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Topic Review: Hatsugai-Kohmoto models: Exactly solvable playground for Mottness and Non-Fermi Liquid

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Pith's one-line read This review shows that the Hatsugai-Kohmoto model is exactly solvable and exhibits both a paramagnetic Mott insulator at half filling and a metal that violates Luttinger's theorem.

desk verdict A genuinely useful review of the HK model with sound exact results in Section 2, but the Hartree-Fock exactness proof in Section 3.1 switches ground states without comment and needs a fix. read the letter →

arxiv 2501.00388 v1 pith:MYY4RWKA submitted 2024-12-31 cond-mat.str-el

Miaomiao Zhao , Wei-Wei Yang , Yin Zhong This is my paper · ORCID
classification cond-mat.str-el
keywords Hatsugai-Kohmotomodelnon-FermiliquidMottinsulatorLuttingertheoremviolationexactlysolvableexclusionstatisticsmomentum-spacefactorizationstronglycorrelatedelectrons
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This pedagogic review establishes that the Hatsugai-Kohmoto (HK) model, an interacting electron model with an infinite-ranged interaction that conserves the center of mass, is exactly solvable in any dimension and at any filling, and that this exact solubility provides a controlled setting for Mottness and non-Fermi-liquid physics. The key step is that Fourier transformation reduces the interaction to $U\sum_k n_{k\uparrow}n_{k\downarrow}$, so the Hamiltonian is a sum of independent four-state momentum atoms and every eigenstate is a product over momenta. In the metallic regime the ground-state occupation has two jumps, defining two quasi-Fermi wavevectors; the electron density is therefore not fixed by one Fermi surface, the Luttinger theorem fails, and the metal is identified as a non-Fermi liquid. At half filling with interaction $U$ larger than bandwidth $W$, the exact ground state is a paramagnetic Mott insulator with gap $U-W$, zero thermodynamic charge susceptibility, and Curie-like spin susceptibility. The review argues that weak short-ranged Hubbard perturbations leave this fixed point stable, so the exact HK solution is a reliable starting point for strongly correlated electron physics beyond the solvable limit.

What carries the argument

The load-bearing mechanism is the center-of-mass conservation $\delta_{j_1+j_3=j_2+j_4}$ in the real-space interaction. After Fourier transformation this delta forces all four momenta of a scattering event to be equal, turning the interaction into $U\sum_k n_{k\uparrow}n_{k\downarrow}$. Each momentum sector then has a four-dimensional Hilbert space, the sectors commute, and the full Hamiltonian is a product over sectors; the exact ground state is built by choosing for each $k$ the lowest of the four local levels. All exact results in the review, including the distribution function, Green function, thermodynamics, susceptibilities, and the phase diagram with Mott insulator and non-Fermi-liquid metal, are derived from this factorization, and it also makes perturbation theory and renormalization-group calculations around the solvable limit well-defined.

What would settle it

Exact-diagonalize the HK model on an open-boundary chain at half filling with $U>W$: if the ground state develops long-range ferromagnetic order rather than the paramagnetic single-occupation product state, then the review's central Mott-insulator picture is a periodic-boundary artifact. Separately, a lattice calculation with one flux quantum per plaquette would settle whether the claimed non-Fermi-liquid quantum oscillations survive beyond the continuum Landau-level approximation.

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Extended reading notes

Core claim

The paper's central claim is that the Hatsugai-Kohmoto Hamiltonian, with the infinite-ranged interaction that conserves the center of mass, becomes $$H_{\mathrm{HK}}=\sum_k \left[(\varepsilon_k-\mu)(n_{k\uparrow}+n_{k\downarrow})+U\,n_{k\uparrow}n_{k\downarrow}\right]$$ after Fourier transformation, so that the problem is a sum of independent momentum sectors, each a four-state momentum atom (empty, spin-up, spin-down, doubly occupied), and every eigenstate is a product over momenta. This exact factorization yields two exact phases: at half filling with $U>W$ (bandwidth $W$), the ground state is a paramagnetic Mott insulator with excitation gap $U-W$, while away from that regime the metal has an electron distribution with two jumps, two quasi-Fermi wavevectors $k_{F1}, k_{F2}$, and a density given by the sum of those wavevectors, violating the Luttinger theorem and identifying the metal as a non-Fermi liquid. The exact single-particle Green function is the sum of holon and doublon contributions, and those quasiparticles obey Haldane exclusion statistics; thermodynamics and charge and spin responses follow exactly. The paper further argues that this solvable fixed point survives weak short-ranged Hubbard perturbations, with forward and exchange channels marginal and the attractive BCS channel driving superconductivity, which is why the model can serve as a playground for Mottness and non-Fermi-liquid physics.

