Representability conditions on the one-and two-particle density matrix for fermion systems are formulated by means of Grassmann integrals.A positivity condition for a certain kind of Grassmann integral is established which by an appropriate choice of the integrand, in turn, induces the well-known G-, P-and Q-Conditions of quantum chemistry.Similarly, the T 1 -and T 2 -Conditions are derived.Furthermore, quasifree Grassmann states are introduced and, for every operator γ ∈ H ⊕ H with 0 ≤ γ ≤ 1, the existence of a unique quasifree Grassmann state whose one-particle density matrix is γ is shown.. More specifically, ifwhere
In the spirit of the generalized one-particle density matrix for fermions, we introduce generalized one- and two-particle density matrices to state representability conditions up to second order for boson systems without assuming particle number-conservation. Furthermore, we show for both particle species that, for a semibounded Hamiltonian, the in?mum of the variation of the energy functional w.r.t quasifree states coincides with the one of a variation over pure quasifree states. Moreover, it is proven for fermions that only pure quasifree states have a generalized 1-pdm that is a projection, and a similar statement for bosons.
We present a constructive solution to the N-representability problem: a full characterization of the conditions for constraining the two-electron reduced density matrix to represent an N-electron density matrix. Previously known conditions, while rigorous, were incomplete. Here, we derive a hierarchy of constraints built upon (i) the bipolar theorem and (ii) tensor decompositions of model Hamiltonians. Existing conditions D, Q, G, T1, and T2, known classical conditions, and new conditions appear naturally. Subsets of the conditions are amenable to polynomial-time computations of strongly correlated systems.
We formulate the density matrix of many-body systems as functional integrals in the holomorphic representation, similar to the S-matrix formulation in field theory. Symmetry and antisymmetry requirements are built into the integration variables: c-numbers for bosons and Grassmann numbers for fermions. The integrals have well-defined normalization. They yield first-order differential equations for semiclassical extremal configurations and corrections to it, and do not require periodicity conditions on the individual integration paths, unlike the formulations found in the literature. We present some applications to simple boson and fermion systems.
We recently presented a constructive solution to the
N-representability problem of the two-electron reduced density matrix (2-RDM)---a systematic approach to constructing complete conditions to ensure that the 2-RDM represents a realistic
N-electron quantum system [D. A. Mazziotti, Phys. Rev. Lett. (to be published)]. In this paper we provide additional details and derive further
N-representability conditions on the 2-RDM that follow from the constructive solution. The resulting conditions can be classified into a hierarchy of constraints, known as the
(2,q)-positivity conditions, where the
q indicates their derivation from the non-negativity of
q-body operators. In addition to the known T1 and T2 conditions, we derive another class of (2,3)-positivity conditions. We also derive 3 classes of (2,4)-positivity conditions, 6 classes of (2,5)-positivity conditions, and 24 classes of (2,6)-positivity conditions. The constraints obtained can be divided into two general types: (i) lifting conditions, that is, conditions which arise from lifting lower
(2,q)-positivity conditions to higher
(2,q+1)-positivity conditions, and (ii) pure conditions, that is, conditions which cannot be derived from a simple lifting of the lower conditions. All of the lifting conditions and the pure
(2,q)-positivity conditions for
q>3 require tensor decompositions of the coefficients in the model Hamiltonians. Subsets of the derived
N-representability conditions can be employed with the previously known conditions to achieve polynomially scaling calculations of ground-state energies and 2-RDMs of many-electron quantum systems even in the presence of strong electron correlation.
Abstract N‐representability conditions for a two‐particle density operator implied by positive‐semidefiniteness of the projection operator PN+1(ϕ1 Λ ΨN) are derived and discussed. The operator PN+1(ϕ1 Λ ΨN) projects onto an (N + 1)‐particle antisymmetric function ϕ1 Λ ΨN, the Grassmann product of a one‐particle factor ϕ1 and an N‐particle factor ΨN. The polar subcone 𝒫2N(g, q) to the set of N‐representable two‐particle density operators 𝒫2N which corresponds to these conditions is found. It is shown that its extreme rays belong to two orbits for the action of the unitary group of transformations in one‐particle Hilbert space. The facial structure of the convex set 𝒫2N exposed by elements of 𝒫2N(g, q) is analyzed. An example of the operator that changes the structure of its bottom eigenspace when the number of fermions N surpasses a certain value is noted. A new approach to the diagonal conditions for N‐representability is found. It consists of the decomposition of the N‐particle antisymmetric identity operator onto the mutually orthogonal projection operators.