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arXiv:2608.18168 · Journal of the Physical Society of Japan

Embedding Paired Free-Fermion Gaussian States into Gutzwiller-Projected Bardeen--Cooper--Schrieffer Wave Functions

Ryui Kaneko

cond-mat.stat-mechcond-mat.str-elquant-phSubmitted Aug 17, 2026• Announced Aug 18, 2026• 0 citations• Revised Sep 28, 2026 (v2)
v1 · Aug 17, 2026v2 · Sep 28, 2026
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Abstract

Gutzwiller-projected Bardeen--Cooper--Schrieffer (BCS) wave functions of Abrikosov fermions are widely used to describe quantum many-body states. Taking the one-dimensional transverse-field Ising model as an example, we exactly embed any even-parity spinless fermionic Gaussian state representable as a paired exponential in the chosen particle basis into a projected BCS state of spinful Abrikosov fermions. The construction provides controlled initial states for variational Monte Carlo studies of nonintegrable models.

Journal reference
J. Phys. Soc. Jpn. 95, 105002 (2026)
DOI
10.7566/JPSJ.95.105002
Comments
2 pages, 1 figure. All codes and data used in this manuscript are available at https://github.com/ryuikaneko/mvmc_1d_transverse_field_ising

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Gutzwiller-projected Bardeen--Cooper--Schrieffer (BCS) wave functions of Abrikosov fermions are widely used to describe quantum many-body states. Taking the one-dimensional transverse-field Ising model as an example, we exactly embed any even-parity spinless fermionic Gaussian state representable as a paired exponential in the chosen particle basis into a projected BCS state of spinful Abrikosov fermions. The construction provides controlled initial states for variational Monte Carlo studies of nonintegrable models.
We investigate the physics of projected ddd-wave pairing states using their fermionic projected entangled pair state (fPEPS) representation. First, we approximate a ddd-wave Bardeen-Cooper-Schrieffer state using the Gaussian fPEPS. Next, we translate the resulting state into fPEPS tensors and implement the Gutzwiller projection which removes double occupancy by modifying the local tensor elements. The tensor network representation of the projected ddd-wave pairing state allows us to evaluate physical quantities in the thermodynamic limit without employing the Gutzwiller approximation. Despite having very few variational parameters, such physically motivated tensor network states are shown to exhibit competitive energies for the doped $t\text{\ensuremath{-}}J$ model. We expect that such construction offers useful initial states and guidance for variational tensor network calculations.
Variational wave functions have enabled exceptional scientific breakthroughs related to the understanding of novel phases of matter. Examples include the Bardeen-Cooper-Schrieffer theory of superconductivity, the description of the fractional quantum Hall effect through the Laughlin state, and Feynman's variational understanding of large-scale quantum effects in liquid helium. More recently, Gutzwiller-projected wave functions, typically constructed from fermionic degrees of freedom, have been employed to examine quantum spin models in the presence of competing interactions, where exotic phases with no spontaneous symmetry breaking and fractional excitations may exist. In this work, we investigate the aforementioned fermionic wave functions supplemented with neural networks, specifically with the so-called restricted Boltzmann machine (RBM), to boost their accuracy and obtain reliable approximations to the ground state of generic spin models. In particular, we apply our neural augmented fermionic construction to the description of both magnetically ordered and disordered phases of increasing complexity, including cases where the ground state displays a nontrivial sign structure. Even though the RBM state is by far more effective for N\'eel states endowed with a particularly simple sign structure, it provides a significant improvement over the original fermionic state in highly frustrated regimes where a complex sign structure is anticipated, thus marking the path to an understanding of strongly correlated spin models on the lattice via neural Gutzwiller-projected variational wave functions.
