Phonons under confinement: Coarse-grained $U(1)$ charge and a new transport mechanism

Abstract

An important feature of solid-state or cold atom systems in low dimensions is the restricted oscillations of ionic/atomic degrees of freedom in the confining directions, for which the conventional phonon from canonical quantization is not an ideal description. In this work we propose a general recipe to introduce this feature to otherwise unrestricted systems by mapping displacement fields to spin degrees of freedom, dubbed soft-corization. In the strong-confinement limit $S=1/2$, we demonstrate the validity of the approach with a harmonic chain; the results lead to a gapped fermionic model showing the absence of acoustic modes as a signature of the confinement of the entire chain. Despite the non-conserved total particle number in the presence of nearest-neighbor parings, the model holds a coarse-grained $U(1)$ symmetry with conserved polarization charge in the absence of nearest-neighbor hopping. This enables a new transport mechanism via "dimer-tumbling" instead of single-particle hopping. We then introduce a short-range interaction via anharmonicity; for energy scales slightly above the gap, it leads to a (quantum) phase transition to a nonreciprocal state carrying polarization charge current and heat current with spontaneous time reversal symmetry breaking (TRSB) of the type ${\hat{T}}^2=+1$. The non-reciprocity can be probed by measuring the change of inductance due to gauge fields in and out of the ordered phase.