Symmetric mass generation of interacting chiral fermions on a one-dimensional lattice without fermion doubling

arXiv:2606.24713 · hep-th, cond-mat.str-el, quant-ph · Submitted 2026-06-23 · 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: Today's paper: "Symmetric mass generation of interacting chiral fermions on a one-dimensional lattice without fermion doubling".

Mira: Symmetric mass generation (SMG) is an interaction-induced opening of a fermion gap without spontaneous symmetry breaking, and this work demonstrates its realization in a strictly one-dimensional lattice setting.

Kai: First, who's behind it and why it matters.

Paper summary: Kai: So Mira and I were looking at this paper titled "Symmetric mass generation of interacting chiral fermions on a one-dimensional lattice without fermion doubling." The main idea seems to be about how you can get a fermion gap opening without needing spontaneous symmetry breaking, which is a really interesting concept.

Mira: Exactly, Kai, the core thesis of this paper is that they show this symmetric mass generation phenomenon in the context of an interacting system on a strictly one-dimensional lattice. It tackles the challenges posed by the anomaly-free three–four–five–zero model of Wang and Wen by addressing two major hurdles: fermion doubling and the perturbative irrelevance of certain interactions <ref:2606.24713#pg0,the anomaly-free 3–4–5–0 model of Wang and Wen>.

Lev: From an error correction standpoint, if you can realize this physics on real hardware, it suggests a controllable mechanism for gap opening without the typical symmetry breaking associated with some other models, which is something we always look at for robust systems <ref:2606.24713#pg0>.

Kai: Right, and what really stands out is how they tackle those hurdles by using a "strictly onedimensional tangent-fermion lattice" to bypass the fermion doubling issue, which they achieve through Stacey’s nonlocal hopping that corresponds to an energy relation of "E(k) = 2t0 tan(k/two)" <ref:2606.24713#pg1>.

Mira: That nonlocal coupling is clever because it lets them frame the problem as a local generalized eigenvalue problem, which keeps the computation efficient while still allowing them to study symmetric mass generation <ref:2606.24713#pg1>. However, they also noted that the six-fermion gapping interaction, H3450, is highly irrelevant in terms of renormalization group scaling with a dimension of "D3450 = five" which would have been an obstacle <ref:2606.24713#pg2>.

Lev: That irrelevance tells me that on real hardware, if we want this mechanism to work without needing extremely strong coupling, we definitely need a way to make that six-fermion interaction relevant, and the paper points toward doing exactly that <ref:2606.24713#pg1>.

Kai: And they introduce an auxiliary Hubbard-type density-density interaction, HHubbard, specifically to tune the scaling dimensions of the gapping terms by introducing effective Luttinger parameters K1 and K2 <ref:2606.24713#pg2>.

Mira: That tuning is crucial because it allows them to make the six-fermion gapping term relevant for symmetric mass generation when "K < two/five" which is what they call Kc <ref:2606.24713#pg2>. This makes the interaction become important in a nonperturbative regime without needing extremely strong coupling initially.

Paper summary: Lev: For us, if we can tune parameters like K1 and K2 to hit that "K < two/five" threshold, it gives us a handle on how easily this gap opens under specific interaction conditions, which is vital for designing error correction codes <ref:2606.24713#pg0>.

Kai: So, the mechanism hinges on this carefully balanced interaction tuning to get that gap opening we're looking for in the system described by "Symmetric mass generation of interacting chiral fermions on a one-dimensional lattice without fermion doubling" <ref:2606.24713#pg0>. Where does this leave us regarding the overall picture?

Mira: The overall picture is that they manage to achieve a gap opening while meticulously preserving the underlying anomaly-free symmetries, which is the defining feature of symmetric mass generation <ref:2606.24713#pg0>. They use bosonization to map this onto two decoupled Tomonaga-Luttinger liquids where the gap opens when "Dp < two" which happens under those specific K1 and K2 conditions <ref:2606.24713#pg2>.

Lev: If the system is gapped, that implies a finite energy scale exists, which is what we need to talk about for physical realization on hardware; but I wonder how stable this gap remains when we move away from the "moderately weak coupling" regime they chose for their RG analysis <ref:2606.24713#pg2>.

Kai: That stability question is exactly where my experimental mindset kicks in; if the gap is opened, does it mean there's a truly distinct ground state configuration we can measure, or is it just a feature of the Hamiltonian description?

Mira: The results from their Density Matrix Renormalization Group simulations on tensor networks confirm that they are seeing three distinct excitation scenarios: a gapless system, one with broken U(one) symmetry, and then the case III where symmetric mass generation occurs <ref:2606.24713#pg0>.

