To our best knowledge there is only one example of a lattice system with long-range two-body interactions whose ground states have been determined exactly: the one-dimensional lattice gas with purely repulsive and strictly convex interactions. Its ground-state particle configurations do not depend on the rate of decay of the interactions and are known as the generalized Wigner lattices or the most homogenenous particle configurations. The question of stability of this beautiful and universal result against certain perturbations of the repulsive and convex interactions seems to be interesting by itself. Additional motivations for studying such perturbations come from surface physics (adsorbtion on crystal surfaces) and theories of correlated fermion systems (recent results on ground-state particle configurations of the one-dimensional spinless Falicov-Kimball model). As a first step we have studied a one-dimensional lattice gas whose two-body interactions are repulsive and strictly convex only from distance 2 on while its value at distance 1 is fixed near its value at infinity. We show that such a modification makes the ground-state particle configurations sensitive to the decay rate of the interactions: if it is fast enough, then particles form 2-particle lattice-connected aggregates that are distributed in the most homogeneous way. Consequently, despite breaking of the convexity property, the ground state exibits the feature known as the complete devil's staircase.
To our best knowledge there is only one example of a lattice system with long-range two-body interactions whose ground states have been determined exactly: the one-dimensional lattice gas with purely repulsive and strictly convex interactions. Its ground-state particle configurations do not depend on the rate of decay of the interactions and are known as the generalized Wigner lattices or the most homogenenous particle configurations. The question of stability of this beautiful and universal result against certain perturbations of the repulsive and convex interactions seems to be interesting by itself. Additional motivations for studying such perturbations come from surface physics (adsorbtion on crystal surfaces) and theories of correlated fermion systems (recent results on ground-state particle configurations of the one-dimensional spinless Falicov-Kimball model). As a first step we have studied a one-dimensional lattice gas whose two-body interactions are repulsive and strictly convex only from distance 2 on while its value at distance 1 is fixed near its value at infinity. We show that such a modification makes the ground-state particle configurations sensitive to the decay rate of the interactions: if it is fast enough, then particles form 2-particle lattice-connected aggregates that are distributed in the most homogeneous way. Consequently, despite breaking of the convexity property, the ground state exibits the feature known as the complete devil's staircase.
The devil's staircase is a fractal structure that characterizes the ground state of one-dimensional classical lattice gases with long-range repulsive convex interactions. Its plateaus mark regions of stability for specific filling fractions which are controlled by a chemical potential. Typically, such a staircase has an explicit particle-hole symmetry; i.e., the staircase at more than half filling can be trivially extracted from the one at less than half filling by exchanging the roles of holes and particles. Here, we introduce a quantum spin chain with competing short-range attractive and long-range repulsive interactions, i.e., a nonconvex potential. In the classical limit the ground state features generalized Wigner crystals that--depending on the filling fraction--are composed of either dimer particles or dimer holes, which results in an emergent complete devil's staircase without explicit particle-hole symmetry of the underlying microscopic model. In our system the particle-hole symmetry is lifted due to the fact that the staircase is controlled through a two-body interaction rather than a one-body chemical potential. The introduction of quantum fluctuations through a transverse field melts the staircase and ultimately makes the system enter a paramagnetic phase. For intermediate transverse field strengths, however, we identify a region where the density-density correlations suggest the emergence of quasi-long-range order. We discuss how this physics can be explored with Rydberg-dressed atoms held in a lattice.
We present and analyze spin models with long-range interactions whose ground state features a so-called devil's staircase and where plateaus of the staircase are accessed by varying two-body interactions. This is in contrast to the canonical devil's staircase, for example, occurring in the one-dimensional Ising model with long-range interactions, where typically a single-body chemical potential is varied to scan through the plateaus. These systems, moreover, typically feature a particle-hole symmetry which trivially connects the hole part of the staircase (filling fraction $f\ensuremath{\ge}1/2$) to its particle part ($f\ensuremath{\le}1/2$). Such symmetry is absent in our models and hence the particle sector and the hole sector can be separately controlled, resulting in exotic hybrid staircases.
