${\mathsf{Z}}_2$-slave-spin theory for strongly correlated fermions

We review a representation of Hubbard-like models that is based on auxiliary pseudospin variables. These pseudospins refer to the local charge modulo two in the original model and display a local ${\mathsf{Z}}_2$ gauge freedom. We discuss the associated mean-field theory in a variety of different contexts which are related to the problem of the interaction-driven metal-insulator transition at half-filling including Fermi surface deformation and spectral features beyond the local approximation. Notably, on the mean-field level, the Hubbard bands are derived from the excitations of an Ising model in a transverse field and the quantum critical point of this model is identified with the Brinkman-Rice criticality of the almost localized Fermi liquid state. Nonlocal correlations are included using a cluster mean-field approximation and the Schwinger boson theory for the auxiliary quantum Ising model.

Figure 1

(Color online) Pictorial illustration of the pseudospin representation discussed in this article.

Figure 2

The nearest-neighbor exchange $\chi t$ and the next-nearest-neighbor exchange ${\chi }^{\prime }{t}^{\prime }$ of the auxiliary pseudospin model as function of $U/t$ for $t^{\prime }=-0.3t$. The inset shows the considered cluster.

Figure 3

(Color online) (a) The quasiparticle weight $Z$ and the nearest-neighbor and next-nearest-neighbor hopping renormalization factors $g_t$ and $g_{t^{\prime }}$, respectively, as function of $U/t$. (b) The Fermi surface for different values of the interaction strength.

Figure 4

(a) The band of pseudospin-wave excitations obtained in the spin-wave analysis of the transverse Ising model in three dimensions as function of $u=U/U_c$. (b) The inverse effective mass $m/m^{\ast }$ and the quasiparticle weight $Z$ as function of $U/U_c$. Note that at the Mott transition $u=1$, $Z$ vanishes in contrast to $m/m^{\ast }$.

Figure 5

(Color online) The contour plot shows the one-particle spectral density $A_{\sigma }\left(\omega \right)$ with the coherent Gutzwiller and the (preformed) Hubbard bands as function of energy $\omega /t$ and interaction strength $u$. On the right-hand side, $A_{\sigma }\left(\omega \right)$ is shown for fixed values of $u$ in the metallic and the insulating phase.

Figure 6

The integrated spectral weight of the one-particle spectral density in the pseudospin-wave analysis. The shaded region denotes the fluctuation regime where the magnitude of the fluctuations is of the same order than the order parameter.

Figure 7

(Color online) The staggered magnetization ${\nu }_s$ at $T=0$ for the square lattice as function of the interaction strength $U/t$ calculated using the local slave-spin approximation of Sec. 4A (dashed dotted) and with fluctuation corrections as discussed in Sec. 5B (solid). For comparison we also show the result obtained in the Hartree-Fock mean-field theory (dashed). In the slave-spin calculation there is a finite range of $U/t$ where there exists two solutions for ${\nu }_s$ which corresponds to a (local) minima of the energy.