Twisting the Hubbard model into the momentum-mixing Hatsugai–Kohmoto model

Nature Physics (2025)Cite this article The Hubbard model is a standard theoretical tool for studying materials with strong electron–electron interactions, such as cuprate superconductors. Unfortunately, interaction-driven phenomena, such as a transition into the strongly correlated Mott insulator phase, are difficult to treat with established theoretical techniques. However, the exactly solvable Hatsugai–Kohmoto model displays similar Mott physics. Here we show how the Hatsugai–Kohmoto model can be deformed continuously into the Hubbard model. The trick is to systematically reintroduce all the momentum mixing that the original Hatsugai–Kohmoto model omits. This can be accomplished by grouping n momenta into a cell and hybridizing them, resulting in the momentum-mixing Hatsugai–Kohmoto model. We recover the Bethe ansatz ground-state energy of the one-dimensional Hubbard model to within 1% from only ten mixed momenta. Overall, the convergence scales as 1/n2 as opposed to the inverse linear behaviour of standard finite-cluster techniques. Our results for a square lattice reproduce all the known features from state-of-the-art simulations also with only a few mixed momenta. Consequently, we believe that the momentum-mixing Hatsugai–Kohmoto model offers an alternative tool for strongly correlated quantum matter.This is a preview of subscription content, access via your institution Access Nature and 54 other Nature Portfolio journals Get Nature+, our best-value online-access subscription $32.99 / 30 days cancel any timeSubscribe to this journal Receive 12 print issues and online access $259.00 per yearonly $21.58 per issueBuy this articleUSD 39.95Prices may be subject to local taxes which are calculated during checkoutThe data and scripts needed to reproduce the figures are openly available via Zenodo at https://doi.org/10.5281/zenodo.17096693 (ref. 64). Source data are provided with this paper.The ED code used for solving the 16-MMHK model is available via GitHub at https://github.com/wztzjhn/quantum_basis. The DMRG algorithm is implemented through TeNPy, which can be accessed at https://tenpy.readthedocs.io/en/v1.0.6/reference/tenpy.algorithms.dmrg.html. All other codes are available from the corresponding authors upon reasonable request.Lieb, E. H. & Wu, F. Y. Absence of Mott transition in an exact solution of the short-range, one-band model in one dimension. Phys. Rev. Lett. 20, 1445–1448 (1968).Article ADS Google Scholar Georges, A., Kotliar, G., Krauth, W. & Rozenberg, M. J. Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions. Rev. Mod. Phys. 68, 13 (1996).Article ADS MathSciNet Google Scholar Zheng, B.-X. et al. Stripe order in the underdoped region of the two-dimensional Hubbard model. Science 358, 1155–1160 (2017).Article ADS MathSciNet Google Scholar Huang, E. W., Mendl, C. B., Jiang, H.-C., Moritz, B. & Devereaux, T. P. Stripe order from the perspective of the Hubbard model. npj Quantum Mater. 3, 22 (2018).Article ADS Google Scholar Qin, M. et al. Absence of superconductivity in the pure two-dimensional Hubbard model. Phys. Rev. X 10, 031016 (2020).

