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arXiv:2606.15588 · Reports on Progress in Physics

Topological tricritical Ising universality class in one dimension

Sheng Yang, Hai-Qing Lin, Xue-Jia Yu

Published Oct 1, 2026• 0 citations
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Abstract

Quantum critical points can host symmetry-protected topological (SPT) edge modes even when the bulk is gapless, giving rise to symmetry-enriched universality classes beyond the conventional Landau-Ginzburg-Wilson paradigm. Here we show that the tricritical Ising (TCI) critical point-described by a paradigmatic conformal field theory (CFT) with emergent supersymmetry-admits a topologically nontrivial, symmetry-enriched realization, which we term the topological TCI. We construct this critical point in the cluster O'Brien-Fendley (OF) spin chain by applying an SPT entangler to the original OF model. The two realizations therefore share the same bulk TCI CFT. Under open boundary conditions, however, the original model exhibits the free conformal boundary condition, whereas the cluster model exhibits a spontaneously fixed boundary condition: an infinitesimal boundary field can select one of the fixed boundary states, yielding spontaneous boundary magnetization. We further show that symmetry enrichment and boundary renormalization-group flow provide two routes to the spontaneously fixed boundary condition. Whereas the free and spontaneously fixed boundary conditions of the original model can be interconverted by tuning a symmetry-preserving boundary term, no such conversion occurs for the cluster model along the boundary deformations studied here, even in the strong-deformation limit. Although they share the same conventional boundary CFT (BCFT) data, including the boundary operator content and boundary-function, the two realizations of the spontaneously fixed boundary condition are distinguished by different time-reversal charges of the disorder field, which impose different symmetry constraints on the allowed boundary conditions under symmetry-preserving boundary deformations. For the tested symmetry-preserving boundary perturbations that lift the exact twofold degeneracy, the finite-size energy splitting is numerically consistent with exponential decay over accessible system sizes, in stark contrast to the algebraic splitting at the symmetry-enriched Ising critical point of the cluster Ising spin chain. We also find that the topological TCI realization persists along a critical line in a two-parameter phase diagram. These results establish a symmetry-enriched TCI universality class and uncover boundary distinctions invisible to conventional BCFT data.

