Hidden Fermion Pfaffian State (HFPS)

• HFPS is a neural quantum state architecture that extends hidden-fermion methods from Slater determinants to Pfaffians, enabling effective representation of both paired and unpaired fermionic phases.
• HFPS integrates hidden fermion sectors and neural-network backflow to implement controlled low-rank corrections, ensuring scalability in variational Monte Carlo simulations.
• HFPS demonstrates state-of-the-art energy accuracy in Hubbard model benchmarks by accurately capturing both s-wave and d-wave pairing channels.

Searching arXiv for recent and foundational papers on Hidden Fermion Pfaffian State and related Pfaffian literature. First, locating the dedicated HFPS paper and closely related Pfaffian references. Hidden Fermion Pfaffian State (HFPS) denotes a neural quantum state architecture in which hidden fermions and neural-network backflow ideas are extended from Slater determinants to Pfaffians, thereby yielding a variational class that can represent both unpaired and superconducting fermionic phases within scalable variational Monte Carlo (Chen et al., 14 Jul 2025). In a broader but less formal sense, the phrase also resonates with several strands of Pfaffian literature in which paired composite-fermion states, Pfaffian daughter hierarchies, particle-hole-symmetric paired states, and geometric response of Pfaffian phases are central, although those works do not explicitly use the term HFPS (Singh et al., 2023, Luo et al., 2017, Dwivedi et al., 2019). The narrow, formal meaning is therefore architectural and computational; the broader significance lies in its relation to Pfaffian pairing as a recurring organizing principle in strongly correlated many-body physics.

1. Formal definition and variational setting

HFPS is introduced as a response to a specific limitation of earlier fermionic neural quantum states. Hidden fermion determinant state (HFDS) and neural network backflow (NNBF) are built around Slater determinants, so they are naturally suited to unpaired fermions and Fermi-liquid-like states. The motivating observation of HFPS is that strongly correlated fermion systems frequently involve pairing, including superconductivity, and that the natural number-projected mean-field language for pairing is the Pfaffian, or equivalently Thouless or BCS form, rather than a determinant (Chen et al., 14 Jul 2025).

The starting point is the standard bilinear mean-field Hamiltonian

$$\hat{\mathcal H}_0=\sum_{p,q}^{M}\left(t_{pq}\hat c_p^\dagger \hat c_q+\frac{1}{2}\Delta_{pq}\hat c_p^\dagger \hat c_q^\dagger+\frac{1}{2}\Delta_{qp}^*\hat c_p\hat c_q\right),$$

whose ground state is a generalized Gaussian state,

$$\ket{\psi_0}=\exp\left(\frac{1}{2}\sum_{p,q}^{M}F_{pq}\hat c_p^\dagger \hat c_q^\dagger\right)\ket{0},$$

with antisymmetric pairing matrix $\mathbf F$. After projection to fixed particle number $N$, the state becomes

$$\ket{\psi_{\mathrm{pf}}}=\hat{\mathcal P}_N\ket{\psi_0} =\frac{1}{(N/2)!}\left(\sum_{p<q}F_{pq}\hat c_p^\dagger \hat c_q^\dagger\right)^{N/2}\ket{0},$$

and for a Fock configuration $\mathbf n$,

$$\psi_{\mathrm{pf}}(\mathbf n)=\mathrm{pf}(\mathbf n\star \mathbf F\star \mathbf n).$$

This is the basic Pfaffian wave-function that HFPS generalizes (Chen et al., 14 Jul 2025).

The formal role of HFPS is therefore not merely to add neural corrections to an existing paired state, but to replace the determinant-centric hidden-fermion architecture by a Pfaffian-centric one. That substitution is substantive because projected BCS states are a special case of Pfaffians, and determinants are contained inside the Pfaffian class rather than the reverse. A plausible implication is that HFPS is aimed at regimes where variational bias toward unpaired reference states would otherwise be problematic.

2. Hidden-sector construction and neural parametrization

HFPS enlarges the physical Hilbert space by adding hidden orbitals and hidden fermions. If the visible sector has $M$ orbitals and $N$ fermions, the hidden sector has $\tilde M$ orbitals and $$\ket{\psi_0}=\exp\left(\frac{1}{2}\sum_{p,q}^{M}F_{pq}\hat c_p^\dagger \hat c_q^\dagger\right)\ket{0},$$0 hidden fermions. In the enlarged space the wave-function is written as

$$\ket{\psi_0}=\exp\left(\frac{1}{2}\sum_{p,q}^{M}F_{pq}\hat c_p^\dagger \hat c_q^\dagger\right)\ket{0},$$1

with $$\ket{\psi_0}=\exp\left(\frac{1}{2}\sum_{p,q}^{M}F_{pq}\hat c_p^\dagger \hat c_q^\dagger\right)\ket{0},$$2 (Chen et al., 14 Jul 2025).

