Holographic Fermi liquids in a spontaneously generated lattice

We discuss fermions in a spontaneously generated holographic lattice background. The lattice structure at the boundary is generated by introducing a higher-derivative interaction term between a $U(1)$ gauge field and a scalar field. We solve the equations of motion below the critical temperature at which the lattice forms and analyze the change in the Fermi surface due to the lattice. The fermion band structure is found to exhibit a gap due to lattice effects.

Figure 1

Numerical solutions for the metric functions, $Q_{tt}(x,z)$, $Q_{zz}(x,z)$, $Q_{xx}(x,z)$, and $Q_{yy}(x,z)$, for $\frac{\eta }{{\mu }^2}=0.41$, $\frac{{\eta }^{\prime }}{{\mu }^4}=0.005$, $q=0$, $\mathrm{\Delta }=1$.

Figure 2

The charge density of the system for $\frac{\eta }{{\mu }_0^2}=0.41$, $\frac{{\eta }^{\prime }}{{\mu }_0^4}=0.005$, $q=0$, $\mathrm{\Delta }=1$, and $\xi =0.1$.

Figure 3

The scalar field of the system with first order contributions included for parameters $\frac{\eta }{{\mu }_0^2}=0.41$, $\frac{{\eta }^{\prime }}{{\mu }_0^4}=0.005$, $q=0$, $\mathrm{\Delta }=1$, and $\xi =0.1$.

Figure 4

The spectral function $A^0(\omega ,k_x,k_y=0)$ at the critical temperature calculated numerically, showing a peak at $k_F=2.573$, for $q_f=1.7$, ${\mu }_0=2.35$, and scalar $\mathrm{\Delta }=1$.

Figure 5

The spectral function $A^0(\omega ,k_x=1.5,k_y=1.9432)$ at the critical temperature calculated numerically, showing a peak at $k_F=\sqrt{k_x^2+k_y^2}=2.4548$, for $q_f=1.7$, ${\mu }_0=2.2507$, and scalar $\mathrm{\Delta }=1$.

Figure 6

The spectral function $A^0(\omega =-0.0001,k_x=1.5,k_y)$ at the critical temperature calculated numerically, showing peaks at $k_y=\pm 1.9432$, implying $k_F=\sqrt{k_x^2+k_y^2}=2.4548$, for $q_f=1.7$, ${\mu }_0=2.2507$, and scalar $\mathrm{\Delta }=1$.

Figure 7

The spectral function $A^l(\omega =-0.0001,k_x,k_y)$ at the critical temperature calculated numerically for $l=-1,0,+1$, $q_f=1.56$, ${\mu }_0=2.2507$, and scalar $\mathrm{\Delta }=1$.

Figure 8

Plot of scaling parameter, ${\nu }_{k_l}$, as a function of fermion charge, $q_f$, at $k_l=k_F$ for parameters $\frac{\eta }{{\mu }_0^2}=0.25$, $\frac{{\eta }^{\prime }}{{\mu }_0^4}=0.005$, $q=0$, $\mathrm{\Delta }=1$, ${\mu }_0=2.2507$, $\frac{T}{{\mu }_0}=0.0613$, and tiny frequecy $\omega =0.0001$ respectively.

Figure 9

Plot of spectral function $A(\omega ,k_x=k,k_y=2.8538)$ vs. $\omega$ for parameters $\frac{\eta }{{\mu }_0^2}=0.25$, $\frac{{\eta }^{\prime }}{{\mu }_0^4}=0.005$, $q=0$, $\mathrm{\Delta }=1$, $q_f=2$, ${\mu }_0=2.2507$, $\frac{T}{{\mu }_0}=0.0613$ and for $\xi =0.0$, 0.05, 0.07, 0.1 respectively.

Figure 10

Plot of spectral function $A(\omega =0.0005,k_x=k,k_y)$ vs. $k_y$ for parameters $\frac{\eta }{{\mu }_0^2}=0.25$, $\frac{{\eta }^{\prime }}{{\mu }_0^4}=0.005$, $q=0$, $\mathrm{\Delta }=1$, $q_f=1.8$, ${\mu }_0=2.2507$, $\frac{T}{{\mu }_0}=0.0613$ and for $\xi =0.0$, 0.07, 0.075, respectively.

Figure 11

Plot of spectral function, $A(\omega ,k_x=k,k_y=0.83)$, as a function of $\omega$ for non-Fermi liquid case, ${\nu }_{k_l}<1/2$, with parameters $\frac{\eta }{{\mu }_0^2}=0.25$, $\frac{{\eta }^{\prime }}{{\mu }_0^4}=0.005$, $q=0$, $\mathrm{\Delta }=1$, $q_f=1.15$, ${\mu }_0=2.2507$, $\frac{T}{{\mu }_0}=0.0613$ and for $\xi =0.0$, 0.25, 0.3, respectively.