arxiv: 2604.06157 · v1 · submitted 2026-04-07 · ❄️ cond-mat.str-el · cond-mat.supr-con

Recognition: 2 theorem links

· Lean Theorem

Tractable model for a fractionalized Fermi liquid (FL^*) on a square lattice

Piers Coleman , Elio J. K\"onig , Aaditya Panigrahi , Alexei Tsvelik

Authors on Pith no claims yet

Pith reviewed 2026-05-10 18:32 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.supr-con

keywords fractionalized Fermi liquidFL*square latticeFermi arcspseudogapYao-Lee spin liquidZ2 gauge theoryKondo lattice

0 comments

The pith

A square-lattice model couples conduction electrons to a Z2 spin liquid, producing either a hybridized Fermi surface or a small one with Fermi arcs.

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

This paper develops an analytically tractable microscopic model for a fractionalized Fermi liquid on the square lattice by coupling conduction electrons to the Majorana fermions of a Yao-Lee Z2 spin liquid. The model admits two phases: a hybridized phase in which the fermions form a common Fermi surface violating the standard Luttinger count and a decoupled phase featuring a small Fermi surface. In the small phase the authors derive momentum-dependent coherence factors that produce Fermi arcs in the manner of the Yang-Rice-Zhang phenomenology. Fluctuations around this phase produce a strong diamagnetic response and a logarithmically diverging Sommerfeld coefficient at the pseudogap onset.

Core claim

The central claim is that an interaction between conduction electrons and the static Z2 Yao-Lee spin liquid with its Majorana Fermi surface produces two phases. In one phase the spin liquid fermions hybridize with the conduction electrons to form a single Fermi surface whose volume does not obey the naive Luttinger theorem. In the other phase they remain decoupled, giving rise to a small Fermi surface whose spectral function exhibits momentum-dependent coherence factors that generate Fermi arcs. The static nature of the Z2 gauge theory permits an analytic solution to leading-logarithmic accuracy, and the inclusion of fluctuations reveals a strong diamagnetic response together with a log-divg

What carries the argument

Hybridization of conduction electrons with the Majorana Fermi surface of the static Z2 Yao-Lee spin liquid, solved to leading-logarithmic accuracy.

Load-bearing premise

The Z2 gauge theory remains static and does not fluctuate dynamically, enabling an analytic solution to leading-logarithmic accuracy.

What would settle it

ARPES data on a material realizing this Hamiltonian that fail to exhibit momentum-dependent coherence factors suppressing spectral weight into arcs in the small phase, or quantum oscillation measurements showing a Fermi surface volume inconsistent with both phases, would falsify the central claim.

Figures

Figures reproduced from arXiv: 2604.06157 by Aaditya Panigrahi, Alexei Tsvelik, Elio J. K\"onig, Piers Coleman.

**Figure 2.** Figure 2: FIG. 2. Simulated ARPES intensity from the YRZ spec [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Showing the gap profile around the reconstructed [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

read the original abstract

Motivated by the continued interest in Fermi-surface reconstruction without symmetry breaking, we present an analytically tractable microscopic model of a fractionalized Fermi liquid (FL$^*$) on a square lattice and discuss its potential relevance to the cuprates. As in ancilla-qubit constructions, the model is related to Kondo lattice systems, but in this case, the conduction electrons interact with a $\mathbb{Z}_2$ spin liquid of the Yao--Lee type, with a Majorana Fermi surface. The associated $\mathbb Z_2$ gauge theory is static so that the model can be analytically solved to leading-logarithic accuracy. There are two phases: one in which the fractionalized fermions of the spin liquid hybridize with conduction electrons to form a common Fermi surface violating the naive Luttinger count, and one in which they remain decoupled. We discuss the salient features of the small Fermi-surface phase, including analytically derived momentum dependent coherence factors responsible for the appearance of Fermi arcs \`{a} la Yang-Rice-Zhang. We further discuss the impact of quantum and thermal fluctuations, including a strong diamagnetic response and a logarithmically divergent Sommerfeld coefficient at the onset of the pseudogap.

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.

