含时数值重整化群方法中连续极限的恢复
Restoring the continuum limit in the time-dependent numerical renormalization group approach
摘要：
量子杂质问题中的连续耦合函数被精确地划分为由有限大小的Wilson链表示的部分和由一组附加的库表示的部分，每个库耦合到一个Wilson链位点。这些额外的水库代表高能量模式的环境中忽略的数值重整化群，并需要恢复原来的问题的连续极限。我们提出了一个混合的时间相关的数值重整化群方法，它结合了一个精确的数值重整化群处理的非平衡动力学的有限大小的威尔逊链与布洛赫Redfield形式主义，包括这些额外的水库的影响。我们的方法克服了含时数值重整化群方法的固有缺点，所导致的浴离散与威尔逊参数$Λ> 1$。我们解析地证明了对于具有单一化学势的系统，热平衡约化密度算符是Bloch-Redfield主方程的稳态解。对于这个主方程的数值解的Lanczos方法耦合的所有能量壳的数值重整化群。将该方法应用于关联费米子量子杂质系统的实时动力学研究。共振能级模型的解析解作为该方法的准确性的基准，然后将其应用于非平凡的模型，如相互作用的共振能级模型和单杂质安德森模型。
英文摘要：
The continuous coupling function in quantum impurity problems is exactly partitioned into a part represented by a finite size Wilson chain and a part represented by a set of additional reservoirs, each coupled to one Wilson chain site. These additional reservoirs represent high-energy modes of the environment neglected by the numerical renormalization group and are required to restore the continuum limit of the original problem. We present a hybrid time-dependent numerical renormalization group approach which combines an accurate numerical renormalization group treatment of the non-equilibrium dynamics on the finite size Wilson chain with a Bloch-Redfield formalism to include the effect of these additional reservoirs. Our approach overcomes the intrinsic shortcoming of the time-dependent numerical renormalization group approach induced by the bath discretization with a Wilson parameter $Λ> 1$. We analytically prove that for a system with a single chemical potential, the thermal equilibrium reduced density operator is the steady-state solution of the Bloch-Redfield master equation. For the numerical solution of this master equation a Lanczos method is employed which couples all energy shells of the numerical renormalization group. The presented hybrid approach is applied to the real-time dynamics in correlated fermionic quantum-impurity systems. An analytical solution of the resonant-level model serves as a benchmark for the accuracy of the method which is then applied to non-trivial models, such as the interacting resonant-level model and the single impurity Anderson model.
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