Flag manifold sigma models from SU(n) chains

Wamer, Kyle; Affleck, Ian

doi:10.1016/j.nuclphysb.2020.115156

Cited by 6 publications

(8 citation statements)

References 20 publications

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“…This fact has led to a renewed theoretical interest in the field of SU(𝑛) spin chains. As a consequence, an SU(𝑛) version of Haldane's conjecture has recently been formulated [161,236,237,234].…”

Section: 𝑖=1mentioning

confidence: 99%

“…In the present section, following [234], we will consider spin chains of the following type: at each site we will place spins in an arbitrary representation R of SU(𝑛), with the condition that this representation is the same for all sites. The representation will be characterized by the lengths of the rows in the Young diagram, 𝑝 1 ≥ 𝑝 2 ≥ • • • ≥ 𝑝 𝑛−1 ≥ 0.…”

Section: More General Representations: Linear and Quadratic Dispersionmentioning

confidence: 99%

“…It was surprising to condensed matter physicists that spin chains were gapped for integer spin and surprising to high energy theorists that the O(3) non-linear sigma model was massless for 𝜃 = 𝜋. Recently, this paradigm of mapping spin chains to relativistic quantum field theories has been generalized to SU(𝑛) chains in various representations [69,73,161,236,237,234]. For chains that have a rank-𝑝 symmetric representation at each site, the corresponding field theory is a sigma model with target space SU(𝑛)/[U(1)] 𝑛−1 .…”

Section: Introductionmentioning

confidence: 99%

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Flag manifold sigma models: spin chains and integrable theories

Affleck,

Bykov,

Wamer

2021

Preprint

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This review is dedicated to two-dimensional sigma models with flag manifold target spaces, which are generalizations of the familiar CP 𝑛−1 and Grassmannian models. They naturally arise in the description of continuum limits of spin chains, and their phase structure is sensitive to the values of the topological angles, which are determined by the representations of spins in the chain. Gapless phases can in certain cases be explained by the presence of discrete 't Hooft anomalies in the continuum theory. We also discuss integrable flag manifold sigma models, which provide a generalization of the theory of integrable models with symmetric target spaces. These models, as well as their deformations, have an alternative equivalent formulation as bosonic Gross-Neveu models, which proves useful for demonstrating that the deformed geometries are solutions of the renormalization group (Ricci flow) equations, as well as for the analysis of anomalies and for describing potential couplings to fermions.

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Section: 𝑖=1mentioning

confidence: 99%

Section: More General Representations: Linear and Quadratic Dispersionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Flag manifold sigma models: spin chains and integrable theories

Affleck,

Bykov,

Wamer

2021

Preprint

Self Cite

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“…2,3 In passing, we mention another motivation to consider the flag-manifold sigma models. In (1 + 1)d, the sigma models with flag-manifold target space have appeared as effective descriptions of the SU(N )-analogue of antiferromagnetic Heisenberg-like chains [15][16][17][18], and are known to show a rich phase diagram thanks to various theta terms [19][20][21][22] (see [23] for a review). Similarly, the (2 + 1)d flag-manifold sigma models describe the 2d quantum SU(3) Heisenberg model on several lattices [24,25], and some of them are equivalent to 1 In this paper, we consider spin bordism groups Ω spin * (X) since the underlying microscopic theories are assumed to contain fermions.…”

Section: Jhep08(2021)075mentioning

confidence: 99%

Topological terms of (2+1)d flag-manifold sigma models

Kobayashi

Lee

Shiozaki

et al. 2021

J. High Energ. Phys.

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We examine topological terms of (2 + 1)d sigma models and their consequences in the light of classifications of invertible quantum field theories utilizing bordism groups. In particular, we study the possible topological terms for the U(N)/U(1)N flag-manifold sigma model in detail. We argue that the Hopf-like term is absent, contrary to the expectation from a nontrivial homotopy group π3(U(N)/U(1)N) = ℤ, and thus skyrmions cannot become anyons with arbitrary statistics. Instead, we find that there exist $$ \frac{N\left(N-1\right)}{2}-1 $$ N N − 1 2 − 1 types of Chern-Simons terms, some of which can turn skyrmions into fermions, and we write down explicit forms of effective Lagrangians.

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“…12 In passing, we mention another motivation to consider the flag-manifold sigma models. In (1 + 1)d, the sigma models with flag-manifold target space have appeared as effective descriptions of the SU (N )-analogue of antiferromagnetic Heisenberg-like chains [15][16][17][18], and are known to show a rich phase diagram thanks to various theta terms [19][20][21][22] (see [23] for a review). Similarly, the (2 + 1)d flag-manifold sigma models describe the 2d quantum SU (3) Heisenberg model on several lattices [24,25], and some of them are equivalent to the Tsunetsugu-Arikawa point [26] of the SU (2) spin-1 model.…”

Section: Introductionmentioning

confidence: 99%

Topological terms of (2+1)d flag-manifold sigma models

Kobayashi,

Lee,

Shiozaki

et al. 2021

Preprint

View full text Add to dashboard Cite

We examine topological terms of (2+1)d sigma models and their consequences in the light of classifications of invertible quantum field theories utilizing bordism groups. In particular, we study the possible topological terms for the U (N )/U (1) N flag-manifold sigma model in detail. We argue that the Hopf-like term is absent, contrary to the expectation from a nontrivial homotopy group π 3 (U (N )/U (1) N ) = Z, and thus skyrmions cannot become anyons with arbitrary statistics. Instead, we find that there exist N (N −1) 2 − 1 types of Chern-Simons terms, some of which can turn skyrmions into fermions, and we write down explicit forms of effective Lagrangians.

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Flag manifold sigma models from SU(n) chains

Cited by 6 publications

References 20 publications

Flag manifold sigma models: spin chains and integrable theories

Flag manifold sigma models: spin chains and integrable theories

Topological terms of (2+1)d flag-manifold sigma models

Topological terms of (2+1)d flag-manifold sigma models

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