Simulating fermionic fractional Chern insulators with infinite projected entangled-pair states

Abstract

Infinite projected entangled-pair states (iPEPS) provide a powerful variational framework for two-dimensional quantum matter and have been widely used to capture bosonic topological order, including chiral spin liquids. Here we extend this approach to \emph{fermionic} topological order by variationally optimizing $U(1)$-symmetric fermionic iPEPS for a fractional Chern insulator (FCI), with bond dimensions up to $D=9$. We find evidence for a critical bond dimension, above which the ansatz faithfully represents the FCI phase. The FCI state is characterized using bulk observables, including the equal-time single-particle Green's function and the pair-correlation function, as well as the momentum-resolved edge entanglement spectrum. To enable entanglement-spectrum calculations for large iPEPS unit cells, we introduce a compression scheme and show that the low-lying part of the spectrum is already well converged at relatively small cutoff dimensions.

Figures (9)

• Figure 1: (a) Distinct hopping processes of the Haldane model given by Eq. \ref{['eq:Hamiltonian']} are shown in different colors, with the arrows for 2nd-NN hopping indicating the phase convention. It is reduced to the corresponding Bravais lattice by grouping the two sublattice sites A and B within each honeycomb unit cell into a single effective site. (b) iPEPS is constructed on the Bravais lattice from rank-5 tensors $A^{[\boldsymbol{r}]}$ defined in Eq. \ref{['eq:onsiteA']}, where the arrows indicate the directions of the $U(1)$ charge flow. We parametrize the ground state with three inequivalent tensors $(A_1,A_2,A_3)$ arranged in $3\times3$ unit cell which periodically tiles the lattice. (c-d) Pair-correlation function from iPEPS $g(\boldsymbol{x})$ for the Chern insulator (CI) and the fractional Chern insulator (FCI), evaluated along the primitive lattice vectors $\boldsymbol{a_1}$ (cross) and $\boldsymbol{a_2}$ (circle). Here $|\boldsymbol{x}|$ is the Cartesian distance and $a$ is the honeycomb lattice constant. Dashed lines are pair-correlations in the continuum for the non-interacting $\nu=1$ IQH state and for the $\nu=1/3$ Laughlin state Fulsebakke2023, respectively, with $\ell_{B}/a=1.63$.
• Figure 2: (a) Ground-state energy density $e_0(D,\chi)$ of the optimized iPEPS for bond dimensions from $D=5$ to $D=9$ and its extrapolations to $\chi\to\infty$. The extrapolated values are plotted in the inset as a function of $D$. (b) Variance of the real-space charge distribution of the optimized states. For comparison, we also show the corresponding charge profiles $\{\langle \hat{n}_{\alpha, \boldsymbol{r}}\rangle\}$ for $D=5$ and $D=9$, plotted with a common color scale.
• Figure 3: Equal-time single-particle Green’s function $|\langle \hat{c}_{\mathrm{A}, 0}\hat{c}^{\dagger}_{\mathrm{A}, \boldsymbol{r}}\rangle|$ between A-A sublattice sites. Distance $|\boldsymbol{r}|$ is calculated in Cartesian coordinates and $a$ is the honeycomb lattice constant. (a) Correlations along the two primitive lattice vector directions $\boldsymbol{a_1}$ and $\boldsymbol{a_2}$, evaluated for the $D=9$ state with $\chi=468$. (b) The Green's function along $\boldsymbol{a_2}$ of the $D=9$ state with increasing $\chi$.
• Figure 4: (a) Left/right dominant eigenvectors $\sigma_{\mathrm L}$ and $\sigma_{\mathrm R}$ of the column-to-column double-layer transfer matrix. For clarity, we omit the arrow directions on tensor legs. The eigenvectors are approximated by MPOs assembled from the CTMRG environment edge tensors $T_{\mathrm L/\mathrm R}^{[i]}$; see Fig. \ref{['fig:CTMRG']} for the definitions of the double-layer tensor and the environment tensors. (b) MPO representation of $\sigma_{\mathrm L}^{T}\sigma_{\mathrm R}$ on a cylinder of width $W=2$. The MPO for $\sigma_{\mathrm L}^{T}\sigma_{\mathrm R}$ has a three-tensor unit cell; small squares indicate fermionic swap gates. (c) MPO unit-cell compression for a 3-site unit cell: at each step, isometries $P_{\mathrm L}^{(i)}$ and $P_{\mathrm R}^{(i)}$ block two neighboring sites within the unit cell, reducing the unit-cell size by one while truncating the MPO "physical" leg to dimension $\le D_{\mathrm c}$.
• Figure 5: Momentum-resolved ES of the $D=7$ state on a cylinder of circumference $W=5$ (15 sites) under APBC. The spectrum is calculated for the $U(1)$ charge sector $n=0$. (a) The cutoff bond dimension $D_{\mathrm{c}}=35$ is fixed, and the environment bond dimension $\chi$ is varied. (b) The environment bond dimension $\chi=35$ is fixed, and the cutoff bond dimension $D_{\mathrm{c}}$ is varied. In both cases, the low-level counting across momentum sectors follows $(1, 1, 2, 3, 5)$, consistent with the chiral $U(1)$ boson edge of the $\nu=1/3$ Laughlin state.