Universal power law in crossover from integrability to quantum chaos

We study models of interacting fermions in one dimension to investigate the crossover from integrability to nonintegrability, i.e., quantum chaos, as a function of system size. Using exact diagonalization of finite-sized systems, we study this crossover by obtaining the energy level statistics and Drude weight associated with transport. Our results reinforce the idea that for system size $L\to \infty$ nonintegrability sets in for an arbitrarily small integrability-breaking perturbation. The crossover value of the perturbation scales as a power law $\sim L^{-\eta }$ when the integrable system is gapless. The exponent $\eta \approx 3$ appears to be robust to microscopic details and the precise form of the perturbation. We conjecture that the exponent in the power law is characteristic of the random matrix ensemble describing the nonintegrable system. For systems with a gap, the crossover scaling appears to be faster than a power law.

Figure 1

(Top and bottom left) Level-spacing distribution $P\left(s\right)$ for the $t\text{−}{t}^{\prime }\text{−}V$ model with $V=1$ at half filling and $L=22$. The values of $t^{\prime }$ are 0.02,0.05,0.1. The dashed (solid) line is the level-spacing distribution for the Poissonian (GOE). (Bottom right) Peak position of $P\left(s\right)$ for the $t\text{−}{t}^{\prime }\text{−}V$ model with $V=2$ at half filling as a function of $t^{\prime }$ for $L=14,16,18$, and 20. The solid lines are the function $0.8tanh(t^{\prime }/t_{cr}^{\prime })$ used to obtain the crossover scale of $t^{\prime }$ as functions of $L$.

Figure 2

(Top and bottom left) Level-spacing distribution $P\left(s\right)$ for the Hubbard model with $U=1$ at half filling and $L=11$. The values of the integrability-breaking parameter $t^{\prime }$ are 0.0,0.05,0.15. As in Fig. 1, the dashed (solid) line is the Poisson (GOE) distribution. (Bottom right) Peak position of $P\left(s\right)$ for the Hubbard model with $U=2$ as a function of $t^{\prime }$ for $L=7,8$, and 9 to obtain the crossover scale of $t^{\prime }$ as a function of $L$. The solid lines are the function $0.8tanh(t^{\prime }/t_{cr}^{\prime })$.

Figure 3

(Left) The change in the variance of the level spacing distribution for the $t-t^{\prime }-V$ model (for $V=1)$ with $t^{\prime }$ for $L=18,19$ and 20. The solid lines are fits to the function $1-0.73tanh(t^{\prime }/t_{cr}^{\prime })$ used to obtain the crossover scale of $t^{\prime }$ as function of $L$. (Right) $T{D}_s\left(T\right)$ as $T\to \infty$ as a function of $t^{\prime }$ for the Hubbard model with $U=1$ for $L=7,8$ and 10 at half-filling. The linear fit for small values of $t^{\prime }$ is also shown.

Figure 4

$T{D}_c\left(T\right)$ as $T\to \infty$ as a function of $t^{\prime }$ for (left) the $t-t^{\prime }-V$ model with $V=1$ for $L=13,15,17$ and 19 and (right) for the Hubbard model with $U=2$ for $L=7,9$ and 11. The linear fit for small values of $t^{\prime }$ is also shown.

Figure 5

$t_{cr}^{\prime }$ as function of $L$ for the $t-t^{\prime }-V$ model with (left) $V=1$ and $V=2$ as obtained from the level spacing distribution and (right) $t-t^{\prime }-V$ model $(V=1$ and $V=1.75)$ using the charge Drude weight. The solid lines are fits to $L^{-3}$.

Figure 6

$t_{cr}^{\prime }$ as function of $L$ for the Hubbard model with (left) $(U=1$ and $U=2)$ as obtained from the level spacing distribution and (right) $(U=1$ and $U=2)$ as obtained from the Drude weight.

Figure 7

$t_{cr}^{\prime }$ as function of $L$ for (left) the $t-t^{\prime }-V$ model with $V=1$ and $V=2$ as obtained from the variance of the level spacing distribution and (right) the Hubbard model $(U=1$ and $U=8)$ at half-filling using the spin Drude weight. The solid lines are fits to $L^{-3}$.

Figure 8

$t_{cr}^{\prime }$ as function of $L$ for the $t-t^{\prime }-V$ model for $V=1,2,3,4$ and 6 as obtained from the level spacing distribution. The system is gapless for $V=1$ and 2 and gapped for $V=3,4$ and $V=6$ and it can be seen that the $t_{cr}^{\prime }$ for the three larger values of $V$ seems to be falling off faster than a power law.