We develop a systematic construction of renormalization group (RG) domain walls between two-dimensional rational conformal field theories. Using phantom currents, we consider an auxiliary chiral extension linked to the folded ultraviolet--infrared theory by a topological interface. The construction of RG walls then reduces to identifying suitable automorphism-twisted Cardy states in the auxiliary theory. We find two such types of automorphisms: outer automorphisms exchanging identical current algebra factors, as in Gaiotto's construction, and continuous inner automorphisms generated by the zero modes of spin-one phantom currents. In the latter case, the RG domain walls generically admit a continuous defect conformal manifold. We further show that, when an RG flow can be orbifolded to another one, the orbifold may lift defect conformal moduli of the RG brane constructed using inner automorphisms. The two types of RG branes can then be mapped to each other via a topological interface defined by the orbifold. We demonstrate the two types of RG brane construction via various examples of flows between non-unitary minimal models, from parafermions to minimal models, and between generalized parafermions. Many physical observables of the RG walls are computed non-perturbatively, including boundary entropies and spectra, and UV--IR operator mixings. We also find interesting nontrivial boundary monodromy: a closed cycle in the defect conformal moduli space can carry an RG domain wall into other distinct non-trivial conformal interfaces.
We develop a systematic construction of renormalization group (RG) domain walls between two-dimensional rational conformal field theories. Using phantom currents, we consider an auxiliary chiral extension linked to the folded ultraviolet--infrared theory by a topological interface. The construction of RG walls then reduces to identifying suitable automorphism-twisted Cardy states in the auxiliary theory. We find two such types of automorphisms: outer automorphisms exchanging identical current algebra factors, as in Gaiotto's construction, and continuous inner automorphisms generated by the zero modes of spin-one phantom currents. In the latter case, the RG domain walls generically admit a continuous defect conformal manifold. We further show that, when an RG flow can be orbifolded to another one, the orbifold may lift defect conformal moduli of the RG brane constructed using inner automorphisms. The two types of RG branes can then be mapped to each other via a topological interface defined by the orbifold. We demonstrate the two types of RG brane construction via various examples of flows between non-unitary minimal models, from parafermions to minimal models, and between generalized parafermions. Many physical observables of the RG walls are computed non-perturbatively, including boundary entropies and spectra, and UV--IR operator mixings. We also find interesting nontrivial boundary monodromy: a closed cycle in the defect conformal moduli space can carry an RG domain wall into other distinct non-trivial conformal interfaces.
Defect Conformal Manifolds along RG Domain Walls between ZN-Parafermions and Minimal Models
We investigate the renormalization group (RG) domain walls interpolating between the ZN parafermion theory (the critical N-state Potts model) and the Virasoro minimal model MN+1. These flows are genuinely non-perturbative and an explicit construction of Gaiotto type RG domain wall remains elusive. We bypass this limitation by employing a bottom-up approach centered on the emergence of ``phantom currents". By tracking the preserved non-invertible symmetries (so(3)N) along the flow, we extract the exact spectrum of these currents localized on the defect. We demonstrate that the presence of a spin-1 phantom current allows the interface to be marginally deformed, dynamically generating a continuous defect conformal manifold. Furthermore, we show that an extra spin-2 operator, crucially as a W(3)-algebra descendant of the spin-1 phantom current, rigidly constrains the UV-IR stress tensor mixing via the cluster decomposition principle. This algebraic framework enables the exact computation of the parameter-dependent transmission rate across the conformal manifold, which we observe strictly vanishes in the large-N limit as a consequence of macroscopic target space collapse.
A bstract We investigate the renormalization group (RG) domain walls interpolating between the ℤ N parafermion theory (the critical N -state Potts model) and the Virasoro minimal model ℳ N +1 . These flows are genuinely non-perturbative and an explicit construction of Gaiotto type RG domain wall remains elusive. We bypass this limitation by employing a bottom-up approach centered on the emergence of “phantom currents”. By tracking the preserved non-invertible symmetries (𝔰𝔬(3) N ) along the flow, we extract the exact spectrum of these currents localized on the defect. We demonstrate that the presence of a spin-1 phantom current allows the interface to be marginally deformed, dynamically generating a continuous defect conformal manifold. Furthermore, we show that an extra spin-2 operator, crucially as a W (3) -algebra descendant of the spin-1 phantom current, rigidly constrains the UV-IR stress tensor mixing via the cluster decomposition principle. This algebraic framework enables the exact computation of the parameter-dependent transmission rate across the conformal manifold, which we observe strictly vanishes in the large- N limit as a consequence of macroscopic target space collapse.
