A new Ising/tricritical-Ising interface: From ${W}_3$ symmetry to Rydberg atoms

Abstract: We consider interfaces between critical spin-chains in different universality classes, described in the continuum limit by defect/interface conformal field theory (DCFT/ICFT). We find a new conformal interface between the Tricritical Ising (TIM) and the Ising CFT. We also explore the possibility of its experimental realizations in the context of Rydberg atom arrays. Our analysis emphasizes non-invertible symmetries and consistency under modular transformations, and uses defect couplings and the defect spectrum -- including in the case of mixed boundary conditions -- to make sharp experimental predictions. The structure of the observables hinges on a newly discovered pattern of emergent ${W}_3$ chiral symmetry for the Tricritical-Ising/Ising interface.

• The paper establishes a new conformal interface between Ising and tricritical Ising CFTs that reveals emergent W3 symmetry using a defect CFT framework.
• It employs a Hamiltonian formulation with finite-size scaling and modular invariance checks to validate the non-invertible Kramers-Wannier duality and operator spectrum.
• Lattice simulations and proposals for Rydberg atom experiments underscore its potential to probe non-invertible quantum symmetries and advance DCFT studies.

A New Conformal Interface between Ising and Tricritical Ising CFTs

Overview

This work establishes a new conformal interface between the Ising and tricritical Ising conformal field theories (CFTs) by leveraging a defect/interface CFT (DCFT/ICFT) framework. This interface is characterized by a non-invertible symmetry structure and exhibits emergent $W_3$ chiral symmetry, generated by a spin-3 current. Explicit lattice realizations are provided using a non-invertible Kramers-Wannier (KW) duality-invariant Hamiltonian. Theoretical predictions are backed by extensive numerical analysis using MPS/DMRG, and prospects for experimental verification via Rydberg atom arrays are discussed.

Theoretical Construction of the Interface

The interface is constructed between two 1D critical quantum spin chains: one governed by the critical Ising CFT ($c=1/2$) and the other by the tricritical Ising CFT ($c=7/10$). The interface is realized through a localized boundary interaction, described within a Hamiltonian formalism that maintains manifest KW duality invariance in the left (tricritical Ising) chain, as per the O'Brien-Fendley construction. The interface coupling is scanned to identify nontrivial fixed points. Three critical points are identified: free boundary (decoupled), fixed spin (anti-ferromagnetic), and a nontrivial interface at $h_b=1.379$, which simultaneously commutes with the Ising $\mathbb{Z}_2$ and KW symmetries.

Spectral analysis via finite-size scaling demonstrates integer spacing consistent with conformal symmetry and reveals the presence of unique operator content inaccessible from standard Gaiotto's RG interface constructions. Notably, the spectrum exhibits features most naturally explained in terms of $W_3$ minimal model representations, rather than the naive tensor product of Virasoro modules for Ising and tricritical Ising. This suggests an emergent, non-diagonal chiral symmetry structure at the interface.

Emergence of $W_3$ Symmetry

The partition function at the interface is precisely matched through modular bootstrap, modular invariance, and explicit calculation in both open-string and closed-string channels. The defect partition functions in the open-string channel are expressed in terms of $W_3$ characters, with several weights ($h=1/9, 7/9, 13/9$) that are not accounted for in the tensor product CFT but arise naturally in the $W_3^{(2)}$ minimal model.

Consistency checks under modular transformations and combinatorics of twisted/untwisted Ishibashi states lead to the conclusion that the $c=1/2$0 algebra is generated by a spin-3 current $c=1/2$1, constructed from fermionic fields in the $c=1/2$2 twisted sector of both Ising and tricritical Ising models. This current becomes a topological line defect that can end on the conformal interface, a structure inaccessible by standard Cardy or Gaiotto's RG interfaces alone.

Lattice Realization and Numerical Validation

The Hamiltonian implementation on the lattice uses parameters finely tuned to equalize the effective "speed of light" in both subsystems, as confirmed by energy scaling. The interface is validated by:

• Finite-size scaling and gap-crossing analyses pinpointing the critical coupling,
• Spectrum classification into $c=1/2$3 even and odd sectors matching CFT predictions,
• Matching of low-lying state degeneracies and scaling dimensions with those predicted by $c=1/2$4 characters, which cannot be explained by the direct sum of Ising and tricritical Ising CFTs,
• Verification of invariance under the KW duality.

For mixed boundary conditions (e.g., FM/factorized), the spectrum shows perfect agreement with the expected "half-spaced" operator spectrum, further corroborating the $c=1/2$5 symmetry and the nature of the interface.

Experimental Realizability in Rydberg Atom Arrays

The work outlines a specific route for experimental realization using programmable Rydberg atom arrays, leveraging the nearly arbitrary control over interaction profiles and detuning. It is shown that by tuning detuning and intersite spacing, one can realize half-chains in Ising and tricritical Ising universality classes, with the interface coupling set by ladder geometry adjustment. Recent advances enable experimentalists to engineer such interface Hamiltonians and directly measure edge spectra, with proposed matching of observed data to the CFT spectra as a practical diagnostic.

Further, the implementation allows preservation of global Ising $c=1/2$6 symmetry and KW duality, which is vital for realizing the intended non-invertible symmetry-protected interface.

Implications and Outlook

The identification of a conformal interface described by $c=1/2$7 symmetry in a system composed of two distinct minimal models has both fundamental and experimental implications:

• Symmetry Enhancement: The findings exemplify emergent symmetry beyond the naive direct product algebras, suggesting new classes of DCFTs with non-diagonal modular invariants and non-invertible symmetry structures.
• Boundary CFT Classification: The construction enriches the taxonomy of conformal boundaries and defects, indicating possible connections to non-rational boundary conditions and irrational CFTs in analogous higher-central-charge or multicritical systems.
• Experimental Probes of Non-Invertible Symmetries: The path toward Rydberg atom implementations constitutes a significant step toward direct laboratory access to nontrivial DCFT phenomena and non-invertible quantum symmetries.
• Future Developments: Generalizations to interfaces involving higher minimal models (e.g., tetracritical Ising, three-state Potts) and more complex geometries (e.g., ladder rings embedding multiple interfaces) are immediate directions, possibly revealing additional symmetry enhancement or irrational boundary phenomena.

Conclusion

This work presents a comprehensive theoretical and numerical construction of a conformal interface between Ising and tricritical Ising CFTs, uncovering emergent $c=1/2$8 symmetry in the defect spectrum. The construction is validated through lattice simulation and modular bootstrap, and a pathway to quantum simulation with Rydberg atom arrays is proposed. The results clarify the structure of non-invertible interfaces, expand the available toolbox for experimental quantum simulations of CFT defects, and motivate further exploration of symmetry-enriched ICFTs in low-dimensional quantum systems.