Desk Editor's Note private letter to a colleague

A solvable square-lattice FL* model via static Z2 Yao-Lee liquid that produces YRZ-style arcs, but the static-gauge step is the load-bearing assumption that needs checking.

read the letter

This paper gives a concrete lattice construction for the fractionalized Fermi liquid by letting conduction electrons hybridize with Majorana fermions from a Yao-Lee Z2 spin liquid. The gauge field is taken static, which lets them solve the hybridization to leading-log accuracy and extract explicit momentum-dependent coherence factors. Those factors produce arc-like segments on a small Fermi surface that violates the naive Luttinger count, while the decoupled phase keeps a large surface. They also track the thermodynamic signatures of fluctuations, including a strong diamagnetic response and a logarithmically diverging Sommerfeld coefficient at the pseudogap onset. That combination of analytic control and direct contact with cuprate phenomenology is the useful part. The construction is a clear extension of earlier ancilla and Kondo-lattice ideas, and the coherence-factor calculation is new enough to be worth testing against other approaches. The central limitation is exactly the static-gauge assumption. The model works because gauge fluctuations are frozen by hand; the paper does not derive a gap for those modes or show that their coupling to the hybridization vertex remains irrelevant at the scales where the arcs form. If the modes stay gapless or only marginally gapped, they can renormalize the vertex and either restore a conventional Luttinger count or smear the arcs. Without those checks the leading-log results stay provisional. The work is aimed at theorists who build solvable models for the pseudogap and Fermi-arc physics. Readers already working on Z2 spin liquids or fractionalized phases will get the most out of it, because they can judge whether the static approximation can be relaxed or justified. It is coherent on its own terms and engages the literature honestly, so it deserves a serious referee to verify the derivations and assess the gauge stability.

Referee Report

1 major / 1 minor

Summary. The paper constructs an analytically tractable model for a fractionalized Fermi liquid (FL*) on the square lattice by coupling conduction electrons to a Yao-Lee Z2 spin liquid possessing a Majorana Fermi surface. With the associated Z2 gauge theory taken to be static, the model is solved to leading-logarithmic accuracy and exhibits two phases: a hybridized phase in which the fractionalized fermions and conduction electrons form a common Fermi surface that violates the naive Luttinger count, and a decoupled phase. In the small-FS hybridized phase the authors derive momentum-dependent coherence factors that produce Yang-Rice-Zhang-like Fermi arcs; they also discuss fluctuation effects including a strong diamagnetic response and a logarithmically divergent Sommerfeld coefficient near the pseudogap onset.

Significance. If the static-gauge assumption can be justified, the construction supplies one of the few microscopic, analytically controlled realizations of an FL* state. The explicit momentum-dependent coherence factors and the thermodynamic signatures (diamagnetism, log-divergent gamma) constitute falsifiable predictions that could be compared with ARPES and specific-heat data in the cuprates. The leading-log solvability and the clean separation into hybridized versus decoupled regimes are genuine strengths.

major comments (1)

[Abstract and model-construction section] Abstract and model-construction section: the central claim that the Z2 gauge theory is static (thereby permitting an analytic solution to leading-log accuracy and the derivation of the hybridized small-FS phase with Luttinger violation) is load-bearing. No explicit argument or calculation is supplied showing that gauge fluctuations acquire a gap or remain irrelevant at the hybridization scale; if the gauge modes are gapless or only marginally gapped they can renormalize the Majorana-conduction vertex and potentially restore a conventional Luttinger count or suppress the reported coherence-factor arcs.

minor comments (1)

[Abstract] Abstract: 'leading-logarithic' is a typographical error and should read 'leading-logarithmic'.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for their constructive feedback. We are pleased that the referee recognizes the potential significance of our analytically tractable model for the FL* state. Below we provide a point-by-point response to the major comment.

read point-by-point responses

Referee: [Abstract and model-construction section] Abstract and model-construction section: the central claim that the Z2 gauge theory is static (thereby permitting an analytic solution to leading-log accuracy and the derivation of the hybridized small-FS phase with Luttinger violation) is load-bearing. No explicit argument or calculation is supplied showing that gauge fluctuations acquire a gap or remain irrelevant at the hybridization scale; if the gauge modes are gapless or only marginally gapped they can renormalize the Majorana-conduction vertex and potentially restore a conventional Luttinger count or suppress the reported coherence-factor arcs.

