Coherent/incoherent metal transition in a holographic model

TL;DR

The paper analyzes AC electric, thermoelectric, and thermal conductivities in a 3+1D holographic Einstein–Maxwell–scalar model with momentum relaxation from massless scalars linear in space. It introduces a robust numerical method for computing retarded Green's functions of coupled bulk fluctuations and derives analytic Drude-like parameters in the coherent regime, identifying a coherent/incoherent metal transition controlled by $β$ relative to the chemical potential $μ$. At low frequencies ($β<μ$) the conductivities follow a modified Drude form $σ(ω) ≈ rac{K τ}{1 - i ω τ} + σ_Q$, with $τ$ approaching $2√3 μ/β^2$ in the appropriate limit, while for $β>μ$ the peaks are non-Drude, indicating incoherence; at intermediate frequencies ($T<ω<μ$) no robust scaling laws emerge. The work provides a practical numerical framework for multi-field holographic transport and offers insights into momentum-dissipation-driven coherent/incoherent behavior relevant to strongly correlated materials like cuprates.

Abstract

We study AC electric($σ$), thermoelectric($α$), and thermal($\barκ$) conductivities in a holographic model, which is based on 3+1 dimensional Einstein-Maxwell-scalar action. There is momentum relaxation due to massless scalar fields linear to spatial coordinate. The model has three field theory parameters: temperature($T$), chemical potential($μ$), and effective impurity($β$). At low frequencies, if $β< μ$, all three AC conductivities($σ, α, \barκ$) exhibit a Drude peak modified by pair creation contribution(coherent metal). The parameters of this modified Drude peak are obtained analytically. In particular, if $β\ll μ$ the relaxation time of electric conductivity approaches to $2\sqrt{3} μ/β^2$ and the modified Drude peak becomes a standard Drude peak. If $β> μ$ the shape of peak deviates from the Drude form(incoherent metal). At intermediate frequencies($T<ω<μ$), we have analysed numerical data of three conductivities($σ, α, \barκ$) for a wide variety of parameters, searching for scaling laws, which are expected from either experimental results on cuprates superconductors or some holographic models. In the model we study, we find no clear signs of scaling behaviour.

Figures (9)

• Figure 1: Electric conductivity without momentum relaxation ($\beta=0$).
• Figure 2: Electric conductivity $\sigma$ with momentum relaxation at fixed $\mu/T = 6$. For larger $\beta$ the Drude-like peak at small $\omega$ becomes broader. As we increase $\beta$, the Drude peak disappears and the transition to incoherent metal is manifest.
• Figure 3: Electric conductivity $\sigma$ with momentum relaxation at fixed $\beta/T = 3$. By comparing with Figure \ref{['beta0']} we may also see how $\beta$ changes conductivity curves since all parameters are the same except $\beta$. As we decrease $\mu$, the Drude peak disappears and the transition to incoherent metal is manifest.
• Figure 4: Relaxation time $\tau$ at small $\omega$ as a function of $\mu/T$ and $\beta/T$. We do not plot the range $\beta/T < 1$ since $\tau$ diverges quickly as $\beta$ goes to zero.
• Figure 5: We compare numerical data(blue dotted lines) with a Drude model(red solid curves)\ref{['Drude00']} of which parameters are fixed analytically in \ref{['Jo0']} and \ref{['tauu']}. ${\mu/T} = 4$. When ${\beta}/{\mu} \le 1/2$ the numerical data agree well to the Drude model. The transition to incoherent metal is around $\beta/\mu \sim 1/2$.