For visible configuration $$\ket{\psi_0}=\exp\left(\frac{1}{2}\sum_{p,q}^{M}F_{pq}\hat c_p^\dagger \hat c_q^\dagger\right)\ket{0},$$3 and hidden configuration $$\ket{\psi_0}=\exp\left(\frac{1}{2}\sum_{p,q}^{M}F_{pq}\hat c_p^\dagger \hat c_q^\dagger\right)\ket{0},$$4,

$$\ket{\psi_0}=\exp\left(\frac{1}{2}\sum_{p,q}^{M}F_{pq}\hat c_p^\dagger \hat c_q^\dagger\right)\ket{0},$$5

To obtain a physical wave-function, the hidden occupations are tied to the visible configuration, yielding the configuration-dependent form

$$\ket{\psi_0}=\exp\left(\frac{1}{2}\sum_{p,q}^{M}F_{pq}\hat c_p^\dagger \hat c_q^\dagger\right)\ket{0},$$6

A practical issue is that $$\ket{\psi_0}=\exp\left(\frac{1}{2}\sum_{p,q}^{M}F_{pq}\hat c_p^\dagger \hat c_q^\dagger\right)\ket{0},$$7 is not directly compatible with standard network outputs. The construction therefore factors it through a unitary matrix $$\ket{\psi_0}=\exp\left(\frac{1}{2}\sum_{p,q}^{M}F_{pq}\hat c_p^\dagger \hat c_q^\dagger\right)\ket{0},$$8,

$$\ket{\psi_0}=\exp\left(\frac{1}{2}\sum_{p,q}^{M}F_{pq}\hat c_p^\dagger \hat c_q^\dagger\right)\ket{0},$$9

and defines

$\mathbf F$0

Using $\mathbf F$1, the state becomes

$\mathbf F$2

where

$\mathbf F$3

is an additional generalized Jastrow factor produced by the ANN. In implementation, $\mathbf F$4 and $\mathbf F$5 are direct variational parameters, while a CNN, GCNN, or other ANN outputs $\mathbf F$6 and $\mathbf F$7 (Chen et al., 14 Jul 2025).

This formulation is also recast as a Pfaffian backflow ansatz,

$\mathbf F$8

with

$\mathbf F$9

The backflow correction has rank bounded by the number of hidden fermions $N$0, which is identified as a key advantage of the formulation (Chen et al., 14 Jul 2025).

3. Expressive hierarchy and relation to determinant-based ansätze

HFPS is designed to represent both unpaired states and paired superconducting states. The argument is structural. A Pfaffian can reduce to a Slater determinant, so determinant states are a subset of Pfaffian states. The paper states this explicitly through

$N$1

for any antisymmetric $N$2 with $N$3 (Chen et al., 14 Jul 2025).

The relation to HFDS is equally explicit. The earlier hidden fermion determinant state is

$N$4

HFPS can reproduce HFDS by choosing

$N$5

but not vice versa. The formal conclusion drawn in the source is that HFPS is strictly more general because some Pfaffians cannot be represented as HFDS, whereas HFDS imposes a determinant structure on the enlarged space (Chen et al., 14 Jul 2025).

Relative to NNBF, HFPS is mapped to a Pfaffian backflow form with a controlled low-rank correction. The emphasis on controlled rank is not incidental: it underwrites both expressivity and computational tractability. This suggests that the architectural novelty of HFPS is not only the replacement of determinants by Pfaffians, but also the replacement of generic backflow by a rank-bounded correction tied directly to hidden-fermion number.

4. Computational scaling and large-system evaluation

The computational claim of HFPS is that it remains compatible with large-scale VMC despite the use of Pfaffians. A naive Pfaffian evaluation costs $N$6 for an $N$7 antisymmetric matrix. If the full enlarged Pfaffian were recomputed directly at every Monte Carlo step, the cost would be substantial. The paper identifies two accelerations that change this conclusion (Chen et al., 14 Jul 2025).

