Spinning the Large-Charge Bootstrap: Parity-Even Operators

Abstract: We study the large-charge bootstrap in three-dimensional CFTs with a global U(1) symmetry using scalar, current, and stress-tensor probes, restricting to parity-even exchanged operators. Under minimal assumptions, the bootstrap requires at least one Regge trajectory with the dispersion relation of the standard Goldstone mode of the conformal superfluid. With additional assumptions, this trajectory is unique, while all non-Goldstone trajectories contribute only to the scalar-scalar channel at this order.

• The paper establishes that the large-charge sector of 3D CFTs necessarily includes a unique superfluid Goldstone mode with dispersion ωₗ² = ℓ(ℓ+1)/2.
• It employs an algebraic bootstrap system to derive infinite moment equations linking excitation energies and OPE coefficients across scalar, current, and tensor channels.
• Additional conditions restrict non-Goldstone contributions to scalar correlators, reinforcing the universal role of the superfluid EFT in large-charge regimes.

Large-Charge Bootstrap with Spinning Parity-Even Operators in 3d CFTs

Introduction and Background

The paper develops an analytic large-charge bootstrap framework for three-dimensional CFTs endowed with a global $U(1)$ symmetry. The analysis incorporates four-point correlators with two large-charge ("heavy") scalars and two "light" probe insertions: specifically scalar, current, and stress-tensor operators. Importantly, the bootstrap restricts the possible intermediate states to parity-even operators in the $s$- and $u$-channel OPEs.

The research aims to elucidate to what extent the low-energy spectrum and OPE coefficients in the large-$Q$ sector are determined by universal principles—crossing symmetry, unitarity, and the presence of conserved operators—versus model-dependent dynamics. The operator-state correspondence connects the large-charge sector to finite-density states on $\mathbb{R} \times S^2$; previous work has argued this often leads to a superfluid EFT with a single Goldstone scalar, but without a fully general bootstrap derivation. The inclusion of spinning probes is central: stress-energy tensor and current insertions impose powerful constraints by unitarity and Ward identities, supplementing those from scalar probes.

Set-up and Minimal Assumptions

The analysis operates under a set of minimal physical and spectral assumptions:

1. In each $U(1)$ sector of fixed charge $Q$, the minimal dimension operator is a unique scalar.
2. The system admits a well-defined macroscopic (thermodynamic) limit with scaling $\Delta_Q \sim \alpha |Q|^{d/(d-1)}$ for $d=3$.
3. At first subleading order in large-$Q$, the $s$0-channel involves the first descendant of the ground state and potentially new primaries.
4. Only finitely many Regge trajectories contribute at this order.
5. Only parity-even exchanges are included.

The technical construction involves a systematic expansion of four-point functions into tensor structures (the $s$1-basis), organized by contractions of the external vectors on the spatial sphere. Each sector—scalar-scalar, scalar-current, current-current, scalar-tensor, current-tensor, tensor-tensor—is decomposed with respect to all available parity-even structures, and large-$s$2 scaling for contributions is fully catalogued.

Algebraic Bootstrap System

Crossing equations are derived for all sectors and recast, via a careful analysis of short-distance OPE limits and spectral representations, into infinite hierarchies of algebraic moment equations in each channel. These relate moments (powers) of excitation energies (relative to the large-charge ground state) of exchanged primaries, weighted by squared OPE coefficients, to universal polynomials fixed by conformal and tensorial kinematics.

A central theoretical tool is the analytic continuation in spin (Regge trajectories), with the organizational principle that each exchanged family yields a "trajectory" $s$3, with $s$4. The generality of this spectral representation is crucial in extracting universal CFT constraints.

The analysis also clarifies, with precise technical treatment, the nature of contact singularities in the cylinder OPE, the role of Ward identities (especially the careful way charge conjugation intertwines normalization choices in the crossed channel), and the projection of spinning structures via differential operators acting on scalar seeds.

Main Results

Goldstone Mode Necessity and Uniqueness

The most robust model-independent outcome is that, under minimal assumptions, the spectrum must contain at least one Regge trajectory with dispersion $s$5—the standard conformal Goldstone of a superfluid EFT. This conclusion emerges from the tensor-tensor channel: any operator that couples to the stress tensor and survives the bootstrap constraints necessarily saturates the Goldstone dispersion. This is independent of detailed dynamics, highlighting the constraint power of crossing, unitarity, and stress-tensor selection rules.

