An Effective Solution to Convex 1-Body N-Representability

Abstract

From a geometric point of view, Pauli’s exclusion principle defines a hypersimplex. This convex polytope describes the compatibility of 1-fermion and N-fermion density matrices; therefore, it coincides with the convex hull of the pure N-representable 1-fermion density matrices. Consequently, the description of ground state physics through 1-fermion density matrices may not necessitate the intricate pure state generalized Pauli constraints. In this article, we study the generalization of the 1-body N-representability problem to ensemble states with fixed spectrum $\varvec{w}$, in order to describe finite-temperature states and distinctive mixtures of excited states. By employing ideas from convex analysis and combinatorics, we present a comprehensive solution to the corresponding convex relaxation, thus circumventing the complexity of generalized Pauli constraints. In particular, we adapt and further develop tools such as symmetric polytopes, sweep polytopes, and Gale order. For both fermions and bosons, generalized exclusion principles are discovered, which we determine for any number of particles and dimension of the 1-particle Hilbert space. These exclusion principles are expressed as linear inequalities satisfying hierarchies determined by the nonzero entries of $\varvec{w}$. The two families of polytopes resulting from these inequalities are part of the new class of so-called lineup polytopes.

Appendices

Appendix A. H-Representation of the Lineup Polytope of the Hypersimplex $\textsf{H}(3,6)$

The H-representation of the 10-lineup polytope of the five-dimensional hypersimplex $\textsf{H}(3,6)$ has 72 inequalities represented below. The inequalities arise in a hierarchy while increasing r from 1 to 10. The weights are taken in the Pauli simplex $\mathsf {\Delta }_{9}$, so that $1\ge w_1 \ge w_2 \ge \cdots \ge w_{10} \ge 0$ and $\sum _{i=1}^{10}w_i=1$. Since $\sum _{i=1}^6x_i=3$ and $\sum _{i=1}^{10}w_i=1$, there are many ways to represent the matrices if one does not use the orthogonal complement of these hyperplanes. We chose the representation where the last coefficient of the rays is zero; as a consequence, all coefficients are nonnegative.

Appendix B. Generalized Exclusion Inequalities for Fermions

The Pauli exclusion principle is equivalent to the first of the following two inequalities which describe the case $r=1$:

$$\begin{aligned} x_1^{\downarrow } \le 1, \sum _{i=1}^{d-1} x_i^{\downarrow } \le N. \end{aligned}$$

The second equation is equivalent to $x_d^{\downarrow }\ge 0$, which implies that all coordinates should be indeed nonnegative, which is inherently true in the physical context. To illustrate larger values, it is more compact to express them in a matrix. Following Theorem 6, after solving the case (r, N, d), when increasing the value of r by 1, the minimal case $(r+1,N',d')$ to consider is such that $N'=N+1$ and $d'=d+2$. Below, we represent this minimal case by the coefficients located between the two vertical bars. The matrix gives the result for $r=8$, so in dimension $d=14$. The right-hand side term involving N is determined using Proposition 6.18.

Appendix C. Generalized Exclusion Inequalities for Bosons

The case $r=1$ has one inequality

$$\begin{aligned} \sum _{i=1}^{d-1} x_i^{\downarrow } \le N. \end{aligned}$$

It is equivalent to $x_d^{\downarrow }\ge 0$, which implies that all coordinates should be indeed nonnegative, which is inherently true in the physical context. To illustrate larger values, we again express the new inequalities in a matrix. Using Theorem 7, after solving the case (r, N, d), when increasing the value of r by 1, the minimal case $(r+1,N',d')$ to consider is such that $N'=N+1$ and $d'=d+1$. Below, we represent this minimal case by the coefficients located on the left of the vertical bar. The matrix gives the result for $r=8$, so in dimension $d=8$. The right-hand side term involving N is determined by adapting Proposition 6.18.