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Matrix-product-state comparison of the numerical renormalization group and the variational formulation of the density-matrix renormalization group

机译：数值重归一化组的矩阵乘积状态比较和密度矩阵重归一化组的变分公式

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Wilson's numerical renormalization group (NRG) method for solving quantum impurity models yields a set of energy eigenstates that have the form of matrix product states (MPS). White's density-matrix renormalization group (DMRG) for treating quantum lattice problems can likewise be reformulated in terms of MPS. Thus, the latter constitute a common algebraic structure for both approaches. We exploit this fact to compare the NRG approach for the single-impurity Anderson model with a variational matrix product state approach (VMPS), equivalent to single-site DMRG. For the latter, we use an "unfolded" Wilson chain, which brings about a significant reduction in numerical costs compared to those of NRG. We show that all NRG eigenstates (kept and discarded) can be reproduced using VMPS, and compare the difference in truncation criteria, sharp vs smooth in energy space, of the two approaches. Finally, we demonstrate that NRG results can be improved upon systematically by performing a variational optimization in the space of variational matrix product states, using the states produced by NRG as input.
机译：用于解决量子杂质模型的威尔逊数值重归一化组（NRG）方法产生了一组具有矩阵乘积态（MPS）形式的能量本征态。怀特的用于处理量子晶格问题的密度矩阵重归一化组（DMRG）同样可以根据MPS重新制定。因此，后者构成了两种方法的通用代数结构。我们利用这一事实将单杂质安德森模型的NRG方法与等效于单站点DMRG的变分矩阵乘积状态方法（VMPS）进行比较。对于后者，我们使用“展开的” Wilson链，与NRG相比，它大大降低了数值成本。我们表明，所有的NRG本征态（保留和丢弃）都可以使用VMPS进行再现，并比较两种方法在截断标准（能量空间中锐利与平滑）的差异。最后，我们证明，通过使用NRG产生的状态作为输入，在变分矩阵乘积状态空间中执行变分优化，可以系统地改善NRG结果。

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