Krylov complexity in the IP matrix model II

In the zero temperature case, one can solve this equation analytically by mapping the equation to the Bessel recursion relation based on the canonical calculation Iizuka:2008hg ,

where $J$ is a Bessel function The spectrum is determined by the pole of $\tilde{G}(\omega)$ which is discrete for nonzero $m>0$. There are infinite poles, which are determined by the zeros of the denominator.

In the infinite temperature limit, $T=\infty$, the recursion relation (11) becomes

This recursion relation is symmetric between $\omega\to\infty$ to $\omega\to-\infty$. Furthermore, there are $m$ shifted structure: if $F(\omega_{0})\neq 0$ at some $\omega=\omega_{0}$, then $F(\omega_{0}\pm m)\neq 0$. In fact, as we increase the temperature, the spectrum change from collections of poles to collections of branch cuts, and finally these cuts merge and the spectrum becomes gapless and continuous Iizuka:2008hg ; Iizuka:2023pov . In large $\omega$, $F(\omega)$ decays exponentially as $F(\omega)\sim|\omega|^{-|\omega|}$. In fact, the asymptotic solution can be obtained as Iizuka:2023pov

This exponential decay of the $F(\omega)$ (with $\log\omega$ correction) leads to the Lanczos coefficients $b_{n}$ growing asymptotically linear in $n$ (with $\log n$ correction) at large $n$.

At zero temperature, let $|v\rangle$ be the free ground state. Then, consider excited states $|j,n\rangle$ with a single excitation by $\hat{a}^{\dagger}_{i}$ and $n$-excitations by $\hat{A}^{\dagger}_{ij}$ such as

In the large $N$ limit, the states $|j,n\rangle$ span an orthonormal basis for the two-point function. Moreover, in the large $N$ limit, we obtain Iizuka:2008hg

Therefore, with $\mathcal{L}=H-M$, the Krylov basis for $\hat{\mathcal{O}}=\hat{a}^{\dagger}_{j}$ is $|\hat{\mathcal{O}}_{n})=|j,n\rangle$ given by eq. (20). Then the Lanczos coefficients are given by

and $b_{n}$ in the IP model does not depend on $n$ because we determine the normalization of (20) in the large $N$ limit.

Given the Lanczos coefficients as eq. (22), we perform numerical calculations of $K(t)$. The resultant Krylov complexity $K(t)$ oscillates due to nonzero $a_{n}$ and does not grow at late times Iizuka:2023pov .

The asymptotic behavior of the spectral density (16) determines the Lanczos coefficients of the IP model at the infinite temperature limit. The spectral density $F(\omega)$ is a positive and smooth function everywhere Iizuka:2008hg and $F(\omega)$ is an even function with respect to $\omega$. Thus $a_{n}=0$. Furthermore the exponential suppression in (16) is known as the slowest decay for the situation where $C(t)$ is analytic in the entire complex time plane Parker:2018yvk ; Avdoshkin:2019trj . From (16), the asymptotic behavior of $b_{n}$ of the IP model in the infinite temperature limit can be determined as

Here $W(n)$ is defined by $z=W(ze^{z})$, called the Lambert W function.

Given the asymptotic behavior of $b_{n}$ with $a_{n}=0$. Then, the late-time behavior of $K(t)$ is given as

This growth behavior is slower than the exponential growth $e^{\mathcal{O}{\left(t\right)}}$, but it is faster than any power low growth behavior for integral systems. The growth of $b_{n}$ as $b_{n}\propto n/{\log n}$ strongly suggests that the IP model in the infinite temperature limit is chaotic.

We would like to understand the Lanczos coefficient of the spectral density which consists of an infinite number of Wigner semicircles as eq. (26). For simplicity, we consider $f(\omega)=f(-\omega)$, which lead $a_{n}=0$. As we mentioned, we choose $A_{j}$ decays exponentially with respect to $|\omega_{j}|$ as eq. (27), since the spectrum of the IP model decays exponentially at large $|\omega|$ including logarithmic correction.

Although the IP model forces the spectral density to have an infinite number of cuts, i.e., $N_{W}\to\infty$, let us examine the finite $N_{W}$ i.e., the number of Wigner semicircles is finite. By increasing the number of cuts, $N_{W}$, we would like to see how the Lanczos coefficients change.

Figure 3 shows Lanczos coefficient $b_{n}$ for the spectral density given by eq. (26), (27) with $\ell=2$, $m=10$, $\Omega=1/12$, (a) $N_{W}=3$ (seven semicircles) and (b) $N_{W}=4$ (nine semicircles) respectively.

Let us comment on properties of $b_{n}$ that can be read from Figure 3.

• •

  Around $n=1$, the Lanczos coefficient is a constant $b_{n}=\ell/2$. This is because $b_{n}$ around $n=1$ is determined from the single semicircle $f_{j=0}(\omega)$ around $\omega=0$.

