Quantum Pfaffian: Theory and Applications

• Quantum Pfaffians are deformations of the classical Pfaffian that encode antisymmetric pairing and non-Abelian topological order, notably in fractional quantum Hall states like the Moore–Read state.
• They underpin rich mathematical structures in quantum algebra, including q-deformations, multiparameter extensions, and braided invariants that extend classical antisymmetry to noncommutative settings.
• Pfaffian formulas enable efficient computation of fermionic Gaussian amplitudes and point processes, linking theoretical advances with experimental studies in synthetic quantum matter.

Searching arXiv for recent and foundational papers on quantum Pfaffians across quantum Hall, algebraic, and fermionic-Gaussian contexts. “Quantum Pfaffian” denotes a family of constructions centered on the Pfaffian of an antisymmetric matrix, appearing in several distinct but connected parts of quantum theory. In fractional quantum Hall physics it most commonly refers to the Moore–Read Pfaffian state and its relatives at half filling, where the Pfaffian encodes chiral $p$-wave pairing, non-Abelian anyons, edge conformal field theory, and geometric response. In quantum algebra it denotes $q$-deformed, multiparameter, and dynamical analogues of the classical Pfaffian. In quantum many-body computation and quantum information it denotes exact Pfaffian formulas for amplitudes, correlators, point processes, and Monte Carlo weights of fermionic Gaussian systems (Zhu et al., 2020, Jing et al., 2013, Rajabpour et al., 3 Jun 2025).

1. Moore–Read Pfaffian as a non-Abelian quantum Hall state

The canonical quantum Hall Pfaffian is the Moore–Read state. In planar complex coordinates it takes the fermionic form

$$\Psi_{\rm Pf}=\mathrm{Pf}\!\left[\frac{1}{z_i-z_j}\right]\prod_{i<j}(z_i-z_j)^2,$$

with

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

The Pfaffian factor describes $p$-wave pairing of composite fermions, while the Jastrow factor attaches two vortices per particle and fixes the sphere flux to $N_\phi=2N-3$ (Yang, 2022, Milovanovic et al., 2021).

This paired state is the standard non-Abelian candidate for $\nu=5/2$. Its edge theory contains one chiral boson and one chiral Majorana fermion,

$$\mathcal L_{\rm edge}=\frac{1}{4\pi}\partial_x\phi(\partial_t-v_c\partial_x)\phi+i\,\psi(\partial_t-v_n\partial_x)\psi,$$

with total chiral central charge $c=3/2$. The charge mode is carried by $\phi$, while $q$0 is the neutral Majorana mode of the Ising CFT. The bulk is characterized by an orbital spin $q$1 entering the guiding-center Hall viscosity

$q$2

so that at $q$3 one has $q$4 (Zhu et al., 2020).

Its fundamental quasiholes carry charge $q$5 and correspond to the Ising field $q$6. Their fusion rules are

$q$7

and $q$8 well-separated charge-$q$9 quasiholes span a $$\Psi_{\rm Pf}=\mathrm{Pf}\!\left[\frac{1}{z_i-z_j}\right]\prod_{i<j}(z_i-z_j)^2,$$0-dimensional non-Abelian Hilbert space. Trial-wavefunction studies of excitations show that quasielectron counting at large angular momentum matches quasihole counting and the Ising$$\Psi_{\rm Pf}=\mathrm{Pf}\!\left[\frac{1}{z_i-z_j}\right]\prod_{i<j}(z_i-z_j)^2,$$1 boundary theory, supporting the statement that the CFT describing both sides of the plateau is the same (Zhu et al., 2020, Rodriguez et al., 2011).

