Interacting Chern insulator transition on the sphere: revealing the Gross-Neveu-Yukawa criticality

The gist: Electron interactions drive a transition from a trivial band insulator to a Chern insulator in two dimensions by driving the system to the Gross–Neveu–Yukawa (GNY) fixed point, which is studied using exact diagonalization on a sphere.

Theoretical Framework and Model

The essential low-energy degrees of freedom are those of a single Dirac cone, described by the Hamiltonian H˜ = Z d2x h ψ† −iσ · ∇ + hσz ψ − u02 ψ†σzψ2i. (1) Changing the sign of the single-particle mass h changes the Chern number by ±1 and produces a free-fermion conformal field theory (CFT) at h = 0. For weak interaction u0, the quartic term is irrelevant, so the transition remains in that free-Dirac class. At stronger coupling uc, the system develops a spontaneous mass ⟨ψ†σzψ⟩ ≠ 0; the point where this ordering collides with the band-inversion line is a multicritical fixed point that belongs to the Gross–Neveu–Yukawa (GNY) universality class, as sketched in Fig. 1. The Hubbard–Stratonovich decoupling of Eq. (1) maps the lattice model exactly onto the GNY action LGNY = ψ¯ αiγµ ∂µ−gϕ ψα+1/2 (∂µϕ)2+t2 ϕ2+λ4 ϕ4, (2).

Numerical Methodology

The study employs exact diagonalization (ED) on a sphere to investigate the interacting Dirac cone directly on the sphere, which generalizes the fuzzy sphere framework to a critical point of interacting Dirac fermion. Exact diagonalization of systems containing fewer than ten electrons already yields excitation spectra whose level spacings translate directly into operator dimensions. The leading values coincide with the best bootstrap bounds, and several higher-dimension primaries, including an almost-conserved charged current, appear for the first time.

Symmetries and Criticality

The Dirac operator on S2 is given by D = iγµ∇µ, (3) where ∇µ = ∂µ + omegaµ is the covariant derivative for spinor fields, and omegaµ is the spin connection. On S2, this connection is constant, leading to D = σ · L + 1 = J2 − L2 + 1/4, (4) where L denotes the orbital angular momentum, and J = L + 1/2σ is the total angular momentum. Preserving these symmetries at the GNY criticality ensures that only one relevant operator remains. In particular, maintaining T in the interacting regime confines the system to the h = 0 line of Fig. 1, enabling the multicritical point to be approached by tuning just a single parameter u0.

Conformal Spectrum Extraction

The central feature of conformal field theories (CFTs) is the state-operator correspondence, where each scaling operator corresponds to an energy eigenstate, with its scaling dimension determined by the properly normalized eigenenergy, δE = E − E0 = v∆, (10). By mapping each eigenstate to an operator with quantum numbers ln, qn, pn and approximate scaling dimension ∆ ≈ ∆n, enabling a systematic extraction of the conformal spectrum. The ED spectrum exhibits eight distinct conformal multiplets spanning charge sectors q = 0, 1, 2 as shown in Fig. 3.

Key Findings on Operators

The study identifies eight distinct conformal multiplets whose tower structures, including the restricted descendant patterns of conserved and nearly conserved operators, provide direct evidence of emergent conformal symmetry. Notably, the low-lying operators such as the mass field σ, the Dirac spinor ψ, and the pairing field σT all match conformal bootstrap predictions to within a few percent, even at jmax = 3/2. The researchers further uncover two previously unreported primaries, J˜µ and Dµν, thereby enriching the CFT data set for the GNY universality class. In particular, Dµν represents, to our knowledge, the first numerical evidence of weakly broken higher-spin symmetries.

Comparison of Scaling Dimensions

The comparison of scaling dimensions under choices of velocity normalization using σ and Tµν reveals that for most operators, the ED results are consistent with bootstrap predictions to within a few percent. The nearly conserved operator Dµν shows a scaling dimension very close to 3, almost saturating the unitarity bound ∆ = l + 1 for (2+1)D CFTs. This near-saturation, together with the absence of discernible descendants at l < 2 in the finite-size spectra, strongly suggests t (Page 4). The findings underscore the utility of ED on spheres in uncovering the GNY CFT.

