Time-evolving matrix product operators for off-diagonal system-bath coupling

Abstract

Based on the process tensor framework, we extend the time-evolving matrix product operator (TEMPO) method to solve bosonic quantum impurity problems (QIPs) with off-diagonal system-bath coupling. Our method is a most generic extension of TEMPO, which applies for any QIPs as long as the bath is noninteracting and the system is linearly coupled to the bath. It naturally contains all the current developments of TEMPO in more restricted settings. As an application, we study the real-time dynamics of a spin that is coupled to a sub-ohmic bath via the Jaynes-Cummings-type system-bath coupling, and compare it against that of the standard spin-boson model. Our results show that the commonly used secular approximation could easily fail in presence of a structural bath. Our method provides a unified framework to understand different variants of TEMPO and directly suggests a fermionic generalization which has not been explored so far, it could also be straightforwardly used as an impurity solver in the bosonic dynamical mean field theory.

Figures (9)

• Figure 1: (a) Discretized time evolution of a system coupled to an environment under an evolutionary operator $e^{\mathcal{L} \delta t}$, starting from a separable initial state $\hat{\rho}_0=\hat{\rho}_S\otimes \hat{\rho}_E$. We have used double leg for the vertical (physical) indices as the Liouville (or more generally Lindblad) equation of motion is used, which evolves the density matrix instead of the pure state. The empty arcuate hat means the inner trace of the double leg. The regime within the dashed box defines the process tensor, which is the tensor network shown in (b), and can be equivalently viewed as an MPO in (c). (d) Trace of the PT, which is $1$ if $\hat{\rho}_0$ is properly normalized. (e) Calculating multi-time correlations based on the PT. In (d,e), we have used a thicker leg to represent the double leg for briefness.
• Figure 2: (a) Schematic illustration of the real-time evolution of the QIP from a separable system-bath initial state as a $1+1$D graph, where the bottom row means the initial state, the rows above mean the propagator $e^{\mathcal{L}_{\rm int} \delta t}$, the red square means the bare system propagator $e^{\mathcal{L}_S \delta t}$. The regime within the dashed box contains all the influence of the bath on the system dynamics, corresponding to that in Fig. \ref{['fig:pt']}(a), which is naturally an MPO as shown in (b).
• Figure 3: The exponential tensor renormalization group algorithm to build the Feynman-Vernon IF as an MPO. By choosing a step size $1/2^m$, the effective thermal state $e^{-\hat{H}_{\rm eff}}$ can be obtained using only $m$ MPO-MPO multiplications.
• Figure 4: (a) Converting a PT of $N$ time steps into an ADT of $N+1$ time steps by applying a 3D copy tensor on each pair of neighbouring site tensors. (b) Calculating the partition function based on the ADT, where the boundaries can be treated in the same way to the case of PT, while the middle indices are simply summed over, indicated by the empty triangular hats. Here we have used thick solid lines for the double legs for briefness.
• Figure 5: Real-time evolution of the JC model with (a) $\lambda^2=0.1$ and (b) $\lambda^2=0.5$. The red dashed lines are extended TEMPO results calculated with $\chi=30$ and $\delta t=0.05$, the solid lines with the same color are the corresponding ED results. The green solid lines are the ED results for the corresponding Rabi model. The other parameters we have chosen are $\Omega=\omega_0=1$ and $\beta=10$. We have used a local Hilbert space cut off $d_c=50$ for the bosonic mode in these ED calculations.