Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2407.12495 · Physical review. B./Physical review. B

Pseudomode expansion of many-body correlation functions

Alexander Evgen’evich Teretenkov, Filipp Gennadevich Uskov, Oleg Valentinovich Lychkovskiy

Published May 22, 2025• 13 citations
arXivHTMLPDF

Abstract

We present an expansion of a many-body correlation function in a sum of pseudomodes: exponents with complex frequencies that encompass both decay and oscillations. The pseudomode expansion emerges in the framework of the Heisenberg version of the recursion method. This method essentially solves Heisenberg equations in a Lanczos tridiagonal basis constructed in the Krylov space of a given observable. To obtain pseudomodes, we first add artificial dissipation satisfying the dissipative generalization of the universal operator growth hypothesis, and then take the limit of the vanishing dissipation strength. Fast convergence of the pseudomode expansion is facilitated by the localization in the Krylov space, which is generic in the presence of dissipation and can survive the limit of the vanishing dissipation strength. As an illustration, we present pseudomode expansions of infinite-temperature autocorrelation functions in the quantum Ising and $XX$ spin-$1/2$ models on the square lattice. It turns out that it is enough to take a few first pseudomodes to obtain a good approximation to the correlation function.

DOI
10.1103/physrevb.111.174308

Related papers

Quantum dynamics in one and two dimensions via the recursion method

Filipp Gennadevich Uskov, Oleg Valentinovich Lychkovskiy

We report an implementation of the recursion method that addresses quantum many-body dynamics in the nonperturbative regime. The method essentially amounts to constructing a Lanczos basis in the space of operators and solving coupled Heisenberg equations in this basis. The reported implementation has two key ingredients: a computer-algebraic routine for the symbolic calculation of nested commutators and a procedure to extrapolate the sequence of Lanczos coefficients according to the universal operator growth hypothesis. We apply the method to calculate infinite-temperature correlation functions for spin-$1/2$ systems on one- and two-dimensional lattices. In two dimensions the accessible timescale is large enough to essentially embrace the relaxation to equilibrium. The method allows one to accurately calculate transport coefficients. As an illustration, we compute the diffusion constant for the transverse-field Ising model on a square lattice.

arXiv:2401.17211•Physical review. B./Physical review. B•Apr 1, 2024•19 citations

Recursion method for quench dynamics: Strengths and limitations

Ilya Shirokov, Viacheslav Khrushchev, Filipp Gennadevich Uskov +3 more

The recursion method, which solves coupled Heisenberg equations in a Lanczos operator basis, has recently emerged as a powerful nonperturbative tool for computing dynamical correlation functions in strongly correlated two- and three-dimensional quantum many-body systems. Motivated by this success, we investigate whether the method can be extended to expectation values of observables following a quantum quench. We find that such an extension encounters an obstacle absent in the computation of dynamical correlation functions. The latter are fully determined by the Lanczos coefficients b_n b n , which in generic systems exhibit universal behavior, enabling reliable extrapolation from the first few dozens of explicitly computed coefficients. In contrast, quench dynamics additionally requires “quench coefficients” c_n c n , defined as overlaps of Lanczos basis operators with the initial state. We show that, unlike the Lanczos coefficients, the quench coefficients exhibit no universal structure and cannot be reliably extrapolated, thereby limiting the time up to which the method yields accurate results. The behavior of quench coefficients is highly state-dependent, ranging from decaying to irregular or even growing sequences; typically, the less regular the sequence c_n c n , the shorter the accessible timescale. Nevertheless, for favorable initial states, the method remains competitive with state-of-the-art approaches. Moreover, its symbolic implementation allows a single computation to be reused across different Hamiltonian parameters and initial states, making it particularly advantageous in studies requiring extensive scans over Hamiltonian parameters or initial states.

arXiv:2503.24362•SciPost Physics Core•Aug 19, 2026•0 citations

Nonequilibrium quantum many-body physics from perspectives of operator dynamics and the Lanczos picture

