The interplay between bulk critical fluctuations and nontrivial topology can enrich defect physics and give rise to novel defect universality classes. Understanding the fate of such defects therefore constitutes an important open problem. In this work, we studied a particularly simple setting: a (0+1)-dimensional pinning-field defect coupled to a (2+1)-dimensional deconfined quantum critical bulk. Using the fuzzy-sphere regularization, we numerically investigated the defect operator spectrum and extracted several universal quantities characterizing the defect conformal fixed point, including the scaling dimensions of defect-changing(creating) operators and the defect \(g\)-function. These results establish the first numerical characterization of line-defect conformal data at deconfined quantum criticality and may stimulate further investigations of defect critical phenomena in topological quantum critical matter.
The interplay between bulk critical fluctuations and nontrivial topology can enrich defect physics and give rise to novel defect universality classes. Understanding the fate of such defects therefore constitutes an important open problem. In this work, we studied a particularly simple setting: a (0+1)-dimensional pinning-field defect coupled to a (2+1)-dimensional deconfined quantum critical bulk. Using the fuzzy-sphere regularization, we numerically investigated the defect operator spectrum and extracted several universal quantities characterizing the defect conformal fixed point, including the scaling dimensions of defect-changing(creating) operators and the defect (g)-function. These results establish the first numerical characterization of line-defect conformal data at deconfined quantum criticality and may stimulate further investigations of defect critical phenomena in topological quantum critical matter.
Monodromy (twist) line defects arise as the boundaries of topological symmetry surfaces, which are among the simplest non-trivial conformal defects with direct realizations as magnetic vortex in quantum magnets, as Wilson and t Hooft lines of gauge theories, and as the replica twist operators in quantum information theory. However, their interacting dynamics remains difficult to characterize non-perturbatively. Here we design a way to realize the monodromy boundary conditions in the fuzzy-sphere regularization by threading a symmetry-selective Aharonov-Bohm flux through the two poles of the sphere. We apply this construction to the Z2 twist of the 3D Ising conformal field theory (CFT), π-twist of the 3D O(2) CFT and three inequivalent monodromies of O(3) (centre twist, z-axis π-twist and the reflection). For each defect, we extract the low-lying defect spectrum together with bulk-to-defect and defect-defect operator product expansion coefficients, which characterize the operators living on the defect line, their mutual interactions, and how they couple to the surrounding critical bulk. Among the low-lying defect primaries resolved in our spectra, we find no relevant scalar perturbation that preserves all the symmetries of the corresponding defect. The relevant deformations identified in our calculations break at least one preserved symmetry, providing evidence that these monodromy defects are stable against symmetry-preserving relevant perturbations. These results establish a systematic set of non-perturbative conformal data for monodromy defects beyond the Ising twist. Our construction therefore turns the symmetry flux into a tunable knob of the fuzzy-sphere regularization, opening a route to exploring the response of critical matter to distinct defects.
The g-function and defect changing operators from wavefunction overlap on a fuzzy sphere
Defects are common in physical systems with boundaries, impurities or extensive measurements. The interaction between bulk and defect can lead to rich physical phenomena. Defects in gapless phases of matter with conformal symmetry usually flow to a defect conformal field theory (dCFT). Understanding the universal properties of dCFTs is a challenging task. In this paper, we propose a computational strategy applicable to a line defect in arbitrary dimensions. Our main assumption is that the defect has a UV description in terms of a local modification of the Hamiltonian so that we can compute the overlap between low-energy eigenstates of a system with or without the defect insertion. We argue that these overlaps contain a wealth of conformal data, including the g g -function, which is an RG monotonic quantity that distinguishes different dCFTs, the scaling dimensions of defect creation operators \Delta^{+0}_\alpha Δ α + 0 and changing operators \Delta^{+-}_\alpha Δ α + − that live on the intersection of different types of line defects, and various OPE coefficients. We apply this method to the fuzzy sphere regularization of 3D CFTs and study the magnetic line defect of the 3D Ising CFT. Using exact diagonalization and DMRG, we report the non-perturbative results g=0.602(2),\Delta^{+0}_0=0.108(5) g = 0.602 ( 2 ) , Δ 0 + 0 = 0.108 ( 5 ) and \Delta^{+-}_0=0.84(5) Δ 0 + − = 0.84 ( 5 ) for the first time. We also obtain other OPE coefficients and scaling dimensions. Our results have significant physical implications. For example, they constrain the possible occurrence of spontaneous symmetry breaking at line defects of the 3D Ising CFT. Our method can be potentially applied to various other dCFTs, such as plane defects and Wilson lines in gauge theories.
