Generalized Pauli constraints in large systems: the Pauli principle dominates

Abstract

Lately, there has been a renewed interest in fermionic 1-body reduced density matrices and their restrictions beyond the Pauli principle. These restrictions are usually quantified using the polytope of allowed, ordered eigenvalues of such matrices. Here, we prove this polytope's volume rapidly approaches the volume predicted by the Pauli principle as the dimension of the 1-body space grows, and that additional corrections, caused by generalized Pauli constraints, are of much lower order unless the number of fermions is small. Indeed, we argue the generalized constraints are most restrictive in (effective) few-fermion settings with low Hilbert space dimension.

Figures (4)

• Figure 1: A contour plot demonstrating the lower bound to $\mathop{\mathrm{Vol}}\nolimits^{d-1}(F_{d,N})/\mathop{\mathrm{Vol}}\nolimits^{d-1}(P_{d,N})$ obtained in Remark \ref{['betterestimate']}. The blue part corresponds to $N>d$, which is not allowed. The convergence happens as orange turns to yellow, and it occurs extremely rapidly if $N\geq80$. Numerical simulations inspired by [walteralgorithm0] (now see [walteralgorithm]) suggest convergence should actually happen more quickly in the region $8\leq N\leq 80$, so that the yellow region extends a long way towards the contour that forms a triangle in the orange region, but this cannot be demonstrated with our method. Similarly, we have no bound for $N,d-N\leq 8$, but numerics suggests rapid convergence in $d$ also for $4\leq N,d-N\leq 8$.
• Figure 3: Bosonic polytopes $B_{2,N}$ for $N=\lambda_1+\lambda_2=1,2,3$ (bold). The unordered polytopes $B^{\text{unord}}_{2,N}$ are $d!=2$ times as large and include the dashed continuations. The $B^{\text{unord}}_{2,N}$ have length $\sqrt{2}N$. One way to calculate this is to note that the lightly shaded region (drawn for $N=2$) has area $N^2/2$. This increases to $(N+\sqrt{2}\epsilon)^2/2$ if we move a distance $\epsilon$ in the normal direction $(1,1)/\sqrt{2}$. Hence the derivative in $\epsilon$ at $\epsilon=0$, which is the surface length, is $\sqrt{2}N$.
• Figure 4: The Pauli polytope $P_{2,1}$ (bold). The unordered polytope $P^{\text{unord}}_{2,1}$ is $d!=2$ times as large and includes the dashed continuation. The method to determine the volume still applies, and we still have $\lambda_1+\lambda_2=N+\sqrt{2}\epsilon$ upon moving distance $\epsilon$ in the normal direction. Naturally, the area of the shaded region is subject to the Pauli bound $\lambda_i\leq1$.
• Figure 5: This image illustrates \ref{['tobound2']} for $d=3$ and $N=2$. Starting with the image at the top, the large triangle represents $B_{3,2}$, containing $P_{3,2}$ as a smaller triangle. The area defined by $\lambda_{[2]}\leq t$ is indicated in grey in the first two images. We then proceed through the steps of \ref{['tobound2']} image by image. These drawings were kindly contributed by an anonymous referee.

Theorems & Definitions (17)

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• proof : Proof of Proposition \ref{['paulipolytope']}: Properties of $P_{d,N}$
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• proof : Proof of Proposition \ref{['AcontainedinF']}
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