Fermion Doubling in Dirac Quantum Walks

TL;DR

The paper investigates fermion doubling and pseudo-doubling in Dirac quantum walks (QWs) and their second-quantized quantum cellular automata (QCAs). It introduces a parametrized family of Dirac walks in $1+1$-D and $3+1$-D that allows a nonzero on-site stay probability and uses projection-based constructions with a tunable angle $\theta$ to suppress doublers while preserving the Dirac continuum limit. In the $1+1$-D case, for $\theta \neq 0$ the model yields Dirac dynamics with no doublers or pseudo-doublers; in $3+1$-D, appropriate choices of $\theta$ can eliminate both, though some extraneous low-energy modes remain. The work further discusses second-quantized QCAs, gauge-invariant extensions to the Schwinger model, and the implications for interacting quantum simulations and vacuum stability, highlighting potential avenues for scattering calculations and higher-dimensional generalizations.

Abstract

We consider discrete spacetime models known as quantum walks, which can be used to simulate Dirac particles. In particular we look at fermion doubling in these models, in which high momentum states yield additional low energy solutions which behave like Dirac particles. The presence of doublers carries over to the `second quantised' version of the walks represented by quantum cellular automata, which may lead to spurious solutions when introducing interactions. Moreover, we also consider pseudo-doublers, which have high energy but behave like low energy Dirac particles, and cause potential problems regarding the stability of the vacuum. To address these issues, we propose a family of quantum walks, that are free of these doublers and pseudo-doublers, but still simulate the Dirac equation in the continuum limit. However, there remain a small number of additional low energy solutions which do not directly correspond to Dirac particles. While the conventional Dirac walk always has a zero probability for the walker staying at the same point, we obtain the family of walks by allowing this probability to be non-zero.

Figures (5)

• Figure 1: Plot of the dispersion relation for the 3+1-D conventional Dirac quantum walk.
• Figure 2: Plot of the dispersion relation of the parametrised 1+1-D Dirac walk with $\theta = 3\pi/4$ and different values of $m$. With this choice of $\theta$, we can see that the energy is always less than $\pi/(2\delta t)$ for the plotted values of mass ($m$).
• Figure 3: We compare the dispersion relation of the parametrised QW ($\theta = 3\pi/4$) with the conventional walk ($\theta = 0$) and the continuum case (QFT) for $mc^2\delta t=0.02$. It is evident that the plots agree for small $p\,\delta x$.
• Figure 4: Plot of the dispersion relation of the parametrised 3-D Weyl walks with $\theta= 2 \pi/3$ and $p_x=p_y=p_z=p$. We can see that both the Weyl walks have the doubler at different points.
• Figure 5: Plot of the dispersion relation for the parametrised 3+1-D Dirac quantum walk with $\theta=\pi/3$. We can see that the energy is never greater than $\pi/(2\,\delta t)$.