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+ + + + +Journal of Open Source Software +JOSS + +2475-9066 + +Open Journals + + + +6263 +10.21105/joss.06263 + +SpiDy.jl: open-source Julia package for the study of +non-Markovian stochastic dynamics + + + +https://orcid.org/0000-0002-8133-1551 + +Scali +Stefano + + +* + + +https://orcid.org/0000-0001-7242-7941 + +Horsley +Simon + + + + +https://orcid.org/0000-0002-9791-0363 + +Anders +Janet + + + + + +https://orcid.org/0000-0003-2961-739X + +Cerisola +Federico + + + + + + +Department of Physics and Astronomy, University of Exeter, +Exeter EX4 4QL, United Kingdom + + + + +Institute of Physics and Astronomy, University of Potsdam, +14476 Potsdam, Germany + + + + +Department of Engineering Science, University of Oxford, +Oxford OX1 3PJ, United Kingdom. + + + + +* E-mail: + + +4 +10 +2023 + +9 +97 +6263 + +Authors of papers retain copyright and release the +work under a Creative Commons Attribution 4.0 International License (CC +BY 4.0) +2022 +The article authors + +Authors of papers retain copyright and release the work under +a Creative Commons Attribution 4.0 International License (CC BY +4.0) + + + +Julia +spin-boson model +harmonic oscillator +stochastic field +non-Markovian memory +colored noise +anisotropic coupling +magnetism + + + + + + Summary +

SpiDy.jl solves the non-Markovian stochastic dynamics of + interacting classical spin vectors and harmonic oscillator networks in + contact with a dissipative environment. The methods implemented allow + the user to include arbitrary memory effects and colored quantum noise + spectra. In this way, SpiDy.jl provides key tools for the simulation + of classical and quantum open systems including non-Markovian effects + and arbitrarily strong coupling to the environment. Among the wide + range of applications, some examples range from atomistic spin + dynamics to ultrafast magnetism and the study of anisotropic + materials. We provide the user with Julia notebooks to guide them + through the various mathematical methods and help them quickly set up + complex simulations.

+ + + Statement of need +

The problem of simulating the dynamics of interacting rotating + bodies and harmonic oscillator networks in the presence of a + dissipative environment can find a vast range of applications in the + modeling of physical systems. This task is rendered particularly + challenging when one desires to capture the non-Markovian effects that + arise in the dynamics due to strong coupling with the environment. + SpiDy.jl is a library that allows the user to efficiently simulate + these systems to obtain both detailed dynamics and steady state + properties.

+

A relevant example of the applicability of SpiDy.jl is the modeling + of spins at low temperatures and at short timescales, which is a + fundamental task to address many open questions in the field of + magnetism and magnetic material modeling + (Halilov + et al., 1998). State-of-the-art tools such as those developed + for atomistic spin dynamics simulations are based on solving the + Landau–Lifshitz–Gilbert (LLG) equation + (Evans + et al., 2014). Despite their massive success, these tools run + into shortcomings in accurately modeling systems at low temperatures + and for short timescales where environment memory effects have been + observed + (Ciornei + et al., 2011; + Neeraj + et al., 2020). Recent work has focused on developing a + comprehensive quantum-thermodynamically consistent framework suitable + to model the dynamics of spins in magnetic materials while addressing + these shortcomings + (Anders + et al., 2022). This framework includes strong coupling effects + to the environment such as non-Markovian memory, colored noise, and + quantum-like fluctuations. At its core, SpiDy.jl implements the + theoretical framework introduced in Anders et al. + (2022), + allowing for the study of environment memory effects and anisotropic + system-environment coupling. SpiDy.jl can be readily adopted for + atomistic spin dynamics simulations + (Barker + & Bauer, 2019; + Evans + et al., 2014), ultrafast magnetism + (Beaurepaire + et al., 1996), and ferromagnetic and semiconductive systems + exhibiting anisotropic damping + (Chen + et al., 2018). A further set of applications stems from the + extension of SpiDy.jl to handle the non-Markovian stochastic dynamics + of harmonic oscillators. This model will be of interest in the field + of quantum thermodynamics where harmonic oscillators play a key role + in modeling open quantum systems. The package is written in pure Julia + and we take advantage of the efficient DifferentialEquations.jl + package + (Rackauckas + & Nie, 2017) by reducing evaluation redundancy, using + callbacks, and pre-allocations.

