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arXiv:2401.10499

Scaling Relations of Spectrum Form Factor and Krylov Complexity at Finite Temperature

Chengming Tan, Zhiyang Wei, Ren Zhang

Published Jan 19, 2024• 1 citation
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Abstract

In the study of quantum chaos diagnostics, considerable attention has been attributed to the Krylov complexity and spectrum form factor (SFF) for systems at infinite temperature. These investigations have unveiled universal properties of quantum chaotic systems. By extending the analysis to include the finite temperature effects on the Krylov complexity and SFF, we demonstrate that the Lanczos coefficients $b_n$, which are associated with the Wightman inner product, display consistency with the universal hypothesis presented in PRX 9, 041017 (2019). This result contrasts with the behavior of Lanczos coefficients associated with the standard inner product. Our results indicate that the slope $α$ of the $b_n$ is bounded by $πk_BT$, where $k_B$ is the Boltzmann constant and $T$ the temperature. We also investigate the SFF, which characterizes the two-point correlation of the spectrum and encapsulates an indicator of ergodicity denoted by $g$ in chaotic systems. Our analysis demonstrates that as the temperature decreases, the value of $g$ decreases as well. Considering that $α$ also represents the operator growth rate, we establish a quantitative relationship between ergodicity indicator and Lanczos coefficients slope. To support our findings, we provide evidence using the Gaussian orthogonal ensemble and a random spin model. Our work deepens the understanding of the finite temperature effects on Krylov complexity, SFF, and the connection between ergodicity and operator growth.

