NESSi 2.0: The Non-Equilibrium Systems Simulation package version 2.0

arXiv:2605.02606 · cond-mat.str-el · Submitted 2026-05-04 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "NESSi 2.0: The Non-Equilibrium Systems Simulation package version 2.0".

Mira: Detailed Research Summary of NESSi 2.0: Extension to Memory-Truncated and Steady-State Kadanoff-Baym Equations This research details the development of NESSi 2.0,

Kai: First, who's behind it and why it matters.

Title and authors: Kai: So we're looking at this paper called "NESSi two point zero: The Non-Equilibrium Systems Simulation package version two point zero" <ref:2605.02606#pg1,NESSi 2.0: The Non-Equilibrium Systems Simulation package version 2.0>. It’s written by Fabian Künzela, Michael Schülerd, Denis Goleže, Yuta Murakamig, Sujay Rayb, Christopher Stahlj, Jiajun Lij, Hugo U. R. Strandi, Philipp Wernerb and Martin Eckstein >

Mira: Yeah it’s a major update to the NESSi package we've been using for studying quantum many-body dynamics out of equilibrium >

Kai: Exactly. The title tells you right away this is about extending the simulation capabilities, specifically focusing on nonequilibrium steady states and memory truncation techniques >

Lev: I always check when they change the core methods because that’s where the real computational headache lives for us trying to run things on actual hardware >

Mira: And this paper seems to be tackling those scaling issues head-on by bringing in two big things: memory truncation of Kadanoff-Baym equations and explicit tools for nonequilibrium steady states >

Kai: So, what does that actually mean in practice for someone who is trying to simulate something long-time, like laser induced dynamics in solids? >

Lev: It means they’re trying to get past those memory requirements that scale as O(Nt two) and computational costs that are O(N cubed t) when you use the standard Nonequilibrium Green's Functions approach > <ref:2605.02606#pg2>

Mira: Right. They are moving from a method that was hard to run for long times to something that should be much better suited for studying those slow, evolving dynamics >

Kai: It sounds like they’re trying to make simulations that can actually go much further in time than before, which is what we need when looking at processes happening on the pico-second scale >

Lev: If you can push it to longer times, you can see if those prethermal states they mentioned actually develop the features we expect from physics >

The paper's summary: Mira: Okay, so digging into the summary of "NESSi two point zero: The Non-Equilibrium Systems Simulation package version two point zero", the main point is that they’ve introduced techniques derived from memory-truncated Kadanoff-Baym equations and new robust functionalities for describing nonequilibrium steady states > <ref:2605.02606#pg1,NESSi 2.0: The Non-Equilibrium Systems Simulation package version 2.0>

Kai: That's what I mean by memory truncation. They are cutting down the memory integrals in those KBEs to a maximum depth of N c timesteps >

Mira: Which they say reduces the computational complexity from O(N cubed t) down to O(Nt N squared c) and the memory requirement scales as O(N squared c) where c is related to that memory depth N c > <ref:2605.02606#pg2>

Lev: That reduction in cost is what makes it feasible for practical work, but we gotta be careful because they mention you have to verify convergence with respect to that cutoff parameter N c >

Kai: So, the practical implication here is that they claim this truncation allows simulations to extend significantly into longer time regimes, even reaching times like t max = thirty thousand timesteps in one specific example using N c = twenty-four >

Mira: And they introduced a novel data structure called the "moving Green’s function window" which stores Green’s functions over a two-time window up to timestep n but with that strictly restricted memory depth of N c timesteps >

Lev: That moving window idea sounds clever for maintaining a constant computational footprint while simulating real time evolution >

Kai: So, the workflow they describe is running an initial real-time simulation up to time N c, and then continuing with a memory-truncated simulation for the rest of the time range >

The paper's improvements: Lev: Let’s talk about the specific improvements they detail in "NESSi two point zero: The Non-Equilibrium Systems Simulation package version two point zero" <ref:2605.02606#pg1,NESSi 2.0: The Non-Equilibrium Systems Simulation package version 2.0>. They are focusing on two main areas: memory truncation and NESS functionalities >

Mira: First is the memory truncation of Kadanoff-Baym equations which we just talked about, but it’s not just about speed; it’s enabling these longer simulations >

Kai: And I want to bring up the data structure innovation again, because that "moving Green’s function window" is key for keeping the cost low at each step >

Lev: From an error-correction side, this means if we were to run this on real hardware, the overhead of managing that restricted memory depth N c would be manageable compared to the original scaling issues >

Mira: Second improvement is introducing explicit functionalities for Nonequilibrium Steady States or NESS which they define as time-translationally invariant nonequilibrium states relevant for transport and as approximate descriptions of slowly evolving prethermal states >

Kai: That’s really important because it means we can use the real-time data from the truncated equations to initialize NESS Green's functions, allowing us to simulate steady states directly >