Authors: We appreciate the referee highlighting this important point. The static gauge approximation is indeed central to the analytic tractability of the model. In the Yao-Lee Z2 spin liquid, the gauge sector is gapped due to the topological order, with vison excitations carrying a finite gap. We expect this gap to persist upon hybridization with the conduction electrons at the mean-field level, rendering gauge fluctuations irrelevant below the vison gap and justifying the static treatment to leading-log accuracy. However, we acknowledge that an explicit calculation of the gauge fluctuation spectrum in the hybridized phase is not provided in the current manuscript. In the revised version, we will add a paragraph in the model-construction section discussing the gapped nature of the Z2 gauge field and its expected irrelevance to the hybridization physics. This addition will not change the main results but will strengthen the justification. revision: yes

Circularity Check

0 steps flagged

Model derivation self-contained under explicit static gauge assumption

full rationale

The paper constructs its FL* model from the established Yao-Lee Z2 spin liquid plus Kondo-like hybridization, then explicitly assumes a static Z2 gauge theory to enable analytic solution to leading-log accuracy. The two phases, Luttinger violation in the hybridized case, and momentum-dependent coherence factors producing YRZ-like arcs all follow directly from diagonalizing or perturbatively solving this Hamiltonian under the stated assumption; no parameter is fitted to the target observables and then relabeled as a prediction, no self-citation supplies a uniqueness theorem that forces the result, and no ansatz is smuggled via prior work. The static-gauge step is an input assumption rather than a derived claim that reduces to itself, so the derivation chain remains non-circular.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on the domain assumption that the Z2 gauge theory remains static and on the prior existence of the Yao-Lee spin liquid; limited information from the abstract prevents exhaustive enumeration of parameters.

free parameters (1)

hybridization coupling strength
A parameter controlling the mixing between conduction electrons and spin-liquid fermions that determines which phase is realized.

axioms (1)

domain assumption The Z2 gauge theory is static
This assumption is invoked to enable the leading-logarithmic analytic solution.

invented entities (1)

hybridized small-Fermi-surface FL* phase no independent evidence
purpose: To realize Luttinger-count violation and Fermi arcs within the model
The phase is defined by the hybridization in the model; no independent falsifiable signature outside the model is stated.

pith-pipeline@v0.9.0 · 5533 in / 1435 out tokens · 94663 ms · 2026-05-10T18:32:09.568694+00:00 · methodology

discussion (0)

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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

The associated Z2 gauge theory is static so that the model can be analytically solved to leading-logarithmic accuracy... analytically derived momentum dependent coherence factors responsible for the appearance of Fermi arcs à la Yang-Rice-Zhang.
IndisputableMonolith/Foundation/DimensionForcing.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the sum runs over eight equivalent hot spots, yielding an overall prefactor 8

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
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.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

39 extracted references · 11 canonical work pages · 3 internal anchors

[1]

DOS” contribution); the phase coherence “Maki-Thompson

The mass acquires a temperature dependence from m(T) =J −1 −Π(0,0;T), and exhibits a crossover from m(T)∝T 2 at the lowest temperatures to an approxi- mately linear-in-Tbehavior at higherT. FIG. 4.f(Zm(T)/2T) in Eq. (31) vs.Tat the QCP. The parameters aret ′ = 0.3,t= 2K= 1 andµ= 0.1 (Zm≈ −1.6 + (2.56 + 100T2)1/2; orange) andµ= 0.2 (Zm≈ −1.5 + (2.25 + 196T...
[2]

Foundations For fixed gauge, e.g.u ij = 1, we obtain the following fermionic mean field Hamiltonian ˆHMF = X k [c† kσϵkck,σ +f † kσKkfk,σ] + X k [V c† kσfk+Q,σ +H.c.] + X k [∆c† k,σ(iσy)σσ ′f+ −k,σ′ +H.c.] (B1) We introduce spinors Ψk =   ck,↑ c+ −(k+Q),↓ fk+Q,↑ f+ −k,↓   (B2) to rewrite ˆH= X k Ψ† k   ϵk 0V∆ 0−ϵ −(k+Q) ∆∗ −V ∗ V ∗ ∆K k+Q 0 ∆∗ ...
[3]

(B13) leading to an approximate critical temperature at Tc = √ Kt 2 e − r π2 (t+K) 8J −ln2 √ K t .(B14)

Quadratic terms in free energy The quadratic terms in Φ in the free energy are f(2) = Φ2 J h 1− X τ Z k J 2(ϵk +τ δ k +K k) ×[tanh( ϵk +τ δ k 2T ) + tanh(Kk 2T )] i .(B8) This expression is evaluated most easily in the case when t′ = 0, in which case we can use the dimensionless density of states ν(x) = 2 π2 K( p 1−x 2) |x|≪1 ≃ 2 π2 ln(1/|x|) (B9) to expr...
[4]

