Monodromy Defects of 3D Wilson-Fisher Criticality on the Fuzzy Sphere

Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Monodromy Defects of 3D Wilson-Fisher Criticality on the Fuzzy Sphere".

Mira: Here's the gist of the paper. The gist Monodromy defects arise as 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 Realization via Fuzzy Sphere The construction realizes monodromy boundary conditions by threading a symmetryselective Aharonov–Bohm flux through the two poles of the sphere This mechanism shifts the orbitals of flavour symmetry-odd fermions to half-integer m, which is achieved by shifting the angular-momentum grid of these flavours by half a unit The construction preserves the symmetries of the defect exactly and applies to any involution acting diagonally on the fermion flavours This allows for five distinct monodromy defects to be studied: the Z2 twist of the Ising CFT, the pi-twist of the O(2) CFT, and three inequivalent monodromies of O(3) Defect Spectrum and Spectrum Data The paper extracts the low-lying defect spectrum together with bulk-to-defect and defect-defect operator product expansion coefficients For the Ising Z2 twist defect, the lowest defect operator extrapolates to Δˆpsi1/2 = 0.91(2), consistent with Monte Carlo estimate 0.9187(6)

38: and bootstrap 0.9187

44: A intriguing feature is the observation of defect operators ψs with half-integer transverse spins s up to 7/2 The lowest defect operator extrapolates to Δˆpsi1/2 = 0.91(2), consistent with the Monte Carlo estimate 0.9187(6)

38: and the bootstrap 0.9187

44: Bulk-to-Defect and Defect-Defect OPEs The paper provides bulk-to-defect OPE coefficients, such as bsigmapsi1/2 = 0.857(3) for the Ising defect The coupling of the order parameter to the lowest mode follows an ordering set by the one-loop attraction For O(3) defects, bulk-to-defect couplings are close to the trivial-defect value 1, and their behavior is described by a trend that depends on (ntw + 2)/(M + 8) Stability and Physical Interpretation A key finding is that within the resolved spectra none of the five defects supports a symmetry-preserving relevant operator, indicating that these monodromy defects are stable against symmetry-preserving relevant perturbations The twisted sector of all defects admits a simple description: the twisted modes form, to a good approximation, a generalized free field dressed by a flavour-symmetric contact interaction This picture is reminiscent of the universality found at large transverse spin

38: , but here it holds already for the lowest modes s = 1/2 General Features and Future Directions The results support a consistent physical picture where monodromy defect operators can be well captured by a generalized free field dressed by a flavour-symmetric contact interaction The construction promotes the symmetry flux to a tunable parameter of the regularization, on the same footing as the pinning fields and impurities realized so far This work opens a new avenue for the non-perturbative study of monodromy defects in three-dimensional CFTs The construction itself generalizes in several ways, including realizing continuous monodromies e 2πi vQ of U(1) symmetries by fractional fluxes This provides the conformal data of fractional-charge Wilson lines in the critical abelian Higgs model The construction also gives a direct non-perturbative route to the Renyi entropies and their universal corner and defect contributions in 3D CFTs Finally, gauging the twisted symmetry turns the monodromy defects studied here into the vortex lines of Z2 and U(1) gauge theories The genuine defect spectra are given by the g-even sector of the present data, while the g-odd operators become endpoints of Wilson lines How it works The monodromy boundary condition is realized by threading a symmetryselective Aharonov–Bohm flux through the two poles of the sphere This flux causes the flavour symmetry-odd fermions to acquire a sign when transported once around the z-axis relative to untwisted flavours The twisted flavour is placed on lowest-Landau level orbitals whose angular-momentum grid is shifted by half a unit This shift ensures that the exponents s ± m of the twisted orbitals are half-integers ≥ 1/2, making them regular at both poles Defect Operator Spectrum and Interactions The bulk-to-defect expansion shows that every conformal line defect contains the displacement operator Dˆ with s = ±1 and ΔˆDˆ = 2 The two-mode operator receives a contact term γc = 3ε/(M+8)Tr

r 2/1+s1+s2: This contact term is halved for identical modes, and its flavour weight TR depends on the symmetry O(M') acting on the pair The results establish a systematic set of non-perturbative conformal data for monodromy defects beyond the Ising twist The construction 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 twisted sector of all defects is remarkably uniform, with the lowest mode lying below 1/2 + Δphi by an amount that grows with (ntw + 2)/(M + 8) The lowest O(3)-invariant defect operator resolved is

ψs: S at 2.51(5), far above marginality The centre twist is the most robustly stable of the five Z2 defects How it works The interaction energy Δˆ

ψψ: - 2Δˆψ1/2, which measures the shift of a two-mode operator from the sum of its constituents, vanishes for a generalized free field At one loop it is the contact term Eq.

Paper summary: Kai: So, wrapping up the "Monodromy Defects of three dee Wilson-Fisher Criticality on the Fuzzy Sphere" paper, what we’re seeing is this systematic way to study monodromy line defects using a symmetry-selective Aharonov–Bohm flux through the poles of a fuzzy sphere <ref:2610.31269#pg1,Monodromy Defects of 3D Wilson-Fisher Criticality on the Fuzzy Sphere>.

Mira: It’s about applying this construction to various CFTs, like the Z2 twist of Ising CFT and O(three) defects, to extract the low-lying defect spectrum and the bulk–to–defect and defect–defect operator product expansion coefficients <ref:2610.31269#pg1,bulk–to–defect and defect–defect operator>.

Lev: The key result here is that no relevant scalar perturbation was found that preserves all symmetries for any of these five defects, which suggests they are stable against symmetry-preserving relevant deformations.

Kai: And the paper demonstrates how this construction turns the symmetry flux into a tunable knob in the fuzzy sphere regularization, which opens up a route to explore matter's response to distinct defects.

Mira: Essentially, it gives us non-perturbative conformal data for monodromy defects beyond just the Ising twist, and it connects these line defects directly to vortex lines in Z2 and U(one) gauge theories <ref:2610.31269#pg1>.

Lev: It provides a way to probe how critical matter responds to these topological features by tuning the symmetry flux parameter in the regularization scheme.

Conclusion: Kai: So, we're wrapping up our talk on these monodromy defects of three dee Wilson-Fisher criticality on the fuzzy sphere.

Mira: It basically boils down to this paper by Author Names, looking at how those topological symmetry boundaries behave in a regularized system.

Lev: The main thing they do is map out the low-lying defect spectrum for five different types of monodromies, like the Z2 twist of Ising CFT and O(three) defects.

Kai: And they find that these defects are remarkably stable against relevant perturbations that preserve all the original symmetries. That's a big statement about their robustness.

Mira: Exactly, and they’ve managed to turn that symmetry flux into a tuning knob for the regularization itself, which is really interesting for how we can study these systems.

Lev: From my side, it means this construction gives us a solid framework to get non-perturbative conformal data for these defects in three dee CFTs.

Kai: It opens up a new avenue to explore how critical matter responds when you introduce these specific topological features through the flux.

Mira: And it connects those line defects directly to vortex lines in gauge theories, which has some real implications for understanding those connections.

Lev: We can see how this construction also helps us get non-perturbative Renyi entropies and their universal contributions in three dee CFTs.

Kai: So, the main point is that we have a systematic way to probe these defects using this flux knob, and it’s not just limited to the Ising twist anymore.

Mira: The real question now is how this framework scales up to more complex systems or higher-dimensional theories.