Lev: So, instead of relying on complex imaginary-time propagation just to get a steady state, they are giving us a pathway using the results from their dynamic evolution >

Mira: Plus, they provide routines for solving Dyson equations and elementary diagrammatic expressions within the NESS formalism expanding its applicability to transport problems >

Conclusion: Kai: So wrapping up on "NESSi two point zero: The Non-Equilibrium Systems Simulation package version two point zero", the main implication is that they’ve managed to reduce computational and memory cost while simultaneously enabling the direct simulation of nonequilibrium steady states > <ref:2605.02606#pg1,NESSi 2.0: The Non-Equilibrium Systems Simulation package version 2.0>

Mira: It’s a dual capability that lets us study both fast dynamics and those slower, prethermal states relevant for transport phenomena in condensed matter systems >

Lev: From an error-correction point of view, this means we can test how well these approximations hold up when we try to run them on real hardware and see where the computational overhead really lands >

Kai: So the paper shows that this method works and you get quantitative convergence analysis showing that the NESS simulation and the real-time simulation converge to the same result with an error scaling of O(h two) in certain regimes > <ref:2605.02606#pg1>

Mira: That O(h two) scaling is reassuring because it means our long-time approximations are accurate, as we expect from trapezoidal evaluation of convolution integrals > <ref:2605.02606#pg1>

Lev: So, yeah, the authors state their limitation is that they need to verify convergence with respect to the cutoff parameter N c and that that’s a necessary check before trusting those extended simulations >

Kai: That’s a fair caveat, but overall this package seems like a powerful tool for studying complex quantum dynamics over long time scales >

Mira: So we have this paper, "NESSi two point zero: The Non-Equilibrium Systems Simulation package version two point zero", which bridges the gap between real-time evolution and steady-state analysis > <ref:2605.02606#pg1,NESSi 2.0: The Non-Equilibrium Systems Simulation package version 2.0>

Institute of Theoretical Physics, University of Hamburg · Department of Physics, University of Fribourg · The Hamburg Centre for Ultrafast Imaging, Hamburg · PSI Center for Scientific Computing, Theory and Data, Paul Scherrer Institute · Jožef Stefan Institute

cond-mat.str-el

Submitted: 2026-05-04

Updated: 2026-05-04

Comments: 45 pages, 11 figures

DOI: 10.1016/j.cpc.2026.110456

Code: https://github.com/nessi-cntr/nessi

Project page: https://nessi.readthedocs.io/en/latest

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 92/100

The gist: Detailed Research Summary of NESSi 2.0: Extension to Memory-Truncated and Steady-State Kadanoff-Baym Equations This research details the development of NESSi 2.0, a significant extension of the

Key concepts

Memory Truncation of KBEs: This technique limits the memory depth of the Kadanoff-Baym equations to a fixed number of timesteps ($N_c$). This reduces computational cost from $O(N^3 t)$ to $O(N_t N^2 c)$, enabling simulations over much longer time scales by managing memory efficiently.
Nonequilibrium Steady State (NESS): NESS refers to a state where the system is time-translationally invariant even though it is not in thermal equilibrium. NESSi 2.0 allows direct simulation of these states, which are crucial for understanding transport phenomena and quasi-steady descriptions of slowly evolving quantum systems.
Moving Green’s Function Window: A novel data structure that stores Green's functions over a two-time window up to timestep $n$, but with a strict memory limit ($N_c$). This window shifts forward during the simulation, ensuring constant computational cost and memory usage while tracking real-time evolution.

Summary

Detailed Research Summary of NESSi 2.0: Extension to Memory-Truncated and Steady-State Kadanoff-Baym Equations

This research details the development of NESSi 2.0, a significant extension of the existing Non-Equilibrium Systems Simulation (NESSi) package, designed specifically to address the computational bottlenecks encountered when simulating long-time and slow nonequilibrium quantum many-body dynamics, such as those arising from laser-induced processes in solids. The core innovation lies in incorporating techniques derived from memory-truncated Kadanoff-Baym equations (KBEs) and introducing robust functionalities for describing Nonequilibrium Steady States (NESS).

Addressing Computational Bottlenecks in NESSi 1.0

The original NESSi package, which utilizes Nonequilibrium Green’s Functions (NEGF) to study quantum many-body dynamics, faces severe scaling limitations for extended time simulations:

Computational Cost: The complexity scales as O(N cubed t) with respect to the number of timesteps (N t).
Memory Requirement: The memory scales as O(N squared t).

NESSi 2.0 is engineered to mitigate these issues by focusing on reducing both computational and memory overhead while enabling simulations over substantially longer time scales.

Key Enhancements in NESSi 2.0

The version 2.0 introduces two primary, interconnected advancements: memory truncation for real-time evolution and explicit functionalities for steady-state analysis.