The existence of additional trajectories is not precluded, but strong numerical or OPE contributions from non-Goldstone states in the tensor or current channels are tightly restricted by the crossing algebra.

Further Reductions Under Additional Assumptions

Under further mild assumptions—no non-Goldstone parity-even trajectory crossing zero energy at $s$6, and non-degeneracy at fixed spin—the Goldstone trajectory is shown to be unique. All other potential trajectories are strictly "invisible" to the current and stress tensor probes: their OPE coefficients in mixed or tensor channels (at this order) vanish. Thus, they only affect the scalar-scalar correlators, manifesting as additional light but "decoupled" fields. An even stronger non-degeneracy requirement, excluding accidental degeneracies of first descendants with new primaries, fixes the contribution of spinning primaries at $s$7 uniquely, entirely eliminating ambiguities in spinning probe sectors.

Explicitly, this yields in the algebraic system that—after subtracting the Goldstone contribution—the remaining freedom in the bootstrap equations is captured by those for the scalar sector, structurally equivalent to the problem considered in earlier scalar-only large-charge bootstrap works, but now "factorized" from the tensor channels.

Implications: Constraints on Large-Charge Sectors

The above results imply that, for generic three-dimensional CFTs with a global $s$8 and under the parity-even, finite-trajectory, and non-degeneracy constraints, the low-energy effective description of the large-charge sector is fixed to be the conformal superfluid EFT, possibly with additional spectator scalar sectors. This is a significant nonperturbative statement, obtained without knowledge of microscopic dynamics, and leverages the full strength of analytic bootstrap machinery.

Crucially, theories where the large-$s$9 sector exhibits fundamentally different low-energy physics—such as conformal solids (requiring parity-odd exchanges associated with transverse phonons) or Fermi liquids (with infinite families of light excitations)—lie completely outside the allowed bootstrap solution space at this order and under these constraints.

Explicit Case Analysis

The paper also offers explicit elucidation for small numbers of trajectories (one or two), detailing how possible extra solutions can or cannot contribute to spinning channels. For instance, with only one Regge trajectory, all probe sectors, including spinning, are entirely determined by the Goldstone contribution. With two, any non-Goldstone contribution can affect only the scalar-scalar channel unless it becomes degenerate with Goldstone states at special spins, in which case non-degeneracy restricts their effect further.

Theoretical and Practical Implications

Theoretical

This work makes rigorous and non-perturbative the connection between large-charge universal behavior (as described by the Goldstone EFT) and the first principles of conformal bootstrap, extending previous results limited to scalar and current probes by including all parity-even spinning probes. The method solidifies the analytic bootstrap as a potent tool for phase classification in CFTs and highlights the power of crossing symmetry in the presence of nontrivial kinematical sectors.

The technical apparatus—the algebraic reduction to moment problems, spectral and analytic continuation analysis, and the finely resolved crossing structure—is broadly applicable, opening avenues for extensions to higher dimensions, non-Abelian symmetries, and sectors with additional protected operators.

Practical/Future Directions

From a practical standpoint, the results supply benchmark constraints for analytic and numerical studies of large-charge CFT sectors, providing a precise criterion for when the EFT description is compelled versus where model dependence can enter. Future work will be needed to incorporate parity-odd exchanges (essential for conformal solids and Chern-Simons-matter theories [cf. “Large Charge Sector of 3d Parity-Violating CFTs” (Cuomo et al., 2021)]), as well as to relax the finite-Regge-trajectory assumption relevant for Fermi liquids and other non-superfluid large-charge regimes.

This framework has potential applications in classifying and distinguishing phases of strongly coupled CFTs, and in constructing CFTs with controlled large-charge expansions, including via bootstrap numerics, effective field theory classification, or lattice approaches.

Conclusion

The analysis provides a comprehensive, algebraic bootstrap derivation of universal features in the large-charge sectors of 3d CFTs with a $u$0 global symmetry, under parity-even, finite-trajectory, and non-degeneracy assumptions. It establishes that the presence and uniqueness of the superfluid Goldstone mode—and the form of its OPE structure in all probe sectors—are dictated by first principles, with all remaining freedom confined to scalar-only channels. These results significantly advance the nonperturbative classification of large-charge dynamics in CFTs and underscore the utility of combining analytic bootstrap techniques with effective field theory reasoning.

Reference:

"Spinning the Large-Charge Bootstrap: Parity-Even Operators" (2607.09550)