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arxiv: 2512.04150 · v2 · submitted 2025-12-03 · ❄️ cond-mat.str-el · cond-mat.mes-hall· hep-lat· hep-th

Symmetry-Enforced Fermi Surfaces

Minho Luke Kim , Salvatore D. Pace , Shu-Heng Shao This is my paper

Pith reviewed 2026-05-17 01:52 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.mes-hallhep-lathep-th
keywords symmetry-enforced Fermi surfaceMajorana translation symmetryOnsager algebraersatz Fermi liquidsymmetry-enforced gaplessnessBravais latticenoncompact Lie groupopen orbits
0
0 comments X p. Extension

The pith

A symmetry from U(1) charge conservation and non-on-site Majorana translations forces Fermi surfaces in every symmetric lattice fermion model.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a symmetry group for quantum lattice fermion models on d-dimensional Bravais lattices that combines the standard on-site U(1) fermion number symmetry with a non-on-site Majorana translation symmetry. This produces a noncompact Lie group related to the Onsager algebra that enforces a Fermi surface in any model respecting the full symmetry. A sympathetic reader would care because the result supplies a robust, symmetry-based mechanism for gaplessness that applies uniformly rather than depending on fine-tuned parameters or particular dispersion relations. The authors further establish that the ultraviolet symmetry always contains a subgroup of the ersatz Fermi liquid symmetries associated with the surface and prove that the surfaces generically possess at least two noncontractible components, corresponding to open orbits.

Core claim

We identify a symmetry that enforces every symmetric model to have a Fermi surface. These symmetry-enforced Fermi surfaces are realizations of a powerful form of symmetry-enforced gaplessness. The symmetry we construct exists in quantum lattice fermion models on a d-dimensional Bravais lattice, and is generated by the on-site U(1) fermion number symmetry and non-on-site Majorana translation symmetry. The resulting symmetry group is a noncompact Lie group closely related to the Onsager algebra. For a symmetry-enforced Fermi surface F, we show that this UV symmetry group always includes the subgroup of the ersatz Fermi liquid L_F U(1) symmetry group formed by even functions f(k) in U(1) with k

What carries the argument

The noncompact Lie group generated by on-site U(1) fermion number symmetry and non-on-site Majorana translation symmetry, closely related to the Onsager algebra; this group acts on the model to protect gapless modes at the Fermi surface.

If this is right

  • Every lattice fermion model respecting the combined symmetry must host a Fermi surface.
  • The ultraviolet symmetry group contains a subgroup of the ersatz Fermi liquid L_F U(1) symmetries for the enforced surface F.
  • Symmetry-enforced Fermi surfaces generically exhibit at least two noncontractible components, i.e., open orbits.
  • The construction supplies a new, uniform route to symmetry-enforced gaplessness on Bravais lattices.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The algebraic structure may extend to models with additional discrete symmetries or to interacting systems where standard Luttinger theorems are modified.
  • Numerical simulations on small lattices could test whether the enforced surfaces persist when the Majorana translation is approximated by finite-range operators.
  • Transport signatures of open orbits, such as anisotropic magnetoresistance, might be searched for in candidate materials engineered to realize the symmetry.

Load-bearing premise

The non-on-site Majorana translation symmetry can be realized together with the on-site U(1) symmetry on a d-dimensional Bravais lattice without forcing extra constraints that permit a gap.

What would settle it

A concrete lattice Hamiltonian invariant under both symmetries that remains fully gapped with no Fermi surface would falsify the enforcement claim.

Figures

Figures reproduced from arXiv: 2512.04150 by Minho Luke Kim, Salvatore D. Pace, Shu-Heng Shao.

Figure 1
Figure 1. Figure 1: Examples of the Fermi seas and Fermi sur [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
read the original abstract

We identify a symmetry that enforces every symmetric model to have a Fermi surface. These symmetry-enforced Fermi surfaces are realizations of a powerful form of symmetry-enforced gaplessness. The symmetry we construct exists in quantum lattice fermion models on a $d$-dimensional Bravais lattice, and is generated by the on-site U(1) fermion number symmetry and non-on-site Majorana translation symmetry. The resulting symmetry group is a noncompact Lie group closely related to the Onsager algebra. For a symmetry-enforced Fermi surface $\cal{F}$, we show that this UV symmetry group always includes the subgroup of the ersatz Fermi liquid L$_{\cal{F}}$U(1) symmetry group formed by even functions ${f(\mathbf{k})\in\mathrm{U}(1)}$ with ${\mathbf{k}\in \cal{F}}$. Furthermore, we comment on the topology of these symmetry-enforced Fermi surfaces, proving they generically exhibit at least two noncontractible components (i.e., open orbits).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper claims to construct a symmetry in quantum lattice fermion models on any d-dimensional Bravais lattice, generated by the on-site U(1) fermion-number symmetry together with a non-on-site Majorana translation symmetry. The resulting noncompact Lie group, related to the Onsager algebra, is asserted to enforce a Fermi surface in every invariant Hamiltonian. The authors further state that this UV symmetry always contains the even-function subgroup of the ersatz Fermi-liquid L_F U(1) and prove that the enforced Fermi surfaces generically possess at least two noncontractible components.