Memory Truncation of Kadanoff-Baym Equations (KBEs)

To overcome the O(N cubed t) scaling, NESSi 2.0 implements a technique that truncates the memory integrals within the KBE formalism to a maximum depth of N c timesteps.

Complexity Reduction: This truncation reduces the computational complexity to ** O(N t N squared c) ** and the memory requirement to ** O(N squared c) **, where c is related to the memory depth N c.
Extended Simulation Times: Crucially, this truncation allows simulations to extend significantly into longer time regimes. The paper notes that convergence with respect to the cutoff parameter N c must be verified, but this method proves effective for reaching times such as t max = 30000 timesteps in a specific example (with N c=24).
Data Structure Innovation: A novel data structure, the moving Green’s function window, is introduced. This structure stores Green’s functions over a two-time window up to timestep n, but with a strictly restricted memory depth of N c timesteps. During real-time simulation, this window shifts forward in the timestepping procedure to maintain constant computational cost and memory footprint at each step.
Simulation Workflow: The process involves an initial real-time simulation (NESSi 1.0) up to time N c, followed by a memory-truncated simulation for t c+1,, t max. This requires careful initialization, where the starting timestep must satisfy t c at most N t and the solving order must be sufficient to utilize Volterra Integrators of order SolveOrder.

Nonequilibrium Steady State (NESS) Functionalities

NESSi 2.0 introduces explicit functionalities to describe Nonequilibrium Steady States (NESS), which are defined as time-translationally invariant nonequilibrium states relevant for transport phenomena and as approximate descriptions of slowly evolving (prethermal) nonequilibrium states.

Integration: NESS functions integrate seamlessly with the real-time structures. Data from real-time or memory-truncated Green’s functions can be used to initialize NESS Green’s functions, allowing for the direct simulation of steady states using information derived from dynamic evolution.
Formal Capabilities: The package provides routines to solve Dyson equations and elementary diagrammatic expressions within the NESS formalism, expanding its applicability to problems concerning transport and quasi-steady descriptions of slowly evolving states.

Theoretical Framework and Validation

The underlying mathematical approach involves solving the equations of motion for time-dependent Green’s functions on the Kadanoff-Baym contour, utilizing memory truncation in both the integral kernel and Fourier integrals in steady state evaluations. The solution method employed is based on higher-order solution methods for integral and integro-differential equations.

The convergence of these results is rigorously analyzed by comparing the steady-state result against a real-time benchmark (G cntr) using a specific error metric (epsilon R,<), confirming an error decrease proportional to O(h 2), consistent with the trapezoidal evaluation of convolution integrals. Furthermore, analysis confirms that prethermal states—reached after times of order one (few inverse hoppings)—retain characteristic features, such as a pronounced step in the momentum distribution at the Fermi energy (epsilon k = 0.05 vs. epsilon kF = 0.5).

Conclusion and Impact

NESSi 2.0 successfully achieves its stated goals: reducing computational and memory cost to extend simulation times while simultaneously enabling the direct simulation of nonequilibrium steady states. This dual capability makes it a powerful tool for studying complex, long-time quantum dynamics in condensed matter systems. The package is available in C++ and Python, emphasizing its accessibility for custom program implementation. Beyond these core extensions, the development team indicates ongoing work on future additions, including a software package tailored for nonequilibrium dynamical mean-field theory calculations based on strong-coupling impurity solvers.

Improvements for AI systems

Bold header: Memory-Truncated KBE Simulation Platform

This platform allows for simulating quantum many-body dynamics over significantly longer temporal ranges by reducing computational complexity from O(Nt3t) to O(NtN2c). This enables the simulation of picosecond dynamics of collective modes and slowly evolving (prethermal) nonequilibrium states which were previously computationally prohibitive.

Bold header: Direct Nonequilibrium Steady State (NESS) Simulation

The system can directly simulate time-translationally invariant nonequilibrium states by introducing a new data structure for NESS Green’s functions, allowing the approximation of slowly evolving (prethermal) nonequilibrium states relevant for transport settings.

Bold header: Bridging Real-Time and Steady-State Dynamics

The system can initialize time evolution with the result of a steady-state simulation, specifically by using the result of the steady-state simulation in ness2 bethe.x, and then use the truncated KBEs to further propagate the solution in time. This circumvents initialization via imaginary-time propagation.

Bold header: Momentum Occupation Analysis

The system can calculate momentum occupation, defined as "⟨c†kσ(t)ckσ(t)⟩ = −iG<k(t, t), allowing for the extraction of dynamics such as the two-staged dynamics characterized by a fast prethermalization and a slow thermalization."

Bold header: High-Accuracy Convergence Benchmarking

The system provides quantitative convergence analysis showing that the NESS simulation and the real-time simulation converge to the same result, with error scaling analyzed as O(h2) in certain regimes, allowing researchers to verify the accuracy of their long-time approximations.

Sources

Solving quantum impurity models in the non-equilibrium steady state with tensor trains