Towards A Holographic Model of D-Wave Superconductors

TL;DR

This work extends holographic superconductor models to D-wave symmetry by introducing a charged symmetric traceless tensor field $B_{\mu\nu}$ in a 3+1D AdS black hole with a U(1) gauge field, analyzed in the probe limit. The condensate forms below a critical temperature $T_c$ with mean-field scaling $\langle O_{ij}\rangle \sim (T_c-T)^{1/2}$, breaking rotational symmetry down to $Z_2$, while preserving translation symmetry. Linear-response transport shows an isotropic AC conductivity with a zero-frequency delta function below $T_c$ and no hard gap, with a soft gap scale $\omega_g/T_c\simeq 13$; above $T_c$ the delta function vanishes and $\mathrm{Re}\,\sigma(\omega)$ is constant. The model provides a holographic realization of D-wave superconductivity, offering insights into nodal excitations and transport in unconventional superconductors within the AdS/CFT framework.

Abstract

The holographic model for S-wave high T_c superconductors developed by Hartnoll, Herzog and Horowitz is generalized to describe D-wave superconductors. The 3+1 dimensional gravitational theory consists a symmetric, traceless second-rank tensor field and a U(1) gauge field in the background of the AdS black hole. Below T_c the tensor field which carries the U(1) charge undergoes the Higgs mechanism and breaks the U(1) symmetry of the boundary theory spontaneously. The phase transition characterized by the D-wave condensate is second order with the mean field critical exponent beta = 1/2. As expected, the AC conductivity is isotropic below T_c and the system becomes superconducting in the DC limit but has no hard gap.

Figures (2)

• Figure 1: (color online) The dimensionless D-wave condensate $f_{1}/\mu ^{2-\Delta _{-}}$ shown as a function of $T/T_{c}$ for $m^{2}=-1/4$. The condensate goes to zero at $T=T_{c}\propto \mu$. The critical exponent $f_{1}\rightarrow c\left( T_{c}-T\right) ^{\beta }$ for $T_{c}-T\rightarrow 0^{+}$ is of the mean field value $\beta =1/2$.
• Figure 2: (color online) The real (left plot) and imaginary (right plot) of conductivity shown as a function for frequency $\omega$ for different temperatures. Above $T_{c}$, $Re[\sigma \left( \omega \right) ]=1$ while $Im[\sigma \left( \omega \right) ]=0$. Below $T_{c}$, $Re[\sigma \left( \omega \right) ]$ has a $\delta \left( \omega \right)$ delta function whose height decrease in $T$ and vanishes at $T_{c}$. The right most curve has the lowest $T$, which implies the zero temperature gap $\omega_g/T_c \simeq 13$. (The construction in Hartnoll:2008vx for S-wave gives the value $8$ for this gap.)