Lev: Case III, the "gapped system with preserved U(one) symmetry," sounds like the ideal scenario for us because it implies we get a gap without that unwanted symmetry breaking that complicates things for error correction <ref:2606.24713#pg0>.

Kai: And they support this by showing the occupation factor nα(k) smoothing out at k=zero which is a clear signature of a gapped system, not something we'd expect in the free limit <ref:2606.24713#pg0>.

Paper summary: Mira: A key distinction they make is that this symmetric mass generation happens while the ground state remains nondegenerate, which they prove using algebraic geometry on the mapping A from T4 to T2 <ref:2606.24713#pg0>.

Lev: The nondegeneracy condition being met because their lattice of interaction vectors Λ is "primitive," meaning no local vertex operator exponent can be a nontrivial fractional linear combination of pinned fields, gives us confidence that the gap doesn't break the protecting U(one) symmetry <ref:2606.24713#pg0>.

Kai: It seems like this paper provides a very clean framework for understanding how interactions can induce mass generation in these chiral systems without resorting to spontaneous symmetry breaking, which is a really interesting pathway for building quantum devices <ref:2606.24713#pg0>.

Mira: Indeed, the implication is that this method of tuning the interaction terms allows us to engineer specific physical phenomena—like a gap opening at "K < two/five"—that are otherwise difficult to access in simpler models <ref:2606.24713#pg1>.

Lev: For hardware, this means we have a clear roadmap: first, use the tangent-fermion lattice to avoid doubling, then tune the Hubbard interaction parameters K1 and K2 to hit that specific regime where H3450 becomes relevant <ref:2606.24713#pg2>.

Kai: So, looking ahead, what does this mean for future work on realizing these states? Are there any immediate next steps suggested by the authors?

Mira: The paper suggests that the tangent-fermion formalism itself offers a way to realize each chiral fermion flavor independently, which provides transparency for the bosonization dictionary <ref:2606.24713#pg0>. This opens up avenues for mapping more complex interactions onto these simpler components.

Lev: If we can map things cleanly, it makes designing error correction protocols much more tractable because we know exactly how the interaction terms influence the low-energy physics <ref:2606.24713#pg1>.

Kai: It's compelling to see this theoretical structure laid out so clearly; it gives us a concrete starting point for what could eventually be built on a quantum simulator or a real chip <ref:2606.24713#pg0>.

Mira: Ultimately, the significance of "Symmetric mass generation of interacting chiral fermions on a one-dimensional lattice without fermion doubling" lies in demonstrating that interaction-induced gapping can occur while upholding fundamental symmetries, suggesting new ways to engineer topological or gapped phases <ref:2606.24713#pg0>.

Lev: It establishes a clear theoretical benchmark for what constitutes a physically realized, symmetry-preserving mass generation mechanism in this specific low-dimensional setting <ref:2606.24713#pg0>.

Conclusion: Kai: So, we've been deep in the mechanics of this paper detailing symmetric mass generation without symmetry breaking on a 1D lattice, and now I want to talk about what that title really means and where this research could take us next <ref:2606.24713#pg0>.

Mira: The authors are tackling the fundamental issue of how interaction terms can induce a gap while keeping the underlying symmetries intact, which is a big theoretical question for condensed matter physics.

Lev: From my side in error correction, I'm thinking about the practical implications—if we can engineer this kind of gapping mechanism through tuning parameters like those Luttinger constants, it gives us a new knob to control the system's energy spectrum.

Kai: Exactly, Lev; that tuning ability is what makes this work interesting because it suggests a pathway to designing systems where you get the desired physical state without having to rely on spontaneous symmetry breaking, which is a huge deal for stable quantum states.

Mira: I agree; essentially, the paper shows that the specific way these chiral fermions interact can create a mass term in an induced way, rather than relying on some inherent asymmetry in the Hamiltonian.

Lev: It's important to remember that this isn't just a theoretical exercise on paper; for this to be useful, we need to know if we can translate these required interaction strengths into actual physical parameters that our current or near-future hardware can generate and cool down to observe.

Kai: That’s the next big question, right? How do we bridge the gap between these abstract lattice models and a tangible experimental setup where you can actually measure this state exist?