The first is low-rank updates. Since VMC changes only a few occupations at each Monte Carlo move, the sliced Pfaffian matrix changes by a low-rank modification. With stored inverse information, a Pfaffian update is reduced from

$N$8

The second is sublattice or translation symmetry. By choosing a unit-cell structure in $N$9, translations that differ only by a sublattice permutation do not require recomputing the Pfaffian from scratch. The effect is to reduce the cost of symmetry projection from scaling with the full system size to scaling with the unit cell (Chen et al., 14 Jul 2025).

The summary scaling given in the source is that, without acceleration, a full VMC step scales like

$$\ket{\psi_{\mathrm{pf}}}=\hat{\mathcal P}_N\ket{\psi_0} =\frac{1}{(N/2)!}\left(\sum_{p<q}F_{pq}\hat c_p^\dagger \hat c_q^\dagger\right)^{N/2}\ket{0},$$0

which for fixed density becomes roughly $$\ket{\psi_{\mathrm{pf}}}=\hat{\mathcal P}_N\ket{\psi_0} =\frac{1}{(N/2)!}\left(\sum_{p<q}F_{pq}\hat c_p^\dagger \hat c_q^\dagger\right)^{N/2}\ket{0},$$1. With low-rank updates and sublattice symmetry, the HFPS forward pass scales as $$\ket{\psi_{\mathrm{pf}}}=\hat{\mathcal P}_N\ket{\psi_0} =\frac{1}{(N/2)!}\left(\sum_{p<q}F_{pq}\hat c_p^\dagger \hat c_q^\dagger\right)^{N/2}\ket{0},$$2 and a full VMC step as $$\ket{\psi_{\mathrm{pf}}}=\hat{\mathcal P}_N\ket{\psi_0} =\frac{1}{(N/2)!}\left(\sum_{p<q}F_{pq}\hat c_p^\dagger \hat c_q^\dagger\right)^{N/2}\ket{0},$$3 (Chen et al., 14 Jul 2025). In the terminology of the abstract, HFPS thus scales to large systems with favorable asymptotic complexity.

5. Numerical performance in Hubbard-model benchmarks

The benchmark model is the square-lattice Hubbard Hamiltonian

$$\ket{\psi_{\mathrm{pf}}}=\hat{\mathcal P}_N\ket{\psi_0} =\frac{1}{(N/2)!}\left(\sum_{p<q}F_{pq}\hat c_p^\dagger \hat c_q^\dagger\right)^{N/2}\ket{0},$$4

and performance is measured using the relative variational energy error

$$\ket{\psi_{\mathrm{pf}}}=\hat{\mathcal P}_N\ket{\psi_0} =\frac{1}{(N/2)!}\left(\sum_{p<q}F_{pq}\hat c_p^\dagger \hat c_q^\dagger\right)^{N/2}\ket{0},$$5

where $$\ket{\psi_{\mathrm{pf}}}=\hat{\mathcal P}_N\ket{\psi_0} =\frac{1}{(N/2)!}\left(\sum_{p<q}F_{pq}\hat c_p^\dagger \hat c_q^\dagger\right)^{N/2}\ket{0},$$6 is the exact or benchmark ground-state energy and $$\ket{\psi_{\mathrm{pf}}}=\hat{\mathcal P}_N\ket{\psi_0} =\frac{1}{(N/2)!}\left(\sum_{p<q}F_{pq}\hat c_p^\dagger \hat c_q^\dagger\right)^{N/2}\ket{0},$$7 is the infinite-temperature reference (Chen et al., 14 Jul 2025).

In the weakly correlated Fermi-liquid regime on a $$\ket{\psi_{\mathrm{pf}}}=\hat{\mathcal P}_N\ket{\psi_0} =\frac{1}{(N/2)!}\left(\sum_{p<q}F_{pq}\hat c_p^\dagger \hat c_q^\dagger\right)^{N/2}\ket{0},$$8 lattice with PBC and $$\ket{\psi_{\mathrm{pf}}}=\hat{\mathcal P}_N\ket{\psi_0} =\frac{1}{(N/2)!}\left(\sum_{p<q}F_{pq}\hat c_p^\dagger \hat c_q^\dagger\right)^{N/2}\ket{0},$$9 at filling $\mathbf n$0, HFPS is compared to determinant+RBM and HFDS. The reported result is that HFPS improves the variational energy significantly over HFDS, reduces the relative error by roughly two orders of magnitude compared with the best previous NQS result, and yields raw energies extremely close to exact diagonalization (Chen et al., 14 Jul 2025).

