We consider the O(6) nonlinear sigma model, with the level-1 Wess-Zumino-Witten term in
3+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.
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.
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
QED theory. In this paper, we propose a dimensional reduction scheme to derive the
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
Nf=4 QED3 with fermions of quadratic dispersion and show that its topological NLSM dual description has level-2 WZW terms on the Grassmannian manifold
U(2)×U(2)U(4) coupled with a dynamical
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) monopole operators and the corresponding quantum number.
I study the renormalization of D-dimensional level-k Wess-Zumino-Witten theory with Stiefel-manifold target space
StN,N−D−1≅SO(N)/SO(D+1), with a particular focus on
D=3. I investigate in particular whether such a theory admits IR-stable fixed points of the renormalization group flow. Such fixed points have been suggested to describe conformal phases of matter that do not have a known dual (super-)renormalizable Lagrangian for
N≥7 in
D=3. They are hence of interest both from the point of view of quantum phases of matter as well as pure field theory. The
D-dimensional expressions enable the computation, by analytic computation, of beta functions in
D=2+ε, at least to first non-trivial order. In
D=2, a stable fixed point is found, serving a generalization of the famed
SU(2)k Wess-Zumino-Witten conformal field theory; it annihilates in
D=2+ε with an unstable fixed point which splits off from the Gaussian one for
ε>0. Although the story is thus qualitatively similar to that of SO(5) deconfined (pseudo-)criticality, for
N⩾6, the annihilation appears to occur only for
ε>1, suggesting the existence of a stable phase in
D=3. Comparisons of the scaling dimension of the lowest singlet operator are made with known results for
N=6, which is dual to QED
3 with
Nf=4 fermion flavors. The predictions for the
N=7 Stiefel liquid represent to my knowledge the first computation of this kind for a Wess-Zumino-Witten theory without a known gauge theory dual.
Motivated by the recent interest in the criticality of open quantum many-body systems, we study nonlinear sigma models with complexified couplings as a general framework for nonunitary field theory. Applying the perturbative renormalization-group analysis to the tenfold symmetric spaces, we demonstrate that fixed points with complex scaling dimensions and critical exponents arise generically, without counterparts in conventional nonlinear sigma models with real couplings. We further clarify the global phase diagrams in the complex-coupling plane and identify both continuous and discontinuous phase transitions. Our work thus identifies nonlinear sigma models as a representative setting for studying complex critical points and elucidates universal aspects of critical phenomena in complexified field theory.