+

The software package has seen a wide range of applications to date. + Firstly, the convenience of three independent environments in SpiDy.jl + finds application in the microscopic modeling of spins affected by + noise due to vibrations of the material lattice + (Anders + et al., 2022). SpiDy.jl also found application in the + demonstration of the quantum-to-classical correspondence at all + coupling strengths between a spin and an external environment + (Cerisola + et al., 2024). Here, the temperature dependence of the spin + steady-state magnetization obtained with SpiDy.jl is successfully + compared with the classical mean force state of the system. Hartmann + et al. + (2023) + take advantage of the customizable coupling tensor in SpiDy.jl to + explore the anisotropic effects of the environment on the system. + Berritta et al. + (2023) + use SpiDy.jl as a sub-routine to build quantum-improved atomistic spin + dynamics simulations. In the paper, the authors take advantage of the + customizable power spectrum to implement ad-hoc simulations matching + known experimental results. Lastly, with an eye to the harmonic + oscillator side, SpiDy.jl is used to match the quantum harmonic + oscillator dynamics with its stochastic counterpart + (Hogg + et al., 2024). Here, the authors exploit the recent + implementation of harmonic oscillator dynamics.

+ + + Overview +

To model a system of interacting classical spin vectors, SpiDy.jl + solves the generalized stochastic LLG equation + (Anders + et al., 2022) + + d𝐒n(t)dt=12𝐒n(t)×[∑m≠nJn,m𝐒m(t)+𝐁+𝐛n(t)+∫t0tdt′Kn(t−t′)𝐒n(t′)],(1) + where + + 𝐒n(t) + represents the + + n-th + spin vector, the interaction matrix + + Jn,m + sets the interaction strength between the + + + n-th + and + + m-th + spins, + + 𝐁 + is the external field, which determines the natural precession + direction and frequency of the spins in the absence of interaction, + and + + bn(t) + is the time-dependent stochastic field induced by the environment. + Finally, the last integral term in Eq.(1) gives the spin dissipation + due to the environment, including non-Markovian effects accounted for + by the memory kernel matrix + + Kn(t). + In addition, SpiDy.jl also allows to study the stochastic dynamics of + coupled harmonic oscillator networks in a form similar to Eq.(1).

+

In conclusion, SpiDy.jl implements the stochastic dynamics of + coupled integro-differential equations to model systems of interacting + spins or harmonic oscillator networks subject to environment noise. + Among others, some of the key features of the package include:

+ + +

Coloured stochastic noise that satisfies the FDR and accounts + for both classical and quantum bath statistics.

+ + +

Simulation of non-Markovian system dynamics due to the memory + kernel + + Kn(t).

+ + +

Custom system-environment coupling tensors, allowing for + isotropic or anisotropic couplings. Both amplitudes and geometry + of the coupling can be specified.

+ + +

Choice between local environments, i.e. distinct baths acting + on the single sub-system, or a single common environment.

+ + +

We show an example run of the stochastic trajectories in the case + of a single spin in + [fig:dynamics]. + The code used to generate this example can be found in the GitHub + repository of SpiDy.jl.

+ +

Single-spin dynamics. Dynamics of the + + + x, + + + y, + and + + z + spin components. The components are normalized against the total + spin length + + S0 + and time axes are expressed in units of the Larmor frequency + + + ωL + ( + + ωL=|𝐁| + in Eq.(1)). We show an example set of 5 stochastic trajectories of + the spin dynamics (colored semi-transparent lines) together with + their stochastic average (gray solid line). Note that, while we show + only 5 trajectories for clarity, the average dynamics is obtained + from 10000 trajectories. We also represent the range of one standard + deviation from the average dynamics (gray-shaded area). In the + inset, we show the convergence of the same dynamics towards the + steady state at longer times. This example is obtained using the + Lorentzian parameters “set 1” obtained from Anders et al. + (2022). + The code used to generate the stochastic trajectories is included in + the GitHub repository. +

+ + + + + Acknowledgements +

SS and FC thank Marco Berritta and Charlie Hogg for insightful + suggestions for the implementation of the spin-spin coupling and the + harmonic oscillator dynamics. SARH and JA thank the Royal Society for + their support. SS is supported by a DTP grant from EPSRC + (EP/R513210/1). SARH acknowledges the Royal Society and TATA for + financial support through the Grant URFR 211033. JA and FC acknowledge + funding from EPSRC (EP/R045577/1). FC gratefully acknowledges funding + from the Foundational Questions Institute Fund (FQXi–IAF19-01).

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