[논문 리뷰] Particle-hole instability in the $AdS_4$ holography

Elena Gubankova|arXiv (Cornell University)|2010. 06. 24.

Black Holes and Theoretical Physics참고 문헌 31인용 수 4

한 줄 요약

이 논문은 전하를 띤 블랙홀 배경과 외부 자기장이 존재하는 조건에서 $AdS_4$ 호로그래피의 한계에서 한계 상태에서의 입자-홀 쌍정 불안정성을 일주기 피드백 효과를 통해 분석한다. 계산 결과, 쌍정의 임계 온도 $T_c$ 는 자기장에 비례하며, 비페르미 액체 상태에 진입할수록 0으로 수렴한다. 이는 순서 매개변수에 의해 피어슨 표면 운동량 $δ k_F$ 가 이동하는 것으로 나타나며, 그래핀에서의 자기장 촉진 현상과 강한 상관계가 있는 캐릭터 스피RAL의 이론적 기반을 제공한다.

ABSTRACT

We show that particle-hole pairing is realized in the background of a charged black hole in magnetic field. The pairing instability occurs for sufficiently large fermion charges, which correspond to the Fermi liquid regime. The critical temperature for Fermi liquids is proportinal to the magnetic field and vanishes as we approach the non-Fermi liquid state. The pairing order parameter leads to a relative shift of the Fermi surfaces corresponding to the bulk fermions with spin up and down. The value of the shift in Fermi momentum $k_F$ and the critical temperature $T_c$ are proportional to the effective density of states at the Fermi surface. Our one-loop calculations provide a dual description of the magnetic catalysis for the lowest Landau level in graphene. This analyses may be relevant for the antiferromagnetic behavior in the cuprate superconductors and for the chiral spirals in the chiral magnetic effect. We also discuss thermodynamic and transport properties of a system at the boundary at zero magnetic field. The scaling behavior of the specific heat is $c\sim T$ for Fermi liquid and $c\sim T^{2ν}$ for non-Fermi liquid, while the behavior of the DC conductivity is the same $σ\sim T^{-2ν}$ in both cases. While it can be difficult to extract transport and hydrodynamic from the lattice, the $AdS/CFT$ approach provides a robust frame for nonperturbative calculation of these properties.

연구 동기 및 목표

• To investigate particle-hole pairing instabilities in $AdS_4$ holography under a magnetic field using one-loop fermionic determinants.
• To determine the conditions under which particle-hole pairing leads to a superconducting-like instability in the boundary Fermi liquid regime.
• To establish a dual description of magnetic catalysis in $(2+1)$-dimensional systems such as graphene via the $AdS/CFT$ correspondence.
• To analyze the scaling behavior of specific heat and DC conductivity in both Fermi and non-Fermi liquid phases.
• To connect the holographic results to real-world phenomena such as antiferromagnetism in cuprates and chiral magnetic effects.

제안 방법

• Utilizes one-loop fermionic effective action via quasinormal mode expansion of the retarded Green's function in a charged black hole background.
• Applies a variational approach to compute the fermion determinant using poles of the retarded Green function, derived from quasinormal modes.
• Constructs a non-local Ginsburg-Landau-type action in the bulk using bulk fermion propagators and radial profile of the pairing order parameter.
• Solves the gap equation in the lowest Landau level approximation, incorporating the magnetic field via the effective coupling $G_{\text{int}}|q\mathcal{H}|$.
• Derives the critical temperature $T_c$ from the condition of negative modes in the one-loop effective action, leading to a transcendental equation involving the digamma function $\Psi(z)$.
• Introduces a filling factor $\eta_{\mathcal{H}} = \mathcal{H}_c / \mathcal{H}$ to characterize the phase transition, with $T_c = 0$ for $\eta_{\mathcal{H}} > 1$.

실험 결과

연구 질문

• RQ1 Under what conditions does particle-hole pairing become unstable in the $AdS_4$ holographic model with a magnetic field?
• RQ2 How does the critical temperature $T_c$ for particle-hole pairing depend on the magnetic field and fermion charge in the Fermi liquid regime?
• RQ3 What is the holographic dual of magnetic catalysis in graphene, and how is it related to the shift in Fermi surface momentum $\delta k_F$?
• RQ4 How do the thermodynamic properties—specific heat and DC conductivity—scale in the Fermi and non-Fermi liquid phases?
• RQ5 What is the role of the radial profile of the order parameter $\Delta(r)$ in determining the critical temperature and the nature of the phase transition?

주요 결과

• The critical temperature $T_c$ for particle-hole pairing is proportional to the magnetic field $\mathcal{H}$ and vanishes as the system approaches the non-Fermi liquid state, with $T_c = 0$ when $\eta_{\mathcal{H}} > 1$.
• The order parameter $\Delta(r)$ induces a relative shift $\delta k_F$ in the Fermi surfaces of spin-up and spin-down fermions, with $\delta k_F \propto \text{Im}\left[\Psi\left(\frac{iz_*}{2\pi T} + \frac{1}{2}\right)\right]$.
• The specific heat scales as $c \sim T$ in the Fermi liquid phase and $c \sim T^{2\nu}$ in the non-Fermi liquid phase, with $\nu$ related to the scaling dimension of the operator.
• The DC conductivity scales as $\sigma \sim T^{-2\nu}$ in both Fermi and non-Fermi liquid regimes, indicating universal hydrodynamic behavior.
• The gap equation yields $T_c = \frac{G_{\text{int}}|q\mathcal{H}|}{2\pi R^8}(1 - \eta_{\mathcal{H}}^2) \int dr \sqrt{-g} (\psi^{0\dagger}\sigma^1\psi^0)^2$, showing explicit dependence on the magnetic field and radial profile.
• The filling factor $\eta_{\mathcal{H}} = \mathcal{H}_c / \mathcal{H}$ determines the phase transition: $T_c > 0$ only when $\eta_{\mathcal{H}} < 1$, corresponding to $n < n_c$ or $\mathcal{H} > \mathcal{H}_c$.