Strong-coupling functional renormalization group: Nagaoka ferromagnetism and non-Fermi liquid physics in the Hubbard model at $ U = \infty $

Abstract

We develop an extension of the fermionic functional renormalization group for systems where strong correlations give rise to projected Hilbert spaces. We use our method to calculate the phase diagram and the electronic spectral function of the Hubbard model at infinite on-site repulsion where many-body states involving doubly occupied lattice sites are eliminated from the physical Hilbert space. For a square lattice with nearest-neighbor hopping we find that the ground state evolves from a paramagnetic Fermi liquid at low densities via a state with antiferromagnetic stripe order at intermediate densities to an extended Nagaoka ferromagnet at high densities. In the strongly correlated magnetic phases, the electrons form an incoherent non-Fermi liquid. Both at high and low densities, the volume of the Fermi surface is not constrained by Luttinger's theorem.

Figures (4)

• Figure 1: Ground state phase diagriam of the $t$ model. With increasing density $n$, the system transitions from a paramagnet (PM) to stripe magnet to ferromagnet (FM). This is accompanied by a crossover from a Fermi liquid at small $n$ to an incoherent non-Fermi liquid at large $n$.
• Figure 2: Graphical representation of the flow equations for (a) holon self-energy $\Sigma_\Lambda ( K )$ and (b) two-body interaction vertex $U_\Lambda ( K_1' , K_2' ; K_2 , K_1 )$. Lines with arrows denote the holon propagator $G_\Lambda ( K )$. An additional dash means the single-scale propagator $\dot{G}_\Lambda ( K ) = G_\Lambda^2 ( K ) \partial_\Lambda ( t_{ \Lambda , \bm{k} } - 2 \delta \mu_\Lambda / 3 )$. Crosses inside loops signify that each $G_\Lambda ( K )$ is subsequently replaced by $\dot{G}_\Lambda ( K )$.
• Figure 3: Brillouin zone plots of the static part of the magnetic channel $M_{ \Lambda = 1 } ( \bm{q} , 0 ; 0 , 0 ) / t$. (a) $n = 0.2$, $T = 0.14 t$ in the paramagnetic state; (b) $n = 0.56$, $T = 0.13 t$ showing the stripe instability; (c) $n = 0.59$, $T = 0.16 t$ where stripe and ferromagnetic instabilities compete; and (d) $n = 0.85$, $T = 0.64 t$ showing the instability towards the Nagaoka ferromagnet.
• Figure 4: Electronic spectral properties for (a) $n = 0.2$ and $T = 0.14 t$ in the paramagnet, (b) $n = 0.56$ and $T = 0.13 t$ coinciding with the stripe instability, and (c) $n = 0.85$ and $T = 0.65 t$ where the system is unstable towards kinetic ferromagnetism. From left to right: Spectral function $t A ( \bm{k} , \omega )$ along a high-symmetry path through the Brillouin zone, with the dashed line marking the Fermi energy; $t A ( \bm{k} , 0 )$ defining the Fermi surface, with the dashed red line indicating the Fermi surface of free electrons with the same $n$ for comparison; the density of states $t \textrm{DOS} ( \omega ) = (1 / N ) \sum_{ \bm{k} } t A ( \bm{k} , \omega )$. For $n = 0.2$, we regularized the DOS with $\eta = 0.1 t$ because in this case the quasi-particle peaks at the Fermi surface are sharper than our $\bm{k}$-resolution.