Research on mixed spacetime symmetry on lattices

Spacetime exchange symmetry ensures that a quantum lattice system remains invariant under the interchange of spatial and temporal coordinates. In this paper, we develop a general framework for constructing Floquet Hamiltonians in multiband lattice models that respect this symmetry, utilizing unitary group representation theory. As an application, we implement this approach in two-band lattice models and analyze their properties. The exchange symmetry enforces a linear dispersion relation, leading to particle transport at a constant velocity. While these transport properties remain robust against weak noise, they break down under strong noise.

We discover a class of spacetime symmetries unique to time-periodic systems, which we term “mixing symmetry” due to its combination of space and time coordinates in the symmetry transformation. We systematically enumerate the symmetry groups and classify the corresponding Floquet-Bloch band theories by utilizing the winding number of quasienergy. Moreover, we provide a comprehensive scheme for the experimental realization of these symmetries. The particle propagator exhibits an intriguing pattern that remains invariant even under transformations mixing space and time coordinates. We anticipate that this distinct feature can be observed in current cold atom experiments.

Quantum Fourier transform (QFT) is a widely used building block for quantum algorithms, whose scalable implementation is challenging in experiments. Here, we propose a protocol of quadratic QFT (QQFT) considering cold atoms confined in an optical lattice. This QQFT is equivalent to QFT in the single-particle subspace and generalizes to a many-body setting through the second quantization formalism. We show this QQFT protocol can be implemented using a programmable laser potential with the digital-micromirror-device techniques recently developed in experiments. The QQFT protocol enables programmable Hamiltonian engineering and allows quantum simulations of Hamiltonian models, which are difficult to realize with conventional approaches. The flexibility of our approach is demonstrated by performing quantum simulations of one-dimensional Poincaré crystal physics and two-dimensional topological flat bands, where the QQFT protocol effectively generates the required long-range tunnelings despite the locality of the cold atom system. We find the discrete Poincaré symmetry and topological properties in the two examples, respectively, have robustness against a certain degree of noise that is potentially existent in the experimental realization. We expect this approach would open wide opportunities for optical lattice-based programmable quantum simulations.

In this paper, we develop the quantum theory that has discrete Poincare symme- try on the one-dimensional Bravais lattice. We review the recently discovered discrete Lorentz symmetry which coexists with the discrete space translational symmetry on a Bravais lattice. The discrete Lorentz transformations and space- time translations form the discrete Poincare group, which are represented by unitary operators in a quantum theory. We find the conditions for the existence of representation, which are expressed as the congruence relation between quasi-momentum and quasi-energy. We then build the Lorentz-invariant many-body theory of indistinguishable particles by expressing both the unitary operators and Floquet Hamiltonians in terms of the field operators. Some typical Hamiltonians include the long-range hopping which fluctuates as the distance between sites increases. We calculate the Green’s function of the theory. The spacetime points where the Green’s function is nonzero display a lattice structure. During the propagation, the particle stays localized on a single or a few sites to preserve the Lorentz symmetry.

As is well known, crystals have discrete space translational symmetry. It was recently noticed that one-dimensional crystals possibly have discrete Poincaré symmetry, which contains discrete Lorentz and discrete time translational symmetry as well. In this paper, we classify the discrete Poincaré groups on two- and three-dimensional Bravais lattices. They are the candidate symmetry groups of two- or three-dimensional crystals, respectively. The group is determined by an integer generator g, and it reduces to the space group of crystals at g = 2.

The Lorentz symmetry and the space and time translational symmetry are fundamental symmetries of nature. Crystals are the manifestation of the continuous space translational symmetry being spontaneously broken into a discrete one. We argue that, following the space translational symmetry, the continuous Lorentz symmetry should also be broken into a discrete one, which further implies that the continuous time translational symmetry is broken into a discrete one. We deduce all the possible discrete Lorentz and discrete time translational symmetries in 1+1-dimensional spacetime, and show how to build a field theory or a lattice field theory that has these symmetries.