Using a Wigner-function-based approach, we study the Renyi entropy of a subsystem $A$ of a system of bosons interacting with a local repulsive potential. The full system is assumed to be in thermal equilibrium at a temperature $T$ and density $\rho$. For a $\mathcal{U}\left(N\right)$-symmetric model, we show that the Renyi entropy of the system in the large-$N$ limit can be understood in terms of an effective noninteracting system with a spatially varying mean field potential, which has to be determined self-consistently. The Renyi entropy is the sum of two terms: (a) the Renyi entropy of this effective system and (b) the difference in thermal free energy between the effective system and the original translation-invariant system, scaled by $T$. We determine the self-consistent equation for this effective potential within a saddle-point approximation. We use this formalism to look at one- and two-dimensional Bose gases on a lattice. In both cases, the potential profile is that of a square well, taking one value in subsystem $A$ and a different value outside it. The potential varies in space near the boundary of subsystem $A$ on the scale of density-density correlation length. The effect of interaction on the entanglement entropy density is determined by the ratio of the potential barrier to the temperature and peaks at an intermediate temperature, while the high- and low-temperature regimes are dominated by the noninteracting answer.

中文翻译：

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大Napproximation中相互作用热玻色子的仁义熵

使用基于 Wigner 函数的方法，我们研究子系统的 Renyi 熵 $\mathrm{一种}$与局部排斥势相互作用的玻色子系统。假设整个系统在某个温度下处于热平衡状态$吨$ 和密度 $\rho$. 为一个$\mathcal{你}\left(N\right)$- 对称模型，我们证明了系统的 Renyi 熵在大 -$N$可以根据具有空间变化的平均场势的有效非相互作用系统来理解极限，该系统必须自洽地确定。Renyi 熵是两项的总和：(a) 该有效系统的 Renyi 熵和 (b) 有效系统和原始平移不变系统之间的热自由能差，按比例缩放$吨$. 我们在鞍点近似值内确定了该有效势的自洽方程。我们使用这种形式来观察晶格上的一维和二维玻色气体。在这两种情况下，电位剖面都是方形井的剖面，在子系统中取一个值$\mathrm{一种}$以及它之外的不同价值。子系统边界附近的空间电位变化$\mathrm{一种}$在密度-密度相关长度的尺度上。相互作用对纠缠熵密度的影响由势垒与温度的比值和中间温度的峰值决定，而高温和低温区域由非相互作用的答案决定。