• •

  As $n$ increases, the other semicircles begin to contribute to $b_{n}$. Since the amplitude of semicircles $A_{j}=e^{-|\omega_{j}|/\Omega}$ decays exponentially with respect to $|\omega_{j}|$ with $\Omega=1/12$, $b_{n}$ increases linearly on average as $b_{n}\sim\Omega\pi n/2={\pi n}/{24}$. More precisely, due to the existence of gaps in the spectra, $b_{n}$ increases in a staircase pattern with fluctuation. In our numerical computations, the fluctuation does not grow much as $n$ increases.

• •

  In Figure 3, we consider only a small number of Wigner semicircles in the spectral density, and thus there exists $\omega_{max}$ such that $f(\omega)=0$ for $|\omega|>\omega_{max}\sim mN_{W}$. At large $n$, $b_{n}$ saturates as $b_{n}\sim\omega_{max}/2$ with fluctuation due to the gaps.

Let us consider further examples with a large number of Wigner semicircles, $N_{W}$. Figure 4 shows Lanczos coefficient $b_{n}$ for the spectral density given by eq. (26), (27) with $\ell=2$, $m=10$, $\Omega=10$, $N_{W}=500,1000,1600$.

• •

  At large $n$, $b_{n}$ in Figure 4 saturates as $b_{n}\sim\omega_{max}/2=mN_{W}/2=2500$ for $N_{W}=500$ and as $b_{n}\sim\omega_{max}/2=mN_{W}/2=5000$ for $N_{W}=1000$. The fluctuation of $b_{n}$ in Figures 4 is small compared to the linear growth of $b_{n}$. The slope of linear growth is $b_{n}\sim\pi\Omega n/2=5\pi n$.

• •

  Comparing Figures 3 and 4, one can check that the linear growth rate of $b_{n}$ becomes smaller as $\Omega$ decreases.

• •

  In the limit that a number of the semicircles is infinite, $N_{W}\to\infty$, the value of $\omega_{max}$ becomes infinite, and $b_{n}$ would increase without the saturation even at large $n$ if the amplitude $A_{j}$ decays with respect to $\omega_{j}$.

Once $b_{n}$ is obtained, we can calculate Krylov complexity numerically. Figure 5 shows log plots of the Krylov complexity $K(t)$ (red solid curves) computed from $b_{n}$ in Figure 4 to confirm the exponential growth for $N_{W}=1600$. For comparison, we also plot $K(t)$ computed from the linear fitting of $b_{n}$ where we exclude the fluctuation by hand (blue dashed curves). We also plot $K(t)$ for the power spectrum

which exponentially decays without gaps (green dotted curves). Their properties are

• •

  In Figure 5(a), the three curves are very similar since the fluctuation of $b_{n}$ for $\Omega=10$ in Figure 4 is very small.

• •

  In Figure 5(b), the slopes of blue and green curves in the log plot at late times are similar. However, the values of $K(t)$ are very different. This is because the Lanczos behaves as $b_{n}\sim\pi\Omega n/2+\gamma$ for these curves are similar but the values of $\gamma$ are different between them. When $K(t)\sim\frac{\eta}{4}e^{\pi\Omega t}$, $\eta$ is related to $\gamma$ as $\eta\sim 4\gamma/(\pi\Omega)+1$ Parker:2018yvk . In Figure 5(b), $\frac{\eta}{4}$ for the blue curve is $\frac{\eta}{4}\sim 2.0$, and $\frac{\eta}{4}$ for the green curve is $\frac{\eta}{4}\sim 0.25$.

• •

  The slope of the red curve in 5(b) at late times is smaller than the one of the blue and green curves. This is because of the fluctuation of $b_{n}$. The fluctuation in $b_{n}$ makes the slope smaller as we will see later in more detail. However, since the red solid curve in 5(b) grows linearly in the log plots at late times, $K(t)$ for the spectra with infinitely many Wigner semicircles would grow exponentially. This leads to the conclusion that the fluctuations of $b_{n}$ make the exponential growth milder, but still, it maintains the exponential form as $K(t)\sim\exp\left(\alpha t\right)$ but with smaller $\alpha$ by fluctuations. Such a mild exponential growth of $K(t)$ was observed in Avdoshkin:2022xuw ; Camargo:2022rnt for the spectrum that has a single mass gap, and our numerical computation suggests that a similar phenomenon happens even when the spectrum has an infinite number of gaps.

• •

  If the number of Wigner semicircles, $N_{W}$ is finite, then $b_{n}$ saturates as in Figure 3 at some $n$, and then the growth of $K(t)$ transitions to the linear growth at late times. Our numerical computations imply that, unless the power spectrum is discrete, the existence of gaps changes the coefficient of exponential growth of $K(t)$, but does maintain the exponential growth of the Krylov complexity.

