We demonstrate the emergence of an intrinsic space-time crystalline order in a long ferromagnetic ${\phi }_0$ Josephson junction on a topological insulator without any external periodic modulation. The presence of the exchange and Dzyaloshinskii-Moriya interactions in a ferromagnetic layer with broken inversion symmetry internally modulates the critical current due to the coupling between the magnetic moment and the Josephson phase. This breaks the time translation symmetry, leading to the appearance of the space-time crystalline pattern in the spatiotemporal dependence of the in-plane current, which oscillates with almost twice the ferromagnetic resonance frequency. In the limit where the critical current is not modulated internally, the space-time crystalline order does not occur. In this case, only when an external parametric modulation is applied, the system exhibits a typical classical discrete space-time crystalline order that oscillates at half of the modulation frequency. Considering the still-pending problem of experimental detection, we demonstrate that a recently developed magnetometry device, which visualizes the supercurrent flow at the nanoscale, can be used to detect space-time crystalline patterns in hybrid Josephson junctions experimentally.

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• Dzyaloshinskii-Moriya interaction
• Time crystals
• Topological insulators
• Josephson junctions
• Sine-Gordon equation
• Landau-Lifschitz-Gilbert equation