Instituut-Lorentz, Universiteit Leiden · Department of Physics and Astronomy, Ghent University · Department of Applied Mathematics and Theoretical Physics, University of Cambridge

hep-th, cond-mat.str-el, quant-ph

Submitted: 2026-06-23

Updated: 2026-09-02

Comments: 11 pages, 5 figures. New appendix on a more general SMG senario. New zenodo version has the DMRG code

Journal ref: Phys. Rev. Research 8, 043009 (2026)

DOI: 10.1103/2v9h-25qs

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

Importance score: 80/100

The gist: Symmetric mass generation (SMG) is an interaction-induced opening of a fermion gap without spontaneous symmetry breaking, and this work demonstrates its realization in a strictly one-dimensional

Key concepts

Symmetric Mass Generation (SMG)
SMG is an interaction-induced opening of a fermion gap that occurs without the need for spontaneous symmetry breaking. It means the system gains an energy gap due to interactions alone, rather than a phase transition that breaks fundamental symmetries.
Tangent-Fermion Lattice
This is a specific way to set up the one-dimensional lattice model to avoid 'fermion doubling,' which is a common problem in such models. It uses non-local hopping defined by Stacey’s nonlocal hopping, leading to a tangent dispersion relation that simplifies the physics while keeping the system strictly one-dimensional.
Luttinger Parameters (K1, K2)
These parameters describe the low-energy behavior of the chiral fermions in a Tomonaga-Luttinger liquid. The paper shows that tuning these parameters allows researchers to control when the six-fermion gapping interaction becomes relevant for opening a gap.
Bosonization
This is a mathematical technique used to simplify complex interacting fermion systems by mapping them onto simpler, non-interacting bosonic fields. This allows for easier analysis of the low-energy theory and understanding how interactions affect the system's behavior.

Terminology

Summary

Symmetric mass generation (SMG) is an interaction-induced opening of a fermion gap without spontaneous symmetry breaking, and this work demonstrates its realization in a strictly one-dimensional lattice setting. The central finding is that by tuning an auxiliary Hubbard-type density-density interaction to make the six-fermion gapping interaction relevant for Luttinger parameter regimes below a critical value, an excitation gap can be opened while preserving the underlying anomaly-free symmetries.

Model Formulation and Obstacle Resolution

The study utilizes the anomaly-free 3–4–5–0 model of Wang and Wen as a minimal setting for SMG in one dimension. To address the obstacles of fermion doubling for local chiral discretizations, the authors formulate the model on a strictly onedimensional tangent-fermion lattice. This approach employs Stacey’s nonlocal hopping, defined by hopping matrix elements (2.3), which corresponds to a tangent dispersion, leading to an energy relation of E(k) = 2t0 tan(k/2). This nonlocal coupling is shown to be equivalent to a local generalized eigenvalue problem, allowing the study of SMG while circumventing the fermion-doubling obstruction and retaining computational efficiency.

Making the Gapping Interaction Relevant

A key challenge addressed is that the six-fermion gapping interaction (H3450) is irrelevant in the sense of the renormalization group (RG) with a scaling dimension of 5 (3.1). To enable an RG scaling analysis to guide numerical simulations, an additional Hubbard-type density-density interaction, HHubbard, is introduced. This term couples the density of left-movers and right-movers. By choosing weight factors corresponding to the interaction vectors (3.3a) and (3.3b), the scaling dimensions D(1) 3450 and D(2) 3450 of the gapping terms in H3450 can be independently tuned by a pair of effective Luttinger parameters K1, K2. The interaction becomes relevant for SMG when "K < 2/5" (Kc).

Bosonization and Scaling Analysis

The paper employs bosonization to analyze the low-energy theory. The four chiral fermions are represented by chiral bosonic fields, leading to a decomposition of the free Hamiltonian into two decoupled Tomonaga-Luttinger liquids: Hfree = X2p=1 v2πZ dx Kp(∂xθp)2 + 1Kp(∂xϕp)2 (4.9). The interaction term H3450 is mapped onto the cosine potential terms, Vp = cos(l(p)Φ), where the scaling dimension is Dp = 5Kp. The gap opens when Dp < 2, which occurs for "K1, K2 < 2/5. Furthermore, the density-density interaction (3.3) is identified as the term that renormalizes the free-fermion scaling dimension D3450 = 5 by making it D(p) 3450 = 5Kp" (4.15).

Symmetric Mass Generation Signatures

The numerical results, obtained via Density Matrix Renormalization Group (DMRG) on tensor networks, confirm the hallmarks of SMG. The excitation spectrum analysis distinguishes between three cases: a gapless system (I), a gapped system with broken U(1) symmetry (II), and a gapped system with preserved U(1) symmetry (III). Case III, representing SMG, is characterized by the opening of an excitation gap without the appearance of a degenerate ground state. This is further supported by examining the occupation factor nα(k), which shows the smoothing of the L-L singularity at k=0, consistent with a gapped system.

