通过BTZ黑洞视界附近的带电标量场研究量子临界性
Investigating quantum criticality through charged scalar fields near the BTZ black hole horizon
AI总结：
本文通过研究BTZ黑洞视界附近具有位置相关质量的带电标量场，在有无外部磁场两种情形下确定了与耦合常数相关的量子临界点，揭示了系统从不稳定耗散态到类超导稳定传播态的转变。
摘要：
我们研究了一个具有位置相关质量 \\( m(ρ) = m_0 + \mathcal{S}(ρ) \\) 的带电标量场，其中 \\(\mathcal{S}(ρ)\\) 表示洛伦兹标量势，该场位于存在外部磁场的BTZ黑洞附近。通过推导该体系的Klein-Gordon方程，我们探讨了两种情形：(i) 质量修正标量场 \\(m(ρ) = m_0 - a/ρ\\)（精确可解情形），以及 (ii) 同时涉及质量修正和外部磁场的情形（条件精确可解）。我们确定了与耦合常数 \\(a\\) 相关的量子临界点（QCPs）。在第一种情形中，对于无质量带电标量场，临界点出现在 \\(a = n + 1/2\\)，适用于所有径向量子数 \\(n \geq 0\\) 和磁量子数 \\(|m| \geq 0\\)。在第二种情形中，这些临界点移动到 \\(a = n + 3/2\\)，适用于 \\(n \geq 0\\) 和 \\(|m| > 0\\)，其中 \\(m = 0\\) 的情形被排除。对于有质量标量场，量子临界点出现在 \\(a = (n + 1/2)/2\\)，导致零频率下的非传播场。在这些量子临界点处，场频率降为零，标志着从稳定振荡模式到非传播状态的转变。在临界点以下，系统表现出不稳定性，其特征为负虚频率，暗示快速衰减和高耗散。在临界点以上，模式稳定并传播，表明向类超导相的转变，其中耗散消失且稳定激发占主导地位。
英文摘要：
We examine a charged scalar field with a position-dependent mass \( m(ρ) = m_0 + \mathcal{S}(ρ) \), where \(\mathcal{S}(ρ)\) represents a Lorentz scalar potential, near a BTZ black hole in the presence of an external magnetic field. By deriving the Klein-Gordon equation for this setup, we explore two scenarios: (i) a mass-modified scalar field with \(m(ρ) = m_0 - a/ρ\) (an exactly solvable case), and (ii) a scenario involving both mass modification and an external magnetic field (conditionally exactly solvable). We identify quantum critical points (QCPs) associated with the coupling constant \(a\). In the first scenario, for massless charged scalar fields, critical points occur at \(a = n + 1/2\) for all radial quantum numbers \(n \geq 0\) and magnetic quantum numbers \(|m| \geq 0\). In the second scenario, these critical points shift to \(a = n + 3/2\) for \(n \geq 0\) and \(|m| > 0\), with the case \(m = 0\) excluded. For massive scalar fields, QCPs emerge at \(a = (n + 1/2)/2\), leading to non-propagating fields at zero frequency. At these QCPs, the field frequencies drop to zero, marking transitions from stable oscillatory modes to non-propagating states. Below the critical points, the system exhibits instability, characterized by negative imaginary frequencies that suggest rapid decay and high dissipation. Above the critical points, the modes stabilize and propagate, indicating a transition to a superconducting-like phase, where dissipation vanishes and stable excitations dominate.
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