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arXiv:2208.10518

Dual topological nonlinear sigma models of $\text{QED}$ theory by dimensional reduction and monopole operators

Wu, Zhengzhi, Linhao Li

Published Aug 22, 2022• 2 citations
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Abstract

Nonlinear $σ$ models (NLSM) with topological terms, i.e., Wess-Zumino-Witten (WZW) terms, or topological NLSM, are potent descriptions of many critical points and phases beyond the Landau paradigm. These critical systems include the deconfined quantum critical points (DQCP) between the Neel order and valance bond solid, and the Dirac spin liquid, in which the topological NLSMs are dual descriptions of the corresponding fermionic models or $\text{QED}$ theory. In this paper, we propose a dimensional reduction scheme to derive the $\text{U}(1)$ gauged topological NLSM in $n$-dimensional spacetime on a general target space represented by a Hermitian matrix from the dual QED theory. Compared with the famous Abanov-Wiegmann (AW) mechanism, which generally requires the fermions to be Dirac fermions in the infrared (IR), our method is also applicable to non-relativistic fermions in IR, which can have quadratic dispersion or even a Fermi surface. As concrete examples, we construct several two dimensional lattice models, whose IR theories are all the $N_f=4$ $\text{QED}_3$ with fermions of quadratic dispersion and show that its topological NLSM dual description has level-2 WZW terms on the Grassmannian manifold $\frac{\text{U(4)}}{\text{U(2)}\times \text{U(2)}}$ coupled with a dynamical $\text{U(1)}$ gauge field. We also study 't Hooft anomaly matching and the same effect of defects in both theories, such as interface, gauge monopoles and skyrmions, which further support our duality. Finally, we discuss how the macroscopic symmetries act on the $\text{U(1)}$ monopole operators and the corresponding quantum number.

