Local quenches in fracton field theory: Lieb-Robinson bound, noncausal dynamics and fractal excitation patterns

We study the out-of-equilibrium dynamics induced by a local perturbation in fracton field theory. For the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi mathvariant="double-struck">Z</a:mi><a:mn>4</a:mn></a:msub></a:math>- and <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"><d:msub><d:mi mathvariant="double-struck">Z</d:mi><d:mn>8</d:mn></d:msub></d:math>-symmetric free fractonic theories, we compute the time dynamics of several observables such as the two-point Green’s function, <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mo stretchy="false">⟨</g:mo><g:msup><g:mi>ϕ</g:mi><g:mn>2</g:mn></g:msup><g:mo stretchy="false">⟩</g:mo></g:math> condensate, energy density, and the dipole momentum. The time-dependent considerations highlight that the free fractonic theory breaks causality and exhibits instantaneous signal propagation, even if an additional relativistic term is included to enforce a speed limit in the system. We show that it is related to the fact that the Lieb-Robinson bound does not hold in the continuum limit of the fracton field theory, and the effective bounded speed of light does not emerge. For the theory in finite volume, we show that the fracton wave front acquires fractal shape with nontrivial Hausdorff dimension and argue that this phenomenon cannot be explained by a simple self-interference effect. Published by the American Physical Society 2024