Load-bearing premise

The exact results all assume the interaction is strictly infinite-ranged and center-of-mass-conserving under periodic boundary conditions; once open boundaries or finite magnetic flux per plaquette are introduced, the solvability is lost and the paper notes ferromagnetic correlations appear, so the clean two-Fermi-surface non-Fermi liquid and paramagnetic Mott insulator may not be generic outside this idealized setting.

Editorial extensions

If this is right

  • If the central claim is correct, a non-Fermi liquid can arise in a translation-invariant interacting electron model without disorder or large-$N$ limits, and the two-jump electron distribution is a sharp exact signature of that state.
  • The interaction-driven Mott transition at $U=W$ and the chemical-potential-driven transition belong to the same free-fermion Lifshitz universality class, with critical exponents controlled by the single-particle density of states.
  • The metallic phase violates Luttinger's theorem because the electron density is set by two quasi-Fermi wavevectors rather than one Fermi surface, so the quasiparticle weight remains finite while the zero-frequency self-energy diverges at the Luttinger surface.
  • Weak short-ranged Hubbard repulsion leaves the HK fixed point stable, while attractive pairing in the BCS channel drives a superconducting instability with doublon-holon pairing, giving a concrete reference for doped Mott insulators.
  • In the non-Fermi-liquid phase, magnetic quantum oscillations survive and at strong fields become aperiodic and violate Onsager's relation, providing an observable distinction from Landau quasiparticle metals.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural extension the review leaves implicit is to use the exact two-jump distribution as a benchmark for proposed non-Fermi-liquid diagnostics, such as Green's-function zeros or entanglement signatures, since all correlation functions are known exactly in this model.
  • The boundary sensitivity reported in the review suggests that in finite clusters or open-boundary settings, HK Mott physics is replaced by ferromagnetism; one could test whether adding a weak antiferromagnetic coupling restores a paramagnetic Mott state and connects the model to Hubbard-model behavior.
  • The unresolved $q\to 0$ singularity of the charge susceptibility hints that infinite-ranged interactions violate the standard $f$-sum rule; a useful test is to compute the $f$-sum rule for a finite-range center-of-mass-conserving interaction and see whether the anomaly disappears smoothly as the range shrinks.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. This is a pedagogic topical review of the Hatsugai-Kohmoto (HK) model and its extensions. The paper derives the momentum-space decoupling of the HK Hamiltonian, the four-state structure of each momentum sector, the ground-state phase diagram, the finite-temperature partition function, the electron distribution function, the single-particle Green function, the zero-temperature charge susceptibility, and the exclusion-statistics interpretation of the thermodynamics. It then surveys perturbative treatments around the HK limit, a renormalization-group analysis of the stability of the HK fixed point, exact-diagonalization results for open boundary conditions, and extensions to Fermi arcs, quantum oscillations, and superconductivity. The central exact statements are that the half-filled U>W state is a paramagnetic Mott insulator and that the metallic phase is a non-Fermi liquid with two quasi-Fermi surfaces and a violation of the Luttinger theorem.

Significance. If the claims are taken as stated, the review is a useful and largely reliable synthesis of an active literature. Its main strengths are the exact, parameter-free derivations in Sections 2.2-2.6 and Appendix A, which are algebraically checkable, and its explicit flagging of open issues: the OBC-induced ferromagnetic correlations (Sections 1.2 and 3.3), the absence of lattice calculations for quantum oscillations (Section 4.2), and the Guerci/Ma controversy over long-wavelength charge response (Sections 2.6 and 5). The review also carefully notes the ground-state degeneracy and the need to average over degenerate states for paramagnetic quantities. However, the Hartree-Fock exactness claim in Section 3.1 contains an unflagged state-selection inconsistency that must be corrected before the review can serve as a reliable pedagogical reference for perturbative treatments of the HK model.

