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arXiv:2111.14133 · Physical Review C

Pfaffian formulation for matrix elements of three-body operators in multiple quasiparticle configurations

Zi-Rui Chen, Long-Jun Wang

Published Mar 30, 2022• 11 citations
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Abstract

We present a Pfaffian formula to calculate matrix elements of three-body operators in symmetry-restoration beyond-mean-field methods, including the case of multiple quasiparticle (qp) configurations. Detailed derivation is provided and the validity of the new formula is checked numerically. The corresponding efficiency is discussed, which turns out to be about five times (one order of magnitude) higher than the conventional method when four (less than four) qp operators are considered.

DOI
10.1103/physrevc.105.034342

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We present a Pfaffian formula to calculate matrix elements of three-body operators in symmetry-restoration beyond-mean-field methods, including the case of multiple quasiparticle (qp) configurations. Detailed derivation is provided and the validity of the new formula is checked numerically. The corresponding efficiency is discussed, which turns out to be about five times (one order of magnitude) higher than the conventional method when four (less than four) qp operators are considered.
In a calculation of rotated matrix elements with angular momentum projection, the generalized Wick's theorem may encounter a practical problem of combinatorial complexity when the configurations have more than four quasiparticles (qp's). The problem can be solved by employing the Pfaffian algorithm generally applicable to calculations of matrix elements for Hartree-Fock-Bogoliubov states with any number of qp's. This breakthrough in many-body techniques enables studies of high-spin states in a shell-model framework. As the first application of the Pfaffian algorithm, the configuration space of the projected shell model is expanded to include 6-qp states for both positive and negative parities. Taking 166Hf^{166}\mathrm{Hf}166Hf as an example, we show that 6-qp states become the main configuration of the yrast band beyond spin $I\ensuremath{\approx}34\ensuremath{\hbar}$, which explains the observed third back-bending in moment of inertia. Structures of multi-qp high-KKK isomers in 176Hf^{176}\mathrm{Hf}176Hf are analyzed as another example.
We derive an expression that allows for the unambiguous evaluation of the overlap between two arbitrary quasiparticle vacua, including its sign. Our expression is based on the Pfaffian of a skew-symmetric matrix, extending the overlap formula recently proposed by Robledo [Phys. Rev. C 79, 021302(R) (2009)] to the most general case of quasiparticle vacua, including the one of the overlap between two different blocked nnn-quasiparticle states for either even or odd systems. The powerfulness of the method is illustrated for a few typical matrix elements that appear in realistic angular-momentum-restored generator coordinate method calculations when breaking time-reversal invariance and using the full model space of occupied single-particle states.
In this Letter, we present a new expression for the overlaps of wave functions in Hartree-Fock-Bogoliubov based theories. Starting from the Pfaffian formula by Bertsch et al. [1], an exact and computationally stable formula for overlaps is derived. We illustrate the convenience of this new formulation with a numerical application in the context of the particle-number projection method. This new formula allows for substantially increased precision and versatility in chemical, atomic, and nuclear physics applications, particularly for methods dealing with superfluidity, symmetry restoration, and uses of nonorthogonal many-body basis states.
Fermionic Gaussian operators are foundational tools in quantum many-body theory, numerical simulation of fermionic dynamics, and fermionic linear optics. While their structure is fully determined by two-point correlations, evaluating their matrix elements in arbitrary local spin bases remains a nontrivial task, especially in applications involving quantum measurements, tomography, and basis-rotated simulations. In this work, we derive a fully explicit and general Pfaffian formula for the matrix elements of fermionic Gaussian operators between arbitrary Pauli product states. Our approach introduces a pair of sign-encoding matrices whose classification leads to a Lie algebra isomorphic to so(2L)\mathfrak{so}(2L)so(2L). This algebraic structure not only guarantees consistency of the Pfaffian signs but also reveals deep connections to Clifford algebras. The resulting framework enables scalable computations across diverse fields -- from quantum tomography and entanglement dynamics to algebraic structure in fermionic circuits and matchgate computation. Beyond its practical utility, our construction sheds light on the internal symmetries of Gaussian operators and offers a new lens through which to explore their role in quantum information and computational models.