[论文解读] Functional renormalization group analysis of Luttinger liquids with impurities

Sabine Andergassen|arXiv (Cornell University)|Jan 1, 2006

Theoretical and Computational Physics参考文献 96被引用 4

一句话总结

该论文将功能重正化群（fRG）方法应用于具有杂质的一维Luttinger液体，将先前针对无自旋费米子的fRG扩展至包含完整两体相互作用重正化的自旋-1/2体系。研究发现，在中间能量尺度下，对数修正和非普遍交叉效应（如瞬态谱权增强和电导率增加）占主导地位，延迟了普遍Luttinger液体渐近行为的出现，尤其在中等杂质强度下更为显著。

ABSTRACT

In one-dimensional quantum wires the interplay of electron correlations and impurities strongly influences the low-energy physics. The diversity of energy scales and the competition of correlations in interacting Fermi systems can be treated very efficiently with the functional renormalization group (fRG), describing the gradual evolution from a microscopic model Hamiltonian to the effective low-energy action as a function of a continuously decreasing energy cutoff. The fRG provides the universal low-energy asymptotics as well as nonuniversal properties, and in particular an answer to the important question at what scale the ultimate asymptotics sets in. The lowest order truncation of the fRG hierarchy of flow equations considered previously for spinless fermions is generalized to spin-1/2 systems and extended including renormalization of the two-particle interaction, in addition to renormalization of the impurity potential. The underlying approximations are devised for weak interactions and arbitrary impurity strengths. A comparison with numerical density-matrix renormalization results for systems with up to 1000 sites shows that the fRG is remarkably accurate even for intermediate interaction strengths. We investigate the influence of impurities on spectral and transport properties of fermionic lattice models with short-range interactions. The results capture relevant energy scales and crossover phenomena, in addition to the universal low-energy asymptotics. For weak and intermediate impurity strengths the asymptotic behavior is approached only at rather low energy scales, accessible only for very large systems. For spin-1/2 systems two-particle backscattering leads to striking effects, which are not captured if the bulk system is approximated by its low-energy fixed point, the Luttinger model. In particular, the expected decrease of spectral weight near the impurity and of the conductance at low energy scales is often preceded by a pronounced increase, and the asymptotic power laws are modified by logarithmic corrections.

研究动机与目标

• 将功能重正化群（fRG）方法扩展至含杂质的自旋-1/2 Luttinger液体体系。
• 研究两体相互作用与杂质势能在不同能量尺度下的重正化行为。
• 确定在杂质存在下普遍低能行为出现的能量尺度。
• 将fRG预测结果与大规模系统下的数值密度矩阵重正化群（DMRG）结果进行比较。
• 理解非普遍交叉现象在谱性质与输运性质中的作用。

提出的方法

• 将功能重正化群（fRG）应用于具有短程相互作用和杂质的微观晶格哈密顿量。
• 采用连续截断流方法，将有效作用量从微观尺度演化至低能尺度。
• 在fRG级联中截断至最低阶，包含杂质势能与两体相互作用的重正化。
• 引入自旋-1/2自由度，并处理自旋依赖的散射过程。
• 通过虚时形式化将形式体系扩展至有限温度。
• 通过大规模DMRG模拟（最多1000个格点）验证结果的定量准确性。

实验结果

研究问题

• RQ1 对于含杂质的体系，普遍Luttinger液体渐近行为在何种能量尺度下开始显现？
• RQ2 与Luttinger模型固定点相比，两体反向散射过程如何改变谱性质与输运性质？
• RQ3 为何谱权与电导率在趋近渐近幂律之前表现出瞬态增加？
• RQ4 对数修正如何影响含杂质的自旋-1/2 Luttinger液体中的普遍标度行为？
• RQ5 fRG在多大程度上准确捕捉了中间能量区域的交叉现象？

主要发现

• 含杂质的Luttinger液体的普遍低能渐近行为仅在极低能量尺度下趋近，尤其在中等杂质强度下更为显著。
• 两体反向散射导致幂律标度的对数修正，使渐近行为超出Luttinger模型的预测。
• 杂质附近谱权与电导率在趋向下达渐近幂律之前表现出显著的瞬态增加。
• fRG准确捕捉了这些非普遍交叉效应，并在中等相互作用强度下仍与DMRG结果高度一致。
• 该方法揭示，将体相近似为Luttinger液体固定点无法描述完整动力学，特别是预渐近区域的行为。
• 普遍行为开始出现的能量尺度显著低于先前假设，因此需要大系统才能在数值上实现访问。