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arxiv: 2602.13050 · v2 · pith:63MBUIOQnew · submitted 2026-02-13 · ❄️ cond-mat.str-el · cond-mat.stat-mech· quant-ph

Topology of the Fermi surface and universality of the metal-metal and metal-insulator transitions: d-dimensional Hatsugai-Kohmoto model as an example

Gennady Y. Chitov This is my paper

Pith reviewed 2026-05-21 12:28 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.stat-mechquant-ph
keywords Fermi surface topologyquantum phase transitionsHatsugai-Kohmoto modelLuttinger theoremLee-Yang zerosMott transitionLifshitz transitionuniversality class
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The pith

The d-volume of the Fermi sea serves as an order parameter for a universality class of Fermi surface topology transitions in interacting fermion models.

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

The paper advances an earlier theory of quantum phase transitions driven by changes in Fermi surface topology. In this framework the Fermi surface is viewed as a quantum critical manifold fixed by Lee-Yang zeros, with an order parameter given by the d-dimensional volume of the enclosed Fermi sea. Analysis of the exactly solvable Hatsugai-Kohmoto model in one, two and three dimensions shows that this order parameter and the associated FST universality class account for both metal-to-insulator transitions and gapless-to-gapless transitions such as Lifshitz and van Hove types. The gapless phases are identified as conventional Landau Fermi liquids, the Luttinger theorem is verified, and the Euler characteristic from homology theory is used to label each phase.

Core claim

The order parameter P, defined as the d-volume of the Fermi sea determined by the Lee-Yang zeros, and the FST universality class describe the transitions between metal and band/Mott insulators as well as the Lifshitz and van Hove gapless-to-gapless transitions in the d-dimensional Hatsugai-Kohmoto model. The gapless phases are Landau Fermi liquids, the Luttinger theorem holds, and the transitions are critical points of the Morse function whose Euler characteristic quantifies the topology change.

What carries the argument

The order parameter P as the d-volume of the Fermi sea together with the FST universality class, where the Fermi surface acts as a quantum critical manifold set by Lee-Yang zeros and the transitions are non-degenerate critical points of a Morse function.

If this is right

  • The FST universality class governs metal to band/Mott insulator transitions.
  • The same class covers Lifshitz and van Hove gapless-to-gapless transitions.
  • Gapless phases remain Landau Fermi liquids.
  • The universality class remains robust against interactions provided critical points stay non-degenerate.
  • The Euler characteristic provides a topological label for each phase of the model.

Where Pith is reading between the lines

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

  • If the non-degenerate condition holds more generally, the FST approach could classify quantum transitions in a wider range of lattice fermion models beyond the Hatsugai-Kohmoto case.
  • Connections may exist between this Morse-function description and other topological invariants used in band theory or topological insulators.
  • Experimental probes of Fermi surface volume changes could directly test the order parameter in real materials near these transitions.

Load-bearing premise

The Fermi surface can be treated as a quantum critical manifold whose topology is fixed by Lee-Yang zeros and whose critical points are non-degenerate Morse critical points.

What would settle it

A numerical or experimental observation of a metal-insulator transition in the Hatsugai-Kohmoto model where the Fermi sea volume remains continuous yet the transition occurs, or where the Euler characteristic fails to change across a claimed topological transition.

Figures

Figures reproduced from arXiv: 2602.13050 by Gennady Y. Chitov.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) The distribution function [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The Matsubara Green’s function [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a,b) One-dimensional spectra [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Path of the LY zeros [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. (a) Consecutive building up the torus [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Two normalized order parameters for the gapless phase, [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The Fermi surfaces of the 3D band ( [PITH_FULL_IMAGE:figures/full_fig_p026_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. (a): circle [PITH_FULL_IMAGE:figures/full_fig_p027_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. (a-c): Evolution of the FS at [PITH_FULL_IMAGE:figures/full_fig_p030_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p031_13.png] view at source ↗
read the original abstract

The earlier theory [1] of the quantum phase transitions related to the change of the Fermi Surface Topology (FST) is advanced. For such transitions the Fermi surface as a quantum critical manifold determined by the Lee-Yang zeros, the order parameter $\mathcal{P}$ as the $d$-volume of the Fermi sea, and the special FST universality class were introduced in [1]. The exactly solvable Hatsugai-Kohmoto (HK) $d$-dimensional ($d=1,2,3$) model of interacting fermions is analyzed. We explore the relation between the Lee-Yang zeros, the Luttinger and the plateau (Oshikawa) theorems. The validity of the Luttinger theorem in the HK model is confirmed. It is shown that the order parameter $\mathcal{P}$ and the FST universality class describe the transitions between metal and band/Mott insulators, as well as the Lifshitz and van Hove gapless-to-gapless transitions. The gapless phases are established to be the Landau Fermi liquids (metals). In addition to the conventional paradigm with a continuous order parameter, we apply the homology theory to analyze the FST transitions. They are critical points of the Morse function. To quantify FST we use the Euler characteristic, which is calculated for each phase of the HK model. We claim that the FST universality class is robust with respect to interactions and other model details, under the condition that the critical points are non-degenerate.

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 manuscript advances the earlier theory of Fermi surface topology (FST) quantum phase transitions by analyzing the exactly solvable d-dimensional (d=1,2,3) Hatsugai-Kohmoto (HK) model of interacting fermions. It confirms the validity of the Luttinger theorem, relates Lee-Yang zeros to Luttinger and Oshikawa (plateau) constraints, identifies gapless phases as Landau Fermi liquids, and argues that the order parameter P (the d-volume of the Fermi sea) together with an FST universality class—viewed through homology theory and the Euler characteristic—describes metal–band/Mott insulator transitions as well as Lifshitz and van Hove gapless-to-gapless transitions. The FST universality class is claimed to be robust with respect to interactions and model details provided the associated Morse critical points remain non-degenerate.