DOI
10.48550/arxiv.2401.10499

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In the study of quantum chaos diagnostics, considerable attention has been attributed to the Krylov complexity and spectrum form factor (SFF) for systems at infinite temperature. These investigations have unveiled universal properties of quantum chaotic systems. By extending the analysis to include the finite temperature effects on the Krylov complexity and SFF, we demonstrate that the Lanczos coefficients bnb_nbn​, which are associated with the Wightman inner product, display consistency with the universal hypothesis presented in PRX 9, 041017 (2019). This result contrasts with the behavior of Lanczos coefficients associated with the standard inner product. Our results indicate that the slope ααα of the bnb_nbn​ is bounded by πkBTπk_BTπkB​T, where kBk_BkB​ is the Boltzmann constant and TTT the temperature. We also investigate the SFF, which characterizes the two-point correlation of the spectrum and encapsulates an indicator of ergodicity denoted by ggg in chaotic systems. Our analysis demonstrates that as the temperature decreases, the value of ggg decreases as well. Considering that ααα also represents the operator growth rate, we establish a quantitative relationship between ergodicity indicator and Lanczos coefficients slope. To support our findings, we provide evidence using the Gaussian orthogonal ensemble and a random spin model. Our work deepens the understanding of the finite temperature effects on Krylov complexity, SFF, and the connection between ergodicity and operator growth.
In the study of quantum chaos diagnostics, considerable attention has been attributed to the Krylov complexity and the spectral form factor (SFF) for systems at infinite temperature. These investigations have unveiled universal properties of quantum chaotic systems. By extending the analysis to include the finite-temperature effects on the Krylov complexity and SFF, we demonstrate that the Lanczos coefficients b_{n}, which are associated with the Wightman inner product, display consistency with the universal hypothesis presented in Parker et al. [Phys. Rev. X 9, 041017 (2019)2160-330810.1103/PhysRevX.9.041017]. This result contrasts with the behavior of Lanczos coefficients associated with the standard inner product. Our results indicate that the slope α of the b_{n} is bounded by πk_{B}T, where k_{B} is the Boltzmann constant and T is the temperature. We also investigate the SFF, which characterizes the two-point correlation of the spectrum and encapsulates an indicator of ergodicity denoted by g in chaotic systems. Our analysis demonstrates that as the temperature decreases, the value of g decreases as well. Considering that α also represents the operator growth rate, we establish a quantitative relationship between the ergodicity indicator and the Lanczos coefficients' slope. To support our findings, we provide evidence using the Gaussian orthogonal ensemble and a random spin model. Our work deepens the understanding of the finite-temperature effects on the Krylov complexity, the SFF, and the connection between ergodicity and operator growth.
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)\mathcal{K}(t)K(t) in Random Matrix Theory (RMT). In large NNN limit: (1) For infinite temperature, we analytically show that the Lanczos coefficient {bn}\{b_n\}{bn​} saturate to constant plateau lim⁡n→∞bn=b\lim\limits_{n\rightarrow\infty}b_n=bn→∞lim​bn​=b, rendering a linear growing complexity K(t)∼t\mathcal{K}(t)\sim tK(t)∼t, in contrast to the exponential-in-time growth in chaotic local systems in thermodynamic limit. After numerically comparing this plateau value bbb to a large class of chaotic local quantum systems, we find that up to small fluctuations, it actually bounds the {bn}\{b_n\}{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 bnb_nbn​ will first exhibit linear growth with nnn, whose slope saturates the famous chaos bound. After hitting the same plateau bbb, bnb_nbn​ will then remain constant. This indicates K(t)∼e2πt/β\mathcal{K}(t)\sim e^{2πt/β}K(t)∼e2πt/β before scrambling time t∗∼O(βlog⁡β)t_*\sim O(β\logβ)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 NNN corrections.
Quantum chaotic systems are conjectured to display a spectrum whose fine-grained features (gaps and correlations) are well described by random matrix theory (RMT). We propose and develop a complementary version of this conjecture: quantum chaotic systems display a Lanczos spectrum whose local means and covariances are well described by RMT. To support this proposal, we first demonstrate its validity in examples of chaotic and integrable systems. We then show that for Haar-random initial states in RMTs the mean and covariance of the Lanczos spectrum suffice to produce the full long-time behavior of general survival probabilities including the spectral form factor, as well as the spread complexity. In addition, for initial states with continuous overlap with energy eigenstates, we analytically find the long-time averages of the probabilities of Krylov basis elements in terms of the mean Lanczos spectrum. This analysis suggests a notion of eigenstate complexity, the statistics of which differentiate integrable systems and classes of quantum chaos. Finally, we clarify the relation between spread complexity and the universality classes of RMT by exploring various values of the Dyson index and Poisson distributed spectra.
Recently, a novel measure for the complexity of operator growth is proposed based on Lanczos algorithm and Krylov recursion method. We study this Krylov complexity in quantum mechanical systems derived from some well-known local toric Calabi–Yau geometries, as well as some nonrelativistic models. We find that for the Calabi–Yau models, the Lanczos coefficients grow slower than linearly for small [Formula: see text]’s, consistent with the behavior of integrable models. On the other hand, for the nonrelativistic models, the Lanczos coefficients initially grow linearly for small [Formula: see text]’s, then reach a plateau. Although this looks like the behavior of a chaotic system, it is mostly likely due to saddle-dominated scrambling effects instead, as argued in the literature. In our cases, the slopes of linearly growing Lanczos coefficients almost saturate a bound by the temperature. During our study, we also provide an alternative general derivation of the bound for the slope.
A bstract Krylov complexity characterizes the operator growth in the quantum many-body systems or quantum field theories. The existing literatures have studied the Krylov complexity in the low temperature limit in the quantum field theories. In this paper, we extend and systematically study the Krylov complexity and Krylov entropy in a scalar field theory with general temperatures. To this end, we propose a new method to calculate the Wightman power spectrum which allows us to compute the Lanczos coefficients and subsequently to study the Krylov complexity (entropy) in general temperatures. We find that the Lanczos coefficients and Krylov complexity (entropy) in the high temperature limit will behave somewhat differently from those studies in the low temperature limit. We give an explanation of why the Krylov complexity does not oscillate in the high-temperature region. Moreover, we uncover the transition temperature that separates the oscillating and monotonic increasing behavior of Krylov complexity.
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.