Pfaffian Trial Wavefunctions: Topological Quantum Phases

• Pfaffian trial wavefunctions are defined via antisymmetrized pairing functions combined with Jastrow factors to capture non-Abelian and clustering properties.
• They serve as variational ansätze for modeling topological phases in fractional quantum Hall systems and Cooper-paired fermion simulations.
• Analytical and numerical studies reveal high state overlaps, distinct entanglement spectra, and mappings between different Landau level projections.

Pfaffian trial wavefunctions are a central class of variational ansätze used in the study of strongly correlated many-body quantum systems, particularly for the fractional quantum Hall effect (FQHE) at half-integer filling and for variational simulations of Cooper-paired fermions in both continuum and lattice models. These wavefunctions utilize the mathematical structure of the Pfaffian—an antisymmetrized pairing function over all pairs of particles—to capture essential non-Abelian, clustering, and topological features not accessible with Slater or Jastrow-only constructions. They form the basis for understanding topological quantum fluids such as the Moore–Read (MR) state, the anti-Pfaffian, and the particle-hole-symmetric PH-Pfaffian state across multiple Landau levels, as well as generalized superconducting and electronic phases.

1. Mathematical Structure of Pfaffian Trial Wavefunctions

Pfaffian trial states describe $N$-body quantum wavefunctions for even $N$ via the Pfaffian of an $N \times N$ antisymmetric matrix $A$, combined with a Jastrow factor and potentially additional correlators: $$\Psi_{\text{Pf}}(z_1,\dots,z_N) = \mathrm{Pf}[g(z_i, z_j)] \prod_{i<j} (z_i - z_j)^M\, e^{-\sum_{k} |z_k|^2/(4\ell^2)}$$ where $g(z_i, z_j)$ encodes a two-particle pairing function and $M$ is fixed by statistics (e.g., $M=2$ for fermionic FQHE at $\nu=1/2$) (Yang, 2023). The Pfaffian itself is defined as

$$\mathrm{Pf}[A] = \frac{1}{2^{N/2}(N/2)!} \sum_{\sigma\in S_N} \mathrm{sgn}(\sigma) \prod_{k=1}^{N/2} A_{\sigma(2k-1),\,\sigma(2k)}$$

and generalizes the determinant, encoding pairwise antisymmetry beyond single-particle occupation.

Concretely, the MR Pfaffian for the FQHE at $N$0 has $N$1 and $N$2, while antiholomorphic versions with $N$3 define the PH-Pfaffian trial state (Yang, 29 Mar 2025, Balram et al., 2018). Additional structures, such as the Hafnian, can be combined to produce further families of paired or clustered states (Yang, 2022).

2. Topological and Physical Interpretation in the Fractional Quantum Hall Effect

Pfaffian wavefunctions provide model gapped ground states for topological phases supporting non-Abelian anyons, edge modes, and quantized thermal Hall conductance. The Moore–Read/fluid is the paradigmatic example at $N$4, supporting Ising anyons with fusion rules $N$5, $N$6 and quasiparticle braiding statistics capturing nontrivial topological order (Yang, 2023, Jackson et al., 2013).

Variants such as the anti-Pfaffian and the PH-Pfaffian, related via particle–hole transformation or angular momentum channel, share similar structures but differ in symmetry and edge content. Minor modifications in pairing function $N$7—e.g., using antiholomorphic variables or introducing higher angular momentum—can reverse chirality or enforce PH-symmetry, leading to new candidate phases for experimental and numerical investigation (Milovanovic et al., 2021, Yang, 29 Mar 2025).

For the anti-Pfaffian at $N$8 (shift $N$9), the state is the particle-hole conjugate of the MR Pfaffian, and explicit constructions with partons have been shown to yield large overlaps and identical entanglement spectra with particle–hole conjugate states (Balram et al., 2018). The PH-Pfaffian, projected into the LLL, is the canonical model for a particle-hole symmetric gapped state consistent with observed thermal Hall conductance at $N \times N$0 (Yang, 2023).

3. Landau Level Projection, Mappings, and Hierarchy

Pfaffian trial states are not always, in their bare form, confined to the lowest Landau level (LLL); LLL or higher Landau level (SLL, TLL, etc.) projection is required for physical relevance in FQHE contexts. Projection is implemented either analytically (e.g., replacing $N \times N$1 in the symmetric gauge) or numerically (e.g., via basis expansions on the sphere) (Yang, 29 Mar 2025, Yang, 2023).

Notably, recent results have established exact unitary mappings between certain SLL- and LLL-projected Pfaffian states. Specifically, the SLL PH-Pfaffian (antiholomorphic pairing projected to SLL) maps exactly to an LLL projected orbital angular momentum $N \times N$2 pairing Pfaffian (the LLL af-Pfaffian), differing only by the action of Landau-level raising operators $N \times N$3 (Yang, 29 Mar 2025). This mapping unifies the understanding of bulk and edge physics: the edge Majorana mode structure is preserved under the mapping, while certain neutral bosonic edge excitations exist only in the LLL af-Pfaffian and are “mapped out” when projecting to SLL.

These connections establish a hierarchy of trial Pfaffian states parameterized by pairing angular momentum $N \times N$4, with different physical edge content and implications for experimental observables, yet unitarily related at the level of bulk topological order (Yang, 29 Mar 2025).

4. Entanglement, Correlation Functions, and Overlaps

A central diagnostic for classifying the topological order encoded by Pfaffian trial states is the orbital entanglement spectrum (OES). For both the SLL PH-Pfaffian and its LLL-mapped antianalytic $N \times N$5 partner—and for the anti-Pfaffian—numerical studies show identical low-lying OES “fingerprints” with level counting sequences $N \times N$6 at consecutive $N \times N$7 values, and extremely large wavefunction overlaps (e.g., $N \times N$8 for $N \times N$9) (Yang, 2023, Yang, 2023).

Density–density correlations further distinguish gapped from gapless phases. For the MR and anti-Pfaffian (as projected paired-CF states), two-point correlation functions decay exponentially with a well-defined correlation length (e.g., $A$0), consistent with a gapped topological phase. By contrast, the simplest projected PH-Pfaffian displays a lack of exponential decay for system sizes up to $A$1 ($A$2), closely mimicking a compressible composite Fermi liquid unless further variational ingredients are added (Yutushui et al., 2020, Mishmash et al., 2018).

Table: Selected Overlaps for SLL Pfaffian Trial States ($A$3 electrons) (Yang, 2023) | State | Overlap with Exact SLL Ground State | Overlap with aPf | |---------------------------------|-------------------------------------|-----------------------| | $A$4 | 0.92–0.94 | 0.95–0.98 | | $A$5 | 0.94–0.96 | 0.95–0.98 | | Anti-Pfaffian ($A$6)| 0.95 | 1 (reference state) |

5. Special Hamiltonians and Parent Operators

Pfaffian wavefunctions are characterized by their vanishing properties when clusters of particles coincide, enabling the construction of model “parent” Hamiltonians for which these trial states are exact zero-energy ground states. For the MR Pfaffian, a three-body delta function Hamiltonian enforces that the wavefunction vanishes as three particles approach coincidence, and the state is the unique densest zero mode (Jackson et al., 2013, Hafezi et al., 2013). Such Hamiltonians may be engineered in systems such as circuit QED arrays using tailored on-site interactions.

Approximate two-body generating interactions for the PH-Pfaffian have also been constructed by optimizing Haldane pseudopotentials, resulting in simple multi-parameter projected Hamiltonians (e.g., CV7, MV3) whose exact ground states have overlap $A$7 with the PH-Pfaffian state for up to $A$8 electrons and reproduce key entanglement spectrum and structure factor properties (Pakrouski, 2021).

In the context of electronic structure theory and variational Monte Carlo/FNDMC, Pfaffian-Jastrow ansätze (allowing for full singlet and triplet pairing) yield greater variational flexibility and can achieve high-accuracy binding energies for strongly correlated molecules—vastly improving over single-determinant wavefunctions at comparable computational cost (Genovese et al., 2020, Chen et al., 14 Jul 2025).

6. Variants and Extensions: Hafnian, Parton, and Neural Network-Augmented Pfaffians

Pfaffian trial states admit a wide family of extensions:

• Hafnian–PH-Pfaffian states: By multiplication of a symmetric Hafnian factor $A$9, one can produce new trial states (Hafnian–PH–Pfaffian) that are mathematically equivalent to uniform $$\Psi_{\text{Pf}}(z_1,\dots,z_N) = \mathrm{Pf}[g(z_i, z_j)] \prod_{i<j} (z_i - z_j)^M\, e^{-\sum_{k} |z_k|^2/(4\ell^2)}$$0 compressed quasiparticle-paired PH–Pfaffian states; these have the same flux shift as the MR Pfaffian and show larger ground-state overlaps for sufficiently strong short-range Coulomb repulsion (Yang, 2022).
• Parton constructions: Parton decompositions, such as $$\Psi_{\text{Pf}}(z_1,\dots,z_N) = \mathrm{Pf}[g(z_i, z_j)] \prod_{i<j} (z_i - z_j)^M\, e^{-\sum_{k} |z_k|^2/(4\ell^2)}$$1, allow for scalable variational ansätze at the anti-Pfaffian shift and demonstrate nearly perfect overlap and phase equivalence with the anti-Pfaffian state. Such forms also enable efficient Monte Carlo evaluation for large systems (Balram et al., 2018).
• Neural-network–augmented Pfaffians: Recent advances generalize the Pfaffian ansatz via “hidden-fermion” constructions and deep neural networks, encoding backflow-like corrections to the pairing function. These methods (e.g., Hidden-Fermion Pfaffian State—HFPS) demonstrate variational energies competitive with state-of-the-art electronic structure methods in two-dimensional Hubbard models and can flexibly interpolate between unpaired and paired phases (Chen et al., 14 Jul 2025).

7. Open Questions and Physical Realization

Despite the mathematical and topological richness of the Pfaffian trial construction, the precise stability of distinct Pfaffian phases under realistic Hamiltonians remains under active investigation. Numerical simulations and entanglement diagnostics consistently show that the simplest projected PH–Pfaffian is unstable to compressible phases in the clean LLL, suggesting a need for nontrivial LL mixing, disorder, or additional interaction terms to stabilize gapped non-Abelian order (Yutushui et al., 2020, Mishmash et al., 2018, Milovanovic et al., 2021). For anti-Pfaffian and SLL PH–Pfaffian projections, evidence strongly favors robust topological order with observed edge structure matching experiments when analyzed via entanglement spectra and wavefunction overlaps (Yang, 2023, Yang, 29 Mar 2025, Yang, 2023).

The mapping between SLL PH–Pfaffian and LLL $$\Psi_{\text{Pf}}(z_1,\dots,z_N) = \mathrm{Pf}[g(z_i, z_j)] \prod_{i<j} (z_i - z_j)^M\, e^{-\sum_{k} |z_k|^2/(4\ell^2)}$$2 AF-Pfaffian wavefunctions empowers variational and numerical investigations to be performed in the most convenient Landau level or angular momentum channel without loss of generality for topological classification, with important caveats regarding the physical accessibility of neutral bosonic edge modes across different projections (Yang, 29 Mar 2025).

A plausible implication is that, although the Pfaffian construction defines a family of trial states interconnected by projections and LL raising/lowering operators, the actual physical realization of the distinct topological sectors may depend delicately on nonuniversal details, including edge reconstruction and external perturbations. The ongoing refinement of neural quantum states and variational composite-fermion approaches continues to broaden the landscape of accessible Pfaffian-like quantum phases.