One-dimensional holographic superconductor from AdS_3/CFT_2 correspondence

TL;DR

This work builds and analyzes a holographic model of a one-dimensional superconductor using the AdS$_3$/CFT$_2$ correspondence, leveraging a BTZ-like gravity dual with a Maxwell field and a charged scalar. A key feature is the degenerate boundary spectrum at $d=2$, necessitating logarithmic regularization of the boundary Green's function and enabling analytic control over current correlators in the normal phase; a scalar hair triggers a superconducting phase with a hard-like gap and a delta function at zero frequency. The study presents exact results for the longitudinal current correlator, dc conductivity in the normal phase, and the full superconducting phase behavior in the probe limit, including condensate scaling, a gap $\omega_g$, and the Ferrell-Glover-Tinkham sum rule, with backreaction analysis clarifying low-frequency scaling and stability. Collectively, the results establish a tractable 1D holographic setup, reveal distinctive 1D transport features, and point to finite-temperature regularization and extensions to other phases and dimensions.

Abstract

We obtain a holographical description of a superconductor by using the d=2 case of the AdS_{d+1}/CFT_d correspondence. The gravity system is a (2+1)-dimensional AdS black hole coupled to a Maxwell field and charged scalar. The dual (1+1)-dimensional superconductor will be strongly correlated. The characteristic exponents for vector perturbations at the boundary degenerate, which implies that d=2 is a critical dimension and the Green's function needs to be regularized. In the normal phase, the current-current correlation function and the conductivity can be analytically solved at zero chemical potential. The dc conductivity can be analytically solved at finite chemical potential. When we add a scalar hair to the black hole, a charged condensate happens at low temperatures. We compare our results with higher-dimensional cases.

Figures (7)

• Figure 1: Real and imaginary parts of the conductivity calculated from AdS$_3$ at finite temperature and zero chemical potential. The horizon is at $z_h=1/(2\pi T)=1$. And $\Lambda=1$. The red, green, and blue lines are the frequency dependence of the conductivity at momentum $k=0$, 4, 8, respectively.
• Figure 2: Real and imaginary parts of the conductivity calculated from AdS$_3$ at finite temperature and finite chemical potential. The horizon is at $z_h=1$. The red, green, and blue solid lines are the frequency dependence of the conductivity at $2\pi T=0.7$, 0.6, 0.1, i.e., $\mu^2=1.2$, 1.6, 3.6, respectively. The dashed lines are the real part of the approximate analytic result at $\mu^2=1.2$, 1.6, 3.6.
• Figure 3: The left panel is the condensate of the operator $\langle{\cal O}_1\rangle$ as a function of temperature, when $m^2=m_{\rm BF}^2$. The right panel is the real part of the conductivity as a function of frequency at low temperatures $T/T_c=0.27$, 0.20, 0.16 (from left to right).
• Figure 4: The left panel is the condensate of the operator $\langle{\cal O}_2\rangle$ as a function of temperature, when $m^2=m_{\rm BF}^2$. The right panel is the real part of the conductivity as a function of frequency at low temperatures $T/T_c=0.21$, 0.17, 0.14 (from left to right).
• Figure 5: The behavior of $\omega{\rm Im}(\sigma)$ as a function of frequency when $\langle{\cal O}_1\rangle$ or $\langle{\cal O}_2\rangle$ condensates. It approaches to a constant as $\omega\to 0$, which implies that there are a pole in Im($\sigma$) and a delta function in Re($\sigma$) at $\omega=0$. The value $\omega_g/T_c$ can also be found in these plots.