We apply recently developed smooth boundary conditions to the quantum Monte Carlo simulation of the two-dimensional Hubbard model. At half-filling, where there is no sign problem, we show that the thermodynamic limit is reached more rapidly with smooth rather than with periodic or open boundary conditions. Away from half-filling, where ordinarily the simulation cannot be carried out at low temperatures due to the existence of the sign problem, we show that smooth boundary conditions allow us to reach significantly lower temperatures. We examine pairing correlation functions away from half-filling in order to determine the possible existence of a superconducting state. On a 10\ifmmode\times\else\texttimes\fi{}10 lattice for
U=4, at a filling of
〈n〉=0.87 and an inverse temperature of $\ensuremath{\beta}=10$, we find enhancement of the
d-wave correlations with respect to the noninteracting case, a possible sign of
d-wave superconductivity.
We investigate the effect of a nonuniform deformation applied to one-dimensional (1D) quantum systems, where the local energy scale is proportional to ${g}_{j}=[\mathrm{sin}(j\ensuremath{\pi}/N)]{}^{m}$ determined by a positive integer
m, site index $1\ensuremath{\leqslant}j\ensuremath{\leqslant}N\ensuremath{-}1$, and system size
N. This deformation introduces a smooth boundary to systems with open-boundary conditions. When $m\ensuremath{\geqslant}2$, the leading
1/N correction to the ground-state energy per bond
e0(N) vanishes and one is left with a
1/N2 correction, the same as with periodic boundary conditions. In particular, when
m=2, the value of
e0(N) obtained from the deformed open-boundary system coincides with the uniform system with periodic boundary conditions. We confirm the fact numerically for correlated systems, such as the extended Hubbard model, in addition to 1D free-fermion models.
We study the problem of the boundary conditions in the numerical simulation of closed and open quantum systems, described by a Schrödinger equation. On one hand, we show that a closed quantum system is defined by local boundary conditions. On the other hand, we argue that, because of the uncertainty principle, no local boundary condition can be defined for open quantum systems. For this reason plane waves or wave packet trains cannot be simulated on a finite numerical lattice with the usual procedures. We suggest a method that avoids these difficulties by using only a small numerical lattice and maintains the correspondence with the physical picture, in which the incident and scattered waves may be infinitely extended.
In quantum lattice models, in the large-
N limit, boundary conditions have little effect upon local observables for sites in the centers of the lattices. In this paper, we will study the boundary effects upon multipartite nonlocality (a kind of multipartite quantum correlation associated with Bell-type inequalities) in one-dimensional finite-size spin chains, both for zero temperature and for finite temperatures. We define a quantity $\frac{\ensuremath{\delta}\mathcal{S}}{\mathcal{S}}$ to characterize the boundary effects, where
S is a measure of global multipartite nonlocality of the entire lattice, and $\ensuremath{\delta}\mathcal{S}$ is the difference of the measure induced by changing the boundary conditions. We find $\frac{\ensuremath{\delta}\mathcal{S}}{\mathcal{S}}$ does not vanish in the large-
N limit. Instead, at zero temperature, with the increase of
N, $\frac{\ensuremath{\delta}\mathcal{S}}{\mathcal{S}}$ would increase steadily in the vicinity of the quantum phase transition point of the models, and converge to a nonzero constant in noncritical regions. It shows clearly that boundary effects generally exist, in the form of multipartite correlations, in long chains. The boundary effects are explained by the competition between the two orders of the models. In addition, based on these numerical results, we construct a Bell inequality, which is violated by chains with periodic (closed) boundary conditions and not violated by chains with open boundary conditions. Furthermore, we study $\frac{\ensuremath{\delta}{\mathcal{S}}_{T}}{{\mathcal{S}}_{T}}$ of finite-size chains at finite temperatures, and show that boundary effects survive in finite temperature regions.
We investigate the mean-field ground states of coupled one-dimensional Bose-Einstein condensates subject to a synthetic magnetic field. The resulting interacting coupled-wire model has one continuous and one discrete spatial direction, providing a controlled way to interpolate between the physics of few-leg ladders and extended vortex lattices. For two wires, we study the vortex-like, biased-density, and Meissner-like states, exploring how the finite longitudinal size of the system modifies the transitions between them. Increasing the number of wires, the ground state evolves towards an extended vortex lattice. We find numerically that periodic boundary conditions in the discrete direction can favour staggered arrays of like-signed vortices resembling an Abrikosov lattice, while open boundaries in small finite-size systems confine the vortices into rows near the centre of the synthetic direction. Our results explore how finite size effects and boundary conditions govern the emergence and spatial organisation of vortices in continuous-discrete quantum fluids, with relevance to future experiments in tunnel-coupled atomic wires or with synthetic dimensions.