Tight-binding models in a magnetic field: Peierls substitution - Topology in condensed matter: tying quantum knots

To understand how the vector potential enters a tight-binding model through the Peierls substitution, let us remind ourselves that the gauge invariance of the Schrodinger equation requires us to transform the wave-function amplitude, or equivalently the creation operator of an electron at a site, as

c_j^\dagger \rightarrow c_j^\dagger \exp\left(-i\frac{e}{\hbar c}\Lambda(\bf r_j)\right),

(1)

where $\Lambda(\bf r)$ generates the gauge transformation of the vector potential $\bf A(\bf r)\rightarrow \bf A(\bf r)+\bf\nabla \Lambda(\bf r)$ . If there is no magnetic field then the vector potential can locally be set to $\bf A=0$ by an appropriate gauge choice of $\bf \Lambda$ . The hopping term in the absence of a vector potential is written as $H_t=t_{jl}c_j^\dagger c_l+h.c$ , which must gauge transform to

H_t=t_{jl} \exp\left(-i\frac{e}{\hbar c}(\Lambda(\bf r_j)-\Lambda(\bf r_l))\right)c_j^\dagger c_l+h.c=t_{jl} \exp\left(-i\frac{e}{\hbar c}(\int_{\bf r_l}^{\bf r_j} d\bf r'\cdot\bf A(\bf r')\right)c_j^\dagger c_l+h.c.

(2)

While this expression is derived for zero magnetic field, by choosing the integration path to be the shortest distance over the nearest-neighbor bond, this expression is used to include magnetic fields in lattice models. This is referred to as the Peierls substitution for lattices.

If we put our topological nanowire in a ring (as with the Aharonov-Bohm effect) with a junction (as in the figure) and concentrate the magnetic field in the center of the ring, the vector potential $\bf A$ is constrained by the magnetic flux $\Phi$ as

\oint d\bf {r'\cdot\bf A(\bf r')}=\int d^2\bf {r'\bf \nabla\times \bf A(\bf r')}=\Phi.

(3)

Choosing a gauge for the vector potential so that it vanishes everywhere except in the junction, the hopping phase $\theta$ for the junction, i.e. $H_t=t_{N,1}e^{i\theta}c_N^\dagger c_1+h.c.$ , is written as

\theta=\int_{\bf r_l}^{\bf r_j} d\bf r'\cdot\bf A(\bf r')=\pi \Phi/\Phi_0,

(4)

where $\Phi_0=hc/2e$ is the superconducting flux quantum.