GKBA — Generalized Kadanoff-Baym Ansatz for quantum transport

Motivation

Understanding spin and charge transport in nanoscale magnetic devices requires solving the real-time quantum dynamics of electrons in an open system: a small correlated region (the central system) connected to macroscopic metallic leads held at equilibrium. The leads drive the system out of equilibrium and act as sources and sinks of electrons and angular momentum.

The exact solution — the two-time Kadanoff-Baym equations (KBE) — scales as $\mathcal{O}(N_t^2)$ in memory and time, making long simulations prohibitively expensive. The Generalized Kadanoff-Baym Ansatz (GKBA) reconstructs the two-time Green function from its equal-time limit, reducing the problem to a set of coupled ODEs that scale as $\mathcal{O}(N_t)$. This makes it possible to simulate spin-transfer torque, spin pumping, and non-equilibrium magnetization dynamics over physically relevant timescales.

Goals

This repository unifies a collection of research notebooks into a reusable Julia module. The core physics is implemented once in src/ and each simulation scenario (geometry, driving protocol, approximation level) becomes a thin driver script in scripts/. Concretely:

• Provide a clean, tested implementation of the GKBA and its extensions for tight-binding systems with s-d coupled local moments.
• Support multiple lead descriptions: k-space embedding (PRB 107, 155141), position-space leads, and the wide-band limit with Ozaki pole expansion.
• Enable coupled electron–spin dynamics (GKBA + Landau-Lifshitz-Gilbert) for studying spin-transfer torque and spin pumping.

System

The central region is a 1D tight-binding chain of $n_x \times n_y$ sites with Rashba spin-orbit coupling $\gamma_{so}$ and local magnetic moments $\mathbf{m}i$ coupled to the electron spin via s-d exchange $j{sd}$:

$$H_s = -\gamma \sum_{\langle ij \rangle} c^\dagger_i c_j + \gamma_{so}(\ldots) - j_{sd} \sum_i \mathbf{m}_i \cdot \hat{\boldsymbol{\sigma}}_i$$

Two semi-infinite leads (modeled as 1D tight-binding chains with hopping $\gamma$ and system-lead coupling $\gamma_c$) are attached to the first and last sites.

---

Julia module for real-time quantum transport simulations of open quantum systems coupled to fermionic leads. Implements several variants of the GKBA and solves the equations of motion for the lesser Green function $G^{<}(t)$.

Physics

The central quantity is the lesser Green function of the system $G^{&lt;}_s(t)$, which encodes the non-equilibrium charge and spin density matrix. The equations of motion couple $G^{<}s$ to the system–lead Green function $G^{<}{e\alpha s}(k,t)$ via the GKBA:

$$i,\partial_t G^{&lt;}_{e\alpha s}(k) = h_{e\alpha}(k),G^{&lt;}_{e\alpha s}(k) - G^{&lt;}_{e\alpha s}(k),h_s + h_{e\alpha s}(k),G^{&lt;}_s - g^{&lt;}_\alpha(k),h_{e\alpha s}(k)$$

$$i,\partial_t G^{&lt;}_s = \left[h_s,, G^{&lt;}_s\right] - \sum_{\alpha,k} \left(h_{se\alpha}(k),G^{&lt;}_{e\alpha s}(k) + \mathrm{h.c.}\right)$$

The first equation holds for each lead mode $k$ independently. $G^{<}{e\alpha s}(k)$ is a row vector in system space — $h{e\alpha}(k)$ (a scalar in k-rep) acts on the lead index while $h_s$ acts on the system index. All equations use natural units with $\hbar = 1$.

Dynamics types

Type	Lead representation	Lead GF
GKBADynamics	k-space (diagonal)	static, Fermi-Dirac
eGKBADynamics	k-space	dynamic (PRL 130, 246301)
PosRepDynamics	position space (full matrix)	static
ePosRepDynamics	position space	dynamic
WBLDynamics	wide-band limit + Ozaki poles	analytic

The WBL dynamics uses the approximation $\Sigma^r = -i\Gamma/2$ (constant self-energy) and expands the Fermi function in Ozaki poles, reducing the problem to a set of auxiliary functions $F_{k\alpha}(t)$ without $k$-space integration.

