Simulating Bulk Gap in Chiral Projected Entangled-Pair States

arXiv:2502.20142 · cond-mat.str-el, quant-ph · Submitted 2025-02-27 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Simulating Bulk Gap in Chiral Projected Entangled-Pair States".

Mira: Here's the gist of the paper. The gist: Projected entangled-pair states (PEPS) can reliably characterize gapped bulk excitations in chiral topological phases by demonstrating that correlation functions decaying faster than r−2 do not necessarily imply gapless modes, which addresses long-standing concerns about their applicability to chiral spin liquids.

Kai: Challenges with Chiral PEPS Projected entangled-pair states (PEPS) have shown success in capturing nonchiral topological phases but face challenges when extended to chiral phases because of the presence of long-range “gossamer” correlation tails and this skepticism stems from the well-established correspondence between the correlation length and the energy gap.

Mira: In gapped phases, correlation functions exhibit exponential decay with distance, reflecting a finite energy cost of excitations, whereas gapless systems are characterized by algebraic decay indicative of long-range correlations associated with low-energy, long-wavelength excitations whose energies approach that of the ground state.

Kai: The relationship between the correlation length ξ and the energy gap ∆ follows the inverse scaling ∆ ∼ 1/ξ, such that a diverging ξ signals a vanishing ∆.

Title and authors: Kai: So we're talking about this paper titled "Simulating Bulk Gap in Chiral Projected Entangled-Pair States." It looks like they're tackling a real headache in the field: how to reliably find the bulk gap excitations when you're working with these chiral systems using PEPS.

Mira: Exactly. The central problem they address is that PEPS have been good for ground states in chiral spin liquids, but there's this issue with long-range "gossamer" correlation tails that makes it hard to tell if there are actually gapless modes hiding there.

Lev: From an error correction standpoint, I'm always worried about those tails because they suggest low-energy excitations that could be problematic for running real hardware, and this paper seems to try and tame that issue.

Kai: What the paper does is set up a variational principle for excited states using a local mode approximation within the PEPS framework. This allows them to calculate excitation gaps directly, which validates PEPS as a useful tool for studying both ground and excited states in these chiral topological systems.

Mira: It’s interesting because they use this approach to show that correlation functions decaying faster than r minus two are not necessarily linked to gapless modes, which is key to ruling out those spurious low-energy excitations.

Lev: So if the tail decays that fast, it means we don't have an algebraic decay, and that points toward a finite energy cost for those excitations. That makes sense for gapped phases.

Kai: Right, and they do this by focusing on the stability of the denominator in their excited state energy calculation. They show that this denominator is guaranteed when correlations decay faster than r minus two, which helps ensure the simulation is stable.

Mira: The results they get are quite telling, particularly when they test it on a specific model, like the Z2 Kitaev model with a chiral term where the PEPS yields excitation gaps that match exact solutions very well.

Lev: When you look at their validation, starting from a Z2 one-form symmetric initial PEPS with bond dimension D equals four and optimizing the ground state energy at kappa over J equals zero point two using CTMRG with an environment truncation dimension chi equals eighty, they get a variational ground state energy of E0 equal to minus one point six nine seven nine J, which is remarkably close to the exact result of minus one point six nine eight zero J.

Title and authors: Kai: That closeness on the ground state energy is encouraging because it shows the variational method works well in practice for these specific setups. How does this relate to what they do for excitations?

Mira: They move on to constructing excited states from a single-site excitation ansatz, which they call Bi> equal W1 W2 W3 W4 B, and these fall into five distinct sectors: zero-vison, single-vison, two-vison, three-vison, and fourvison.

Lev: The constraint they put on the nonzero-vison sectors is that they enforce orthogonality for states between different sites because the expectation value of H times those states is zero for i not equal to j. This orthogonality means not all correlation functions in these chiral PEPS will show long-range tails at all.

Kai: And based on that, the energy of these excited states is determined by the local value E equal to the expectation value of H times B zero over B zero squared. For their Z2 model, this leads them to determine the gap of the Z2 chiral spin liquid as delta equals zero point five two J in the singlevison sector.

Mira: Then they extend this framework to a more complex case, like the Z3 Kitaev model with one-form symmetry where there isn't an exact solution for it <ref:2502.20142#pg1>. Optimizing the ground state PEPS with bond dimension D equal to six and environment bond dimension chi equal to one hundred eight gives a ground-state energy per unit cell of E0 equal to minus two point nine six zero zero J, which is close to the exact diagonalization results of minus three point zero one three J for an eighteen site torus.

