多重猝灭的含时数值重整化群方法：热力学观测量和谱函数长时间极限的精确结果
Time-dependent numerical renormalization group method for multiple quenches: towards exact results for the long time limit of thermodynamic observables and spectral functions
摘要：
我们发展了另一种含时的数值重整化群（TDNRG）形式主义的多重猝灭，并实现它来研究量子杂质系统的一般脉冲的响应。在这种方法中，我们通过一种公式减少了NRG近似对观测量时间演化中数值误差的贡献，该公式避免了在我们以前的多淬熄TDNRG形式主义中使用广义重叠矩阵元素[Nghiem { \em et al.，}物理修订版B { \bf 89}，075118（2014）;物理修订版B { \bf 90}，035129（2014）]。我们表明，该形式主义产生一个较小的累积误差的轨迹的投影密度矩阵作为时间的函数和一个较小的不连续性的局部观测淬火之间比我们以前的方法。此外，通过增加接通时间，即离散化脉冲的第一次和最后一次猝灭之间的时间，可观测量的长时间极限系统地收敛到其最终状态下的期望值，即，切换越绝热，恢复的长时间限制就越精确。目前的形式主义可以直接扩展到无限的开关时间。我们表明，这对热力学观测值和光谱函数的长时间限制产生了高度精确的结果，并克服了单猝灭形式主义[Anders { \em et al.}，物理修订版信函{ \bf 95}，196801（2005）; Nghiem { \em等人}，物理修订信函{ \bf 119}，156601（2017）]。这种改进为精确描述量子杂质系统的非平衡稳态迈出了第一步，在散射状态NRG方法内[Anders，Phys. Rev. Lett. { \bf 101}，066804（2008）]。
英文摘要：
We develop an alternative time-dependent numerical renormalization group (TDNRG) formalism for multiple quenches and implement it to study the response of a quantum impurity system to a general pulse. Within this approach, we reduce the contribution of the NRG approximation to numerical errors in the time evolution of observables by a formulation that avoids the use of the generalized overlap matrix elements in our previous multiple-quench TDNRG formalism [Nghiem {\em et al.,} Phys. Rev. B {\bf 89}, 075118 (2014); Phys. Rev. B {\bf 90}, 035129 (2014)]. We demonstrate that the formalism yields a smaller cumulative error in the trace of the projected density matrix as a function of time and a smaller discontinuity of local observables between quenches than in our previous approach. Moreover, by increasing the switch-on time, the time between the first and last quench of the discretized pulse, the long-time limit of observables systematically converges to its expected value in the final state, i.e., the more adiabatic the switching, the more accurately is the long-time limit recovered. The present formalism can be straightforwardly extended to infinite switch-on times. We show that this yields highly accurate results for the long-time limit of both thermodynamic observables and spectral functions, and overcomes the significant errors within the single quench formalism [Anders {\em et al.}, Phys. Rev. Lett. {\bf 95}, 196801 (2005); Nghiem {\em et al.}, Phys. Rev. Lett. {\bf 119}, 156601 (2017)]. This improvement provides a first step towards an accurate description of nonequilibrium steady states of quantum impurity systems, e.g., within the scattering states NRG approach [Anders, Phys. Rev. Lett. {\bf 101}, 066804 (2008)].
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