- Physical Review Journals
All JournalsPhysics Magazine

Physical Review B

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Large isolated stripes on short 18-leg t−J cylinders

Tizian Blatz1,2,*, Sebastian Paeckel1,2, Ulrich Schollwöck1,2, Fabian Grusdt1,2, and Annabelle Bohrdt1,2,†

  • *Contact author: blatz.tizian@physik.uni-muenchen.de
  • †Contact author: a.bohrdt@lmu.de
PDF

Phys. Rev. B 113, 245138 – Published 18 June, 2026

DOI: https://doi.org/10.1103/3jnq-y7gc

Abstract

Spin-charge stripes are among the most prominent competing orders in high-temperature superconductors. However, this phase is particularly challenging to study numerically due to finite-size effects. Here, we study the formation of long isolated stripes in the t−t′−J model using density-matrix renormalization group simulations on an unusually wide cylindrical strip geometry. This approach allows us to probe the physics of a single stripe largely free from stripe-stripe interactions and commensurability effects. We find that the wide range of stripe filling fractions reported in previous numerical studies is well explained by the physics of an individual stripe. Taking a microscopic look at stripe formation, we reveal two separate regimes—a high-filling regime captured by a simplified squeezed-space model and a low-filling regime characterized by the real-space structure of individual pairs of dopants. Our results connect the phenomenology of stripe order to its microscopic constituents and provide a unified perspective that highlights the different challenges for observing the two regimes in quantum simulation experiments.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (48)

