Sparse Polynomial Space Approach to Dissipative Quantum Systems: Application to the Sub-Ohmic Spin-Boson Model

Abstract

We propose a general numerical approach to open quantum systems with a coupling to bath degrees of freedom. The technique combines the methodology of polynomial expansions of spectral functions with the sparse grid concept from interpolation theory. Thereby we construct a Hilbert space of moderate dimension to represent the bath degrees of freedom, which allows us to perform highly accurate and efficient calculations of static, spectral, and dynamic quantities using standard exact diagonalization algorithms. The strength of the approach is demonstrated for the phase transition, critical behavior, and dissipative spin dynamics in the spin-boson model.

Generalized Polaron Ansatz for the Ground State of the Sub-Ohmic Spin-Boson Model: An Analytic Theory of the Localization Transition

“…(3) that the magnetisation obtained from the energy minimization does indeed behave like M ∝ (α − α c ) 1 2 close to the transition. The magnetisation data are again in good agreement with the QMC and SPSA results [16,17]. Figure 4 shows the behaviour of the cohrence σ x as a function of α for ∆/ω c = 0.1.…”

“…(9). Fortunately, in the scaling limit∆ c can also be found analytically, leading to the final prediction, previous seen in NRG and other approaches [11,16,17,20,22,23].…”

“…Therefore, by varying the value of M used in the unitary transformation, we can seek out the true value by looking for the value with the fastest convergence time. This idea of removing the effect of the mean-field displacment was orginally used in the SPSA study where it was found to yield an efficient and accurate simulation of the magnetic phase [17]. However, this method will only be applicable to the case described here (s < 0.5), where a mean-field bias arises from the bath.…”

“…However, it is now believed that the nonmean-field results found by NRG in 0 < s < 0.5 are incorrect, and arise from the truncation of the number of states N b used to describe each oscillator in the Wilson chain [14,15]. Recent studies of the sub-Ohmic QPT using quantum monte carlo (QMC) [16], sparse polynomial space approach (SPSA) [17], and an extended coherent state technique have indeed found mean-field critical exponents for 0 < s < 0.5 [18].In this article we propose a variational ansatz for the ground state of the sub-Ohmic SBM for 0 < s < 0.5 which does not require any truncation of the environment. This is an important feature, as we shall show that the number of environmental bosons diverges above the transition.…”

Quantum critical point of spin-boson model and infrared catastrophe in bosonic bath

“…In last ten years, various numerical techniques were used for calculation of the QCP in the SBM, such as the numerical renormalization group (NRG) [3,4,5], the quantum Monte Carlo (QMC) [8], the method of sparse polynomial space representation [9], the extended coherent state approach [10], and the variational matrix product state approach [11]. Besides, recently an extension of the Silbey-Harris ground state was proposed by Zhao et al [12] and Chin et al [13] to study the QPT in the s = 1/2 sub-Ohmic SBM.…”