2. Pfaffian, anti-Pfaffian, and PH-Pfaffian competition at $$\Psi_{\rm Pf}=\mathrm{Pf}\!\left[\frac{1}{z_i-z_j}\right]\prod_{i<j}(z_i-z_j)^2,$$2

The Pfaffian admits two closely related competitors. Its particle–hole conjugate is the anti-Pfaffian, while the PH-Pfaffian is obtained by replacing $$\Psi_{\rm Pf}=\mathrm{Pf}\!\left[\frac{1}{z_i-z_j}\right]\prod_{i<j}(z_i-z_j)^2,$$3 inside the Pfaffian,

$$\Psi_{\rm Pf}=\mathrm{Pf}\!\left[\frac{1}{z_i-z_j}\right]\prod_{i<j}(z_i-z_j)^2,$$4

Under exact particle–hole symmetry in the half-filled second Landau level, $$\Psi_{\rm Pf}=\mathrm{Pf}\!\left[\frac{1}{z_i-z_j}\right]\prod_{i<j}(z_i-z_j)^2,$$5 is self-conjugate. It has the same non-Abelian statistics as the Pfaffian but reversed chirality; correspondingly, the total central charge is $$\Psi_{\rm Pf}=\mathrm{Pf}\!\left[\frac{1}{z_i-z_j}\right]\prod_{i<j}(z_i-z_j)^2,$$6 rather than $$\Psi_{\rm Pf}=\mathrm{Pf}\!\left[\frac{1}{z_i-z_j}\right]\prod_{i<j}(z_i-z_j)^2,$$7 (Yang, 2022).

Field-theoretical analyses based on Son’s Dirac composite fermion description tie the competition among these phases to Landau-level mixing. A Dirac mass term $$\Psi_{\rm Pf}=\mathrm{Pf}\!\left[\frac{1}{z_i-z_j}\right]\prod_{i<j}(z_i-z_j)^2,$$8 breaks particle–hole symmetry, with small positive $$\Psi_{\rm Pf}=\mathrm{Pf}\!\left[\frac{1}{z_i-z_j}\right]\prod_{i<j}(z_i-z_j)^2,$$9 selecting the Pfaffian and small negative $$\mathrm{Pf}(A_{ij})=\frac{1}{2^{N/2}(N/2)!}\sum_{\sigma\in S_N}\mathrm{sgn}(\sigma)\prod_{k=1}^{N/2}A_{\sigma(2k-1),\sigma(2k)}.$$0 selecting the anti-Pfaffian. The PH-Pfaffian channel appears only when Landau-level mixing is strong and cannot be treated perturbatively; in that regime a non-perturbative admixture of at least two Landau levels is required. In the bilayer analogy at total filling factor one, the intermediate phase is described as coexisting intra-layer and inter-layer negative-flux $$\mathrm{Pf}(A_{ij})=\frac{1}{2^{N/2}(N/2)!}\sum_{\sigma\in S_N}\mathrm{sgn}(\sigma)\prod_{k=1}^{N/2}A_{\sigma(2k-1),\sigma(2k)}.$$1-wave pairing, and this analogy motivates a two-Landau-level PH-Pfaffian wave function (Milovanovic et al., 2021, Milovanovic et al., 2021).

Several trial constructions were introduced to reconcile thermal Hall data with microscopic numerics. The compressed PH-Pfaffian removes two flux quanta from the parent PH-Pfaffian, creates two abelian Laughlin-type quasiparticles of maximum avoidance, and lives at the same flux $$\mathrm{Pf}(A_{ij})=\frac{1}{2^{N/2}(N/2)!}\sum_{\sigma\in S_N}\mathrm{sgn}(\sigma)\prod_{k=1}^{N/2}A_{\sigma(2k-1),\sigma(2k)}.$$2 as the ordinary Pfaffian. Exact diagonalization for $$\mathrm{Pf}(A_{ij})=\frac{1}{2^{N/2}(N/2)!}\sum_{\sigma\in S_N}\mathrm{sgn}(\sigma)\prod_{k=1}^{N/2}A_{\sigma(2k-1),\sigma(2k)}.$$3 with varying $$\mathrm{Pf}(A_{ij})=\frac{1}{2^{N/2}(N/2)!}\sum_{\sigma\in S_N}\mathrm{sgn}(\sigma)\prod_{k=1}^{N/2}A_{\sigma(2k-1),\sigma(2k)}.$$4 finds that the overlap with the exact ground state crosses from the Pfaffian to the compressed PH-Pfaffian at $$\mathrm{Pf}(A_{ij})=\frac{1}{2^{N/2}(N/2)!}\sum_{\sigma\in S_N}\mathrm{sgn}(\sigma)\prod_{k=1}^{N/2}A_{\sigma(2k-1),\sigma(2k)}.$$5, while the neutral gap eventually closes for $$\mathrm{Pf}(A_{ij})=\frac{1}{2^{N/2}(N/2)!}\sum_{\sigma\in S_N}\mathrm{sgn}(\sigma)\prod_{k=1}^{N/2}A_{\sigma(2k-1),\sigma(2k)}.$$6 (Yang, 2020). The Hafnian PH-Pfaffian state,

