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arxiv: 2604.11054 · v1 · submitted 2026-04-13 · ❄️ cond-mat.str-el · hep-ph

Topological charge of fermions and Landau theory of Fermi liquid

G.E. Volovik This is my paper

Pith reviewed 2026-05-10 16:06 UTC · model grok-4.3

classification ❄️ cond-mat.str-el hep-ph
keywords topological chargeFermi liquidGreen's functionLuttinger theoremtopological invariantnon-Fermi liquidfermion charge
0
0 comments X p. Extension

The pith

The electric charge of a fermion is identical to its topological charge.

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

The paper argues that a topological invariant extracted from the Green's function at the Fermi surface is the same as the physical charge of the fermion particle. This makes charge conservation a topological statement, which protects the Fermi surface and explains why the Landau Fermi liquid description remains valid when the Green's function has a pole with nonzero residue. The same invariant governs non-Landau cases such as Luttinger liquids and extends to crystalline insulators through the Luttinger theorem. A reader would care because the result supplies a topological reason for the robustness of Fermi-liquid behavior in real materials even under strong interactions.

Core claim

In fermionic liquids the Fermi surface is topologically stable. This stability comes from a topological invariant in the Green's function. The paper shows that this invariant is a property of the fermionic particle itself, so the particle charge (for example the electric charge of an electron) equals the topological charge of the fermion. Consequently the conservation of fermionic charge is equivalent to the conservation of topological charge. The same framework is applied to the Landau theory of Fermi liquids, to non-Fermi liquids, and to crystalline insulators in connection with the Luttinger theorem.

What carries the argument

The topological invariant of the Green's function, which remains nonzero at the Fermi surface and encodes the winding or pole residue that counts the enclosed states.

If this is right

  • The Landau Fermi liquid exists under the condition that the Green's function has a pole with nonzero residue Z.
  • Non-Landau Fermi liquids such as the Luttinger liquid are still described by the same topological invariant.
  • Conservation of particle charge follows directly from conservation of the topological charge.
  • The Luttinger theorem for crystalline insulators is a direct consequence of this topological charge.

Where Pith is reading between the lines

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

  • This equivalence implies that charge conservation remains protected even when well-defined quasiparticles cease to exist.
  • The topological charge supplies a unified counting rule that connects metallic Fermi surfaces to the gapped states of insulators.

Load-bearing premise

That the topological invariant obtained from the Green's function is numerically and physically identical to the electric charge of the fermion, without extra coefficients or dependence on interaction strength or lattice details.

What would settle it

An experimental measurement, such as quantum oscillations or ARPES, showing that the volume enclosed by a Fermi surface does not match the expected particle density given by the topological invariant.

Figures

Figures reproduced from arXiv: 2604.11054 by G.E. Volovik.

Figure 1
Figure 1. Figure 1: FIG. 1: Topological invariant [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Distribution of fermions in Landau theory. The role of the particle number [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Illustration of the real occupation number [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Illustration of Landau theory for Fermi liquid which contains the flat band. Variation of the phenomenological Landau [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Illustration of the momentum-space topology of the Khodel-Shaginyan flat band on example of the two-dimensional [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
read the original abstract

In the fermionic liquids, the Fermi surface is topologically stable,\cite{Volovik2003} which is at the origin of the applicability of the Landau theory of Fermi liquid (LFL). The LFL exists under special condition, when the Green's function has a pole with nonzero residue $Z$. Otherwise one has non-Landau Fermi liquid (NLFL), such as Luttinger liquid, which is described by the same topological invariant. It appears that in general this topological invariant is the property of the fermionic particle, i.e. the particle charge (or the electric charge of electron) is equivalent to the topological charge of the fermion. The conservation of the fermionic charge is equivalent to the conservation of the topological charge. We consider the application of this topological charge to the Landau theory of Fermi liquids. We also consider the application to non-Fermi liquids and crystalline insulators in relation to the Luttinger theorem.

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

2 major / 1 minor

Summary. The manuscript claims that the topological invariant of the Fermi surface (extracted from the Green's function pole structure) is identical to the conserved U(1) charge of the fermion. This identification is said to explain the stability of Landau Fermi liquid theory when the residue Z is nonzero, to apply equally to non-Landau Fermi liquids such as the Luttinger liquid, and to extend to crystalline insulators via the Luttinger theorem. Conservation of the topological charge is asserted to be equivalent to conservation of electric charge.

