Multisite Tensor Network Path Integral

Fri, 04 Nov 2022 20:27:50 +0530

Simulating the dynamics of a quantum system coupled to a dissipative environment gets challenging because of the non-Markovian memory that needs to be accounted for. While iterative propagation and basic tensor network path integral (TNPI) enable simulation of longer memory times, the dimensionalities of systems that can be simulated are still very limited. For extended systems the dimensionality grows exponentially with the number of sites or monomers, making these techniques untenable. Consider $50$ monomers described by two levels each. That implies that the Hilbert space has a dimensionality of $2^{50}$. This exponential growth of the dimensionality of the Hilbert space may be controlled if the dynamics can be restricted to a significantly smaller subspace. This is the case when considering the single particle subspace in a Frenkel-like process which conserves the number of particles. However, solving the 50-dimensional problem, while more feasible than the $2^{50}$ dimensional problem, is still a significant challenge.

The standard TNPI decomposes the path integral expression only along the temporal dimension. However, to account for these extended systems, one needs to decompose it also along a spatial dimension, separating out the individual sites. Thus a two-dimensional tensor network decomposition is achieved. Contraction of this 2D tensor network after incorporation of the influence functional yields the time-dependent reduced density operator corresponding to the extended system in the presence of a dissipative environment. The simulations happen in the full Hilbert space allowing for a transparent inclusion of more involved effects like multi-photon process, multi-dimensional spectra, etc.

Tensor Network Path Integral

Fri, 04 Nov 2022 16:34:44 +0530

Simulations of real-time dynamics of quantum systems coupled with dissipative media is plagued by the curse of dimensionality. As a way to avoid the problem, many methods integrate out the bath and simulate the dynamics of the reduced density matrix. Such simulations are characterized by the presence of non-Markovian memory. Path integrals, through the use of Feynman-Vernon’s influence functional, provides a rigorous way of capturing this non-Markovian effect of the environment on the system.

The most challenging aspect is that the cost of these calculations grow exponentially with the memory length. We classify approaches that utilize tensor networks to make path integral simulations more efficient as belonging to the tensor network path integral (TNPI) family of methods. There can be many different kinds of tensor networks that are used. Time-evolved matrix product operators (TEMPO) is a particular one that uses matrix product states and matrix product operators for simulating real-time dynamics with Feynman-Vernon influence functional. We have developed a method called the pairwise-connected tensor network path integral (PC-TNPI) as a generalization on these approaches. Additionally, the TNPI framework allows for further factorization of the system to deal with extended systems. This extension, called the multisite tensor network path integral (MS-TNPI), is described in its own section.

Path Integrals | Bose Research Group

Multisite Tensor Network Path Integral

Tensor Network Path Integral