DOI
10.48550/arxiv.2208.10518

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Dual topological nonlinear sigma models of QED\text{QED}QED theory by dimensional reduction and monopole operators
Nonlinear σσσ models (NLSM) with topological terms, i.e., Wess-Zumino-Witten (WZW) terms, or topological NLSM, are potent descriptions of many critical points and phases beyond the Landau paradigm. These critical systems include the deconfined quantum critical points (DQCP) between the Neel order and valance bond solid, and the Dirac spin liquid, in which the topological NLSMs are dual descriptions of the corresponding fermionic models or QED\text{QED}QED theory. In this paper, we propose a dimensional reduction scheme to derive the U(1)\text{U}(1)U(1) gauged topological NLSM in nnn-dimensional spacetime on a general target space represented by a Hermitian matrix from the dual QED theory. Compared with the famous Abanov-Wiegmann (AW) mechanism, which generally requires the fermions to be Dirac fermions in the infrared (IR), our method is also applicable to non-relativistic fermions in IR, which can have quadratic dispersion or even a Fermi surface. As concrete examples, we construct several two dimensional lattice models, whose IR theories are all the Nf=4N_f=4Nf​=4 QED3\text{QED}_3QED3​ with fermions of quadratic dispersion and show that its topological NLSM dual description has level-2 WZW terms on the Grassmannian manifold U(4)U(2)×U(2)\frac{\text{U(4)}}{\text{U(2)}\times \text{U(2)}}U(2)×U(2)U(4)​ coupled with a dynamical U(1)\text{U(1)}U(1) gauge field. We also study 't Hooft anomaly matching and the same effect of defects in both theories, such as interface, gauge monopoles and skyrmions, which further support our duality. Finally, we discuss how the macroscopic symmetries act on the U(1)\text{U(1)}U(1) monopole operators and the corresponding quantum number.
In this paper, we propose using the nonlinear sigma model (NLSM) with the Wess-Zumino-Witten (WZW) term as a general description of deconfined quantum critical points that separate two spontaneously symmetry-breaking (SSB) phases in arbitrary dimensions. In particular, we discuss the suitable choice of the target space of the NLSM, which is in general the homogeneous space G/K, where G G is the UV symmetry and K K is generated by {\mathfrak k}={\mathfrak h}_1\cap {\mathfrak h}_2 𝔨 = 𝔥 1 ∩ 𝔥 2 , and {\mathfrak h}_i 𝔥 i is the Lie algebra of the unbroken symmetry in each SSB phase. With this specific target space, the symmetry defects in both SSB phases are on equal footing, and their intertwinement is captured by the WZW term. The DQCP transition is then tuned by proliferating the symmetry defects. By coupling the G/K G / K NLSM with the WZW term to the background gauge field, the ’t Hooft anomaly of this theory can be determined. The bulk symmetry-protected topological (SPT) phase that cancels the anomaly is described by the relative Chern-Simons term in odd spacetime dimensions or mixed \theta θ term in even dimensions. We construct and discuss a series of models with Grassmannian symmetry defects in 3+1d. We also provide the fermionic model that reproduces the G/K G / K NLSM with the WZW term.
Global symmetries can be generalised to transformations generated by topological operators, including cases in which the topological operator does not have an inverse. A family of such topological operators are intimately related to dualities via the procedure of half-space gauging. In this work we discuss the construction of non-invertible defects based on T-duality in two dimensions, generalising the well-known case of the free compact boson to any Non-Linear Sigma Model with Wess-Zumino term which is T-dualisable. This requires that the target space has an isometry with compact orbits that acts without fixed points. Our approach allows us to include target spaces without non-trivial 1-cycles, does not require the NLSM to be conformal, and when it is conformal it does not need to be rational; moreover, it highlights the microscopic origin of the topological terms that are responsible for the non-invertibility of the defect. An interesting class of examples are Wess-Zumino-Witten models, which are self-dual under a discrete gauging of a subgroup of the isometry symmetry and so host a topological defect line with Tambara-Yamagami fusion. Along the way, we discuss how the usual 0-form symmetries match across T-dual models in target spaces without 1-cycles, and how global obstructions can prevent locally conserved currents from giving rise to topological operators.
We consider the O(6) nonlinear sigma model, with the level-1 Wess-Zumino-Witten term in 3+13+13+1 space-time dimensions. If we divide the O(6) order parameter into two sets of three components, the solitons (hedgehogs) in the first and second set play the role of charge and monopole in the electromagnetic duality. The condensation of these solitons can trigger a direct transition between orders characterized by the two sets of order parameters. We conjecture that the critical point occurs when the theory is self-dual. Moreover, the theory possesses two deconfined phases, where finite-energy hedgehogs coexist with gapless U(1) gauge bosons.
We grow the web of dualities for conformal field theories (CFTs) in 2+1 spacetime dimensions to include quantum critical transitions of Dirac fermions. Our construction uses the seed duality, equating a free Dirac fermion with a complex boson coupled to a Chern-Simons gauge field, to express various Gross-Neveu-Yukawa (GNY) critical points of N fermions in terms of the same number of complex bosons and partner gauge fields. In some instances, monopole operators appear explicitly in the dual action, such as for the superconducting transition of a single Dirac fermion, which is a CFT with emergent supersymmetry. We match phase diagrams, symmetries and their anomalies. By further gauging certain symmetries, we derive offspring dualities. For example, starting with the duality for the regular GNY transition of 2 Dirac fermions, we arrive at the duality between a Neel-to-VBS deconfined quantum critical point and a gauged-Gross-Neveu transition. Finally, we use a large-N expansion for monopole operators, and compare with known scaling dimensions in the simplest single-flavour GNY transition.
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We develop an optimized continuous-field quantum Monte Carlo (QMC) algorithm to investigate the projected SO(5) nonlinear sigma model with a Wess-Zumino-Witten term, which describes half-filled Dirac fermions in 2+1 space-time dimensions akin to graphene and Yukawa coupled to a quintuplet of compatible mass terms. Our algorithm reduces the computational complexity to O(βNqNϕ2)O(βN_{\mathbf{q}} N_ϕ^2)O(βNq​Nϕ2​), yielding a speedup of a factor of NϕN_ϕNϕ​ (the number of magnetic fluxes, i.e., system size) relative to prior works [1-4]. This advance enables us to simulate system sizes up to Nϕ=140N_ϕ=140Nϕ​=140 on the torus and Nϕ=59N_ϕ=59Nϕ​=59 on the sphere, far exceeding the maximum sizes previously accessed, and to map out the universal phase diagram of the model on both geometries. Most notably, we identify and characterize a critical point that separates an SO(5)-broken ordered phase at small coupling from an SO(5)-symmetric disordered phase at large coupling. The critical point becomes multicritical upon the inclusion of terms that break the SO(5) symmetry down to U(1)×SU(2)\mathrm{U}(1) \times \mathrm{SU}(2)U(1)×SU(2), relevant for the deconfined phase transition between Néel antiferromagnetic and valence-bond-solid orders in quantum magnets. Our finding of a multicritical point in the phase diagram of the SO(5) nonlinear sigma model with Wess-Zumino-Witten term resolves the long-standing open question of its global structure, and our QMC algorithm opens a new avenue for systematic studies of projected Hamiltonians, ranging from correlated flat bands to fractional quantum (anomalous) Hall systems.