$$\mathrm{Pf}(A_{ij})=\frac{1}{2^{N/2}(N/2)!}\sum_{\sigma\in S_N}\mathrm{sgn}(\sigma)\prod_{k=1}^{N/2}A_{\sigma(2k-1),\sigma(2k)}.$$7

is mathematically identical to the compressed PH-Pfaffian and shares the same $$\mathrm{Pf}(A_{ij})=\frac{1}{2^{N/2}(N/2)!}\sum_{\sigma\in S_N}\mathrm{sgn}(\sigma)\prod_{k=1}^{N/2}A_{\sigma(2k-1),\sigma(2k)}.$$8, thereby remaining consistent with $$\mathrm{Pf}(A_{ij})=\frac{1}{2^{N/2}(N/2)!}\sum_{\sigma\in S_N}\mathrm{sgn}(\sigma)\prod_{k=1}^{N/2}A_{\sigma(2k-1),\sigma(2k)}.$$9 (Yang, 2022).

A separate route to PH-symmetric physics appears in systems with strong screening and small Landau-level gaps. In ZnO, where $p$0–$p$1, screened exact-diagonalization studies found that the conventional Pfaffian overlap becomes strongly size-dependent and suggested a PH-symmetric Pfaffian candidate at shift $p$2. The same work identified a possible topological phase transition in the range $p$3 (Luo et al., 2017).

3. Interface physics, Hall viscosity mismatch, and mesoscopic structure

The Pfaffian and anti-Pfaffian need not appear only as uniform bulk phases. Thermal Hall experiments motivated pictures in which mesoscopic puddles of Pfaffian and anti-Pfaffian order coexist. At a sharp Pf–APf domain wall, the two edges meet and their counter-propagating charge modes can gap out by tunneling, leaving only four co-propagating neutral Majorana modes on the interface. Density-matrix renormalization group simulations on cylinder geometry found that the edge modes strongly hybridize and that the interface develops an intrinsic electric dipole moment (Zhu et al., 2020).

The dipole has a topological origin. The two bulks carry opposite Hall viscosities,

$p$4

so that

$p$5

Thermodynamic force balance between Hall-viscosity-driven stress and the interface dipole requires

$p$6

or equivalently

$p$7

Cylinder DMRG found a robust $p$8 charge on the Pf side and $p$9 on the APf side, exactly counterbalancing the Hall-viscosity difference (Zhu et al., 2020).

A distinct mesoscopic regime arises in the weak-coupling Pfaffian viewed as a $N_\phi=2N-3$0 superconductor of composite fermions. In the BCS limit $N_\phi=2N-3$1, the coherence length $N_\phi=2N-3$2 becomes much larger than the screening length $N_\phi=2N-3$3, giving $N_\phi=2N-3$4. In this regime the system behaves as a Type I quantum Hall liquid: the charge-$N_\phi=2N-3$5 vortices attract, and density deviations away from $N_\phi=2N-3$6 produce Coulomb-frustrated phase separation into droplets, stripes, or bubbles rather than a dilute gas of isolated quasiparticles (Parameswaran et al., 2010). This suggests that “quantum Pfaffian” physics includes not only isolated anyons and edges, but also mesoscale textures controlled by pairing scale, screening, and long-range Coulomb repulsion.