DOI
10.1088/1361-6633/aeb3b3

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Quantum critical points can host symmetry-protected topological (SPT) edge modes even when the bulk is gapless, giving rise to symmetry-enriched universality classes beyond the conventional Landau-Ginzburg-Wilson paradigm. Here we show that the tricritical Ising (TCI) critical point-described by a paradigmatic conformal field theory (CFT) with emergent supersymmetry-admits a topologically nontrivial, symmetry-enriched realization, which we term the topological TCI. We construct this critical point in the cluster O'Brien-Fendley (OF) spin chain by applying an SPT entangler to the original OF model. The two realizations therefore share the same bulk TCI CFT. Under open boundary conditions, however, the original model exhibits the free conformal boundary condition, whereas the cluster model exhibits a spontaneously fixed boundary condition: an infinitesimal boundary field can select one of the fixed boundary states, yielding spontaneous boundary magnetization. We further show that symmetry enrichment and boundary renormalization-group flow provide two routes to the spontaneously fixed boundary condition. Whereas the free and spontaneously fixed boundary conditions of the original model can be interconverted by tuning a symmetry-preserving boundary term, no such conversion occurs for the cluster model along the boundary deformations studied here, even in the strong-deformation limit. Although they share the same conventional boundary CFT (BCFT) data, including the boundary operator content and boundary-function, the two realizations of the spontaneously fixed boundary condition are distinguished by different time-reversal charges of the disorder field, which impose different symmetry constraints on the allowed boundary conditions under symmetry-preserving boundary deformations. For the tested symmetry-preserving boundary perturbations that lift the exact twofold degeneracy, the finite-size energy splitting is numerically consistent with exponential decay over accessible system sizes, in stark contrast to the algebraic splitting at the symmetry-enriched Ising critical point of the cluster Ising spin chain. We also find that the topological TCI realization persists along a critical line in a two-parameter phase diagram. These results establish a symmetry-enriched TCI universality class and uncover boundary distinctions invisible to conventional BCFT data.
We introduce topological invariants for gapless systems and study the associated boundary phenomena.More generally, the symmetry properties of the low-energy conformal field theory (CFT) provide discrete invariants establishing the notion of symmetry-enriched quantum criticality.The charges of nonlocal scaling operators, or more generally, of symmetry defects, are topological and imply the presence of localized edge modes.We primarily focus on the 1 þ 1d case where the edge has a topological degeneracy, whose finite-size splitting can be exponential or algebraic in system size depending on the involvement of additional gapped sectors.An example of the exponential case is given by tuning the spin-1 Heisenberg chain to a symmetry-breaking Ising phase.An example of the algebraic case arises between the gapped Ising and cluster phases: This symmetry-enriched Ising CFT has an edge mode with finite-size splitting scaling as 1=L 14 .In addition to such new cases, our formalism unifies various examples previously studied in the literature.Similar to gapped symmetry-protected topological phases, a given CFT can split into several distinct symmetry-enriched CFTs.This raises the question of classification, to which we give a partial answer-including a complete characterization of symmetry-enriched 1 þ 1d Ising CFTs.Nontrivial topological invariants can also be constructed in higher dimensions, which we illustrate for a symmetry-enriched 2 þ 1d CFT without gapped sectors.
Gapless quantum phases can become distinct when internal symmetries are enforced, in analogy with gapped symmetry-protected topological (SPT) phases. However, this distinction does not always lead to protected edge modes, raising the question of how the bulk-boundary correspondence is generalized to gapless cases. We propose that the spatial interface between gapless phases -- rather than their boundaries -- provides a more robust fingerprint. We show that whenever two 1+1d conformal field theories (CFTs) differ in symmetry charge assignments of local operators or twisted sectors, any symmetry-preserving spatial interface between the theories must flow to a non-invertible defect. We illustrate this general result for different versions of the Ising CFT with Z2×Z2T\mathbb{Z}_2 \times \mathbb{Z}_2^TZ2​×Z2T​ symmetry, obtaining a complete classification of allowed conformal interfaces. When the Ising CFTs differ by nonlocal operator charges, the interface hosts 0+1d symmetry-breaking phases with finite-size splittings scaling as 1/L31/L^31/L3, as well as continuous phase transitions between them. For general gapless phases differing by an SPT entangler, the interfaces between them can be mapped to conformal defects with a certain defect 't Hooft anomaly. This classification also gives implications for higher-dimensional examples, including symmetry-enriched variants of the 2+1d Ising CFT. Our results establish a physical indicator for symmetry-enriched criticality through symmetry-protected interfaces, giving a new handle on the interplay between topology and gapless phases.
Some quantum critical states cannot be smoothly deformed into each other without either crossing some multicritical points or explicitly breaking certain symmetries even if they belong to the same universality class. This brings up the notion of "symmetry-enriched" quantum criticality. While recent works in the literature focused on critical states with robust degenerate edge modes, we propose that the conformal boundary condition (B.C.) is a more generic characteristic of such quantum critical states. We show that in two families of quantum spin chains, which generalize the Ising and the three-state Potts models, the quantum critical point between a symmetry-protected topological phase and a symmetry-breaking order realizes a conformal B.C. distinct from the simple Ising and Potts chains. Furthermore, we argue that the conformal B.C. can be derived from the bulk effective field theory, which realizes a novel bulk-boundary correspondence in symmetry-enriched quantum critical states.
Two-dimensional topological paramagnets protected by Z3\mathbb{Z}_3Z3​ symmetry: Properties of the boundary Hamiltonian
We systematically study gapless edge modes corresponding to \mathbb{Z}_3 ℤ 3 symmetry-protected topological (SPT) phases of two-dimensional three-state Potts paramagnets on a triangular lattice. First, we derive microscopic lattice models for the gapless edge and, using the density-matrix renormalization group (DMRG) approach, investigate the finite-size scaling of the low-lying excitation spectrum and the entanglement entropy. Based on the obtained results, we identify the universality class of the critical edge, namely the corresponding conformal field theory and the central charge. Finally, we discuss the inherent symmetries of the edge models and the emergent winding number symmetry. As a result, one-dimensional chains with this symmetry form a model that supports gapless excitations due to its tricritical symmetry. Numerically, we show that low-energy states in the continuous limit of the edge model can be described by conformal field theory (CFT) with central charge c=1 c = 1 , given by the coset SU_k(3)/SU_k(2) S U k ( 3 ) / S U k ( 2 ) CFT at level k=1.
Tricritical Ising (TCI) phase transition is known to occur in several interacting spin and Majorana fermion models and is described in terms of a supersymmetric conformal field theory (CFT) with central charge c=7/10c=7/10c=7/10. The field content of this CFT is highly nontrivial and includes among its primary fields the Fibonacci anyon, making it of potential interest to strategies seeking to implement fault-tolerant topological quantum computation with non-Abelian phases of matter. In this paper we explore the possibility that a TCI CFT can occur at the edge of a gapped two-dimensional topological state as a stable phase. We discuss a possible realization of this 2D phase based on a coupled-wire construction using the Grover-Sheng-Vishwanath chain model of Majorana zero modes coupled to Ising spins which is known to undergo the TCI phase transition. From the combined analysis using mean-field theory, conformal field theory, and density matrix renormalization group (DMRG) on two- and four-leg ladders, we find that the left- and right-moving gapless TCI modes become spatially separated and reside on two opposite edges of the system, forming a precursor of the required 2D topological phase.
Abstract Quantum phase transitions out of a symmetry-protected topological (SPT) phase in (1 + 1) dimensions into an adjacent, topologically distinct SPT phase protected by the same symmetry or a trivial gapped phase, are typically described by a conformal field theory (CFT). At the same time, the low-lying entanglement spectrum of a gapped phase close to such a quantum critical point is known (Cho et al arXiv:1603.04016), very generally, to be universal and described by (gapless) boundary conformal field theory. Using this connection we show that symmetry properties of the boundary conditions in boundary CFT can be used to characterize the symmetry-protected degeneracies of the entanglement spectrum, a hallmark of non-trivial symmetry-protected topological phases. Specifically, we show that the relevant boundary CFT is the orbifold of the quantum critical point with respect to the symmetry group defining the SPT, and that the boundary states of this orbifold carry a quantum anomaly that determines the topological class of the SPT. We illustrate this connection using various characteristic examples such as the time-reversal breaking ‘Kitaev chain’ superconductor (symmetry class D), the Haldane phase, and the Z 8 classification of interacting topological superconductors in symmetry class BDI in (1 + 1) dimensions.