We construct conformal interfaces having defect conformal manifolds between two-dimensional rational CFTs, at least one of which is non-unitary. In a continuous family, interfaces with real and complex spectra can be marginally deformed to each other via their defect moduli parameters. An exceptional point then arises along the trajectory in the conformal manifold, where two interface operators with complex conjugate weights coalesce into a logarithmic pair. We reconstruct the entire trajectory of this novel spectral collision, independently, from self-annulus partition functions and bulk two point correlators in the presence of the interfaces.
We construct conformal interfaces having defect conformal manifolds between two-dimensional rational CFTs, at least one of which is non-unitary. In a continuous family, interfaces with real and complex spectra can be marginally deformed to each other via their defect moduli parameters. An exceptional point then arises along the trajectory in the conformal manifold, where two interface operators with complex conjugate weights coalesce into a logarithmic pair. We reconstruct the entire trajectory of this novel spectral collision, independently, from self-annulus partition functions and bulk two point correlators in the presence of the interfaces.
In parity-time (PT)-symmetric non-Hermitian systems, an exceptional point, where eigenvalues and eigenvectors coalesce, typically separates a real spectrum from complex-conjugate eigenvalues. In logarithmic conformal field theories (LCFTs), such an exceptional point appears in the spectrum of scaling dimensions. Holographic models of LCFTs, such as the dipole model, are built exactly at this point. Here, we deform the holographic dipole model so that its two coalescing scaling dimensions split into either two real ones or a complex-conjugate pair. We obtain the renormalized two-point functions in closed form. They change continuously from a linear combination of two power laws, through the logarithmic form of the LCFT at the exceptional point, to log-periodic oscillations. For any pair of scalar primary operators with real mixing under dilatations, the conformal Ward identities fix this form up to two normalization constants, which the holographic dictionary determines. In PT-symmetric theories with such a dilatation operator, real and complex-conjugate scaling dimensions correspond to the PT-unbroken and PT-broken phases. In the BTZ black hole at finite temperature, the difference between the quasinormal frequencies associated with the two scaling dimensions changes from purely imaginary to purely real across the real-to-complex spectral transition. Correspondingly, the linear response changes from overdamped to underdamped, with critical damping at the exceptional point.
We construct directly on the lattice a class of non-unitary interfaces that are both exactly conformal and exactly solvable, and establish their corresponding boundary and interface conformal field theory (CFT) descriptions. The construction is obtained by analytically continuing the scattering data of known exact unitary conformal interfaces on the lattice, yielding an SL(2,C)-parametrized family, which is non-compact and breaks probability-current conservation. Exploiting the exact lattice-continuum correspondence, we derive the conformal boundary states in the folded picture. We show that a proper definition of the Hilbert space in the closed-string channel requires the incoming and outgoing boundary states to be specified independently by boundary data associated with a pair of dual biorthogonal bases, in close analogy with the right and left eigenvectors of a non-Hermitian Hamiltonian. This requirement determines a consistent CFT construction of non-unitary boundaries and interfaces, and leads to a non-unitary generalization of the conventional Cardy's condition for unitary boundary CFT. Beyond their formal construction, these non-unitary interfaces are shown to exhibit logarithmic entanglement scaling governed by an effective central charge that is generally complex. For the SU(1,1) subclass, the effective central charge remains real but grows without bound as the transmission coefficient increases. This result is demonstrated through analytical and numerical lattice calculations, as well as an interface CFT analysis in the unfolded picture. Finally, we present a general CFT analysis of a class of global quantum quenches whose initial states are prepared with non-unitary boundaries. We relate their effective temperature to the conformal dimension of the boundary-condition-changing operators associated with non-unitary boundary conditions.
We present a model-independent study of boundary states in the Cardy case that covers all conformal field theories for which the representation category of the chiral algebra is a – not necessarily semisimple – modular tensor category. This class, which we call finite CFTs, includes all rational theories, but goes much beyond these, and in particular comprises many logarithmic conformal field theories. We show that the following two postulates for a Cardy case are compatible beyond rational CFT and lead to a universal description of boundary states that realizes a standard mathematical setup: First, for bulk fields, the pairing of left and right movers is given by (a coend involving) charge conjugation; and second, the boundary conditions are given by the objects of the category of chiral data. For rational theories our proposal reproduces the familiar result for the boundary states of the Cardy case. Further, with the help of sewing we compute annulus amplitudes. Our results show in particular that these possess an interpretation as partition functions, a constraint that for generic finite CFTs is much more restrictive than for rational ones.
Symmetry breaking of continuous symmetries by extended dynamical defects entails the existence of defect families, which form conformal manifolds in a critical setup. In the presence of bulk 't Hooft anomalies, defects are in fact required to break the symmetry: defect conformal manifolds are anomaly-enforced. We show that, by coupling the system to a modulated deformation parameter, the geometric structure of the conformal manifold is sensitive to the 't Hooft anomaly. This leads to measurable effects in the presence of a boundary/defect: in (1+1)d the anomaly predicts a quantized boundary charge pumping, while in higher dimensions it gives rise to non-dissipative boundary Hall currents.
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.
A bstract We study two-dimensional conformal field theories (CFTs) with boundaries via the conformal bootstrap. We derive a positive semi-definite program from crossing symmetry of three observables: the annulus partition function, the two-point function of identical operators in the presence of a boundary, and the four-point function of the same operators on the infinite plane. The mixed-correlator system allows the numerical bootstrap to access new data, like the bulk-to-boundary Operator Product Expansion coefficients, and to strengthen the bounds on observables already contained in the partition function on the annulus, such as the boundary entropy. We test the method on the free boson CFT; then, as a first application, we produce new non-perturbative bounds on the entropy and the gaps in boundary CFTs with central charge c = 3/2, with special emphasis on the 𝔰𝔲(2) 2 WZW model.