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Xsede: accelerating scientific discovery. Comput. Sci. Eng. 16, 62–74 (2014).Article Google Scholar Mai, P. Data for Mai etal Twisting Hubbard into the Momentum-Mixing Hatsugai-Kohmoto Model. Zenodo https://doi.org/10.5281/zenodo.17096693 (2025).Download referencesWe thank B. Bradlyn for helpful comments, S. Zhang for sending us his double-occupancy data (Fig. 4a, green dots), G. Sawatzky for a helpful e-mail exchange and Z. Wang for his help in ED. This work was supported by the Center for Quantum Sensing and Quantum Materials, a Department of Energy, Energy Frontier Research Center, grant number DE-SC0021238 (P.M. and P.W.P.). P.W.P. also acknowledges NSF DMR-2111379 for partial funding of the HK work, which led to these results. P.M. was also supported by the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant number GBMF 8691. The DQMC calculation of this work used the Advanced Cyberinfrastructure Coordination Ecosystem: Services & Support (ACCESS) Expanse supercomputer through the research allocation TG-PHY220042, which is supported by National Science Foundation via grant number ACI-1548562 (ref. 63).Department of Physics and the Anthony J. Leggett Institute of Condensed Matter Theory, University of Illinois at Urbana-Champaign, Urbana, IL, USAPeizhi Mai, Jinchao Zhao, Gaurav Tenkila, Nico A. Hackner, Dhruv Kush, Derek Pan & Philip W. PhillipsDepartment of Physics, Hong Kong University of Science and Technology, Hong Kong, ChinaJinchao ZhaoSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarP.M. performed the calculations for the 4- and 16-MMHK models in two dimensions and analysed the data. J.Z. performed the calculations for the 2-MMHK model and carried out the analytic derivation. G.T. performed the DMRG calculations for MMHK in one dimension and ladder. N.A.H. performed the ED calculations for MMHK in one dimension. D.K. performed the calculations for local correlations in the band HK model. D.P. performed the calculations for spin correlations in the band HK model. P.W.P. supervised the project. P.M., J.Z., G.T. and P.W.P. wrote the paper with input from all authors.Correspondence to Peizhi Mai or Philip W. Phillips.The authors declare no competing interests.Nature Physics thanks the anonymous reviewers for their contribution to the peer review of this work.Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.a is reproduced from Fig. 1b for convenience. Alternative real-space construction by grouping the atoms into cells (encircled by green/red lines) with an updated lattice constant. b, Original unit cell in the band HK model with primitive unit cell in dashed line. c, The 2-MMHK model leads to a doubling of the unit cell size and the lattice constant (dashed line) d, The 4-MMHK model results in a quadrupling of the unit cell size. When n = N, the unit cell contains all the sites.Source dataComparison of the deviation in ground-state energy relative to the Bethe ansatz result for the TD limit (\({E}_{{\rm{HB}}}^{\infty }\)) among different model settings. a shows the deviation of the n-MMHK model (\({E}_{{\rm{MMHK}}}^{n}\)), which is consistently less than 1% for n≥10. b gives the deviation of a periodic n-site Hubbard chain (\({E}_{{\rm{HB}}-{\rm{PBC}}}^{n}\)), solved by ED, with the same legend as a. The scattered U-dependence for various n indicates non-systematic finite-size effects. c displays the deviation of an open Hubbard chain with length L (\({E}_{{\rm{HB}}-{\rm{OBC}}}^{L}\)), obtained from standard DMRG. d is similar to a but with larger n (the number of mixed momenta) and the n-MMHK models are solved by DMRG for n > 12. Comparing c and d, we find that n-MMHK exhibits more stable accuracy across the range of U, while open Hubbard chains show larger errors for small U.Source dataStandard DMRG results for an open finite Hubbard chain at U = 4 are shown as deviations from the Bethe-ansatz ground-state energy. The deviation scales linearly with 1/L.Source dataThe hybrid ladder consists of sites along along the y-direction which was solved using MMHK and sites along the x-direction described by Hubbard. We used open boundary conditions along the x-direction. The physical extent along (a) the y-direction and (d) the x-direction is clearly shown. The ground-state energy of the hybrid ladder is shown for U = 6 (smaller than bandwidth w = 8) in (b, e) and for U = 10 (larger than w) in (c, f). b, c display the ground-state energy as a function of ny, the number of MMHK sites along the y-direction, corresponding to the setup in (a). e, f show the energy as a function of Lx, the length along the x-direction, corresponding to the setup in (d). These results