Level statistics of the one-dimensional ionic Hubbard model

In this paper we analyze the spectral level statistics of the one-dimensional ionic Hubbard model, the Hubbard model with an alternating on-site potential. In particular, we focus on the statistics of the gap ratios between consecutive energy levels. This quantity is often used in order to signal whether a many-body system is integrable or chaotic. A chaotic system has typically the statistics of a Gaussian ensemble of random matrices while the spectral properties of the integrable system follow a Poisson statistics. We find that whereas the Hubbard model without alternating potential is known to be integrable and its spectral properties follow a Poissonian statistics, the presence of an alternating potential causes a drastic change in the spectral properties, which resemble the one of a Gaussian ensemble of random matrices. However, to uncover this behavior one has to separately consider the blocks of all symmetries of the ionic Hubbard model.

Figure 1

Distribution of the ratios of consecutive level spacings for (a) the standard Hubbard model ($\eta /J=0$) and (b) the ionic Hubbard model at $\eta /J=1$ in a chain with $N=6, L=12$ at $U/J=2$ combining all $s$ and $k,p$ symmetry sectors for $m_z=0$. We use 30 bins to represent the distribution.

Figure 2

Evolution of the mean value $\langle r\rangle$ of the ratios of consecutive level spacings with the strength of the alternating potential $\eta /J$ for (a) $U/J=2$ and (b) $U/J=4$. The expected mean value for the Poissonian distribution and the GOE are marked with the horizontal-dashed lines. For system size $L=12$ combining all $s$ and $k,p$ symmetry sectors, for $L=16$ combining $s=3,4$ and all $k,p$ for $m_z=0$.

Figure 3

Distribution of the ratios of consecutive level spacings for $N=6, L=12$ for the symmetry sector $m_z=0$ combining $s=0,1$ and all $k,p$ symmetry sectors. The parameters of the system are $U/J=2$ and $\eta /J=0.001,0.01,0.05,0.1$, and 100 bins. The mean values are given by $\langle r\rangle =0.391,0.405,0.450,0.482$.

Figure 4

Distribution of the ratios of consecutive level spacings for $N=6, L=12, N=8, L=16$. The parameters of the system are $U/J=1$ and $\eta /J=0.1$, 30 bins. The mean value is given by $\langle r\rangle =0.474,0.502$. For system size $N=6$ combining all $s$ and $k,p$ symmetry sectors, for $N=8$ combining $s=3,4$ and all $k,p$ for $m_z=0$.

Figure 5

Evolution of the mean value $\langle r\rangle$ of the ratios of consecutive level spacings with the interaction strength $U/J$ for (a) $\eta /J=1$ and (b) $\eta /J=2$. The value of the Poisson distribution and the GOE are marked by horizontal-dashed lines. For system size $N=6$ combining all $s$ and $k,p$ symmetry sectors for $m_z=0$ and for $N=8$ combining $s=3,4$ and all $k,p$ for $m_z=0$. The statistical standard error of the weighted mean of $r$ over all computed symmetry sectors was found to be of order $3\times {10}^{-4}$ or lower for all shown system sizes. This is below the symbol sizes.

Figure 6

Evolution of the mean value $\langle r\rangle$ of the ratios of consecutive level spacings (a) with the alternating potential $\eta /J$ at $U/J=2$, and (b) with the interaction strength $U/J$ at $\eta /J=1$. The value for the Poissonian distribution and the GOE are marked by horizontal-dashed lines. For system size $N=8$ and $L=12$ combining $s=2,3,4$ and all $k,p$ for $m_z=0$.

Figure 7

Distribution of the ratios of consecutive level spacings for $N=10, L=10$ for the symmetry sector $m_z=0$ combining all $s$ and $k,p$ symmetry sectors. The parameters of the system are $U/J=2$ and $\eta /J=1$. The mean value is given by $\langle r\rangle =0.426$.

Figure 8

Evolution of the mean value $\langle r\rangle$ of the ratios of consecutive level spacings with (a) the alternating potential at $U/J=1$ and (b) the interaction strength $U/J$ at $\eta /J=1$. The value for the Poissonian distribution and the GOE for different $m$ are marked by horizontal-dashed lines (see Table 1).

Figure 9

Distribution of the ratios of consecutive level spacings for $N=10, L=10$ for the symmetry sector $m_z=0$ combining all $s$ and $k,p$ symmetry sectors. The parameters of the system are $U_0/J=1 U_1/J=2$ and $\eta /J=1$, and 30 bins. The mean value is given by $\langle r\rangle =0.527$.

Figure 10

Evolution of the mean value $\langle r\rangle$ of the ratios of consecutive level spacings with the difference of the interaction strengths $(U_1-U_0)/J$ at $\eta /J=1, U_0/J=1$. The horizontal-dashed lines mark the expected values for the Poisson distribution and for the GOE with different $m$ (see Table 1). System size $N=10, L=10$ considered symmetry sector $m_z=0$ combining all $s$ and $k,p$ symmetry sectors.

Figure 11

Distribution of the ratios of consecutive level spacings for $N=10, L=10$ for the symmetry sector $m_z=0$, but not separating for the $s$ quantum number and combining data from all $k,p$ sectors. The parameters of the system are $U/J=2$ and $\eta /J=1$. The mean value is given by $\langle r\rangle =0.392$.