强相互作用Dirac费米子系统的非微扰Dyson-Schwinger方程方法
Nonperturbative Dyson-Schwinger equation approach to strongly interacting Dirac fermion systems
摘要：
研究相互作用狄拉克费米子系统中的强关联效应是现代凝聚态物理中最具挑战性的问题之一。长程库仑相互作用和费米子-声子相互作用可以导致各种有趣的性质。在强耦合区域，弱耦合微扰理论失效。由于$N$在实际系统中等于$2$或$4$，因此以$N$作为费米子味的$1/N$展开的有效性也是值得怀疑的。本文研究了（1+2）维和（1+3）维无质量狄拉克费米子与一般标量玻色子的相互作用，并发展了一种有效的非微扰方法来研究强耦合机制.我们首先推导出一些自洽耦合沃德-高桥身份的基础上仔细的对称性分析，然后使用这些身份证明，完全费米子玻色子顶点函数是完全由费米子传播子。利用这一结果，我们严格地证明了全费米子传播子满足一个精确的自封闭的Dyson-Schwinger积分方程，该方程可以用数值方法求解。我们的非微扰方法的一个主要优点是，没有必要采用任何小的扩展参数。我们的方法提供了一个统一的理论框架，研究强库仑和费米子声子相互作用。它也可以用来近似处理连续量子临界点附近费米子与序参量涨落之间的汤川耦合。我们的方法被应用到处理未掺杂的石墨烯中的库仑相互作用。我们发现重整化费米子速度表现出动量的对数依赖性，但几乎与能量无关，并且库仑相互作用没有产生激子能隙。这些理论结果与石墨烯中的实验一致。
英文摘要：
Studying the strong correlation effects in interacting Dirac fermion systems is one of the most challenging problems in modern condensed matter physics. The long-range Coulomb interaction and the fermion-phonon interaction can lead to a variety of intriguing properties. In the strong-coupling regime, weak-coupling perturbation theory breaks down. The validity of $1/N$ expansion with $N$ being the fermion flavor is also in doubt since $N$ equals to $2$ or $4$ in realistic systems. Here, we investigate the interaction between (1+2)- and (1+3)-dimensional massless Dirac fermions and a generic scalar boson, and develop an efficient non-perturbative approach to access the strong-coupling regime. We first derive a number of self-consistently coupled Ward-Takahashi identities based on a careful symmetry analysis and then use these identities to show that the full fermion-boson vertex function is solely determined by the full fermion propagator. Making use of this result, we rigorously prove that the full fermion propagator satisfies an exact and self-closed Dyson-Schwinger integral equation, which can be solved by employing numerical methods. A major advantage of our non-perturbative approach is that there is no need to employ any small expansion parameter. Our approach provides a unified theoretical framework for studying strong Coulomb and fermion-phonon interactions. It may also be used to approximately handle the Yukawa coupling between fermions and order-parameter fluctuations around continuous quantum critical points. Our approach is applied to treat the Coulomb interaction in undoped graphene. We find that the renormalized fermion velocity exhibits a logarithmic momentum-dependence but is nearly energy independent, and that no excitonic gap is generated by the Coulomb interaction. These theoretical results are consistent with experiments in graphene.
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