1. 1.

   Small $\ell\to 0$
   Let us consider the small $\ell$ and also the limit $\ell\to 0$. We consider the spectral density given by eq. (26), (27) with $m=10$, $\Omega=1/12$, $N_{W}=25$ (51 semicircles), and $\ell=1/10$ and $\ell=1/100$ respectively.

   The smaller $\ell$ is, the closer the spectrum is to a discrete spectrum. Figure 6 show the Lanczos coefficient $b_{n}$ for the above parameters. Comparing Figures 6(a) and 6(b), the smaller $\ell$ is, the larger fluctuation of $b_{n}$. This implies that the fluctuation is large if the spectrum is closer to a discrete spectrum. Figure 7 shows the Krylov complexity $K(t)$ computed from $b_{n}$ for various small $\ell$. We plot two figures with different horizontal scales, where the maximum value of $t$ in the left figure is $t_{max}=20000$, and $t_{max}=200000$ in the right figure. One can see that $K(t)$ with the same value of $\ell t_{max}$ behaves similarly in these figures. This is because (33) depends on $\ell$ in the form of $\ell t$. From Figures 6 and 7, we can see that the larger the fluctuation of $b_{n}$, the smaller the slope of $\log K(t)$. Thus, in the limit $\ell\to 0$, huge fluctuations kill the exponential growth of $K(t)$ and we have a transition to the oscillation behavior of $K(t)$ in that limit.

   

   

   

   

2. 2.

   Large $\ell\to m/2$
   Let us study the fluctuation of $b_{n}$ when the Wigner semicircles in the spectrum touch with each other as in Figure 8(a), where $\ell\to m/2$. For visibility, we choose $\Omega=10$ in Figure 8(a). In Figure 8(b) with $\Omega=1/12$ for the strong exponential decay of $A_{j}$, we can see that the fluctuation of $b_{n}$ exists even where $\ell\to m/2$. This is because the spectrum is not smooth between the Wigner semicircles.

   

   

At finite temperature $y\neq 1$, our key equation eq. (11) is not symmetric under $\omega\to-\omega$. Due to this non-symmetric property of the equation, the decay rate of $F(\omega)$ is not symmetric under $\omega\to-\omega$. To see this non-symmetric effect, let us consider the spectrum (26) with

For example, Figure 9 is a plot of the non-symmetric spectrum with $m=2,\ell=1/2,\Omega_{+}=20,\Omega_{-}=10,N_{W}=50$.

We numerically compute the Lanczos coefficients $a_{n}$ and $b_{n}$ of this non-symmetric spectrum with $m=10$, $\ell=2$, $\Omega_{+}=1$, $\Omega_{-}=1/12$, $N_{W}=200$ as shown in Figure 10. In this numerical computation, the Lanczos coefficients increase linearly with fluctuation. If $N_{W}$ is small, as shown in Figure 11, $a_{n}$ fluctuates around $a_{n}=0$ at large $n$, and $b_{n}$ saturates to a nonzero value with fluctuation. Figure 12 is a log plot of Krylov complexity for the non-symmetric spectrum with $N_{W}=200$ computed from the Lanczos coefficients in Figure 10. It shows the exponential growth of $K(t)$ at late times. Therefore, our numerical computation of the non-symmetric model indicates that the non-symmetric decay rate of $F(\omega)$ does not change the exponential growth of Krylov complexity.

In summary, our study of the low-temperature gapped spectral density model shows that

1. 1.

   At $\ell=0$ corresponds to $T=0$, the Krylov complexity oscillates and does not grow.

2. 2.

   However once $\ell>0$ corresponds to $T>0$, the Lanczos coefficients grow linearly in $n$ with fluctuations. Fluctuations of $b_{n}$ are huge near $\ell\to 0$ but as we increase $\ell$, the fluctuations become smaller.

3. 3.

   Fluctuations of $b_{n}$ makes the slope of $\log K(t)$ smaller. However, as long as $\ell\neq 0$, the Lancoz coefficients $b_{n}$ grow linearly with fluctuations and the Krylov complexity grow exponentially as $K(t)\sim\exp\left(\alpha t\right)$. In the limit $\ell\to 0$, $\alpha\to 0$ and as we increases $\ell$, $\alpha$ also increases.

4. 4.

   It is crucial for the $K(t)$ grows exponentially in time, or equivalently for $b_{n}$ grows linearly in $n$, there are infinite cuts, i.e., $N_{W}\to\infty$. For finite $N_{W}$, the exponential growth of $K(t)$ saturates at some time. $N_{W}\to\infty$ is the nature of the key equation eq. (11) obtained from the Schwinger-Dyson equation of the IP model in the large $N$ limit.