Past exam of the theoretical physics II course of the University of Cambridge / 2018 / 1 / c / Solution

Use $c_j=({\gamma }_j^A+i{\gamma }_j^B)/2$ and $c_j^{†}=({\gamma }_j^A-i{\gamma }_j^B)/2$. Moving operators from different modes past one another introduces a minus sign. Direct expansion givesThus, for real hopping and pairing parameters,At $\Delta =t\ne 0$ this becomesUse $\tau$ for time to distinguish it from the hopping amplitude $t$. Each unused Majorana operator anticommutes with both factors in $H$, so commuting it through their product gives two minus signs. Hence $[H,{\gamma }_1^A]=[H,{\gamma }_2^B]=0$. The Heisenberg picture equation $dO/d\tau =(i/\hslash )[H,O]$ therefore givesThese are the two uncoupled Majorana zero modes.