4. Quantum Pfaffians in quantum group theory and noncommutative geometry

In quantum algebra, the quantum Pfaffian is a deformation of the classical Pfaffian for matrices with noncommuting entries. A basic setting is a $N_\phi=2N-3$7 $N_\phi=2N-3$8-skew-symmetric matrix $N_\phi=2N-3$9 satisfying

$\nu=5/2$0

together with $\nu=5/2$1-Plücker relations such as

$\nu=5/2$2

The quantum Pfaffian is then defined by a weighted sum over pairings,

$\nu=5/2$3

and in low dimension

$\nu=5/2$4

Its defining properties include skew-symmetry up to powers of $\nu=5/2$5, multilinearity, and the factorization

$\nu=5/2$6

for compatible quantum matrices $\nu=5/2$7 (Jing et al., 2014).

The quantum exterior-algebra approach gives a structural interpretation. If

$\nu=5/2$8

in $\nu=5/2$9, then

$$\mathcal L_{\rm edge}=\frac{1}{4\pi}\partial_x\phi(\partial_t-v_c\partial_x)\phi+i\,\psi(\partial_t-v_n\partial_x)\psi,$$0

This same framework yields complete families of quantum Plücker relations, proves that any $$\mathcal L_{\rm edge}=\frac{1}{4\pi}\partial_x\phi(\partial_t-v_c\partial_x)\phi+i\,\psi(\partial_t-v_n\partial_x)\psi,$$1 quantum determinant can be realized as a quantum Pfaffian, and extends to quantum hyper-Pfaffians when indices are grouped into blocks of size $$\mathcal L_{\rm edge}=\frac{1}{4\pi}\partial_x\phi(\partial_t-v_c\partial_x)\phi+i\,\psi(\partial_t-v_n\partial_x)\psi,$$2 (Jing et al., 2013).

The construction has several variants. Multiparameter quantum Pfaffians replace the single deformation parameter by families $$\mathcal L_{\rm edge}=\frac{1}{4\pi}\partial_x\phi(\partial_t-v_c\partial_x)\phi+i\,\psi(\partial_t-v_n\partial_x)\psi,$$3 and satisfy the covariance law

$$\mathcal L_{\rm edge}=\frac{1}{4\pi}\partial_x\phi(\partial_t-v_c\partial_x)\phi+i\,\psi(\partial_t-v_n\partial_x)\psi,$$4

with corresponding hyper-Pfaffians and minor expansions (Jing et al., 2017). Two-parameter formulations relate quantum Pfaffians, determinants, Hafnians, and permanents; in particular settings one finds $$\mathcal L_{\rm edge}=\frac{1}{4\pi}\partial_x\phi(\partial_t-v_c\partial_x)\phi+i\,\psi(\partial_t-v_n\partial_x)\psi,$$5 and a closely related quantum Hafnian equals a quantum permanent (Jing et al., 2015). Dynamical quantum Pfaffians introduce meromorphic weight factors $$\mathcal L_{\rm edge}=\frac{1}{4\pi}\partial_x\phi(\partial_t-v_c\partial_x)\phi+i\,\psi(\partial_t-v_n\partial_x)\psi,$$6 and obey the transformation law

$$\mathcal L_{\rm edge}=\frac{1}{4\pi}\partial_x\phi(\partial_t-v_c\partial_x)\phi+i\,\psi(\partial_t-v_n\partial_x)\psi,$$7

together with hyper-dynamical generalizations (Jing et al., 2020). In this algebraic sense, the quantum Pfaffian is a braided volume invariant, not a quantum Hall wave function.

