10.4 Berry phase

Concept of Berry phase

Berry phase captures the geometric structure hidden in quantum states as they evolve through parameter space. When a quantum system is slowly guided around a closed loop in its parameter space, it picks up a phase factor that depends only on the geometry of the path, not on how fast the system traversed it. This geometric phase, distinct from the ordinary dynamical phase, turns out to encode deep information about the topology of band structures, polarization in crystals, and the existence of protected edge states in topological materials.

Geometric phase in quantum mechanics

The Berry phase arises whenever a quantum system undergoes cyclic evolution in parameter space. What makes it "geometric" is that it depends only on the shape of the path traced out, not on the speed of traversal.

• It appears as an additional phase factor in the wavefunction, on top of the familiar dynamical phase (which depends on energy and time).
• Systems with degeneracies or near-degeneracies in their energy spectrum are natural settings where Berry phase becomes physically significant.
• A classic example: a spin-1/2 particle in a slowly rotating magnetic field. As the field direction traces a closed loop on the unit sphere, the spin state accumulates a Berry phase equal to half the solid angle subtended by the loop.

The key distinction to remember: the dynamical phase tells you how long the system evolved, while the Berry phase tells you where it went in parameter space.

Adiabatic evolution of quantum states

Berry phase emerges most naturally under adiabatic evolution, where the system's parameters change slowly compared to its internal timescales.

• The adiabatic theorem guarantees that if parameters change slowly enough, a system initially in an eigenstate of the Hamiltonian will remain in the corresponding instantaneous eigenstate throughout the evolution.
• "Slowly enough" means the rate of parameter change must be much smaller than the energy gap between the occupied state and other states.
• Under these conditions, the state tracks the instantaneous eigenstate but accumulates both a dynamical phase and a Berry phase.

A concrete example: a spin in a magnetic field whose direction is rotated very slowly. The spin follows the field direction (adiabatic tracking), but after a full rotation, it has picked up a geometric phase that depends on the solid angle swept by the field vector.

Cyclic evolution and phase accumulation

When the parameters return to their starting values after a closed loop, the total phase acquired by the quantum state splits into two parts:

1. Dynamical phase: $\gamma_{\text{dyn}} = -\frac{1}{\hbar}\int_0^T E_n(t)\, dt$, which depends on the energy eigenvalue and the total evolution time.
2. Berry phase: $\gamma_n = \oint \mathbf{A}_n(\mathbf{R}) \cdot d\mathbf{R}$, which depends only on the geometry of the closed path in parameter space.

The Berry phase is physically measurable through interferometric techniques (splitting a quantum state, sending the two parts along different paths, and observing the resulting interference pattern). It's closely analogous to the Aharonov-Bohm effect, where a charged particle acquires a phase from an electromagnetic vector potential even in regions where the fields vanish. In both cases, a gauge potential that might seem like a mathematical convenience produces real, observable consequences.

Mathematical formulation

The mathematical framework of Berry phase draws directly from differential geometry, using the language of connections and curvature on fiber bundles. For condensed matter, this machinery provides concrete, calculable quantities that characterize the topology of band structures.

Berry connection and curvature

The Berry connection (also called the Berry vector potential) is defined for a band $n$ as:

$A_n(\mathbf{R}) = i\langle n(\mathbf{R})|\nabla_{\mathbf{R}}|n(\mathbf{R})\rangle$

This is a vector in parameter space that plays the role of a gauge potential, analogous to the electromagnetic vector potential.

The Berry curvature is the curl of the Berry connection:

$\mathbf{F}_n(\mathbf{R}) = \nabla_{\mathbf{R}} \times \mathbf{A}_n(\mathbf{R})$

Think of Berry curvature as a "magnetic field" in parameter space. Just as a real magnetic field is the curl of the vector potential, Berry curvature is the curl of the Berry connection.

The Berry phase for a closed loop is then the flux of this curvature through any surface bounded by the loop:

$\gamma_n = \oint \mathbf{A}_n \cdot d\mathbf{R} = \iint \mathbf{F}_n \cdot d\mathbf{S}$

A physically important consequence: non-zero Berry curvature in crystal momentum space gives electrons an anomalous velocity perpendicular to an applied electric field. This is the microscopic origin of the anomalous Hall effect.

Parallel transport in parameter space

Parallel transport provides the geometric intuition behind Berry phase.

• As a quantum state evolves along a path in parameter space, "parallel transport" means adjusting the state at each point so that it has no local phase twist relative to its neighbors.
• After completing a closed loop, the parallel-transported state generally does not return to its original phase. The mismatch is exactly the Berry phase.

