Abstract
We address the question whether time translation symmetry can be spontaneously broken in a quantum many-body system. One way of detecting such a symmetry breaking is to examine the time-dependence of a correlation function. If the large-distance behavior of the correlation function exhibits a nontrivial time-dependence in the thermodynamic limit, the system would develop a temporal long-range order, realizing a time crystal. In an earlier publication, we sketched a proof for the absence of such time dependence in the thermal equilibrium described by the Gibbs state (Watanabe and Oshikawa in Phys Rev Lett 114:251603, 2015). Here we present a complete proof and extend the argument to a more general class of stationary states than the Gibbs states.
Highlights
Time crystals are a newly proposed state of matter that spontaneously breaks the time translation symmetry
The idea of time crystals in the case of the continuous time translation symmetry was first proposed by Wilczek [1], the validity of the concrete model in this original proposal was soon questioned in Ref. [2]
In a more general setting, the absence of time crystalline orders in the ground state or in the Gibbs state was proven in Ref. [4] without specifying the Hamiltonian but assuming only its locality
Summary
Time crystals are a newly proposed state of matter that spontaneously breaks the time translation symmetry. [4] without specifying the Hamiltonian but assuming only its locality These developments triggered further investigation of so-called Floquet time crystals or discrete time. [4], Fourier transformation of the correlation function was performed with respect to an infinitely long time, out of the validity of the constraint. This issue was recently pointed out by Ref. We examine the conditions on the density operator to which our argument can be straightforwardly extended Clarifying these subtleties and settling down the limitations on what the Gibbs state and similar type of stationary states can do should in turn accelerate our exploration of new states that exhibit nontrivial temporal orders
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