Information measures for a local quantum phase transition: Lattice fermions in a one-dimensional harmonic trap

We use quantum information measures to study the local quantum phase transition that occurs for trapped spinless fermions in one-dimensional lattices. We focus on the case of a harmonic confinement. The transition occurs upon increasing the characteristic density and results in the formation of a band-insulating domain in the center of the trap. We show that the ground-state bipartite entanglement entropy can be used as an order parameter to characterize this local quantum phase transition. We also study excited eigenstates by calculating the average von Neumann and second Renyi eigenstate entanglement entropies, and compare the results with the thermodynamic entropy and the mutual information of thermal states at the same energy density. While at low temperatures we observe a linear increase of the thermodynamic entropy with temperature at all characteristic densities, the average eigenstate entanglement entropies exhibit a strikingly different behavior as functions of temperature below and above the transition. They are linear in temperature below the transition but exhibit activated behavior above it. Hence, at nonvanishing energy densities above the ground state, the average eigenstate entanglement entropies carry fingerprints of the local quantum phase transition.

Figure 1

Properties of one-dimensional lattice fermions in a harmonic trap. (a) Second derivative of the total energy vs $\rho$, ${\overline{E}}^{″}\left(\rho \right)=R\mathrm{\Delta }{\varepsilon }_N$, which is proportional to the level spacing $\mathrm{\Delta }{\varepsilon }_N={\varepsilon }_{N+1}-{\varepsilon }_N$ of the single-particle energy spectrum of $\widehat{H}$ in Eq. (1). The dashed line denotes the critical characteristic density ${\rho }_c=8/\pi$ [see Eq. (10)]. We choose the parity-breaking parameter $\eta =1/N_c$ throughout the paper [see text after Eq. (1)]. (b) Site occupations at (from top to bottom) $\rho =3$, $\rho ={\rho }_c=8/\pi$, and $\rho =2$. Solid lines are exact numerical results $n_x$ (with integer $x$) and the overlapping dashed lines are the LDA results $n\left(x\right)$ from Eq. (9). (c) Single-particle density of states $D\left({\mu }_0\right)/R$, where ${\mu }_0$ is the single-particle energy (the Fermi energy of the many-body ground state). The solid and dashed lines are the LDA results from Eq. (11) and the corresponding leading term from Eq. (12), respectively. The symbols are the numerical exact diagonalization results with $(\eta =1/N_c)$ and without $(\eta =0)$ parity breaking. The results in panels (b) and (c) were obtained for $R=500\pi$.

Figure 2

Ground-state entanglement entropy. (a) von Neumann entanglement entropy $S_{\mathrm{vN}}$ vs $\rho$ about the critical point (main panel), and in a wider range (inset, results for $R=500\pi$). The vertical dashed line in the main panel denotes the critical point ${\rho }_c=8/\pi$. The dashed line in the inset is $S_{\mathrm{vN}}=(1/6)[lnN+ln8+1.485]$ from Ref. [37]. (b) Rescaling of the $x$ axis, $\rho \to (\rho -{\rho }_c)R$, which results in data collapse about the critical point. We plot $S_{\mathrm{vN}}$ (main panel) and $S_2$ (inset) for different values of $R$, as indicated in the legend.

Figure 3

Numerical results for (a) the thermodynamic entropy density, $S_{\mathrm{GE}}/R$, and (b) the mutual information $I$ as a function of $\rho$ and $T$ for trapped systems with $R=125\pi$.

Figure 4

Thermodynamic entropy density $S_{\mathrm{GE}}/\mathcal{N}$. (a) Entropy density of harmonically trapped fermions in a one-dimensional lattice, Eq. (1), with $\mathcal{N}=4R$, as a function of the characteristic density $\rho =\langle N\rangle /R$ for $R=250\pi$. (b) Entropy density of fermions in a translationally invariant square lattice, Eq. (14), with $\mathcal{N}=L^2$, as a function of the site occupation $r=\langle N\rangle /L$ for $L=2000$.

Figure 5

(main panel) Average von Neumann entanglement entropy ${\overline{S}}_{\mathrm{vN}}/R$ (solid lines) and second Renyi entanglement entropy ${\overline{S}}_2/R$ (dashed lines) in eigenstates of the Hamiltonian plotted as functions of $\rho$ for two temperatures. The inset shows the mutual information $I$ vs $\rho$ for five temperatures. All results were obtained for $R=7.5\pi$.

Figure 6

Average eigenstate entanglement entropy density $s_{\mathrm{n}}$, as obtained from fits to Eq. (15), and the thermodynamic entropy density $s_{\mathrm{GE}}={lim}_{R\to \infty }{S}_{\mathrm{GE}}/\left(2R\right)$. The entropy densities are plotted as functions of temperature for (a) $\rho =2$ and (b) $\rho =3$. (c) Data collapse of the von Neumann entanglement entropy density for different values of $\rho \gtrsim {\rho }_c$. The symbols depict the numerical results. The dashed lines in panel (a) are linear fits according to Eq. (16), while the solid lines in panels (a) and (b) are guides to the eye. The dashed line in panel (c) is the function $1.295(\mathrm{\Delta }/T)+1.288$.

Figure 7

Site occupations as functions of $x/R$ for different values of $\rho$. Solid lines are exact numerical results $n_x=\langle {\widehat{n}}_x\rangle$ (with integer $x$ and $R=500\pi$), and the overlapping dashed lines are the LDA results $n\left(x\right)$ from Eq. (8). The dashed-dotted lines are the results for the continuum (limit $\rho \to 0$ in a lattice), given by Eq. (A2).

Figure 8

Accuracy of the truncation used to evaluate the average von Neumann eigenstate entanglement entropy in Eq. (4), for $\rho =2$ and $T=0.1$. (a) Difference between the exact average ${\overline{S}}_{\mathrm{exact}}$ and the result after the truncation $\overline{S}\left(\mathrm{\Lambda }\right)$, for $R=5$ and $L=24$. (b) $\overline{S}\left(40\right)-\overline{S}\left(\mathrm{\Lambda }\right)$ for $R=20$ and $L=150$.

Figure 9

Comparison between the canonical and grand canonical averages of the von Neumann eigenstate entanglement entropy ${\overline{S}}_{\mathrm{vN}}$. The former is calculated by using Eq. (4), while the latter one is calculated by using Eq. (C1). (a) Results for $R=7.5\pi$ as a function of $\rho$, for two temperatures. (b) Absolute difference between ${\overline{S}}_{\mathrm{vN}}^{\left(\mathrm{CE}\right)}$ and ${\overline{S}}_{\mathrm{vN}}^{\left(\mathrm{GE}\right)}$, at $\rho =2$ and $T=0.1$, as a function of $R$.

Figure 10

Volume-law scaling of the average eigenstate entanglement entropies. (a) $\rho =2$ and (b) $\rho =3$. Symbols display the von Neumann entropy ${\overline{S}}_{\mathrm{vN}}$ and the second Renyi entropy ${\overline{S}}_2$ for different temperatures. Dashed lines are linear fits using the ansatz in Eq. (15).