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  • Letter
  • Open Access

Stacked tree construction for free-fermion projected entangled pair states

Yuman He, Kangle Li, Yanbai Zhang, and Hoi Chun Po*

  • Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China; IAS Center for Quantum Technologies, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China; and Center for Theoretical Condensed Matter Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China

  • *hcpo@ust.hk
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Phys. Rev. Research 6, L022016 – Published 17 April, 2024

DOI: https://doi.org/10.1103/PhysRevResearch.6.L022016

Abstract

The tensor network representation of a state in higher dimensions, say a projected entangled-pair state (PEPS), is typically obtained indirectly through variational optimization or imaginary-time Hamiltonian evolution. Here, we propose a divide-and-conquer approach to directly construct a PEPS representation for free-fermion states admitting descriptions in terms of filling exponentially localized Wannier functions. Our approach relies on first obtaining a tree tensor network description of the state in local subregions. Next, a stacking procedure is used to combine the local trees into a PEPS. Lastly, the local tensors are compressed to obtain a more efficient description. We demonstrate our construction for states in one and two dimensions, including the ground state of an obstructed atomic insulator on the square lattice.

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References (49)

  1. U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Ann. Phys. 326, 96 (2011).
  2. R. Orús, A practical introduction to tensor networks: Matrix product states and projected entangled pair states, Ann. Phys. 349, 117 (2014).
  3. G. Vidal, Efficient classical simulation of slightly entangled quantum computations, Phys. Rev. Lett. 91, 147902 (2003).
  4. G. Vidal, Efficient simulation of one-dimensional quantum many-body systems, Phys. Rev. Lett. 93, 040502 (2004).
  5. S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992).
  6. S. Liang and H. Pang, Approximate diagonalization using the density matrix renormalization-group method: A two-dimensional-systems perspective, Phys. Rev. B 49, 9214 (1994).
  7. T. Xiang, J. Lou, and Z. Su, Two-dimensional algorithm of the density-matrix renormalization group, Phys. Rev. B 64, 104414 (2001).
  8. E. Stoudenmire and S. R. White, Studying two-dimensional systems with the density matrix renormalization group, Annu. Rev. Condens. Matter Phys. 3, 111 (2012).
  9. V. Murg, F. Verstraete, Ö. Legeza, and R. M. Noack, Simulating strongly correlated quantum systems with tree tensor networks, Phys. Rev. B 82, 205105 (2010).
  10. Y.-Y. Shi, L.-M. Duan, and G. Vidal, Classical simulation of quantum many-body systems with a tree tensor network, Phys. Rev. A 74, 022320 (2006).
  11. L. Tagliacozzo, G. Evenbly, and G. Vidal, Simulation of two-dimensional quantum systems using a tree tensor network that exploits the entropic area law, Phys. Rev. B 80, 235127 (2009).
  12. N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, Computational complexity of projected entangled pair states, Phys. Rev. Lett. 98, 140506 (2007).
  13. M. P. Zaletel and F. Pollmann, Isometric tensor network states in two dimensions, Phys. Rev. Lett. 124, 037201 (2020).
  14. M. S. J. Tepaske and D. J. Luitz, Three-dimensional isometric tensor networks, Phys. Rev. Res. 3, 023236 (2021).
  15. N. Schuch, I. Cirac, and D. Pérez-García, PEPS as ground states: Degeneracy and topology, Ann. Phys. 325, 2153 (2010).
  16. F. Verstraete, M. M. Wolf, D. Perez-Garcia, and J. I. Cirac, Criticality, the area law, and the computational power of projected entangled pair states, Phys. Rev. Lett. 96, 220601 (2006).
  17. H. He, Y. Zheng, B. A. Bernevig, and N. Regnault, Entanglement entropy from tensor network states for stabilizer codes, Phys. Rev. B 97, 125102 (2018).
  18. C. V. Kraus, N. Schuch, F. Verstraete, and J. I. Cirac, Fermionic projected entangled pair states, Phys. Rev. A 81, 052338 (2010).
  19. T. B. Wahl, H.-H. Tu, N. Schuch, and J. I. Cirac, Projected entangled-pair states can describe chiral topological states, Phys. Rev. Lett. 111, 236805 (2013).
  20. Q. Mortier, N. Schuch, F. Verstraete, and J. Haegeman, Tensor networks can resolve Fermi surfaces, Phys. Rev. Lett. 129, 206401 (2022).
  21. J. Dubail and N. Read, Tensor network trial states for chiral topological phases in two dimensions and a no-go theorem in any dimension, Phys. Rev. B 92, 205307 (2015).
