By using the quantum Ising chain as a test bed and treating the spin polarization along the external transverse field as the generalized density, we examine the performance of different levels of density functional approximations parallel to those widely used for interacting electrons, such as local density approximation (LDA) and generalized-gradient approximation (GGA). We show that by adding the lowest-order and nearest-neighbor density variation correction to the simple LDA, a semilocal energy functional in the spirit of GGA is almost exact over a wide range of inhomogeneous density distribution. In addition, the LDA and GGA error structures bear a high level of resemblance to the quantum phase diagram of the system. These results provide insights into the triumph and failure of these approximations in a general context.

中文翻译：

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在量子伊辛链中测试密度泛函理论

通过使用量子 Ising 链作为试验台，并将沿外部横向场的自旋极化视为广义密度，我们检查了与广泛用于相互作用电子的密度函数近似平行的不同水平密度泛函近似的性能，例如局部密度近似(LDA) 和广义梯度近似 (GGA)。我们表明，通过将最低阶和最近邻密度变化校正添加到简单的 LDA，本着 GGA 精神的半局域能量泛函在广泛的非均匀密度分布范围内几乎是精确的。此外，LDA 和 GGA 误差结构与系统的量子相图高度相似。这些结果提供了对这些近似在一般情况下的成功和失败的见解。