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The feasibility of path integral Monte Carlo ground state calculations with very few beads using a high-order short-time Greens function expansion is discussed. An explicit expression of the evolution operator which provides dramatic enhancements in the quality of ground-state wave-functions is examined. The efficiency of the method makes possible to remove the trial wave function and thus obtain completely model-independent results still with a very small number of beads. If a single iteration of the method is used to improve a given model wave function, the result is invariably a shadow-type wave function, whose precise content is provided by the high-order algorithm employed.
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High order actions proposed by Chin have been used for the first time in path integral Monte Carlo simulations. Contrarily to the Takahashi-Imada action, which is accurate to fourth order only for the trace, the Chin action is fully fourth order, with the additional advantage that the leading fourth and sixth order error coefficients are finely tunable. By optimizing two free parameters entering in the new action we show that the time step error dependence achieved is best fitted with a sixth order law. The computational effort per bead is increased but the total number of beads is greatly reduced, and the efficiency improvement with respect to the primitive approximation is approximately a factor of ten. The Chin action is tested in a one-dimensional harmonic oscillator, a H$_2$ drop, and bulk liquid $^4$He. In all cases a sixth-order law is obtained with values of the number of beads that compare well with the pair action approximation in the stringent test of superfluid $^4$He.
The local order around alkali (Li$^+$ and Na$^+$) and alkaline-eath (Be$^+$, Mg$^+$ and Ca$^+$) ions in $^4$He clusters has been studied using ground-state path integral Monte Carlo calculations. We apply a criterion based on multipole dynamical correlations to discriminate between solid-like versus liquid-like behavior of the $^4$He shells coating the ions. As it was earlier suggested by experimental measurements in bulk $^4$He, our findings indicate that Be$^+$ produces a solid-like (snowball) structure, similarly to alkali ions and in contrast to the more liquid-like $^4$He structure embedding heavier alkaline-earth ions.
We develop an approach of calculating the many-body path integral based on the linked cluster expansion method. First, we derive a linked cluster expansion and we give the diagrammatic rules for calculating the free-energy and the pair distribution function $g(r)$ as a systematic power series expansion in the particle density. We also generalize the hypernetted-chain (HNC) equation for $g(r)$, known from its application to classical statistical mechanics, to a set of quantum HNC equations (QHNC) for the quantum case. The calculated $g(r)$ for distinguishable particles interacting with a Lennard-Jones potential in various attempted schemes of approximation of the diagrammatic series compares very well with the results of path integral Monte Carlo simulation even for densities as high as the equilibrium density of the strongly correlated liquid $^4$He. Our method is applicable to a wide range of problems of current general interest and may be extended to the case of identical particles and, in particular, to the case of the many-fermion problem.
Path integrals constitute powerful representations for both quantum and stochastic dynamics. Yet despite many decades of intensive studies, there is no consensus on how to formulate them for dynamics in curved space, or how to make them covariant with respect to nonlinear transform of variables. In this work, we construct rigorous and covariant formulations of time-slicing path integrals for quantum and classical stochastic dynamics in curved space. We first establish a rigorous criterion for correct time-slice actions of path integrals (Lemma 1). This implies the existence of infinitely many equivalent representations for time-slicing path integral. We then show that, for any dynamics with second order generator, all time-slice actions are asymptotically equivalent to a Gaussian (Lemma 2). Using these results, we further construct a continuous family of equivalent actions parameterized by an interpolation parameter $alpha in [0,1]$ (Lemma 3). The action generically contains a spurious drift term linear in $Delta boldsymbol x$, whose concrete form depends on $alpha$. Finally we also establish the covariance of our path-integral formalism, by demonstrating how the action transforms under nonlinear transform of variables. The $alpha = 0$ representation of time-slice action is particularly convenient because it is Gaussian and invariant, as long as $Delta boldsymbol x$ transforms according to Itos formula.
Majorana fermions are currently of huge interest in the context of nanoscience and condensed matter physics. Different to usual fermions, Majorana fermions have the property that the particle is its own anti-particle thus, they must be described by real fields. Mathematically, this property makes nontrivial the quantization of the problem due, for instance, to the absence of a Wick-like theorem. In view of the present interest on the subject, it is important to develop different theoretical approaches in order to study problems where Majorana fermions are involved. In this note we show that Majorana fermions can be studied in the context of field theories for constrained systems. Using the Faddeev-Jackiw formalism for quantum field theories with constraints, we derived the path integral representation for Majorana fermions. In order to show the validity of the path integral we apply it to an exactly solvable problem. This application also shows that it is rather simple to perform systematic calculations on the basis of the present framework.
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