Imaginary time evolution

Since [H, P] = 0, the imaginary time evolution preserves the symmetry and the fermion thermal density matrix takes the form... 

For quantum mechanical systems, G usually is the imaginary-time evolution operator exp( — ). As mentioned above, a technical problem in that case is that an explicit expression is known only asymptotically for short times r. In practice, this asymptotic expression is used for a small but finite and this leads to systematic, time-step errors. We shall deal with this problem below at length, but ignore it for the time being. 

Instead of having imaginary time evolution as in DMC, one keeps the entire path in memory and moves it around. PIMC uses a sophisticated Metropolis Monte Carlo method to move the paths. One trades off the complexity of the trial function for more complex ways to move the paths [23]. One gains in this trade-off because the former changes the answer while the latter changes only the computational cost. 

So for a given realization of the random potential the polymer partition sum can be expressed as a matrix element of the imaginary-time evolution operator. The matrix elements can be expanded in eigenfunctions of the Hamiltonian operator to yield... 

The DMC method simulates this (imaginary) time evolution by repeating the short time operation ... 

Figure 11 World-line configuration for the XXZ Hamiltonian of [38]. The world lines (thick lines) connect space-time points where the component of the spin points up. They can be either straight or cross the shaded squares, which show where the imaginary time evolution operators and act. The dotted line shows the configuration change after a local Monte Carlo update.

In the diflfiision QMC (DMC) method [114. 119], the evolution of a trial wavefiinction (typically wavefiinctions of the Slater-Jastrow type, for example, obtained by VMC) proceeds in imaginary time, i = it, according to the time-dependent Schrodinger equation, which then becomes a drfifiision equation. All... 

Abstract The theoretical basis for the quantum time evolution of path integral centroid variables is described, as weU as the motivation for using these variables to study condensed phase quantum dynamics. The equihbrium centroid distribution is shown to be a well-defined distribution function in the canonical ensemble. A quantum mechanical quasi-density operator (QDO) can then be associated with each value of the distribution so that, upon the application of rigorous quantum mechanics, it can be used to provide an exact definition of both static and dynamical centroid variables. Various properties of the dynamical centroid variables can thus be defined and explored. Importantly, this perspective shows that the centroid constraint on the imaginary time paths introduces a non-stationarity in the equihbrium ensemble. This, in turn, can be proven to yield information on the correlations of spontaneous dynamical fluctuations. This exact formalism also leads to a derivation of Centroid Molecular Dynamics, as well as the basis for systematic improvements of that theory. 

According to Doi [107], Zeldovich and Ovchinnikov [35], the evolution of the state of a system given by the vector (t)) obeys the Schrodinger equation with imaginary time and non-Hermitian Hamiltonian. The averaging procedure also differs from that generally-accepted in quantum mechanics. 

The DMC method achieves the lowest-energy eigenfunction by employing the quantum mechanical evolution operator in imaginary time [25], For an initial function expanded in eigenstates, one finds that contributions of the excited states decay exponentially fast with respect to the ground state. 

Time evolution of c m(t) coefficients at different detunings from the center of far a Gaussian pulse with fall-width at half maximum (FWHM) of 120 cm-1. (a) <4(f), ( )] .(c) hn c m(t). [E is to be replaced by — co0)]. Note that Re denotes the real and Im denotes the imaginary part, of the argument that follows. 

Note that as is just the Larmor frequency and, because real numbers are associated with the x axis and imaginary numbers with the y axis, time evolution is simply rotation in the x-y plane at the offset frequency. For double-quantum transitions, > = a> + >s, and for zero-quantum transitions co = coi — cos. For example, a 90° pulse on the y axis followed by a delay r would give... 

Let us go very briefly to the mathematics of DQMC-FSGO and then to the results. Let us have a transformed imaginary time operator, which determines the evolution of distribution of psips,... 

Note that the density matrix operator e H is the same as the time evolution operator c Wt// if we assign the imaginary value x = // ( to the time interval... 

We see that calculating the thermodynamics of a quantum system is the same as calculating transition amplitudes for its evolution in imaginary time [83]. 

The analogy of the time-evolution operator in quantum mechanics on the one hand, and the transfer matrix and the Markov matrix in statistical mechanics on the other, allows the two fields to share numerous techniques. Specifically, a transfer matrix G of a statistical mechanical lattice system in d dimensions often can be interpreted as the evolution operator in discrete, imaginary time t of a quantum mechanical analog in d — 1 dimensions. That is, G exp(—tJf), where is the Hamiltonian of a system in d — 1 dimensions, the quantum mechanical analog of the statistical mechanical system. From this point of view, the computation of the partition function and of the ground-state energy are essentially the same problems finding... 

The nonlocality, however, is a problem for the DMC simulations because the matrix element for the evolution of the imaginary-time diffusion is not necessarily positive. For realistic pseudopotentials the matrix elements are indeed negative and thus create a sign problem (even for one electron), with consequences similar to those of the fermion sign problem (see, e.g., work of Bosin et al. [49]). 

To make it possible to deal with systems with many degrees of freedom, the Boltzmann operator/time evolution operators are represented by a Feynman path integraland the path integral evaluated by a Monte Carlo random walk method. It is in general not feasible to do this for real values of the time t, however, because the integrand of the path integral would be oscillatory. We thus first calculate for real values of t it, i.e., pure imaginary time,... 

In this path-integral context, going to find the evolution amplitude of a particle with mass M moving in D dimension under the influence of an external potential V x) its path integral representation within the imaginary-time approach may be assumed (Putz, 2009, 2011b Putz Ori, 2014) ... 

Both DQMC and GFQMC provide the lowest energy solution to the Schrodinger equation subject to any constraints that may be imposed on the solution. For excited states, one must impose the necessary constraints. In some cases, this is relatively easy to do, but in others it is difficult or as yet impossible. For many cases, alternate methods are available in particular, a matrix procedure may be applied to the simultaneous evolution of several states in imaginary time. ... 

Quantum dynamics can be treated equally easily with this path integral formalism. By making the replacement -> it/h, the Boltzmann operator exp(—yS//) turns into the time-evolution operator exp —itH/h). In other words, yS can be considered to be an imaginary time and the time-evolution operator can be written as the product of Q short-time propagators as in equation (3). With this, the Green s function is... 

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