Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Moscow, Russia.
J Phys Condens Matter. 2012 Feb 22;24(7):075701. doi: 10.1088/0953-8984/24/7/075701.
It is known that solutions of Richardson equations can be represented as stationary points of the 'energy' of classical free charges on the plane. We suggest considering the 'probabilities' of the system of charges occupying certain states in the configurational space at the effective temperature given by the interaction constant, which goes to zero in the thermodynamical limit. It is quite remarkable that the expression of 'probability' has similarities with the square of the Laughlin wavefunction. Next, we introduce the 'partition function', from which the ground state energy of the initial quantum-mechanical system can be determined. The 'partition function' is given by a multidimensional integral, which is similar to the Selberg integrals appearing in conformal field theory and random-matrix models. As a first application of this approach, we consider a system with the constant density of energy states at arbitrary filling of the energy interval where potential acts. In this case, the 'partition function' is rather easily evaluated using properties of the Vandermonde matrix. Our approach thus yields a quite simple and short way to find the ground state energy, which is shown to be described by a single expression all over from the dilute to the dense regime of pairs. It also provides additional insight into the physics of Cooper-paired states.
已知理查森方程的解可以表示为平面上自由电荷的“能量”的稳定点。我们建议考虑在给定的有效温度下，系统电荷占据构型空间中某些状态的“概率”，该有效温度由相互作用常数给出，当热力学极限时，相互作用常数趋近于零。值得注意的是，“概率”的表达式与 Laughlin 波函数的平方具有相似性。接下来，我们引入“配分函数”，可以通过该配分函数确定初始量子力学系统的基态能量。“配分函数”由多维积分给出，该多维积分类似于在共形场论和随机矩阵模型中出现的 Selberg 积分。作为这种方法的首次应用，我们考虑了在任意填充能量间隔的情况下具有常数能量态密度的系统，其中势能起作用。在这种情况下，使用 Vandermonde 矩阵的性质可以很容易地评估“配分函数”。因此，我们的方法提供了一种非常简单和简短的方法来找到基态能量，结果表明该基态能量在从稀疏到密集的对区中都可以用单个表达式来描述。它还为库珀配对态的物理性质提供了更多的见解。
