The paradigmatic example of deconfined quantum criticality is the Neel to valence bond solid phase transition. The continuum description of this transition is the N=2 case of the CP^{N-1} model, which is a field theory of N complex scalars in 3D coupled to an Abelian gauge field with SU(N)×U(1) global symmetry. Lattice studies and duality arguments suggest the global symmetry of the CP^{1} model is enhanced to SO(5). We perform a conformal bootstrap study of SO(5) invariant fixed points with one relevant SO(5) singlet operator, which would correspond to two relevant SU(2)×U(1) singlets, i.e., a tricritical point. We find that the bootstrap bounds are saturated by four different predictions from the large N computation of monopole operator scaling dimensions, which were recently shown to be very accurate even for small N. This suggests that the Neel to valence bond solid phase transition is described by this bootstrap bound, which predicts that the second relevant singlet has dimension ≈2.36.
The deconfined quantum critical point (DQCP) provides a prominent example of the unconventional phase transitions beyond the Landau-Ginzburg-Wilson paradigm and its nature has been controversial for decades. The DQCP has been extensively studied and the results lead to two opposite scenarios with pseudo-criticality or multicriticality. The pseudo-criticality is a prevailing scenario of DQCP which interprets the approximately scale invariant numerical results with the walking behavior near complex fixed points. In contrast, the multicriticality scenario conjectures the DQCP is a unitary fixed point with a relevant
SO(5) singlet scalar. In this work we provide substantial evidence for the multicriticality scenario using conformal bootstrap. We start with the observation that the large scale Quantum Monte Carlo (QMC) results nearly saturate the bootstrap bounds. After imposing suitable sparseness condition the bootstrap bound forms a sharp cone in the three-dimensional parameter space. The bootstrap cone is close to the QMC data. We use the navigator algorithm to locate the apex of the cone and extract the extremal solutions. We find striking consistencies between the bootstrap solutions and the fuzzy sphere data of DQCP, including the coefficients in the operator product expansions (OPEs) and the higher spectrum! The bootstrap cone unifies the QMC and the fuzzy sphere data into a unitary conformal field theory with a relevant
SO(5) singlet scalar, thus strongly supporting the multicriticality scenario of DQCP. The agreement between the conformal bootstrap, QMC and fuzzy sphere results is a surprise towards solving DQCP and decoding the profound phase diagram of the two-dimensional quantum magnets.
It has been proposed that the deconfined criticality in
(2+1)d---the quantum phase transition between a N\'eel antiferromagnet and a valence-bond solid (VBS)---may actually be pseudocritical, in the sense that it is a weakly first-order transition with a generically long correlation length. The underlying field theory of the transition would be a slightly complex (nonunitary) fixed point as a result of fixed points annihilation. This proposal was motivated by existing numerical results from large scale Monte Carlo simulations as well as a conformal bootstrap. However, an actual theory of such a complex fixed point, incorporating key features of the transition such as the emergent SO(5) symmetry, is so far absent. Here we propose a Wess-Zumino-Witten (WZW) nonlinear sigma model with level
k=1, defined in $2+\ensuremath{\epsilon}$ dimensions, with target space ${S}^{3+\ensuremath{\epsilon}}$ and global symmetry $\text{SO}(4+\ensuremath{\epsilon})$. This gives a formal interpolation between the deconfined criticality at
d=3 and the
SU(2)1 WZW theory at
d=2 describing the spin-
1/2 Heisenberg chain. The theory can be formally controlled, at least to leading order, in terms of the inverse of the WZW level
1/k. We show that at leading order there is a fixed point annihilation at ${d}^{*}\ensuremath{\approx}2.77$, with complex fixed points above this dimension including the physical
d=3 case. The pseudocritical properties such as correlation length, scaling dimensions, and the drifts of scaling dimensions as the system size increases, calculated crudely to leading order, are qualitatively consistent with existing numerics.
