Holographic endpoint of spatially modulated phase transition

In a previous paper [S. Nakamura, H. Ooguri, and C. S. Park, Phys. Rev. D 81, 044018 (2010)], we showed that the Reissner-Nordström black hole in the five-dimensional anti–de Sitter space coupled to the Maxwell theory with the Chern-Simons term is unstable when the Chern-Simons coupling is sufficiently large. In the dual conformal field theory, the instability suggests a spatially modulated phase transition. In this paper, we construct and analyze nonlinear solutions which describe the endpoint of this phase transition. In the limit where the Chern-Simons coupling is large, we find that the phase transition is of the second order with the mean field critical exponent. However, the dispersion relation with the Van Hove singularity enhances quantum corrections in the bulk, and we argue that this changes the order of the phase transition from the second to the first. We compute linear response functions in the nonlinear solution and find an infinite off-diagonal DC conductivity in the new phase.

Figure 1

Double well potential $U$ for a classical particle with coordinate $h$. If the particle starts slightly outside of the origin, say, at $A$, then the particle will oscillate between $A$ and another point $B$ with the same potential energy.

Figure 2

The amplitude $h$ and the energy density $\mathcal{H}$ as a function of $k$.

Figure 3

Critical temperature as a function of the Chern-Simons coupling $\alpha$. The shaded region indicates a phase with a nonzero expectation value of the conserved current $\overrightarrow{J}$ which is helical and position dependent.

Figure 4

The amplitude $h_0$ and the energy density difference $\Delta \tilde{\mathcal{H}}$ from the homogeneous phase as functions of $k$.

Figure 5

The expectation value of the order parameter as a function of the temperature. The dotted curve is the numerical result and the solid curve is its fit with $(1-\tau /{\tau }_c{)}^{1/2}$.

Figure 6

A loop diagram that contributes to the two-point correlation function.

Figure 7

Different shapes of the graphs of the free energy $\mathcal{F}$ as a function of $a$ as the parameter $m^2$ changes.

Figure 8

The relation between $m^2$ and $M^2$. Note that for any value of $m^2$, there is always a solution with $M^2>0$.

Figure 9

The real and the imaginary parts of $G_{TT}^R$.

Figure 10

The real and the imaginary parts of ${\sigma }_{TL}$ and ${\sigma }_{LT}$.

Figure 11

The real parts of the diagonal components of the diagonalized conductivity matrix.