Continuous extension of the discrete shift translations on one-dimensional quantum lattice systems (Q6602173)

Quantum systems on the infinite one-dimensional lattice have a natural operation of a discrete shift translation of the lattice \(\{\tau_i\, \colon i \in \mathbb{Z}\}\). This paper deals with the problem of finding continous extensions of such discrete shifts, i.e., to construct strongly continuous one-parameter groups of automorphisms \(\{\tau_t\, \colon t \in \mathbb{R}\}\) that, at integer values, coincide with the discrete shifts. The authors consider the case of a system of spinless fermions, represented by the CAR algebra, and construct a quasi-free continuous extension of the translation group via the Fourier transform on the underlying \(\ell_2(\mathbb{Z})\) Hilbert space. In a formal but non-rigorous sense, this can be interpreted as constructing the spatial translations as the time evolution generated by the momentum operator.\N\NThe rest of the paper deals with the connection with the same problem on spin systems, related to the fermionic chain via the Jordan-Wigner transformation. This is of interest since, in a recent work [\textit{D. Ranard} et al., Ann. Henri Poincaré 23, No. 11, 3905--3979 (2022; Zbl 1514.81095)] it has been shown, using an index theory, that it is not possible to construct a continuous translation group for a spin chain which is generated by a local momentum operator, or more generally, any translation group satisfying the locality estimates of Lieb-Robinsons bounds. The latter results generalises a similar one on quantum cellular automatas, also for spin systems [\textit{D. Ranard} et al., Ann. Henri Poincaré 23, No. 11, 3905--3979 (2022; Zbl 1514.81095)].\N\NThe authors show that the continuous translation group they constructed cannot be extended to the spin lattice via the Jordan-Wigner transformation, and that in fact this time evolution does not satisfy the locality property of locally generated evolutions, since at non-integer times it can map local operators to non-local ones. This is consistent with the fact that the formal momentum operator has interaction decaying as \(1/r\), which is outside the decay rate for which there are known Lieb-Robinson bounds.