We continue the analysis of the Krylov complexity in the IP matrix model. In a previous paper, for a fundamental operator, it was shown that at zero temperature, the Krylov complexity oscillates and does not grow, but in the infinite temperature limit, the Krylov complexity grows exponentially in time as $\sim \exp\left( {\mathcal{O}\left( {\sqrt{t}}\right)} \right)$. We study how the Krylov complexity changes from a zero-temperature oscillation to an infinite-temperature exponential growth. At low temperatures, the spectral density is approximated as collections of infinite Wigner semicircles. We showed that this infinite collection of branch cuts yields linear growth to the Lanczos coefficients and gives exponential growth of the Krylov complexity. Thus the IP model for any nonzero temperature shows exponential growth for the Krylov complexity even though the Green function decays by a power law in time. We also study the Lanczos coefficients and the Krylov complexity in the IOP matrix model taking into account the $1/N^2$ corrections. There, the Lanczos coefficients are constants and the Krylov complexity does not grow exponentially as expected.
We continue the analysis of the Krylov complexity in the IP matrix model. In a previous paper, for a fundamental operator, it was shown that at zero temperature, the Krylov complexity oscillates and does not grow, but in the infinite temperature limit, the Krylov complexity grows exponentially in time as ∼exp(O(t)). We study how the Krylov complexity changes from a zero-temperature oscillation to an infinite-temperature exponential growth. At low temperatures, the spectral density is approximated as collections of infinite Wigner semicircles. We showed that this infinite collection of branch cuts yields linear growth to the Lanczos coefficients and gives exponential growth of the Krylov complexity. Thus the IP model for any nonzero temperature shows exponential growth for the Krylov complexity even though the Green function decays by a power law in time. We also study the Lanczos coefficients and the Krylov complexity in the IOP matrix model taking into account the 1/N2 corrections. There, the Lanczos coefficients are constants and the Krylov complexity does not grow exponentially as expected.
Abstract The IP matrix model is a simple large N quantum mechanical model made up of an adjoint harmonic oscillator plus a fundamental harmonic oscillator. It is a model introduced previously as a toy model of the gauge theory dual of an AdS black hole. In the large N limit, one can solve the Schwinger-Dyson equation for the fundamental correlator, and at sufficiently high temperature, this model shows key signatures of thermalization and information loss; the correlator decay exponentially in time, and the spectral density becomes continuous and gapless. We study the Lanczos coefficients bn in this model and at sufficiently high temperature, it grows linearly in n with logarithmic corrections, which is one of the fastest growth under certain conditions. As a result, the Krylov complexity grows exponentially in time as ∼exp(O(t)) ∼ exp O t . These results indicate that the IP model at sufficiently high temperature is chaotic.
Krylov complexity, as a novel measure of operator complexity under Heisenberg evolution, exhibits many interesting universal behaviors and also bounds many other complexity measures. In this work, we study Krylov complexity K(t) in Random Matrix Theory (RMT). In large N limit: (1) For infinite temperature, we analytically show that the Lanczos coefficient {bn} saturate to constant plateau n→∞limbn=b, rendering a linear growing complexity K(t)∼t, in contrast to the exponential-in-time growth in chaotic local systems in thermodynamic limit. After numerically comparing this plateau value b to a large class of chaotic local quantum systems, we find that up to small fluctuations, it actually bounds the {bn} in chaotic local quantum systems. Therefore we conjecture that in chaotic local quantum systems after scrambling time, the speed of linear growth of Krylov complexity cannot be larger than that in RMT. (2) For low temperature, we analytically show that bn will first exhibit linear growth with n, whose slope saturates the famous chaos bound. After hitting the same plateau b, bn will then remain constant. This indicates K(t)∼e2πt/β before scrambling time t∗∼O(βlogβ), and after that it will grow linearly in time, with the same speed as in infinite temperature. We finally remark on the effect of finite N corrections.
