Holographic approach to phase transitions

We provide a description of phase transitions at finite temperature in strongly coupled field theories using holography. For this purpose, we introduce a general class of gravity duals to superconducting theories that exhibit various types of phase transitions (first or second order with both mean and non-mean field behavior) as parameters in their Lagrangian are changed. Moreover the size and strength of the conductivity coherence peak can also be controlled. Our results suggest that certain parameters in the gravitational dual control the interactions responsible for binding the condensate and the magnitude of its fluctuations close to the transition.

Figure 1

Upper: Normalized condensate $⟨{\mathcal{O}}_2⟩$ as a function of temperature at fixed $\rho =1$ for $\mathcal{F}$ given by (5) with $c_3=0$, and different $c_4$’s. A jump in the condensate at the critical temperature is clearly observed for $c_4⪆1$. Dashed lines delimit the metastable region typical in first-order phase transitions. Lower: Free energy difference $\Delta W(T)=W-W_0$ with $W$ given by Eq. (6), $W_0=1/2r_H$ the free energy in the uncondensed phase, $c_3=0$, and different $c_4$’s. $\Delta W$ is not analytic at $T_c$ ($\Delta W(T_c)=0$) for $c_4⪆1$. This is a signature of a first-order phase transition.

Figure 2

$⟨{\mathcal{O}}_2⟩$ condensate as a function of temperature at fixed $\rho =1$ for $\mathcal{F}$ given by (5) with $c_3=-1$ and $c_4=1/2$.

Figure 3

Condensate in the proximities of $T_c$ for $c_{\alpha }=-1$ and different values of the exponent $\alpha$ in (7). The critical exponent $\beta$ (the slope of the curves) clearly depends on $\alpha$. The best linear fit yields $\beta \approx 1$, 0.80, 0.65, 0.5 for $\alpha =3$, 3.25, 3.5, 4, respectively. In fact, solving the equations of motion in power series around the horizon, it is possible to show that the critical exponent $\beta$ is related to $\alpha$ by $\beta =(\alpha -2{)}^{-1}$.

Figure 4

Upper: $\mathrm{Re}[\sigma ]$ (9) as a function of $\omega$ for $T\sim 0.84T_c$, $c_3=0$ and different values of $c_4$. We observe the spectroscopic gap and the coherence peak that becomes narrower as $c_4$ increases. These features are similar to those of certain strongly coupled superconductors [8, 9]. Lower: A similar plot for $c_{\alpha }=1$ and different values of $\alpha$ in (7). The coherence peak is more pronounced as $\alpha$ increases. This is an indication that condensate fluctuations also increase with $\alpha$.