标题： 玻色子的具有对称适应变分波函数的哈特里-福克近似 显示英文标题

标题： Hartree-Fock approximation for bosons with symmetry-adapted variational wave functions

摘要： 对于玻色子的Hartree-Fock近似使用的是永久式的变分波函数。这些是费米子斯莱特行列式的玻色子对应物，但具有显著的区别，即用于构建永久式的单粒子轨道可以是任意的，不需要彼此正交。通常，变分波函数可能会破坏哈密顿量的对称性，导致物理可观测量出现定性和定量的误差。一种直接恢复对称性的方法是在变分之后进行投影，即将变分波函数投影到所需的对称性子空间。然而，更有效的方法是在投影之后进行变分，这涉及首先创建一个适应对称性的变分波函数，然后优化其参数。我们设计了一个实现此策略的方案，并在各种具有从$\mathbb{Z}_2$、$\text{C}_L$到$\text{D}_L$对称群的模型上进行了测试。在所有研究过的模型和对称子空间中，变分波函数不仅准确估计了最低本征态的能量，还准确估计了单粒子关联函数，因为它在波函数层面上很好地近似了目标本征态。我们将这种方法应用于研究少数粒子束缚态、超流分数以及一些玻色-哈伯德模型的Yrast线。这种方法应有助于研究少数粒子或介观玻色子系统。 显示英文摘要

摘要： The Hartree-Fock approximation for bosons employs variational wave functions that are a combination of permanents. These are bosonic counterpart of the fermionic Slater determinants, but with the significant distinction that the single-particle orbitals used to construct a permanent can be arbitrary and do not need to be orthogonal to each other. Typically, the variational wave function may break the symmetry of the Hamiltonian, resulting in qualitative and quantitative errors in physical observables. A straightforward method to restore symmetry is projection after variation, where we project the variational wave function onto the desired symmetry sector. However, a more effective strategy is variation after projection, which involves first creating a symmetry-adapted variational wave function and then optimizing its parameters. We have devised a scheme to realize this strategy and have tested it on various models with symmetry groups ranging from $\mathbb{Z}_2$, $\text{C}_L$, to $\text{D}_L$. In all the models and symmetry sectors studied, the variational wave function accurately estimates not only the energy of the lowest eigenstate but also the single-particle correlation function, as it approximate the target eigenstate very well on the wave function level. We have applied this method to study few-body bound states, superfluid fraction, and Yrast lines of some Bose-Hubbard models. This approach should be valuable for studying few-body or mesoscopic bosonic systems.