Senthil, S

T. Senthil, S. Sachdev, and M. Vojta, Fractionalized fermi liquids, Physical Review Letters90, 216403 (2003)

2003
[5]

P. M. Bonetti, M. Christos, A. Nikolaenko, A. A. Patel, and S. Sachdev, Critical quantum liquids and the cuprate high temperature superconductors, arXiv preprint (2025), 2508.20164

work page internal anchor Pith review Pith/arXiv arXiv 2025
[6]

H.-B. Yang, J. D. Rameau, P. D. Johnson, T. Valla, A. Tsvelik, and G. D. Gu, Emergence of preformed cooper pairs from the doped mott insulating state in bi2sr2cacu2o8+δ, Nature456, 77 (2008)

2008
[7]

M. K. Chan, K. A. Schreiber, O. E. Ayala-Valenzuela, E. D. Bauer, A. Shekhter, and N. Harrison, Observation of the yamaji effect in a cuprate superconductor, Nature Physics21, 1753 (2025)

2025
[8]

Senthil, S

T. Senthil, S. Sachdev, and M. Vojta, Fractionalized fermi liquids, Physical Review Letters90, 216403 (2003), cond-mat/0209144

work page internal anchor Pith review arXiv 2003
[9]

Senthil, M

T. Senthil, M. Vojta, and S. Sachdev, Weak mag- netism and non-fermi liquids near heavy-fermion criti- cal points, Physical Review B69, 035111 (2004), cond- mat/0305193

work page arXiv 2004
[10]

Paramekanti and A

A. Paramekanti and A. Vishwanath, Extending lut- tinger’s theorem to z2 fractionalized phases of mat- ter, Physical Review B70, 245118 (2004), cond- mat/0406619

work page arXiv 2004
[11]

Grover and T

T. Grover and T. Senthil, Quantum phase transition from an antiferromagnet to a spin liquid in a metal, Physical Review B81, 205102 (2010), 0910.1277

work page arXiv 2010
[12]

2016.arXiv e-prints:arXiv:1601.07902

P. Bonderson, M. Cheng, K. Patel, and E. Plamadeala, Topological enrichment of luttinger’s theorem, arXiv e- prints (2016), 1601.07902

work page arXiv 2016
[13]

A. M. Tsvelik, Fractionalized fermi liquid in a kondo- heisenberg model, Physical Review B94, 165114 (2016), 1604.06417

work page arXiv 2016
[14]

Affleck and J

I. Affleck and J. B. Marston, Large-n limit of the heisenberg-hubbard model: Implications for high-tc su- perconductors, Physical Review B37, 3774 (1988)

1988
[15]

Read and S

N. Read and S. Sachdev, Spin-peierls, valence-bond solid, and n´ eel ground states of low-dimensional quantum an- tiferromagnets, Physical Review B42, 4568 (1990)

1990
[16]

Kitaev, Anyons in an exactly solved model and be- yond, Annals of Physics321, 2 –111 (2006)

A. Kitaev, Anyons in an exactly solved model and be- yond, Annals of Physics321, 2 –111 (2006)

2006
[17]

Trebst and C

S. Trebst and C. Hickey, Kitaev materials, Physics Re- ports950, 1 (2022)

2022
[18]

A microscopic model of fractionalized Fermi liquid

P. Coleman, A. Panigrahi, and A. Tsvelik, A microscopic model of a fractionalized fermi liquid, arXiv2511.01115

work page internal anchor Pith review Pith/arXiv arXiv
[19]

Coleman, A

P. Coleman, A. Panigrahi, and A. Tsvelik, Solvable 3d kondo lattice exhibiting pair density wave, odd-frequency pairing, and order fractionalization, Physical Review Let- ters129, 177601 (2022). 11

2022
[20]

Yao and D.-H

H. Yao and D.-H. Lee, Fermionic magnons, non-abelian spinons, and the spin quantum hall effect from an ex- actly solvable spin-1/2 kitaev model with su(2) symme- try, Phys. Rev.Lett107, 087205 (2011)

2011
[21]

Chulliparambil, U

S. Chulliparambil, U. F. P. Seifert, M. Vojta, L. Janssen, and H.-H. Tu, Microscopic models for kitaev’s sixteenfold way of anyon theories, Physical Review B102, 201111 (2020)

2020
[22]

K.-Y. Yang, T. M. Rice, and F.-C. Zhang, Phenomeno- logical theory of the pseudogap state, Phys. Rev. B73, 174501 (2006)

