Quantitative convergence and error control of particle-number-projected BCS wavefunctions versus exact many-body solutions for pairing correlations

The Bardeen-Cooper-Schrieffer (BCS) mean field describes superconductivity and nuclear pairing through a state that breaks the global U(1) gauge symmetry (particle-number conservation) of the underlying Hamiltonian. Restoring the symmetry by particle-number projection yields the number-conserving projected-BCS (PBCS) variational class. The open question, raised by a 2014 two-level-model study of symmetry restoration and quantum correlations and by a 2023 quantum-ring symmetry-restoration benchmark, is whether this class can quantitatively reproduce the pairing correlations of the exact many-body solution: correlation-observable-level agreement (correlation energy, pairing gap, occupation numbers, pair-transfer amplitudes, and mode-pair entanglement and discord) across system size and level degeneracy, and the still-unperformed benchmark of U(1) breaking and restoration against exact or error-controlled references in spatially continuous model systems. The existing literature provides numerical benchmarks only: PBCS is accurate in small-to-medium discrete pairing spaces but demonstrably fails in large weakly-paired spaces, while projected configuration-interaction, projected coupled-cluster, and quartet-condensate generalizations approach exact accuracy on Richardson-solvable Hamiltonians; the quantitative reconciliation of number-conserving symmetry-restored wavefunctions with the exact pairing solution — a systematic substantiation or refutation of the observed degeneracy trend at the correlation-observable level, together with the deferred continuous-space U(1) benchmark — remains open.

Background

In the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity, and in its nuclear-physics counterpart describing pairing between nucleons, the ground state of an attractively interacting fermion system is approximated by a product over single-particle levels $k$ of independently occupied, time-reversed pairs $(k,\bar k)$ — a condensate of Cooper pairs. This quasiparticle vacuum is not an eigenstate of the particle-number operator: it breaks the global U(1) gauge symmetry (invariance of the Hamiltonian under a common phase rotation of all fermion creation and annihilation operators) that expresses charge or nucleon-number conservation. In bulk superconductors the resulting particle-number fluctuation is negligible, but in finite systems — atomic nuclei, ultrasmall superconducting grains with fully discrete level spectra — it is spurious, and observables such as pair-transfer amplitudes between even-even systems are only defined for states of sharp particle number.

The standard remedy is symmetry restoration by projection: applying the number-projection operator to the BCS state selects its fixed-$N$ component, giving the number-conserving projected-BCS (PBCS) state. The variational parameters may be kept from the unprojected solution (projection after variation) or re-optimized on the projected state (variation after projection). The projected state retains a product-of-collective-pairs structure while conserving $N$, and is the archetype of the symmetry-restored variational wavefunctions used throughout nuclear energy-density-functional (EDF) theory — where Hartree-Fock-Bogoliubov (HFB, the generalized pairing mean field) states are routinely projected — and of modern number-conserving variational ansätze. Beyond plain PBCS, the projected class has been generalized along several directions: projected configuration interaction, which adds selected two- and four-quasiparticle excitations on top of the projected state; symmetry-projected coupled-cluster theory, which combines number projection with coupled-cluster theory, a standard correlated-wavefunction parametrization built as an exponential of pair-excitation operators; and quartet-condensate states, which build the variational state from collective four-particle (two-pair) quartets rather than Cooper pairs.

The accuracy question is sharp because exact references exist. The reduced (constant-strength attractive) pairing Hamiltonian acting on doubly degenerate time-reversed levels is exactly solvable by Richardson's method — an exact solution expressing the eigenstates through a set of self-consistently determined collective-pair amplitudes — and spaces of moderate size admit exact diagonalization. Natural tests of a pairing wavefunction are the correlation energy (the gap between the exact ground-state energy and the mean-field energy), the effective pairing gap (the energy cost of adding or removing a pair, which signals superconducting order), level occupation numbers, pair-transfer amplitudes, and two-mode quantum correlations.

