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arXiv:2401.00039 · SciPost Physics

The $g$-function and defect changing operators from wavefunction overlap on a fuzzy sphere

Zheng Zhou, Davide Gaiotto A, Yin-Chen He, Yijian Zou

Published Jul 25, 2024• 33 citations
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Abstract

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.

DOI
10.21468/scipostphys.17.1.021

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The ggg-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.
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\mathbb{Z}_2Z2​ twist of the 3D Ising conformal field theory (CFT), πππ-twist of the 3D O(2) CFT and three inequivalent monodromies of O(3) (centre twist, z-axis πππ-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 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 ggg-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.
Abstract We explore a C-theorem in defect conformal field theories (DCFTs) that unify all the known conjectures and theorems until now. We examine as a candidate C-function the additional contributions from conformal defects to the sphere free energy and the entanglement entropy across a sphere in a number of examples including holographic models. We find the two quantities are equivalent, when suitably regularized, for codimension-one defects (or boundaries), but differ by a universal constant term otherwise. Moreover, we find in a few field theoretic examples that the sphere free energy decreases but the entanglement entropy increases along a certain renormalization group (RG) flow triggered by a defect localized perturbation which is assumed to have a trivial IR fixed point without defects. We hence propose a C-theorem in DCFTs stating that the increment of the regularized sphere free energy due to the defect does not increase under any defect RG flow. We also provide a proof of our proposal in several holographic models of defect RG flows.
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+11+11+1-dimensional boundaries of the 2+12+12+1-dimensional O(2)O(2)O(2) and O(3)O(3)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 aσa_σaσ​ and btb_tbt​ extracted from one- and two-point functions agree quantitatively with Monte Carlo benchmarks where available. For both N=2N=2N=2 and N=3N=3N=3, we find a positive extraordinary-log exponent ααα, 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)O(N)O(N) symmetry.
We consider defect conformal field theories (DCFTs) at finite temperature, where a ppp-dimensional conformal defect wraps the thermal circle. Extending the previous study of line defects to defects of general dimensions, we determine the general forms of the thermal one-point functions of bulk scalars, conserved currents and the stress tensor by imposing the residual symmetry and the conservation laws. The low temperature expansion of the thermal one-point functions is shown to be reproduced by the bulk-defect operator expansion, allowing us to read off the thermal one-point coefficients of defect operators. We examine the general results in four classes of examples with free bulk theories: the trivial defect, free scalar and fermion theories with a boundary, the free scalar theory with a localized ϕϕϕ deformation in d=2 p+2d=2\,p+2d=2p+2 dimensions and the free O(N)\mathrm{O}(N)O(N) model with a localized ϕ2ϕ^2ϕ2 deformation in d=p+2−εd=p+2-εd=p+2−ε dimensions. In the last example with ε=1ε= 1ε=1, where the defect becomes an interface, we find that the thermal one-point functions coincide with those of the free scalar theory with the Dirichlet boundary condition, in accordance with the conjectured factorization of an interface CFT into two decoupled boundary CFTs.
Numerical studies of phase transitions in statistical and quantum lattice models provide crucial insights into the corresponding Conformal Field Theories (CFTs). In higher dimensions, comparing finite-volume numerical results to infinite-volume CFT data is facilitated by choosing the sphere S^{d-1} S d − 1 as the spatial manifold. Recently, the fuzzy sphere regulator [Zhu et al, Phys. Rev. X 13, 021009 (2023)] has enabled such studies with exact rotational invariance, yielding impressive agreement with known 3D Ising CFT predictions, as well as new results. However, systematic improvements and a deeper understanding of finite-size corrections remain essential. In this work, we revisit the fuzzy sphere regulator, focusing on the original Ising model, with two main goals. First, we assess the robustness of this approach using Conformal Perturbation Theory (CPT), to which we provide a detailed guidebook. We demonstrate how CPT provides a unified framework for determining the critical point, the speed of light, and residual deviations from CFT predictions. Applying this framework, we study finite-size corrections and clarify the role of tuning the model in minimizing these effects. Second, we develop a novel method for extracting Operator Product Expansion (OPE) coefficients from fuzzy sphere data. This method leverages the sensitivity of energy levels to detuning from criticality, providing new insights into level mixing and avoided crossings in finite systems. Our work also includes validation of CPT in a 1+1D Ising model away from the integrable limit.