Ground State Nondegeneracy

A distinctive feature distinguishing SMG from spontaneous symmetry breaking is that the ground state remains nondegenerate. This is proven through algebraic geometry applied to the mapping A from T4 to T2. The ground state is nondegenerate if the kernel of the map A, M, is singly connected on T4. This condition is satisfied because the lattice of interaction vectors, Λ = spanZ[l(1), l(2)], is primitive, meaning no local vertex operator exponent can be a nontrivial fractional linear combination of pinned fields. This primitivity ensures that the gap opening does not break the protecting U(1) symmetry.

Conclusion and Significance

The study concludes that for K < Kc = 2/5, the 3–4–5–0 interaction becomes relevant, leading to a gap opening without spontaneous symmetry breaking. The tangent-fermion formalism provides a framework where each chiral fermion flavor is realized independently, offering transparency for the bosonization dictionary.

Improvements for AI systems

As a fastidious researcher, I have analyzed this work on Symmetric Mass Generation (SMG) in 1D chiral fermions. The core findings relate to using interaction-induced gaps without spontaneous symmetry breaking, specifically leveraging a tangent fermion lattice formulation and controlling the relevance of the gapping interaction via an auxiliary Hubbard term.

Here are specific improvements for AI systems derived from this research:

AI System Improvements Derived from SMG Research:

  1. Enhanced Robustness in Quantum Simulation/Modeling (Physics-Informed Neural Networks - PINNs):

Based on the paper's successful use of Matrix Product Operators (MPO) and DMRG to model complex, interacting 1D systems (like the 3-4-5-0 model), AI systems can be improved by incorporating these structure-preserving representations.

  1. Improved Feature Extraction for Topological Phases:

The paper demonstrates a mechanism where the ground state remains nondegenerate (Case III) despite an excitation gap, distinguishing it from spontaneous symmetry breaking (Case II). AI models trained on these DMRG results can be optimized to recognize the specific signatures of SMG versus standard Higgs mechanisms.

  1. Design of Synthetic Interaction Hamiltonians for Quantum Simulators:

The paper provides a rigorous framework for constructing effective, anomaly-free interaction terms (like the 3-4-5-0 model) that are relevant only under specific conditions (e.g., Luttinger parameter K < 2/5). This knowledge can be used to train generative AI models to propose novel, physically relevant Hamiltonians for quantum simulators.

  1. Advanced Data Analysis for Correlators and Occupation Factors:

The work shows how the occupation factor's singularity is smoothed by the gap opening. AI systems designed to analyze experimental data (e.g., from quantum Hall edge states or synthetic condensed matter experiments) can be improved to accurately model these non-trivial, exponentially decaying correlator behaviors rather than relying on simple power-law fits.

  1. Development of Adaptive Renormalization Schemes for Machine Learning:

The paper details how an auxiliary interaction (Hubbard term) can tune the scaling dimension of the primary gap-opening interaction. This concept can be translated into adaptive learning architectures where a secondary, auxiliary tuning loss function is used to dynamically adjust the relevance or strength of the primary physics being modeled, allowing AI to explore parameter spaces more efficiently than standard gradient descent.

  1. Discovery of New Interaction Classes via Algebraic Constraints:

The Appendix A analysis shows that the specific structure of interaction vectors (like those in Eq. 2.6) dictates whether the system decouples into independent Luttinger liquids or remains coupled, and it defines the critical threshold for relevance (e.g., determining if K1 < 1/5). AI can be used to search for novel interaction structures that satisfy these specific algebraic constraints, potentially leading to the discovery of new physical phenomena in strongly correlated systems.

Improved AI System Capabilities:

The improved AI system can perform the following tasks:

  1. Predictive Modeling of Non-Spontaneous Symmetry Breaking (SSB) States: The system can reliably distinguish between systems that require spontaneous symmetry breaking (which lead to degenerate ground states) and those exhibiting Symmetric Mass Generation, providing a ground state fingerprint for interaction-induced mass generation.

  2. Designing Novel Quantum Interaction Hamiltonians: The AI can generate complex, anomaly-free interaction terms that are tuned to be relevant for specific regimes (e.g., K < 2/5), which is crucial for building accurate digital simulators or quantum device architectures.

  3. High-Fidelity Simulation of Edge States: The system can accurately model the momentum-dependent occupation factors and excitation spectra of 1D chiral systems, allowing for precise prediction of transport properties (like conductance) in synthetic materials or condensed matter simulations.

  4. Automated Search for Physically Relevant Interaction Vectors: The AI can search vast spaces of possible interaction vectors to find those that satisfy the necessary algebraic conditions (like the null condition and orthogonality constraints), thereby automating the discovery of new, physically viable mass-generating interactions.

Sources

Related papers