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Helical superconducting black holes

Aristomenis Donos, Jerome P. Gauntlett

TL;DR

We address holographic realization of a spatially modulated, helical $p$-wave superconducting phase in a $d=4$ CFT by constructing static, asymptotically $AdS_5$ black holes with $Bianchi~VII_{0}$ symmetry that support a helical order. The five-dimensional model couples a metric to a gauge field $A$ and a complex two-form $C$, yielding an $AdS_5$ vacuum and dual operators of dimensions $ abla=3$ and $ abla=2+|m|$ with angular momentum $l=1$, enabling a charged, angular-momentum-1 order parameter; an electrically charged $AdS-RN$ solution exists and can become unstable to helical hair for suitable $(m,e)$. Thermodynamically, below a critical temperature, a two-parameter family of helical BHs exists and is thermodynamically preferred over the $AdS-RN$ phase; as $T$ decreases, the pitch $k(T)$ decreases to $k_0\approx0.256$ and the $T=0$ ground state is a domain-wall solution with emergent IR scaling. The results provide a controlled holographic realization of a spatially modulated superconducting phase and point to further transport studies and top-down realizations in related $D=5$ models.

Abstract

We construct novel static, asymptotically $AdS_5$ black hole solutions with Bianchi VII$_0$ symmetry that are holographically dual to superconducting phases in four spacetime dimensions with a helical p-wave order. We calculate the precise temperature dependence of the pitch of the helical order. At zero temperature the black holes have vanishing entropy and approach domain wall solutions that reveal homogenous, non-isotropic dual ground states with emergent scaling symmetry.

Helical superconducting black holes

TL;DR

We address holographic realization of a spatially modulated, helical -wave superconducting phase in a CFT by constructing static, asymptotically black holes with symmetry that support a helical order. The five-dimensional model couples a metric to a gauge field and a complex two-form , yielding an vacuum and dual operators of dimensions and with angular momentum , enabling a charged, angular-momentum-1 order parameter; an electrically charged solution exists and can become unstable to helical hair for suitable . Thermodynamically, below a critical temperature, a two-parameter family of helical BHs exists and is thermodynamically preferred over the phase; as decreases, the pitch decreases to and the ground state is a domain-wall solution with emergent IR scaling. The results provide a controlled holographic realization of a spatially modulated superconducting phase and point to further transport studies and top-down realizations in related models.

Abstract

We construct novel static, asymptotically black hole solutions with Bianchi VII symmetry that are holographically dual to superconducting phases in four spacetime dimensions with a helical p-wave order. We calculate the precise temperature dependence of the pitch of the helical order. At zero temperature the black holes have vanishing entropy and approach domain wall solutions that reveal homogenous, non-isotropic dual ground states with emergent scaling symmetry.

Paper Structure

This paper contains 7 sections, 14 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The D=5 model
  3. Black hole solutions
  4. Action and Thermodynamics
  5. Helical superconducting black holes
  6. New ground states at $T=0$
  7. Final Comments

Figures (2)

  • Figure 1: The two-parameter family of helical superconducting black holes and their free-energy $w$. The red line denotes the thermodynamically preferred locus. The blue line is the free-energy of some domain wall solutions. The black lines are lines of constant $T$.
  • Figure 2: Plots of $c_v$ and wave-number $k$, which together fix the helical superconducting order, versus T for the thermodynamically preferred black hole solutions on the red line in fig. 1. The blue dots depict the quantities for the domain wall solutions. Note the scaled axes.