Lev: That finite-chi effect they mention is considerable, but it prevents them from reaching large correlation lengths and accurately determining the functional form of that long-distance tail because the slowest exponential decay carries only a very small weight, less than w equal to one point three thousand.

Kai: So even though those finite-chi effects are bigger, they can safely ignore the contribution of that slow decay behavior because its weight is so small. That's a big win for their simulation approach.

Title and authors: Mira: In conclusion, this paper establishes a robust framework for simulating gapped bulk excitations in chiral spin liquids using PEPS, effectively addressing the concerns about their applicability to chiral topological phases. They conclude that when dealing with phase transitions in these systems, the finite correlation length scaling in PEPS might need refinement to properly distinguish between those "gossamer" tails and physical correlations.

Lev: For me, what this means is that this method gives us a concrete way to get excitation gaps from PEPS without getting lost in unphysical long-range tails, which is a necessary first step before we even think about error correction overheads for real hardware.

Kai: Exactly. This study on simulating bulk gap in chiral projected entangled-pair states shows that PEPS can reliably capture gapped excitations, and it gives us a solid way to compute those gaps in complex models like the Z2 and Z3 Kitaev models.

Mira: It’s about establishing a variational principle for excited states within a local mode approximation, showing that correlation functions decaying faster than r minus two are not necessarily related to gapless modes. This is really important for understanding the physics of these systems beyond just the ground state.

Lev: And in terms of what this means for running things, it means we have a method to find the excitation spectrum that respects the gap structure, which is what we need when translating these ideas into something you can actually build on a quantum computer.

Kai: So this paper on simulating bulk gap in chiral projected entangled-pair states gives us a solid tool to characterize gapped excitations in chiral spin liquids, and it does that by showing how to manage those correlation function tails through a variational approach.

Mira: It really shows the potential for PEPS not just as ground state simulators, but as tools for understanding the full excitation spectrum in these challenging chiral topological phases.

Lev: And I think this work on simulating bulk gap in chiral projected entangled-pair states gives us a concrete way to get excitation gaps from PEPS without getting lost in unphysical long-range tails, which is a necessary first step before we even think about error correction overheads for real hardware.

The paper's summary: Kai: So, this paper is basically saying that for these chiral systems using PEPS, you can actually find out what the bulk excitation gap is reliably by setting up a variational principle for excited states with some local mode approximation.

Mira: Right, and it’s a big deal because it tackles that old worry about long-range "gossamer" correlation tails in chiral phases and shows they don't automatically mean there are gapless modes hiding there.

Lev: For me, the main thing is how they handle those correlation functions decaying faster than r minus two, because that’s where you usually see the algebraic decay for gapless systems.

Kai: Exactly, and they argue that if you enforce a condition on those tails—specifically focusing on making sure the denominator in the energy formula stays stable—you can rule out those spurious low-energy states.

Mira: So, it gives us a way to be confident that when we calculate an excitation gap with PEPS for these chiral systems, we’re seeing a real gap and not just some long-range artifact.

Lev: If you were running this on actual hardware, the math they use for the excitation spectrum—the five sectors they defined like zero-vison up to fourvison—that would tell you exactly what kind of energy cost you’re looking at for a given mode.

Kai: And when they tested it on the Z2 Kitaev model, we got an excitation gap of zero point five two J, which actually matched the exact solution pretty well.

Mira: That matching is key because it proves that this variational approach works for calculating those excited state energies in chiral spin liquids where we usually don't have exact solutions.

Lev: But I gotta point out that they also had to deal with finite-chi effects when they looked at the Z3 Kitaev model, which means their simulation isn't perfect yet because of how much environment truncation matters.

Kai: Yeah, and even though those finite-chi errors were bigger in that case, the authors say you can basically ignore the contribution of that slow decay behavior because its weight is tiny.

Mira: So what this really changes for us is that PEPS isn't just a ground state tool anymore; it’s becoming a practical way to study the full excitation spectrum in these notoriously tricky chiral phases.

Lev: It gives us a concrete path on how to get those excitation gaps from PEPS without getting trapped by unphysical long-range tails, which is pretty essential before we even think about error correction overheads for real hardware.

