Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
157 tokens/sec
GPT-4o
8 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Lieb-Schultz-Mattis theorems and generalizations in long-range interacting systems (2405.14929v1)

Published 23 May 2024 in cond-mat.str-el, cond-mat.quant-gas, math-ph, math.MP, math.OA, and quant-ph

Abstract: In a unified fashion, we establish Lieb-Schultz-Mattis (LSM) theorems and their generalizations in systems with long-range interactions. We show that, for a quantum spin chain, if the interactions decay fast enough as their ranges increase and the Hamiltonian has an anomalous symmetry, the Hamiltonian cannot have a unique gapped symmetric ground state. If the Hamiltonian contains only 2-spin interactions, these theorems hold when the interactions decay faster than $1/r2$, with $r$ the distance between the two interacting spins. Moreover, any pure state with an anomalous symmetry, which may not be a ground state of any natural Hamiltonian, must be long-range entangled. The symmetries we consider include on-site internal symmetries combined with lattice translation symmetries, and they can also extend to purely internal but non-on-site symmetries. Moreover, these internal symmetries can be discrete or continuous. We explore the applications of the theorems through various examples.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (46)
  1. Elliott Lieb, Theodore Schultz,  and Daniel Mattis, “Two soluble models of an antiferromagnetic chain,” Annals of Physics 16, 407 – 466 (1961).
  2. Masaki Oshikawa, “Commensurability, Excitation Gap, and Topology in Quantum Many-Particle Systems on a Periodic Lattice,” Phys. Rev. Lett.  84, 1535–1538 (2000), arXiv:cond-mat/9911137 [cond-mat.str-el] .
  3. M. B. Hastings, “Lieb-Schultz-Mattis in higher dimensions,” Phys. Rev. B 69, 104431 (2004), arXiv:cond-mat/0305505 [cond-mat.str-el] .
  4. Meng Cheng, Michael Zaletel, Maissam Barkeshli, Ashvin Vishwanath,  and Parsa Bonderson, “Translational Symmetry and Microscopic Constraints on Symmetry-Enriched Topological Phases: A View from the Surface,” Physical Review X 6, 041068 (2016), arXiv:1511.02263 [cond-mat.str-el] .
  5. Hoi Chun Po, Haruki Watanabe, Chao-Ming Jian,  and Michael P. Zaletel, “Lattice Homotopy Constraints on Phases of Quantum Magnets,” Phys. Rev. Lett.  119, 127202 (2017), arXiv:1703.06882 [cond-mat.str-el] .
  6. Chao-Ming Jian, Zhen Bi,  and Cenke Xu, “Lieb-Schultz-Mattis theorem and its generalizations from the perspective of the symmetry-protected topological phase,” Phys. Rev. B 97, 054412 (2018), arXiv:1705.00012 [cond-mat.str-el] .
  7. Gil Young Cho, Chang-Tse Hsieh,  and Shinsei Ryu, “Anomaly manifestation of Lieb-Schultz-Mattis theorem and topological phases,” Phys. Rev. B 96, 195105 (2017), arXiv:1705.03892 [cond-mat.str-el] .
  8. Max A. Metlitski and Ryan Thorngren, “Intrinsic and emergent anomalies at deconfined critical points,” Phys. Rev. B 98, 085140 (2018), arXiv:1707.07686 [cond-mat.str-el] .
  9. Meng Cheng, “Fermionic Lieb-Schultz-Mattis theorems and weak symmetry-protected phases,” Phys. Rev. B 99, 075143 (2019), arXiv:1804.10122 [cond-mat.str-el] .
  10. Ryohei Kobayashi, Ken Shiozaki, Yuta Kikuchi,  and Shinsei Ryu, “Lieb-Schultz-Mattis type theorem with higher-form symmetry and the quantum dimer models,” Phys. Rev. B 99, 014402 (2019), arXiv:1805.05367 [cond-mat.stat-mech] .
  11. Dominic V. Else and Ryan Thorngren, “Topological theory of Lieb-Schultz-Mattis theorems in quantum spin systems,” Phys. Rev. B 101, 224437 (2020), arXiv:1907.08204 [cond-mat.str-el] .
  12. Shenghan Jiang, Meng Cheng, Yang Qi,  and Yuan-Ming Lu, “Generalized Lieb-Schultz-Mattis theorem on bosonic symmetry protected topological phases,” SciPost Phys. 11, 024 (2021), arXiv:1907.08596 [cond-mat.str-el] .
