Combined use of translational and spin-rotational invariance for spin systems
Combined use of translational and spin-rotational invariance for spin systems
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DOI：
10.1103/physrevb.99.134405
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发表时间：
2019-02
影响因子：
3.7
通讯作者：
T. Heitmann;J. Schnack
中科院分区：
文献类型：
--
作者：
T. Heitmann;J. Schnack
众所周知，量子自旋系统的精确对角化和其他数值研究受到希尔伯特空间维数随系统大小呈指数增长的限制。一个常见的和众所周知的做法，以减少这种不断增加的计算工作量是利用平移对称$C_N$在周期性系统。这代表了群论对称投影算子技术的一个相当简单而优雅的应用。对于各向同性交换相互作用，可以使用自旋-旋转对称SU（2），其中根据总自旋和磁化量子数，哈密顿矩阵是块结构的。重写海森堡哈密顿量的不可约张量算子允许一个有效的和高度并行的实现，以计算其矩阵元素递归的自旋耦合的基础。当结合$C_N$和SU（2）时，在数学上，对称投影技术导致现成的公式。然而，这些公式的评估是非常苛刻的计算时间和内存消耗，据说这些问题超过了对称性减少矩阵形状的好处。我们展示了一种方法来最大限度地减少选定系统的计算工作量，并提出了最大的数值可访问的情况下。
Exact diagonalization and other numerical studies of quantum spin systems are notoriously limited by the exponential growth of the Hilbert space dimension with system size. A common and well-known practice to reduce this increasing computational effort is to take advantage of the translational symmetry $C_N$ in periodic systems. This represents a rather simple yet elegant application of the group theoretical symmetry projection operator technique. For isotropic exchange interactions, the spin-rotational symmetry SU(2) can be used, where the Hamiltonian matrix is block-structured according to the total spin- and magnetization quantum numbers. Rewriting the Heisenberg Hamiltonian in terms of irreducible tensor operators allows for an efficient and highly parallelizable implementation to calculate its matrix elements recursively in the spin-coupling basis. When combining both $C_N$ and SU(2), mathematically, the symmetry projection technique leads to ready-to-use formulas. However, the evaluation of these formulas is very demanding in both computation time and memory consumption, problems which are said to outweigh the benefits of the symmetry reduced matrix shape. We show a way to minimize the computational effort for selected systems and present the largest numerically accessible cases.