For the attractive Hubbard model on an $\mathbf n$1 lattice at $\mathbf n$2, where the system exhibits s-wave superconductivity interpolating from BCS at weak coupling to BEC-like local pairs at stronger coupling, the diagnostic is the pair-pair correlator

$\mathbf n$3

with

$\mathbf n$4

used as the ODLRO diagnostic. HFPS, initialized from a BCS mean-field state, produces pair correlations that match DQMC within error bars and reaches energy errors of order

$\mathbf n$5

while the BCS wave-function remains around $\mathbf n$6. This is presented as a clear demonstration that HFPS accurately captures s-wave superconductivity across the BCS–BEC crossover (Chen et al., 14 Jul 2025).

At half-filling on the repulsive $\mathbf n$7 Hubbard model, the benchmarks are against AFQMC, with comparisons to PP+RBM + Lanczos, HFDS, fPEPS ($\mathbf n$8), and Transformer-NNBF. The reported conclusion is that HFPS is more accurate than these methods, with relative energy error at the $\mathbf n$9 to $$\psi_{\mathrm{pf}}(\mathbf n)=\mathrm{pf}(\mathbf n\star \mathbf F\star \mathbf n).$$0 scale and often better than the competing NQSs. The same source notes that benchmark uncertainty from AFQMC becomes non-negligible when the variational error becomes this small (Chen et al., 14 Jul 2025).

The most demanding test is the stripe phase on $$\psi_{\mathrm{pf}}(\mathbf n)=\mathrm{pf}(\mathbf n\star \mathbf F\star \mathbf n).$$1 systems at $$\psi_{\mathrm{pf}}(\mathbf n)=\mathrm{pf}(\mathbf n\star \mathbf F\star \mathbf n).$$2 and $$\psi_{\mathrm{pf}}(\mathbf n)=\mathrm{pf}(\mathbf n\star \mathbf F\star \mathbf n).$$3. HFPS is reported to outperform MLP-NNBF, HFDS, TBF, TBF+Lanczos, and Transformer-NNBF; to achieve a nearly exact energy on $$\psi_{\mathrm{pf}}(\mathbf n)=\mathrm{pf}(\mathbf n\star \mathbf F\star \mathbf n).$$4 with

$$\psi_{\mathrm{pf}}(\mathbf n)=\mathrm{pf}(\mathbf n\star \mathbf F\star \mathbf n).$$5

and to reach

$$\psi_{\mathrm{pf}}(\mathbf n)=\mathrm{pf}(\mathbf n\star \mathbf F\star \mathbf n).$$6

on $$\psi_{\mathrm{pf}}(\mathbf n)=\mathrm{pf}(\mathbf n\star \mathbf F\star \mathbf n).$$7, better than the reported Transformer-NNBF result

$$\psi_{\mathrm{pf}}(\mathbf n)=\mathrm{pf}(\mathbf n\star \mathbf F\star \mathbf n).$$8

In that regime the diagnostics show spin density wave order with period 16, charge density wave order with period 8, and finite short-range d-wave pairing correlations. The paper does not claim a d-wave superconductor in the $$\psi_{\mathrm{pf}}(\mathbf n)=\mathrm{pf}(\mathbf n\star \mathbf F\star \mathbf n).$$9-doped stripe phase; rather, it argues that d-wave pairing is an important short-range ingredient of the energetics and that HFPS represents it accurately (Chen et al., 14 Jul 2025).

The d-wave pairing operator used in that analysis is

$M$0

with $M$1 for horizontal bonds and $M$2 for vertical bonds. The mean-field Hamiltonian used to seed that structure is

$M$3

Taken together, these experiments support the claim that HFPS captures both s-wave and d-wave pairing channels and provides state-of-the-art variational accuracy in different regimes of both the attractive and repulsive Hubbard models (Chen et al., 14 Jul 2025).

6. Relation to Pfaffian topological phases and broader usage

The formal HFPS architecture should be distinguished from the broader Pfaffian literature in quantum Hall and synthetic platforms, where the phrase “Hidden Fermion Pfaffian State” is usually absent even when the phenomenology is close. The following works are central points of contact.

