Gapless topological phases and symmetry-enriched quantum criticality

TL;DR

The paper develops a general framework for symmetry-enriched quantum criticality by defining symmetry fluxes as the central topological invariants at gapless points described by CFTs. It shows how these fluxes carry charges under other symmetries, yielding edge modes and finite-size splittings that depend on the symmetry content and the underlying universality class. The work provides complete results for the Ising CFT in 1+1D, partial results for Gaussian (c=1) cases, and illustrative higher-dimensional generalizations via twisted sectors, unifying many prior observations of edge phenomena at criticality. It also connects these invariants to phase-diagram constraints and presents multiple explicit lattice realizations, including Majorana-edge scenarios and fermionic/bosonic Z_3×Z_3 examples. The framework promises a broad applicability to gapless topological phenomena and offers avenues for further classification and experimental exploration of symmetry-enriched criticality across dimensions.

Abstract

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 $1+1d$ 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 former is given by tuning the spin-1 Heisenberg chain to a symmetry-breaking Ising phase. An example of the latter arises between the gapped Ising and cluster phases: this symmetry-enriched Ising CFT has an edge mode with finite-size splitting $\sim 1/L^{14}$. 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 $1+1d$ Ising CFTs. Non-trivial topological invariants can also be constructed in higher dimensions, which we illustrate for a symmetry-enriched $2+1d$ CFT without gapped sectors.

Figures (18)

• Figure 1: Persistent edge mode in a spin-1 XXZ chain at criticality. (a) The bulk correlation length $\xi$ blows up, whereas the edge mode localization length $\xi_\textrm{loc}$ stays finite. (b) The latter can be measured in two ways: the two ground states $|\uparrow_l \downarrow_r \rangle \pm |\downarrow_l \uparrow_r \rangle$ have a splitting which is exponentially small in system size $L$, or, equivalently, the two states differ by an operator which is exponentially localized near the edges ($L$ is the region we trace out near an edge before calculating the distance of the two density matrices). The dashed lines are proportional to $\exp(-L/\xi_\textrm{loc})$ with $\xi_\textrm{loc}\approx 3.3$.
• Figure 2: Order parameters and symmetry fluxes for the spin-$1$ XXZ chain. (a) The Haldane string order parameter and Ising order parameter have long-range order in the SPT and symmetry-breaking phase, respectively. (b) At criticality, the long-range order is replaced by algebraic decay. Both operators have the same scaling dimension (i.e., their correlators decay as $\sim 1/|n-m|^{2\Delta}$ with $\Delta = 1/8$; the dashed lines are a guide to the eye) and in the continuum limit correspond to the $\mu$ and $\sigma$ operators of the Ising CFT, respectively. We conclude that $\mu$ has non-trivial charge under other symmetries, which functions as a topological invariant.
• Figure 3: Transitions between topologically distinct Ising universality classes. (a) Phase diagram of the bond-alternating $S=1$ XXZ chain. There is a $c=1$ transition between the topologically distinct $c=1/2$ transitions. The tricritical point (hollow marker) is a WZW $SU(2)_1$ CFT, where the trivial and Haldane string order parameters both have scaling dimension $1/8$. (b) Similar phase diagram for an exactly solvable $S=1/2$ model; in this case, the $c=1$ boson CFT is in the free Dirac universality class (inset: analogue of Fig. \ref{['fig:xiloc']}(a)).
• Figure 4: Topologically protected edge modes in the Ising CFT. Boundary RG flow for the Ising CFT: usually the free boundary condition is stable (when preserving global $\mathbb Z_2$), but it can be prevented when $\mu$ is charged under additional symmetries. In that case, the spontaneously fixed boundary condition (with a global twofold degeneracy) is stable!
• Figure 5: Finite-size splitting of edge mode in the Ising CFT. If a unitary symmetry protects the edge mode, finite-size splitting is exponentially small in system size. For an anti-unitary symmetry, however, the field theory analysis in the main text suggests a splitting $\sim 1/L^{14}$. (a) We confirm the latter numerically in a spin chain model where $\mu$ is charged under $\mathbb Z_2^T$. Note that for $L=15$, the gap is around machine precision. (b) To separate out higher-order corrections present in (a), we extrapolate the leading exponent of $1/L$; the data agrees with the predicted splitting $\sim 1/L^{14} \times \left(1 + \alpha/L^{2} \right)$.