Kai: It’s a solid framework for simulating gapped bulk excitations in chiral spin liquids, and they suggest that maybe when we look at phase transitions, we need to refine how we handle that finite correlation length scaling in PEPS.

Mira: So the next thing I want to look at is how this specific variational principle translates into something you can actually implement on a larger quantum computer architecture.

The paper's improvements: Tom: So, we're looking at what the authors suggest to do next with this PEPS framework for chiral systems that they just tested on.

Kai: They aren't just stopping there; they suggest that when we deal with phase transitions in these chiral phases, we might need to refine how we handle the finite correlation length scaling in PEPS.

Mira: Right, because right now the authors are saying that while their method works for finding gaps, you still have to be careful about those "gossamer" tails when you look at a phase transition point.

Lev: It means they're hinting that the current way they measure those correlation lengths in PEPS isn't quite sharp enough to tell the difference between a physical gap and just some slow decay.

Kai: So what this implies for us is that we need a more precise way to measure those length scales if we want to map out exactly where the system transitions from gapped to gapless behavior.

Mira: It points toward needing a better way to quantify that scaling so we can distinguish between true physical correlations and just mathematical artifacts of the PEPS approximation.

Lev: From an error correction standpoint, that refinement is crucial because if we can’t reliably characterize the correlation length near a transition, we can’t accurately predict how much noise or overhead we'll need to manage when trying to simulate it on a real machine.

Kai: So essentially, they are saying the next step is improving the tools themselves—the way PEPS measures distance—to get a clearer picture of those long-range behaviors in chiral environments.

Mira: It means the theoretical challenge isn't just finding the gap anymore; it's making sure our simulation method can actually resolve subtle changes in correlation structure near critical points.

Lev: That refinement would translate into needing better tools for characterizing entanglement, which is always a pain when you’re building fault-tolerant hardware.

Conclusion: Kai: So, to wrap up on "Simulating Bulk Gap in Chiral Projected Entangled-Pair States," we established that PEPS can reliably find excitation gaps in chiral spin liquids using a variational principle for excited states with some local mode approximation.

Mira: Right, and they showed that correlation functions decaying faster than r minus two don't necessarily mean there are gapless modes hiding there, which is really important for ruling out those spurious low-energy states.

Lev: For me, the biggest win here is getting a method to compute those excitation gaps directly from the PEPS structure without getting lost in unphysical long-range tails that would make running this on real hardware impossible.

Kai: It means we have a solid tool now for studying both ground and excited states in these chiral topological systems, validated by matching results on the Z2 Kitaev model.

Mira: And the extension to the Z3 Kitaev model, even with those finite-chi caveats, still gave us a determined gap value of zero point seven one J in that specific sector.

Lev: Those numbers are good for theory but they don't tell me much about how much noise or how high the required fidelity needs to be for any real-world implementation of this simulation method.

Kai: Exactly, and it tells us the framework is robust enough to handle these complex chiral models, even when they aren't exactly solvable.

Mira: So the implication is that PEPS isn't just a ground state tool anymore; it’s becoming a practical way to probe the full excitation spectrum in these tricky chiral phases.

Kai: That's the core idea of this work on "Simulating Bulk Gap in Chiral Projected Entangled-Pair States."

Mira: And it suggests that when we look at phase transitions in these chiral systems, maybe we need to refine how we handle that finite correlation length scaling in PEPS.

Lev: I just think getting a reliable gap determination mechanism for chiral models is a necessary first step before any real quantum computation can actually use this approach effectively.

Ji-Yao Chen, *, Yi Tan, *, Sylvain Capponi, Didier Poilblanc, Fei Ye, Jia-Wei Mei

Center for Neutron Science and Technology, Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices, School of Physics, Sun Yat-sen University, Guangzhou 510275, China · Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China · Laboratoire de Physique Théorique, Université de Toulouse, CNRS

cond-mat.str-el, quant-ph

Submitted: 2025-02-27

Updated: 2025-02-27

Comments: 4 figures in 6 pages

Journal ref: Phys. Rev. B 113, 085130 (2026)

DOI: 10.1103/g8cb-fz29

Code: https://github.com/Jutho/TensorKit.jl

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 91/100

The gist: The gist: Projected entangled-pair states (PEPS) can reliably characterize gapped bulk excitations in chiral topological phases by demonstrating that correlation functions decaying faster than r−2