are all obtained using DMRG and the dashed lines represent a fitting curve. b, c, The energy converges faster than the 1/n2 behavior observed in the purely one-dimensional n-MMHK cases. e, f, The energy converges linearly as Lx increases.Source dataThe single-particle charge gaps are shown for (a) the n-MMHK model, (b) a periodic Hubbard chain with length L, and (c) an open Hubbard chain with length L, all obtained from DMRG simulations. In all panels, the dashed line denotes the exact result from Bethe ansatz. a, n-MMHK captures the vanishing gap at U = 0 (even for small n) and slowly converges to the exponential singularity (Bethe ansatz) at small U as n increases. b, Periodic Hubbard chain exhibits non-systematic L-dependence, with large L behaviors close to MMHK. c, Open Hubbard chain captures the larger-U regime accurately but slowly converges to the correct U = 0 limit due to finite length.a, Evolution of the csingle-particle charge gap Δ with U at half-filling for \(t^{\prime} =0\) and \(t^{\prime} /t=-0.25\). At \(t^{\prime} =0\), the gap opens for any finite U, while in the frustrated case (\(t^{\prime} /t=-0.25\)), the gap opens only for U > Uc = 1.62, indicating a Mott transition. b, Antiferromagnetic spin susceptibility at \(t^{\prime} =-0.25\) and β = 100 as a function of U, showing a cusp at Uc. c The critical interaction strength Uc for the Mott transition as a function of \(\left\vert \right.t^{\prime} \left\vert \right.\)(the sign of \(t^{\prime}\) does not make a difference in Uc). Initially, Uc grows almost linearly with \(| t^{\prime} |\). It saturates around \(| t^{\prime} | \approx 0.55\) and then decreases with further increasing \(| t^{\prime} |\).Source dataa, Equal-time spin correlation as a function of inverse temperature for different transfer momenta q for 4-MMHK at U = 10. The q = (π, π) case dominates the spin correlation, indicating the leading magnetic fluctuation is of the antiferromagnetic type. b, Equal-time antiferromagnetic correlation as a function of U for different n-MMHK models. As the number of mixed momenta n increases from 1 (band HK), the antiferromagnetic correlation dominates. c, Equal-time antiferromagnetic correlation as a function of density at U = 10, βt = 20 for 2- and 4-MMHK models. The antiferromagnetic correlation remains for a wide range of doping, essential for capturing the more complicated doping physics in the Hubbard model.Source dataWe compare the half-filled (U = 8) spectral function A(k, ω) among n-MMHK models (at zero temperature with a Lorentzian broadening of 0.2) and other methods for solving the Hubbard model. (a) 4-MMHK (or 2 × 2-MMHK model) and (b) 2 × 2 cluster perturbation theory (reprinted from34) show qualitatively similar leading features, including peak positions and gap size. The (c) 8-MMHK and (e) 16-MMHK results progressively improve the description of weaker sub-leading features, becoming quantitatively comparable to (d) state-of-the-art determinant quantum Monte-carlo and/or cluster perturbation theory (reprinted from35) at β = 16. This underscores that as n → N, MMHK becomes the Hubbard model.Source dataWe present the density of states at varying hole-doped densities x with U = 10, β = 30. The cutoff ωg is set to the gap for calculating the low energy spectral weight with Eq. (10).Source dataa-c are for high temperatures, while d-f are for low temperatures.Source dataSupplementary Fig. 1, Equations (1)–(23) and Sections 1–5.Simulation data used to plot the figure.Simulation data used to plot the figure.Simulation data used to plot the figure.Simulation data used to plot the figure.Simulation data used to plot the figure.Simulation data used to plot the figure.Simulation data used to plot the figure.Simulation data used to plot the figure.Simulation data used to plot the figure.Simulation data used to plot the figure.Simulation data used to plot the figure.Simulation data used to plot the figure.Simulation data used to plot the figure.Simulation data used to plot the figure.Simulation data used to plot the figure.Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.Reprints and permissionsMai, P., Zhao, J., Tenkila, G. et al. Twisting the Hubbard model into the momentum-mixing Hatsugai–Kohmoto model. Nat. Phys. (2025). https://doi.org/10.1038/s41567-025-03095-1Download citationReceived: 04 September 2024Accepted: 08 October 2025Published: 27 November 2025Version of record: 27 November 2025DOI: https://doi.org/10.1038/s41567-025-03095-1Anyone you share the following link with will be able to read this content:Sorry, a shareable link is not currently available for this article. Provided by the Springer Nature SharedIt content-sharing initiative