Fermi arcs and hidden zeros of the Green function in the pseudogap state

We investigate the low-energy properties of a correlated metal in the proximity of a Mott insulator within the Hubbard model in two dimensions. We introduce a version of the cellular dynamical mean-field theory using cumulants as the basic irreducible objects. The cumulants are used for reconstructing the lattice quantities from their cluster counterparts. The zero-temperature one-particle Green function is characterized by the appearance of lines of zeros, in addition to a Fermi surface which changes topology as a function of doping. We show that these features are intimately connected to the opening of a pseudogap in the one-particle spectrum and provide a simple picture for the appearance of Fermi arcs.

Figure 1

(Color online) Kinetic energy of the half-filled 1D Hubbard model as a function of the on-site interaction $U$ at zero temperature using: the Bethe ansatz (blue/dark gray line), the cluster Green function (black line with circles), and the lattice Green function obtained by periodizing the self-energy (green/light gray line) and the irreducible cumulant (dashed red line). The inset shows the kinetic energy for the 2D case (same color code). The Green function periodization scheme (not shown) produces results that are almost identical with those given by the cumulant scheme. The results were obtained using CDMFT with an exact diagonalization (ED) impurity solver.

Figure 2

(Color online) Cluster cumulant as a function of the imaginary frequency for the half-filled 2D Hubbard model with $U=8t$. We compare the on-site cumulant $M_{11}$ (blue/dark gray) with the next-nearest-neighbor cumulant $M_{13}$ (orange/light gray), as both components are purely imaginary. The nearest-neighbor cumulant $M_{12}$ (not shown) is real. Notice that $\mathrm{Im}\left[M_{13}\right]⪡\mathrm{Im}\left[M_{11}\right]$ for all frequencies, as a result of the local nature of the cumulant. The inset shows the imaginary components of the cluster self-energy, ${\Sigma }_{11}$ (blue/dark gray) and ${\Sigma }_{13}$ (orange/light gray). Notice that (i) both components diverge as ${\omega }_n\to 0$, and (ii) the two components become comparable at small energies, as proved by the vanishing of the sum ${\Sigma }_{11}+{\Sigma }_{13}$ (dashed red line). This behavior is a result of the nonlocal nature of the self-energy.

Figure 3

(Color online) Comparison of the local cluster Green function (green line) with the lattice local Green function calculated using the cumulant periodization scheme (blue circles) and the self-energy periodization scheme (orange triangles). The good agreement between the cluster $G_{11}$ and the lattice Green function obtained using cumulants is a result of the local nature of this irreducible quantity. In contrast, the self-energy is nonlocal and the reconstruction scheme based on $\Sigma$ fails. The results are for a 2D half-filled Hubbard model with $U=8t$ and were obtained using a four-site cluster CDMFT with an ED impurity solver.

Figure 4

(Color online) Renormalized energy $r\left(\mathbf{k}\right)$ (upper panels) and spectral function $A\left(\mathbf{k}\right)$ (lower panels) for the 2D Hubbard model with $U=8t$ and $T=0$. The color code for the upper panels is green/gray $(r<0)$, blue/dark gray line $(r=0)$, yellow/light gray $(r>0)$, red dashed line $(r\to \infty )$. The frequency dependence of the spectral function for the points marked by $A$, $B$, and $C$ is shown in Fig. 5.

Figure 5

(Color online) Frequency dependence of the spectral function for three points in the Brillouin zone marked by A, B, and C in Fig. 4. Point $A$ (blue line with triangles) is on the Fermi surface, close to $(\pi ∕2,\pi ∕2)$; point $B$ (green squares) is on the “dark side” of the Fermi surface, in the vicinity of the zero line; and point $C$ (red circles) is in the pseudogap region on the line and corresponding to the maxima of the spectral function (see Fig. 4). Notice that the leading edge gap is quantitatively much smaller than the distance between the peaks at positive and negative energy.