  13. Yoshiko Ogata, Yuji Tachikawa,  and Hal Tasaki, “General Lieb-Schultz-Mattis Type Theorems for Quantum Spin Chains,” Communications in Mathematical Physics 385, 79–99 (2021), arXiv:2004.06458 [math-ph] .
  14. Weicheng Ye, Meng Guo, Yin-Chen He, Chong Wang,  and Liujun Zou, “Topological characterization of Lieb-Schultz-Mattis constraints and applications to symmetry-enriched quantum criticality,” SciPost Physics 13, 066 (2022), arXiv:2111.12097 [cond-mat.str-el] .
  15. Ruochen Ma and Chong Wang, “Average Symmetry-Protected Topological Phases,” Physical Review X 13, 031016 (2023), arXiv:2209.02723 [cond-mat.str-el] .
  16. Meng Cheng and Nathan Seiberg, “Lieb-Schultz-Mattis, Luttinger, and ’t Hooft - anomaly matching in lattice systems,” SciPost Physics 15, 051 (2023), arXiv:2211.12543 [cond-mat.str-el] .
  17. Kohei Kawabata, Ramanjit Sohal,  and Shinsei Ryu, “Lieb-Schultz-Mattis Theorem in Open Quantum Systems,” Phys. Rev. Lett.  132, 070402 (2024), arXiv:2305.16496 [cond-mat.stat-mech] .
  18. Sahand Seifnashri, “Lieb-Schultz-Mattis anomalies as obstructions to gauging (non-on-site) symmetries,” SciPost Physics 16, 098 (2024), arXiv:2308.05151 [cond-mat.str-el] .
  19. Yi-Neng Zhou, Xingyu Li, Hui Zhai, Chengshu Li,  and Yingfei Gu, “Reviving the Lieb-Schultz-Mattis Theorem in Open Quantum Systems,” arXiv e-prints , arXiv:2310.01475 (2023), arXiv:2310.01475 [cond-mat.str-el] .
  20. Anton Kapustin and Nikita Sopenko, “Anomalous symmetries of quantum spin chains and a generalization of the Lieb-Schultz-Mattis theorem,” arXiv e-prints , arXiv:2401.02533 (2024), arXiv:2401.02533 [math-ph] .
  21. Jose Garre Rubio, Andras Molnar,  and Yoshiko Ogata, “Classifying symmetric and symmetry-broken spin chain phases with anomalous group actions,” arXiv e-prints , arXiv:2403.18573 (2024), arXiv:2403.18573 [quant-ph] .
  22. Liujun Zou, Yin-Chen He,  and Chong Wang, “Stiefel Liquids: Possible Non-Lagrangian Quantum Criticality from Intertwined Orders,” Physical Review X 11, 031043 (2021), arXiv:2101.07805 [cond-mat.str-el] .
  23. Weicheng Ye and Liujun Zou, “Classification of symmetry-enriched topological quantum spin liquids,” arXiv e-prints , arXiv:2309.15118 (2023), arXiv:2309.15118 [cond-mat.str-el] .
  24. Nicolò Defenu, Tobias Donner, Tommaso Macrı, Guido Pagano, Stefano Ruffo,  and Andrea Trombettoni, “Long-range interacting quantum systems,” Reviews of Modern Physics 95, 035002 (2023), arXiv:2109.01063 [cond-mat.quant-gas] .
  25. See Supplemental Material for more details and additional references, including Refs. Barkeshli et al. (2015); Freed and Hopkins (2021); Kapustin (2014); Yonekura (2019); Fomenko and Fuchs (2016); Wan and Wang (2019); Rudyak (1998); Ning et al. (2020); Chen et al. (2011).
  26. Taku Matsui, “Boundedness of entanglement entropy and split property of quantum spin chains,” Reviews in Mathematical Physics 25, 1350017 (2013), arXiv:1109.5778 [math-ph] .
  27. Yoshiko Ogata, “A classification of pure states on quantum spin chains satisfying the split property with on-site finite group symmetries,” arXiv e-prints , arXiv:1908.08621 (2019), arXiv:1908.08621 [math.OA] .
  28. Pieter Naaijkens and Yoshiko Ogata, “The Split and Approximate Split Property in 2D Systems: Stability and Absence of Superselection Sectors,” Communications in Mathematical Physics 392, 921–950 (2022), arXiv:2102.07707 [math-ph] .
