AI 中文总结
本文研究带指数熵修正的五维爱因斯坦-杨-米尔斯黑洞，解析推导其热力学量，发现指数修正移动热容发散点，数值求解磁杨-米尔斯微扰得到纯虚数不稳定模及相关塔状结构。
AI 中文摘要
由高维吴-杨 ansatz 得到的五维爱因斯坦-杨-米尔斯黑洞，其度规函数带有对数形式的非阿贝尔贡献，为研究四维以上规范电荷如何重塑视界热力学提供了可解析处理的场景。我们推导了质量-半径关系、霍金温度、熵、杨-米尔斯势及热力学第一定律的表达式，并记录了曲率不变量，结果显示该对数项会强化中心奇点而非使其平滑。随后加入非微扰的指数熵修正，解析得到修正后的热容与自由能。该几何在 $r_h=|Q|$ 处为极值，热容在 $r_h=\sqrt{3}\\,|Q|$（戴维斯型点）处发散，修正会使该点向内移动，$r_{\rm tr}/|Q|\simeq\sqrt{3}-9\lambda\alpha\eta\\,e^{-3\sqrt{3}\lambda}$。利用广义离壳自由能，我们发现稳定-不稳定缺陷对携带守恒拓扑荷 $W=0$，仅其内部分子会对修正产生响应。接着求解球对称磁杨-米尔斯微扰部分（探测规范场本身而非标量测试场），两种独立数值方案的吻合度优于 $1.2\times10^{-7}$，得到纯虚数的不稳定模，其增长率随 $Q/r_h$ 从 $0$ 时的 $\u0393_0r_h=0.414753$ 单调降至 $Q/r_h=0.99$ 时的 $0.317520$，且在戴维斯标度处无特征。由于有效势随 $-9/(4r_*^2)$ 衰减，系数高于临界值 $1/4$，该部分支持无穷多几何塔状不稳定模，在 $\u0393\to0$ 处积累，且电荷无关的比率 $\u0393_{n+1}/\u0393_n\to e^{-\pi/\sqrt{2}}\simeq0.108453$，该结果也得到数值验证。
英文摘要
The five-dimensional Einstein--Yang--Mills black hole obtained from the higher-dimensional Wu--Yang ansatz carries a logarithmic non-Abelian contribution to the metric function, making it an analytically tractable setting for studying how gauge charge reshapes horizon thermodynamics beyond four dimensions. We derive expressions for the mass--radius relation, the Hawking temperature, the entropy, the Yang--Mills potential and the first law, and record the curvature invariants, which show that the logarithm sharpens the central singularity rather than smoothing it. A non-perturbative exponential entropy correction is then added, and the corrected heat capacity and free energies are obtained analytically. The geometry is extremal at $r_h=|Q|$ and the heat capacity diverges at $r_h=\sqrt{3}\,|Q|$, a Davies-type point. The correction moves that point inward, $r_{\rm tr}/|Q|\simeq\sqrt{3}-9λαη\,e^{-3\sqrt{3}λ}$. Using the generalized off-shell free energy we find a conserved topological charge $W=0$ carried by a stable--unstable defect pair whose inner member alone responds to the correction. We then solve the spherically symmetric magnetic Yang--Mills perturbation sector, which probes the gauge field itself rather than a test scalar. Two independent numerical schemes agree to better than $1.2\times10^{-7}$ and give a purely imaginary unstable mode whose growth rate falls monotonically from $Γ_{0}r_h=0.414753$ at $Q/r_h\to0$ to $0.317520$ at $Q/r_h=0.99$, passing through the Davies scale without any feature. Because the effective potential decays as $-9/(4r_{*}^{2})$, with a coefficient above the critical value $1/4$, the sector supports an infinite geometric tower of unstable modes accumulating at $Γ\to0$ with the charge-independent ratio $Γ_{n+1}/Γ_{n}\to e^{-π/\sqrt{2}}\simeq0.108453$, which we confirm numerically.
