Free fermions beyond Jordan and Wigner (2310.19897v3)
Abstract: The Jordan-Wigner transformation is frequently utilised to rewrite quantum spin chains in terms of fermionic operators. When the resulting Hamiltonian is bilinear in these fermions, i.e. the fermions are free, the exact spectrum follows from the eigenvalues of a matrix whose size grows only linearly with the volume of the system. However, several Hamiltonians that do not admit a Jordan-Wigner transformation to fermion bilinears still have the same type of free-fermion spectra. The spectra of such ``free fermions in disguise" models can be found exactly by an intricate but explicit construction of the raising and lowering operators. We generalise the methods further to find a family of such spin chains. We compute the exact spectrum, and generalise an elegant graph-theory construction. We also explain how this family admits an N=2 lattice supersymmetry.
- P. Jordan and E. P. Wigner, “About the Pauli exclusion principle,” Z. Phys. 47 (1928) 631–651.
- E. H. Lieb, T. Schultz, and D. Mattis, “Two soluble models of an antiferromagnetic chain,” Annals Phys. 16 (1961) 407–466.
- A. Chapman and S. T. Flammia, “Characterization of solvable spin models via graph invariants,” Quantum 4 (2020) 278, arXiv:2003.05465.
- M. Ogura, Y. Imamura, N. Kameyama, K. Minami, and M. Sato, “Geometric Criterion for Solvability of Lattice Spin Systems,” Phys. Rev. B 102 (2020) no. 24, 245118, arXiv:2003.13264 [cond-mat.stat-mech].
- P. Fendley and K. Schoutens, “Cooper pairs and exclusion statistics from coupled free-fermion chains,” J. Stat. Mech. 2007 (2007) no. 2, 02017, arXiv:cond-mat/0612270 [cond-mat.stat-mech].
- E. Witten, “Constraints on supersymmetry breaking,” Nuclear Physics B 202 (1982) no. 2, 253 – 316.
- P. Fendley, K. Schoutens, and J. de Boer, “Lattice Models with N=2 Supersymmetry,” Phys. Rev. Lett. 90 (2003) no. 12, 120402, arXiv:hep-th/0210161 [hep-th].
- Springer International Publishing, Cham, 2019. arXiv:1710.02658 [cond-mat.stat-mech].
- J. de Gier, G. Z. Feher, B. Nienhuis, and M. Rusaczonek, “Integrable supersymmetric chain without particle conservation,” J. Stat. Mech. (2016) 023104, arXiv:1510.02520 [cond-mat.quant-gas].
- P. Fendley, “Free fermions in disguise,” Journal of Physics A Mathematical General 52 (2019) no. 33, 335002, arXiv:1901.08078 [cond-mat.stat-mech].
- S. J. Elman, A. Chapman, and S. T. Flammia, “Free fermions behind the disguise,” Commun. Math. Phys. 388 (2021) 969–1003, arXiv:2012.07857 [quant-ph].
- A. Chapman, S. J. Elman, and R. L. Mann, “A Unified Graph-Theoretic Framework for Free-Fermion Solvability,” arXiv:2305.15625 [quant-ph].
- F. C. Alcaraz and R. A. Pimenta, “Free fermionic and parafermionic quantum spin chains with multispin interactions,” Phys. Rev. B 102 (2020) no. 12, 121101, arXiv:2005.14622.
- T. Gombor and B. Pozsgay, “Integrable spin chains and cellular automata with medium-range interaction,” Phys. Rev. E 104 (2021) no. 5, 054123, arXiv:2108.02053.
- Cambridge University Press, 1993.
- M. P. Grabowski and P. Mathieu, “Integrability test for spin chains,” J. Phys. A 28 (1995) no. 17, 4777–4798, arXiv:hep-th/9412039 [hep-th].
- M. de Leeuw, A. Pribytok, and P. Ryan, “Classifying two-dimensional integrable spin chains,” J. Phys. A 52 (2019) no. 50, 505201, arXiv:1904.12005 [math-ph].
- M. de Leeuw, C. Paletta, A. Pribytok, A. L. Retore, and P. Ryan, “Classifying Nearest-Neighbor Interactions and Deformations of AdS,” Phys. Rev. Lett. 125 (2020) no. 3, 031604, arXiv:2003.04332 [hep-th].
- M. de Leeuw, C. Paletta, and B. Pozsgay, “Constructing Integrable Lindblad Superoperators,” Phys. Rev. Lett. 126 (2021) no. 24, 240403, arXiv:2101.08279 [cond-mat.stat-mech].
- T. Gombor and B. Pozsgay , to appear.
- B. Pozsgay, A. Hutsalyuk, L. Pristyák, and G. Takács, “Sublattice entanglement in an exactly solvable anyonlike spin ladder,” Phys. Rev. E 106 (2022) no. 4, 044120, arXiv:2205.01465 [cond-mat.stat-mech].
- P. Fendley, “Free parafermions,” J. Phys. A 47 (2014) no. 7, 075001, arXiv:1310.6049 [cond-mat.stat-mech].
- F. C. Alcaraz and R. A. Pimenta, “Integrable quantum spin chains with free fermionic and parafermionic spectrum,” Phys. Rev. B 102 (2021) no. 23, 235170, arXiv:2010.01116 [cond-mat.stat-mech].
- F. C. Alcaraz, J. A. Hoyos, and R. A. Pimenta, “A random free-fermion quantum spin chain with multi-spin interactions,” arXiv:2308.16249 [cond-mat.dis-nn].
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.