Significance. If the asserted numerical and physical identity between the topological winding number (or pole residue) and the electric charge holds without renormalization factors or additional assumptions, the work would supply a topological underpinning for both the applicability of Landau theory and the Luttinger sum rule in interacting systems. It would also unify the description of LFL and NLFL states under a single invariant.

major comments (2)
  1. [Abstract] Abstract: The central claim that 'the particle charge ... is equivalent to the topological charge of the fermion' is introduced with the qualifier 'it appears that' and is not accompanied by an explicit operator mapping or step-by-step calculation showing that the topological invariant N_top extracted from G(k,ω) equals the conserved charge Q exactly (rather than N_top = Z Q or a lattice-dependent multiple). This equivalence is load-bearing for all subsequent applications to LFL, NLFL, and the Luttinger theorem.
  2. The manuscript states that the same topological invariant describes both LFL (pole with Z ≠ 0) and NLFL states, yet provides no derivation demonstrating that the invariant remains numerically equal to the charge when the quasiparticle residue vanishes or when interactions are strong. Without this step, the extension to non-Fermi liquids rests on identification rather than equality.
minor comments (1)
  1. [Abstract] The abstract cites Volovik2003 but does not indicate which specific result from that reference is being used to define the topological invariant in the interacting case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for greater rigor in establishing the central equivalence. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that 'the particle charge ... is equivalent to the topological charge of the fermion' is introduced with the qualifier 'it appears that' and is not accompanied by an explicit operator mapping or step-by-step calculation showing that the topological invariant N_top extracted from G(k,ω) equals the conserved charge Q exactly (rather than N_top = Z Q or a lattice-dependent multiple). This equivalence is load-bearing for all subsequent applications to LFL, NLFL, and the Luttinger theorem.

    Authors: We agree that the abstract phrasing is tentative and that an explicit demonstration is required. In the revised manuscript we will insert a dedicated subsection (immediately following the definition of N_top) that derives the equality N_top = Q from the analytic continuation of the Green's function and the Ward identity associated with U(1) particle-number conservation. The derivation shows that the winding number around the singularity equals the conserved charge without a residue factor Z or lattice-dependent prefactor, because the phase accumulation is fixed by the residue theorem applied to the closed contour in the complex frequency plane. This step will be placed before the applications to LFL, NLFL and the Luttinger theorem. revision: yes

  2. Referee: The manuscript states that the same topological invariant describes both LFL (pole with Z ≠ 0) and NLFL states, yet provides no derivation demonstrating that the invariant remains numerically equal to the charge when the quasiparticle residue vanishes or when interactions are strong. Without this step, the extension to non-Fermi liquids rests on identification rather than equality.

    Authors: The topological invariant is defined via the Green's function G(k,ω) for arbitrary interaction strength; its value is the winding number of arg G around the Fermi surface in the (k,ω) plane. When Z = 0 the singularity changes from a simple pole to a branch point, yet the contour integral that extracts the winding number remains unchanged because it depends only on the existence of the singularity and the asymptotic behavior at infinity, both of which are protected by particle-number conservation. We will add an explicit calculation for the Luttinger-liquid case (using the known form of G) that confirms N_top = 1 per spin per branch, identical to the conserved charge. This derivation will be inserted in the section discussing NLFL states. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper builds its claims on the topological stability of the Fermi surface (cited to Volovik2003) and the standard pole/winding structure of the Green's function to identify the topological invariant with fermionic charge and conservation laws. This identification is presented as an observation ('it appears that') rather than a closed mathematical reduction or fitted prediction. No equations are shown to equal their inputs by construction, no parameters are fitted then relabeled as predictions, and the applications to LFL/NLFL/Luttinger theorem follow from the premises without self-referential loops or load-bearing self-citation chains that lack external content. The derivation remains self-contained against the definitions of Green's functions and U(1) conservation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard many-body definitions of the Green's function and topological invariants in fermionic systems; no new free parameters, invented particles, or ad-hoc entities are introduced. The key step is the proposed equivalence, which is presented as following from conservation rather than from new postulates.

axioms (2)
  • domain assumption The Fermi surface is topologically stable, as established in Volovik2003
    Invoked at the opening of the abstract as the origin of Landau theory applicability.
  • domain assumption A pole in the Green's function with nonzero residue Z defines the Landau Fermi liquid regime
    Stated explicitly as the special condition separating LFL from NLFL while sharing the same topological invariant.

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discussion (0)

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Reference graph

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