Key concepts

Projected Entangled-Pair States (PEPS): PEPS are a specific type of tensor network used to represent quantum states. They are powerful because they can capture the complex entanglement found in topological phases, allowing researchers to simulate both the ground state and excited states of these systems.
Correlation Length and Energy Gap: In gapped systems, correlation functions decay exponentially with distance, which is directly related to a finite energy gap. Conversely, gapless systems show algebraic decay. The paper uses this relationship to ensure that fast correlation decay in chiral PEPS does not falsely suggest the presence of low-energy excitations.
Variational Principle for Excited States: This principle uses an excitation ansatz based on the tangent space of PEPS to find the energy of excited states. This approach helps manage long-range correlation tails by focusing on the stability of a specific structure factor, ensuring accurate gap determination.

Summary

The gist: Projected entangled-pair states (PEPS) can reliably characterize gapped bulk excitations in chiral topological phases by demonstrating that correlation functions decaying faster than r−2 do not necessarily imply gapless modes, which addresses long-standing concerns about their applicability to chiral spin liquids.

Challenges with Chiral PEPS

Projected entangled-pair states (PEPS) have shown success in capturing nonchiral topological phases but face challenges when extended to chiral phases because of the presence of long-range “gossamer” correlation tails and this skepticism stems from the well-established correspondence between the correlation length and the energy gap. In gapped phases, correlation functions exhibit exponential decay with distance, reflecting a finite energy cost of excitations, whereas gapless systems are characterized by algebraic decay indicative of long-range correlations associated with low-energy, long-wavelength excitations whose energies approach that of the ground state. The relationship between the correlation length ξ and the energy gap ∆ follows the inverse scaling ∆ ∼ 1/ξ, such that a diverging ξ signals a vanishing ∆.

Variational Principle for Excited States

The work establishes a variational principle for excited states within a local mode approximation using recently developed tangent space based excitation ansatz. The translationally invariant PEPS for the honeycomb lattice is expressed by the formula Ψ(A)⟩ = X sαi,···,sβN tTr(A sαi abcA sβi a′b′c)sα1 · · · sαi sβi · · · sβN ⟩. The tangent space of PEPS provides the foundation for variational ground state and excited states. The energy of an excited state is given by Eex = ⟨Φ(B; A)(H − E0)Φ(B; A)⟩ / ⟨Φ(B; A)Φ(B; A)⟩. The primary challenge in managing the “long-range” correlation tails shifts to ensuring the stability of the denominator, ⟨Φ(B; A)Φ(B; A)⟩ = P ij ⟨BiBj⟩, which corresponds to the structure factor of the zero-momentum correlation function of Bi⟩.

Validation with Z2 Kitaev Model

The framework is validated in the spin-1/2 Kitaev model with a chiral term, where PEPS yields excitation gaps that agree well with exact solutions. Starting from a Z2 1-form symmetric initial PEPS of virtual bond dimension D = 4 and optimizing the ground state energy at κ/J = 0.2 using CTMRG with environment truncation dimension χ = 80, a variational ground state energy of E0 = −1.6979J was achieved, remarkably close to the exact result E0 = −1.6980J. The connected dimer-dimer correlation function for the Z2 model exhibits an exponential decay at short distances with a correlation length ξ = 0.26, and at larger distances, it transitions into a power-law form, r−α, with α ≃ 2 as χ increases. The paper concludes that r−2 represents the slowest decay behavior observed for the long-range correlation tails in chiral PEPS and its overall contribution is negligible because the tail has an extremely small weight, less than w = 10−5.

Excitation Spectrum Analysis

The excited states are constructed from the single-site excitation ansatz Bi⟩ = W1 W2 W3 W4 B. (4). These excited states fall into five distinct sectors: zero-vison, single-vison, two-vison, three-vison, and fourvison. The nonzero-vison sectors enforce orthogonality for states between different sites ensuring that ⟨BiHBj⟩ = ⟨BiBj⟩ = 0 for i ≠ j. This orthogonality guarantees that not all correlation functions exhibit long-range tails in chiral PEPS, and their energy is determined by the on-site value E = ⟨B0HB0⟩ / ⟨B0B0⟩. For the Z2 model, the gap of the Z2 chiral spin liquid is determined by the singlevison sector with a gap ∆ = 0.52J [49].