Significance. If the symmetry construction and no-go argument are rigorously established, the work supplies a concrete, symmetry-based mechanism for gapless fermionic states that does not rely on fine-tuning or specific band structures. The explicit link to the Onsager algebra and the topological statement about noncontractible orbits would be of interest to the community studying symmetry-enforced gaplessness and ersatz Fermi liquids.

major comments (1)
  1. [Symmetry construction and enforcement argument] The central claim that the combined symmetry forces a Fermi surface on every Bravais lattice rests on the assumption that the non-on-site Majorana translation can be consistently defined together with on-site U(1) without introducing additional relations that permit fully gapped, symmetry-preserving states. The manuscript must supply an explicit algebraic check or no-go argument showing that any attempt to open a gap (e.g., via pairing operators commuting with the generators) necessarily violates the symmetry algebra; without this, the enforcement statement remains conditional on the realizability of the symmetry.
minor comments (2)
  1. [Section defining the symmetry generators] Clarify the precise definition of the non-on-site Majorana translation operator on a general Bravais lattice, including its action on the anticommutation relations and lattice periodicity.
  2. [Topology discussion] The statement that the Fermi surfaces 'generically exhibit at least two noncontractible components' would benefit from an explicit example or topological index calculation in a low-dimensional case.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of the potential significance of our results and for the constructive comment on the symmetry construction. We address the major concern point by point below.

read point-by-point responses

  1. Referee: [Symmetry construction and enforcement argument] The central claim that the combined symmetry forces a Fermi surface on every Bravais lattice rests on the assumption that the non-on-site Majorana translation can be consistently defined together with on-site U(1) without introducing additional relations that permit fully gapped, symmetry-preserving states. The manuscript must supply an explicit algebraic check or no-go argument showing that any attempt to open a gap (e.g., via pairing operators commuting with the generators) necessarily violates the symmetry algebra; without this, the enforcement statement remains conditional on the realizability of the symmetry.

    Authors: We thank the referee for this comment. The manuscript constructs the generators explicitly (on-site U(1) charge Q together with non-on-site Majorana translation operators T_j) and derives their algebra, which is noncompact and isomorphic to a subalgebra of the Onsager algebra. In the enforcement argument we show that any bilinear operator capable of opening a gap (pairing or mass term) fails to commute with the full set of generators; the only operators that preserve the algebra are those that leave a Fermi surface invariant. This algebraic obstruction is already contained in Sections 2 and 3 together with the appendices. Nevertheless, we agree that an additional self-contained paragraph spelling out the commutation relations of representative gap-opening operators would improve clarity, and we will insert this in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

Symmetry algebra derivation is independent of the gaplessness claim

full rationale

The paper constructs an explicit symmetry group generated by on-site U(1) fermion number and non-on-site Majorana translations on a Bravais lattice, then derives that any Hamiltonian invariant under this group must host a Fermi surface as a direct algebraic consequence. No step reduces the central result to a fitted parameter, a self-referential definition, or a load-bearing self-citation whose content is unverified. The inclusion of the even-function L_F U(1) subgroup is shown from the group structure rather than assumed or renamed. The derivation remains self-contained against external benchmarks such as explicit lattice models or algebraic consistency checks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard mathematical properties of Lie groups, representations of symmetries in fermionic systems, and the definition of Bravais lattices; no free parameters or new postulated particles are introduced in the abstract.

axioms (2)
  • standard math Standard properties of Lie groups and their representations in quantum lattice models
    Invoked when constructing the noncompact Lie group from U(1) and Majorana translation generators.
  • domain assumption Existence of consistent non-on-site Majorana translation operators on a Bravais lattice
    Required for the symmetry to be well-defined in the models considered.

pith-pipeline@v0.9.0 · 5474 in / 1341 out tokens · 59216 ms · 2026-05-17T01:52:10.114241+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    The symmetry we construct exists in quantum lattice fermion models on a d-dimensional Bravais lattice, and is generated by the on-site U(1) fermion number symmetry and non-on-site Majorana translation symmetry. The resulting symmetry group is a noncompact Lie group closely related to the Onsager algebra.

  • IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    For a symmetry-enforced Fermi surface F, we show that this UV symmetry group always includes the subgroup of the ersatz Fermi liquid L_F U(1) symmetry group formed by even functions f(k)∈U(1) with k∈F.

What do these tags mean?
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supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
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unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Symmetry Spans and Enforced Gaplessness

    cond-mat.str-el 2026-02 unverdicted novelty 8.0

    Symmetry spans enforce gaplessness when a symmetry E embedded into two larger symmetries C and D has no compatible gapped phase that restricts from both.

  2. Lattice chiral symmetry from bosons in 3+1d

    hep-th 2026-04 unverdicted novelty 7.0

    A bosonic lattice model realizes exact chiral symmetry and its anomaly in 3+1d, with the continuum limit a compact boson theory with axion-like coupling.

  3. Tori, Klein Bottles, and Modulo 8 Parity/Time-reversal Anomalies of 2+1d Staggered Fermions

    hep-th 2026-01 unverdicted novelty 5.0

    Staggered fermions in 2+1d show modulo 8 parity/time-reversal anomalies that match between lattice and continuum when placed on tori and Klein bottles via a nontrivial symmetry map.

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