Variational Monte Carlo method for the Baeriswyl wave function: Application to the one-dimensional bosonic Hubbard model

A variational Monte Carlo method for bosonic lattice models is introduced. The method is based on the Baeriswyl projected wave function. The Baeriswyl wave function consists of a kinetic energy based projection applied to the wave function at infinite interaction, and is related to the shadow wave function already used in the study of continuous models of bosons. The wave function at infinite interaction, and the projector, are represented in coordinate space, leading to an expression for expectation values which can be evaluated via Monte Carlo sampling. We calculate the phase diagram and other properties of the bosonic Hubbard model. The calculated phase diagram is in excellent agreement with known quantum Monte Carlo results. We also analyze correlation functions.

Figure 1

Kinetic energy propagator as a function of distance for different values of $\alpha$.

Figure 2

Representation of our variational algorithm. In our method there are three replicas of the system, labeled left, center, and right. The black squares represent the lattice sites of the one-dimensional system for each such replica. The blue filled circles represent particles. Each particle is represented by one replica on the left, center, and right lattices. The dashed red line represents the kinetic energy projection operator [Eq. (8)]. The left and right replicas correspond to the infinitely interacting system. Since in the case there would be an infinite energy cost, there are no sites with more than one particle among the left and right replicas. In the center replica of the lattice, two or more particles can be on the same lattice.

Figure 3

Representation of a configuration with virtual hopping which contributes to the one-particle reduced density matrix [Eq. (13)]. The virtual hopping takes place from site $y$ to site $x$, indicated by the black semicircular arrow. The contribution to the average reduced density matrix is the proportion of the Baeriswyl propagators denoted by the straight solid red lines [see also Eq. (14)]. The propagators connect the left coordinate of the particle considered (in this case site 4) to the central coordinate after virtual hopping (in this case site 1, particle represented by green circle), and the propagator between the left coordinate to the unmoved central coordinate of the particle (site 4, particle represented by blue circle).

Figure 4

Representation of a configuration with virtual hopping which contributes to the one-particle reduced density matrix [Eq. (13)]. The virtual hopping takes place from site $y$ to site $x$, indicated by the black semicircular arrow. At the same time, a virtual hopping takes place among the left coordinates, also indicated by a black semicircular arrow. The contribution to the average reduced density matrix is the proportion of the Baeriswyl propagators denoted by the straight solid red lines [see also Eq. (15)]. The propagators connect the left coordinate of the particle (site 1, particle represented by green circle) to the central coordinate of the particle (site 1, particle represented by green circle), both after the virtual hopping, and the propagator left coordinate (site 4, represented by a blue circle) and the central coordinate of the particle (site 4, represented by a blue circle) both before the virtual hopping.

Figure 5

Energy per lattice site for fillings of one and one-half calculated by our variational Monte Carlo method, and mean-field theory. Lines with symbols represent variational Monte Carlo calculations, lines without symbols are the results of mean-field theory.

Figure 6

Phase diagram of the bosonic Hubbard model according to the Baeriswyl wave function (full circles connected by solid line) compared to quantum Monte Carlo results of Rousseau et al. [26] (open diamonds) recalculated via QMC [35, 36]. $n$ denotes the filling factor. The error bars in the calculations are smaller than the thickness of the symbols.

Figure 7

One-particle reduced density matrix for systems at filling one with $J/U=0.1,0.2,0.3,0.4$. The bottom panel is a semilog plot. The bottom panel is a semilog plot, the $y$ axis is labeled according to powers of ten. The decay of the one-body reduced density matrices is nearly exponential, implying the absence of a condensate.