The analogy from differential geometry is helpful here: if you parallel-transport a vector around a closed loop on the surface of a sphere, the vector rotates by an angle equal to the solid angle enclosed by the loop. The vector "remembers" the curvature of the surface it traveled on. Berry phase works the same way, except the "surface" is the parameter space of the quantum Hamiltonian, and the "curvature" is the Berry curvature.

Gauge transformations and invariance

Under a local gauge transformation $|n(\mathbf{R})\rangle \to e^{i\alpha(\mathbf{R})}|n(\mathbf{R})\rangle$, the Berry connection shifts:

$A_n \to A_n - \nabla_{\mathbf{R}}\alpha$

This is exactly how the electromagnetic vector potential transforms under a U(1) gauge transformation. However, the Berry curvature and the Berry phase around any closed loop are gauge-invariant, meaning they don't depend on your choice of phase convention for the wavefunctions.

This gauge invariance is what makes Berry curvature and Berry phase physically meaningful. You can pick any convenient basis for your states, and the observable quantities come out the same. The formal structure mirrors U(1) gauge theory in electromagnetism, which is one reason Berry phase ideas connect naturally to gauge theories in other areas of physics.

Berry phase in crystals

Applying Berry phase to electrons in periodic crystals is where these geometric ideas become most powerful for condensed matter physics. The crystal momentum $\mathbf{k}$ plays the role of the parameter, and the Brillouin zone becomes the parameter space.

Bloch waves and Brillouin zone

Electrons in a periodic crystal potential are described by Bloch states $|\psi_{n\mathbf{k}}\rangle = e^{i\mathbf{k}\cdot\mathbf{r}}|u_{n\mathbf{k}}\rangle$, where $|u_{n\mathbf{k}}\rangle$ has the periodicity of the lattice and $n$ is the band index.

• The Brillouin zone (BZ) is the fundamental cell in $\mathbf{k}$-space. Because of the lattice periodicity, $\mathbf{k}$-values differing by a reciprocal lattice vector are equivalent, so the BZ has the topology of a torus.
• Berry connection and curvature can be defined for the cell-periodic part $|u_{n\mathbf{k}}\rangle$ as $\mathbf{k}$ varies across the BZ.
• The periodic boundary conditions on the BZ constrain the Berry phase to take quantized values in certain symmetry settings.

A striking example: in graphene's honeycomb lattice, the two inequivalent Dirac points ($K$ and $K'$) each carry a Berry phase of $\pi$. This non-trivial Berry phase is directly responsible for the suppression of backscattering and the unusual quantum Hall effect in graphene.

k-space topology and band structure

The distribution of Berry curvature across the Brillouin zone determines the topological classification of energy bands.

• Integrating the Berry curvature of band $n$ over the entire 2D Brillouin zone gives the Chern number:

$C_n = \frac{1}{2\pi}\iint_{\text{BZ}} F_n(\mathbf{k})\, d^2k$

• The Chern number is always an integer. A non-zero Chern number signals a topologically non-trivial band.
• Band crossings and degeneracies act as sources or sinks of Berry curvature (analogous to magnetic monopoles in $\mathbf{k}$-space).
• The bulk-boundary correspondence connects the Chern number to the number of topologically protected edge states: a material with Chern number $C$ hosts $|C|$ chiral edge channels.

In quantum Hall systems, the quantized Hall conductance $\sigma_{xy} = \frac{e^2}{h}C$ is a direct physical manifestation of the Chern number. This is the TKNN (Thouless-Kohmoto-Nightingale-den Nijs) result, one of the landmark connections between topology and measurable transport properties.

Zak phase in one-dimensional systems

The Zak phase is the 1D version of the Berry phase, accumulated as $k$ traverses the entire one-dimensional Brillouin zone from $-\pi/a$ to $\pi/a$:

$\gamma_{\text{Zak}} = \int_{-\pi/a}^{\pi/a} A_n(k)\, dk$

• The Zak phase is directly related to the electric polarization of a 1D insulator through the modern theory of polarization.
• In systems with inversion symmetry or time-reversal symmetry, the Zak phase is quantized to either $0$ or $\pi$ (modulo $2\pi$), making it a $\mathbb{Z}_2$ topological invariant.
• A Zak phase of $\pi$ signals a topologically non-trivial phase, which by bulk-boundary correspondence implies the existence of protected edge states.

The Su-Schrieffer-Heeger (SSH) model of polyacetylene is the textbook example. This 1D chain has two atoms per unit cell with alternating hopping amplitudes $t_1$ and $t_2$. When $t_1 < t_2$, the Zak phase is $\pi$ and zero-energy edge states appear at the chain's ends. When $t_1 > t_2$, the Zak phase is $0$ and no edge states exist. The topological phase transition occurs at $t_1 = t_2$, where the bulk gap closes.