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Algebraic Varieties in Second Quantization

Svala Sverrisdóttir

TL;DR

The paper develops an algebraic-geometric framework for second-quantized coupled cluster theory by embedding the Fock space into the exterior algebra and encoding fermionic operators via the Fermi-Dirac (Clifford) algebra. It constructs a non-commutative Gröbner basis for Wick's theorem, introduces the exponential parametrization with a polynomial-inverse map, and defines Fock-space truncation varieties that include Grassmannians, partial flags, and spinor varieties as natural CC truncations. It then analyzes the coupled cluster equations on these varieties, deriving CC-degree bounds and proving a key link between CC degree and the graph of the parametric map, with concrete classifications for linear cases and examples via EOM-CC. Finally, it validates the theory numerically, computing CC degrees for small instances and showing feasibility of algebraic methods for ionized systems. The work provides a unifying, computationally tractable algebraic framework for fixed-N and FSCC/EOM-CC, offering new geometric insight into solution multiplicities and potential toric-degenerations for degree computations.

Abstract

We develop an algebraic geometric framework for Fock space coupled cluster theory in second quantization. In quantum chemistry, many-electron states are represented as elements of the exterior algebra. The fermionic creation and annihilation operators generate the Fermi-Dirac algebra, which can be realized as a Clifford algebra acting on the exterior algebra. We present a non-commutative Gröbner basis for the Fermi-Dirac algebra; offering an alternative proof of Wick's theorem, a fundamental result in quantum field theory. In coupled cluster theory, eigenpairs of the Schrödinger equation are approximated by a hierarchy of polynomial equations corresponding to different levels of truncation. The coupled cluster exponential parameterization of quantum states gives rise to Fock space truncation varieties. This reveals well-known varieties, such as the Grassmannian, flag varieties and spinor varieties. We offer a detailed study of the truncation varieties, providing an explicit description of their defining equations and dimension. Furthermore, we classify all cases in which their coupled cluster degree coincides with the degree of the graph of the exponential parameterization - most notably for singleton truncations such as CCD and for the Schubert like truncation varieties such as the Grassmannian, flag variety and the spinor variety.

Algebraic Varieties in Second Quantization

TL;DR

The paper develops an algebraic-geometric framework for second-quantized coupled cluster theory by embedding the Fock space into the exterior algebra and encoding fermionic operators via the Fermi-Dirac (Clifford) algebra. It constructs a non-commutative Gröbner basis for Wick's theorem, introduces the exponential parametrization with a polynomial-inverse map, and defines Fock-space truncation varieties that include Grassmannians, partial flags, and spinor varieties as natural CC truncations. It then analyzes the coupled cluster equations on these varieties, deriving CC-degree bounds and proving a key link between CC degree and the graph of the parametric map, with concrete classifications for linear cases and examples via EOM-CC. Finally, it validates the theory numerically, computing CC degrees for small instances and showing feasibility of algebraic methods for ionized systems. The work provides a unifying, computationally tractable algebraic framework for fixed-N and FSCC/EOM-CC, offering new geometric insight into solution multiplicities and potential toric-degenerations for degree computations.

Abstract

We develop an algebraic geometric framework for Fock space coupled cluster theory in second quantization. In quantum chemistry, many-electron states are represented as elements of the exterior algebra. The fermionic creation and annihilation operators generate the Fermi-Dirac algebra, which can be realized as a Clifford algebra acting on the exterior algebra. We present a non-commutative Gröbner basis for the Fermi-Dirac algebra; offering an alternative proof of Wick's theorem, a fundamental result in quantum field theory. In coupled cluster theory, eigenpairs of the Schrödinger equation are approximated by a hierarchy of polynomial equations corresponding to different levels of truncation. The coupled cluster exponential parameterization of quantum states gives rise to Fock space truncation varieties. This reveals well-known varieties, such as the Grassmannian, flag varieties and spinor varieties. We offer a detailed study of the truncation varieties, providing an explicit description of their defining equations and dimension. Furthermore, we classify all cases in which their coupled cluster degree coincides with the degree of the graph of the exponential parameterization - most notably for singleton truncations such as CCD and for the Schubert like truncation varieties such as the Grassmannian, flag variety and the spinor variety.

Paper Structure

This paper contains 7 sections, 13 theorems, 97 equations, 2 tables.

Table of Contents

  1. Introduction
  2. The Fermi-Dirac algebra
  3. Exponential parameterization
  4. Fock space truncation varieties
  5. Flag varieties and spinor varieties
  6. The coupled cluster equations
  7. Numerical solutions

Key Result

Proposition 2.2

The creation and annihilation operators span the Fermi-Dirac algebra, which is a special Clifford algebra on $\mathbb{C}\langle a_1, \dots, a_n, a_1^\dagger, \dots a_n^\dagger \rangle$, associated with the quadratic form

Theorems & Definitions (42)

  • Remark 2.1: Jordan-Wigner transformation
  • Proposition 2.2
  • Proof 1
  • Corollary 2.3
  • Proof 2
  • Remark 2.4: Exterior algebras in the Fermi-Dirac algebra
  • Theorem 2.5: Wick's theorem wick1950evaluation
  • Proof 3
  • Proposition 2.6
  • Example 2.7: $a_ia_p^\dagger a_j a_q^\dagger$
  • ...and 32 more