Symmetry-breaking line defects embedded to a 3D O(N) critical bulk
While spontaneous breaking of a discrete symmetry in one-dimensional classical systems with short-range interactions is absent, it is expected that a line defect embedded in a bulk criticality exhibits a stable discrete symmetry spontaneous breaking. Here, we investigate the behavior of a pinning-field line defect immersed in a 3D bulk that remains tuned to the O(N) Wilson-Fisher critical point. Employing the fuzzy sphere technique, we provide convincing evidence of the existence of stable defect conformal fixed points, and we demonstrate their renormalization group stability by showing no relevant operator and less effective degrees of freedom than that at bulk fixed point via g-function. Moreover, we investigate the defect domain wall operator for various N, and we identify that it becomes irrelevance for N≳3 but it is relevant for N<3.These evidence indicate that a one-dimensional defect coupled to a critical bulk cannot support a stable symmetry spontaneously broken defect fixed point due to domain wall proliferation for N<3 Wilson-Fisher universality, while in the case of N≳3 a symmetry broken defect is possible.
Boundary conformal field theory (BCFT) provides a universal framework for critical phenomena in the presence of boundaries. We determine BCFT data for the normal and ordinary boundary universality classes of the 1+1-dimensional boundaries of the 2+1-dimensional O(2) and O(3) Wilson-Fisher fixed points, realized microscopically by a bilayer Heisenberg model on the fuzzy sphere. Using the fuzzy-sphere state-operator correspondence, we obtain boundary operator spectra, identify low-lying boundary primary operators, extract operator-product-expansion (OPE) data, and estimate the boundary central charges for both boundary conditions. For the normal boundary condition, the universal amplitudes aσ and bt extracted from one- and two-point functions agree quantitatively with Monte Carlo benchmarks where available. For both N=2 and N=3, we find a positive extraordinary-log exponent α, providing independent microscopic evidence for extraordinary-log boundary criticality. Our results extend fuzzy-sphere BCFT spectroscopy beyond the Ising universality class to continuous O(N) symmetry.
A challenge in the study of conformal field theory (CFT) is to characterize the possible defects in specific bulk CFTs. Given the success of numerical bootstrap techniques applied to the characterization of bulk CFTs, it is desirable to develop similar tools to study conformal defects. In this work, we successfully demonstrate this possibility for endable conformal line defects. We achieve this by incorporating the endpoints of a conformal line defect into the numerical four-point bootstrap and exploit novel crossing symmetry relations that mix bulk and defect CFT data in a way that further possesses positivity, so that rigorous numerical bootstrap techniques are applicable. We implement this approach for the pinning field line defect of the 3d Ising CFT, obtaining estimates of its defect CFT data that agree well with other recent estimates, particularly those obtained via the fuzzy sphere regularization. An interesting consequence of our bounds is nearly rigorous evidence that the Z2-symmetric defect exhibiting long range order obtained as a direct sum of two conjugate pinning field defects is unstable to domain wall proliferation.
Conformal field theory (CFT) offers a modern viewpoint for understanding phase transitions. However, directly accessing the conformal algebra and microscopically uncovering the emergent conformal symmetry, especially in higher dimensions, remains a significant challenge. Motivated by recent advances in revealing CFT features via the fuzzy two-sphere, here we generalize this approach to higher dimensions and aim to expose the conformality at the (3+1)-D quantum critical point. We demonstrate this framework by investigating quantum phase transitions belonging to the Ising and Yang-Lee universality classes in a (3+1)-D quantum model (equivalent to a classical four-dimensional system), realized via Landau level projection on the fuzzy three-sphere. By computing the energy spectra at criticality, we explicitly verify the state-operator correspondence, a hallmark of conformal invariance. Together with prior advances, this work establishes a new pathway for the microscopic study of emergent conformality in higher-dimensional phase transitions.