Phase diagram and quasiparticle properties of the Hubbard model within cluster two-site dynamical mean-field theory

We present a cluster dynamical mean-field treatment of the Hubbard model on a square lattice to study the evolution of magnetism and quasiparticle properties as the electron filling and interaction strength are varied. Our approach for solving the dynamical mean-field equations is an extension of Potthoff’s “two-site” method [Phys. Rev. B. 64, 165114 (2001)], where the self-consistent bath is represented by a highly restricted set of states. As well as the expected antiferromagnetism close to half-filling, we observe distortions of the Fermi surface. The proximity of a van Hove point and the incipient antiferromagnetism lead to the evolution from an electronlike Fermi surface away from the Mott transition, to a holelike one near half-filling. Our results also show a gap opening anisotropically around the Fermi surface, close to the Mott transition (reminiscent of the pseudogap phenomenon seen in the cuprate high-$T_c$ superconductors). This leaves Fermi arcs that are closed into pockets by lines with very small quasiparticle residue.

Figure 1

In our pair-cluster approach we identify a small cluster in the lattice (a), and treat this cluster as an “impurity” connected to a bath. A highly truncated basis is chosen for this bath, leading to an impurity model with four sites (b). To reconstruct the extended system, the cluster is reembedded in the lattice periodically (c). The bath is then adjusted to ensure it is self-consistent with this embedding. Although our cluster shape manifestly breaks the $x\text{−}y$ symmetry, there are self-consistent solutions that reflect the original square lattice symmetry.

Figure 2

The schematic ground-state phase diagram $(n,U∕t)$ for the square lattice Hubbard model (bandwidth $8t$) for our cluster DMFT, showing a conventional paramagnetic phase (a), a paramagnetic phase with a distorted Fermi surface (b), a Slaterlike antiferromagnetic phase (c), another antiferromagnetic phase (d), a pure Slater nested antiferromagnetic phase (e), and a ferromagnetic phase (f). A dashed line indicates a second-order transition. The phases are characterized further in the text. Solutions for the hatched region, $0.98<n<1$, are not considered (see text).

Figure 3

At $U=20t$, $n=0.8$, the ground state is paramagnetic [Fig. 2, phase (a)]. The smoothed local density of states for this solution is shown. The central quasiparticle peak shows a single-particle density of states, with a bandwidth $8t*\sim 3t$ significantly renormalized from the original $8t$ due to correlations. Above and below the peak there are upper and lower Hubbard bands, separated by a Mott gap of size $\sim U$, and both Hubbard bands show imprints of the single-particle dispersion.

Figure 4

At $U=2t$, $n=0.90$, the ground state is paramagnetic but the Fermi surface has spontaneously distorted [Fig. 2, phase (b)]. Here the electron-filling $n_{\mathbf{k}}$ is shown in grayscale across the Brillouin zone: black corresponds to fully occupied states, and white corresponds to completely empty states. The Fermi surface is well defined, but has undergone a Pomeranchuk transition: electrons have moved from $(\pi ,0)$ to $(0,\pi )$, exploiting the flat dispersion near the van Hove point (compare with the $U=0.5t$ Fermi surface indicated with the dashed line). Note the appearance of a faint “ghost” Fermi surface, displaced from the original by a wave vector $(\pi ∕a,\pi ∕a)$. This is due to the effect of antiferromagnetic fluctuations.

Figure 5

At $U=6t$, $n=0.88$, the ground state is an antiferromagnetic metal with a distorted Fermi surface [Fig. 2 phase (c)]. (a) The total local density of states is shown for this state (smoothed and combining up and down spins). The magnetic gap is apparent, and correlations cause further partitioning. In this region, the gap size increases with increasing $U$. The densities of states for the neighboring paramagnetic solution [Fig. 2, phase (b)] are insignificantly different, although the sublattice- and spin-resolved densities of states are, of course, unbalanced for antiferromagnetic solutions. (b) The evolution of the electron spectral function is shown for various straight-line sections through the Brillouin zone. The plot is grayscale with darker regions representing states with higher spectral weight. As well as weakly dispersing bands at high energies, the quasiparticle dispersion can be seen for energies between $\pm 4t$. The form of the dispersion is that expected for a mean-field spin-density wave state. Hence we identify the gap in the density of states as being Slaterlike.

Figure 6

At $U=20t$, $n=0.9$ the ground state is also an antiferromagnetic metal with little $x\text{−}y$ symmetry breaking [Fig. 2, phase (d)]. (a) Here we show the local density of states (smoothed, up and down spins combined). The central quasiparticle peak is separated from upper and lower Hubbard bands and is heavily renormalized due to correlations. It is also gapped; this gap decreases with increasing $U$. (b) From the spectral density we identify the dispersing quasiparticle near the chemical potential. The form of the dispersion is very similar to that of Fig. 5b, so Slaterlike, but the degree of dispersion is very much reduced.

Figure 7

At $U=5t$, $n=0.9$, the ground state is paramagnetic and breaks $x\text{−}y$ symmetry. It also shows the breakup of the Fermi surface into hole pockets by the formation of an anisotropic gap around the Fermi surface. (a) shows $n_{\mathbf{k}}$ in grayscale across the Brillouin zone $(\text{black}\leftrightarrow 1,\text{white}\leftrightarrow 0)$. A distinct Fermi surface can be seen along some directions but not others. (b) illustrates the absence of a Fermi surface discontinuity in $n_{\mathbf{k}}$ along the line $(0,0)\to (\pi ,\pi )$ (solid line). This should be compared to the form of $n_{\mathbf{k}}$ in a phase without such a gap [$(U=8t,n=0.81)$ shown by the dotted line]. The nature of the anisotropy around the Fermi surface can be seen in the electron spectral functions: (c) A grayscale plot of the spectral function $(\text{dark}\leftrightarrow \text{larger}\text{weight})$ along the line $(0,0)\to (\pi ,0)$. The chemical potential $\mu$ crosses a modified dispersion twice, creating the hole pocket—though with little weight at the second crossing point. (d) The spectral function $(\text{dark}\leftrightarrow \text{larger}\text{weight})$ along the line $(0,0)\to (\pi ,\pi )$ shows that $\mu$ falls within a gap; there is no Fermi surface along this line. The dashed line in (c) and (d) shows the noninteracting dispersion. Figure 8 shows, schematically, the origin of this electronic structure.

Figure 8

A schematic view of the origin of the structure of the electron spectral function in the pseudogap region [as seen within our approach in Figs. 7c, 7d]. Darker lines carry more spectral weight. (a) Distinct fluctuations on the $A$ and $B$ sublattices result in a second dispersion of low weight (a “ghost” dispersion). This is formed by the folding of the original, possibly renormalized, dispersion in the antiferromagnetic Brillouin zone (defining $g_m$). (b) This hybridizes with the original dispersion and forms a gap. Along directions in momentum space where the chemical potential lies in the gap [line (i)], there is a gap in the excitation spectrum and no Fermi surface. Along directions where the chemical potential lies away from the gap, it crosses the dispersion at two points [line (ii)]. A hole pocket forms with low spectral weight at the second crossing.