Exact integral equation for the renormalized Fermi surface

The true Fermi surface of a fermionic many-body system can be viewed as a fixed point manifold of the renormalization group (RG). Within the framework of the exact functional RG we show that the fixed point condition implies an exact integral equation for the counterterm which is needed for a self-consistent calculation of the Fermi surface. In the simplest approximation, our integral equation reduces to the self-consistent Hartree-Fock equation for the counterterm.Comment: 5 pages, 1 figur

d-wave superconductivity and Pomeranchuk instability in the two-dimensional Hubbard model

We present a systematic stability analysis for the two-dimensional Hubbard model, which is based on a new renormalization group method for interacting Fermi systems. The flow of effective interactions and susceptibilities confirms the expected existence of a d-wave pairing instability driven by antiferromagnetic spin fluctuations. More unexpectedly, we find that strong forward scattering interactions develop which may lead to a Pomeranchuk instability breaking the tetragonal symmetry of the Fermi surface.Comment: 4 pages (RevTeX), 4 eps figure

Renormalized perturbation theory for Fermi systems: Fermi surface deformation and superconductivity in the two-dimensional Hubbard model

Divergencies appearing in perturbation expansions of interacting many-body systems can often be removed by expanding around a suitably chosen renormalized (instead of the non-interacting) Hamiltonian. We describe such a renormalized perturbation expansion for interacting Fermi systems, which treats Fermi surface shifts and superconductivity with an arbitrary gap function via additive counterterms. The expansion is formulated explicitly for the Hubbard model to second order in the interaction. Numerical soutions of the self-consistency condition determining the Fermi surface and the gap function are calculated for the two-dimensional case. For the repulsive Hubbard model close to half-filling we find a superconducting state with d-wave symmetry, as expected. For Fermi levels close to the van Hove singularity a Pomeranchuk instability leads to Fermi surfaces with broken square lattice symmetry, whose topology can be closed or open. For the attractive Hubbard model the second order calculation yeilds s-wave superconductivity with a weakly momentum dependent gap, whose size is reduced compared to the mean-field result.Comment: 18 pages incl. 6 figure

Deformation of the Fermi surface in the extended Hubbard model

The deformation of the Fermi surface induced by Coulomb interactions is investigated in the t-t'-Hubbard model. The interplay of the local U and extended V interactions is analyzed. It is found that exchange interactions V enhance small anisotropies producing deformations of the Fermi surface which break the point group symmetry of the square lattice at the Van Hove filling. This Pomeranchuck instability competes with ferromagnetism and is suppressed at a critical value of U(V). The interaction V renormalizes the t' parameter to smaller values what favours nesting. It also induces changes on the topology of the Fermi surface which can go from hole to electron-like what may explain recent ARPES experiments.Comment: 5 pages, 4 ps figure

Dynamic scaling in the vicinity of the Luttinger liquid fixed point

We calculate the single-particle spectral function A (k, omega) of a one-dimensional Luttinger liquid by means of a functional renormalization group (RG) approach. Given an infrared energy cutoff Lambda = Lambda_0 e^{- l}, our approach yields the spectral function in the scaling form, A_{\Lambda} (k_F + p, omega) = tau Z_l tilde{A}_l (p xi, omega tau), where k_F is the Fermi momentum, Z_l is the wave-function renormalization factor, tau = 1 / \Lambda is the time scale and xi = v_F / \Lambda is the length scale associated with Lambda. At the Luttinger liquid fixed point (l rightarrow infty) our RG result for A (k, omega) exhibits the correct anomalous scaling properties, and for k = \pm k_F agrees exactly with the well-known bosonization result at weak coupling. Our calculation demonstrates that the field rescaling is essential for obtaining the crossover from Fermi liquid behavior to Luttinger liquid behavior from a truncation of the hierarchy of exact RG flow equations as the infrared cutoff is reduced.Comment: 15 pages, 5 figure

