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arXiv:2407.04365 · Journal of Chemical Theory and Computation

Simulation of Spin Chains with Off-Diagonal Coupling Using the Inchworm Method

Yixiao Sun, Geshuo Wang, Zhenning Cai

Published Oct 18, 2024• 3 citations
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Abstract

We study the dynamical simulation of an open quantum spin chain with nearest neighboring coupling, where each spin in the chain is associated with a harmonic bath. This is an extension of our previous work (Wang, G.; Cai, Z. J. Chem. Theory Comput. 2023, 19, 8523–8540) by generalizing the application of the inchworm method and the technique of modular path integrals from diagonally coupled cases to off-diagonally coupled cases. Additionally, to reduce computational and memory cost in long time simulation, we apply tensor-train representation to efficiently represent the reduced density matrix of the spin chains, and employ the transfer tensor method (TTM) to avoid exponential growth of computational cost with respect to time. Abundant numerical experiments are performed to validate our method.

DOI
10.1021/acs.jctc.4c00864

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We study the dynamical simulation of an open quantum spin chain with nearest neighboring coupling, where each spin in the chain is associated with a harmonic bath. This is an extension of our previous work (Wang, G.; Cai, Z. J. Chem. Theory Comput. 2023, 19, 8523–8540) by generalizing the application of the inchworm method and the technique of modular path integrals from diagonally coupled cases to off-diagonally coupled cases. Additionally, to reduce computational and memory cost in long time simulation, we apply tensor-train representation to efficiently represent the reduced density matrix of the spin chains, and employ the transfer tensor method (TTM) to avoid exponential growth of computational cost with respect to time. Abundant numerical experiments are performed to validate our method.
This article is devoted to the development of analytical and numerical approaches to the problem of the end-to-end quantum state transfer along the spin-1/2 chain using two methods: (a) a homogeneous spin chain with week end bonds and equal Larmor frequencies and (b) a homogeneous spin chain with end Larmor frequencies different from inner ones. A tridiagonal matrix representation of the XY\mathit{XY}XY Hamiltonian with nearest neighbor interactions relevant to the quantum state transfer is exactly diagonalized for a combination of the preceding two methods. In order to take into account interactions of the remote spins, we used numerical simulations of the quantum state transfer in 10-node chains. We compare the state transfer times obtained using the two preceding methods for chains governed by the both XY\mathit{XY}XY and XXZ\mathit{XXZ}XXZ Hamiltonians and using both nearest neighbor and all-node interactions.
In this second paper of a two part series, we present extensive benchmark results for two different inchworm Monte Carlo expansions for the spin-boson model. Our results are compared to previously developed numerically exact approaches for this problem. A detailed discussion of convergence and error propagation is presented. Our results and analysis allow for an understanding of the benefits and drawbacks of inchworm Monte Carlo compared to other approaches for exact real-time non-adiabatic quantum dynamics.
We study the real-time simulation of open quantum systems, where the system is modeled by a spin chain with each spin associated with its own harmonic bath. Our method couples the inchworm method for the spin-boson model and the modular path integral methodology for spin systems. In particular, the introduction of the inchworm method can significantly suppress the numerical sign problem. Both methods are tweaked to make them work seamlessly with each other. We represent our approach in the language of diagrammatic methods and analyze the asymptotic behavior of the computational cost. Extensive numerical experiments are performed to validate our method.
A recent article by Ivander, Lindoy and Lee [Nature Communications 15, 8087 (2024)] claims to discover the relationship between the generalized quantum master equation (GQME) and the path integral for a system coupled to a harmonic bath. However, this relationship was already established in 2020 by Makri in the context of the small matrix decomposition of the path integral (SMatPI) [J. Chem. Theory and Comput. 16, 4038 (2020)]. The procedure that this article uses in its Supplementary Information (SI) to obtain the various matrices follows the SMatPI decomposition steps for the alternative Trotter ordering. The absence of endpoint effects in the kernel matrices of the discretized GQME expression for the reduced density matrix (RDM) is the consequence of a crude GQME discretization and is not consistent with the SMatPI decomposition of an auxiliary matrix presented in the SI. This form is identical to the transfer tensor method (TTM) of Cerrillo and Cao [Phys. Rev. Lett. 112, 110401 (2014)]. Further, the Dyck path section of this article follows precisely the diagrammatic analysis developed by Wang and Cai in a recent paper [Communications in Computational Physics 36, 389 (2024)]. We elaborate on these three critiques in this Comment.