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The bulk scaling limit of eigenvalue distribution on the complex plane${\mathbb{C}}$ of the complex Ginibre random matrices provides a determinantalpoint process (DPP). This point process is a typical example of disorderedhyperuniform system characterized by an anomalous suppression of large-scaledensity fluctuations. As extensions of the Ginibre DPP, we consider a family ofDPPs defined on the $D$-dimensional complex spaces ${\mathbb{C}}$, $D \in{\mathbb{N}}$, in which the Ginibre DPP is realized when $D=1$. Thisone-parameter family ($D \in {\mathbb{N}}$) of DPPs is called the Heisenbergfamily, since the correlation kernels are identified with the Szeg\H{o} kernelsfor the reduced Heisenberg group. For each $D$, using the modified Besselfunctions, an exact and useful expression is shown for the local numbervariance of points included in a ball with radius $R$ in ${\mathbb{R}}^{2D}\simeq {\mathbb{C}}^D$. We prove that any DPP in the Heisenberg family is inthe hyperuniform state of Class I, in the sense that the number variancebehaves as $R^{2D-1}$ as $R \to \infty$. Our exact results provide asymptoticexpansions of the number variances in large $R$.

Local number variances and hyperuniformity of the Heisenberg family of determinantal point processes