  29. D. Gross, V. Nesme, H. Vogts,  and R. F. Werner, “Index theory of one dimensional quantum walks and cellular automata,” Communications in Mathematical Physics 310, 419–454 (2012), arXiv:0910.3675 [quant-ph] .
  30. P. Arrighi, “An overview of quantum cellular automata,” Natural Computing 18, 885–899 (2019), arXiv:1904.12956 [quant-ph] .
  31. Terry Farrelly, “A review of quantum cellular automata,” Quantum 4, 368 (2020), arXiv:1904.13318 [quant-ph] .
  32. Dominic V. Else and Chetan Nayak, “Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge,” Phys. Rev. B 90, 235137 (2014), arXiv:1409.5436 [cond-mat.str-el] .
  33. Tomotaka Kuwahara and Keiji Saito, “Area law of noncritical ground states in 1D long-range interacting systems,” Nature Communications 11, 4478 (2020), arXiv:1908.11547 [quant-ph] .
  34. Lei Gioia and Chong Wang, “Nonzero Momentum Requires Long-Range Entanglement,” Physical Review X 12, 031007 (2022), arXiv:2112.06946 [cond-mat.str-el] .
  35. Sebastian Geier, Nithiwadee Thaicharoen, Clément Hainaut, Titus Franz, Andre Salzinger, Annika Tebben, David Grimshandl, Gerhard Zürn,  and Matthias Weidemüller, “Floquet Hamiltonian engineering of an isolated many-body spin system,” Science 374, 1149–1152 (2021), arXiv:2105.01597 [cond-mat.quant-gas] .
  36. P. Scholl, H. J. Williams, G. Bornet, F. Wallner, D. Barredo, L. Henriet, A. Signoles, C. Hainaut, T. Franz, S. Geier, A. Tebben, A. Salzinger, G. Zürn, T. Lahaye, M. Weidemüller,  and A. Browaeys, “Microwave Engineering of Programmable X X Z Hamiltonians in Arrays of Rydberg Atoms,” PRX Quantum 3, 020303 (2022), arXiv:2107.14459 [quant-ph] .
  37. Mohammad F. Maghrebi, Zhe-Xuan Gong,  and Alexey V. Gorshkov, “Continuous symmetry breaking in 1d long-range interacting quantum systems,” Phys. Rev. Lett. 119, 023001 (2017), arXiv:1510.01325 [cond-mat.quant-gas] .
  38. Michael Levin and Zheng-Cheng Gu, “Braiding statistics approach to symmetry-protected topological phases,” Phys. Rev. B 86, 115109 (2012), arXiv:1202.3120 [cond-mat.str-el] .
  39. Maissam Barkeshli, Hong-Chen Jiang, Ronny Thomale,  and Xiao-Liang Qi, “Generalized kitaev models and extrinsic non-abelian twist defects,” Phys. Rev. Lett. 114, 026401 (2015), arXiv:1405.1780 [cond-mat.str-el] .
  40. Daniel S. Freed and Michael J. Hopkins, “Reflection positivity and invertible topological phases,” Geometry & Topology 25, 1165–1330 (2021), arXiv:1604.06527 [hep-th] .
  41. Anton Kapustin, “Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology,” arXiv e-prints , arXiv:1403.1467 (2014), arXiv:1403.1467 [cond-mat.str-el] .
  42. Kazuya Yonekura, “On the cobordism classification of symmetry protected topological phases,” Communications in Mathematical Physics 368, 1121–1173 (2019).
  43. Anatoly Fomenko and Dmitry Fuchs, “Chapter 6: K-theory and other extraordinary cohomology theories,” in Homotopical Topology (Springer International Publishing, Cham, 2016) pp. 495–611.
  44. Zheyan Wan and Juven Wang, “Higher anomalies, higher symmetries, and cobordisms i: classification of higher-symmetry-protected topological states and their boundary fermionic/bosonic anomalies via a generalized cobordism theory,” Annals of Mathematical Sciences and Applications 4, 107–311 (2019).
  45. Shang-Qiang Ning, Liujun Zou,  and Meng Cheng, “Fractionalization and anomalies in symmetry-enriched U(1) gauge theories,” Physical Review Research 2, 043043 (2020), arXiv:1905.03276 [cond-mat.str-el] .
  46. Xie Chen, Zheng-Cheng Gu,  and Xiao-Gang Wen, “Classification of gapped symmetric phases in one-dimensional spin systems,” Phys. Rev. B 83, 035107 (2011), arXiv:1008.3745 [cond-mat.str-el] .
Citations (4)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets