Hall viscosity and putative quantum Hall states without positive-definite K-matrix

We investigate putative quantum Hall effect states, labeled by their K-matrix equal to (1 1 3), by defining them on the torus and computing their Hall viscosity. Such states have been introduced on the sphere as a phase distinct from Pfaffian and anti-Pfaffian ones. This was done in order to explain certain results on thermal Hall conductivity in favor of particle-hole symmetric Pfaffian topological order in presence of Landau level mixing. The requirements of boundary conditions, modular invariance and ground state degeneracy are enough to uniquely fix the form of the proposed wave functions. We generalize a method to enforce them which we call monodromy matching and check our results on wave functions and Hall viscosity against realizations on the torus of Laughlin and hierarchical states. We highlight the issues in the realization of these states, which turn out to exhibit the formation of clusters. We show that the effect of anti-symmetrization on the system is not enough to prevent clustering; we compute the Hall viscosity for the Halperin version of these states and the fully anti-symmetrized one and we find them being dependent on the geometry and the particle number.

💡 Research Summary

The authors address the long‑standing puzzle of a putative fractional quantum Hall (FQH) state characterized by the non‑positive‑definite K‑matrix K = (1 1 3). This “113 state” was originally proposed on the sphere as a candidate distinct from the Pfaffian, anti‑Pfaffian, and particle‑hole‑symmetric Pfaffian phases, motivated by thermal Hall conductivity measurements that suggested a new topological order in the ν = 5/2 plateau when Landau‑level mixing is strong. Because the K‑matrix is not positive‑definite, standard Abelian Chern‑Simons field theory and the conformal‑field‑theory (CFT) construction of wave functions break down. The paper therefore tackles the problem from a geometric and many‑body perspective by placing the state on a torus, where boundary conditions, modular invariance, and ground‑state degeneracy (|det K| g) provide enough constraints to uniquely determine the trial wave functions.

The authors first review the general formalism for constructing FQH wave functions on a torus: a Gaussian factor fixed by the magnetic gauge, a holomorphic Jastrow factor, and a center‑of‑mass (CoM) factor built from theta functions. They emphasize that the magnetic translation operators generate a Heisenberg algebra, and that any admissible wave function must furnish a representation of this algebra consistent with the chosen boundary phases. For the 113 state, the usual CFT correlator approach fails because the underlying charge lattice is indefinite; instead the authors introduce a “monodromy matching” procedure. This method enforces the correct monodromy (phase acquired under particle exchanges and torus cycles) by matching the transformation properties of the trial wave function to those required by the K‑matrix and the torus modular group SL(2,ℤ). In practice, they construct a Halperin‑type (1,1,3) multicomponent wave function, then apply full antisymmetrization over all electrons to obtain a fermionic state. They also consider the un‑antisymmetrized Halperin version for comparison.

To probe the physical content of these trial states, the authors compute the Hall viscosity η_H, a topological transport coefficient that in many Abelian states equals (ħ ρ s)/2, where s is the average orbital spin. They perform Monte‑Carlo sampling of the overlap ⟨Ψ(τ)|Ψ(τ+δτ)⟩ after infinitesimal deformations of the torus complex structure τ. By extracting the Berry curvature associated with τ‑variations, they obtain η_H for several particle numbers (N = 6–12) and for both the Halperin and fully antisymmetrized 113 wave functions. The key finding is that η_H is not a universal constant for these states; it depends explicitly on the geometry (both Re τ and Im τ) and on N. Moreover, the antisymmetrization does not eliminate clustering: density–density correlations show that particles still tend to form bound clusters, confirming earlier numerical observations on the sphere. Consequently, the Hall viscosity of the 113 state deviates from the quantized value expected for a topologically protected phase.

The paper contrasts these results with benchmark states—Laughlin ν = 1/3, ν = 1/5, and hierarchical Jain states—where η_H remains invariant under modular transformations and matches the topological prediction. The authors argue that the dependence of η_H on geometry signals a breakdown of the usual bulk‑edge correspondence for non‑positive‑definite K‑matrices. The clustering and geometry‑dependent viscosity suggest that the 113 state, if realized, would require additional mechanisms (strong Landau‑level mixing, non‑standard electron‑electron interactions, or external strain) to stabilize a genuine topological order.

In conclusion, the work provides a concrete torus construction of the 113 state, introduces the monodromy‑matching technique for handling indefinite K‑matrices, and demonstrates through Hall viscosity calculations that such states exhibit non‑universal, geometry‑sensitive responses and persistent clustering. These findings cast doubt on the viability of the 113 state as a stable topological phase in realistic quantum Hall systems, while also offering a methodological framework for exploring other exotic states with non‑positive‑definite K‑matrices.