Abstract
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.
Highlights
StableUnstable parts of A, around the ground state described by i.e. cos(pcx)-pattern correspond to the fluctuations of the amplitude or the phase of the order parameter, see figure 2: A0 cos(pcx + δθ), (A0 + δA) cos(pcx),⇔ δIm[A] = A0δθ ⇔ δRe[A] = δA (2.15)In other words, the imaginary part of A, which multiplies a sin(pcx) pattern in (2.14) represents the phason mode around the cos(pcx) ground state, see figure 2.Let’s consider what happens if we introduce a periodic explicit potential with wavevector k, commensurate with pc [54]
We show that the damped Goldstone mode, characterized by finite phase relaxation rate Ω is a generic feature of dynamical spontaneous symmetry breaking in presence of weak explicit sources, and it can be described via a dissipative time dependent Ginzburg-Landau equation (TDGL)
We investigated the nature of the damped Goldstone in the spectrum of holographic models with spontaneous translation symmetry breaking and the associated phase relaxation rate Ω, which has previously been observed in several homogeneous holographic models with broken translations
Summary
Let us first consider the simplest model for spontaneous breaking of global U(1) symmetry described by a complex scalar field Φ with Ginzburg-Landau free energy. The explicit breaking changes the temperature of the phase transition Another important observation here is that at the moment when δφ becomes unstable, the other mode δφ is still decaying with the rate δφ2 ∼ e−2ft. ∂tδφ1 = − (α − f ) + 3β(φf ) δφ1 = 2(α − f )δφ1, ∂tδφ2 = − (α + f ) + β(φf ) δφ2 = −2f δφ, δφ1(t) ∼ e2(α−f)t, δφ2(t) ∼ e−2ft This is the key result of our treatment: in the broken phase the explicit symmetry breaking leads to a damping of “would be” Goldstone mode with the rate Ω = −2f. With, again, the sign of parameter α governing the phase transition This is, identical to the time dependent Ginzburg-Landau equation of the U(1) case (2.3).
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