5. Pfaffian formulas for fermionic Gaussian amplitudes and point processes

In fermionic Gaussian theory, Pfaffians provide exact amplitudes and correlators beyond number-conserving settings. For an $$\mathcal L_{\rm edge}=\frac{1}{4\pi}\partial_x\phi(\partial_t-v_c\partial_x)\phi+i\,\psi(\partial_t-v_n\partial_x)\psi,$$8-qubit fermionic Gaussian pure state $$\mathcal L_{\rm edge}=\frac{1}{4\pi}\partial_x\phi(\partial_t-v_c\partial_x)\phi+i\,\psi(\partial_t-v_n\partial_x)\psi,$$9, the amplitude in an arbitrary local Pauli product basis has the explicit form

$c=3/2$0

where the antisymmetric matrix $c=3/2$1 depends on the covariance data and on the local basis angles $c=3/2$2. A companion recursive relation relates amplitudes for different qubit numbers. The formalism was presented as a scalable route to formation probabilities, Shannon–Rényi entropies, global entanglement, negative log-likelihood objectives in Gaussian tomography, and post-measurement entanglement entropy (Rajabpour et al., 7 Feb 2025).

A more general result gives a single-Pfaffian formula for matrix elements of fermionic Gaussian operators between arbitrary Pauli product states,

$c=3/2$3

Its construction introduces sign-encoding matrices $c=3/2$4 whose commutators generate the full Lie algebra $c=3/2$5, while the associated Clifford embedding explains Pfaffian-sign consistency under basis rotations and Wick contractions. The resulting evaluations are available in $c=3/2$6 time per Pfaffian (Rajabpour et al., 3 Jun 2025).

Pfaffians also define point processes associated with superconducting Gaussian states. For a Bogoliubov–de Gennes Hamiltonian,

$c=3/2$7

the occupation correlators satisfy

$c=3/2$8

which defines a Pfaffian point process. In the zero-temperature projective case, samples have fixed parity almost surely. Quantum circuits based on Jordan–Wigner mapping and Bogoliubov transforms prepare these states with gate count $c=3/2$9 and depth $\phi$0 or $\phi$1 depending on connectivity constraints (Bardenet et al., 2023).

6. Numerical many-body methods and experimental realization

Pfaffians now function as direct computational objects in interacting many-body simulation. In Pfaffian quantum Monte Carlo, each auxiliary-field configuration in a Majorana representation carries a Boltzmann weight given by a closed-form Pfaffian rather than an ambiguous square root of a determinant. This resolves the sign ambiguity of earlier Majorana-based QMC by working on $\phi$2 rather than only its projection to $\phi$3. The resulting algorithm applies to generic interacting fermion models, retains polynomial-time fast updates, and was used to study the interacting Kitaev chain, where the average sign is smallest near the transition $\phi$4 and the edge-Majorana correlation agrees with the exact formula $\phi$5 in each parity sector (Han et al., 2024).

Experimental realization of a Pfaffian state has also been reported in synthetic matter. A three-particle bosonic Pfaffian state of ultracold $\phi$6 atoms was prepared in a $\phi$7 optical lattice subject to a Floquet-engineered synthetic magnetic field. The continuum bosonic Moore–Read state at $\phi$8,

$\phi$9

was adapted to a finite-size three-particle setting, and a Bayesian-optimized adiabatic protocol was used to reach the target regime (Kwan et al., 10 Jun 2026).

The measured observables were explicitly Pfaffian in character. Site-resolved multi-point density correlations showed that $q$00 and $q$01 exhibit a pronounced dip at short distance only in the Pfaffian regime, while the connected three-point correlator $q$02 takes finite negative values at $q$03. Hall drift measurements yielded $q$04, in semi-quantitative agreement with a finite-size expectation $q$05 for the $q$06 Pfaffian (Kwan et al., 10 Jun 2026). This suggests that the quantum Pfaffian is no longer only a theoretical wave function or algebraic invariant, but also an experimentally addressable paired topological state.

Across these settings, a common principle persists: antisymmetric pairing data are encoded in a Pfaffian, and that Pfaffian controls either topological order, braided algebraic invariants, or computationally tractable many-body amplitudes. The term therefore names not one object but a coherent class of quantum constructions in which fermionic pairing, noncommutativity, and exact antisymmetric combinatorics are inseparable.