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arXiv:0710.4938 · Physical Review B

Breakdown of the Fermi-liquid regime in the two-dimensional Hubbard model from a two-loop field-theoretical renormalization group approach

Hermann Freire, Eberth de Almeida Correa, Álvaro Antônio Bandeira Ferraz

Published Sep 29, 2008• 17 citations
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Abstract

We analyze the particle-hole symmetric two-dimensional Hubbard model on a square lattice starting from weak-to-moderate couplings by means of the field-theoretical renormalization group approach up to two-loop order. This method is essential in order to evaluate the effect of the momentum-resolved anomalous dimension $\ensuremath{\eta}(\mathbf{p})$ which arises in the normal phase of this model on the corresponding low-energy single-particle excitations. As a result, we find important indications pointing to the existence of a non-Fermi-liquid (NFL) regime at temperature $T\ensuremath{\rightarrow}0$ displaying a truncated Fermi surface (FS) for a doping range exactly in between the well-known antiferromagnetic insulating and the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave singlet superconducting phases. This NFL evolves as a function of doping into a correlated metal with a large FS before the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave pairing susceptibility finally produces the dominant instability in the low-energy limit.

DOI
10.1103/physrevb.78.125114

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We analyze the particle-hole symmetric two-dimensional Hubbard model on a square lattice starting from weak-to-moderate couplings by means of the field-theoretical renormalization group approach up to two-loop order. This method is essential in order to evaluate the effect of the momentum-resolved anomalous dimension \ensuremathη(p)\ensuremath{\eta}(\mathbf{p})\ensuremathη(p) which arises in the normal phase of this model on the corresponding low-energy single-particle excitations. As a result, we find important indications pointing to the existence of a non-Fermi-liquid (NFL) regime at temperature T\ensuremath→0T\ensuremath{\rightarrow}0T\ensuremath→0 displaying a truncated Fermi surface (FS) for a doping range exactly in between the well-known antiferromagnetic insulating and the dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​-wave singlet superconducting phases. This NFL evolves as a function of doping into a correlated metal with a large FS before the dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​-wave pairing susceptibility finally produces the dominant instability in the low-energy limit.
We study the normal-state, doping-driven phase diagram of the square-lattice Hubbard model using the dynamical cluster approximation combined with the numerical renormalization group as a cluster solver, which gives direct access to real-frequency dynamics at essentially zero temperature. In a parameter regime relevant for cuprates, U=7tU=7tU=7t and t′=−0.3tt'=-0.3tt′=−0.3t, we find a critical doping p∗p^{\ast}p∗ that marks a continuous quantum phase transition between a pseudogap metal and a normal Fermi liquid. The transition is identified by a continuous collapse, from both sides, of the Fermi-liquid scale extracted from charge, spin, and dx2−y2d_{x^2-y^2}dx2−y2​-wave pairing susceptibilities. This collapse produces a non-Fermi-liquid regime at intermediate energy scales, which appears to extend to arbitrarily low scales at p∗p^{\ast}p∗. As p∗p^{\ast}p∗ is crossed from the normal Fermi liquid at p>p∗p>p^{\ast}p>p∗ into the pseudogap metal at $p
Using perturbation theory and the field theoretical renormalization group approach we consider a two-dimensional anisotropic truncated Fermi Surface (FS) with both flat and curved sectors which approximately simulates the ``cold'' and ``hot'' spots in the cuprate superconductors. We calculate the one-particle two-loop irreducible functions ${\ensuremath{\Gamma}}^{(2)}$ and ${\ensuremath{\Gamma}}^{(4)}$ as well as the spin, the charge and pairing response functions up to one-loop order. We find nontrivial infrared stable fixed points and we show that there are important effects produced by the mixing of the existing scattering channels in higher order of perturbation theory. Our results indicate that the ``cold'' spots are turned into a non-Fermi liquid with divergents $\ensuremath{\partial}{\ensuremath{\Sigma}}_{0}/\ensuremath{\partial}{p}_{0}$ and $\ensuremath{\partial}{\ensuremath{\Sigma}}_{0}/\ensuremath{\partial}\overline{p},$ a vanishing Z, and either a finite or zero ``Fermi velocity'' at the FS when the effects produced by the flat portions are taken into account.
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.
Abstract We present a T = 0 K renormalization group (RG) phase diagram for the electronic Hubbard model in two dimensions on the square lattice at half filling. The RG procedure treats quantum fluctuations in the single particle occupation number nonperturbatively via the unitarily decoupling of one electronic state at every RG step. The resulting phase diagram thus possesses the quantum fluctuation energy scale (ω) as one of its axes. A relation is derived between ω and the effective temperature scale upto which gapless, as well as emergent gapped phases can be obtained. We find that the normal and insulating phases of the half-filled Hubbard model correspond, for any on-site repulsion, to a marginal Fermi liquid normal phase and a topologically-ordered gapped Mott insulating liquid respectively. The marginal Fermi liquid is found to arise from singular forward scattering in directions normal to the nested Fermi surface, while singular backscattering events lead to Mott liquid state. The transition between these two phases involves passage through a pseudogapped phase bookended by Fermi surface topology-changing Lifshitz transitions. The pseudogap phase is observed to arise from the electronic differentiation encoded within the nested Fermi surface, and involves the gradual gapping of the Fermi surface from antinodes to nodes via charge and spin excitations that are mutually entangled. We obtain effective Hamiltonians for various phases, as well as wavefunctions for the low-energy many-body eigenstates of the Mott liquid. Benchmarking of the ground-state energy per particle and the double-occupancy fraction for the Mott liquid against existing numerical results yields excellent agreement. Presence of a Néel ordering symmetry-breaking perturbation in the RG leads to an antiferromagnetic spin-ordered charge insulating Mott state. Our results thus offer novel insights on a variety of aspects of the Mott–Hubbard problem, and can be extended to the doped system.
In the present work, we implement an explicit two-loop renormalization of a two-dimensional flat Fermi surface (FS) in the framework of a field theoretical renormalization group approach (RG). In our scheme, we derive the RG equations for both the coupling functions and Fermi energy. In this way, we are able to probe the existence of spin-charge separation by showing that the low-energy sector of the system is in fact a non-Fermi liquid. In addition, associating the true interacting FS to the infrared stable (IR) fixed point of the Fermi energy, we demonstrate here that it either acquires a small curvature and behaves as a "Luttinger liquid" or it suffers a truncation in k-space depicting an insulating spin liquid.
We present the formalism for a two-loop renormalization group (RG) calculation of some order-parameter susceptibilities associated with a two-dimensional (2D) flat Fermi-surface model. In this order of perturbation theory, one must take into account the self-energy effects directly in all RG flow equations. In one-loop order, our calculation reproduces the well-known results obtained previously by other RG schemes. That is, for repulsive interactions all susceptibilities diverge in the low-energy limit and the antiferromagnetic (AF) spin-density-wave correlations produce indeed the leading instability in the system. In contrast, in two-loop order, only the AF susceptibility diverges for this model. However, even this divergence takes place at a much slower rate than in the one-loop RG approach. The purpose of this paper is to show in a very simple setting how to assess the importance of two-loop quantum fluctuations in 2D interacting fermionic models. With some modifications, the present formalism can also be extended to discuss more realistic models such as the paradigmatic 2D Hubbard model.