非零温度下的共形场论：算子乘积展开、蒙特卡罗和全息术
Conformal field theories at non-zero temperature: operator product expansions, Monte Carlo, and holography
摘要：
本文计算了2+1维时空中共形场论中守恒味流的非零温度电导率。在比温度$\hbarω>> k_B T$大得多的频率处，可以从电流与在T > 0处获得非零期望值的算子之间的算子乘积展开（OPE）来计算$ω$相关性。这些结果与O（2）Wilson-Fisher CFT的量子MonteCarlo研究结果非常吻合。电导率和其他观测量的结果也得到矢量1/N展开。我们将这些大的$ω$结果与CFT的全息表示的相应的放大器相匹配：全息方法然后允许我们外推到小的$\hbar ω/(k_B T)$。其他的全息研究只隐含地使用了电流和能量-动量张量之间的OPE，这就为一大类CFTs产生了正确的大$ω$行为。然而，对于威尔逊-费舍尔CFT的一个相关的“热”运营商也必须考虑，然后与蒙特卡洛结果的一致性，而无需事先需要的特设重新缩放的T值。我们还建立了广泛的一类CFTs的电导率服从的求和规则。
英文摘要：
We compute the non-zero temperature conductivity of conserved flavor currents in conformal field theories (CFTs) in 2+1 spacetime dimensions. At frequencies much greater than the temperature, $\hbarω>> k_B T$, the $ω$ dependence can be computed from the operator product expansion (OPE) between the currents and operators which acquire a non-zero expectation value at T > 0. Such results are found to be in excellent agreement with quantum Monte Carlo studies of the O(2) Wilson-Fisher CFT. Results for the conductivity and other observables are also obtained in vector 1/N expansions. We match these large $ω$ results to the corresponding correlators of holographic representations of the CFT: the holographic approach then allows us to extrapolate to small $\hbar ω/(k_B T)$. Other holographic studies implicitly only used the OPE between the currents and the energy-momentum tensor, and this yields the correct leading large $ω$ behavior for a large class of CFTs. However, for the Wilson-Fisher CFT a relevant "thermal" operator must also be considered, and then consistency with the Monte Carlo results is obtained without a previously needed ad hoc rescaling of the T value. We also establish sum rules obeyed by the conductivity of a wide class of CFTs.
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