Bypassing the Chiral Obstruction in Two-Dimensional Tensor Networks

Infinite projected entangled pair states (PEPS) are typically unable to faithfully describe chiral gapped phases due to spurious long-range power-law correlations, but this work introduces an auxiliary-assisted framework that bypasses this obstruction by embedding the physical system with a time-reversed partner, yielding a non-chiral enlarged representation whose physical chiral sector is recovered with controlled decoupling.

The Gist

An auxiliary-assisted PEPS framework is proposed to access gapped chiral phases by embedding the physical system into an enlarged non-chiral Hilbert space using a decoupled auxiliary anti-chiral model, which cures the artificial long-range correlations characteristic of direct chiral PEPS representations while retaining access to universal chiral information.

Framework: Auxiliary-Assisted Wavefunction

The construction embeds the physical chiral system into an enlarged non-chiral Hilbert space by introducing a decoupled auxiliary time-reversal partner. Specifically, the total Hamiltonian is defined as:

Htotal = Hphy ⊗ Iaux + Iphy ⊗ Haux, where Hphy is the original physical chiral system and Haux = ΘHphyΘ−1 is its anti-chiral time-reversal partner. This enlarges the local Hilbert space from (Hphy, dphy) to (Htot = Hphy ⊗ Haux, dtotal = dauxdphy). Although the exact ground state factorizes as ψtot⟩ = ψphy⟩ ⊗ ψaux⟩, the enlarged system possesses no net chirality and can be efficiently represented by a finite-D PEPS.

Variational Optimization and Residual Couplings

The optimization is performed on an infinite PEPS for Htotal with local tensor A saux sphy, where LDRU denote common virtual indices of a finite bond dimension D. Consequently, the optimized state generally satisfies ψtot⟩ = ψphy⟩ ⊗ ψaux⟩ and contains weak residual correlations between the two layers. These residual physical–auxiliary correlations are described by Heff = Htotal + X i λ i O i phy ⊗ O i aux, where λ i are effective local couplings. The paper asserts that these weak couplings only affect short-distance correlations and decrease systematically when enlarging entanglement space, thus not affecting long-range low-energy quantities and universal topological features of the target gapped chiral system.

Gapped Correlations and Correlation Length

The auxiliary construction effectively removes the artificial long-range correlations seen in direct finite-D PEPS representations. For both a free-fermion Chern insulator and an interacting chiral spin liquid, the auxiliary-assisted PEPS exhibits clean short-range correlations and a finite transfer-matrix correlation length. For the interacting chiral spin liquid, this approach shows that the direct chiral PEPS develops a characteristic gossamer tail, whereas the auxiliary construction shows a purely short-ranged decay with no visible powerlaw-like tail. Furthermore, the correlation length ξ of the auxiliary-extended state extrapolates to a genuinely short correlation length ξ ∼ 0.826, confirming an exponential real-space decay.

Resolving Chiral Edge Physics

The universal chiral edge structure is encoded in the lowlying entanglement spectrum. By placing the state on an infinite cylinder and analyzing its bipartite entanglement Hamiltonian HE, the paper shows that while the enlarged boundary is described by a non-chiral SU(2)1 × SU(2)1 theory, the physical and auxiliary degrees of freedom remain explicitly distinguished. The key finding is that the momentum operator and the entanglement Hamiltonian are actually the same one up to non-universal normalization factors, allowing for independent control over momenta in each layer. This enables the resolution of universal chiral entanglement data by using subsystem-momentum resolution, where the energy spectrum takes the structure q ∼ hi + Ni − k a0 ∆ ∼ hi + Ni + k a0, confirming that the auxiliary construction removes the finite-D chiral obstruction without erasing the chiral topological information.

Residual Physical-Auxiliary Coupling

The residual physical-auxiliary coupling is shown to be strongly suppressed compared to physical–physical spin correlations and decays exponentially with distance for all bond dimensions studied. A systematic measure of decoupling, the factorization error ∆R = ρ aux,phy R - ρ aux R ⊗ ρ phy R F, demonstrates that this coupling decreases rapidly with increasing bond dimension, confirming that the auxiliary system introduces only weak and short-ranged correlations. These results establish a mechanism where the auxiliary layer provides the necessary entanglement structure to remove the chiral PEPS obstruction while introducing only effects irrelevant to the universal low-energy organization.