We consider a single-component gas of dipolar bosons confined in a one-dimensional optical lattice, where the dipoles are aligned such that the long-ranged dipolar interactions are maximally repulsive. In the limit of zero inter-site hopping and sufficiently large on-site interaction, the phase diagram is a complete devil's staircase for filling fractions between 0 and 1: every commensurate state at a rational filling is stable over a finite interval in chemical potential, and for every chemical potential the system is in a gapped commensurate phase. We perturb away from this limit in two experimentally motivated directions involving the addition of hopping and a reduction of the onsite interaction. The addition of hopping alone yields a phase diagram, which we compute in perturbation theory in the hopping, where the commensurate Mott phases now compete with the superfluid. We capture the physics of the Mott-superfluid phase transitions via bosonization. In a finite trap, we argue using an LDA and simulated annealing that this results in regions of commensurate states separated by patches with dominant superfluid correlations. Further softening of the onsite interaction yields alternative commensurate states with double occupancies which can form a devil's staircase of their own; we describe these states and discuss the possible transitions between them. Adding a hopping term in this case produces one-dimensional "supersolids" which simultaneously exhibit discrete broken symmetries and superfluidity. The contents of this work was first published as a chapter in the doctoral thesis `On Exotic Orders in Strongly Correlated Systems' (Princeton University, 09/09, supervised by S. L. Sondhi), and constitutes a considerably more detailed version of [PRB 80: 174519 (2009)].
The spinless Falicov-Kimball model on a two-dimensional square lattice is studied using the method of restricted phase diagrams constructed in the grand canonical ensemble. The results are compared with the one-dimensional model. Although the two-dimensional phase diagrams are more complex, with several distinct families of ion configurations occurring as ground states, there are surprising similarities with the one-dimensional case. Within each family of configurations, the ground states form a devil`s staircase structure and the configurations are constructed according to a composition rule identical to that in one dimension. It is also found that, as in one dimension, segregation occurs in the non-neutral model for large ion-electron interaction strength. Some features of the phase diagrams are understood by examining the effective two-body ion interaction.
The thermodynamic characteristics of one-dimensional lattice electron systems with long-range interelectron interactions are explicated and studied at arbitrary temperature T and pressure P. Only interactions between neighboring particles are taken into account. Electron-electron correlations are found to be controlled only by the ratio P/T, increases in which produce a continuous transition from a weakly nonideal gas to an electron ``crystal'' with filling factor 1/q,q=1,2, . . .. This is a remnant of the devil's staircase which obtains at T=O. It is shown that with a decrease in P the crystal periods, q, increase, the transitions occurring via narrow regions of a liquidlike state. Charge carriers in the ``crystals'' are discrete solitons with a fractional charge e∗=\ifmmode\pm\else\textpm\fi{}e/q. \textcopyright{} 1996 The American Physical Society.
Abstract Motivated by the realization of hard-wall boundary conditions in experiments with ultracold atoms, we investigate the ground-state properties of spin-1/2 fermions with attractive interactions in a one-dimensional box. We use lattice Monte Carlo methods to determine essential quantities like the energy, which we compute as a function of coupling strength and particle number in the regime from few to many particles. Many-fermion systems bound by hard walls display non-trivial density profiles characterized by so-called Friedel oscillations (which are similar to those observed in harmonic traps). In non-interacting systems, the characteristic length scale of the oscillations is set by ( 2 k F ) − 1 , where k F is the Fermi momentum, while repulsive interactions tend to generate Wigner-crystal oscillations of period ( 4 k F ) − 1 . Based on the non-interacting result, we find a remarkably simple parametrization of the density profiles of the attractively interacting case, which we generalize to the one-body density matrix. While the total momentum is not a conserved quantity in the presence of hard walls, the magnitude of the momentum does provide a good quantum number. We are therefore able to provide a detailed characterization of the (quasi-)momentum distribution, which displays rather robust discontinuity at the Fermi surface. In addition, we determine the spatially varying on-site density–density correlation, which in turn yields Tan’s contact density and, upon integration, Tan’s contact. As is well known, the latter fully determines the short-range correlations and plays a crucial role in a multitude of equilibrium and non-equilibrium sum rules.