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 ${\ensuremath{\varphi}}^{4}$ scalar field theory. A brief introduction to ${\ensuremath{\varphi}}^{4}$ 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 cutoff $\ensuremath{\Lambda}$ are chosen for study, in analogy with the modes near the origin in ${\ensuremath{\varphi}}^{4}$ theory, and a path integral is written to describe them. Next, an RG transformation that 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 arbitrarily weak attraction, the renormalization-group approach correctly yields a scale-invariant system (Luttinger liquid) by taking into account both instabilities. Application of the renormalization group in
d=2and3, for rotationally invariant Fermi surfaces, automatically leads to Landau's Fermi-liquid theory, which appears as a fixed point characterized by an effective mass and 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 and separated into an infinite number of flows, one for each angular momentum. It is shown that similar results hold for rotationally noninvariant (but time-reversal-invariant) Fermi surfaces also, with obvious loss of rotational invariance in the parametrization of the fixed-point interactions. A study of a nested Fermi surface shows an additional relevant flow leading to charge-density-wave formation. It is pointed out that, for small $\frac{\ensuremath{\Lambda}}{{K}_{F}}$, a
N1 expansion emerges, with $N=\frac{{K}_{F}}{\ensuremath{\Lambda}}$, which explains why one is able to solve the narrow-cutoff theory. The search for non-Fermi liquids in
d=2 using the RG is discussed. Bringing a variety of phenomena (Landau theory, charge-density waves, BCS instability, nesting, etc.) under the one unifying principle of the RG not only allows us to better understand and unify them, but also paves the way for generalizations and extensions. The article is pedagogical in nature and is expected to be accessible to any serious graduate student. On the other hand, its survey of the vast literature is mostly limited to the RG approach.
The renormalization-group (RG) approach proposed earlier by Shankar for interacting spinless fermions at T=0 is extended to the case of nonzero temperature and spin. We study a model with SU(N)-invariant short-range effective interaction and rotationally invariant Fermi surface in two and three dimensions. We show that the Landau interaction function of the Fermi liquid, constructed from the bare parameters of the low-energy effective action, is RG invariant. On the other hand, the physical forward-scattering vertex is found as a stable fixed point of the RG flow. We demonstrate that in d=2 and 3, the RG approach to this model is equivalent to Landau's mean-field treatment of the Fermi liquid. We discuss subtleties associated with the symmetry properties of the scattering amplitude, the Landau function, and the low-energy effective action. Applying the RG to response functions, we find the compressibility and the spin susceptibility as fixed points.
We formulate the exact Wilsonian renormalization group for a system of interacting fermions on a lattice. The flow equations for all vertices of the Wilson effective action are expressed in form of the Polchinski equation. We apply this method to the Hubbard model on a square lattice using both zero- and finite- temperature methods. Truncating the effective action at the sixth term in fermionic variables we obtain the one-loop functional renormalization equations for the effective interaction. We find the temperature of the instability Tc^{RG} as function of doping. We calculate furthermore the renormalization of the angle-resolved correlation functions for the superconductivity (SC) and for the antiferromagnetism (AF). The dominant component of the SC correlations is of the type d while the AF fluctuations are of the type s Following the strength of both SC and AF fluctuation along the instability line we obtain the phase diagram. The temperature Tc^{RG} can be identified with the crossover temperature T{co} found in the underdoped regime of the high-temperature superconductors, while in the overdoped regime Tc^{RG} corresponds to the superconducting critical temperature.
We formulate the exact Wilsonian renormalization group for a system of interacting fermions on a lattice. The flow equations for all vertices of the Wilson effective action are expressed in the form of the Polchinski equation. The advantage of this renormalization scheme is that the flow itself has a physical interpretation, i.e., the cutoff has the meaning of the temperature. We apply this method to the Hubbard model on a square lattice using both zero- and finite-temperature methods. Truncating the effective action at the sixth term in fermionic variables and neglecting self-energy renormalization, we obtain the one-loop functional renormalization equations for the effective interaction. We find the temperature of the instability
TcRG as a function of doping. Furthermore we calculate the renormalization of the angle-resolved correlation functions for the superconductivity (SC) and for the antiferromagnetism (AF). The dominant component of the SC correlations is of the type ${d}_{{x}^{2}\ensuremath{-}{y}^{2}},$ while the AF fluctuations are of the type s. Following the strength of both SC and AF fluctuations along the instability line, we obtain the phase diagram. The temperature
TcRG can be identified with the crossover temperature
Tco found in the underdoped regime of the high-temperature superconductors, while in the overdoped regime
TcRG corresponds to the superconducting critical temperature.
High-temperature superconductivity in copper oxides (cuprates, e.g. La2−xSrxCuO4, YBa2Cu3O6+δ or Bi2Sr2CaCu2O8+δ) remains among the most spectacular phenomena in condensed matter physics. Since its discovery in 1986, an enormous number (∼ 105) of papers on the subject have appeared. So far, there is no single, commonly accepted theory of high-temperature superconductivity. However, it is widely believed that a basic description of this phenomenon can be provided by a single-band Hubbard model or its derivative, the t-J model. The latter model is regarded as a minimal microscopic model, capable of describing the essential aspects of the complex physics of the cuprates. Unfortunately, in general case, neither of those two models can be solved exactly and therefore various approximate methods are used. Among them, the so-called mean-field methods provide a simple, yet fairly reasonable description of the cuprates. In particular, some of the main qualitative features of the phase diagram and the essential features of electronic spectrum are roughly reproduced. A standard mean-field approach to the t-J model, known under the name of renormalized mean-field theory (RMFT) goes beyond the Hartree-Fock approximation. Consequently, its fully consistent treatment requires a novel theoretical approach. This has been our original motivation to develop a general approach to the mean-field models, which is based on the maximum entropy (MaxEnt) principle. The method is presented in detail in Part II of this Thesis, and in Part III it is applied to study RMFT of the t-J model. First, we compare the results obtained within our formalism with those of the frequently used non-variational approach based entirely on the self-consistent equations. Also, various versions of RMFT are compared, and the most satisfactory of them is selected. This optimal version is subsequently used to study different versions of the original t-J Hamiltonian. As a result, upper critical concentration and doping dependence of the selected physical quantities (e.g. the superconducting gap and the Fermi velocity) is determined at low temperatures and in the absence of external magnetic field. We compare our findings both with theoretical results obtained from the Variational Monte Carlo (VMC) methods, as well as with the experimental data for selected cuprates. We show that the version of RMFT approach formulated in this Thesis provides a reasonable qualitative and in some cases semiquantitative rationalization of the principal characteristics of the hole-doped high-temperature superconductors at the optimal doping and in the overdoped regime. Possible extensions of the proposed analysis are mentioned at the end.
We use renormalization group (RG) analysis and dimensional regularization techniques to study potential superconductivity-inducing four-fermion interactions in systems with critical Fermi surfaces of general dimensions
(m) and codimensions $(d\ensuremath{-}m)$, arising as a result of quasiparticle interaction with a gapless Ising-nematic order parameter. These are examples of non-Fermi liquid states in
d spatial dimensions. Our formalism allows us to treat the corresponding zero-temperature low-energy effective theory in a controlled approximation close to the upper critical dimension
d=dc(m). The fixed points are identified from the RG flow equations, as functions of
d and
m. We find that the flow toward the non-Fermi liquid fixed point is preempted by Cooper pair formation for both the physical cases of
(d=3,m=2) and
(d=2,m=1). In fact, there is a strong enhancement of superconductivity by the order parameter fluctuations at the quantum critical point.