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Computing quantum dynamics in condensed matter systems is an open challenge due to the exponential scaling of exact algorithms with the number of degrees of freedom. Current methods try to reduce the cost of the calculation using classical dynamics as the key ingredient of approximations of the quantum time evolution. Two main approaches exist, quantum classical and semi-classical, but they suffer from various difficulties, in particular when trying to go beyond the classical approximation. It may then be useful to reconsider the problem focusing on statistical time-dependent averages rather than directly on the dynamics. In this paper, we discuss a recently developed scheme for calculating symmetrized correlation functions. In this scheme, the full (complex time) evolution is broken into segments alternating thermal and real-time propagation, and the latter is reduced to classical dynamics via a linearization approximation. Increasing the number of segments systematically improves the result with respect to full classical dynamics, but at a cost which is still prohibitive. If only one segment is considered, a cumulant expansion can be used to obtain a computationally efficient algorithm, which has proven accurate for condensed phase systems in moderately quantum regimes. This scheme is summarized in the second part of the paper. We conclude by outlining how the cumulant expansion formally provides a way to improve convergence also for more than one segment. Future work will focus on testing the numerical performance of this extension and, more importantly, on investigating the limit for the number of segments that goes to infinity of the approximate expression for the symmetrized correlation function to assess formally its convergence to the exact result.

Exact simulation methods to compute either the evolution of the wave function or dynamical statistical averages for quantum systems in the condensed phase are currently restricted to small sizes and short times. The exponential scaling of available algorithms with the number of degrees of freedom, in fact, limits calculations to ten–twenty particles (and this for Hamiltonians of relatively simple form) and to time scales of at most a few picoseconds. This situation is in striking contrast with analogous classical calculations, which, when empirical potentials are adopted, are nowadays routinely used to study high dimensional, complex systems for times reaching, on dedicated machines, microseconds. (

In semi-classical schemes, originally developed for approximating wave function propagation, all degrees of freedom are treated on equal footings. To begin with, the quantum time propagator is expressed, in the path integral formalism [

In the expression above, |^{−γ(q−r)2+ip(q−r)/ℏ}), _{cl}_{sc}_{B}T_{B}

The problems and numerical cost of semi-classical calculations justify the development of the second, alternative approximation scheme mentioned at the beginning of this section: mixed quantum classical dynamics. In this approach, the degrees of freedom of the system are partitioned into two sets, usually based on their mass ratio. The first set (called the subsystem) is composed of a few degrees of freedom and is treated quantum mechanically; the second (called the environment or the bath) is often high dimensional and is treated classically. Existing quantum classical methods differ in the way in which the coupling among the classical evolution of the bath and the quantum propagation of the subsystem is taken into account. The first approach of this kind, still very popular due to its efficiency and ease of implementation, is Tully’s surface hopping [

While application driven calculations might not be paralyzed by the state of affairs described above, in particular, if and when it is possible to verify that these well-known pathologies have no uncontrolled effects on the results, it is important to pursue alternative approaches in an effort to derive more general schemes allowing for systematic improvement and/or assessment of the approximations employed. Indeed, a critical stumbling block common to semi-classical and mixed quantum classical methods is that it is essentially impossible to go beyond classical trajectories to approximate the quantum evolution of the full system (semi-classical) or of the bath (mixed). In the semi-classical case, including terms of higher order, the expansion of the action along the paths makes it impossible to obtain calculable expressions for the pre-factor in the expression of the wave function (already at third order, the integral corresponds to intractable Airy functions [

Let us begin by expressing the symmetrized correlation function,

The structure of the integrand is represented in _{0} to _{c}_{c}_{c}/L_{c}_{L}_{tc}

The expression above is an exact, incalculable, expression of the time correlation function. In the following, we will work on the generic _{β}_{t}_{t}/n

The linearization approximation then has two crucial consequences: (1) by allowing the integration over the difference paths, it transforms the quantum expression of the correlation function, which, in the beginning, includes two propagators and, therefore, two paths, into a formula where only the semi-sum path appears, thus leading to a structure more similar to classical time correlations in which only one propagation is present; (2) (perhaps more importantly) it forces the semi-sum path to follow a, classical, Hamiltonian trajectory, as identified by the arguments of the delta functions.

The final step to obtain a suitable expression for ^{ν}^{ν}^{ν}

The expression above is interesting. First of all, assuming that the linearization approximation of each short time propagator improves when the propagation time goes to zero, there is potential for systematic improvement with increasing _{J}_{J= 0,...,L−1} variables) as
^{l}_{t}_{1}. The integrals over
_{1} in the expression for
_{1} = Δ_{tc}_{tc}

A Monte Carlo algorithm to sample Π for different values of

The expression for the

We are now going to simplify the expression above using four steps: (1) observe that the integral over Δ_{tc}_{tc}^{(ν+l)} (^{l}_{tc}_{t}_{t}_{t}_{t}_{0}) in the evaluation of the matrix elements, allows one to integrate also over Δ_{0}. The surviving variables (^{1}, ^{0}, …, ^{ν}^{1}, …, Δ^{ν−1}}; (4) simplify the notation by dropping the bar from the semi-sum variables and the subscript, which identifies the

