[论文解读] Many-body effects and ultraviolet renormalization in 3D Dirac materials

Robert E. Throckmorton, Johannes Hofmann+2 more|arXiv (Cornell University)|May 19, 2015

Topological Materials and Phenomena参考文献 5被引用 1

一句话总结

本文为三维狄拉克和外尔半金属建立了多体理论，计算了有效精细结构常数二阶以内的电子-电子相互作用对极化率、自能和顶点函数的修正。结果表明，高阶修正抵消了主导阶的重整化群流，使得在临界耦合以上，相互作用在低能下增强，并诱导出非单调的费米速度行为——尤其在狄拉克锥简并度高的材料中尤为显著，与RPA或大N近似形成对比。

ABSTRACT

We develop a theory for electron-electron interaction-induced many-body effects in threedimensional (3D) Weyl or Dirac semimetals, including interaction corrections to the polarizability, electron self-energy, and vertex function, up to second order in the effective fine structure constant of the Dirac material. These results are used to derive the higher-order ultraviolet renormalization of the Fermi velocity, effective coupling, and quasiparticle residue, revealing that the corrections to the renormalization group (RG) flows of both the velocity and coupling counteract the leading-order tendencies of velocity enhancement and coupling suppression at low energies. This in turn leads to the emergence of a critical coupling above which the interaction strength grows with decreasing energy scale. In addition, we identify a range of coupling strengths below the critical point in which the Fermi velocity varies non-monotonically as the low-energy, non-interacting fixed point is approached. Furthermore, we find that while the higher-order correction to the flow of the coupling is generally small compared to the leading order, the corresponding correction to the velocity flow carries an additional factor of the Dirac cone flavor number (the multiplicity of electron species, e.g. ground-state valley degeneracy arising from the band structure) relative to the leading-order result. Thus, for materials with a larger multiplicity, the regime of velocity non-monotonicity is reached for modest values of the coupling strength. This is in stark contrast to an approach based on a large-N expansion or the random phase approximation (RPA), where higher-order corrections are strongly suppressed for larger values of the Dirac cone multiplicity. This suggests that perturbation theory in the coupling constant (i.e. the loop expansion) and the RPA/large-N expansion are complementary in the sense that they are applicable in different parameter regimes of the theory. We show how our results for the ultraviolet renormalization of quasiparticle properties can be tested experimentally through measurements of quantities such as the optical conductivity or dielectric function (with carrier density or temperature acting as the scale being varied to induce the running coupling). Although experiments typically access the finite-density regime, we show that our zero-density results still capture clear many-body signatures that should be visible at higher temperatures even in real systems with disorder and finite doping.

研究动机与目标

• 开发一种超越主导阶的系统性多体理论，用于描述三维狄拉克和外尔半金属中的电子-电子相互作用。
• 计算有效精细结构常数二阶以内的极化率、电子自能和顶点函数的相互作用修正。
• 确定高阶修正如何影响费米速度、耦合常数和准粒子权重的重整化群流。
• 识别非单调速度演化与临界耦合出现的参数区域，尤其在狄拉克锥简并度高的材料中。
• 将理论预测与实验可观测量（如光学电导率和介电函数）联系起来。

提出的方法

• 使用有效精细结构常数的微扰圈展开，计算极化率、自能和顶点函数的二阶修正。
• 从计算得到的顶点和自能修正中，推导出费米速度、有效耦合和准粒子权重的高阶紫外重整化。
• 分析速度和耦合的重整化群流，识别高阶项如何修正主导阶趋势。
• 将狄拉克锥种类数（电子种类的简并度）作为关键参数，表明其在速度修正中起增强作用。
• 将微扰圈展开结果与RPA和大N近似进行比较，突出其在不同参数区域中的互补适用性。
• 提出通过调节载流子浓度或温度，探测光学电导率和介电函数中运行耦合效应的实验验证方案。

实验结果

研究问题

• RQ1 在三维狄拉克材料中，二阶电子-电子相互作用修正如何改变费米速度和耦合常数的重整化群流？
• RQ2 狄拉克锥简并度在非相互作用固定点附近增强费米速度非单调行为中起什么作用？
• RQ3 是否存在一个临界耦合，使得在该耦合以上，相互作用强度随能量尺度降低而增强？高阶修正如何影响该临界行为？
• RQ4 速度和耦合流的高阶修正在量级上如何比较？简并度因子在速度修正中的意义是什么？
• RQ5 在存在杂质和有限掺杂的情况下，哪些实验参数区域可观察到预测的多体信号（如非单调速度演化）？

主要发现

• 费米速度和耦合常数的重整化群流的高阶修正，抵消了主导阶在低能下增强速度和抑制耦合的倾向。
• 存在一个临界耦合，当超过该值时，有效相互作用强度随能量尺度降低而增强，表明弱耦合行为的破坏。
• 对于狄拉克锥简并度高的材料，由于速度修正中简并度因子的增强贡献，非单调费米速度演化在适中耦合强度下即出现。
• 速度流的高阶修正相比主导阶结果，额外包含一个狄拉克锥种类数因子，而耦合修正相比之下仍较小。
• 微扰理论与RPA/大N展开互补：微扰理论适用于简并度增强非单调性的区域，而RPA在大简并度下抑制高阶效应。
• 零密度体系的理论预测仍能在光学电导率和介电函数中产生清晰的多体信号，这些信号可在较高温度和有限掺杂下通过实验探测。