Abstract Wigner’s seminal work on the Poincaré group revealed one of the fundamental principles of quantum theory: symmetry groups are projectively represented. The condensed-matter counterparts of the Poincaré group could be the spacetime groups of periodically driven crystals or spacetime crystals featuring spacetime periodicity. In this study, we establish the general theory of projective spacetime symmetry algebras of spacetime crystals and reveal their intrinsic connections to gauge structures. As important applications, we exhaustively classify (1,1)D projective symmetry algebras and systematically construct spacetime lattice models for them all. Additionally, we present three consequences of projective spacetime symmetry that surpass ordinary theory: the electric Floquet-Bloch theorem, Kramers-like degeneracy of spinless Floquet crystals, and symmetry-enforced crossings in the Hamiltonian spectral flows. Our work provides both theoretical and experimental foundations to explore novel physics protected by projective spacetime symmetry of spacetime crystals.
Symmetry groups are projectively represented in quantum mechanics, and crystalline symmetries are fundamental in condensed matter physics. Here, we systematically present a unified theory of quantum mechanical space groups from two complementary aspects. First, we provide a decomposition form for the space-group factor systems to characterize all quantum space groups. It consists of three factors, the factor system for the translation subgroup
L, an in-homogeneous factor system for the point group
P, and a factor connecting
L and
P. The three factors satisfy three consistency equations, which are exactly solvable and can completely exhaust all factor systems for space groups. Second, since factors systems are classified by the second cohomology group, we show the (co)homology groups for space groups can be derived from Borel's equivariant (co)homology theory, which leads to an algorithm that can compute all (co)homology groups for space groups. To demonstrate the general theory, we explicitly present quantum wallpaper groups with the
Z2 gauge group. Furthermore, as a primitive application, we find the time-reversal invariant quantum space groups with inversion symmetry can lead to a novel clifford band theory, where each band is fourfold degenerate to represent certain real Clifford algebras with topologically nontrivial pinor structures over the Brillouin zone. Our work serves as a foundation for exploring quantum mechanical space groups, and can find applications in spin liquids, unconventional superconductors, and artificial lattice systems, including cold atoms, photonic and phononic crystals, and even LC electric circuit networks.
Abstract In this paper, we develop the quantum theory that has discrete Poincaré symmetry 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 spacetime translations form the discrete Poincaré 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.
Quantum theory and relativity are the pillar theories on which our understanding of physics is based. Poincar\'e invariance is a fundamental physical principle stating that the experimental results must be the same in all inertial reference frames in Minkowski spacetime. It is a basic condition imposed on quantum theory in order to construct quantum field theories, hence, it plays a fundamental role in the standard model of particle physics too. As is well known, Minkowski spacetime follows from clear physical principles, like the relativity principle and the invariance of the speed of light. Here we reproduce such a derivation but leave the number of spatial dimensions
n as a free variable. Then, assuming that spacetime is Minkowski in
1+n dimensions and within the framework of general probabilistic theories, we reconstruct the qubit Bloch ball and finite dimensional quantum theory, and obtain that the number of spatial dimensions must be
n=3, from Poincar\'e invariance and other physical postulates. Our results suggest a fundamental physical connection between spacetime and quantum theory.
Abstract Time crystals are physical systems whose time translation symmetry is spontaneously broken. Although the spontaneous breaking of continuous time-translation symmetry in static systems is proved impossible for the equilibrium state, the discrete time-translation symmetry in periodically driven (Floquet) systems is allowed to be spontaneously broken, resulting in the so-called Floquet or discrete time crystals. While most works so far searching for time crystals focus on the symmetry breaking process and the possible stabilising mechanisms, the many-body physics from the interplay of symmetry-broken states, which we call the condensed matter physics in time crystals, is not fully explored yet. This review aims to summarise the very preliminary results in this new research field with an analogous structure of condensed matter theory in solids. The whole theory is built on a hidden symmetry in time crystals, i.e., the phase space lattice symmetry, which allows us to develop the band theory, topology and strongly correlated models in phase space lattice. In the end, we outline the possible topics and directions for the future research.
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.
Time crystals are physical systems whose time translation symmetry is spontaneously broken. Although the spontaneous breaking of continuous time-translation symmetry in static systems is proved impossible for the equilibrium state, the discrete time-translation symmetry in periodically driven (Floquet) systems is allowed to be spontaneously broken, resulting in the so-called Floquet or discrete time crystals. While most works so far searching for time crystals focus on the symmetry breaking process and the possible stabilising mechanisms, the many-body physics from the interplay of symmetry-broken states, which we call the condensed matter physics in time crystals, is not fully explored yet. This review aims to summarise the very preliminary results in this new research field with an analogous structure of condensed matter theory in solids. The whole theory is built on a hidden symmetry in time crystals, i.e., the phase space lattice symmetry, which allows us to develop the band theory, topology and strongly correlated models in phase space lattice. In the end, we outline the possible topics and directions for the future research.