Imaginary-time correlation functions

In mean field approximation we obtain for the imaginary-time correlation functions [296]... 

Dynamical quantities are harder to obtain, since the QMC representations only give access to imaginary-time correlation function. With the exception of measurements of spin gaps, which can be obtained from an exponential decay of the spin-spin correlation function in imaginary time, the measurement of real-time or real-frequency correlation functions requires an ill-posed analytical continuation of noisy Monte Carlo data, for example using the Maximum Entropy Method [46-48]. 

It should be noted that the imaginary time correlation function in Eq. (2.4) provides a measure of the localization of quantum particles in condensed media [17-19,52]. From this point on, the notation denotes an averaging by integrating some centroid-dependent function over the centroid position q weighted by the normalized centroid density p q ) Z. An alternative method for defining the correlation function... 

A general imaginary-time correlation function for coordinate-dependent operators is defined as [17-19]... 

Figure 3. Calculation of the imaginary-time correlation function (v4(t)B(0)) in the discretized Feynman path-integral picture. The operators A[g(r )] and B[ (r")] can be evaluated at any two points on the thermal loop subject to the constraint 0 t = t — hp. The centroid variable, which is off the loop, is also shown.

In Paper I, general imaginary-time correlation functions were expressed in terms of an averaging over the coordinate-space centroid density p (qj and the centroid-constrained imaginary-time-position correlation function Q(t, qj. This formalism was extended in Paper III to the phase-space centroid picture so that the momentum could be treated as an independent variable. The final result for a general imaginary-time correlation function is found to be given approximately by [5,59]... 

In Fig. 4 the imaginary-time correlation function ( (t)9 (0)) is plotted for a temperature of 8 = 5 as a function of the dimensionless variable m = t/ /3. The centroid-based formalism is seen to be in very good agreement with the numerically exact result, confirming the validity of the various approximations for this completely nonquadratic example. 

Figure 4. Plot of the imaginary-time correlation function q T)q 0)) for the nonquadratic potential described in Section II.E [Eq. (2.66)]. The correlation function is plotted as a function of the dimensionless variable u = T/h with /3 = 5. The solid circles are the numerically exact results, while the solid line is for the optimized LHO theory in Eqs. (2.24)-(2.28) used in the centroid-based formulation of the correlation function in Eq. (2.50).

Centroid-Constrained Imaginary Time Correlation Function... 

By definition, the centroid variable occupies a central role in the behavior of the centroid-constrained imaginary-time correlation function in Eq. (2.1). However, it is even more interesting to analyze the role of the centroid variable in the real-time quantum position correlation function [4, 8]. This information can in principle be extracted from the exact centroid-constrained correlation function C (t, q ) through the analytic continuation t— if. Such a procedure, however, is generally not tractable unless there is some prior simplification of the problem. One such simplification is achieved [4, 8] through use of the optimized reference quadratic action functional, given by [3, 21-23]... 

The case of coordinate-dependent operators A and B will first be described for simplicity. In this approach, the general imaginary-time correlation function Qb(t) = (A(t)B(O) is expressed as... 

After performing the k integrals in Eq. (3.46), the expression for the general imaginary time correlation function is given by the double-Gaussian average... 

Reptation quantum Monte Carlo (RQMC) [15,16] allows pure sampling to be done directly, albeit in common with DMC, with a bias introduced by the time-step (large, but controllable in DMC e.g. [17]) and the fixed-node approach (small, but not controllable e.g. [18]). Property estimation in this manner is free from population-control bias that plagues calculation of properties in diffusion Monte Carlo (e.g. [19]). Inverse Laplace transforms of the imaginary time correlation functions allow simulation of dynamic structure factors and other properties of physical interest. 

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