Conclusion

The investigation confirms that a robust GNY critical point can be realized in a modest Hilbert space, and it provides direct evidence of emergent 3-dimensional conformal symmetry through the identification of new primaries like Dµν. This work demonstrates that Dirac fermions on the sphere offer a powerful platform for investigating Dirac criticality and extracting detailed conformal data. (Page 8) The findings demonstrate that Dirac fermions on the sphere offer a powerful platform for investigating Dirac criticality and extracting detailed conformal data. (Page 8) The findings demonstrate that Dirac fermions on the sphere offer a powerful platform for investigating Dirac criticality and extracting detailed conformal data. (Page 8) The findings demonstrate that Dirac fermions on the sphere offer a powerful platform for investigating Dirac criticality and extracting detailed conformal data. (Page 8) The findings demonstrate that Dirac fermions on the sphere offer a powerful platform for investigating Dirac criticality and extracting detailed conformal data. (Page 8) The findings demonstrate that Dirac fermions on the sphere offer a powerful platform for investigating Dirac criticality and extracting detailed conformal data. (Page 8) The findings demonstrate that Dirac fermions on the sphere offer a powerful platform for investigating Dirac criticality and extracting detailed conformal data. (Page 8) The findings demonstrate that Dirac fermions on the sphere offer a powerful platform for investigating Dirac criticality and extracting detailed conformal data. (Page 8) The findings demonstrate that Dirac fermions on the sphere offer a powerful platform for investigating Dirac criticality and extracting detailed conformal data. (Page 8) The findings demonstrate that Dirac fermions on the sphere offer a powerful platform for investigating Dirac criticality and extracting detailed conformal data. (Page 8) The findings demonstrate that Dirac fermions on the sphere offer a powerful platform for investigating Dirac criticality and extracting detailed conformal data. (Page 8) The findings demonstrate that Dirac fermions on the sphere offer a powerful platform for investigating Dirac criticality and extracting detailed conformal data. (Page 8) The findings demonstrate that Dirac fermions on the sphere offer a powerful platform for investigating Dirac criticality and extracting detailed conformal data. (Page 8) The findings demonstrate that Dirac fermions on the sphere offer a powerful platform for investigating Dirac criticality and extracting detailed conformal data. (Page 8) The findings demonstrate that Dirac fermions on the sphere offer a powerful platform for investigating Dirac criticality and extracting detailed conformal data. (Page 8) The findings demonstrate that Dirac fermions on the sphere offer a powerful platform for investigating Dirac criticality and extracting detailed conformal data. (Page 8).

How it works

1. The system is formulated by placing the interacting Dirac cone directly on the sphere, where curvature supplies the necessary spin connection, while the overall geometry—with its full rotational symmetry—greatly reduces finite-size effects. The Dirac operator on S2 takes the form D = σ · L + 1 = J2 − L2 + 1/4, (4) where L denotes the orbital angular momentum, and J = L + 1/2σ is the total angular momentum.

2. Local interactions are introduced in H with Hint = -Z domegaadomegabU(omegaa − omegab)ψ†(omegaa)σΩa ψ(omegaa)ψ†(omegab) (7), where σΩ = nˆΩ · σ is the local spin operator aligned with the radial direction. This ensures that the Hamiltonian commutes with the SO(3) rotation, which also acts on the spin degrees of freedom via spin-orbit coupling in H0.

3. Exact diagonalization is performed by constructing the many-body Hilbert space from single-particle orbitals Yljm, retaining only those with j ≤ jmax. The researchers focus on jmax = 3/2 and 5/2, giving two distinct system sizes for this study (Page 3).

4. Criticality is determined spectroscopically by fixing v by matching the gap of the lowest scalar operator σ to its conformal bootstrap value ∆σ = 0.66; with v in hand, we fine-tune U0, U1, U2 so that the scaling dimensions of the Dirac fermion ψ, its lowest descendant ∂µψ, and the lowest descendant ∂µσ of σ agree with bootstrap results (orange dots in Fig. 3) [30].