Merlin Füllgraf

This cumulative doctoral thesis, based on the works [P1–P5], addresses various topical concepts as well as foundational questions in the nonequilibrium physics of quantum many-body systems. Publication [P1] concerns itself with the question of time’s arrow in quantum mechanics. Based on the recursion method, we attribute autocorrelation functions of few-body observables at infinite temperature (almost) monotonously decreasing functions, constructed from temporal derivatives of the correlation function that indicate a directed approach to equilibrium. We numerically find that in chaotic models, these constructions comprise a notion of simplicity of the underlying dynamics. Addressing the ubiquity of simple relaxation dynamics in otherwise complex chaotic systems, in Ref. [P3] we introduce a scheme based on Lanczos coefficients to approximate correlation functions in the thermodynamic limit. Examining several pertinent condensed-matter type models and benchmarking with state-of-the-art numerics, we find the method to yield good results already for a low number of included Lanczos coefficients, provided that the latter exhibit a smooth and increasing structure. We underpin these findings with analytical arguments on the method’s applicability. Ref. [P2] provides an estimate on the equilibration time of such infinite-temperature correlation functions in the thermodynamic limit. Similarly to Ref. [P3], we find the estimate to capture the equilibration times very reasonably, given that the corresponding Lanczos coefficients exhibit a smooth growth. Substantiated by analytical considerations, the convergence of the estimate implies that equilibration timescales of the relaxation processes are finite and conceivably short. In Ref. [P5] we scrutinize the interpretative power of the Lanczos coefficients concerning the system’s integrability or chaoticity, operator growth and the indicativeness of physical behavior. To this end, we analytically deter- mine the Lanczos coefficients for a Majorana observable in a strictly quadratic model and illustrate that within this model any structure of the Lanczos coefficients may be realised. Different to the other works, Ref. [P4] is not cast in the language of the recursion method. Focusing on the full eigenstate thermalization hypothesis, it explores the connection between thermalization dynamics of multi-point correlation functions and the scaling of thermal free cumulants in setups mimicking a system-bath situation.

•Jul 6, 2026•0 citations

Quasi-Lindblad pseudomode theory for open quantum systems

Gunhee Park, Zhen Huang, Yuanran Zhu +3 more

We introduce a new framework to study the dynamics of open quantum systems with linearly coupled Gaussian baths. Our approach replaces the continuous bath with an auxiliary discrete set of pseudomodes with dissipative dynamics, but we further relax the complete positivity requirement in the Lindblad master equation and formulate a quasi-Lindblad pseudomode theory. We show that this quasi-Lindblad pseudomode formulation directly leads to a representation of the bath correlation function in terms of a complex weighted sum of complex exponentials, an expansion that is known to be rapidly convergent in practice and thus leads to a compact set of pseudomodes. The pseudomode representation is not unique and can differ by a gauge choice. When the global dynamics can be simulated exactly, the system dynamics is unique and independent of the specific pseudomode representation. However, the gauge choice may affect the stability of the global dynamics, and we provide an analysis of why and when the global dynamics can retain stability despite losing positivity. We showcase the performance of this formulation across various spectral densities in both bosonic and fermionic problems, finding significant improvements over conventional pseudomode formulations.

arXiv:2408.15529•Physical review. B./Physical review. B•Nov 25, 2024•32 citations

Universal hypothesis of autocorrelation function from Krylov complexity

Ren Zhang, Hui Zhai

Abstract In a quantum many-body system, autocorrelation functions can determine linear responses nearby equilibrium and quantum dynamics far from equilibrium. In this letter, we bring out the connection between the operator complexity and the autocorrelation function. In particular, we focus on a particular kind of operator complexity called the Krylov complexity. We find that a set of Lanczos coefficients $\{b_{n}\}$ { b n } computed for determining the Krylov complexity can reveal the universal behaviors of autocorrelations, which are otherwise impossible. When the time axis is scaled by $b_{1}$ b 1 , different autocorrelation functions obey a universal function form at short time. We further propose a characteristic parameter deduced from $\{b_{n}\}$ { b n } that can largely determine the behavior of autocorrelations at the intermediate time. This parameter can also largely determine whether the autocorrelation function oscillates or monotonically decays in time. We present numerical evidences and physical intuitions to support these universal hypotheses of autocorrelations. We emphasize that these universal behaviors are held across different operators and different physical systems.

