Abstract
There has been a surge of interest in effective non-Lorentzian theories of excitations with restricted mobility, known as fractons. Examples include defects in elastic materials, vortex lattices or spin liquids. In the effective theory novel coordinate-dependent symmetries emerge that shape the properties of fractons. In this review we will discuss these symmetries, cover the effective description of gapless fractons via elastic duality, and discuss their hydrodynamics.
Highlights
Fractons are usually identified with excitations of a system that are immobile or have restricted mobility - they can propagate along some spatial directions but not along others
Following Pretko and Radzihovsky we show that symmetric elasticity in two spatial dimensions can be mapped via a duality transformation to the symmetric tensor gauge theories, whose geometric structure corresponds precisely to the Heisenberg group
This latter observation leads to an extension of the celebrated Coleman-Hohenberg-Mermin-Wagner (CHMW) theorem, which forbids the spontaneous breaking of continuous internal symmetries in equilibrium field theories and their corresponding Nambu-Goldstone bosons (NGBs) in dimensions one and two [15,16,17]
Summary
Fractons are usually identified with excitations of a system that are immobile or have restricted mobility - they can propagate along some spatial directions but not along others. Gromov initiated a systematic classification of fracton phases of matter based on symmetry principles privileging the charge and dipole symmetries and their higher-order generalizations–the multipole algebra [5] He noted that the multipole algebra for a scalar field theory was on-the-nose the same as the so-called polynomial shift symmetries that had been studied previously by Griffin, Grosvenor, Hořava and Yan (GGHY) in the context of technical naturalness in non-relativistic quantum field theories [6, 7]. The best understood example consist of the symmetric tensor gauge theories, whose geometric nature can be understood in terms of the Heisenberg symmetry group and the associated group manifold.
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