Conclusions

The framework successfully demonstrates that adding an anti-chiral auxiliary layer allows for a variational optimization of PEPS that describes genuine gapped phases by curing spurious gapless long-range correlations. The physical and auxiliary sectors remain asymptotically decoupled, ensuring that the universal chiral information of the physical system remains resolvable

Here are the specific improvements and capabilities for AI systems derived from this scientific paper:

Improvements for AI Systems

1. Robust Simulation of Chiral Topological Phases (PEPS): The core improvement is the ability to simulate two-dimensional, gapped chiral phases (like Chern insulators and chiral spin liquids) using finite-bond-dimension Projected Entangled Pair States (PEPS) without incurring spurious long-range power-law correlations.

2. Accurate Representation of Topological Information: The auxiliary-assisted framework ensures that the essential topological information encoded in the entanglement structure is preserved, even when using a nonchiral enlarged representation.

3. Precise Characterization of Entanglement Spectra: The system allows for the extraction and analysis of the low-lying entanglement spectrum, which serves as a fingerprint for bulk topological order (specifically SU(2)1 chiral topological order).

4. Quantification of Correlation Lengths: The method enables the calculation of a genuine, short correlation length from the transfer matrix spectrum, distinguishing it from artificial divergences found in direct PEPS representations.

What the Improved AI System Can Do

Based on these improvements, an AI system utilizing this framework can perform the following specific tasks:

1. Predictive Modeling for Novel Chiral Materials:

Perturb or design Hamiltonians corresponding to known and hypothetical chiral topological insulators or chiral spin liquids. The AI can then use the auxiliary-assisted PEPS method to determine if a finite bond-dimension representation exists that accurately captures the true gapped bulk physics, bypassing the limitations of standard PEPS simulations.

2. Topological Phase Classification via Entanglement Signatures:

The AI can analyze simulated or experimentally derived entanglement spectra (obtained via layer-resolved momentum projection) to classify the underlying topological order. Specifically, it can distinguish between a true chiral topological phase and one exhibiting spurious gapless behavior (i.e., identifying the presence or absence of the gossamer tail).

3. Verification of Topological Order in Lattice Models:

The system can rigorously test whether lattice models (like the free-fermion Chern insulator or interacting spin liquids) possess genuine topological order when represented by finite-D PEPS. It can confirm that for a given bond dimension, the simulation yields clean exponential decay (short-range correlations) rather than power-law decay.

4. Extraction of Universal Boundary Theory Parameters:

The AI can use the momentum resolution techniques to calculate the low-energy entanglement boundary theory parameters (like chiral conformal weights and level degeneracies). This allows for a direct comparison between the simulated lattice state and established Conformal Field Theory (CFT) predictions for chiral topological orders, providing high-fidelity verification of bulk-boundary correspondence.

5. System Design Optimization:

The framework can be used in an inverse design capacity: given a desired low-energy entanglement spectrum or a specific boundary CFT structure, the AI can suggest modifications to the physical Hamiltonian that would yield the required finite-D PEPS representation.

Gapped chiral phases pose an intrinsic obstruction to projected entangled pair state (PEPS) representations with finite bond dimension D, which generically develop spurious long-range correlations despite the gapped nature of the target state. To address this long-standing problem, we take a qualitatively different route by considering the target chiral system together with a completely decoupled time-reversed auxiliary copy. Constraining the finite- D PEPS to the factorized form would simply reduce the problem to representing each chiral layer independently, leaving the original finite- D obstruction unchanged. Unexpectedly, however, we find that finite- D variational optimization generates an emergent residual interlayer entanglement, despite the absence of any explicit interlayer coupling in the Hamiltonian. This emergent entanglement is the key mechanism that bypasses the obstruction, qualitatively changing the PEPS from an effectively gapless representation with long-range correlation tails to one with a finite correlation length. We demonstrate this mechanism for both a free-fermion Chern insulator and an interacting chiral spin liquid. The residual entanglement is systematically suppressed with increasing D, approaching the decoupled limit, while the chiral topological information remains accessible through layer-resolved entanglement spectra. Our results thus uncover a previously unrecognized mechanism for bypassing the chiral obstruction through variationally generated auxiliary entanglement.