A bstract We investigate Krylov complexity of the fermion chain operator which consists of multiple Majorana fermions in the double-scaled SYK (DSSYK) model with finite temperature. Using the fact that Krylov complexity is computable from two-point functions, the analysis is performed in the limit where the two-point function becomes simple and we compare the results with those of other previous studies. We confirm the exponential growth of Krylov complexity in the very low temperature regime. In general, Krylov complexity grows at most linearly at very late times in any system with a bounded energy spectrum. Therefore, we have to focus on the initial growth to see differences in the behaviors of systems or operators. Since the DSSYK model is such a bounded system, its chaotic nature can be expected to appear as the initial exponential growth of the Krylov complexity. In particular, the time at which the initial exponential growth of Krylov complexity terminates is independent of the number of degrees of freedom. More generally, and not limited to the DSSYK model, we systematically and specifically study the Lanczos coefficients and Krylov complexity using a toy power spectrum and deepen our understanding of those initial behaviors. In particular, we confirm that the overall sech-like behavior of the power spectrum shows the initial linear growth of the Lanczos coefficient, even when the energy spectrum is bounded.
We study the operator growth problem and its complexity in the many-body localization (MBL) system from the Lanczos algorithm perspective. Using the Krylov basis, the operator growth problem can be viewed as a single-particle hopping problem on a semi-infinite chain with the hopping amplitudes given by the Lanczos coefficients. We find that, in the MBL systems, the Lanczos coefficients scale as ∼ n/ ln(n) asymptotically, same as in the ergodic systems, but with an additional even-odd alteration and an effective randomness. We use a simple linear extrapolation scheme as an attempt to extrapolate the Lanczos coefficients to the thermodynamic limit. With the original and extrapolated Lanczos coefficients, we study the properties of the emergent single-particle hopping problem via its spectral function, integrals of motion, Krylov complexity, wavefunction profile and return probability. Our numerical results of the above quantities suggest that the emergent single-particle hopping problem in the MBL system is localized when initialized on the first site. We also study the operator growth in the MBL phenomenological model, whose Lanczos coefficients also have an even-odd alteration, but approach constants asymptotically. The Krylov complexity grows linearly in time in this case.
We study Krylov complexity in BMN Plane Wave Matrix Model at large mass deformation. We consider various consistent reductions of the matrix model that allow us to perform a Hamiltonian analysis which leads to different notions of the Krylov complexity. In the first part of the paper, we study the Krylov state complexity considering systematic reduction of N=3 and N=4 representations of the matrix model, which reveals a universal characteristic scaling for the Lanczos coefficients and fix them completely in terms of the mass deformation parameter. In the second part of the paper, we study the Krylov operator growth in the matrix model and compute the corresponding Lanczos coefficients. In both cases, we observe a \emph{linear} scaling of Lanczos coefficients with the mass parameter. The early time growth in Krylov complexity receives quadratic correction due to the presence of the massive deformation in the matrix model. Our analysis reveals that such massive corrections appear at same order in time for both the notion of the Krylov complexity.
We study the operator growth problem and its complexity in the many-body localization (MBL) system from the Lanczos algorithm perspective. Using the Krylov basis, the operator growth problem can be viewed as a single-particle hopping problem on a semi-infinite chain with the hopping amplitudes given by the Lanczos coefficients. We find that, in the MBL systems, the Lanczos coefficients scale as \sim n/\ln(n) ∼ n / ln ( n ) asymptotically, same as in the ergodic systems, but with an additional even-odd alteration and an effective randomness. We use a simple linear extrapolation scheme as an attempt to extrapolate the Lanczos coefficients to the thermodynamic limit. With the original and extrapolated Lanczos coefficients, we study the properties of the emergent single-particle hopping problem via its spectral function, integrals of motion, Krylov complexity, wavefunction profile and return probability. Our numerical results of the above quantities suggest that the emergent single-particle hopping problem in the MBL system is localized when initialized on the first site. We also study the operator growth in the MBL phenomenological model, whose Lanczos coefficients also have an even-odd alteration, but approach constants asymptotically. The Krylov complexity grows linearly in time in this case.