arXiv:2305.02356•Quantum Frontiers•Apr 11, 2024•14 citations

Universal properties of the many-body Lanczos algorithm at finite size

Luca Capizzi, Leonardo Mazza, Sara Murciano

We study the universal properties of the Lanczos algorithm applied to finite-size many-body quantum systems. Focusing on autocorrelation functions of local operators and on their infinite-time behaviour at finite size, we conjecture that in the large $n$ limit, the ratios between consecutive Lanczos coefficients should have specific scalings with the size of the lattice that we make precise and that depend on the hydrodynamic tail of the autocorrelation function. The scaling associated with strong or approximate zero-modes is also discussed. We support our conjecture with a numerical study of different models.

arXiv:2507.17424•Physical review. B./Physical review. B•Mar 26, 2026•2 citations
We present an expansion of a many-body correlation function in a sum of pseudomodes: exponents with complex frequencies that encompass both decay and oscillations. The pseudomode expansion emerges in the framework of the Heisenberg version of the recursion method. This method essentially solves Heisenberg equations in a Lanczos tridiagonal basis constructed in the Krylov space of a given observable. To obtain pseudomodes, we first add artificial dissipation satisfying the dissipative generalization of the universal operator growth hypothesis, and then take the limit of the vanishing dissipation strength. Fast convergence of the pseudomode expansion is facilitated by the localization in the Krylov space, which is generic in the presence of dissipation and can survive the limit of the vanishing dissipation strength. As an illustration, we present pseudomode expansions of infinite-temperature autocorrelation functions in the quantum Ising and XXXXXX spin-1/21/21/2 models on the square lattice. It turns out that it is enough to take a few first pseudomodes to obtain a good approximation to the correlation function.
We report an implementation of the recursion method that addresses quantum many-body dynamics in the nonperturbative regime. The method essentially amounts to constructing a Lanczos basis in the space of operators and solving coupled Heisenberg equations in this basis. The reported implementation has two key ingredients: a computer-algebraic routine for the symbolic calculation of nested commutators and a procedure to extrapolate the sequence of Lanczos coefficients according to the universal operator growth hypothesis. We apply the method to calculate infinite-temperature correlation functions for spin-1/21/21/2 systems on one- and two-dimensional lattices. In two dimensions the accessible timescale is large enough to essentially embrace the relaxation to equilibrium. The method allows one to accurately calculate transport coefficients. As an illustration, we compute the diffusion constant for the transverse-field Ising model on a square lattice.
The recursion method, which solves coupled Heisenberg equations in a Lanczos operator basis, has recently emerged as a powerful nonperturbative tool for computing dynamical correlation functions in strongly correlated two- and three-dimensional quantum many-body systems. Motivated by this success, we investigate whether the method can be extended to expectation values of observables following a quantum quench. We find that such an extension encounters an obstacle absent in the computation of dynamical correlation functions. The latter are fully determined by the Lanczos coefficients b_n b n , which in generic systems exhibit universal behavior, enabling reliable extrapolation from the first few dozens of explicitly computed coefficients. In contrast, quench dynamics additionally requires “quench coefficients” c_n c n , defined as overlaps of Lanczos basis operators with the initial state. We show that, unlike the Lanczos coefficients, the quench coefficients exhibit no universal structure and cannot be reliably extrapolated, thereby limiting the time up to which the method yields accurate results. The behavior of quench coefficients is highly state-dependent, ranging from decaying to irregular or even growing sequences; typically, the less regular the sequence c_n c n , the shorter the accessible timescale. Nevertheless, for favorable initial states, the method remains competitive with state-of-the-art approaches. Moreover, its symbolic implementation allows a single computation to be reused across different Hamiltonian parameters and initial states, making it particularly advantageous in studies requiring extensive scans over Hamiltonian parameters or initial states.
