Thirring quantum cellular automaton

We analytically diagonalize a discrete-time on-site interacting fermionic cellular automaton in the two-particle sector. Important features of the solutions sensibly differ from those of analogous Hamiltonian models. In particular, we find a wider variety of scattering processes, we have bound states for every value of the total momentum, and there exist bound states also in the free case, where the coupling constant is null.

Figure 1

The top plot shows $k_R$ and $k_I$ representing the real and the imaginary part of the relative momentum $k$ in the two-fermion state. The highlighted regions collect the values of $k\in \mathbb{C}$ providing a real value of the quasienergy $\omega$. On the bottom the disjoint subregions of the unit circle are the images under $k\mapsto e^{i{\omega }_{rs}(p,k)}$ of the disjoint regions in the top figure, for fixed values of the total momentum $p=0.55$ and mass $\mu =0.8$. Here ${\mathrm{\Omega }}_f$ coincides with the continuous spectrum of $U_2(\chi ,p)$ [see Eq. (4)]. The discrete spectrum lies in the other regions and for a fixed value of the coupling constant $\chi$ it consists of a single point. Varying the value of $\chi$, the unit circle is covered and the boundary points of the arcs depend on $p$.

Figure 2

Spectrum of the two-particle automaton of Eq. (4): The continuous spectrum bands are depicted in red (inner band) and yellow (outer band); depicted in black is the discrete band for different values of the coupling: ${\chi }_1=-\frac{\pi }{5}, {\chi }_2=-\frac{\pi }{2}, {\chi }_3=-\frac{4\pi }{5}, {\chi }_4=\frac{4\pi }{5}, {\chi }_5=\frac{\pi }{2}$, and ${\chi }_6=\frac{\pi }{5}$.