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Physical review. B, Condensed matter

Weakly correlated electrons on a square lattice: Renormalization-group theory

Dražen Zanchi, Heinz J. Schulz

Published May 15, 2000• 222 citations
Publisher

Abstract

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 ${T}_{c}^{\mathrm{RG}}$ 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 ${T}_{c}^{\mathrm{RG}}$ can be identified with the crossover temperature ${T}_{\mathrm{co}}$ found in the underdoped regime of the high-temperature superconductors, while in the overdoped regime ${T}_{c}^{\mathrm{RG}}$ corresponds to the superconducting critical temperature.

DOI
10.1103/physrevb.61.13609

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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{T}_{c}^{\mathrm{RG}}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 dx2\ensuremath−y2,{d}_{{x}^{2}\ensuremath{-}{y}^{2}},dx2\ensuremath−y2​, 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{T}_{c}^{\mathrm{RG}}TcRG​ can be identified with the crossover temperature Tco{T}_{\mathrm{co}}Tco​ found in the underdoped regime of the high-temperature superconductors, while in the overdoped regime TcRG{T}_{c}^{\mathrm{RG}}TcRG​ 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 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.
The problem of weakly correlated electrons on a square lattice is formulated in terms of a one-loop renormalization group. Starting from the action for the entire Brillouin zone we reduce successively the cutoff \ensuremath{\Lambda} about the Fermi surface and follow the renormalization of the coupling U as a function of three energy-momenta. We calculate the intrinsic temperature scale Tco{\mathit{T}}_{\mathit{co}}Tco​ where the renormalization-group flow crosses over from the regime (\ensuremath{\Lambda}>Tco{\mathit{T}}_{\mathit{co}}Tco​) where the electron-electron (e-e) and electron-hole (e-h) terms are equally important to the regime (\ensuremath{\Lambda}Tco{\mathit{T}}_{\mathit{co}}Tco​) where only the e-e term plays a role. In the low-energy regime only the pairing interaction V is marginally relevant, containing contributions from all renormalization-group steps of the regime \ensuremath{\Lambda}>Tco{\mathit{T}}_{\mathit{co}}Tco​. We identify the most attractive eigenvalue ${\ensuremath{\lambda}}_{\mathrm{min}}$ of ${\mathit{V}}_{\mathrm{\ensuremath{\Lambda}}={\mathit{T}}_{\mathit{co}}}$. At low filling, ${\ensuremath{\lambda}}_{\mathrm{min}}$ corresponds to the B2{\mathit{B}}_{2}B2​ representation (dxy{\mathit{d}}_{\mathit{xy}}dxy​ symmetry), while near half filling the strongest attraction occurs in the B1{\mathit{B}}_{1}B1​ representation (${\mathit{d}}_{{\mathit{x}}^{2}\mathrm{\ensuremath{-}}{\mathit{y}}^{2}}$ symmetry). In the direction of the van Hove singularities, the order parameter shows peaks with increasing strength as one approaches half filling. We also give a possible interpretation of angle-resolved photoemission spectroscopy experiments trying to determine the symmetry of the order parameter in the high-Tc{\mathit{T}}_{\mathit{c}}Tc​ compound Bi2{\mathrm{Bi}}_{2}Bi2​Sr2{\mathrm{Sr}}_{2}Sr2​CaCu2{\mathrm{CaCu}}_{2}CaCu2​O8{\mathrm{O}}_{8}O8​. \textcopyright{} 1996 The American Physical Society.
Despite its apparent simplicity, the two-dimensional Hubbard model for locally interacting fermions on a square lattice is widely considered as a promising approach for the understanding of Cooper pair formation in the quasi two-dimensional high-Tc cuprate materials. In the present work this model is investigated by means of the functional renormalization group, based on an exact flow equation for the effective average action. In addition to the fermionic degrees of freedom of the Hubbard Hamiltonian, bosonic fields are introduced which correspond to the different possible collective orders of the system, for example magnetism and superconductivity. The interactions between bosons and fermions are determined by means of the method of “rebosonization” (or “flowing bosonization”), which can be described as a continuous, scale-dependent Hubbard-Stratonovich transformation. This method allows an efficient parameterization of the momentumdependent effective two-particle interaction between fermions (four-point vertex), and it makes it possible to follow the flow of the running couplings into the regimes exhibiting spontaneous symmetry breaking, where bosonic fluctuations determine the types of order which are present on large length scales. Numerical results for the phase diagram are presented, which include the mutual influence of different, competing types of order.
We derive a differential equation for the one-particle-irreducible vertex functions of interacting fermions as a function of the temperature. Formally, these equations correspond to a Wilsonian renormalization group (RG) scheme that uses the temperature as an explicit scale parameter. Our method allows us to analyze the competition between superconducting and various magnetic Fermi surface instabilities in the one-loop approximation. In particular, this includes ferromagnetic fluctuations, which are difficult to treat on an equal footing in conventional Wilsonian momentum space techniques. Applying the scheme to the two-dimensional $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model we investigate the RG flow of the interactions at the van Hove filling with varying next-nearest-neighbor hopping ${t}^{\ensuremath{'}}.$ Starting at ${t}^{\ensuremath{'}}=0$ we describe the evolution of the flow to strong coupling from an antiferromagnetic nesting regime over a d-wave regime at moderate ${t}^{\ensuremath{'}}$ to a ferromagnetic region at larger absolute values of ${t}^{\ensuremath{'}}.$ Upon increasing the particle density in the latter regime the ferromagnetic tendencies are cut off and the leading instability occurs in the triplet superconducting pairing channel.
We describe a Wick ordered functional renormalization group method for interacting Fermi systems, where the complete flow from the bare action of the microscopic model to the effective low-energy action is obtained from a differential flow equation. We apply this renormalization group approach to a prototypical two-dimensional lattice electron system, the Hubbard model on a square lattice. The flow equation for the effective interactions is evaluated numerically on 1-loop level. The effective interactions diverge at a finite energy scale which is exponentially small for small bare interactions. To analyze the nature of the instabilities signalled by the diverging interactions we compute the flow of the singlet superconducting susceptibilities for various pairing symmetries and also charge and spin density susceptibilities. Depending on the choice of the model parameters (hopping amplitudes, interaction strength and band-filling) we find antiferromagnetic order or d-wave superconductivity as leading symmetry breaking instability.
We describe a Wick ordered functional renormalization group method for interacting Fermi systems, where the effective low-energy action is generated from the bare action of the microscopic model by a differential flow equation. We apply this renormalization group approach to a prototypical two-dimensional lattice electron system, the Hubbard model on a square lattice. The flow equation for the effective interactions is evaluated numerically on 1-loop level. The effective interactions diverge at a finite energy scale which is exponentially small for small bare interactions. To analyze the nature of the instabilities signaled by the diverging interactions, the flow of the singlet superconducting susceptibilities for various pairing symmetries and also charge and spin density susceptibilities are computed. This and similar RG calculations by others conclusively establish the existence of d-wave superconductivity in the Hubbard model.