Abstract:
We present a renormalization group (RG) analysis of a fermionic "hot spot" model of interacting electrons on the square lattice. We truncate the Fermi surface excitations to linearly dispersing quasiparticles in the vicinity of eight hot spots on the Fermi surface, with each hot spot separated from another by the wavevector $(\pi, \pi)$. This motivated by the importance of these Fermi surface locations to the onset of antiferromagnetic order; however, we allow for all possible quartic interactions between the fermions, and also for all possible ordering instabilities. We compute the RG equations for our model, which depend on whether the hot spots are perfectly nested or not, and relate our results to earlier models. We also compute the RG flow of the relevant order parameters for both Hubbard and $J$, $V$ interactions, and present our results for the dominant instabilities in the nested and non-nested cases. In particular, we find that non-nested hot spots with $J$, $V$ interactions have competing singlet $d_{x^2-y^2}$ superconducting and $d$-form factor incommensurate density wave instabilities. We also investigate the enhancement of incommensurate density waves near experimentally observed wavevectors, and find dominant $d$-form factor enhancement for a range of couplings.

Abstract:
A renormalization group (RG) analysis of the superconductive instability of an anisotropic fermionic system is developed at a finite temperature. The method appears a natural generalization of Shankar's approach to interacting fermions and of Weinberg's discussion about anisotropic superconductors at T=0. The need of such an extension is fully justified by the effectiveness of the RG at the critical point. Moreover the relationship between the RG and a mean-field approach is clarified, and a scale-invariant gap equation is discussed at a renormalization level in terms of the eigenfunctions of the interaction potential, regarded as the kernel of an integral operator on the Fermi surface. At the critical point, the gap function is expressed by a single eigenfunction and no symmetry mixing is allowed. As an illustration of the method we discuss an anisotropic tight-binding model for some classes of high $T_c$ cuprate superconductors, exhibiting a layered structure. Some indications on the nature of the pairing interaction emerge from a comparison of the model predictions with the experimental data.

Abstract:
We demonstrate, in the context of quadratic fermion lattice models in one and two spatial dimensions, the potential of entanglement renormalization (ER) to define a proper real-space renormalization group transformation. Our results show, for the first time, the validity of the multi-scale entanglement renormalization ansatz (MERA) to describe ground states in two dimensions, even at a quantum critical point. They also unveil a connection between the performance of ER and the logarithmic violations of the boundary law for entanglement in systems with a one-dimensional Fermi surface. ER is recast in the language of creation/annihilation operators and correlation matrices.

Abstract:
We build the two dimensional Gross-Neveu model by a new method which requires neither cluster expansion nor discretization of phase-space. It simply reorganizes the perturbative series in terms of trees. With this method we can for the first time define non perturbatively the renormalization group differential equations of the model and at the same time construct explicitly their solution.

Abstract:
The Polchinski version of the exact renormalization group equation is discussed and its applications in scalar and fermionic theories are reviewed. Relation between this approach and the standard renormalization group is studied, in particular the relation between the derivative expansion and the perturbation theory expansion is worked out in some detail.

Abstract:
In a recent contribution [arXiv:0904:4151] entanglement renormalization was generalized to fermionic lattice systems in two spatial dimensions. Entanglement renormalization is a real-space coarse-graining transformation for lattice systems that produces a variational ansatz, the multi-scale entanglement renormalization ansatz (MERA), for the ground states of local Hamiltonians. In this paper we describe in detail the fermionic version of the MERA formalism and algorithm. Starting from the bosonic MERA, which can be regarded both as a quantum circuit or in relation to a coarse-graining transformation, we indicate how the scheme needs to be modified to simulate fermions. To confirm the validity of the approach, we present benchmark results for free and interacting fermions on a square lattice with sizes between $6 \times 6$ and $162\times 162$ and with periodic boundary conditions. The present formulation of the approach applies to generic tensor network algorithms.

Abstract:
We introduce a family of states, the fPEPS, which describes fermionic systems on lattices in arbitrary spatial dimensions. It constitutes the natural extension of another family of states, the PEPS, which efficiently approximate ground and thermal states of spin systems with short-range interactions. We give an explicit mapping between those families, which allows us to extend previous simulation methods to fermionic systems. We also show that fPEPS naturally arise as exact ground states of certain fermionic Hamiltonians. We give an example of such a Hamiltonian, exhibiting criticality while obeying an area law.

Abstract:
The exact renormalization group approach (ERG) is developed for the case of pure fermionic theories by deriving a Grassmann version of the ERG equation and applying it to the study of fixed point solutions and critical exponents of the two-dimensional chiral Gross-Neveu model. An approximation based on the derivative expansion and a further truncation in the number of fields is used. Two solutions are obtained analytically in the limit $N\to \infty $, with N being the number of fermionic species. For finite N some fixed point solutions, with their anomalous dimensions and critical exponents, are computed numerically. The issue of separation of physical results from the numerous spurious ones is discussed. We argue that one of the solutions we find can be identified with that of Dashen and Frishman, whereas the others seem to be new ones.

Abstract:
We present a comprehensive analysis of quantum fluctuation effects in the superfluid ground state of an attractively interacting Fermi system, employing the attractive Hubbard model as a prototype. The superfluid order parameter, and fluctuations thereof, are implemented by a bosonic Hubbard-Stratonovich field, which splits into two components corresponding to longitudinal and transverse (Goldstone) fluctuations. Physical properties of the system are computed from a set of approximate flow equations obtained by truncating the exact functional renormalization group flow of the coupled fermion-boson action. The equations capture the influence of fluctuations on non-universal quantities such as the fermionic gap, as well as the universal infrared asymptotics present in every fermionic superfluid. We solve the flow equations numerically in two dimensions and compute the asymptotic behavior analytically in two and three dimensions. The fermionic gap \Delta is reduced significantly compared to the mean-field gap, and the bosonic order parameter \alpha, which is equivalent to \Delta in mean-field theory, is suppressed to values below \Delta by fluctuations. The fermion-boson vertex is only slightly renormalized. In the infrared regime, transverse order parameter fluctuations associated with the Goldstone mode lead to a strong renormalization of longitudinal fluctuations: the longitudinal mass and the bosonic self-interaction vanish linearly as a function of the scale in two dimensions, and logarithmically in three dimensions, in agreement with the exact behavior of an interacting Bose gas.

Abstract:
We argue that incoherent pair tunneling in a cuprate superconductor junction with an optimally doped superconducting and an underdoped normal lead can be used to detect the presence of pairing correlations in the pseudogap phase of the underdoped lead. We estimate that the junction characteristics most suitable for studying the pair tunneling current are close to recently manufactured cuprate tunneling devices.