Simulating triangle Hofstadter-Hubbard model with fermionic projected entangled simplex states

Abstract

The triangular Hofstadter-Hubbard model, realizable in moiré bilayers, provides a fertile ground for discovering correlated topological states. We investigate this model in the grand canonical ensemble by introducing a fermionic infinite projected entangled simplex state (iPESS) approach, which offers direct access to the stability of the emergent correlated states at the thermodynamic limit. Through numerically optimizing fermionic iPESS, we accurately capture the chiral spin liquid (CSL) phase in the Mott insulating regime, characterized by a uniform chiral order, entanglement spectrum and the appearance of gossamer correlation tails in spin channel. The intermediate-$U$ CSL is separated from the weak-$U$ Chern insulator by a Mott transition at $U_{c_1} \approx 11.5$, signaled by changes in the charge fluctuation and compressibility. Finite-correlation-length scaling of the magnetization reveals a transition into a large-$U$ $120^\circ$ Néel phase at $U_{c_2} \approx 22.5$. Remarkably, with finite hole doping $δ$, we identify a uniform superconducting state with a finite pairing amplitude, whose order parameter displays a nearly universal phase winding across the $U$-$δ$ phase diagram. Our work demonstrates robust chiral superconductivity in the thermodynamic limit through doping Chern insulator and CSL.

Figures (5)

• Figure 1: Fermionic iPESS ansatz and phase diagram. (a) Fermionic simplex tensors for a single site. (b) Fermionic iPESS on the infinite lattice. (c) Symmetry quality of optimized fermionic iPESS for $D=8$: bond thickness and color indicate hopping amplitudes and phases; black circle arrows indicate scalar chirality magnitudes; dashed lines mark the $2\times2$ cell of iPESS. (d) Phase diagrams for the insulating $\delta=0$ and hole-doped $\delta>0$ regimes.
• Figure 2: Weak-$U$ CI and intermediate-$U$ CSL phases. (a)-(b) Energy convergence for finite-$D$ iPESS and iDMRG on cylinders of width $L_y$. (c)-(d) iPESS entanglement spectrum for $(D,\chi_{\rm{iPESS}})=(12,60)$. (e)-(f) iPESS correlation functions for $(D,\chi_{\rm{iPESS}})=(12,160)$.
• Figure 3: Signatures of Mott transition between CI and CSL. (a) Deviation from half-filling $n=1$ for negative chemical potential $\mu$. (b)-(c) Double occupancy $\langle n_{i\uparrow}n_{i\downarrow} \rangle$ and scalar chirality $\langle \chi_{ijk} \rangle$ and their derivatives versus $U$ at half filling.
• Figure 4: Magnetic transition at half-filling via finite-correlation-length scaling.$Z_2$ symmetric $6\times 6$ iPESS ansatz is used. The square, down-triangle, up-triangle, right-triangle, diamond, star symbols correspond to bond dimensions $D=6,7,8,9,10,12$. (a)-(b) Energy and magnetization comparison between simple-update and variational optimization. (c) Extrapolated magnetization vs. $U$ from variational optimization.
• Figure 5: SC pairing symmetry and strength versus hole doping $\delta$. (a)-(c) Phases (color) and magnitudes (bond thickness) of pairing $\Delta$ under $C_6$ imaginary gauge with $U=12, D=12$. The bond thickness is proportional to square root of pairing amplitude $\sqrt{|\Delta|}$. Black circular arrows indicate the phase winding. (d) Spin chirality and $\Delta$ strength vs. $\delta$, with $1/D$ extrapolation for $U=9$.