As anticipated, both in the numerator and denominator of this expression, a phase factor appears, which, for high dimensional systems, hinders an efficient convergence of the calculation. To alleviate this problem, we proposed a method, described in detail in [^{0}, …, ^{ν}^{1}, ^{(ν−1)}}. This joint probability (whose form can be inferred from _{c}_{m}^{(ν−1)} with respect to the conditional probability, _{c}^{(ν−1)}. Importantly, the conditional probability density is an even function of the difference variables, implying that only even order terms in the series above are non-zero and that the series corresponds to a real function that we will denote in the following with ^{−E(p,r)} and _{m}

To simplify the discussion, we introduce some notation. Let us indicate the coordinate-dependent Gaussian terms in ^{0} = 0) and write the potential term as:

We also rewrite the marginal probability, _{m}

The Monte Carlo scheme to sample

(1) A move on

choose a new momentum according to ^{p}_{p}_{s}

The expression for the acceptance probability differs from the standard Metropolis prescription for the presence of

(2) A move on

in this case, indicating with ^{r}^{r}_{G}

The structure of this relationship is analogous to the one considered by Kennedy ^{r}

Above, ^{r}^{−E(r′,p)} is only known with noise. To solve this problem, we employ the penalty method to obtain configurations distributed according to ^{−Vr(r′)} and acceptance probability
_{r}

This concludes the description of our Monte Carlo moves. The practical implementation of this algorithm requires the definition of the numerical estimators, _{p}_{r}_{i}_{c}_{c}_{Δ}(Δ_{β}_{a}_{t}_{m}_{a}_{m}_{a}

The computational overhead introduced by the auxiliary Monte Carlo calculation increases the cost of our calculation, but it is very small compared to the number of moves necessary to converge the estimate of

The results show a rather pronounced asymmetry around zero, due to detailed balance, that indicates the presence of relevant quantum effects in the system. The agreement between our calculations and experiments is very good, as it is the agreement with the standard linearized calculation by Poulsen (a state-of-the-art reference in the field). The numerical cost of the two calculations is very similar (about a million Monte Carlo steps in total for initial condition sampling), showing that the auxiliary steps, due to the noisy distributions in our approach, are essentially irrelevant. Indeed, other tests indicate that, depending on the system, the overall cost of our method can be less than that of alternative schemes with comparable or better accuracy. The approach described in this section, for example, has also been used to obtain the infrared spectra of simple models of molecules in the gas phase [

In this subsection, we present a new development of the approach summarized above that extends the use of cumulants to pre-average the phase factors in the expression of the symmetrized correlation function to the case

In the equation above, (
_{0}) was defined in the previous section (see _{c}_{1}|_{1}), probabilities are defined in analogy with the expressions introduced in Section 3, with the caveat that for _{J}_{J}

For _{1}, _{2}} is a vector of positive integers (including zero), |_{1}! _{2}! and:

As in the previous subsection, the conditional distribution density is even with respect to the difference variables, implying that only even terms are non-zero in _{J}_{J}_{J}_{J}^{−E(πJ,rJ)} and define, in analogy with

Substitution of the definition above in the expression for the symmetrized correlation function shows that we can write the two-segment approximation as the following expectation value:
_{m}_{J}

In this paper, we summarized a recently developed method to approximate symmetrized quantum time correlation functions. The method recasts the problem as the calculation of averages over a stochastic process based on a linearized approximation of the complex time propagators in the correlation function. This approximation can be enforced either on the full length of the evolution (fully linearized approach) or in an iterative form obtained via the (complex) time composition property of the evolution operators. Thanks to the use of a cumulant expansion, which tames the phase factors present in the observable, the fully linearized approach has proven efficient and accurate in calculations on moderately quantum systems in the condensed phase. The iterative form offers, in principle, a way to improve the accuracy of the results with respect to the fully linearized case and may be useful when higher order quantum effects must be kept into account. While the potential for systematic improvement with respect to the fully classical limit for the dynamics is indeed the most interesting feature of the approach (and the one that distinguishes it from other available methods for which there is no way to improve upon the classical or semi-classical approximation), the practical use of the approach for

The authors are grateful to C. Pierleoni and M. Monteferrante for their substantial contributions to the earlier methods for symmetrized correlation functions summarized in this work. Funding from IIT-SEED grant No 259 “SIMBEDD” is also acknowledged.

The authors declare no conflict of interest.

Schematic representation of the integrand in the coordinate representation of the symmetrized time correlation function; see the text.

Schematic representation of the break up of the propagators in complex time: the short complex time propagators are represented as the segments with arrows along the forward and backward path, and the pairing mentioned in the text to obtain the

Graphic representation of the propagators in real and imaginary times contributing to the approximate Schofield function for the case

Dynamic structure factor for liquid neon (see the text). The solid green line shows the experimental curve, our results (with error bars) are the red triangles. We also report for comparison results obtained with the linearized IVRmethod by Polusen