Finite Difference Time Domain Methods for the Dirac Equation Coupled with the Chern–Simons Gauge Field

Abstract

We introduce finite difference time domain (FDTD) methods for solving the Dirac equation coupled with the Chern–Simons gauge field and provide their error estimates. To discretize the spinor equation, we utilize well-known FDTD schemes, including the Crank–Nicolson method, leap-frog method, and semi-implicit methods. On the other hand, to discretize the gauge equations, we mainly employ the Lorenz gauge condition and derive Crank–Nicolson or leap-frog type finite difference methods for the gauge equations. We establish the error estimates for the introduced FDTD methods and prove the second-order accuracy both in space and time. Numerical examples are also presented to validate the second-order convergence and illustrate the dependencies of the numerical solutions on the parameters or gauge conditions.

Appendix A. Proof of the Conservation of the Energy

In this appendix, we provide a proof for the conservation of the energy E(t) of the CSD equation. Recall that the energy E(t) is given by

$$\begin{aligned} E(t)=\int _{\Omega }-i \sum _{i=1}^2\psi ^\dagger \alpha ^i(\partial _i-i A_i)\psi +\psi ^\dagger \beta \psi \,d x\,d y.\end{aligned}$$

To show that $\frac{d }{d t}E(t)=0$, we first multiply (1.2)$_1$ by $(\partial _t\psi -i A_0\psi )^\dagger$ and take the real part to obtain

$$\begin{aligned} 0&=Re \left[ \sum _{i=1}^2 (\partial _t\psi -i A_0\psi )^\dagger \alpha ^i(-i \partial _i\psi -A_i \psi )+(\partial _t\psi -i A_0\psi )^\dagger \beta \psi \right] \\&=Re \left[ \sum _{i=1}^2 \left( (\partial _t\psi )^\dagger \alpha ^i(-i \partial _i\psi -A_i\psi )+i A_0\psi ^\dagger \alpha ^i(-i \partial _i\psi )\right) +(\partial _t\psi )^\dagger \beta \psi \right] \\&=\frac{1}{2}\sum _{i=1}^2\left( (\partial _t\psi )^\dagger \alpha ^i(-i \partial _i\psi -A_i\psi )+(i \partial _i\psi ^\dagger -A_i\psi ^\dagger )\alpha ^i(\partial _t\psi )\right. \\&\left. \quad +A_0(\psi ^\dagger \alpha ^i\partial _i\psi +(\partial _i\psi )^\dagger \alpha ^i\psi )\right) \\&\quad +\frac{1}{2}\left( (\partial _t\psi )^\dagger \beta \psi +\psi ^\dagger \beta (\partial _t\psi )\right) . \end{aligned}$$

Then, integrating over the domain $\Omega$ yields the following:

$$\begin{aligned} \frac{1}{2}\frac{d E(t)}{d t}&=\frac{1}{2}\frac{d }{d t}\int _{\Omega }-i \sum _{i=1}^2 \psi ^\dagger \alpha ^i(\partial _i-i A_i)\psi +\psi ^\dagger \beta \psi \,d x\,d y\\&=\frac{1}{2}\sum _{i=1}^2\int _{\Omega } (\partial _t\psi )^\dagger \alpha ^i(-i \partial _i\psi -A_i\psi )\\&\quad +\psi ^\dagger \alpha ^i(-i \partial _i\partial _t\psi -A_i\partial _t\psi )-(\partial _t A_i)\psi ^\dagger \alpha ^i\psi \,d x\,d y\\&\quad +\frac{1}{2}\int _{\Omega }(\partial _t\psi )^\dagger \beta \psi +\psi ^\dagger \beta (\partial _t\psi )\,d x\,d y\\&=\frac{1}{2}\sum _{i=1}^2\int _{\Omega }-A_0\partial _i(\psi ^\dagger \alpha ^i\psi )-(\partial _tA_i)\psi ^\dagger \alpha ^i\psi \,d x\,d y\\&=\frac{1}{2}\sum _{i=1}^2\int _{\Omega }(\partial _iA_0-\partial _t A_i)(\psi ^\dagger \alpha ^i\psi )\,d x\,d y=0, \end{aligned}$$

where we used the gauge equation in the last equality. Therefore, the energy E(t) for the CSD equation is conserved, regardless of the gauge condition.