimal change in the parameters, also the states change by small amounts. The notion of parallel transport may be implemented by imposing that the inner product between a given eigenstate at some point q, and the evolved eigenstate at a following neighboring point q+Δq, be as close to 1 as possible (leading changes of order Δq^{2}).
Note that each stationary state "rotates" (in the complex sense) as
exp(-i E_{k} t/ h) [where
E_{k} is the energy of the state, t is time and
h is Planck's constant], because of its Schroedinger time
evolution. On top of this, since at each point the eigensystem is
unrelated to that at a different point, nothing forbids to multiply each
eigenstate by an additional arbitrary phase factor
exp[-iΦ_{k}(q)] (a "gauge choice")!
The correct rule for the "parallel transport",
as sketched above, permits to calculate the
geometrical phase, such that it ignores completely any dynamical and gauge
phase. The result depends only on the geometry of the loop. In this
sense, Berry's phase is said to be "gauge-invariant".
Two classical examples of Berry phase in quantum mechanics (known long before Berry' s systematization) are:
As you must have guessed, however, in most cases the Berry phase vanishes. As one might expect, usually coming back after a loop to the starting point, makes nothing important, and the vector quantities go back to exactly what they used to be before the loop, indicating that the space where those vectors are moved around is flat. So, nonzero phases are clues pointing to nontrivial topological properties underlying the relation between the vector quantities and the adiabatic parameters q.
In physical applications, one often encounters Hamiltonian operators depending on several external parameters, thus potentially affected by Berry phases. Two general kinds of such q-dependences can be distinguished:
This granted, it's clear that Berry's recipe for the phase at the end of any loop can only yield 0 or π, since any (nondegenerate) wave function must come back either to itself or to minus itself.
As usual, in 99% of the cases, the Berry phase turns out to be zero. However, it was clear right at the beginning that when some point of degeneracy is enclosed in the loop, the phase can be nonzero. By "point of degeneracy" we mean a q-point where the energy of the state under consideration becomes exactly the same as that of the state immediately above or below it. By "enclosed" we mean that the loop cannot be smoothly deformed to avoid surrounding the degenerate point.
For example, in a 2-dimensional space of q-parameters, a circle centered in the origin does enclose the origin (as well as any other internal point...). To unpin it from that point, there is no solution but cutting the circle (obviously not a smooth thing to do to a circle). Instead, if a third axis is added to the q-space, the circle could be raised a little into the third dimension, thus getting free from the degenerate point (it may afterwards be displaced away from the origin as indicated in the figure. A whole line of degenerate points (such as the vertical axis) would be needed for the circle not to be contractable. In general, for to a D-dimensional manifold of q-points, the sub-manifold of singular points needs to be at least (D-2)-dimensional, to pin some loops. Loops in q-space can be divided into classes according to the possibility of either being smoothly contracted to one point, avoiding the degenerate points, or being intrinsically entangled in them.
Nontrivial Berry phases of π can therefore appear only when the q-space with the degenerate q-points removed is multiply connected (an exquisitely geometric-topologic concept).
In conclusion, in the case of real Hamiltonians, the Berry phase along some path equals the number of level degenerate points that this path encloses times π.
To fix the ideas, we consider here in some detail the instructive example of the triangular molecule (E × e JT model). The equilibrium geometry of a triangular cluster such as Xi_{3} is not the equilateral triangle, but a distorted isosceles geometry. Indeed in the equilateral configuration, the electronic ground state is twofold degenerate (E). These 2 electronic states split for any displacement away from the equilateral geometry. Due to symmetry, it is clear that equivalent isosceles distortions where any two Li atoms get closer are energetically equivalent. Actually, it may be verified that (for linear coupling) the equilibrium configurations constitute a flat circular valley - or trough - when the Born-Oppenheimer potential energy is plotted against the q coordinates. Note the degeneracy of the two Born-Oppenheimer potential energy sheets at the central q=0 high-symmetry point in the figure. So, this "JT valley" encircles a degeneracy point for the electronic (real) Hamiltonian: a Berry phase should therefore be present and, moreover, physically relevant, since this loop is frequently accessed by the low-energy dynamics of the q-distortions.
We can get ourselves convinced that the electronic wave functions indeed undergoes a Berry phase change of π while the distortions loop around the trough, by a little meditation on the changes in the electronic wave function during a loop in the distortions space illustrated in the figure above. The real adiabatic choice of the electronic state is indicated, in terms of the three atomic s orbitals of the three Li atoms. Notice that we choose carefully the phase at each point in order to simulate "parallel transport", i.e. to have positive overlap to the immediately preceding configuration. A 2π loop along the JT valley is completed going from one configuration to that pictured at the opposite side: the electronic state changes sign. Another 2π loop is needed to restore the original sign.
In our example for the triangular molecule, the minus sign acquired by the electronic wave function after one loop needs to be compensated by another minus sign of the vibrational counterpart, since the overall (vibronic) molecular state needs to be represented by genuine single-valued wave function. The low-energy pseudo-rotational motion (described by a free-particle wave function of type exp(-i k Φ), where Φ is the angle parameterizing the trough) along the JT valley is dramatically affected by this Berry-phase-induced anti-periodic boundary condition: the quantization rule selects half odd integer pseudorotational "momenta" k, rejecting ordinary integer ones. Substantial consequences follow for the vibronic spectral properties, both in energetics (the JT energy gain is reduced, due to the larger amplitude of ''zero-point'' motion of the lowest k=±½ rotor state, instead of k=0 as would be without Berry phase), and in symmetry (the ground state would be singly degenerate without a Berry phase).
The role of the geometrical phase in determining the symmetry of the vibronic ground state is not restricted to the E × e model presented here, being instead very general in this kind of coupled electron-distortion systems.
These two conceptual steps are actually approximations (the classical description of the vibrons is quickly released, as soon as the phases on the loops have been computed). They could in principle be avoided from the beginning, by diagonalizing the full e-v quantum Hamiltonian. In this sense, the Berry phase is an unnecessary semiclassical concept. However, as long as a Born-Oppenheimer factorization is considered useful, the Berry-phase concept is also a very useful framework for understanding, classifying and computing spectral properties of DJT systems. There are relevant examples of JT systems showing changes in ground state symmetry, that would be unexplained if the Berry-phase analysis were ignored.
Also, why restricting to a loop at all? Indeed it is possible to define and measure Berry phases also for arbitrary open paths, provided that the evolved state at the end of the path is not orthogonal to the initial one.
Furthermore, does the path really need to be continuous? People like to compute quantities numerically, so there must be some discrete version of Berry's integral. Indeed there is a very natural one (in terms of Bargmann invariants), and it turns out so simple and pretty that it may be conveniently taken as an alternative fundamental definition of Berry's phase.
Also, why considering one single state? n degenerate states may evolve together, acquiring not just a phase factor [an element of the group U(1)], but a whole matrix [an element of the group U(n)]: this leads to the non-abelian phases of F. Wilczek and A. Zee, Phys. Rev. Lett. 52, 2111 (1984).
Finally, in-house research in Grenoble/Trieste/Milano lead us to one extra exciting generalization: off-diagonal geometrical phases.
Some of these generalization come out naturally if one considers the vectors as objects evolving in the ray space... but this is a rather complicate story: I suggest reading the cited articles by Aharonov, Simon, and Resta.
created: 2 Aug. 1999 | last modified: 18 Mar 2013 by Nicola Manini |