Berry Phase

The general definition of the Berry phase is a phase angle (from 0 to $2\pi$) that a complex vector acquires as it is transported through a path in its complex vector space. The idea was popularized by Sir Michael Berry in 1984.

To demonstrate the Berry phase, we consider some state in Hilbert space (e.g. the ground state) of some system parameterized by a continuously varying parameter (e.g magnetic field, nuclear coordinates etc.). We then vary the parameter in a closed path, such that the system returns to where it started.

Adiabatic Approximation

If the parameters very “slowly” with time, then energy eigenvalues should just follow the change in parameters. The eigenvectors will acquire an additional geometrical phase associated with the change in parameter.

Assuming no degeneracy (so as to have constant ordering of states),the energy eigenstates are found from the time-independent Schrodinger equation

$$H(t)|n;t⟩=E_n(t)|n;t⟩$$

noting that at any time $t$, the states and eigenvalues may change. The time-dependent Schrodinger equation for some arbitrary ket expanded in the energy eigenstates is

$$i\hslash \frac{\partial }{\partial t}|\alpha ;t⟩=H(t)|\alpha ;t⟩$$

where

$$|\alpha ;t⟩=\sum _n{c}_n(t)e^{i{\theta }_n(t)}|n;t⟩$$

and $\theta (t)$ is the typical dynamical phase acquired by the energy eigenstates

$${\theta }_n(t)=-\frac{1}{\hslash }{\int }_0^t{E}_n(t^{\prime })d{t}^{\prime }$$

Through a straightforward substitution into the TDSE, taking the time-derivative of the TDSE, acting from the left with an equal time eigenstate, and using orthogonality of equal time eigenstates, one arrives at a differential equation for the coefficients,

$${\dot{c}}_m(t)=-c_m(t)⟨m;t|\frac{\partial }{\partial t}|m;t⟩-\sum _n{c}_n(t)e^{i({\theta }_n-{\theta }_m)}\frac{⟨m;t|\dot{H}|n;t⟩}{E_n(t)-E_m(t)}$$

The second term shows mixing with other states due to the time dependence of the Hamiltonian. This result is exact. The adiabatic approximation amounts to neglecting the second term, meaning

$$\frac{⟨m;t|\dot{H}|n;t⟩}{E_n(t)-E_m(t)}\equiv \frac{1}{\tau }\ll ⟨m;t|\frac{\partial }{\partial t}|m;t⟩\approx \frac{E_m}{\hslash }$$

The solution for the coefficients is then

Discrete formulation

Consider N vectors $|u_0⟩$ to $|u_{N-1}⟩$ that describe the ground state of a system at each discrete step in parameter space, where the last state $|u_N⟩$ is identified with $|u_0⟩$. The Berry phase is defined as

$$\varphi =-Imln[⟨u_0|u_1⟩⟨u_1|u_2⟩\cdots ⟨u_{N-1}|u_0⟩]$$

The reason for the imaginary natural log is that for some complex number $z=|z|e^{i\varphi }$, $Imlnz=\varphi$ , so we only retain the information about its phase. This definition chooses a branch of $Imlnz$ such that $-\pi <\varphi \le \pi$. The minus sign is conventional. This formulation has the benefit that it is gauge independent.

Parallel-Transport Gauge

We are free to choose any gauge we like for each of the $|u_j⟩$ . We can define new gauges s.t. $|{\bar{u}}_0⟩=|u_0⟩$ , then choose the phase of $|u_1⟩$ such that $⟨{\bar{u}}_0|{\bar{u}}_1⟩$ is real and positive. If we propagate this down the line of states, then

$$\text{Im}\text{ln}⟨{\bar{u}}_j|{\bar{u}}_{j+1}⟩=0$$

This generates what is known as the parallel transport gauge. In this gauge choice,$|{\bar{u}}_0⟩$ is no longer equivalent to $|{\bar{u}}_N⟩$, and they differ by exactly the Berry phase

$$\varphi =\text{Im}\text{ln}⟨{\bar{u}}_N|{\bar{u}}_0⟩$$

The phase difference of the states are exactly zero until an abrupt discontinuity at the end of the loop.

Twisted Parallel-Transport Gauge

We may like to have a choice of phase that is continuous throughout the evolution of the state.

Spin-1/2 in a magnetic field

A quantum mechanical system that exhibits holonomic properties is a spin-1/2 particle in a rotating magnetic field, $\vec{B}(t)$. A spin-1/2 particle has a magnetic moment $\vec{\mu }=g\frac{q\hslash }{2m}\vec{S}$, where $g$ is the Landé factor, $q$ and $m$ are the charge and mass of the particle respectively, and $\vec{S}=\frac{1}{2}\vec{\sigma }$, with $\vec{\sigma }=({\sigma }_x,{\sigma }_y,{\sigma }_z)$ being the Pauli matrices. The magnetic field precesses around the z-axis at a polar angle $\theta$ at a constant angular frequency $\omega$. The direction of the magnetic field at some time, t, is $\hat{\mathbf{R}}=\mathbf{B}/B$ defined in spherical coordinates by the angles $\theta$ and $\varphi =\omega t$. In the laboratory frame the Hamiltonian of the system is

$$H(t)=\hslash {\omega }_B\hat{\mathbf{R}}(\theta ,\omega t)\cdot \mathbf{S}$$

where ${\omega }_B=\frac{gqB}{2m}$ is the Larmor frequency. The time evolution of this system is dictated by the time-dependent Schrödinger equation

$$i\hslash \frac{\partial }{\partial t}|\psi (t)⟩=H(t)|\psi (t)⟩$$

The states that are of interest are cyclic states such that, $|\psi (0)⟩=|\psi (\tau )⟩$ up to some phase factor, where $\tau$ is the period of the cyclic state. In this case, $\tau =\frac{2\pi }{\omega }$. To solve this problem, it is convenient to change reference frames such that the Hamiltonian becomes time independent. We can do this by rotating the state vector by an angle $\varphi (t)=-\omega t$. The new state vector ${|\psi (t)⟩}^{{}^{\prime }}$, evolves according to the time independent Hamiltonian

$$H^{{}^{\prime }}=\hslash \Omega \hat{\mathbf{R}}({\theta }^{{}^{\prime }},0)\cdot \mathbf{S}$$