Competing orders in the generalized Hund chain model at half filling

By using a combination of several nonperturbative techniques—a one-dimensional field theoretical approach together with numerical simulations using density-matrix renormalization group—we present an extensive study of the phase diagram of the generalized Hund model at half filling. This model encloses the physics of various strongly correlated one-dimensional systems, such as two-leg electronic ladders, ultracold degenerate fermionic gases carrying a large hyperfine spin $\frac{3}{2}$, other cold gases such as ytterbium 171 or alkaline-earth condensates. A particular emphasis is laid on the possibility to enumerate and exhaust the eight possible Mott-insulating phases by means of a duality approach. We exhibit a one-to-one correspondence between these phases and those of the two-leg electronic ladders with interchain hopping. Our results obtained from a weak-coupling analysis are in remarkable quantitative agreement with our numerical results carried out at moderate coupling.

Figure 1

Energy-level diagram for the one-site Hamiltonian (1) with $t=0$.

Figure 2

(Color online) Low-energy phase diagram of the generalized Hund model at half filling for $U=-0.005t$ $\left(t=1\right)$. ($\text{SP}=\text{spin}$ Peierls, $\text{CDW}=\text{charge}$-density wave, $\text{ODW}=\text{orbital}$ density wave, $\text{RS}=\text{rung}$ singlet, $\text{HC}=\text{Haldane}$ charge, $\text{HO}=\text{Haldane}$ orbital, $\text{RT}=\text{rung}$ triplet). At the intersection of the highly symmetric lines lie points with even larger symmetry (see Table ).

Figure 3

Quantum phase transitions that can occur in the generalized Hund model. The letter next to the arrows indicates which degrees of freedom are critical: $c=\text{charge}$, $s=\text{spin}$, and $o=\text{orbital}$.

Figure 4

(Color online) Phase diagram of the spin-$\frac{3}{2}$ cold fermions model at half filling $\left(t=1\right)$.

Figure 5

(Color online) Phase diagram of the SZH model (8) with the fine tuning $J_{\text{SZH}}=4\left(U_{\text{SZH}}+V_{\text{SZH}}\right)$ at half filling $\left(t=1\right)$.

Figure 6

(Color online) Phase diagram of the SO(4) model (5) with $J_H=0$ $\left(t=1\right)$ at half filling. The continuous symmetry group is $\text{U}{\left(1\right)}_c\times \text{U}{\left(1\right)}_o\times \text{SO}{\left(4\right)}_s$.

Figure 7

(Color online) Phase diagram of the half-filled SO(4) model with $J_t=2U$ $\left(t=1\right)$ which unifies charge and orbital degrees of freedom.

Figure 8

(Color online) Phase diagram of the spin-orbital model at half filling with $J_t=-3J_H/4$ $\left(t=1\right)$.

Figure 9

(Color online) Phase diagram of the spin-charge model at half filling with $J_H=-8U/3$ $\left(t=1\right)$. The symmetry is $\text{SU}{\left(2\right)}_s\times \text{SU}{\left(2\right)}_c\times \text{U}{\left(1\right)}_o$.

Figure 10

(Color online) The eight fully gapped phases of the generalized Hund chain with a transverse hopping at half filling $\left(U=-0.005t\right)$.

Figure 11

(Color online) Numerical phase diagram of the generalized Hund model at half filling for $U=-t$. Notations are the same as in Fig. 2.

Figure 12

(Color online) Numerical phase diagram of the SZH SO(5) model with the fine tuning $J_{\text{SZH}}=4\left(U_{\text{SZH}}+V_{\text{SZH}}\right)$ at half filling. Notations are the same as in Fig. 5. Dashed lines indicate models with higher symmetry (see Table ).

Figure 13

(Color online) Numerical phase diagram of the spin-orbital model $\left(J_t=-3J_H/4\right)$ at half filling. Notations are the same as in Fig. 8. Dashed lines indicate models with higher symmetry (see Table ).

Figure 14

(Color online) Phase diagram of the generalized Hund ladder with interchain hopping. The coordinate on the horizontal axis is proportional to $t_{\perp }$ whereas that on the vertical axis is the distance $D_o$ to the SU(2) orbital symmetric model. Bold (blue) dashed lines indicate $c=\frac{1}{2}$ critical lines, while the bold (red) one is a $c=1$ critical line. See the main text for the definition of the phases. The dotted line, with equation $m^2{\delta }^2=h_o^2\left(h_o^2-m^2\right)$, indicates the location where incommensuration appears: on the right of this curve, the lower band ${ϵ}_{-}$ displays a minimum for a wave vector $k_{min}$. The two insets display the typical spectrum in the commensurate and incommensurate regions, respectively.