Extension to Z3 Kitaev Model

The framework is extended to the Z3 Kitaev model with 1-form symmetry, which lacks an exact solution. Optimizing the ground-state PEPS with bond dimension D = 6 and environment bond dimension χ = 108 yields a ground-state energy per unit cell of E0 = −2.9600J, which is close to the exact diagonalization results E0 = −3.013J for an 18-site torus. The connected dimer-dimer correlation function for the Z3 model exhibits a quick exponential decay at short distance. Although finite-χ effects are considerably larger, preventing the attainment of large correlation lengths and accurate determination of the long-distance tail’s functional form, because the slowest exponential decay carries only a very small weight (w ∼ 10−3), its contribution can be safely neglected. The lowest system gap is determined to be ∆ = 0.71J in the 1ω-1ω2-vison sector.

Conclusion and Future Directions

This work establishes a robust framework for simulating gapped bulk excitations in chiral spin liquids using projected entangled-pair states. The study conclusively demonstrates the capability of PEPS to simulate gapped bulk excitations in chiral spin liquids, effectively addressing longstanding concerns regarding their applicability to chiral topological phases. The authors suggest that when dealing with chiral phases and their phase transitions, the finite correlation length scaling in PEPS may need refinement to distinguish “gossamer” tail from truly physical correlation.

Acknowledgment – We thank Jize Zhao, Yongshi Wu, Hong-Hao Tu, Donghoon Kim, and Guang-Ming Zhang for useful discussions. The work is supported by the National Natural Science Foundation of China (Grant No. 12474143), the National Key Research and Development Program of China (Grant No. 2021YFA1400400), and Shenzhen Fundamental Research Program (Grant No. JCYJ20220818100405013 and JCYJ26). Part of the calculations reported were performed on resources provided by the Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices, No. 2022B1212010008, and at CALMIP (grant 2024-P0677). The work was also supported by the TNTOP ANR-18-CE30-0026-01 grant awarded by the French Research Council. The authors contributed equally to this work.

References

[1] R. B. Laughlin, Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations, Phys. Rev. Lett. 50, 1395 (1983).

[2] V. Kalmeyer and R. B. Laughlin, Equivalence of the resonating-valence-bond and fractional quantum hall states, Phys. Rev. Lett. 59, 2095 (1987).

[3] X. G. Wen, F. Wilczek, and A. Zee, Chiral spin states and superconductivity, Phys. Rev. B 39, 11413 (1989).

[4] X. G. Wen, Topological orders in rigid states, International Journal of Modern Physics B 04, 239 (1990).

[5] F. Verstraete and J. I. Cirac, Renormalization algorithms for quantum-many body systems in two and higher dimensions, arXiv preprint (2004), cond-mat/0407066.

[6] N. Schuch, I. Cirac, and D. P´erez-Garc´ıa, PEPS as ground states: Degeneracy and topology, Annals of Physics 325, 2153 (2010).

[7] J. I. Cirac, D. P´erez-Garc´ıa, N. Schuch, and F. Verstraete, Matrix product states and projected entangled pair states: Concepts, symmetries, theorems.

[8] J. Dubail and N. Read, Tensor network trial states for chiral topological phases in two dimensions and a nogo theorem in any dimension, Phys. Rev. B 92, 205307 (2015).

[9] T. B. Wahl, H.-H. Tu, N.

Improvements for AI systems

textbf Rough Simulation of Gapped Excitations in Chiral Phases with PEPS Architecture: The improved AI system can reliably characterize gapped bulk excitations in chiral topological phases by utilizing a variational principle for excited states within a local mode approximation, as established in the paper. This allows the system to accurately compute excitation gaps, validating PEPS as a powerful tool for studying both ground and excited states in chiral topological systems.
textbf Managing Correlation Function Tails: The improved AI system can distinguish between physical correlation decay and unphysical long-range tails by enforcing that correlation functions decaying faster than r−2 are not necessarily related to gapless modes. This ensures the simulation avoids spurious gapless modes by focusing on the stability of the denominator, which is guaranteed when correlations decay faster than r−2.
textbf Precise Gap Determination for Z2 and Z3 Models: The improved AI system can accurately resolve gapped excitations in complex chiral models like the Z2 Kitaev model and its Z3 counterpart. For instance, it can compute a gap of ∆ = 0.52J for the Z2 chiral spin liquid by utilizing the variational approach, which closely matching the exact result reported in Ref. [36].