Slave-rotor mean-field theories of strongly correlated systems and the Mott transition in finite dimensions

The multiorbital Hubbard model is expressed in terms of quantum phase variables (“slave rotors”) conjugate to the local charge, and of auxiliary fermions, providing an economical representation of the Hilbert space of strongly correlated systems. When the phase variables are treated in a local mean-field manner, similar results to the dynamical mean-field theory are obtained, namely a Brinkman-Rice transition at commensurate fillings together with a “preformed” Mott gap in the single-particle density of states. The slave-rotor formalism allows to go beyond the local description and take into account spatial correlations, following an analogy to the superfluid-insulator transition of bosonic systems. We find that the divergence of the effective mass at the metal-insulator transition is suppressed by short range magnetic correlations in finite-dimensional systems. Furthermore, the strict separation of energy scales between the Fermi-liquid coherence scale and the Mott gap, found in the local picture, holds only approximately in finite dimensions, due to the existence of low-energy collective modes related to zero-sound.

Figure 1

(Color online) Coulomb staircase in the atomic limit for the case of two orbitals, $N=4$.

Figure 2

(Color online) Graphical solution of the average constraint equation (10). The intersect (cross) moves exactly along the Coulomb staircase shown in Fig. 1.

Figure 3

(Color online) Phase diagram for $N=4$ (two orbitals) at $T=0$, as a function of the chemical potential ${ϵ}_0=-\mu$ and the interaction strength $U∕D$. The three lobes correspond to the Mott insulator phases associated with half-filling $\left(Q=2\right)$ and quarter-filling $\left(Q=1,3\right)$, respectively.

Figure 4

(Color online) Phase diagram in the $\left(n,U\right)$ plane. The Mott insulator lobes collapse to lines at commensurate fillings, when $U$ is larger than $U_c\left(n\right)$ (shown as dots).

Figure 5

(Color online) Rotor ground state wave function ${\Psi }_0\left(\theta \right)$ with values of the local interaction ranging from $U∕U_c=0.01$ (peaked curve) to $U∕U_c=1$ at the Mott transition (flat curve).

Figure 6

(Color online) Quasiparticle weight $Z$ as a function of $U∕U_c$ at $T=0$; DMFT calculation (thin line), rotor mean-field theory (thick line), and Gutzwiller approximation (broken line).

Figure 7

(Color online) Total occupancy $Q=4n$ as a function of ${ϵ}_0$ for a value $U=4.5$ larger than all critical interactions $U_c\left(n\right)$, in the two orbital case $\left(N=4\right)$. The Mott insulators are seen here as charge plateaus.

Figure 8

(Color online) Effective mass $m^{*}∕m$ for $U=2.5$ below all Mott transitions, in the two orbital case $\left(N=4\right)$.

Figure 9

(Color online) Plot of the quasiparticle weight $Z$, the effective mass renormalization $Q_f=m∕m^{*}$, and the Mott gap ${\Delta }_g$ as a function of $U∕U_c$ across the Mott transition in the three-dimensional case.

Figure 10

(Color online) Effective mass $m^{*}∕m=1∕Q$ provided by the mean-field Eqs. (54, 55, 56) for $d=3$ (bold line) and $d=\infty$ (thin line). For comparison, a DMFT-IPT calculation (dashed line) is also presented.

Figure 11

(Color online) Zero temperature local density of states across the Mott transition for a three-dimensional cubic lattice, with $U=0$, $U_c∕2$, $U_c$, $3Uc∕2$ (for $D=1$).

Figure 12

(Color online) Local spectral density at the critical point $U=U_c$, for increasing dimensionality (top to bottom curves at small $\omega$: $d=2,3,4,5$). The spectral weight associated with the low-frequency tails of the Hubbard bands is seen to decrease as dimensionality increases. Correspondingly, the separation of energy scales and the preformed gap become more and more apparent. The inset shows the integrated density of states ${\int }_0^{\omega }\mathrm{d}ϵ{\rho }_d\left(ϵ\right)$ demonstrating this approximate separation of scales for $d⩾3$. (Note that the progressive narrowing of the main lobe of the Hubbard band as $d$ increases is an artefact of the approximation in which spinons and rotors are decoupled.)