Observables

compute_observables! fills an ObservablesVar with:

• curr_α — charge current $J_\alpha$ per lead
• scurr_xα — spin current $J^x_\alpha$ (x,y,z) per lead
• sden_i1x — site-resolved spin density $\langle S^x_i \rangle$
• cden_i — charge density $\langle n_i \rangle$ per orbital

Structure

src/
  GKBA.jl          module entry point and exports
  constants.jl     physical constants (ħ=1, k_B, μ_B, …)
  types.jl         dynamics and observables structs
  hamiltonians.jl  build_hs, build_heα, build_hseα, ozaki_poles, …
  eom.jl           equations of motion + unpack!
  observables.jl   compute_observables!
  init.jl          init_gkba, init_egkba, init_gkba_wbl, …
  llg.jl           Landau-Lifshitz-Gilbert spin dynamics

scripts/           one driver script per simulation variant (see table below)
test/
  runtests.jl      18 unit tests (init, EOM step, conservation laws)
  benchmark.jl     EOM vs direct reference + physical invariant checks

Scripts

Script	Method	Notes
run_gkba_krep_static.jl	GKBA k-rep	static spins
run_gkba_krep_free.jl	GKBA k-rep	free precessing spins
run_gkba_krep_prece.jl	GKBA k-rep	driven precession
run_gkba_krep_prece_single_lead.jl	GKBA k-rep	1 lead
run_gkba_krep_leviton.jl	GKBA k-rep	Leviton pulse
run_egkba_krep_static.jl	eGKBA k-rep	static spins
run_egkba_krep_free.jl	eGKBA k-rep	free precession
run_egkba_krep_prece.jl	eGKBA k-rep	driven precession
run_egkba_krep_prece_extra_bath.jl	eGKBA k-rep	extra bath lead
run_gkba_posrep_static.jl	GKBA pos-rep	nx=5 chain
run_gkba_posrep_large_static.jl	GKBA pos-rep	nx=2, large lead
run_egkba_posrep_static.jl	eGKBA pos-rep	static spins
run_egkba_posrep_prece.jl	eGKBA pos-rep	driven precession
run_egkba_posrep_prece_single_lead.jl	eGKBA pos-rep	1 lead
run_gkba_wbl_free.jl	GKBA WBL	free precession
run_gkba_wbl_prece.jl	GKBA WBL	driven precession

Usage

using DifferentialEquations
import GKBA: init_gkba_wbl, eom_gkba_wbl!, compute_observables!, unpack!

dv, ov = init_gkba_wbl(; nx=2, ny=1, nk=20, γ=1.0, γso=0.0,
                          γc=1.0, j_sd=0.1, Temp=300.0,
                          vm_i1x = vm_init)

prob       = ODEProblem(eom_gkba_wbl!, dv.rkvec, (0.0, 60.0), dv)
integrator = init(prob, RK4(); dt=0.1, adaptive=true, save_everystep=false)

for t in 0.0:0.1:59.9
    step!(integrator, 0.1, true)
    unpack!(dv, integrator.u)
    compute_observables!(ov, dv)
    # access ov.curr_α, ov.sden_i1x, …
end

References

• PRB 107, 155141 — Core theory: GKBA with embedding for leads with internal structure (basis for k-rep and pos-rep dynamics).
• PRL 130, 246301 — Extended GKBA: time-linear transport with dynamical lead correlators (basis for eGKBADynamics and ePosRepDynamics).

Reference solvers

KBE-Test2-prece.ipynb — full two-time Kadanoff-Baym equations via KadanoffBaym.jl. Useful as an exact reference to benchmark GKBA accuracy.

Krylov.ipynb — exact non-interacting evolution via matrix exponentiation (Expokit.jl). Propagates the single-particle correlation matrix $$C(t+\delta t) = e^{-iH,\delta t},C(t),e^{iH,\delta t}$$

Running the tests

julia --project test/runtests.jl
julia --project test/benchmark.jl