Abstract
The Bloch theorem is a general theorem restricting the persistent current associated with a conserved U(1) charge in a ground state or in a thermal equilibrium. It gives an upper bound of the magnitude of the current density, which is inversely proportional to the system size. In a recent paper, Else and Senthil applied the argument for the Bloch theorem to a generalized Gibbs ensemble, assuming the presence of an additional conserved charge, and predicted a nonzero current density in the nonthermal steady state [D. V. Else and T. Senthil, Phys. Rev. B 104, 205132 (2021)]. In this work, we provide a complementary derivation based on the canonical ensemble, given that the additional charge is strictly conserved within the system by itself. Furthermore, using the example where the additional conserved charge is the momentum operator, we discuss that the persistent current tends to vanish when the system is in contact with an external momentum reservoir in the co-moving frame of the reservoir.
Highlights
The Bloch theorem is a fundamental theorem stating that the current density of a conserved U(1) charge vanishes in thermodynamically large systems [1,2,3,4,5,6]
The theorem was originally derived for a ground state [1], it has been generalized to a thermal equilibrium described by a canonical ensemble and a grand canonical ensemble [2,4,5]
When the system is in contact with an external momentum reservoir, we find that the velocities of the system and the reservoir must coincide in the generalized Gibbs ensemble, implying that the persistent current density vanishes in the laboratory frame where the reservoir is assumed to be stationary
Summary
The Bloch theorem is a fundamental theorem stating that the current density of a conserved U(1) charge vanishes in thermodynamically large systems [1,2,3,4,5,6]. The generalized Gibbs ensemble usually describes the quench dynamics of isolated systems with an extensive number of conserved charges [8,9], here the system has only a few number of conserved quantities: the Hamiltonian H , the U(1) charge Qthat defines the current density, and the additional charge. Whenis the momentum operator, a nonzero persistent current implies a relative motion between the system and the reservoir, and whether such a motion persists after equilibration needs to be clarified. We address these issues in this paper. When the system by itself is playing the role of the reservoir for its subsystem [8], this argument implies a uniform flow over the entire system
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