Antiferromagnetism of the 2D Hubbard Model at Half Filling: Analytic Ground State at Weak Coupling

We introduce a local formalism to deal with the Hubbard model on a N times N square lattice (for even N) in terms of eigenstates of number operators, having well defined point symmetry. For U -> 0, the low lying shells of the kinetic energy are filled in the ground state. At half filling, using the 2N-2 one-body states of the partially occupied shell S_{hf}, we build a set of (2N-2 N-1)^{2} degenerate unperturbed ground states with S_{z}=0 which are then resolved by the Hubbard interaction \hat{W}=U\sum_{r}\hat{n}_{r\ua}\hat{n}_{r\da}. In S_{hf} we study the many-body eigenstates of the kinetic energy with vanishing eigenvalue of the Hubbard repulsion (W=0 states). In the S_{z}=0 sector, this is a N times degenerate multiplet. From the singlet component one obtains the ground state of the Hubbard model for U=0^{+}, which is unique in agreement with a theorem by Lieb. The wave function demonstrates an antiferromagnetic order, a lattice step translation being equivalent to a spin flip. We show that the total momentum vanishes, while the point symmetry is s or d for even or odd N/2, respectively.Comment: 13 pages, no figure

Functional renormalization group for d-wave superconductivity in Hubbard type models

The temperature dependence of d-wave superconducting order for two dimensional fermions with d-wave attraction is investigated by means of the functional renormalization group with partial bosonization. Below the critical temperature T_c we find superconductivity, a gap in the electron propagator and a temperature dependent anomalous dimension. At T_c the renormalized "superfluid density" jumps and the approach to T_c from above is characterized by essential scaling. These features are characteristic for a phase transition of the Kosterlitz-Thouless (KT) type.Comment: 5 pages, 4 figures, references added, discussion improve

Magnetic and superconducting instabilities of the Hubbard model at the van Hove filling

We use a novel temperature-flow renormalization group technique to analyze magnetic and superconducting instabilities in the two-dimensional t-t' Hubbard model for particle densities close to the van Hove filling as a function of the next-nearest neighbor hopping t'. In the one-loop flow at the van Hove filling, the characteristic temperature for the flow to strong coupling is suppressed drastically around t'_c approx. -0.33t, suggesting a quantum critical point between d-wave pairing at moderate t'>t'_c and ferromagnetism for t'<t'_c. Upon increasing the particle density in the latter regime the leading instability occurs in the triplet pairing channel.Comment: 4 pages, to appear in Physical Review Letter

Superconducting and pseudogap phases from scaling near a Van Hove singularity

We study the quantum corrections to the Fermi energy of a two-dimensional electron system, showing that it is attracted towards the Van Hove singularity for a certain range of doping levels. The scaling of the Fermi level allows to cure the infrared singularities left in the BCS channel after renormalization of the leading logarithm near the divergent density of states. A phase of d-wave superconductivity arises beyond the point of optimal doping corresponding to the peak of the superconducting instability. For lower doping levels, the condensation of particle-hole pairs due to the nesting of the saddle points takes over, leading to the opening of a gap for quasiparticles in the neighborhood of the singular points.Comment: 4 pages, 6 Postscript figures, the physical discussion of the results has been clarifie

Fermi surface renormalization in Hubbard ladders

We derive the one-loop renormalization equations for the shift in the Fermi-wavevectors for one-dimensional interacting models with four Fermi-points (two left and two right movers) and two Fermi velocities v_1 and v_2. We find the shift to be proportional to (v_1-v_2)U^2, where U is the Hubbard-U. Our results apply to the Hubbard ladder and to the t_1-t_2 Hubbard model. The Fermi-sea with fewer particles tends to empty. The stability of a saddle point due to shifts of the Fermi-energy and the shift of the Fermi-wavevector at the Mott-Hubbard transition are discussed.Comment: 5 pages, 4 Postscript figure