major comments (1)
  1. [§3.1, Eqs. (46)-(48)] The proof that the Hartree-Fock Hamiltonian H0 reproduces the exact single-particle Green function silently replaces the paramagnetic mixed ground state used in Section 2.3 and Eq. (45) by a fully spin-polarized pure state in the singly occupied regime εk<μ<εk+U. For the paramagnetic ensemble, ⟨n_{k↑}⟩=⟨n_{k↓}⟩=1/2 and the exact Green function has two poles each with weight 1/2, whereas the paramagnetic H0 would give a single pole at ω=εk−μ+U/2. Setting ⟨n_{k↑}⟩=1 and ⟨n_{k↓}⟩=0 in Eq. (48) makes G0 equal to the exact Green function only for one particular degenerate ferromagnetic member of the ground-state manifold. The subsection should be rewritten to state explicitly that the HF starting point is defined at a symmetry-broken pure state, not at the paramagnetic ensemble used elsewhere in the review, and the subsequent perturbation theory should be identified as a perturbation around that member.
minor comments (4)
  1. [§2.5] In the operator definitions preceding Eq. (33), ĥ_{k↓} is defined as ĉ_{k↑}(1−n̂_{k↑}); this appears to be a typo and should read ĉ_{k↓}(1−n̂_{k↑}) or the analogous correct operator.
  2. [§2.4.1, Eq. (26)] The symbol LI is used without definition; please define it as a Luttinger integral so that the equality with the right-hand side of Eq. (15) is transparent.
  3. [§2.3.1, Eq. (14)] The displayed density formula is typeset in a way that obscures the factors of 2 and the momentum-space measure; adding a sentence that the double-occupied interval carries two electrons per mode and the single-occupied interval carries one electron per mode would greatly improve readability.
  4. [§3.1, Eq. (43)] The definitions of H1 and H2 with opposite signs are central to the claimed cancellation of the Hartree contributions; a short explanation of the sign choice in H1 would help readers who are not already familiar with the Wang-Yang construction.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the exact HK-model derivations are self-contained and externally checkable; the Section 3.1 state-selection gap is a consistency issue rather than a circular reduction.

full rationale

The paper is a review that re-derives, rather than imports, the central exact results of the HK model. The diagonalization in Eqs. (1)-(3) is a direct Fourier transform of the center-of-mass-conserving interaction; Eq. (3) is a sum over independent four-state momentum sectors. From this alone the eigenstates, T=0 distribution function (Eq. 11), thermodynamics (Eqs. 16-21), and single-particle Green function (Eq. 25) follow by closed algebra with no adjustable parameters. The two-particle susceptibility is re-derived in Appendix A by equations of motion and reduces to Eq. (38), so the earlier statement that Wick's theorem effectively holds is a derived result, not an input. The only self-citations (Refs. [81], [98], [102]) point to the authors' earlier papers, but the load-bearing content of [98] is re-derived here and [102] is presented as reviewed work with an independent companion [103]; no central premise rests on an unverified self-citation. The noted state-selection issue in Section 3.1 is real: exactness of the Hartree-Fock starting point is demonstrated using a Zeeman-polarized member of the degenerate singly-occupied manifold (⟨n↑⟩=1, ⟨n↓⟩=0), whereas Sections 2.3-2.6 use the paramagnetic mixed state with ⟨n↑⟩=⟨n↓⟩=1/2. This is an internal logical inconsistency in the perturbation-theory setup, but it is not circularity: the exact Green function (45) is derived independently, and Eqs. (46)-(48) verify an identity for one member of the degenerate manifold rather than producing the paper's main results from fitted inputs. Limitations (OBC effects, missing lattice flux calculations) are explicitly acknowledged and do not affect the self-contained nature of the exact solution.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The review introduces no new free parameters or invented entities. All numbers are model inputs (t, U, μ) or previously derived critical values such as U_c = W. The central results rest on standard exact-diagonalization algebra plus background assumptions about Luttinger theorem, ground-state averaging, and exclusion statistics.

assumptions (4)
  • domain assumption Luttinger theorem: for a Fermi liquid the volume enclosed by the Fermi surface is fixed by particle density and is encoded in Re G(k,0)=0; its violation signals non-Fermi liquid behavior.
    Invoked in Section 2.3.1 (Eq. 15) and Section 2.4.1 to classify the HK metallic state as non-Fermi liquid through the two-jump distribution and the Green-function zeros.
  • domain assumption Zero-temperature thermodynamic quantities are obtained by averaging over all degenerate ground states, i.e., treating the ground state as a mixed state.
    Section 2.3.1 uses this averaging to get n_{kσ}=1/2 in singly occupied regions; the distribution function, density, and charge susceptibility depend on this choice.
  • domain assumption The Haldane-Wu exclusion statistics formalism, with the statistical matrix from Ref. [68], maps the HK Hilbert space to non-interacting holon, spin-down holon, and doublon species and reproduces the free energy.
    Section 2.5 relies on this mapping to interpret the thermodynamics; the review cites the original references rather than proving the general Wu equation.
  • standard math Equation-of-motion hierarchies for retarded Green functions close for charge correlations because products of occupation operators have closed commutators under the HK interaction.
    Appendix A derives χ_c(q,ω) using closure of the hierarchy; this is an algebraic property of the model, presented as an exact calculation.