Significance. If the non-degeneracy condition holds and the topological classification is internally consistent, the work supplies a concrete, solvable-model test of a topological framework that unifies metal–insulator and gapless-to-gapless transitions via an order parameter P and Morse-theoretic critical points. The confirmation of the Luttinger theorem in the HK model and the explicit computation of the Euler characteristic per phase constitute verifiable strengths that ground the abstract construction introduced in prior work.

major comments (1)
  1. [Abstract and FST universality discussion] Abstract and the section discussing the FST universality class: the central claim that the FST universality class is robust under the non-degeneracy condition is load-bearing for the classification of metal–insulator, Lifshitz, and van Hove transitions, yet the manuscript supplies no explicit verification (Hessian eigenvalues, higher-order derivatives, or local normal-form analysis) that the Morse critical points at these loci are non-degenerate in the HK model for d=1,2,3. Degeneracy would invalidate the claimed universality and the topological distinction between gapless phases.
minor comments (2)
  1. The relation between the order parameter P and the Euler characteristic could be stated more explicitly when the homology analysis is introduced, to clarify how the volume of the Fermi sea is extracted from the topological invariants.
  2. Notation for the Lee-Yang zeros and their connection to the Luttinger volume should be cross-referenced consistently between the model analysis and the general FST framework.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The single major comment raises an important point about explicit verification of the non-degeneracy condition for the Morse critical points in the Hatsugai-Kohmoto model. We address it directly below and will incorporate the requested analysis in the revised manuscript.

read point-by-point responses

  1. Referee: [Abstract and FST universality discussion] Abstract and the section discussing the FST universality class: the central claim that the FST universality class is robust under the non-degeneracy condition is load-bearing for the classification of metal–insulator, Lifshitz, and van Hove transitions, yet the manuscript supplies no explicit verification (Hessian eigenvalues, higher-order derivatives, or local normal-form analysis) that the Morse critical points at these loci are non-degenerate in the HK model for d=1,2,3. Degeneracy would invalidate the claimed universality and the topological distinction between gapless phases.

    Authors: We agree that an explicit check of non-degeneracy is necessary to substantiate the robustness claim. In the revised manuscript we will add a dedicated subsection (likely in Sec. III or IV) that computes the Hessian matrix of the dispersion at each relevant critical point for d=1,2,3. For the HK model the single-particle dispersion is piecewise linear or quadratic in the appropriate coordinates, allowing direct evaluation of the second derivatives; we will report the eigenvalues and confirm that none vanish at the loci where the Fermi-sea volume P changes. If any higher-order terms appear we will also supply the leading normal form. This verification will be model-specific yet sufficient to justify applying the Morse-theoretic classification to the HK phases. revision: yes

Circularity Check

1 steps flagged

FST order parameter and universality class imported via self-citation [1] with non-degeneracy of Morse points assumed but unverified in HK model

specific steps
  1. self citation load bearing [Abstract]
    "The earlier theory [1] of the quantum phase transitions related to the change of the Fermi Surface Topology (FST) is advanced. For such transitions the Fermi surface as a quantum critical manifold determined by the Lee-Yang zeros, the order parameter P as the d-volume of the Fermi sea, and the special FST universality class were introduced in [1]. ... We claim that the FST universality class is robust with respect to interactions and other model details, under the condition that the critical points are non-degenerate."

    The defining elements of the FST approach (quantum critical manifold, order parameter P, universality class) and the non-degeneracy condition required for robustness are taken directly from self-cited prior work [1]. The manuscript's strongest claim—that these describe the HK model's transitions—therefore reduces to the assumptions of that earlier theory without independent derivation or explicit non-degeneracy checks (Hessian or higher derivatives) in the current analysis.

full rationale

The paper's core framework (Fermi surface as quantum critical manifold via Lee-Yang zeros, order parameter P as d-volume of Fermi sea, and FST universality class) is explicitly introduced in prior work [1] by the same author. The claim that this framework describes metal-insulator, Lifshitz, and van Hove transitions in the HK model, and is robust under non-degenerate critical points, relies on that self-citation without new explicit verification (e.g., Hessian eigenvalues or normal forms) for the model's phases. However, the paper performs independent calculations on the solvable HK model, confirms the Luttinger theorem, and computes Euler characteristics per phase using homology, providing some external grounding. This yields moderate circularity (score 5) rather than full reduction, as the application and confirmations add content beyond the citation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The paper rests on concepts from prior work on Lee-Yang zeros and Fermi surface topology, plus standard theorems in many-body physics. No explicit free parameters are introduced in the abstract, but the non-degeneracy condition functions as a key modeling assumption.

axioms (2)
  • domain assumption The Luttinger theorem remains valid in the interacting HK model
    Stated as confirmed in the analysis of the model.
  • ad hoc to paper Critical points of the Morse function for FST are non-degenerate
    Explicit condition required for the robustness claim of the FST universality class.
invented entities (2)
  • Fermi surface as quantum critical manifold determined by Lee-Yang zeros no independent evidence
    purpose: To define the manifold whose topology changes at the transitions
    Introduced in the earlier theory [1] and used as the basis for the order parameter.
  • Order parameter P as the d-volume of the Fermi sea no independent evidence
    purpose: To serve as a continuous order parameter for FST transitions
    Defined within the FST framework to quantify the Fermi sea.

pith-pipeline@v0.9.0 · 5820 in / 1601 out tokens · 65776 ms · 2026-05-21T12:28:34.185877+00:00 · methodology

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Reference graph

Works this paper leans on

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