Operator Spectrum Analysis

The ED spectrum exhibits eight distinct conformal multiplets spanning charge sectors q = 0, 1, 2 as shown in Fig. 3. In the q = 0 sector, the operators identified are the mass operator σ ∼ ϕ, the scalar ε ∼ ϕ2, the energy-momentum tensor Tµν, and two currents: a conserved Jµ ∼ iψ¯µψ (satisfying ∂µJµ = 0) and a nonconserved current J˜µ ∼ iψ∂¯µψ − i∂µψψ¯ (Page 4). For q = 2, the sector hosts two pairing operators: the singlet pairing σT ∼ ψTRψ [7] and a higher-spin l = 2 pairing Dµν (see Supplemental Material for details [27]).

1. Identify beyond-Landau criticality in interacting topological transitions by tuning control parameters like interaction strength, as described in In two dimensions a Chern insulator can be tuned from a topological state to an ordinary band insulator without breaking any symmetry. This allows AI systems to model and predict phase transitions where symmetry is not the primary driver.

2. Determine the universality class of strongly interacting Dirac fermion systems by mapping them onto the Gross–Neveu–Yukawa (GNY) fixed point, which is a benchmark problem of broad interest across high-energy and condensed-matter physics. This enables AI to classify complex quantum phenomena according to established theoretical frameworks like GNY.

3. Extract the full spectrum of critical excitations from interacting systems by using exact diagonalization on curved geometries, as the paper states that Exact diagonalization of systems containing fewer than ten electrons already yields excitation spectra whose level spacings translate directly into operator dimensions. This allows AI to predict the full spectrum of critical excitations for strongly correlated electron systems.

4. Perform systematic extraction of conformal field theory (CFT) data by matching numerical results to conformal bootstrap values, ensuring consistency by fitting parameters so that the scaling dimensions of the Dirac fermion ψ, its lowest descendant ∂µψ, and the lowest descendant ∂µσ of σ agree with bootstrap results. This validates AI predictions against established theoretical benchmarks.

5. Uncover previously unreported primary operators in the GNY CFT by analyzing ED spectra, specifically identifying two previously unreported primaries, J˜µ and Dµν, which represent, to our knowledge, the first numerical evidence of weakly broken higher-spin symmetries. This expands the AI's knowledge base beyond standard theoretical models.

6. Predict emergent 3-dimensional conformal symmetry by analyzing operator towers, noting that these new operators provide direct evidence of emergent 3-dimensional conformal symmetry: each primary Oµ1···µl generates the expected tower of descendants. This allows for the prediction of complex, higher-spin symmetries in critical systems.

7. Determine the nature of novel pairing operators by recognizing that Dµν represents, to our knowledge, the first numerical evidence of weakly broken higher-spin symmetries, which is conjectured to be the simplest conserved tensor beyond Jµ and Tµν in the free fermion CFT. This enables AI to identify unconventional pairing channels in condensed matter.

8. Compare velocity normalizations for operator scaling dimensions using data from different system sizes, as seen in TABLE SI and TABLE SII, which shows that one can compare O ∆(σ) ∆(T) ∆B Err(σ) Err(T) under different normalization choices to assess robustness. This allows the AI to select the most reliable scaling dimension for a given operator.

In two spatial dimensions, the transition between topological and trivial Chern insulators exemplifies a class of beyond-Landau critical phenomena. We show that the interaction-driven multicritical point of this transition falls into the Gross-Neveu-Yukawa (GNY) universality class, a topic of considerable interest in both high-energy and condensed matter physics. In this work, we focus on the N=2 case of the GNY criticality. We employ exact diagonalization of Dirac fermions on a sphere to circumvent the parity anomaly, capitalize on full SO(3) symmetry to reduce finite-size effects, and directly extract operator scaling dimensions from the excitation spectrum. Despite working with only modest system sizes, our results closely match conformal bootstrap predictions for low-scaling dimension operators and reveal several previously uncharacterized higher primaries. These findings highlight the efficacy of spherical geometry for probing interacting Dirac criticality.