The dynamical image potential: a comparison of path-integral and matrix methods

We propose a path-integral variant of the density-matrix renormalization group method to calculate real-time correlation functions at arbitrary finite temperatures. To illustrate the method we study the longitudinal autocorrelation function of the $XXZ$ chain. By comparison with exact results at the free fermion point, we show that our method yields accurate results up to a limiting time which is determined by the spectrum of the reduced density matrix.We propose a path-integral variant of the density-matrix renormalization group method to calculate real-time correlation functions at arbitrary finite temperatures. To illustrate the method we study the longitudinal autocorrelation function of the $XXZ$ chain. By comparison with exact results at the free fermion point, we show that our method yields accurate results up to a limiting time which is determined by the spectrum of the reduced density matrix.

arXiv:cond-mat/0504091Physical Review BJun 2, 200566 citations

Path Integrals and Complex Dynamics

Springer eBooksJan 1, 20070 citations

Tree-Based Implementation of the Small Matrix Path Integral for System-Bath Dynamics

Geshuo Wang, Zhenning Cai

The small matrix path integral (SMatPI) method is an efficient numerical approach to simulating the evolution of a quantum system coupled to a harmonic bath. The method relies on a sequence of kernel matrices that defines the non-Markovian dynamics of the quantum system. In the original SMatPI method, these kernels are computed indirectly through the QuAPI method. Instead, we focus on the definition of the kernel matrices and reveal a recurrence relation in these matrices. Using such a relationship, a tree based algorithm (t-SMatPI) is developed, which is shown to be much faster than straightforward computation of the kernel matrices based on their definitions. This algorithm bypasses the step to compute the SMatPI matrices by other path integral methods and provides more understanding of the SMatPI matrices themselves. Meanwhile, it keeps the memory cost and computational cost low. Numerical experiments show that the t-SMatPI algorithm gives exactly the same result as i-QuAPI and SMatPI. In spite of this, our method may indicate some new properties of open quantum systems, and has the potential to be generalized to higher-order numerical schemes.The small matrix path integral (SMatPI) method is an efficient numerical approach to simulating the evolution of a quantum system coupled to a harmonic bath. The method relies on a sequence of kernel matrices that defines the non-Markovian dynamics of the quantum system. In the original SMatPI method, these kernels are computed indirectly through the QuAPI method. Instead, we focus on the definition of the kernel matrices and reveal a recurrence relation in these matrices. Using such a relationship, a tree based algorithm (t-SMatPI) is developed, which is shown to be much faster than straightforward computation of the kernel matrices based on their definitions. This algorithm bypasses the step to compute the SMatPI matrices by other path integral methods and provides more understanding of the SMatPI matrices themselves. Meanwhile, it keeps the memory cost and computational cost low. Numerical experiments show that the t-SMatPI algorithm gives exactly the same result as i-QuAPI and SMatPI. In spite of this, our method may indicate some new properties of open quantum systems, and has the potential to be generalized to higher-order numerical schemes.

arXiv:2207.11830Communications in Computational PhysicsSep 5, 202410 citations

Path integral solution for some time-dependent potential

S. N. Storchak

Physics Letters AJan 1, 19925 citations

XXZ

chain. By comparison with exact results at the free fermion point, we show that our method yields accurate results up to a limiting time which is determined by the spectrum of the reduced density matrix.