Imaginary Time Crystal of Thermal Quantum Matter

Temperature is a fundamental thermodynamic variable for matter. Physical observables are often found to either increase or decrease with it, or show a non-monotonic dependence with peaks signaling underlying phase transitions or anomalies. Statistical field theory has established connection between temperature and time: a quantum ensemble with inverse temperature \beta is formally equivalent to a dynamic system evolving along an imaginary time from 0 to i\beta in the space one dimension higher. Here we report that a gas of hard-core bosons interacting with a thermal bath manifests an unexpected temperature-periodic oscillation of its macroscopic observables, arising from the microscopic origin of space-time locked translational symmetry breaking and crystalline ordering. Such a temperature crystal, supported by quantum Monte Carlo simulation, generalizes the concept of purely spatial density-wave order to the imaginary time axis for Euclidean action.