Monte Carlo simulations of the SU(2)-symmetric deconfined critical point action reveal strong violations of scale invariance for the deconfinement transition. We find compelling evidence that the generic runaway renormalization flow of the gauge coupling is to a weak first-order transition, similar to the case of U(1) x U(1) symmetry. Our results imply that recent numeric studies of the Nèel antiferromagnet to valence bond solid quantum phase transition in SU(2)-symmetric models were not accurate enough in determining the nature of the transition.
We study the critical properties of the
QED3-Gross-Neveu model with
2N flavors of two-component Dirac fermions coupled to a massless scalar field and a U(1) gauge field. For
N=1, this theory has recently been suggested to be dual to the SU(2) noncompact
CP1 model that describes the deconfined phase transition between the N\'eel antiferromagnet and the valence bond solid on the square lattice. For
N=2, the theory has been proposed as an effective description of a deconfined critical point between chiral and Dirac spin liquid phases, and may potentially be realizable in spin-
1/2 systems on the kagome lattice. We demonstrate the existence of a stable quantum critical point in the
QED3-Gross-Neveu model for all values of
N. This quantum critical point is shown to escape the notorious fixed-point annihilation mechanism that renders plain
QED3 (without scalar-field coupling) unstable at low values of
N. The theory exhibits an upper critical space-time dimension of four, enabling us to access the critical behavior in a controlled expansion in the small parameter $\ensuremath{\epsilon}=4\ensuremath{-}D$. We compute the scalar-field anomalous dimension ${\ensuremath{\eta}}_{\ensuremath{\phi}}$, the correlation-length exponent $\ensuremath{\nu}$, as well as the scaling dimension of the flavor-symmetry-breaking bilinear $\overline{\ensuremath{\psi}}{\ensuremath{\sigma}}^{z}\ensuremath{\psi}$ at the critical point, and compare our leading-order estimates with predictions of the conjectured duality.
The deconfined quantum critical point (DQCP) is an example of phase transitions beyond the Landau symmetry-breaking paradigm that attracts wide interest. However, its nature has not been settled after decades of study. In this paper, we apply the recently proposed fuzzy-sphere regularization to study the SO(5) nonlinear sigma model with a topological Wess-Zumino-Witten term, which serves as a dual description of the DQCP with an exact SO(5) symmetry. We demonstrate that the fuzzy sphere functions as a powerful microscope, magnifying and revealing a wealth of crucial information about the DQCP, ultimately paving the way toward its final answer. In particular, through exact diagonalization, we provide clear evidence that the DQCP exhibits approximate conformal symmetry. The evidence includes the existence of a conserved SO(5) symmetry current, a stress tensor, and integer-spaced levels between conformal primaries and their descendants. Most remarkably, we identify 23 primaries and 76 conformal descendants. Furthermore, by examining the renormalization group flow of the lowest symmetry singlet as well as other primaries, we provide numerical evidence in favor of DQCP being pseudo-critical, with the approximate conformal symmetry plausibly emerging from nearby complex fixed points. The primary spectrum we compute also has important implications, including the conclusion that the SO(5) DQCP cannot describe a direct transition from the Néel to valence bond solid phase on the honeycomb lattice. Published by the American Physical Society 2024
We show numerically that the "deconfined" quantum critical point between the Néel antiferromagnet and the columnar valence-bond solid, for a square lattice of spin 1/2, has an emergent SO(5) symmetry. This symmetry allows the Néel vector and the valence-bond solid order parameter to be rotated into each other. It is a remarkable (2+1)-dimensional analogue of the SO(4)=[SU(2)×SU(2)]/Z(2) symmetry that appears in the scaling limit for the spin-1/2 Heisenberg chain. The emergent SO(5) symmetry is strong evidence that the phase transition in the (2+1)-dimensional system is truly continuous, despite the violations of finite-size scaling observed previously in this problem. It also implies surprising relations between correlation functions at the transition. The symmetry enhancement is expected to apply generally to the critical two-component Abelian Higgs model (noncompact CP(1) model). The result indicates that in three dimensions there is an SO(5)-symmetric conformal field theory that has no relevant singlet operators, so is radically different from conventional Wilson-Fisher-type conformal field theories.