Renormalization group for 2D fermions with a flat Fermi surface

Numerous correlated electron systems exhibit a strongly scale-dependent behavior. Upon lowering the energy scale, collective phenomena, bound states, and new effective degrees of freedom emerge. Typical examples include (i) competing magnetic, charge, and pairing instabilities in two-dimensional electron systems, (ii) the interplay of electronic excitations and order parameter fluctuations near thermal and quantum phase transitions in metals, (iii) correlation effects such as...

We review recent developments in functional renormalization group (RG) methods for interacting fermions. These approaches aim at obtaining an unbiased picture of competing Fermi liquid instabilities in the low-dimensional models like the two-dimensional Hubbard model. We discuss how these instabilities can be approached from various sides and how the fermionic RG flow can be continued into phases with broken symmetry.

Motivated by a recent experimental observation of a nodal liquid on both single crystals and thin films of Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ by Chatterjee \emph{et al.} [Nature Physics \textbf{6}, 99 (2010)], we perform a field-theoretical renormalization group (RG) analysis of a two-dimensional model consisting of eight points located near the "hot spots" on the Fermi surface which are directly connected by spin density wave ordering wave vector. We derive RG equations up t...

We apply the renormalization-group (RG) approach to two model systems where the two-dimensional Fermi surface has portions which give rise to the logarithmically singular two-loop self-energy process.

We describe a new formulation of the functional renormalization group (RG) for interacting fermions within a Wilsonian momentum-shell approach. We show that the Luttinger-Ward functional is a fixed point of the RG, and derive the infinite hierarchy of flow equations satisfied by the two-particle-irreducible (2PI) vertices. In the one-loop approximation, this hierarchy reduces to two equations that determine the self-energy and the 2PI two-particle vertex $\Phi^{(2)}$. Suscept...

We consider a local effective model for fermionic low lying excitations in a metal. Introducing a boson auxiliary field and taking into account that the most significant interactions between quasiparticles arise for those which are near a given point on the Fermi surface, we find that the effective boson-fermion interaction is singular for a spatial dimension d=2. We regularize the theory and apply the renormalization group method. As we vary the coupling constant as a functi...

A phenomenological theory is presented for two-dimensional quantum liquids in terms of the Fermi surface geometry. It is shown that there is a one-to-one correspondence between the properties of an interacting electron system and its corresponding Fermi surface. By doing this, the concept of Fermi surface is generalized to include different topologies. It is shown that for a Fermi liquid the corresponding Fermi surface is rough. In the presence of a condensate, the Fermi surf...

This is the second part of the notes to the course on quantum theory of large systems of non-relativistic matter taught by J. Fr\"{o}hlich at the 1994 Les Houches summer school. It is devoted to a sketchy exposition of some of the beautiful and important, recent results of J.Feldman and E.Trubowitz, and J. Feldman, H. Kn\"{o}rrer, D. Lehmann, J. Magnen, V. Rivasseau and E. Trubowitz. Their results are about a mathematical analysis of non-relativistic many-body theory, in part...

In this thesis, I study a two-dimensional extended Hubbard model in the weak coupling limit. Quite generally, the electron gas is unstable towards a superconducting state even in the absence of phonons. However in the special case of a half-filled band, the Fermi surface is nested and the system is at a Van Hove singularity. In this situation, there are six competing instabilities: $s$- and d-wave superconductivity, spin-and charge-density waves and two phases with circulatin...

We present an analytic theory unraveling the microscopic mechanism of high-temperature superconductivity posing a longstanding challenge in condensed matter physics. Our model consists of a $D$-dimensional electron gas with electron and hole Fermi surfaces, subject to a repulsive electron-electron interaction of a finite range exceeding the average inter-particle distance. Evaluating susceptibilities in a systematic perturbative approach and via one-loop renormalization group...