Conformal interfaces separating two conformal field theories (CFTs) provide maps between different CFTs, and naturally exist in nature as domain walls between different phases. One particularly interesting construction of a conformal interface is the renormalization group (RG) domain wall between CFTs. For a given Virasoro minimal model \mathcal{M}_{k+3,k+2} ℳ k + 3 , k + 2 , an RG domain wall can be generated by a specific deformation which triggers an RG flow towards its adjacent Virasoro minimal model \mathcal{M}_{k+2,k+1} ℳ k + 2 , k + 1 with the deformation turned on over part of the space. An algebraic construction of this domain wall was proposed by Gaiotto in [J. High Energy Phys. 12, 103 (2012)]. In this paper, we will provide a study of this RG domain wall for the minimal case k=2 k = 2 , which can be thought of as a nonperturbative check of the construction. In this case the wall is separating the Tricritical Ising Model (TIM) CFT and the Ising Model (IM) CFT. We will check the analytical results of correlation functions from the RG brane construction with the numerical density matrix renormalization group (DMRG) calculation using a lattice model proposed in [arXiv.1206.1332, Science 344, 280 (2014)], and find a perfect agreement. We comment on possible experimental realizations of this RG domain wall.
Recently, Gaiotto (2012 arXiv:1201.0767) considered conformal defects which produce an expansion of infrared local fields in terms of the ultraviolet (UV) ones for a given renormalization group flow. In this paper, we propose that for a boundary RG flow in two dimensions there exist boundary condition changing fields (RG defect fields) linking the UV and the infrared (IR) conformal boundary conditions which carry similar information on the expansion of boundary fields at the fixed points. We propose an expression for a pairing between IR and UV operators in terms of a four-point function with two insertions of the RG defect fields. For the boundary flows in minimal models triggered by ψ13 perturbation, we make an explicit proposal for the RG defect fields. We check our conjecture by a number of calculations done for the example of (p, 2) → (p − 1, 1)⊕(p + 1, 1) flows.
Recently, Gaiotto (2012 arXiv:1201.0767) considered conformal defects which produce an expansion of infrared local fields in terms of the ultraviolet (UV) ones for a given renormalization group flow. In this paper, we propose that for a boundary RG flow in two dimensions there exist boundary condition changing fields (RG defect fields) linking the UV and the infrared (IR) conformal boundary conditions which carry similar information on the expansion of boundary fields at the fixed points. We propose an expression for a pairing between IR and UV operators in terms of a four-point function with two insertions of the RG defect fields. For the boundary flows in minimal models triggered by ψ 13 perturbation, we make an explicit proposal for the RG defect fields. We check our conjecture by a number of calculations done for the example of ( p , 2) → ( p − 1, 1)⊕( p + 1, 1) flows.
We formulate the ZN group gauging operations (or group extensions) in (smeared) boundary conformal field theories (BCFTs) and bulk conformal field theories, and study their applications to various phenomena in topologically ordered systems. We apply the resultant theories to analyze the correspondence between the renormalization group (RG) flow of CFTs and the classification of topological quantum field theories using the testable information of general classes of partition functions. One can obtain the bulk topological properties of the 2+1 dimensional topological ordered phase corresponding to the massive RG flow of the 1+1 dimensional systems equivalent to the smeared BCFT. We present an obstruction to mass condensation for the smeared BCFT analogous to the Lieb-Shultz-Mattis theorem for noninvertible symmetry. Corresponding to the bulk topological degeneracies in 2+1 dimensions and gapped quantum phases in 1+1 dimensions, we construct a new series of BCFT. We also study the implications of the massless RG flow of the 1+1 dimensional CFTs corresponding to the domain wall or coupling problem in 2+1 dimensional topological orders, which corresponds to the earlier proposal by L. Kong and H. Zheng [Nucl. Phys. B 966, 115384 (2021)] closely related to the integer-spin simple current by Schellekens and Gato-Rivera. We reveal that two distinct kinds of massless RG flow exist, coming from anomaly cancellation and anomaly matching related to Witt equivalence of bulk CFTs. The anomaly cancellation flows produce a new series of modular invariants as coupled models, and the anomaly matching flows produce nontrivial domain walls changing the chirality of anyons. Our paper gives a unified direction for future theoretical and numerical studies of the topological phase based on the established data of classifications of conformal field theories or modular invariants (and the corresponding category theories).