Paper	System	Relevance to HFPS
(Singh et al., 2023)	Wide GaAs quantum wells near $M$4	One-component Pfaffian parent and Pfaffian daughter fractions
(Dwivedi et al., 2019)	Pfaffian state on curved Riemann surfaces	Universal geometric and anomaly response
(Luo et al., 2017)	ZnO with strong Landau level mixing	PH-symmetric Pfaffian-like paired state
(Apalkov et al., 2022)	AB-stacked bilayer graphene	Tunable half-filled Pfaffian regime
(Hayward et al., 2016)	Coupled atom-cavity arrays	Bosonic Pfaffian-like analogue with effective three-body interaction

In wide GaAs quantum wells, the observed $M$5 FQHS is argued to be a one-component Pfaffian rather than a competing two-component Jain/331 state. The core evidence is the abrupt emergence of unusually strong $M$6 and $M$7 states at the same tuning point where $M$8 becomes especially robust. Since $M$9 and $N$0 are identified as precisely the theoretically predicted, simplest daughter states of a one-component Pfaffian $N$1 FQHS, the data are interpreted as showing a topological phase transition from ordinary Jain-sequence fractions to Pfaffian hierarchical daughter states (Singh et al., 2023). That work does not use the term HFPS, but it is explicitly described as closely aligned with the idea of a robust $N$2 parent whose side fractions behave like Pfaffian daughters rather than standard Jain states.

In the geometric-response literature, the Pfaffian is formulated on arbitrary compact Riemann surfaces with curved metric and inhomogeneous magnetic field through an Ising-CFT correlator composed of a free compact boson and a free Majorana fermion. The resulting generating functional encodes linear response to metric and magnetic-field variations, while the effective action contains Aubin–Yau, Mabuchi, and Liouville terms. The density expansion,

$N$3

and the long-wavelength static structure factor,

$N$4

exhibit the gravitational-anomaly contribution associated with central charge $N$5 (Dwivedi et al., 2019). If HFPS is interpreted as a physical Pfaffian state rather than only a variational architecture, this suggests inheritance of the Pfaffian’s shift or spin-response structure, Hall-viscosity-like curvature coupling, and $N$6 anomaly signature.

In ZnO, where Landau level mixing is non-perturbatively strong, the screened Coulomb interaction approach yields evidence for a particle-hole-symmetric Pfaffian-like incompressible state. The overlap with the Pfaffian trial state is strongly system-size dependent and never very high, reaching at most about $N$7 in the calculations summarized, while the range $N$8 is identified as a possible topological phase-transition regime (Luo et al., 2017). This work also does not use HFPS terminology, but it is described as conceptually close because it moves beyond a simple Pfaffian versus anti-Pfaffian dichotomy toward a PH-symmetric paired phase.

In AB-stacked bilayer graphene, the Pfaffian problem is recast in a tunable Dirac-fermion setting. The special bilayer Landau levels $N$9 and $\tilde M$0 can host a $\tilde M$1 Pfaffian state, and the overlap with the Pfaffian trial state together with the collective gap shows a nonmonotonic dependence on magnetic field, with a representative maximum near $\tilde M$2 T for $\tilde M$3 meV and $\tilde M$4 (Apalkov et al., 2022). The same source emphasizes quasiparticles of fractional charge $\tilde M$5 and non-Abelian statistics. Here again, the paper does not label the phase HFPS, but it does describe a paired composite-fermion Pfaffian state of the type to which the phrase may be applied informally.

Finally, a synthetic analogue appears in coupled atom-cavity arrays. By replacing the two-level atom in the Jaynes–Cummings–Hubbard model with a three-level atom and tuning

$\tilde M$6

the effective two-body interaction can be suppressed while a three-body repulsion remains comparatively dominant. On a torus, the reported signatures are a quasi-gap, a three-dimensional quasi-degenerate ground-state manifold on a $\tilde M$7 lattice with 4 particles and 4 flux quanta, a total Chern number of 3, and significant overlap with the Pfaffian trial wavefunction (Hayward et al., 2016). The authors describe the state as “Pfaffian-like,” not as a rigorous proof of the exact Moore–Read phase. The connection to HFPS is therefore analogical rather than terminological.

A recurrent misconception is that HFPS is simply another name for the Moore–Read Pfaffian. The literature provided here does not support that equivalence. The explicit formal term “Hidden Fermion Pfaffian State” names a neural-network-augmented Pfaffian variational ansatz for interacting lattice fermions (Chen et al., 14 Jul 2025). By contrast, the fractional quantum Hall, graphene, ZnO, curved-space, and cavity-QED papers discuss physical Pfaffian or Pfaffian-like phases, and several state explicitly that they do not use the term HFPS (Singh et al., 2023, Luo et al., 2017, Apalkov et al., 2022, Hayward et al., 2016). The more precise statement is that HFPS, in its formal sense, is a computational architecture whose conceptual center is Pfaffian pairing, while several neighboring literatures supply the topological, geometric, and experimental contexts in which Pfaffian structure acquires physical meaning.