Renormalization Group Approach to Interacting Fermions
Renormalization Group Approach to Interacting Fermions
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DOI：
10.1103/revmodphys.66.129
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发表时间：
1993-07
期刊：
影响因子：
--
通讯作者：
P. Kopietz
中科院分区：
文献类型：
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作者：
P. Kopietz
在重整化群（RG）框架内研究了非相对论性费米子系统对相互作用的稳定性或缺乏稳定性，这与使用φ标量场理论研究临界现象非常相似。在讨论当前的问题之前，先简要介绍了四维φ理论和费米子的路径积分公式。对于后者，使用以下程序。首先，选取费米面两侧的模，在截断Λ内，与φ理论中原点附近的模进行类比，并用路径积分来描述它们。接下来，RG变换，消除了这些模式的一部分，但保留了非相互作用系统的行动被确定。最后，这个自由场不动点的可能的扰动被分类为相关的，无关的或边缘的。一个d = 1的费米子系统的预热计算表明，与平均场理论相反，重整化群方法正确地产生了一个尺度不变系统（Luttinger液体），通过考虑两种不稳定性，平均场理论预测了任意弱排斥的电荷密度波，以及任意弱吸引的超导性。在d = 2和3的情况下，对于旋转不变的费米面，应用重整化群，自动导致朗道的费米液体理论，它表现为一个由有效质量和朗道函数F表征的不动点，唯一相关的扰动是超导（BCS）型的。推导了BCS耦合的泛函流方程，并将其分解为无穷多个流，每个流对应一个角动量。它表明，类似的结果举行的旋转非不变（但时间反转不变）
The stability or lack thereof of nonrelativistic fermionic systems to interactions is studied within the Renormalization Group (RG) framework, in close analogy with the study of critical phenomena using φ scalar field theory. A brief introduction to φ theory in four dimensions and the path integral formulation for fermions is given before turning to the problem at hand. As for the latter, the following procedure is used. First, the modes on either side of the Fermi surface within a cut-off Λ are chosen for study in analogy with the modes near the origin in φ theory and a path integral is written to describe them. Next, an RG transformation which eliminates a part of these modes, but preserves the action of the noninteracting system is identified. Finally the possible perturbations of this free-field fixed point are classified as relevant, irrelevant or marginal. A d = 1 warmup calculation involving a system of fermions shows how , in contrast to mean-field theory, which predicts a charge density wave for arbitrarily weak repulsion, and superconductivity for arb itrarily weak attraction, the renormalization group approach correct ly yields a scale invariant system (Luttinger liquid) by taking int o account both instabilities. Application of the renormalization gr oup in d = 2 and 3, for rotationally invariant Fermi surfaces, automatically leads to Landau’s Fermi liquid theory, which appears as a fixed point characterized by an effective mass an d a Landau function F , with the only relevant perturbations being of the superconducting (BCS) type. The functional flow equations for the BCS couplings are derived an d separated into an infinite number of flows, one for each angular momentum. It is shown that similar results hold for rotationally non-invari ant (but time-reversal invariant)