arXiv subjects
Publications and source records attributed to Svala Sverrisdóttir.
Any linear space of square matrices has an associated eigenvector variety. Its points are eigenvectors of matrices from that linear space. We present a systematic study of eigenvector varieties, with focus on Lie algebras and Hamiltonians of quantum systems.
We investigate determinantal varieties for symmetric matrices that have zero blocks along the main diagonal. In theoretical physics, these arise as Gram matrices for kinematic variables in quantum field theories. We study the ideals of relations among functions in the matrix entries that serve as building blocks for conformal correlators.
We develop and numerically analyze an algebraic-geometric framework for spin-adapted coupled-cluster (CC) theory. Since the electronic Hamiltonian is SU(2)-invariant, physically relevant quantum states lie in the spin singlet sector. We give an explicit description of the SU(2)-invariant (spin singlet) many-body space by identifying it with an Artinian commutative ring, called the excitation ring, whose dimension is governed by a Narayana number. We define spin-adapted truncation varieties via embeddings of graded subspaces of this ring, and we identify the CCS truncation variety with the Veronese square of the Grassmannian. Compared to the spin-generalized formulation, this approach yields a substantial reduction in dimension and degree, with direct computational consequences. In particular, the CC degree of the truncation variety -- governing the number of homotopy paths required to compute all CC solutions -- is reduced by orders of magnitude. We present scaling studies demonstrating asymptotic improvements and we exploit this reduction to compute the full solution landscape of spin-adapted CC equations for water and lithium hydride.
We describe an explicit basis for the $\operatorname{SU}(2)$-invariant space of the exterior power $\wedge_{2k} \mathbb{C}^{2m}$ via the combinatorics of plane partitions. In quantum chemistry, this is the space of spin adapted quantum states of an electronic system with $m$ spin orbitals and $k$ electron pairs. We construct our basis by identifying the invariant space with an Artinian commutative ring called the excitation ring. We compute a Gröbner basis and enumerate its standard monomials via an explicit bijection to Dyck paths counted by the Narayana numbers.
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.
We study algebraic varieties that encode the kinematic data for $n$ massless particles in $d$-dimensional spacetime subject to momentum conservation. Their coordinates are spinor brackets, which we derive from the Clifford algebra associated to the Lorentz group. This was proposed for $d=5$ in the recent physics literature. Our kinematic varieties are given by polynomial constraints on tensors with both symmetric and skew symmetric slices.
The exploration of the root structure of coupled cluster equations holds both foundational and practical significance for computational quantum chemistry. This study provides insight into the intricate root structures of these non-linear equations at both the CCD and CCSD level of theory. We utilize computational techniques from algebraic geometry, specifically the monodromy and parametric homotopy continuation methods, to calculate the full solution set. We compare the computed CC roots against various established theoretical upper bounds, shedding light on the accuracy and efficiency of these bounds. We hereby focus on the dissociation processes of four-electron systems such as (H$_2$)$_2$ in both D$_{2{\rm h}}$ and D$_{\infty {\rm h}}$ configurations, H$_4$ symmetrically distorted on a circle, and lithium hydride. We moreover investigate the ability of single-reference coupled cluster solutions to approximate excited state energies. We find that multiple CC roots describe energies of excited states with high accuracy. Our investigations reveal that for systems like lithium hydride, CC not only provides high-accuracy approximations to several excited state energies but also to the states themselves.
We determine the number of complex solutions to a nonlinear eigenvalue problem on the Grassmannian in its Plücker embedding. This is motivated by quantum chemistry, where it represents the truncation to single electrons in coupled cluster theory. We prove the formula for the Grassmannian of lines which was conjectured in earlier work with Fabian Faulstich. This rests on the geometry of the graph of a birational parametrization of the Grassmannian. We present a squarefree Gröbner basis for this graph, and we develop connections to toric degenerations from representation theory.
We develop algebraic geometry for coupled cluster (CC) theory of quantum many-body systems. The high-dimensional eigenvalue problems that encode the electronic Schrödinger equation are approximated by a hierarchy of polynomial systems at various levels of truncation. The exponential parametrization of the eigenstates gives rise to truncation varieties. These generalize Grassmannians in their Plücker embedding. We explain how to derive Hamiltonians, we offer a detailed study of truncation varieties and their CC degrees, and we present the state of the art in solving the CC equations.
The projective variety of Lie algebra structures on a 4-dimensional vector space has four irreducible components of dimension 11. We compute their prime ideals in the polynomial ring in 24 variables. By listing their degrees and Hilbert polynomials, we correct an earlier publication and we answer a 1987 question by Kirillov and Neretin.