arXiv subjects

Merlin Füllgraf

Publications and source records attributed to Merlin Füllgraf.

6 recordsLinked to original sources

arXiv2026

Resonant level model from a Krylov perspective: Lanczos coefficients in a quadratic model

We study the Lanczos coefficients in a quadratic model given by an impurity interacting with a multi-mode field of fermions, also known as resonant level model. We analytically derive closed expressions for the Lanczos coefficients of Majorana fermion operators of the impurity for different structures of the coupling to the hybridization band at zero temperature. While the model remains quadratic, we find that the growth of the Lanczos coefficients structurally depends strongly on the chosen coupling. Concretely, we find $(i)$ approximately constant, $(ii)$ exactly constant, $(iii)$ square root-like as well as $(iv)$ linear growth in the same model. We further argue that in fact through suitably chosen couplings, essentially arbitrary Lanczos coefficients can be obtained in this model. These altogether evince the inadequacy of the Lanczos coefficients as a reliable criterion for classifying the integrability or chaoticity of the systems. Eventually, in the wide-band limit, we find exponential decay of autocorrelation functions in all the settings $(i)-(iv)$, which demonstrates the different structures of the Lanczos coefficients not being indicative of different physical behavior.

cond-mat.str-el↗

arXiv2025

Scaling of free cumulants in closed system-bath setups

The Eigenstate Thermalization Hypothesis (ETH) has been established as a cornerstone for understanding thermalization in quantum many-body systems. Recently, there has been growing interest in the full ETH, which extends the framework of the conventional ETH and postulates a smooth function to describe the multi-point correlations among matrix elements. Within this framework, free cumulants play a central role, and most previous studies have primarily focused on closed systems. In this paper, we extend the analysis to a system-bath setup, considering both an idealized case with a random-matrix bath and a more realistic scenario where the bath is modeled as a defect Ising chain. In both cases, we uncover a universal scaling of the microcanonical free cumulants of observables associated with the central system Hamiltonian with respect to the interaction strength. Furthermore we establish a connection between this scaling behavior and the thermalization dynamics of the thermal free cumulants of corresponding observables.

cond-mat.stat-mech↗

arXiv2025

Lanczos-Pascal approach to correlation functions in chaotic quantum systems

We suggest a method to compute approximations to temporal correlation functions of few-body observables in chaotic many-body systems in the thermodynamic limit based on the respective Lanczos coefficients. Given the knowledge of these Lanczos coefficients, the method is very cheap. Usually accuracy increases with more Lanczos coefficients taken into account, however, we numerically find and analytically argue that the convergence is rather quick, if the Lanczos coefficients exhibit a smoothly increasing structure. For pertinent examples we compare with data from dynamical typicality computations for large but finite systems and find good agreement if few Lanczos coefficients are taken into account. From the method it is evident that in these cases the correlation functions are well described by a low number of damped oscillations.

cond-mat.stat-mech↗

arXiv2024

Estimate of equilibration times of quantum correlation functions in the thermodynamic limit based on Lanczos coefficients

We study the equilibration times $T_\text{eq}$ of local observables in quantum chaotic systems by considering their auto-correlation functions. Based on the recursion method, we suggest a scheme to estimate $T_\text{eq}$ from the corresponding Lanczos coefficients that is expected to hold in the thermodynamic limit. We numerically find that if the observable eventually shows smoothly growing Lanczos coefficients, a finite number of the former is sufficient for a reasonable estimate of the equilibration time. This implies that equilibration occurs on a realistic time scale much shorter than the life of the universe. The numerical findings are further supported by analytical arguments.

cond-mat.stat-mech↗

arXiv2024

Evidence for simple "arrow of time functions" in closed chaotic quantum systems

Through an explicit construction, we assign to any infinite temperature autocorrelation function $C(t)$ a set of functions $\alpha^n(t)$. The construction of $\alpha^n(t)$ from $C(t)$ requires the first $2n$ temporal derivatives of $C(t)$ at times $0$ and $t$. Our focus is on $\alpha^n(t)$ that (almost) monotonously decrease, we call these ``arrows of time functions" (AOTFs). For autocorrelation functions of few body observables we numerically observe the following: An AOTF featuring a low $n$ may always be found unless the the system is in or close to a nonchaotic regime with respect to a variation of some system parameter. All $\alpha^n(t)$ put upper bounds to the respective autocorrelation functions, i.e. $\alpha^n(t) \geq C^2(t)$. Thus the implication of the existence of an AOTF is comparable to that of the H-Theorem, as it indicates a directed approach to equilibrium. We furthermore argue that our numerical finding may to some extent be traced back to the operator growth hypothesis. This argument is laid out in the framework of the so-called recursion method.

cond-mat.stat-mech↗

arXiv2024

Maximum Expected Reward Lines If Non Specialist (for darts)

Owning up to the authors' occasional mixed performances when playing darts we follow up on the ingenious work to determine the optimal aim to score high by Tibshirani et. al [JRS A 174, 213 (2011)] and expand on maximal expected reward lines assuming spread profiles beyond the standard case of an isotropic distribution by investigating situations in which a players' aiming quality differs in horizontal and vertical direction, arguably representing more realistic profiles. To render the theoretical findings applicable, we determine regions $Γ$ on the dart board that may serve as an aid to players in order to maximise their score depending on whether said areas are hit sufficiently often. Lastly we compare all three strategies with the simple approaches to always aim for triple 20 or bulls eye respectively, probing their usefulness.

physics.pop-ph↗