Crystalline geometries from fermionic vortex lattice with hyperscaling violation

We analytically consider the spontaneous formation of a fermionic crystalline geometry in a gravity background with Lifshitz scaling and/or hyperscaling violation. A fermionic vortex lattice solution sourced by the lowest Landau level has been obtained. Thermodynamic analysis shows that the fermionic vortex lattice favors an equilateral triangular configuration, regardless of the values of the Lifshitz scaling $z$ and the hyperscaling violation exponent $\theta$. Our results also show that the larger $z$ or lower $\theta$ leads to deeper minima in the free energy.

Figure 1

The zeroth-order free energy plotted as a function of the Lifshitz scaling $z$ (left plot) and hyperscaling violation exponent $\theta$ (right plot). It decreases (almost linearly) with increasing $z$ as shown in the left plot and increases with increasing $\theta$ as shown in the right plot.

Figure 2

The fourth-order free energy plotted as a function of $a_1$ with $a_2=\frac{a_1^2}{2}$. The left plot corresponds to curves of fourth-order free energy vs $a_1$ for different $\theta$, while the right one refers to the corresponding curves for different $z$. Both plots show a minimum located at the neighbor of $a_1=2$.

Figure 3

The fourth-order free energy plotted as a function of $a_1$ with $a_2=\frac{a_1^2}{2}$ for different $z$, but with fixed exponent $\theta =0$.

Figure 4

The fourth-order free energy plotted as a function of $a_1$ with $a_2=\frac{a_1^2}{2}$. The left plot corresponds to curves of full fourth-order free energy vs $a_1$ for different $\theta$ with fixed $z=3$, while the right one refers to the corresponding curves for different $z$ with fixed $\theta =3$. Both plots show a minimum located in the neighborhood of $a_1\approx 2.69$, which implies an equilateral triangular configuration.

Figure 5

The contour plot of the fourth-order free energy as a function of $a_1$ and $a_2$ for fixed $z=3$, $\theta =3$. The minimum is located around $a1=2.69$ and $a_2=a_1^2/2\simeq 3.61$ as expected.