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Physical Review Letters

Simulating Fermionic Fractional Chern Insulators with Infinite Projected Entangled-Pair States

Hao Chen, Titus Neupert, Juraj Hašík

Published May 28, 2026• 0 citations
Publisher

Abstract

Infinite projected entangled-pair states (iPEPSs) 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 fermionic topological order by variationally optimizing U(1)-symmetric fermionic iPEPSs 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.

DOI
10.1103/c9dr-bpn4

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Infinite projected entangled-pair states (iPEPSs) 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 fermionic topological order by variationally optimizing U(1)-symmetric fermionic iPEPSs 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.
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)U(1)U(1)-symmetric fermionic iPEPS for a fractional Chern insulator (FCI), with bond dimensions up to D=9D=9D=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.
Chiral spin liquids (CSL) based on spin-1/2 fermionic projected entangled pair states (fPEPS) are considered on the square lattice. First, fPEPS approximants of Gutzwiller-projected Chern insulators (GPCI) are investigated by variational Monte Carlo (VMC) techniques on finite-size tori. We show that such fPEPS of finite bond dimension can correctly capture the topological properties of the chiral spin liquid, as the exact GPCI, with the correct topological ground-state degeneracy on the torus. Further, more general fPEPS are considered and optimized (on the infinite plane) to describe the CSL phase of a chiral frustrated Heisenberg antiferromagnet. The chiral modes are computed on the edge of a semi-infinite cylinder (of finite circumference) and shown to follow the predictions from conformal field theory. In contrast to their bosonic analogs the (optimized) fPEPS do not suffer from the replication of the chiral edge mode in the odd topological sector.
We study the parton construction of chiral spin liquids (CSLs) using projected Gaussian fermionic entangled pair states (GfPEPSs). First, we show that GfPEPSs can represent generic spinless Chern insulators faithfully with finite bond dimensions. Then by applying the Gutzwiller projection to a bilayer GfPEPSs, spin-12\frac{1}{2}21​ Abelian and non-Abelian CSLs are obtained for Chern number C=1C=1C=1 and 2, respectively. As a consequence of the topological obstruction for GfPEPSs, very weak Gossamer tails are observed in the correlation functions of the fermionic projected entangled pair state (PEPS) Ans\"atze, suggesting that the no-go theorem for chiral PEPS is universal but does not bring any practical limitation. Remarkably, without finetuning, all topological sectors can be constructed showing the expected number of chiral branches in the respective entanglement spectra, providing a sharp improvement with respect to the known bosonic PEPS approach.
Doubts have been raised on the representation of chiral spin liquids exhibiting topological order in terms of projected entangled pair states (PEPSs). Here, starting from a simple spin-1/2 chiral frustrated Heisenberg model, we show that a faithful representation of the chiral spin liquid phase is in fact possible in terms of a generic PEPS upon variational optimization. We find a perfectly chiral gapless edge mode and a rapid decay of correlation functions at short distances consistent with a bulk gap, concomitant with a gossamer long-range tail originating from a PEPS bulk-edge correspondence. For increasing bond dimension, (i) the rapid decrease of spurious features-SU(2) symmetry breaking and long-range tails in correlations-together with (ii) a faster convergence of the ground state energy as compared to state-of-the-art cylinder matrix-product state simulations involving far more variational parameters, prove the fundamental relevance of the PEPS ansatz for simulating systems with chiral topological order.
Abelian and non-Abelian topological phases exhibiting protected chiral edge modes are ubiquitous in the realm of the fractional quantum Hall (FQH) effect. Here, we investigate a spin-1 Hamiltonian on the square lattice which could, potentially, host the spin liquid analog of the (bosonic) non-Abelian Moore-Read FQH state, as suggested by exact diagonalization of small clusters. Using families of fully SU(2)-spin symmetric and translationally invariant chiral projected entangled pair states (PEPS), variational energy optimization is performed using infinite-PEPS methods, providing good agreement with density matrix renormalization group (DMRG) results. A careful analysis of the bulk spin-spin and dimer-dimer correlation functions in the optimized spin liquid suggests that they exhibit long-range ``gossamer tails''. From the investigation of the entanglement spectrum, we observe sharply defined chiral edge modes following the prediction of the SU(2)2{}_{2}2​ Wess-Zumino-Witten theory and exhibiting a conformal field theory (CFT) central charge c=3/2c=3/2c=3/2, as expected for a Moore-Read chiral spin liquid. Using the PEPS bulk-edge correspondence, we argue the ``weak'' criticality of the bulk is in fact a finite-DDD artifact of the chiral PEPS, which quickly becomes (practically) irrelevant as the PEPS bond dimension DDD is increased. We conclude that the PEPS formalism offers an unbiased and efficient method to investigate non-Abelian chiral spin liquids in quantum antiferromagnets.
An important class of model Hamiltonians for investigation of topological phases of matter consists of mobile, interacting particles on a lattice subject to a semiclassical gauge field, as exemplified by the bosonic Harper-Hofstadter model. A unique method for investigations of two-dimensional quantum systems are the infinite projected-entangled pair states, as they avoid spurious finite-size effects that can alter the phase structure. However, due to no-go theorems in related cases, this was often conjectured to be impossible in the past. In this Letter, we show that upon variational optimization, the infinite projected-entangled pair states can be used to this end by identifying fractional Hall states in the bosonic Harper-Hofstadter model. The obtained states are characterized by showing exponential decay of bulk correlations, as dictated by a bulk gap, as well as chiral edge modes via the entanglement spectrum.