Phase relaxation and pattern formation in holographic gapless charge density waves

We study the dynamics of spontaneous translation symmetry breaking in holographic models in presence of weak explicit sources. We show that, unlike conventional gapped quantum charge density wave systems, this dynamics is well characterized by the effective time dependent Ginzburg-Landau equation, both above and below the critical temperature, which leads to a “gapless” algebraic pattern of metal-insulator phase transition. In this framework we elucidate the nature of the damped Goldstone mode (the phason), which has earlier been identified in the effective hydrodynamic theory of pinned charge density wave and observed in holographic homogeneous lattice models. We follow the motion of the quasinormal modes across the dynamical phase transition in models with either periodic inhomogeneous or helical homogeneous spatial structures, showing that the phase relaxation rate is continuous at the critical temperature. Moreover, we find that the qualitative low-energy dynamics of the broken phase is universal, insensitive to the precise pattern of translation symmetry breaking, and therefore applies to homogeneous models as well.

Rigorous bounds on dynamical response functions and time-translation symmetry breaking

Dynamical response functions are standard tools for probing local physics near the equilibrium. They provide information about relaxation properties after the equilibrium state is weakly perturbed. In this paper we focus on systems which break the assumption of thermalization by exhibiting persistent temporal oscillations. We provide rigorous bounds on the Fourier components of dynamical response functions in terms of extensive or local dynamical symmetries, i.e., extensive or local operators with periodic time dependence. Additionally, we discuss the effects of spatially inhomogeneous dynamical symmetries. The bounds are explicitly implemented on the example of an interacting Floquet system, specifically in the integrable Trotterization of the Heisenberg XXZ model.

Branched Hamiltonians and time translation symmetry breaking in equations of the Liénard type

Shapere and Wilczek [Phys. Rev. Lett. 109, 160402 and 200402 (2012)] have recently described certain singular Lagrangian systems which display spontaneous breaking of time translation symmetry. We begin by considering the standard Liénard equation for which a Lagrangian is constructed by using the method of Jacobi Last Multiplier. The velocity dependence of the Lagrangian is such that the momentum may exhibit multi-valuedness, thereby leading to the so-called branched Hamiltonian. Next, with a quadratic velocity dependence in the Liénard equation, one can construct a Hamiltonian description involving a position-dependent mass. We compute the Lagrangian and Hamiltonian of this system and show that the canonical Hamiltonian is single valued. However, we find that up to a constant shift, the square of this Hamiltonian describes systems giving rise to spontaneous time translation symmetry breaking provided the potential function is negative.