2006
[23]

Michon, C

B. Michon, C. Girod, S.Badoux, J. Kaˇ cmarˇ c´ ık, Q. Ma, M. Dragomir, H. A. Dabkowska, B. D. Gaulin, J.- S. Zhou, S. Pyon, T. Takayama, H. Takagi, S. Ver- ret, N. Doiron-Leyraud, C. Marcenat, L. Taillefer, and T. Klein, Thermodynamic signatures of quantum criti- cality in cuprates, Nature567, 218 (2018)

2018
[24]

Zhang and S

Y.-H. Zhang and S. Sachdev, From the pseudogap metal to the fermi liquid using ancilla qubits, Physical Review Research2, 023172 (2020), 2001.09159

work page arXiv 2020
[25]

Flint, P

R. Flint, P. Coleman, and P. Coleman, Symplectic nand time reversal in frustrated magnetism, Phys. Rev. B79, 014424 (2009)

2009
[26]

Khodas, H.-B

M. Khodas, H.-B. Yang, J. Rameau, P. D. Johnson, and A. Tsvelik, Analysis of the quasiparticle spectral function in the underdoped cuprates, arXiv1007.4837(2010)

work page arXiv 2010
[27]

Abrikosov, Aleksei, L

A. Abrikosov, Aleksei, L. P. Gorkov, I. E. Dzyaloshinski, R. A. Silverman, and G. H. Weiss, Methods of quantum field theory in statistical physics (1964)

1964
[28]

Dzyaloshinskii, Some consequences of the luttinger the- orem: The luttinger surfaces in non-fermi liquids and mott insulators, Phys

I. Dzyaloshinskii, Some consequences of the luttinger the- orem: The luttinger surfaces in non-fermi liquids and mott insulators, Phys. Rev. B68, 085113 (2003)

2003
[29]

Fabrizio, Spin-liquid insulators can be landau’s fermi liquids, Phys

M. Fabrizio, Spin-liquid insulators can be landau’s fermi liquids, Phys. Rev. Lett.130, 156702 (2023)

2023
[30]

Coleman, A

P. Coleman, A. Kuklov, and A. M. Tsvelik, Dual view of the z2-gauged xy model in 3d, Phys. Rev. Lett.134, 236001 (2025)

2025
[31]

Panigrahi, A

A. Panigrahi, A. Tsvelik, and P. Coleman, Breakdown of order fractionalization in the cpt model, Physical Review B110, 104520 (2024)

2024
[32]

Panigrahi, A

A. Panigrahi, A. Tsvelik, and P. Coleman, Microscopic theory of pair density waves in spin-orbit coupled kondo lattice., Physical review letters135, 046504 (2025)

2025
[33]

A. J. Millis, Effect of a nonzero temperature on quantum critical points in itinerant fermion systems, Phys. Rev. B 48, 7183 (1993)

1993
[34]

Abanov, A

A. Abanov, A. V. Chubukov, and J. Schmalian, Quantum-critical theory of the spin-fermion model and its application to cuprates: Normal state analysis, Ad- vances in Physics52, 119 (2003)

2003
[35]

M. A. Metlitski and S. Sachdev, Quantum phase transi- tions of metals in two spatial dimensions. ii. spin density wave order, Phys. Rev. B82, 075128 (2010)

2010
[36]

Zhang and S

Y.-H. Zhang and S. Sachdev, Deconfined criticality and ghost fermi surfaces at the onset of antiferromagnetism in a metal, Phys. Rev. B102, 155124 (2020), 2006.01140

work page arXiv 2020
[37]

Sakai, M

S. Sakai, M. Civelli, and M. Imada, Hidden fermionic excitation boosting high-temperature superconductivity in cuprates, Phys. Rev. Lett116, 057003 (2016)

2016
[38]

Imada and T

M. Imada and T. J. Suzuki, Excitons and dark fermions as origins of mott gap, pseudogap and superconductivity in cuprate superconductors - general concept and basic formalism based on gap physics, J. Phys. Soc. Jpn.88, 024701 (2019)

2019
[39]

Rossi, F

R. Rossi, F. S. IV, M. Ferrero, A. Georges, A. M. Tsvelik, N. V. Prokof’ev, and I. S. Tupitsyn, Interaction-enhanced nesting in spin-fermion and fermi-hubbard models, Phys. Rev. Research6, L032058 (2024)

2024