Spectral and Krylov complexity in billiard systems

In this work, we investigate spectral complexity and Krylov complexity in quantum billiard systems at finite temperature. We study both circle and stadium billiards as paradigmatic examples of integrable and nonintegrable quantum-mechanical systems, respectively. We show that the saturation value and timescale of spectral complexity may be used to probe the nonintegrability of the system since we find that when computed for the circle billiard, it saturates at a later timescale compared to the stadium billiards. This observation is verified for different temperatures. Furthermore, we study the Krylov complexity of the position operator and its associated Lanczos coefficients at finite temperature using the Wightman inner product. We find that the growth rate of the Lanczos coefficients saturates the conjectured universal bound at low temperatures. Additionally, we also find that even a subset of the Lanczos coefficients can potentially serve as an indicator of integrability, as they demonstrate erratic behavior specifically in the circle billiard case, in contrast to the stadium billiard. Finally, we also study Krylov entropy and verify its early-time logarithmic relation with Krylov complexity in both types of billiard systems.

Figure 1

Geometry of the stadium billiard.

Figure 2

Eigenvalues of the quantum billiard systems. (a) the stadium billiard ($a/R=1$). (b) the circle billiard ($a/R=0$).

Figure 3

Eigenfunctions of the quantum billiard systems for $n=1$, 7, 50. Upper panels (a)–(c): the stadium billiard ($a/R=1$). Lower panels (d)–(f): the circle billiard ($a/R=0$).

Figure 4

Spectral complexity at infinite temperature with $a/R=0$, 0.1, 0.3, 0.5, 0.7, 0.9, 1 (black, purple, blue, green, yellow, orange, red). (a) Spectral complexity. (b) Spectral complexity at $t={10}^{10}$.

Figure 5

Temperature dependence of spectral complexity at $T=10$, 20, 40, 100 (blue, green, orange, red). (a) Stadium billiard $(a/R=1)$. (b) Circle billiard $(a/R=0)$.

Figure 6

Lanczos coefficients $b_n$ at $T=10$, 20, 40, 100 (blue, green, orange, red). The solid lines are linear fitting curves (3.6). (a) Stadium billiard $(a/R=1)$. (b) Circle billiard $(a/R=0)$.

Figure 7

The temperature dependence of the growth rate of $b_n$. The solid line is $\alpha =\pi T$ and dots are obtained from the linear fitting of $b_n$. The bound is saturated at low temperature.

Figure 8

The normalization condition at $T=10$, 20, 40, 100 (blue, green, orange, red). The insets show near $t\approx 0.1$. (a) Stadium billiard $(a/R=1)$. (b) Circle billiard $(a/R=0)$.

Figure 9

Krylov complexity at $T=10$, 20, 40, 100 (blue, green, orange, red). Dots are numerical data and the dashed lines are the analytic early-time results (2.15). (a) Stadium billiard $(a/R=1)$. (b) Circle billiard $(a/R=0)$.

Figure 10

Krylov entropy at $T=10$, 20, 40, 100 (blue, green, orange, red). Dots are numerical data and the dashed lines are the analytic early-time results (2.15). (a) Stadium billiard $(a/R=1)$. (b) Circle billiard $(a/R=0)$.

Figure 11

The logarithmic relation between Krylov complexity and entropy at $T=10$, 20, 40, 100 (blue, green, orange, red). We find our data is consistent with (2.16) at early time. (a) Stadium billiard $(a/R=1)$. (b) Circle billiard $(a/R=0)$.

Figure 12

Both sides of the Ehrenfest theorem (3.8) for stadium and circle billiards at $T=10$. (a) Left hand side of the Ehrenfest theorem for stadium billiard $(a/R=1)$ at $T=10$. (b) Right hand side of the Ehrenfest theorem for stadium billiard $(a/R=1)$ at $T=10$. (c) Left hand side of the Ehrenfest theorem for circle billiard $(a/R=0)$ at $T=10$. (d) Right hand side of the Ehrenfest theorem for circle billiard $(a/R=0)$ at $T=10$.

Figure 13

Numerically obtained Lanczos coefficient at $\omega =1$ and $\beta =(1,0.5)$ (a), (b).

Figure 14

Numerically obtained Krylov complexity and entropy at $\omega =\beta =1$ (red dashed line). The black solid line is analytic formula (a21).

Figure 15

Krylov complexity at $\omega =1,\beta =1/2$. Black dashed lines are analytic result (a24) and red solid lines are numerical result. (a) single operator case, $x$ or $p$. (b) quadratic operator case, $x^2$ or $p^2$. (c) crossed operator case, $xp$ or $px$.

Figure 16

Lanczos coefficients at $T=10$, 20, 40, 100 (blue, green, orange, red). Dots are obtained by Lanczos algorithm, while the solid lines are evaluated by the moment method. (a) Stadium billiard $(a/R=1)$. (b) Circle billiard $(a/R=0)$.

Figure 17

Stadium billiard ($a/R=1$) at $T=100$. (a) $b_n$. (b) Normalization condition. (c) Krylov complexity.