Fig. 2: Projective P1 and Pgt symmetry model.
a Spacetime tight-binding model with projective P1 symmetry. The projective algebraic relation is [Lx: LT] = −1, where Lx and LT are gauge-modified space and time primitive translations, respectively. The dashed lines indicate a continuous evolution of hopping amplitudes. The three hopping amplitudes are explicitly shown at t = 0, T, 2T. The different thicknesses and lengths of hoppings mean different hopping strengths. The red hoppings at t = T differ from the corresponding blue hoppings at t = 0 by a minus sign. Each spacetime plaquette has flux 3π. The signs marked in red at each site at t = T describe the gauge transformation GT needed to restore the initial connections. b Spacetime tight-binding model with projective Pgt symmetry. The projective algebraic relation is \({{\mathsf{g}}}_{t}{{\mathsf{L}}}_{T}{{\mathsf{g}}}_{t}^{-1}{{\mathsf{L}}}_{T}=-1\), where gt is the gauge-modified time-glide reflection. Glide axes are plotted as dark horizontal lines. The distribution of hopping amplitudes is explicitly marked at five times, respectively. The different thicknesses and lengths of hoppings mean different hopping strengths. A red hopping differs from its blue counterpart with the same thickness and length by a minus sign. Each loop formed by \({{\mathsf{g}}}_{t}{{\mathsf{L}}}_{T}{{\mathsf{g}}}_{t}^{-1}{{\mathsf{L}}}_{T}\) contains π flux. We observe that the glide time-reflection through t = T/2 is manifestly preserved, while that for t = 0 or T is preserved up to a gauge transformation \({{\mathsf{G}}}_{{g}_{t}}\). The signs for \({{\mathsf{G}}}_{{g}_{t}}\) at a given time are marked in red. c Quasibands of U(2T) for the model in (a). d Quasibands of U(2T) for the model in (b). e Quasibands of Floquet model with ordinary Pgt symmetry. The parameter values for (c), (d) and (e) are given in Supplementary Note 5.
