Observing constituent particles with fractional quantum numbers in confined and deconfined states is an interesting and challenging problem in quantum many-body physics. Here we further explore a computational scheme [Y. Tang and A. W. Sandvik, Phys.
Rev. Lett. {bf 107}, 157201 (2011)] based on valence-bond quantum Monte Carlo simulations of quantum spin systems. Using several different one-dimensional models, we characterize $S=1/2$ spinon excitations using the spinon size and confinement length (the size of a bound state). The spinons have finite size in valence-bond-solid states, infinite size in the critical region, and become ill-defined in the Neel state. We also verify that pairs of spinons are deconfined in these uniform spin chains but become confined upon introducing a pattern of alternating coupling strengths (dimerization) or coupling two chains (forming a ladder). In the dimerized system an individual spinon can be small when the confinement length is large---this is the case when the imposed dimerization is weak but the ground state of the corresponding uniform chain is a spontaneously formed valence-bond-solid (where the spinons are deconfined). Based on our numerical results, we argue that the situation $lambda ll Lambda$ is associated with weak repulsive short-range spinon-spinon interactions. In principle both the length-scales can be individually tuned from small to infinite (with $lambda le Lambda$) by varying model parameters. In the ladder system the two lengths are always similar, and this is the case also in the dimerized systems when the corresponding uniform chain is in the critical phase. In these systems the effective spinon-spinon interactions are purely attractive and there is only a single large length scale close to criticality, which is reflected in the standard spin correlations as well as in the spinon characteristics.
We use Monte Carlo methods to study spinons in two-dimensional quantum spin systems, characterizing their intrinsic size $lambda$ and confinement length $Lambda$. We confirm that spinons are deconfined, $Lambda to infty$ and $lambda$ finite, in a res
onating valence-bond spin-liquid state. In a valence-bond solid, we find finite $lambda$ and $Lambda$, with $lambda$ of a single spinon significantly larger than the bound-state---the spinon is soft and shrinks as the bound state is formed. Both $lambda$ and $Lambda$ diverge upon approaching the critical point separating valence-bond solid and Neel ground states. We conclude that the spinon deconfinement is marginal in the lowest-energy state in the spin-1 sector, due to weak attractive spinon interactions. Deconfinement in the vicinity of the critical point should occur at higher energies.
Global dynamical behaviors of the competitive Lotka-Volterra system even in 3-dimension are not fully understood. The Lyapunov function can provide us such knowledge once it is constructed. In this paper, we construct explicitly the Lyapunov function
in three examples of the competitive Lotka-Volterra system for the whole state space: (1) the general 2-dimensional case; (2) a 3-dimensional model; (3) the model of May-Leonard. The dynamics of these examples include bistable case and cyclical behavior. The first two examples are the generalized gradient system defined in the Appendixes, while the model of May-Leonard is not. Our method is helpful to understand the limit cycle problems in general 3-dimensional case.
We develop a technique to directly study spinons (emergent spin S = 1/2 particles) in quantum spin models in any number of dimensions. The size of a spinon wave packet and of a bound pair (a triplon) are defined in terms of wave-function overlaps tha
t can be evaluated by quantum Monte Carlo simulations. We show that the same information is contained in the spin-spin correlation function as well. We illustrate the method in one dimension. We confirm that spinons are well defined particles (have exponentially localized wave packet) in a valence-bond-solid state, are marginally defined (with power-law shaped wave packet) in the standard Heisenberg critical state, and are not well defined in an ordered Neel state (achieved in one dimension using long-range interactions).
