Many-body entanglement and spectral clusters in the extended hard-core bosonic Hatano-Nelson model

We study many-body entanglements and spectra of the extended bosonic Hatano-Nelson model in the hard-core limit. We show that the system undergoes a phase transition from a gapless phase to a charge-density-wave phase accompanied by a parity-time transition in the first-excited state. The phase transition is characterized by the crossing of the ground-state biorthogonal-order parameter and the sudden change of the first-excited-state entanglement entropy. The gapless phase is verified by the logarithmic scaling of the ground-state entanglement entropy with the central charge $c=1$. Furthermore, we show that all energy spectral clusters would form ellipses in strong nearest-neighbor interactions, for which we establish a universal scaling law. The lengths of the major and minor axes are shown to obey power laws with respect to the nearest-neighbor interaction. The exact expressions are derived for the numbers of energy levels on the outermost elliptic ring of each clusters.

Figure 1

Many-body phase diagram of the half-filled hard-core BHN model with respect to $\gamma$ and $V$ at $t=1$ up to $L=24$ lattice sites in PBCs. (a) The phase diagram obtained from the change of the first-excited energy as shown in panels (c) and (d), in which the red solid line denotes critical points in the thermodynamic limit derived by extrapolations. (b) The biorthogonal correlation function $C_{1,L/2}$ at $\gamma =0.6$, where the crossing point denotes the critical point. (c) The four lowest real parts of eigenenergies at $\gamma =0.6$. (d) The imaginary parts of the four eigenenergies shown in panel (c).

Figure 2

Half-chain entanglement entropy of the BHN model at $t=1$ and $\gamma =0.6$ in PBCs. (a)–(c) The biorthogonal entanglement entropy in the half filling, the self-norm entanglement entropy in the half filling, and the biorthogonal entanglement entropy of the ground states with a quarter of the filling as the function of $V$. (d)–(f) The scaling of the biorthogonal entanglement entropy from panel (a), the self-norm entanglement entropy from panel (b) at $V=2$ and $V=4$, and the biorthogonal entanglement entropy from panel (c) at $V=2$ and $V=7$.

Figure 3

Entanglement entropy of the first-excited states at $L=24$ at $t=1$ and $\gamma =0.6$ in PBCs. (a) The biorthogonal entanglement entropy with respect to $V$. (b) The self-norm entanglement entropy as the function of $V$.

Figure 4

The spectral clusters of the BHN model with $L=10$ and $N=3$ at $t=1$ and $\gamma =0.2$. (a)–(c) Spectral clusters for $n_s=1$, 2, and 3 and $V=20$. (d)–(f) Spectral clusters for $n_s=1$, 2, and 3 and $V=200$.

Figure 5

The energy shift in the half-filled BHN model with $L=12$ at $t=1$ in PBCs. (a) and (b) The energy shift $\varepsilon$ for the cluster $n_s=N$ and $n_s=1$ with respect to $\gamma$ at $V=40$, respectively. (c) and (d) The coefficients $C$ in $\varepsilon$ as the function of $n_s$ for $V=40$ and $V=200$ at $\gamma =0.2$, respectively.

Figure 6

The scaling parameters of the major and minor axes at $t=1$ and $\gamma =0.2$. (a) The scaling of major and minor axes of the ellipse for the cluster $n_s=3$ in the BHN model with $L=10$ and $N=3$ particles. (b) and (c) The exponent $p_a$ and coefficients $C_a$ and $C_b$ with respect to $N$ for cluster $n_s=N$ up to $L=14$. (d) The exponent $p_a$ as the function $L$ for cluster $n_s=1$. The star symbols in panels (a)–(c) denote the data in the half filling $N=L/2$.