Analytic continuation average spectrum method for transport in quantum liquids Orly Kletenik-Edelman a , Eran Rabani a, * , David R. Reichman b a School of Chemistry, The Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel b Department of Chemistry, Columbia University, 3000 Broadway, New York, NY 10027, United States article info Article history: Received 11 October 2009 In final form 20 January 2010 Available online 28 January 2010 Keywords: Quantum dynamics PIMC Analytic continuation para-Hydrogen Normal helium Self-diffusion abstract Recently, we have applied the analytic continuation averaged spectrum method (ASM) to calculate col- lective density fluctuations in quantum liquid [27]. Unlike the maximum entropy (MaxEnt) method, the ASM approach is capable of revealing resolved modes in the dynamic structure factor in agreement with experiments. In this work we further develop the ASM to study single-particle dynamics in quantum liquids with dynamical susceptibilities that are characterized by a smooth spectrum. Surprisingly, we find that for the power spectrum of the velocity autocorrelation function there are pronounced differences in comparison with the MaxEnt approach, even for this simple case of smooth unimodal dynamic response. We show that for liquid para-hydrogen the ASM is closer to the centroid molecular dynamics (CMD) result while for normal liquid helium it agrees better with the quantum mode coupling theory (QMCT) and with the MaxEnt approach. Ó 2010 Elsevier B.V. All rights reserved. 1. Introduction The description of the dynamics of quantum liquids is an impor- tant and challenging problem. Phenomena ranging from the rich collective dynamics of superfluid helium [1] to electron transport in liquids [2,3] have long gained the attention of physical scientists. The recent experimental and theoretical attention focus on new questions, such as the interplay between the glass transition and superfluidity [4–6], illustrates the enduring interest in the dynam- ics of quantum liquids. In principle simulating the dynamics of quantum liquid should be simple. Essentially any imaginary-time quantum correlation function may be computed [7,8] and then a well-defined analytical continuation to the real-time domain may be carried out [9]. The difficulty with this procedure is that the continuation is numeri- cally unstable, making the results unreliable [9]. While there are several different numerical protocols for performing analytical continuation, the most commonly use method is the maximum en- tropy (MaxEnt) approach [9–24]. MaxEnt is generally assumed to be accurate in describing spectral functions that have one domi- nant feature, but is known to have difficulty in resolving fine struc- ture [25–27]. Recently, several related analytical continuation approaches have been developed. The averaged spectrum method (ASM) [28–31], the Genetic Inversion via Falsification of Theories (GIFT) method of Vitali et al. [32], and the Generalized Doetsch formula [33] are examples of such approaches. These methods have been argued to be superior to MaxEnt at least in their ability to resolve sharp spectral features [27,32]. In a recent paper we applied the ASM to the problem of density fluctuations in liquid para-hydrogen and ortho-deuterium [27]. We found that the ASM gave a reason- able description of the coherent scattering function Sðq; xÞ and was superior to MaxEnt in describing the sharp quasi-particle peaks revealed in experiments [34–38] and predicted by computa- tional techniques such as the centroid molecular dynamics (CMD) approach [39], and theoretical approaches such as the quantum mode-coupling theory (QMCT) [40]. In this work we investigate single-particle diffusive dynamics as exhibited by the frequency dependent diffusion constant DðxÞ via the ASM approach. The systems we consider are liquid para-hydro- gen and normal liquid helium above the lambda transition. These systems have been extensively studied by a variety of techniques [19,21–23,38,41–52]. Such studies have all consistently found that DðxÞ in these systems exhibits only one main broad maximum [19]. In this regard it is expected that MaxEnt should be accurate in its description of DðxÞ. Rather surprisingly, we find that while the ASM yields diffusion coefficients consistent with MaxEnt, the spectral features are noticeably different. We demonstrate that for liquid helium the ASM yields a spectrum in close agreement with MaxEnt and QMCT [21], while for para-hydrogen the ASM re- sult is closer to that described by CMD [51,24] and possibly exper- iments [38]. The paper is organized as follows: in Section 2 we describe the average spectrum method and its implementation to the transport problem. In Section 3 we describe the model for liquid para-hydro- gen and normal liquid helium and the computational details, and discuss the results for self-diffusion in these systems. Comparisons 0301-0104/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2010.01.012 * Corresponding author. Tel.: +972 36407599; fax: +972 36407042. E-mail address: rabani@tau.ac.il (E. Rabani). Chemical Physics 370 (2010) 132–136 Contents lists available at ScienceDirect Chemical Physics journal homepage: www.elsevier.com/locate/chemphys