Accessing long timescales in the relaxation dynamics of spins coupled to a conduction-electron system using absorbing boundary conditions

The relaxation time of a classical spin interacting with a large conduction-electron system is computed for a weak magnetic field, which initially drives the spin out of equilibrium. We trace the spin and the conduction-electron dynamics on a timescale which exceeds the characteristic electronic scale that is set by the inverse nearest-neighbor hopping by more than five orders of magnitude. This is achieved with a construction of absorbing boundary conditions, which employs a generalized Lindblad master-equation approach to couple the edge sites of the conduction-electron tight-binding model to an external bath. The failure of the standard Lindblad approach to absorbing boundaries is traced back to artificial excitations initially generated due to the coupling to the bath. This can be cured by introducing Lindblad parameter matrices and by fixing those matrices to perfectly suppress initial-state artifacts as well as reflections of physical excitations propagating to the system boundaries. Numerical results are presented and discussed for generic one-dimensional models of the electronic structure.

Figure 1

Relaxation of a single spin or a few spins interacting with a large conduction-electron system after an initial local excitation. In the long-time limit, the spin-electron system is expected to reach its ground state locally, i.e., in the vicinity of the impurity spin(s), since the excitation energy is completely dissipated to the bulk.

Figure 2

Sketch of the system geometry: A classical spin $\mathit{S}$ of length $\left|\mathit{S}\right|=\frac{1}{2}$ is coupled via a local antiferromagnetic exchange interaction $J$ to a noninteracting system of electrons on a chain of $L$ sites. The hopping between nearest-neighboring sites is $-T$. $L_B$ sites on the left and $L_B$ sites on the right edge are coupled to a bath. The spin is located at the chain center and subjected to a local magnetic field $\mathit{B}$. Suddenly flipping the field direction induces the real-time dynamics.

Figure 3

Time evolution of the $z$ and the $x$ component of the classical spin coupled to an electron system with NN hopping $-T$ at half filling after a sudden switch of the local magnetic field from the $x$ to $z$ direction (see text for details). Red/orange lines: Standard theory for a chain with open boundary conditions (open BCs) with $L=1001$ sites ($i_0=501$, $J=1$, $B=1$). Green/blue lines: Calculation with absorbing boundaries (absorbing BCs) [Eqs. (2), (14), (15), and (17)] for $L=47$ ($i_0=24$, $J=1$, $B=1$, $L_B=5$, ${\mathrm{\Gamma }}_{\text{min}}=0.2$). Energy and timescales set by $T=1$, $\hslash =1$.

Figure 4

Initial one-particle reduced density matrix at time $t=0$ for a system with $L=47$ sites, open BCs, and the impurity spin at the central site $i_0=(L+1)/2$ pointing in the $x$ direction. The color coding is indicated by the bar on the right side. Exchange coupling $J=1$. We display the elements ${\rho }_{\alpha \beta }$ of $\mathit{\rho }$ using the combined site-spin (“orbital”) index $\alpha \equiv 2i-\frac{1}{2}(1+z_{\sigma })=1,...,2L$ with $z_{\uparrow }=+1$, $z_{\downarrow }=-1$.

Figure 5

Time dependence of the one-particle reduced density matrix for a system of $L=47$ sites. The color code (see bottom) quantifies the real part of the difference to the initial density matrix, $\text{Re}\left[\mathit{\rho }\right(t)-\mathit{\rho }(0\left)\right]$, at selected instants of time (see the time labels at the top). Representation of the elements ${\rho }_{\alpha \beta }$ as in Fig. 4 using the orbital index $\alpha =2i-\frac{1}{2}(1+z_{\sigma })=1,...,2L$. Middle panel: System with open BCs. Upper panel: Same system but with absorbing BCs based on the standard Lindblad approach [Eqs. (14), (15), and (17)]. Lower panel: Same system but with modified absorbing BCs (see text). Other parameters as in Figs. 3 or 6, respectively.

Figure 6

Time evolution of the $z$ and the $x$ component of the classical spin as in Fig. 3 but here the results of the standard theory (red/orange lines) for open BCs and $L=1001$ are compared to those obtained for $L=47$ sites (green/blue lines) with modified BCs (linear profile and ${\gamma }_{\text{min}}=0.2$). Other parameters as in Fig. 3.

Figure 7

Relaxation time $\tau$ as a function of $1/B$. Calculations for $i_0=1$ (spin couples to the “left” edge), $L=46$, $J=1$, and modified BCs for the “right” edge (linear profile, ${\gamma }_{\min }=0.2$, $L_B=5$).

Figure 8

Relaxation time $\tau$ as function of $1/B$ as in Fig. 7 but for an insulator [see Eq. (24)] and for different values of the on-site potential $\varepsilon$ as indicated.

Figure 9

Relaxation time $\tau$ as function of the system size $L=6$, 16, 26, 36, 46, 56, 66 and $L_B=5=\text{const}$ as in Fig. 7 for $B=0.001$.