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Cite this review

Pith. "Pith review of Topic Review: Hatsugai-Kohmoto models: Exactly solvable playground for Mottness and Non-Fermi Liquid." pith.science (2026). https://pith.science/paper/MYY4RWKA

@misc{pith2026250100388,
  author       = {Pith},
  title        = {Pith review of: Topic Review: Hatsugai-Kohmoto models: Exactly solvable playground for Mottness and Non-Fermi Liquid},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MYY4RWKA}},
  note         = {Machine review of arXiv:2501.00388}
}
read the original abstract

This pedagogic review aims to give a gentle introduction to an exactly solvable model, the Hatsugai-Kohmoto (HK) model, which has infinite-ranged interaction but conserves the center of mass. Although this model is invented in 1992, intensive studies on its properties ranging from unconventional superconductivity, topological ordered states to non-Fermi liquid behaviors are made since 2020. We focus on its emergent non-Fermi liquid behavior and provide discussion on its thermodynamics, single-particle and two-particle correlation functions. Perturbation around solvable limit has also been explored with the help of perturbation theory, renormalization group and exact diagonalization calculation. We hope the present review will be helpful for graduate students or researchers interested in HK-like models or more generic strongly correlated electron systems.

Figures

Figures reproduced from arXiv: 2501.00388 by the authors.

