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Physical review. B, Condensed matter

Commensurate and incommensurate ground states in a one-dimensional model

Béla Joós, B. Bergersen, Robert J Gooding, Michael Plischke

Published Jan 1, 1983• 12 citations
Publisher

Abstract

We present a renormalization argument for the discrete linear chain with a harmonic interparticle interaction and a sinusoidal substrate potential. We find that there exists a critical value of the substrate potential above which the incommensurate phase lying in between two successive registered phases is a complete devil's staircase of commensurate states. For weaker substrate potentials, the devil's staircase is partially incomplete. Near the commensurate-incommensurate transition it consists only of commensurate states, while at higher soliton densities the commensurate states are separated by truly incommensurate states. These results are in agreement with the findings of previous authors using different methods.

DOI
10.1103/physrevb.27.467

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We present a renormalization argument for the discrete linear chain with a harmonic interparticle interaction and a sinusoidal substrate potential. We find that there exists a critical value of the substrate potential above which the incommensurate phase lying in between two successive registered phases is a complete devil's staircase of commensurate states. For weaker substrate potentials, the devil's staircase is partially incomplete. Near the commensurate-incommensurate transition it consists only of commensurate states, while at higher soliton densities the commensurate states are separated by truly incommensurate states. These results are in agreement with the findings of previous authors using different methods.
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)].
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 intersite hopping and sufficiently large on-site interaction, the phase diagram is a complete devil's staircase for filling fractions between 0 and 1, wherein every commensurate state at a rational filling is stable over a finite interval in chemical potential. We perturb away from this limit in two experimentally motivated directions involving the addition of hopping and a reduction in the on-site 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. Further softening of the on-site interaction yields alternative commensurate states with double occupancies which can form a staircase of their own, as well as one-dimensional ``supersolids'' which simultaneously exhibit discrete broken symmetries and superfluidity.
In this paper we study the critical exponents of the quantum sine-Gordon model and U(1) Thirring models in the incommensurate phase. This phase appears when the chemical potential h exceeds a critical value and is characterized by a finite density of solitons. The low-energy sector of this phase is critical and is described by the Gaussian model (Tomonaga-Luttinger liquid) with the compactification radius dependent on the soliton density and the sine-Gordon model coupling constant $\ensuremath{\beta}.$ For a fixed value of $\ensuremath{\beta},$ we find that the Luttinger parameter K is equal to 1/2 at the commensurate-incommensurate transition point and approaches the asymptotic value ${\ensuremath{\beta}}^{2}/8\ensuremath{\pi}$ away from it. We describe a possible phase diagram of the model consisting of an array of weakly coupled chains. The possible phases are Fermi liquid, spin density wave, spin-Peierls, and Wigner crystal.
We present a renormalization-group-approach analysis of the commensurate-incommensurate transition of the two-dimensional classical sine-Gordon system which includes incommensurability as a new renormalizable parameter. The critical behavior of various quantities shows corrections with respect to the simpler self-consistent harmonic approximation (SCHA). However, our result for the phase boundary still has a SCHA-type TTT dependence, suggesting that incommensurability tends to suppress the role of fluctuations.
We study fermions on a finite chain, interacting repulsively when residing on the same and on nearest-neighbor sites, and subjected to a Wannier-Stark linearly varying potential. Using the density matrix renormalization-group numerical technique to solve this generalized extended Hubbard model, the ground state exhibits a staircase of (quasi) plateaus in the average local site density along the chain, decreasing from being doubly filled to empty as the potential increases. These ``plateaus'' represent locked-in commensurate phases of charge density waves together with band and Mott insulators. These phases are separated by incompressible regions with incommensurate fillings. These results differ from the many-body localization proposed for this model earlier. It is suggested that experimental variations of the slope of the potential and the range of the repulsive interactions will produce such a coexistence of phases which have been individually expected theoretically and observed experimentally for uniform systems.
A modified density-matrix renormalization-group (DMRG) method is introduced and applied to classical two-dimensional models: the anisotropic triangular nearest-neighbor Ising model and the anisotropic triangular next-nearest-neighbor Ising model. Phase diagrams of both models have complex structures and exhibit incommensurate phases. It was found that the incommensurate phase completely separates the disordered phase from one of the commensurate phases, i.e., the nonexistence of the Lifshitz point is conjectured in phase diagrams of both models with the DMRG method. It is shown that the DMRG method is also an effective-field approximation, but in spite of that it yields the correct value of the domain wall density critical exponent.