Mott physics and spin fluctuations: A unified framework

We present a formalism for strongly correlated electron systems which consists in a local approximation of the dynamical three-leg interaction vertex. This vertex is self-consistently computed using a quantum impurity model with dynamical interactions in the charge and spin channels, similar to dynamical mean field theory approaches. The electronic self-energy and the polarization are both frequency and momentum dependent. The method interpolates between the spin-fluctuation or GW approximations at weak coupling and the atomic limit at strong coupling. We apply the formalism to the Hubbard model on a two-dimensional square lattice and show that as interactions are increased towards the Mott insulating state, the local vertex acquires a strong frequency dependence, driving the system to a Mott transition, while at low enough temperatures the momentum dependence of the self-energy is enhanced due to large spin fluctuations. Upon doping, we find a Fermi arc in the one-particle spectral function, which is one signature of the pseudogap state.

Figure 1

The TRILEX self-consistency loop.

Figure 2

$(T,U)$ phase diagram (half filling, $t^{\prime }=0$). The green squares denote converged TRILEX solutions. A, B, and C are defined as (A) $\beta D=96, U/D=0.5$, (B) $\beta D=24, U/D=2$, and (C) $\beta D=48, U/D=4$. The red dotted line denotes $T_{\mathrm{N}\acute{\mathrm{e}}\mathrm{el}}^{\text{DMFT}}$ for the square lattice (from Ref. [68]). The black squares denote $\overline{T}$ (temperature below which one cannot obtain stable solutions, hatched region); the black dashed line is a guide to the eye.

Figure 3

Left: Evolution of the local vertex $\mathrm{Re}{\mathrm{\Lambda }}_{\mathrm{imp}}^{\eta }(i{\omega }_n,i{\mathrm{\Omega }}_m)$ as a function of ${\omega }_n$ (half filling, $t^{\prime }=0$). A, B, and C are defined in Fig. 2. The dashed lines denote the atomic vertex ${\mathrm{\Lambda }}_{\mathrm{at}}^{\eta }$ [Eq. (A1)]. Right: $\mathrm{Im}{\mathrm{\Sigma }}_{\mathrm{loc}}\left(i{\omega }_n\right)$ for TRILEX and DMFT (paramagnetic phase).

Figure 4

Momentum dependence of the self-energy and polarization (half filling, $t^{\prime }=0$). A, B, and C are defined in Fig. 2. Left: $\mathrm{Re}{P}^{\mathrm{ch}}(\mathbf{q},\omega =0)$. Middle: $\mathrm{Re}{P}^{\mathrm{sp}}(\mathbf{q},\omega =0)$. Right: $\mathrm{Im}\mathrm{\Sigma }(\mathbf{k},i{\omega }_0)$.

Figure 5

From left to right: $A(\mathbf{k},\omega =0), \chi (\mathbf{q},i\mathrm{\Omega }=0)$, and $\mathrm{Im}\mathrm{\Sigma }(\mathbf{k},i{\omega }_0)$ in the doped case: $U/D=1.8, t^{\prime }=-0.4t, \beta D=96, \delta =10\%$.