  22. N. Schuch and B. Bauer, Matrix product state algorithms for Gaussian fermionic states, Phys. Rev. B 100, 245121 (2019).
  23. S. Boutin and B. Bauer, Quantum impurity models using superpositions of fermionic Gaussian states: Practical methods and applications, Phys. Rev. Res. 3, 033188 (2021).
  24. By a “free-fermion” state, we refer to a fermionic Gaussian state with a definite particle number.
  25. Q. Yang, X.-Y. Zhang, H.-J. Liao, H.-H. Tu, and L. Wang, Projected d-wave superconducting state: A fermionic projected entangled pair state study, Phys. Rev. B 107, 125128 (2023).
  26. J.-W. Li, J. von Delft, and H.-H. Tu, U(1)-symmetric Gaussian fermionic projected entangled paired states and their Gutzwiller projection, Phys. Rev. B 107, 085148 (2023).
  27. S. Yang, T. B. Wahl, H.-H. Tu, N. Schuch, and J. I. Cirac, Chiral projected entangled-pair state with topological order, Phys. Rev. Lett. 114, 106803 (2015).
  28. Y.-H. Wu, L. Wang, and H.-H. Tu, Tensor network representations of parton wave functions, Phys. Rev. Lett. 124, 246401 (2020).
  29. I. Peschel and V. Eisler, Reduced density matrices and entanglement entropy in free lattice models, J. Phys. A 42, 504003 (2009).
  30. S. Bravyi, Lagrangian representation for fermionic linear optics, Quantum Info. Comput. 5, 216 (2005).
  31. N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, Maximally localized wannier functions: Theory and applications, Rev. Mod. Phys. 84, 1419 (2012).
  32. More generally, basis choice within degenerate subspaces of the singular values.
  33. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.6.L022016 for details of the constructed models and operations on fermionic Gaussian states.
  34. Mapping the fermion problem to a bosonic one is one possible (and perhaps standard) way, but this would invoke unwieldy Jordan-Wigner strings and does not manifestly preserve the free-fermion nature of the present problem. While this problem is arguably mild for an MPS or a tree TN, it is much more severe for the subsequent steps in our problem concerning a 2D state. As such, a fully fermionic formulation is desired, as we discuss in Supplemental Material [33].
  35. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, Cambridge, 2010).
  36. J. I. Latorre and A. Riera, A short review on entanglement in quantum spin systems, J. Phys. A 42, 504002 (2009).
  37. G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Entanglement in quantum critical phenomena, Phys. Rev. Lett. 90, 227902 (2003).
  38. J. I. Latorre, E. Rico, and G. Vidal, Ground state entanglement in quantum spin chains, Quantum Info. Comput. 4, 48 (2004).
  39. M. T. Fishman and S. R. White, Compression of correlation matrices and an efficient method for forming matrix product states of fermionic Gaussian states, Phys. Rev. B 92, 075132 (2015).
  40. Y. Xu, L. Elcoro, Z.-D. Song, M. G. Vergniory, C. Felser, S. S. P. Parkin, N. Regnault, J. L. Mañes, and B. A. Bernevig, Filling-enforced obstructed atomic insulators, arXiv:2106.10276.
  41. H. C. Po, A. Vishwanath, and H. Watanabe, Symmetry-based indicators of band topology in the 230 space groups, Nat. Commun. 8, 50 (2017).
  42. B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, and B. A. Bernevig, Topological quantum chemistry, Nature (London) 547, 298 (2017).
  43. A. M. Alhambra and J. I. Cirac, Locally accurate tensor networks for thermal states and time evolution, PRX Quantum 2, 040331 (2021).
  44. Y. Huang, Computing energy density in one dimension, arXiv:1505.00772.
  45. N. Schuch and F. Verstraete, Matrix product state approximations for infinite systems, arXiv:1711.06559.
  46. A. M. Dalzell and F. G. S. L. Brandão, Locally accurate MPS approximations for ground states of one-dimensional gapped local Hamiltonians, Quantum 3, 187 (2019).
  47. Y. Huang, Approximating local properties by tensor network states with constant bond dimension, arXiv:1903.10048.
  48. I. H. Kim, Entropy scaling law and the quantum marginal problem, Phys. Rev. X 11, 021039 (2021).
  49. I. H. Kim, Entropy scaling law and the quantum marginal problem: simplification and generalization, arXiv:2109.11688.

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Phys. Rev. Research 6, L022016– Published 17 April, 2024
  • Received 16 October 2023
  • Revised 15 January 2024
  • Accepted 29 January 2024
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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.

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