二维和三维短程相互作用量子力学少体和多体问题的精确关系
Exact relations for quantum-mechanical few-body and many-body problems with short-range interactions in two and three dimensions
摘要：
我们推导出N粒子的各种观测值之间的关系与零程或短程相互作用，在连续空间或晶格上，在二维或三维，在一个任意的外部潜力。我们的某些结果推广了动量分布的大动量行为、对关联函数和单体密度矩阵的短程行为、能量对散射长度或时间的导数以及波函数正则部分的范数之间的已知关系;在有限程相互作用的情况下，相互作用能也与dE/da有关。能量与动量分布函数的关系式也得到了推广，并发现，打破Efimov状态与零程相互作用，由于一个subleading振荡尾巴的动量分布。我们还获得了新的表达式的一个普遍的状态的能量的导数相对于的有效范围，efimovian状态的能量的导数相对于的三体参数，和二阶导数的能量相对于倒数（或在二维情况下的对数）的散射长度。后者在固定熵下为负。我们使用精确的关系来计算修正，完全可解的三体问题，并找到与现有的数值协议。对于幺正气体，我们比较精确的关系到现有的固定节点的Monte Carlo数据，我们测试，与现有的量子Monte Carlo结果在不同的有限范围模型，我们的预测，领先的偏离临界温度从其零范围值是线性的相互作用有效范围r_e与模型无关的数值系数。
英文摘要：
We derive relations between various observables for N particles with zero-range or short-range interactions, in continuous space or on a lattice, in two or three dimensions, in an arbitrary external potential. Some of our results generalise known relations between large-momentum behavior of the momentum distribution, short-distance behavior of the pair correlation function and of the one-body density matrix, derivative of the energy with respect to the scattering length or to time, and the norm of the regular part of the wavefunction; in the case of finite-range interactions, the interaction energy is also related to dE/da. The expression relating the energy to a functional of the momentum distribution is also generalised, and is found to break down for Efimov states with zero-range interactions, due to a subleading oscillating tail in the momentum distribution. We also obtain new expressions for the derivative of the energy of a universal state with respect to the effective range, the derivative of the energy of an efimovian state with respect to the three-body parameter, and the second order derivative of the energy with respect to the inverse (or the logarithm in the two-dimensional case) of the scattering length. The latter is negative at fixed entropy. We use exact relations to compute corrections to exactly solvable three-body problems and find agreement with available numerics. For the unitary gas, we compare exact relations to existing fixed-node Monte-Carlo data, and we test, with existing Quantum Monte Carlo results on different finite range models, our prediction that the leading deviation of the critical temperature from its zero range value is linear in the interaction effective range r_e with a model independent numerical coefficient.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
