Integrable Floquet Time Crystals in One Dimension

We demonstrate the realization of a Discrete Time-Crystal (DTC) phase in a family of periodically driven, one-dimensional quadratic lattice Hamiltonians that can be obtained using spin chains. These interactions preserve integrability while opening controllable gaps at resonant quasienergies and pinning the emergent quasienergy modes that are responsible for subharmonics. We demonstrate that the DTC phase is rigid in the parameter space of transverse field and an additional interaction like Next-Nearest-Neighbor (NNN) coupling strength, with the drive frequency optimized to produce the strongest subharmonic response. We also provide a detailed phase diagram of the model, exhibiting a Floquet Paramagnet (FPM) phase, as well as sharp quantum phase transitions between the FPM and the DTC. Finite-size scaling of the Floquet quasienergy splitting between the emergent subharmonic mode and its conjugate shows that the DTC lifetime diverges exponentially with system size. Thus, our work establishes a novel mechanism for achieving robust long-lived DTCs in one dimension. Motivation for this work stems from the limitations of disorder-based stabilization schemes that rely on many-body localization and exhibit only prethermal or finite-lived plateaus, eventually restoring ergodicity. Disorder-free routes are therefore highly desirable. Integrable (or Floquet-integrable) systems provide an attractive alternative because their extensive set of conserved quantities and constrained scattering strongly restrict thermalization channels. Our construction exploits these integrable restrictions together with longer-range NNN engineering to produce a clean, robust DTC that avoids the prethermal fragility of disordered realizations.