Electron-Phonon Interactions, Metal-Insulator Transitions, and Holographic Massive Gravity

Massive gravity is holographically dual to “realistic” materials with momentum relaxation. The dual graviton potential encodes the phonon dynamics, and it allows for a much broader diversity than considered so far. We construct a simple family of isotropic and homogeneous materials that exhibit an interaction-driven metal-insulator transition. The transition relates to the formation of polarons—phonon-electron quasibound states that dominate the conductivities, shifting the spectral weight above a mass gap. We characterize the polaron gap, width, and dispersion.

Figure 1

Development of polaron formation as temperature is decreased. $T=1,0.46(T_2),0.35,0.3(T_1),0.27,0.22,0.17,0.04$. Inset: motion of the lightest quasi-normal-mode (QNM) for the same temperatures (corresponding colors) and some intermediate additional values (purple dots). At large $T$, the QNM separates from the real axis with decreasing $T$, until it collides with the next QNM (near $T_1$) and forms a pair of conjugated poles with positive and negative real parts—the polaron particle-antiparticle poles. (Similar QNM collisions have been observed, e.g., in [27, 28]. In our case, it is crucial that, after collision, the QNM approaches the real axis.)

Figure 2

(a) The polaron energy-gap ${\omega }_0$ as a function of $T$, extracted numerically as the peak position in $\mathrm{Abs}[\sigma (\omega )]$. ${\omega }_0$ detaches from 0 at $T_1$. (b) dc conductivity ${\sigma }_{\text{dc}}(T)$ as a function of $T$. The continuous line is the analytic result (11) and the dots are the numerical computation. One observes two critical temperatures: the maximum in ${\sigma }_{\text{dc}}(T)$ marking the separation between metallic-insulating behavior ($T_2$), and the polaron gap formation ($T_1$).

Figure 3

Motion of the polaron peak with wave number. $T=0.04$, $k=0,0.5,0.8,1.2,2$. Inset: the extracted dispersion relation.