Electronic instabilities of a Hubbard model approached as a large array of coupled chains: Competition between $d$-wave superconductivity and pseudogap phase

We study the electronic instabilities in a two-dimensional (2D) Hubbard model where one of the dimensions has a finite width, so that it can be considered as a large array of coupled chains. The finite transverse size of the system gives rise to a discrete string of Fermi points, with respective electron fields that, due to their mutual interaction, acquire anomalous scaling dimensions depending on the point of the string. Using bosonization methods, we show that the anomalous scaling dimensions vanish when the number of coupled chains goes to infinity, implying the Fermi liquid behavior of a 2D system in that limit. However, when the Fermi level is at the Van Hove singularity arising from the saddle points of the 2D dispersion, backscattering and Cooper-pair scattering lead to the breakdown of the metallic behavior at low energies. These interactions are taken into account through their renormalization group scaling, studying in turn their influence on the nonperturbative bosonization of the model. We show that, at a certain low-energy scale, the anomalous electron dimension diverges at the Fermi points closer to the saddle points of the 2D dispersion. The $d$-wave superconducting correlations also become large at low energies, but their growth is cut off as the suppression of fermion excitations takes place first, extending progressively along the Fermi points toward the diagonals of the 2D Brillouin zone. We stress that this effect arises from the vanishing of the charge stiffness at the Fermi points, characterizing a critical behavior that is well captured within our nonperturbative approach.

Figure 1

Schematic representation of the Fermi points for a 2D model made of a finite number of coupled chains, when the Fermi level is at the Van Hove singularity. The horizontal lines correspond to quantized transverse momenta for twisted boundary conditions, $k_a=\pi (2a+1)∕N$, with $a=-N∕2,-N∕2+1,\dots ,N∕2-1$. Energy contour lines are also represented for the cutoff at $E_c$ and $-E_c$ about the Fermi level.

Figure 2

Interaction processes that survive at low energies within the cutoff $E_c$ about the Fermi level. The solid and dashed lines correspond to the two different chiralities (right- and left-moving character) of the electron fields.

Figure 3

Dependence of the anomalous electron dimension ${\gamma }_a$ on the number $N$ of coupled chains in a Hubbard model with periodic boundary conditions, when the Fermi point is taken close to the diagonal of the Brillouin zone (full line), with a constant transverse momentum $k_a=\pi ∕64$ (dashed line), and at the nearest location to the saddle point with $k_a=2\pi ∕N$ (dotted line).

Figure 4

Plot of the dominant response functions for each different kind of order as a function of $l=-\log (\Lambda ∕E_c)$. From top to bottom, we have SS $d_{x^2-y^2}$ (solid line), CDW (dotted line), SDW (dashed line), and TS (dotted-dashed line) response functions. The data correspond to a Hubbard model with $N=48$, $t^{\prime }=-0.2t$, and $U=t$, at the Van Hove filling. The point where the renormalization group flow diverges is at $l^{*}\approx 6.96$. The vertical line at $l_{c1}\approx 6.40$ indicates the energy scale at which the Fermi line starts to collapse.

Figure 5

Similar plot as in Fig. 4 for $N=76$, with $l^{*}\approx 4.99$ and $l_{c1}\approx 4.36$.

Figure 6

Diagrams contributing to the electron self-energy which renormalize to lowest order the electron quasiparticle weight.

Figure 7

Angle-resolved quasiparticle weight $Z_a$ as a function of the transverse momentum $k_a$ for the first four iterations of the combined renormalization group and bosonization approaches for an array with $N=48$. From top to bottom, $Z_a$ is shown at the points $l_{c1}$, $l_{c2}$, $l_{c3}$, and $l_{c4}$, where a new set of Fermi points decouples from the spectrum. The parameters are the same as in Fig. 4.

Figure 8

Plot of the critical scales $l^{*}$ (dotted line) and $l_{c1}$ (full line) as a function of $N$. The parameters are the same as in Fig. 4.