Phases and criticality of the triangular lattice SU(N) Hofstadter-Hubbard model

Abstract

We report the study of phases and transitions of SU(N) Hofstadter-Hubbard model subject to commensurate magnetic field on the triangular lattice. At filling one fermion per site, for the number of fermion flavors 2 <= N <= 8, we identify three distinct phases and calculate critical interaction strength from parton large-N mean-field approximation. Integer quantum Hall, chiral spin liquid, and valence bond solid states could be realized upon varying the Hubbard interaction U and the number of flavor N . We construct the critical theory for the putative continuous transition from quantum Hall states to chiral spin liquid and calculate the critical transport behavior using quantum Boltzmann equations for general N . These results could be validated in synthetic systems such as moire superlattices and cold atom platforms.

Figures (4)

• Figure 1: Left: the triangular lattice with gauge field $A_{ij}$ defined on the bond. Flux $\Phi_\Delta$ piercing each triangle is $\pi/N$. The electrons feel an on-site Hubbard interaction $U$, which is fractionalized into the rotor and spinon degrees of freedom. In the large $U$ limit, the rotor is gapped. Middle: phase diagram of the Hofstadter-Hubbard model of coupling strength $U/t$ and the $N$. It is calculated using the slave rotor mean-field in the weak coupling limit and spinon parton in the strong coupling limit. Starting from weak interaction limit, the system hosts IQH states(blue) and transitions into CSL (red) as $U/t$ increases. The grey region indicates VBS phases. The critical points $U_{c_1}$ and $U_{c_2}$ are the critical interaction strength for IQH-CSL and CSL-VBS transition, respectively. Right: the schematic representation of the ansatz $\chi_{ij} = \sum_{\alpha}\langle f_{i\alpha}^\dagger f_{j\alpha} \rangle/N$ of chiral spin liquid(CSL) states, valence bond solid(VBS) as a function of $N$. The thickness of the link schematically represents the magnitude of $\chi_{ij}$.
• Figure 2: The ground state energy per site as a function of interaction strength $\frac{U}{t}$. The critical interaction strength is marked by the dotted vertical line. The red marker represents the phase of CSL and gray markers represent the VBS phase. The energy is defined in the unit of $10N^2N_sJ$.
• Figure 3: Top: schematic plot of the finite temperature phase diagram from the scaling theory ($\delta$ is the tuning parameter). The phase boundary(red line) between the quantum critical region and the topological phases is determined by $T = \delta^{\nu z}$. Bottom: the numerical result of the longitudinal(left) and transverse(right) conductivity when $N=2$ by tuning the parameter $\delta$ at various temperatures $T = 0.1,0.3,0.5,0.7$. The temperature $T$ and $\delta$ are both given in a common and arbitrary unit of energy. In the quantum critical fan, the conductivity curve is universal and can be obtained from quantum Boltzmann equation.
• Figure 4: The order parameter of CSL $\chi_{ij}$(left) and VBS(right) for different $N$ and interaction strength $U$, where we set $t = 1$. The blackness of the bond represent the absolute value of $\chi_{ij}$. The redness of the dot represent the occupation of spinon $\langle f^\dagger_{i\alpha}f_{i\alpha}\rangle/N$ at each site. Thus the occupation is $1$ at each site in all cases. For $N\geq 5$, the ground state is always chiral spin liquid.