  1. J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Nakamura, and S. Uchida, Evidence for stripe correlations of spins and holes in copper oxide superconductors, Nature (London) 375, 561 (1995).
  2. M. von Zimmermann, A. Vigilante, T. Frello, J. Madsen, D. J. Buttrey, N. H. Andersen, J. R. Schneider, B. Gibs, and J. M. Trancpada, X-ray scattering study of charge scattering associated with stripe order in La2NiO4, J. Supercond. 10, 447 (1997).
  3. V. J. Emery, S. A. Kivelson, and J. M. Tranquada, Stripe phases in high-temperature superconductors, Proc. Natl. Acad. Sci. USA 96, 8814 (1999).
  4. D. J. Scalapino, A common thread: The pairing interaction for unconventional superconductors, Rev. Mod. Phys. 84, 1383 (2012).
  5. B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida, and J. Zaanen, From quantum matter to high-temperature superconductivity in copper oxides, Nature (London) 518, 179 (2015).
  6. V. J. Emery, S. A. Kivelson, and O. Zachar, Spin-gap proximity effect mechanism of high-temperature superconductivity, Phys. Rev. B 56, 6120 (1997).
  7. S. A. Kivelson, E. Fradkin, and V. J. Emery, Electronic liquid-crystal phases of a doped Mott insulator, Nature (London) 393, 550 (1998).
  8. A. A. Kordyuk, Pseudogap from ARPES experiment: Three gaps in cuprates and topological superconductivity (review article), Low Temp. Phys. 41, 319 (2015).
  9. M. Qin, C.-M. Chung, H. Shi, E. Vitali, C. Hubig, U. Schollwöck, S. R. White, and S. Zhang, Absence of superconductivity in the pure two-dimensional Hubbard model, Phys. Rev. X 10, 031016 (2020).
  10. S. Jiang, D. J. Scalapino, and S. R. White, Ground-state phase diagram of the t−t′−J model, Proc. Natl. Acad. Sci. USA 118, e2109978118 (2021).
  11. H. Xu, C.-M. Chung, M. Qin, U. Schollwöck, S. R. White, and S. Zhang, Coexistence of superconductivity with partially filled stripes in the Hubbard model, Science 384, eadh7691 (2024).
  12. Y.-F. Jiang, T. P. Devereaux, and H.-C. Jiang, Ground-state phase diagram and superconductivity of the doped Hubbard model on six-leg square cylinders, Phys. Rev. B 109, 085121 (2024).
  13. F. Chen, F. D. M. Haldane, and D. N. Sheng, Global phase diagram of d-wave superconductivity in the square-lattice t-J model, Proc. Natl. Acad. Sci. USA 122, e2420963122 (2025).
  14. X. Lu, F. Chen, W. Zhu, D. N. Sheng, and S.-S. Gong, Emergent superconductivity and competing charge orders in hole-doped square-lattice t−J model, Phys. Rev. Lett. 132, 066002 (2024).
  15. H. Xu, H. Shi, E. Vitali, M. Qin, and S. Zhang, Stripes and spin-density waves in the doped two-dimensional Hubbard model: Ground state phase diagram, Phys. Rev. Res. 4, 013239 (2022).
  16. Y. Shen, X. Qian, and M. Qin, Ground state of electron-doped t−t′−J model on cylinders: An investigation of finite size and boundary condition effects, Chin. Phys. B 34, 087105 (2025).
  17. B.-X. Zheng, C.-M. Chung, P. Corboz, G. Ehlers, M.-P. Qin, R. M. Noack, H. Shi, S. R. White, S. Zhang, and G. Kin-Lic Chan, Stripe order in the underdoped region of the two-dimensional Hubbard model, Science 358, 1155 (2017).
  18. C. Roth, A. Chen, A. Sengupta, and A. Georges, Superconductivity in the two-dimensional Hubbard model revealed by neural quantum states, arXiv:2511.07566.
  19. H.-C. Jiang, T. P. Devereaux, and S. A. Kivelson, Competition between charge-density-wave and superconducting orders on eight-leg square Hubbard cylinders, arXiv:2511.18644.
  20. Y. Gu, W. Li, H. Lin, B. Zhan, R. Li, Y. Huang, D. He, Y. Wu, T. Xiang, M. Qin, L. Wang, and D. Lv, Solving the Hubbard model with neural quantum states, arXiv:2507.02644.
  21. W.-Y. Liu, H. Zhai, R. Peng, Z.-C. Gu, and G. K.-L. Chan, Accurate simulation of the Hubbard model with finite fermionic projected entangled pair states, Phys. Rev. Lett. 134, 256502 (2025).
  22. J. Hubbard, Electron correlations in narrow energy bands, Proc. R. Soc. Lond. Ser. A 276, 238 (1963).
  23. K. A. Chao, J. Spalek, and A. M. Oles, Kinetic exchange interaction in a narrow S-band, J. Phys. C 10, L271 (1977).
  24. K. A. Chao, J. Spałek, and A. M. Oleś, Canonical perturbation expansion of the Hubbard model, Phys. Rev. B 18, 3453 (1978).
  25. J. E. Hirsch, Attractive interaction and pairing in fermion systems with strong on-site repulsion, Phys. Rev. Lett. 54, 1317 (1985).
  26. C.-M. Chung, M. Qin, S. Zhang, U. Schollwöck, and S. R. White, Plaquette versus ordinary d-wave pairing in the t′-Hubbard model on a width-4 cylinder, Phys. Rev. B 102, 041106(R) (2020).
  27. T. Blatz, U. Schollwöck, F. Grusdt, and A. Bohrdt, Two-dopant origin of competing stripe and pair formation in Hubbard and t−J models, Phys. Rev. X 15, 031074 (2025).
  28. M. Ogata and H. Shiba, Bethe-ansatz wave function, momentum distribution, and spin correlation in the one-dimensional strongly correlated Hubbard model, Phys. Rev. B 41, 2326 (1990).
  29. H. Schlömer, A. Bohrdt, L. Pollet, U. Schollwöck, and F. Grusdt, Robust stripes in the mixed-dimensional t−J model, Phys. Rev. Res. 5, L022027 (2023).
  30. D. Bourgund, T. Chalopin, P. Bojović, H. Schlömer, S. Wang, T. Franz, S. Hirthe, A. Bohrdt, F. Grusdt, I. Bloch, and T. A. Hilker, Formation of individual stripes in a mixed-dimensional cold-atom Fermi–Hubbard system, Nature (London) 637, 57 (2025).
  31. F. C. Zhang and T. M. Rice, Effective Hamiltonian for the superconducting Cu oxides, Phys. Rev. B 37, 3759 (1988).
  32. A. Auerbach, Interacting Electrons and Quantum Magnetism, Graduate Texts in Contemporary Physics (Springer, New York, 1994).
  33. M. S. Hybertsen, M. Schlüter, and N. E. Christensen, Calculation of Coulomb-interaction parameters for La2CuO4 using a constrained-density-functional approach, Phys. Rev. B 39, 9028 (1989).
  34. E. Dagotto, Correlated electrons in high-temperature superconductors, Rev. Mod. Phys. 66, 763 (1994).
  35. O. K. Andersen, A. I. Liechtenstein, O. Jepsen, and F. Paulsen, LDA energy bands, low-energy hamiltonians, t′, t′′, t⊥(k), and J⊥, J. Phys. Chem. Solids 56, 1573 (1995).
  36. M. Hirayama, Y. Yamaji, T. Misawa, and M. Imada, Ab initio effective Hamiltonians for cuprate superconductors, Phys. Rev. B 98, 134501 (2018).
  37. S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992).
  38. S. R. White, Density-matrix algorithms for quantum renormalization groups, Phys. Rev. B 48, 10345 (1993).
  39. U. Schollwoeck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. (NY) 326, 96 (2011).
  40. S-W. Cheong, G. Aeppli, T. E. Mason, H. Mook, S. M. Hayden, P. C. Canfield, Z. Fisk, K. N. Clausen, and J. L. Martinez, Incommensurate magnetic fluctuations in La2−xSrxCuO4, Phys. Rev. Lett. 67, 1791 (1991).
  41. R. H. Wilke, T. Köhler, F. A. Palm, and S. Paeckel, Symmetry-protected Bose-Einstein condensation of interacting hardcore bosons, Commun. Phys. 6, 182 (2023).
  42. M. Xu, L. H. Kendrick, A. Kale, Y. Gang, C. Feng, S. Zhang, A. W. Young, M. Lebrat, and M. Greiner, A neutral-atom Hubbard quantum simulator in the cryogenic regime, Nature (London) 642, 909 (2025).
  43. https://github.com/TizianBlatz/stripes_tJ.
  44. S. Zhang, J. Carlson, and J. E. Gubernatis, Constrained path Monte Carlo method for fermion ground states, Phys. Rev. B 55, 7464 (1997).
  45. F. Verstraete and J. I. Cirac, Renormalization algorithms for quantum-many body systems in two and higher dimensions, arXiv:cond-mat/0407066.
  46. G. Knizia and G. K.-L. Chan, Density matrix embedding: A simple alternative to dynamical mean-field theory, Phys. Rev. Lett. 109, 186404 (2012).
  47. C. Hubig, F. Lachenmaier, N.-O. Linden, T. Reinhard, L. Stenzel, A. Swoboda, M. Grundner, S. Mardazad, and S. Paeckel, The SyTen toolkit, https://syten.eu/docs/.
  48. M. Yang and S. R. White, Time dependent variational principle with ancillary Krylov subspace, Phys. Rev. B 102, 094315 (2020).

Outline

Information

Phys. Rev. B 113, 245138– Published 18 June, 2026
  • Received 19 December 2025
  • Revised 5 May 2026
  • Accepted 2 June 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation