nLab spin chain

Idea

(…)

Examples

• Heisenberg model

• Kitaev spin chain

• (…)

Related concepts

• solid state physics

• Bethe ansatz

• integrable system

• BMN limit

References

General

The original articles on the Heisenberg spin chain model:

• Werner Heisenberg: Zur Theorie des Ferromagnetismus, Zeitschrift für Physik 49 (1928) 619–636 (1928) [doi:10.1007/BF01328601]

• Hans Bethe: Zur Theorie der Metalle, Zeitschrift für Physik 71 (1931) 205 [doi:10.1007/BF01341708]

Extension to the AKLT model:

• Ian Affleck, Tom Kennedy, Elliott H. Lieb, Hal Tasaki: Rigorous results on valence-bond ground states in antiferromagnets, Phys. Rev. Lett. 59 (1987) 799 [doi:10.1103/PhysRevLett.59.799]

Observation of the “Haldan gap phase” in antiferromagnetic integer Heisenberg spin chains (and relation to the SO(3) sigma-model):

• F. Duncan M. Haldane: Nonlinear Field Theory of Large-Spin Heisenberg Antiferromagnets: Semiclassically Quantized Solitons of the One-Dimensional Easy-Axis Néel State, Phys. Rev. Lett. 50 (1983) 1153 [doi:10.1103/PhysRevLett.50.1153]

• Duncan Haldane, Continuum dynamics of the 1-D Heisenberg antiferromagnet: Identification with the $O(3)$ nonlinear sigma model, Physics Letters A 93 9 (1983) 464-468 [doi:10.1016/0375-9601(83)90631-X]

• Duncan Haldane: Ground State Properties of Antiferromagnetic Chains with Unrestricted Spin: Integer Spin Chains as Realisations of the $O(3)$ Non-Linear Sigma Model [arXiv:1612.00076]

• T. Jolicoeur, Olivier Golinelli: Physics of integer-spin antiferromagnetic chains: Haldane gaps and edge states, Comptes Rendus Chimie 22 (2019) 445-451 [doi:10.1016/j.crci.2019.05.005, arXiv:1906.12207]

Discussion of symmetry protected topological phases:

• Pierre Fromholz, Philippe Lecheminant, Symmetry-protected topological phases in the $SU(N)$ Heisenberg spin chain: a Majorana-fermion approach, Phys. Rev. B 102 094410 (2020) [arXiv:2006.13553, doi:10.1103/PhysRevB.102.094410]

Review:

• Andreas Schadschneider, Götz S. Uhrig: Part II of: Strongly Correlated Systems in Solid State Physics, lecture notes (2004) [pdf, pdf]

• Ingmar Saberi, An introduction to spin systems for mathematicians [arXiv:1801.07270]

• Amanda Young: Quantum Spin Systems [arXiv:2308.07848, spire:2688309]

See also:

• Wikipedia: Quantum Heisenberg model

• Wikipedia: AKLT model

Relation to tensor networks:

• Mari Carmen Banuls, Michal P. Heller, Karl Jansen, Johannes Knaute, Viktor Svensson, From spin chains to real-time thermal field theory using tensor networks (arXiv:1912.08836)

Claim of supersymmetric spin chains carrying braid group representations (“anyons”):

• Indrajit Jana, Filippo Montorsi, Pramod Padmanabhan, Diego Trancanelli, Topological Quantum Computation on Supersymmetric Spin Chains [arXiv:2209.03822]

On topological phases (i.e. gapped ground states) of spin chains:

• Houssam Abdul-Rahman, Marius Lemm, Angelo Lucia, Bruno Nachtergaele, Amanda Young, A class of two-dimensional AKLT models with a gap, in: Analytic Trends in Mathematical Physics, Contemporary Mathematics 741, AMS (2020) [arXiv:1901.09297, doi:10.1090/conm/741/14917]

On simulation of quantum spin chains on quantum computers:

• M. Gluza, M. Kliesch, Jens Eisert, Leandro Aolita, Fidelity witnesses for fermionic quantum simulations, Phys. Rev. Lett. 120 190501 (2018) [arXiv:1703.03152, doi:10.1103/PhysRevLett.120.190501]

A holographic CMT model for spin chains:

• Naoto Yokoi, Yasuyuki Oikawa, Eiji Saitoh, Holographic Dual of Quantum Spin Chain as Chern-Simons-Scalar Theory [arXiv:2310.01890&rbrack:

For single trace operators in super Yang-Mills theory

Identifiication of spin chain dynamics in the action of the dilatation operator in super Yang-Mills theory, specifically D=4 N=4 super Yang-Mills theory, on single trace operators/BMN operators and correspondence to Green-Schwarz superstrings on AdS5 under the AdS-CFT correspondence:

• J. A. Minahan, Konstantin Zarembo, The Bethe-Ansatz for $N=4$ Super Yang-Mills, JHEP 0303 (2003) 013 (arXiv:hep-th/0212208)

• Niklas Beisert, Matthias Staudacher, The $\mathcal{N}=4$ SYM Integrable Super Spin Chain,

  Nucl. Phys. B670:439-463, 2003 (arXiv:hep-th/0307042)

• Niklas Beisert, Sergey Frolov, Matthias Staudacher, Arkady Tseytlin, Precision Spectroscopy of AdS/CFT, JHEP 0310:037, 2003 (arXiv:hep-th/0308117)

Review includes

• A. V. Belitsky, Volker Braun, A. S. Gorsky, G. P. Korchemsky, Integrability in QCD and beyond, Int. J. Mod. Phys. A19:4715-4788, 2004 (arXiv:hep-th/0407232)

• Niklas Beisert, Luis Alday, Radu Roiban, Sakura Schafer-Nameki, Matthias Staudacher, Alessandro Torrielli, Arkady Tseytlin, et. al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99, 3 (2012) (arXiv:1012.3982)

For D3-D7 brane bound states and D3-D5 brane bound states:

• Marius de Leeuw, Tamás Gombor, Charlotte Kristjansen, Georgios Linardopoulos, Balázs Pozsgay, Spin Chain Overlaps and the Twisted Yangian (arXiv:1912.09338)

See also:

• Taro Kimura, Rui-Dong Zhu, Bethe/Gauge Correspondence for $SO$/$Sp$ Gauge Theories and Open Spin Chains, JHEP 2021 227 (2021) [arXiv:2012.14197, doi:10.1007/JHEP03(2021)227]

• Ziwei Wang, Rui-Dong Zhu, Bethe/Gauge Correspondence for $A_N$ Spin Chains with Integrable Boundaries [arXiv:2401.00764]

Analogous spin chain aspects claimed to appear in D=6 N=(2,0) SCFT:

• Florent Baume, Jonathan J. Heckman, Craig Lawrie, 6D SCFTs, 4D SCFTs, Conformal Matter, and Spin Chains, Nuclear Physics B 967 (2021) 115401 [doi:10.1016/j.nuclphysb.2021.115401, arXiv:2007.07262]

• Jonathan J. Heckman, Qubit construction in 6D SCFTs, Physics Letters B 811 (2020) 135891 [doi:10.1016/j.physletb.2020.135891, arXiv:2007.08545]