Flag manifold sigma models: spin chains and integrable theories

Affleck, Ian; Bykov, Dmitri; Wamer, Kyle

doi:10.48550/arxiv.2101.11638

Cited by 11 publications

(15 citation statements)

References 252 publications

(440 reference statements)

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“…The 4-dimensional gauge theoretic origin of 2-dimensional integrable field theories, as proposed by Costello and Yamazaki in [CY], has been extensively studied over the past couple of years, see for instance [DLMV3,Sc1,FSY1,FSY2,CS,T,THC,BSV,BS,LV,CSV,FSY3,ABW,FSY4,D,St,Sc2,L2,FSY5].…”

Section: Quantum Gaudin Modelsmentioning

confidence: 99%

3-dimensional mixed BF theory and Hitchin's integrable system

Vicedo,

Winstone

2022

Preprint

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The affine Gaudin model, associated with an untwisted affine Kac-Moody algebra, is known to arise from a certain gauge fixing of 4-dimensional mixed topological-holomorphic Chern-Simons theory in the Hamiltonian framework. We show that the finite Gaudin model, associated with a finite-dimensional semisimple Lie algebra, or more generally the tamely ramified Hitchin system on an arbitrary Riemann surface, can likewise be obtained from a similar gauge fixing of 3-dimensional mixed BF theory in the Hamiltonian framework.

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Section: Quantum Gaudin Modelsmentioning

confidence: 99%

3-dimensional mixed BF theory and Hitchin's integrable system

Vicedo,

Winstone

2022

Preprint

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“…Here, we show how to produce the Lax pair and Lagrangian for the model discussed in [ABW,Section 16.2] (see also [By] and references therein for the particular case of rank M = 1) as a particular case of our construction. In fact, more than just this single Lagrangian and its Lax pair, we can in principle generate all the elementary Lagrangians in the whole Lagrangian multiform and all the elementary Lax pairs for the hierarchy containing this model as its main representative.…”

Section: Deformed Sigma and Gross-neveu Modelsmentioning

confidence: 99%

Classical Yang-Baxter equation, Lagrangian multiforms and ultralocal integrable hierarchies

Caudrelier¹,

Stoppato²

2022

Preprint

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We cast the classical Yang-Baxter equation (CYBE) in a variational context for the first time, by relating it to the theory of Lagrangian multiforms, a framework designed to capture integrability in a variational fashion. This provides a significant connection between Lagrangian multiforms and the CYBE, one of the most fundamental concepts of integrable systems. This is achieved by introducing a generating Lagrangian multiform which depends on a skew-symmetric classical r-matrix with spectral parameters. The multiform Euler-Lagrange equations produce a generating Lax equation which yields a generating zero curvature equation. The CYBE plays a role at three levels: 1) It ensures the commutativity of the flows of the generating Lax equation; 2) It ensures that the generating zero curvature equation holds; 3) It implies the closure relation for the generating Lagrangian multiform. The specification of an integrable hierarchy is achieved by fixing certain data: a finite set S ∈ CP 1 , a Lie algebra g, a g-valued rational function with poles in S and an r-matrix. We show how our framework is able to generate a large class of ultralocal integrable hierarchies by providing several known and new examples pertaining to the rational or trigonometric class. These include the Ablowitz-Kaup-Newell-Segur hierarchy, the sine-Gordon (sG) hierarchy and various hierachies related to Zakharov-Mikhailov type models which contain the Faddeev-Reshetikhin (FR) model and recently introduced deformed sigma/Gross-Neveu models as particular cases. The versatility of our method is illustrated by showing how to couple integrable hierarchies together to create new examples of integrable field theories and their hierarchies. We provide two examples: the coupling of the nonlinear Schrödinger system to the FR model and the coupling of sG with the anisotropic FR model.

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“…In this paper, we consider the SU (3) antiferromagnetic spins on the triangular lattice, and generalize the Néel-VBS transition to SU (3) spin systems. Such a generalization from SU (2) spins to SU (N ) spins has been recently considered in the context of spin chains [5][6][7][8][9][10][11][12], which leads to the SU (N ) generalization of the Haldane conjecture. On the triangular lattice, the classical SU (3) spins can be set to the Néel order without any frustrations thanks to the tripartite lattice structure.…”

Section: Introductionmentioning

confidence: 99%

Sigma-model analysis of $SU(3)$ antiferromagnetic spins on the triangular lattice

Takahashi,

Tanizaki

2021

Preprint

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Using field-theoretic techniques, we study the SU (3) analogue of anti-ferromagnetic Heisenberg spin model on the triangular lattice putting the p-box symmetric representation on each site. Taking the large-p limit, we show that the low-energy effective theory is described by a (2 + 1)-dimensional relativistic SU (3)/U (1) 2 nonlinear sigma model. Since the target space has a nontrivial homotopy π2(SU (3)/U (1) 2 )Z 2 , this model has two kinds of magnetic skyrmions, which can be created and annihilated by monopole instantons. By careful analysis of the Wess-Zumino term in the spin coherent path integral, we compute the Berry phase for these monopoles and it produces the destructive interference. This restricts possible perturbations of the effective Lagrangian by monopole operators, and we see that the valence-bond-solid (VBS) phase should have degenerate ground states when p ∈ 3Z. We also compute 't Hooft anomalies to constrain possible phases of this system, and a direct phase transition between Néel and VBS phases is supported from the anomaly matching.

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Flag manifold sigma models: spin chains and integrable theories

Cited by 11 publications

References 252 publications

3-dimensional mixed BF theory and Hitchin's integrable system

3-dimensional mixed BF theory and Hitchin's integrable system

Classical Yang-Baxter equation, Lagrangian multiforms and ultralocal integrable hierarchies

Sigma-model analysis of $SU(3)$ antiferromagnetic spins on the triangular lattice

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