Dynamics and decoherence in the central spin model in the low-field limit

We present a combination of analytic calculations and a powerful numerical method for large spin baths in the low-field limit. The hyperfine interaction between the central spin and the bath is fully captured by the density matrix renormalization group. The adoption of the density matrix renormalization group for the central spin model is presented and a proper method for calculating the real-time evolution at infinite temperature is identified. In addition, we study to which extent a semiclassical model, where a quantum spin-1/2 interacts with a bath of classical Gaussian fluctuations, can capture the physics of the central spin model. The model is treated by average Hamiltonian theory and by numerical simulation.

Figure 1

Exemplary realization of the DMRG algorithm for the central spin model with 10 bath spins. For clarity, we refrain from showing the central spin as a part of the environment. The dashed and solid lines mark the interaction between the central spin and the system and the environment block, respectively.

Figure 2

Exemplary realization of the DMRG algorithm for the central spin model for a purified system. The real sites are represented by solid dots, the auxiliary ones by circles. The dashed and solid lines mark the interaction between the central site and the system and the environment block, respectively. In the lower setup, the central site is purified as well.

Figure 3

Time evolution of the central spin interacting with a bath of 19 spins (upper panel) for 1024 states. The inset is a magnification of the autocorrelation function for $t\ge 15$. Both the result for the Krylov and Chebychev method deviate from the Trotter-Suzuki result for $t>20$. The deviation $\Delta S\left(t\right)=|S_{\mathrm{a}}\left(t\right)-S_{\mathrm{b}}\left(t\right)|$ is shown for selected pairs of methods $\mathrm{a}$ and $\mathrm{b}$ in the lower panel.

Figure 4

Truncation error for the real-time evolution shown in Fig. 3. For the Trotter-Suzuki and Krylov approach, the truncation error is plotted as a function of time $t$ (lower $x$ axis), while for the Chebychev approach it is plotted in dependence of the number of contributing polynomials (upper $x$ axis). The vertical dashed line marks the number of required Chebychev polynomials up to $t=20$. While the truncation error for the Krylov and the Trotter-Suzuki method is still acceptable at $t=20$, it is too large for the Chebychev polynomials.

Figure 5

Autocorrelation function of the bath spins for various system sizes calculated with tDMRG. With increasing number of bath spins, the fluctuations become more and more static. Note the scale of the $y$ axis in the lower panel.

Figure 6

Random noise simulation and first-order AHT (40) result for the autocorrelation function of the central spin in the semiclassical model (28). The correlation of the random noise is constant.

Figure 7

Comparison between the tDMRG result for 49 bath spins, the random noise simulation, and the AHT result in first and second order. The AHT and the random noise result have both been calculated with the tDMRG autocorrelation function (tDMRG AC) and a constant autocorrelation function (constant AC) as input. The AHT result is fairly robust. The plateau of $1/12$ in the autocorrelation of the central spin occurs in the random noise simulation only when the bath fluctuations are completely frozen. There is no noticeable difference between the AHT in first and second order.

Figure 8

Convergence of the tDMRG calculation and the random noise simulation towards the AHT result for large system sizes. For the random noise simulation, the autocorrelation functions obtained by tDMRG are used to sample the fluctuations.

Figure 9

AHT in first and second order for the exemplary correlation function $g\left(t\right)=1/4e^{-\left|t\right|/\left(8\tau \right)}$.