Numerical algorithm for the double-orbital Hubbard model: Hund-coupled pairing symmetry in the doped case

In order to numerically study electron correlation effects in multiorbital systems, we propose a new type of discrete transformation for the exchange (Hund’s coupling) and pair-hopping interactions to be used in the dynamical mean field $\text{theory}+\text{quantum}$ Monte Carlo method. The transformation, which is real and exact, turns out to suppress the sign problem in a wide parameter region including non-half-filled bands. This enables us to obtain the dominant pairing symmetry in the double-orbital Hubbard model, which shows that the spin-triplet, orbital-antisymmetric pairing that exploits Hund’s coupling is stable in a wide region of the band filling.

Figure 1

The average sign plotted as a function of band filling at $\beta \equiv 1∕T=2$ for $U=U^{\prime }=1$, $J=0.5$ and at $\beta =10$ for $U=U^{\prime }=1$, $J=0.2$ calculated with the present algorithm (circles) and with the one due to Ref. 9 (squares). The average sign for the latter is complex, so we have plotted its absolute value for this method. In this particular plot we have set $U=U^{\prime }$ to facilitate comparison with the Motome- Imada method (Ref. 11). Curves are guides to the eye here and in subsequent figures.

Figure 2

The average sign plotted as a function of inverse temperature for $n=1.8$, $U=U^{\prime }=1$, $J=0.2$ calculated with the present algorithm (circles) and with the one due to (Ref. 9) (squares).

Figure 3

Temperature dependence of the pairing susceptibilities for $n=1.6$, $U^{\prime }=0.7$, $J=0.4$, $U=U^{\prime }+2J=1.5$. The symbols denote pairing symmetries (1: spin singlet, 3: triplet; ${\mathrm{S}}^{\mathrm{a},\mathrm{b},\mathrm{c}}$: orbital-symmetric [Eq. (4)], A: antisymmetric; E: even, O: odd frequency).

Figure 4

Band-filling dependence of the pairing susceptibilities for $\beta =60$, $U^{\prime }=0.7$, $J=0.4$, $U=U^{\prime }+2J=1.5$ with the same abbreviation for pairing symmetries as in Fig. 3.