Among correlation measures borrowed from quantum information, concurrence is an entanglement monotone for a pair of qubits, here applied to the two single-particle modes $k,\bar k$ treated as one qubit, and quantum discord is the difference between the total correlation and the maximal classical correlation of that pair of modes. The plain BCS state, being a product over pairs, erases both. Zeng, Wu and Xing (2014) found that PBCS restores them and, in a two-level pairing model, that the differences between PBCS and exact diagonalization diminish as the level degeneracy $\Omega$ grows — a purely numerical observation, established in the two-level (and analytically solvable one-level) model only, with no convergence rate and no error bound.

The benchmark context of the second source is the systematic validation of symmetry-restoration methodology on solvable or controllable systems. Cesca and Simenel (2023) benchmarked rotational-symmetry breaking followed by angular-momentum projection against variational Monte Carlo (a stochastic, systematically improvable variational method) for identical fermions confined to a one-dimensional quantum ring, and, in their conclusions, explicitly deferred the U(1) analogue — pairing correlations obtained by breaking and restoring gauge symmetry with particle-number projection — to future work. Whether the number-projected BCS variational class reproduces the exact pairing correlations quantitatively is thus the open question that would reconcile the exact, number-conserving treatment with the symmetry-broken mean field that manifestly exhibits the superconducting state.

Problem Statement

Determine, quantitatively, whether the number-conserving symmetry-restored variational wavefunction — the particle-number-projected BCS (PBCS) state, with projection after variation or variation after projection — reproduces the pairing correlations of the exact many-body solution, and characterize when it does. For the purposes of (i), the comparison is required for the full set of observables enumerated there, each over the stated ranges; and since projection after variation and variation after projection are the two members of the projected-BCS class, any conclusion stated for the class must be supported for both variants, or the variant to which it applies must be stated explicitly. Concretely: (i) for exactly solvable reduced pairing Hamiltonians, quantify the discrepancy between the projected-BCS class and the exact solution for the correlation energy (relative to the unprojected mean-field energy), the effective pairing gap, the occupation numbers, the pair-transfer amplitudes, and the mode-pair quantum correlations (concurrence and quantum discord between time-reversed mode pairs), across the normal-to-superfluid coupling range and as a function of the parameters of two distinct models whose level structures must not be conflated: (a) the reduced pairing Hamiltonian with constant pairing strength on $L$ doubly degenerate time-reversed levels, each level holding one time-reversed pair (so the per-level degeneracy is fixed at two), with Richardson's exact solution and exact diagonalization as references, where system size is varied through the particle number $N$ and the number of levels $L$; and (b) the two-level (plus analytically solvable one-level) degenerate pairing model of Zeng, Wu and Xing (2014), in which the two levels carry the equal degeneracy $\Omega$ (each level holding $\Omega$ time-reversed pairs, so that the per-level degeneracy $\Omega$ is a parameter distinct from the number of levels), where the varied axis is that per-level degeneracy $\Omega$ — in particular substantiating or refuting, in model (b) by varying $\Omega$, the numerical observation of Zeng, Wu and Xing (2014) that the differences between PBCS and exact diagonalization diminish as the level degeneracy grows and tend to vanish; and (ii) carry out the deferred U(1) gauge-symmetry benchmark: break particle-number symmetry at the BCS/HFB mean-field level in a spatially continuous one-dimensional quantum-ring system of identical fermions with attractive pairing interactions, and quantify how much of the correlation energy — defined as in (i), relative to the unprojected mean-field energy — and of the pairing observables is recovered by particle-number projection, relative either to an exact reference or to a numerically controllable reference whose own error is quantified and separated from the error of the projection. The common objective of these two linked deliverables is a quantitative characterization of the accuracy of the number-projected BCS class — for the discrete models of (i), an explicit substantiation or refutation of the degeneracy trend at the level of the correlation observables, and, for (ii), a projection error separated from a quantified reference error — complete enough that a domain expert can decide, for each model class and coupling regime treated, whether the symmetry-restored number-conserving wavefunction reproduces the exact pairing correlations — thereby reconciling the exact number-conserving treatment with the symmetry-broken mean field that manifestly exhibits the superconducting state.

References

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