This cumulative doctoral thesis, based on the works [P1–P5], addresses various topical concepts as well as foundational questions in the nonequilibrium physics of quantum many-body systems. Publication [P1] concerns itself with the question of time’s arrow in quantum mechanics. Based on the recursion method, we attribute autocorrelation functions of few-body observables at infinite temperature (almost) monotonously decreasing functions, constructed from temporal derivatives of the correlation function that indicate a directed approach to equilibrium. We numerically find that in chaotic models, these constructions comprise a notion of simplicity of the underlying dynamics. Addressing the ubiquity of simple relaxation dynamics in otherwise complex chaotic systems, in Ref. [P3] we introduce a scheme based on Lanczos coefficients to approximate correlation functions in the thermodynamic limit. Examining several pertinent condensed-matter type models and benchmarking with state-of-the-art numerics, we find the method to yield good results already for a low number of included Lanczos coefficients, provided that the latter exhibit a smooth and increasing structure. We underpin these findings with analytical arguments on the method’s applicability. Ref. [P2] provides an estimate on the equilibration time of such infinite-temperature correlation functions in the thermodynamic limit. Similarly to Ref. [P3], we find the estimate to capture the equilibration times very reasonably, given that the corresponding Lanczos coefficients exhibit a smooth growth. Substantiated by analytical considerations, the convergence of the estimate implies that equilibration timescales of the relaxation processes are finite and conceivably short. In Ref. [P5] we scrutinize the interpretative power of the Lanczos coefficients concerning the system’s integrability or chaoticity, operator growth and the indicativeness of physical behavior. To this end, we analytically deter- mine the Lanczos coefficients for a Majorana observable in a strictly quadratic model and illustrate that within this model any structure of the Lanczos coefficients may be realised. Different to the other works, Ref. [P4] is not cast in the language of the recursion method. Focusing on the full eigenstate thermalization hypothesis, it explores the connection between thermalization dynamics of multi-point correlation functions and the scaling of thermal free cumulants in setups mimicking a system-bath situation.
We introduce a new framework to study the dynamics of open quantum systems with linearly coupled Gaussian baths. Our approach replaces the continuous bath with an auxiliary discrete set of pseudomodes with dissipative dynamics, but we further relax the complete positivity requirement in the Lindblad master equation and formulate a quasi-Lindblad pseudomode theory. We show that this quasi-Lindblad pseudomode formulation directly leads to a representation of the bath correlation function in terms of a complex weighted sum of complex exponentials, an expansion that is known to be rapidly convergent in practice and thus leads to a compact set of pseudomodes. The pseudomode representation is not unique and can differ by a gauge choice. When the global dynamics can be simulated exactly, the system dynamics is unique and independent of the specific pseudomode representation. However, the gauge choice may affect the stability of the global dynamics, and we provide an analysis of why and when the global dynamics can retain stability despite losing positivity. We showcase the performance of this formulation across various spectral densities in both bosonic and fermionic problems, finding significant improvements over conventional pseudomode formulations.
Abstract In a quantum many-body system, autocorrelation functions can determine linear responses nearby equilibrium and quantum dynamics far from equilibrium. In this letter, we bring out the connection between the operator complexity and the autocorrelation function. In particular, we focus on a particular kind of operator complexity called the Krylov complexity. We find that a set of Lanczos coefficients {bn}\{b_{n}\}{bn​} { b n } computed for determining the Krylov complexity can reveal the universal behaviors of autocorrelations, which are otherwise impossible. When the time axis is scaled by b1b_{1}b1​ b 1 , different autocorrelation functions obey a universal function form at short time. We further propose a characteristic parameter deduced from {bn}\{b_{n}\}{bn​} { b n } that can largely determine the behavior of autocorrelations at the intermediate time. This parameter can also largely determine whether the autocorrelation function oscillates or monotonically decays in time. We present numerical evidences and physical intuitions to support these universal hypotheses of autocorrelations. We emphasize that these universal behaviors are held across different operators and different physical systems.
We study the universal properties of the Lanczos algorithm applied to finite-size many-body quantum systems. Focusing on autocorrelation functions of local operators and on their infinite-time behaviour at finite size, we conjecture that in the large nnn limit, the ratios between consecutive Lanczos coefficients should have specific scalings with the size of the lattice that we make precise and that depend on the hydrodynamic tail of the autocorrelation function. The scaling associated with strong or approximate zero-modes is also discussed. We support our conjecture with a numerical study of different models.