Abstract:
In the present paper we revisit the so-called Haldane limit, i.e. a particular continuum limit, which leads from a spin chain to a sigma model. We use the coherent state formulation of the path integral to reduce the problem to a semiclassical one, which leads us to the observation that the Haldane limit is closely related to a Lagrangian embedding into the classical phase space of the spin chain. Using this property, we find a spin chain whose limit produces a relativistic sigma model with target space the manifold of complete flags U(3)/U(1)3. We discuss possible other future applications of Lagrangian/isotropic embeddings in this context.
My work was supported by the Irish Research Council for Science, Engineering and Technology, in part by grants RFBR 11-01-00296-a, 11-01-12037-ofi-m-2011 and in part by grant for the Support of Leading Scientific Schools of Russia NSh-8265.2010.1.
This publication is cited in the following 27 articles:
Yuki Amari, Toshiaki Fujimori, Muneto Nitta, Keisuke Ohashi, “String junctions in flag manifold sigma models”, Phys. Rev. D, 111:6 (2025)
Dmitri Bykov, Andrew Kuzovchikov, “The classical and quantum particle on a flag manifold”, Class. Quantum Grav., 41:20 (2024), 205009
Mendel Nguyen, Yuya Tanizaki, Mithat Ünsal, “Winding θ and destructive interference of instantons”, J. High Energ. Phys., 2023:9 (2023)
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Heisenberg spin chain”, Phys. Rev. B, 107:20 (2023)
Ian Affleck, Dmitri Bykov, Kyle Wamer, “Flag manifold sigma models: Spin chains and integrable theories”, Phys. Rep., 953 (2022), 1–93
Yahya Alavirad, Maissam Barkeshli, “Anomalies and unusual stability of multicomponent Luttinger liquids in
Zn×Zn
spin chains”, Phys. Rev. B, 104:4 (2021)
Ryohei Kobayashi, Yasunori Lee, Ken Shiozaki, Yuya Tanizaki, “Topological terms of (2+1)d flag-manifold sigma models”, J. High Energ. Phys., 2021:8 (2021)
Itsuki Takahashi, Yuya Tanizaki, “Sigma-model analysis of SU(3) antiferromagnetic spins on the triangular lattice”, Phys. Rev. B, 104:23 (2021)
Tin Sulejmanpasic, Daniel Göschl, Christof Gattringer, “First-Principles Simulations of
1+1D
Quantum Field Theories at
θ=π
and Spin Chains”, Phys. Rev. Lett., 125:20 (2020)
P. Fromholz, P. Lecheminant, “Symmetry-protected topological phases in the
SU(N)
Heisenberg spin chain: A Majorana fermion approach”, Phys. Rev. B, 102:9 (2020)
Kyle Wamer, Ian Affleck, “Mass generation by fractional instantons in SU(
n
) chains”, Phys. Rev. B, 101:24 (2020)
Kyle Wamer, Miklós Lajkó, Frédéric Mila, Ian Affleck, “Generalization of the Haldane conjecture to SU(n) chains”, Nuclear Physics B, 952 (2020), 114932
Kyle Wamer, Ian Affleck, “Flag manifold sigma models from SU(n) chains”, Nuclear Physics B, 959 (2020), 115156
D. Pavshinkin, “Grassmannian and flag sigma models on interval: phase structure and L-dependence”, J. High Energ. Phys., 2019:12 (2019)
Kyle Wamer, Francisco H. Kim, Miklós Lajkó, Frédéric Mila, Ian Affleck, “Self-conjugate representation SU(3) chains”, Phys. Rev. B, 100:11 (2019)
Dmitri Bykov, “Flag manifold σ-models: The 1N-expansion and the anomaly two-form”, Nuclear Phys. B, 941 (2019), 316–360
Miklós Lajkó, Kyle Wamer, Frédéric Mila, Ian Affleck, “Corrigendum to “Generalization of the Haldane conjecture to SU(3) chains” [Nucl. Phys. B 924 (2017) 508–577]”, Nuclear Physics B, 949 (2019), 114781
Masaru Hongo, Tatsuhiro Misumi, Yuya Tanizaki, “Phase structure of the twisted SU(3)/U(1)2 flag sigma model on ℝ × S1”, J. High Energ. Phys., 2019:2 (2019)
Tatsuhiro Misumi, Yuya Tanizaki, Mithat Ünsal, “Fractional θ angle, 't Hooft anomaly, and quantum instantons in charge-q multi-flavor Schwinger model”, J. High Energ. Phys., 2019:7 (2019)
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