Quantum simulation requires highly accurate input states. Gutzwiller-projected Bardeen-Cooper-Schrieffer (BCS) states provide physically motivated input states for solving strongly correlated lattice models, but their preparation on a quantum computer is hindered by the non-trivial nature of the Gutzwiller projection. We construct scalable quantum algorithms for this task by combining a circuit construction for arbitrary BCS states with the amplitude amplification for Gutzwiller projection (AAGP) procedure. AAGP yields a quadratic reduction in the number of projection queries compared with measurement-based postselection and leads to substantially improved fault-tolerant resource scaling. For projected BCS states optimized for the square-lattice ttt-JJJ model, we find that the projected-state weight decreases exponentially with system size, but the quadratic improvement is still large enough at physically relevant finite sizes to make a decisive practical difference. In particular, for a 100-site benchmark, AAGP reduces the required number of projection queries by about seven orders of magnitude. These results establish AAGP as an enabling input-state preparation protocol for projected BCS states in quantum simulation.
We demonstrate that the ground state of the fermionic Tonks-Girardeau gas is a number-conserving Bardeen-Cooper-Schrieffer (BCS) state. This offers a compelling new perspective on describing the ground-state wave function of the fermionic Tonks-Girardeau gas, enabling a complete description in second quantization. The proposed wave function is constructed based on the occupation numbers and natural orbitals of the one-body density matrix and is valid under any external potential. Specifically, we provide the exact coefficients that allow one to describe the fermionic Tonks-Girardeau gas as a number-conserving BCS state. Furthermore, the proposed wave function's form in second quantization allows us to derive the necessary expectation values to detect pairing in the fermionic Tonks-Girardeau gas. Through this approach, we demonstrate and show how to detect that the fermionic Tonks-Girardeau gas not only exhibits nontrivial quantum correlations but also is a paired state.
Topological pairing of composite fermions has led to remarkable ideas, such as excitations obeying non-Abelian braid statistics and topological quantum computation. We construct a ppp-wave paired Bardeen-Cooper-Schrieffer (BCS) wave function for composite fermions in the torus geometry, which is a convenient geometry for formulating momentum space pairing as well as for revealing the underlying composite-fermion Fermi sea. Following the standard BCS approach, we minimize the Coulomb interaction energy at half filling in the lowest and the second Landau levels, which correspond to filling factors $\ensuremath{\nu}=1/2$ and $\ensuremath{\nu}=5/2$ in GaAs quantum wells, by optimizing two variational parameters that are analogous to the gap and the Debye cutoff energy of the BCS theory. Our results show no evidence for pairing at $\ensuremath{\nu}=1/2$ but a clear evidence for pairing at $\ensuremath{\nu}=5/2$. To a good approximation, the highest overlap between the exact Coulomb ground state at $\ensuremath{\nu}=5/2$ and the BCS state is obtained for parameters that minimize the energy of the latter, thereby providing support for the physics of composite-fermion pairing as the mechanism for the 5/25/25/2 fractional quantum Hall effect. We discuss the issue of modular covariance of the composite-fermion BCS wave function, and calculate its Hall viscosity and pair correlation function. By similar methods, we look for but do not find an instability to sss-wave pairing for a spin-singlet composite-fermion Fermi sea at half-filled lowest Landau level in a system where the Zeeman splitting has been set to zero.
Recent work by Wu et al. [arXiv:1910.11011] proposed a numerical method, so-called matrix product operator--matrix product state (MPO-MPS) method, by which several types of quantum many-body wave functions, in particular, the projected Fermi sea state, can be efficiently represented as a tensor network. In this paper, we generalize the MPO-MPS method to study Gutzwiller projected paired states of fermions, where the maximally localized Wannier orbitals for Bogoliubov quasiparticles/quasiholes have been adapted to improve the computational performance. The study of SO(3)SO(3)SO(3)-symmetric spin-1 chains reveals that this new method has better performance than variational Monte Carlo for gapped states and similar performance for gapless states. Moreover, we demonstrate that dynamic correlation functions can be easily evaluated by this method cooperating with other MPS-based accurate approaches, such as the Chebyshev MPS method.