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.
It is known that the U ( 2 ) Wess-Zumino-Witten model is dual to the free fermion theory in two dimensions via non-Abelian bosonization. While it is decomposed into the S U ( 2 ) Wess-Zumino-Witten model and a free compact boson, the former is believed to be equivalent to the O ( 3 ) nonlinear sigma model with the theta term at θ = π . In this work, we reexamine this duality through the lens of nonperturbative renormalization group (RG) flow. We analyze the RG flow structure of the O ( 3 ) nonlinear sigma model with the theta term in two dimensions using the functional renormalization group. Our results reveal a nontrivial fixed point with a nonzero value of the topological coupling. The scaling dimensions (critical exponents) at this fixed point suggest the realization of a duality between the O ( 3 ) nonlinear sigma model with the theta term and the free fermion theory, indicating that these models belong to the same universality class.
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.
We suggest the possibility that the two-dimensional
SU(2)k Wess-Zumino-Witten (WZW) theory, which has global
SO(4) symmetry, can be continued to $2+\ensuremath{\epsilon}$ dimensions by enlarging the symmetry to $\mathrm{SO}(4+\ensuremath{\epsilon})$. This is motivated by the three-dimensional sigma model with
SO(5) symmetry and a WZW term, which is relevant to deconfined criticality. If such a continuation exists, the structure of the renormalization group flows at small $\ensuremath{\epsilon}$ may be fixed by assuming analyticity in $\ensuremath{\epsilon}$. This leads to the conjecture that the WZW fixed point annihilates with a new, unstable fixed point at a critical dimensionality
dc>2. We suggest that
dc<3 for all
k, and we compute
dc in the limit of large
k. The flows support the conjecture that the deconfined phase transition in
SU(2) magnets is a ``pseudocritical'' point with approximate
SO(5), controlled by a fixed point slightly outside the physical parameter space.
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), yielding a speedup of a factor of
Nϕ (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=140 on the torus and
Nϕ=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), 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.
We study
3d (or
(3+1)d) Quantum Electrodynamics (QED) realized on the boundary of
4d (or
(4+1)d) bosonic symmetry protected topological (BSPT) states, using a systematic nonlinear sigma model (NLSM) field theory description of BSPT states. We demonstrate that many of these QED states have an exact electric-magnetic duality due to the symmetry of the BSPT states in the
4d bulk. The gauge charge and Dirac monopole both carry projective representations of the bulk symmetry, and the emergent gapless photons of the QED phase also transform nontrivially under the bulk symmetry. Some of these QED boundary states can be further driven into a
3d Z2 topological order, and the statistics and symmetry transformation of its point particle and vison loop excitations guarantee that this topological order cannot be driven into a trivial confined or Higgs phase. With a finite fourth dimension, the entire system becomes a
3d lattice, the self-dual QED and the
Z2 topological order can coexist on two opposite boundaries respectively, which together constitute an exotic
3d self-dual "topological photon phase".