contributions to the gap equation originate from the derivative of the anomalous polarization function, and the crucial process is an interaction of quasiparticles through the exchange of fluctuations.'
author:
- Mareike Hoyer
- Jörg Schmalian
title: 'Role of fluctuations for density-wave instabilities: Failure of the mean-field description'
---

Introduction
============

In condensed matter systems, the interaction between electrons often gives rise to an instability of the Fermi-liquid ground state, resulting in the formation of a low-temperature ordered phase. Examples of such ground states include superconductivity, charge-density wave order and magnetically ordered states, and are expected in all materials at least in the clean limit. Often, the low-temperature ordered phase can be described in terms of an effective mean field that characterizes the novel ground state and serves as an order parameter (Fig. \[fig:visu-fluctuations\]). The poster child of such a mean-field theory is the celebrated BCS theory [@BardeenCooperSchrieffer-PR106.1957; @BardeenCooperSchrieffer-PR108.1957] of superconductivity, which offers arguably the single most successful mean-field description of an interacting many-body system. The principal reasons for the success of this theory are the following:

1.  the instability towards the BCS state occurs for arbitrarily small interactions,

2.  fluctuations beyond mean-field theory can be neglected for small interaction strength 