Figure 1
Figure 1. The Luttinger theorem for one and two-dimensional fermion systems, which is valid for FL and Luttinger liquid. 1. Introduction 1.1. Landau’s Fermi liquid and non-Fermi liquid The Landau’s Fermi liquid (FL) theory has been the cornerstone of modern condensed matter physics for several decades. [1–4] In this paradigm, the low-energy physics of interacting many fermions system is described by weakly interacting quasipa… view at source ↗
Figure 2
Figure 2. (Left) The HK model defined on one-dimensional chain and square lattice, noting the conservation of center of mass before and after electrons interact. (Right) Ground-state phase diagram of HK model on hypercubic lattice, Uc = W and W denotes the non-interacting electron band-width. To give more insight on the above interaction, consider just two-site case, Hˆ U = U 2  cˆ † 1↑ cˆ1↑cˆ † 1↓ cˆ1↓ + ˆc † 1↑ cˆ1↑cˆ † 2↓… view at source ↗
Figure 3
Figure 3. The empty (Ω0), single (Ω1) and double (Ω2) occupation regime for HK model in momentum space. Here, a d = 1 HK system is assumed and Ek+ = εk − µ + U, Ek− = εk − µ are quasi-particle bands. degenerated, which means the eigenstates are highly-degenerated, including the ground￾states. The degeneracy can be lifted due to Zeeman energy if an infinitesimal uniform magnetic field is applied, then a unique ferromagnetic gr… view at source ↗
Figures from the paper (16 more)
Figure 4
Figure 4. Figure 4: Band structure of HK model for different parameters, (A) U > W = Uc, µ = U/2,(B) U < W, µ = U/2,(C) U < W, µ = 0 and (D) U < W, µ = U. The Mott insulator has definite gap between two bands while the bands in metallic state are partially occupied. which does not depend …
Figure 5
Figure 5. Figure 5: The ground-state electron occupation nkσ = ⟨cˆ † kσcˆkσ⟩. (a) Mott insulator with U > W = 4t; (b) Metal state for U < W = 4t. For comparison, Fermi gas with U/t = 0 is also shown. From [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]
Figure 6
Figure 6. Figure 6: (Left) The metallic state’s electron distribution nkσ = ⟨cˆ † kσcˆkσ⟩ of HK model on square lattice with two jumps; (Right) The corresponding quasi-Fermi surface structure. (U/t = 2, µ = U/2) electron density cannot be uniquely determined by quasi-Fermi wavevector kF2,…
Figure 7
Figure 7. Figure 7: (Left) The specific heat Cv and (Right) spin susceptibility χs of one￾dimensional HK model with µ = U/2 [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: (Left) The one-dimensional HK model’s electron density n and (Right) the charge susceptibility χc at T = 0. At T = 0, the ground-state energy is contributed from two spectrums, E → Eg = X k [(εk − µ)θ(µ − εk) + (εk − µ + U)θ(µ − εk − U)] . (18) Then, the electron densi…
Figure 9
Figure 9. Figure 9: (a) Electron distribution function nkσ for d = 1 HK model; (b) The corresponding real and imaginary part of Green function. 2.4.2. holon and doublon Because of the divergency of zero-frequency self-energy, FL weight Z loses its meaning, and the system is in NFL, namely…
Figure 10
Figure 10. Figure 10: Density of state of half-filled HK model on square lattice for different interaction U. 2.4.3. Density of state The density of state of electrons is obtained by the single￾particle Green function as N(ω) = − 1 πN P k ImGσ(k, ω) [PITH_FULL_IMAGE:figures/full_fig_p022_…
Figure 11
Figure 11. Figure 11: Density of state of half-filled HK model on square lattice for different interaction U. Here, the quasiparticle-like peak is more visible though it does not come from the Kondo effect. into non-interacting models with the elementary excitations obeying generalized Pau…
Figure 12
Figure 12. Figure 12: δni calculated from the linear-response theory for U/t = 0, 1, 2, 3, 4, 5, 6, 8 with V /t = 0.1 and µ = U/2. − 1 4Ns X k,σ ∂fF (εk − µ + U) ∂εk . (39) For metallic state with U < W, all terms in χc are nonzero, thus χc is finite. For Mott insulator, the lower band is …
Figure 13
Figure 13. Figure 13: Interaction vertex induced by HK and Hubbard interaction. (a) H1, (b) H2, (c) HH. The self-energy correction of (a) and Hartree term (b) cancel out, which is expressed in (d). It is seen that these two ones are identical. For εk > µ, εk + U > µ, namely εk > µ, we have…
Figure 14
Figure 14. Figure 14: Ladder diagrams in the particle-particle channel. + + + 𝑝1 0𝐩 ↑ + 𝑝3 0𝐩 ↑ 𝑝3 0𝐩 ↑ 𝑝1 0 − 𝑠 0𝐩 ↑ 𝑝1 0𝐩 ↑ 𝑝1 0𝐩 ↑ 𝑝3 0𝐩 ↑ 𝑝2 0𝐩 ↓ 𝑝4 0𝐩 ↓ Γ𝐩𝐡 = = + ⋯ = + ⋯ 𝑝1 0 − 𝑝3 0 𝑝2 0𝐩 ↓ 𝑝4 0 + 𝑠 0𝐩 ↓ 𝑝4 0𝐩 ↓ = 𝑝4 0𝐩 ↓ 𝑝2 0 𝑈 𝐩 ↓ 𝑠 0 𝑈 + [PITH_FULL_IMAGE:figures/full_fig_p030_14.png]
Figure 15
Figure 15. Figure 15: Ladder diagrams in the particle-hole channel. = − U⟨nˆkσ¯⟩ + U =⟨nˆkσ¯ ⟩ z }| { θ(µ − εk − U⟨nˆkσ⟩) = 0. (49) Furthermore, Wang and Yang find the contribution of Hˆ 2 can be written as the summation of ladder diagram in particle-particle and particle-hole channel. It …
Figure 16
Figure 16. Figure 16: Second-order self-energy diagrams with Hubbard interaction UH. Γpp(p 0 1 − p 0 2 , p) = U(1 − ⟨nˆpσ⟩)(1 − ⟨nˆpσ¯⟩) 1 − U i(p 0 1+p 0 2 )−(Epσ+Epσ¯ ) + U⟨nˆpσ⟩⟨nˆpσ¯⟩ 1 + U i(p 0 1+p 0 2 )−(Epσ+Epσ¯ ) , Γph(p 0 1 − p 0 4 , p) = U(1 − ⟨nˆpσ⟩)⟨nˆpσ¯⟩ 1 − U i(p 0 1−p 0 4 …
Figure 17
Figure 17. Figure 17: (a) Electrons near pseudo-Fermi surface with Fermi wavevector kL. (b) The RG flow of HK model with local interaction in BCS-pairing channel. The starting point of Phillips group is the path integral formalism of HK model, Z = Z DcDce ¯ −S , S = Z dτ X k "X σ c¯kσ(∂τ +…
Figure 18
Figure 18. Figure 18: Spin correlation function ⟨ ˆ S⃗ γ · ˆ S⃗ 0⟩ for half-filled HK and Hubbard model with OBC. (adapted from Ref. [101]) To discuss the stability of HK physics, we can inspect the system with Hubbard interaction. Particularly, because the strong coupling system has domin…
Figure 19
Figure 19. Figure 19: (Left) Electron distribution function nk; (Middle) Spectral function A(k, 0); (Right) ξk = 0, ξk+Uk = 0 and 2ξk+Uk = 0. (tx = ty = 1, µ = −0.1, U = 0.6, Tx = 1, Ty = 0.2) 4. Extension of HK model 4.1. Fermi arc A modified HK model proposed by Yang is able to generate …

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Forward citations

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