We prove a lower bound for the free energy (per unit volume) of the two-dimensional Bose gas in the thermodynamic limit. We show that the free energy at density$\unicode[STIX]{x1D70C}$and inverse temperature$\unicode[STIX]{x1D6FD}$differs from the one of the noninteracting system by the correction term$4\unicode[STIX]{x1D70B}\unicode[STIX]{x1D70C}^{2}|\ln \,a^{2}\unicode[STIX]{x1D70C}|^{-1}(2-[1-\unicode[STIX]{x1D6FD}_{\text{c}}/\unicode[STIX]{x1D6FD}]_{+}^{2})$. Here,$a$is the scattering length of the interaction potential,$[\cdot ]_{+}=\max \{0,\cdot \}$and$\unicode[STIX]{x1D6FD}_{\text{c}}$is the inverse Berezinskii–Kosterlitz–Thouless critical temperature for superfluidity. The result is valid in the dilute limit$a^{2}\unicode[STIX]{x1D70C}\ll 1$and if$\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D70C}\gtrsim 1$.

中文翻译：

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二维稀玻色气体的自由能。一、下限

我们证明了二维玻色气体在热力学极限下的自由能（每单位体积）的下限。我们证明了密度下的自由能$\unicode[STIX]{x1D70C}$和逆温度$\unicode[STIX]{x1D6FD}$通过修正项与非交互系统之一不同$4\unicode[STIX]{x1D70B}\unicode[STIX]{x1D70C}^{2}|\ln \,a^{2}\unicode[STIX]{x1D70C}|^{-1}(2-[1 -\unicode[STIX]{x1D6FD}_{\text{c}}/\unicode[STIX]{x1D6FD}]_{+}^{2})$. 这里，$a$是相互作用势的散射长度，$[\cdot ]_{+}=\max \{0,\cdot \}$和$\unicode[STIX]{x1D6FD}_{\text{c}}$是超流体的逆 Berezinskii-Kosterlitz-Thouless 临界温度。结果在稀释限度内有效$a^{2}\unicode[STIX]{x1D70C}\ll 1$而如果$\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D70C}\gtrsim 1$.