Detecting dimensional crossover and finite Hilbert space through entanglement entropies

The information content of the two-particle one- and two-dimensional Calogero model is studied by using the von Neumann and Rényi entropies. The one-dimensional model is shown to have nonmonotonic entropies with finite values in the large-interaction-strength limit. On the other hand, the von Neumann entropy of the two-dimensional model with isotropic confinement is a monotone increasing function of the interaction strength which diverges logarithmically. By considering an anisotropic confinement in the two-dimensional case we show that the one-dimensional behavior is eventually reached when the anisotropy increases. The crossover from two to one dimensions is demonstrated by using the harmonic approximation and it is shown that the von Neumann divergence only occurs in the isotropic case. The Rényi entropies are used to highlight the structure of the model spectrum. In particular, it is shown that these entropies have a nonmonotonic and nonanalytical behavior in the neighborhood of the interaction strength parameter values where the Hilbert space and, consequently, the spectrum of the reduced density matrix are both finite.

Figure 1

Eigenvalues of the 1-RDM for the ground state of a one-dimensional Calogero model from a variational calculation, using 50 one-particle basis functions. Panel (a) shows the results for fermions and panel (b) shows those for bosons. The abrupt drop to zero of the eigenvalues at certain integer values (odd for fermions, even for bosons) indicate that the number of natural orbitals is finite. In panel (a), each eigenvalue is doubly degenerate.

Figure 2

(a) Bosons (red line) and fermions (blue dashed line) larger eigenvalues of the 1-RDM and (b) von Neumann entropy for the two-particle ground state of a one-dimensional Calogero model from a variational calculation using 50 one-particle basis functions. Note that the eigenvalues of fermions and bosons become degenerate in the large-interaction limit.

Figure 3

One-dimensional bosonic von Neumann entropy (black full line) and Rényi entropies as a function of the interaction parameter $\nu$, for $\alpha =0.1,0.2,0.3,0.4,0.5,0.6,2$ in red, dark green, blue, orange, magenta, light green, and cyan full line, respectively. The exact values of the entropies for $\nu =2n$ are depicted as gray points.

Figure 4

One-dimensional bosonic Rényi entropies (upper panels) and their derivatives (lower panels) as a function of the interaction parameter near $\nu =4$, for $\alpha =0.4,0.5,0.6$ from top to bottom.

Figure 5

(a) The von Neumann entropy as a function of the interaction strength $\nu$ for the rotationally invariant fermion wave function ${\psi }_S^{+}$ (green line) and for the fermion wave function ${\psi }_S^x=(x_1-x_2)$ (blue line). Note the logarithmic divergence for large interaction strength. Any point in the gray shaded area is a $(\nu ,S_{vN})$ pair, which can be obtained from a particular choice of the linear combination parameter $\beta$ defined in Eq. (23). The inset shows that the vNE for the rotationally-invariant-fermion case (green line) is asymptotically equal to the vNE for bosons (red line). (b) The fermion vNE of ${\psi }_{lc}\left(\beta \right)$ as a function of ${\beta }^2$ across the shaded region for two values of the interacting parameter, $\nu =2$ (orange-dashed line) and $\nu =(1+\sqrt{33})/2$ (black-dashed line). The horizontal lines correspond to the vNE of ${\psi }_S^{+},{\psi }_S^x$.

Figure 6

von Neumann (vNE) and linear entropy (LE) in the large-interaction limit computed by using finite sums of the exact eigenvalues (full lines) and the exact vNE (magenta dot-dashed line) as a function of the anisotropy parameter $\varepsilon$ [Eq. (45)]. The number of included eigenvalues [Eq. (42)] are 50 (black), 100 (red), 150 (green), and 200 (blue) respectively. The leading term of the exact vNE, Eq. (52), is depicted as a yellow dashed line (shifted to make it visible).

Figure 7

(a) The data corresponds to the ratio between the variational energy and the energy of the relative Hamiltonian in the large-interaction limit, $E_{00}^{\infty }$ [see Eq. (53)]. The variational energies were calculated for, from bottom to top, $\nu (\nu -1)=20,50,100,500,2000$. The black dashed line corresponds to the exact limit. (b) The numerical derivative of the variational ground-state energy times ${({\varepsilon }^2-1)}^{1/2}$ vs $\varepsilon$; see Eq. (55) for the precise definition of the function $\mathrm{\Delta }{E}_{00}^{\mathrm{var}}$. This function was chosen to show the derivative of the ground-state energy of the relative Hamiltonian in the limit $\nu \to \infty$ as the black dashed straight line with slope one-half. The other curves correspond to the data shown in panel (a) using the same color convention for the different values of $\nu (\nu -1)$.