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arXiv:0802.2498 · Physical Review A

Ultracold bosons in disordered superlattices: Mott insulators induced by tunneling

Dominik Muth, Alexander Mering, Michael Fleischhauer

Published Apr 15, 2008• 13 citations
arXivHTMLPDF

Abstract

We analyze the phase diagram of ultracold bosons in a one-dimensional superlattice potential with disorder, using the time-evolving block decimation algorithm for infinite-sized systems. For degenerate potential energies within the unit cell of the superlattice, loophole-shaped insulating phases with noninteger filling emerge with a particle-hole gap proportional to the boson hopping. Addition of a small amount of disorder destroys this gap. For not too large disorder, the loophole Mott regions detach from the axis of vanishing hopping, giving rise to insulating islands. Thus the system shows a transition from a compressible Bose glass to a Mott-insulating phase with increasing hopping amplitude. We present a straightforward effective model for the dynamics within a unit cell which provides a simple explanation for the emergence of Mott-insulating islands. In particular, it gives rather accurate predictions for the inner critical point of the Bose glass to Mott insulator transition.

DOI
10.1103/physreva.77.043618

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We analyze the phase diagram of ultracold bosons in a one-dimensional superlattice potential with disorder, using the time-evolving block decimation algorithm for infinite-sized systems. For degenerate potential energies within the unit cell of the superlattice, loophole-shaped insulating phases with noninteger filling emerge with a particle-hole gap proportional to the boson hopping. Addition of a small amount of disorder destroys this gap. For not too large disorder, the loophole Mott regions detach from the axis of vanishing hopping, giving rise to insulating islands. Thus the system shows a transition from a compressible Bose glass to a Mott-insulating phase with increasing hopping amplitude. We present a straightforward effective model for the dynamics within a unit cell which provides a simple explanation for the emergence of Mott-insulating islands. In particular, it gives rather accurate predictions for the inner critical point of the Bose glass to Mott insulator transition.
We study the superfluid-insulator transition in a one-dimensional system of interacting bosons, modeled as a disordered Josephson array, using a strong-randomness real-space renormalization-group technique. Unlike perturbative methods, this approach does not suffer from run-away flows and allows us to study the complete phase diagram. We show that the superfluid-insulator transition is always Kosterlitz-Thouless like in the way that length and time scales diverge at the critical point. Interestingly however, we find that the transition at strong disorder occurs at a nonuniversal value of the Luttinger parameter, which depends on the disorder strength. This result places the transition in a universality class different from the weak disorder transition first analyzed by Giamarchi and Schulz [Europhys. Lett. 3, 1287 (1987)]. While the details of the disorder potential are unimportant at the critical point, the type of disorder does influence the properties of the insulating phases. We find three classes of insulators which arise for different classes of disorder potential. For disorder only in the charging energies and Josephson coupling constants, at integer filling we find an incompressible but gapless Mott-glass phase. If both integer and half-integer filling factors are allowed then the corresponding phase is a random-singlet insulator, which has a divergent compressibility. Finally in a generic disorder potential the insulator is a Bose glass with a finite compressibility.
The phase diagrams and phase transitions of bosons with short-ranged repulsive interactions moving in periodic and/or random external potentials at zero temperature are investigated with emphasis on the superfluid-insulator transition induced by varying a parameter such as the density. Bosons in periodic potentials (e.g., on a lattice) at T=0 exhibit two types of phases: a superfluid phase and Mott insulating phases characterized by integer (or commensurate) boson densities, by the existence of a gap for particle-hole excitations, and by zero compressibility. Generically, the superfluid onset transition in d dimensions from a Mott insulator to superfluidity is ``ideal,'' or mean field in character, but at special multicritical points with particle-hole symmetry it is in the universality class of the (d+1)-dimensional XY model. In the presence of disorder, a third, ``Bose glass'' phase exists. This phase is insulating because of the localization effects of the randomness and analogous to the Fermi glass phase of interacting fermions in a strongly disordered potential.The Bose glass phase is characterized by a finite compressibility, no gap, but an infinite superfluid susceptibility. In the presence of disorder the transition to superfluidity is argued to occur only from the Bose glass phase, and never directly from the Mott insulator. This zero-temperature superfluid-insulator transition is studied via generalizations of the Josephson scaling relation for the superfluid density at the ordinary \ensuremath{\lambda} transition, highlighting the crucial role of quantum fluctuations. The transition is found to have a dynamic critical exponent z exactly equal to d and correlation length and order-parameter correlation exponents \ensuremath{\nu} and \ensuremath{\eta} which satisfy the bounds \ensuremath{\nu}\ensuremath{\ge}2/d and \ensuremath{\eta}\ensuremath{\le}2-d, respectively. It is argued that the superfluid-insulator transition in the presence of disorder may have an upper critical dimension dc{d}_{c}dc​ which is infinite, but a perturbative renormalization-group calculation wherein the critical exponents have mean-field values for weak disorder above d=4 is also discussed. Many of these conclusions are verified by explicit calculations on a model of one-dimensional bosons in the presence of both random and periodic potentials. The general results are applied to experiments on 4He^{4}\mathrm{He}4He absorbed in porous media such as Vycor. Some measurable properties of the superfluid onset are predicted exactly [e.g., the exponent x relating the \ensuremath{\lambda} transition temperature to the zero-temperature superfluid density is found to be d/2(d-1)], while stringent bounds are placed on others. Analysis of preliminary data is consistent with these predictions.
We analyze various quantum phases of ultracold bosonic atoms in a periodic one-dimensional optical superlattice. Our studies have been performed using the finite-size density-matrix renormalization group method in the framework of the Bose-Hubbard model. Calculations have been carried out for a wide range of densities and the energy shifts due to the superlattice potential. At commensurate fillings, we find the Mott insulator and the superfluid phases as well as Mott insulators induced by the superlattice. At a particular incommensurate density, the system is found to be in the superfluid phase coexisting with density oscillations for a certain range of parameters of the system.
We investigate the behavior of the quasiparticle energy gap near quantum phase transitions in a two-dimensional disordered boson Hubbard model at a commensurate filling. Via Monte Carlo simulations of ensembles with fixed numbers of particles, we observe the behavior of the gap as a function of the tuning parameter for various strengths of diagonal disorder. For weak disorder, we find that gapped Mott insulating phase is sustained up to the transition point and disappears only in a superfluid, strongly supporting a direct Mott-insulator-to-superfluid transition. Bose glass behavior, insulating with vanishing gap, appears only when the strength of disorder is bigger than a critical value.
We establish the phase diagram of the disordered three-dimensional Bose-Hubbard model at unity filling which has been controversial for many years. The theorem of inclusions, proven by Pollet et al. [Phys. Rev. Lett. 103, 140402 (2009)] states that the Bose-glass phase always intervenes between the Mott insulating and superfluid phases. Here, we note that assumptions on which the theorem is based exclude phase transitions between gapped (Mott insulator) and gapless phases (Bose glass). The apparent paradox is resolved through a unique mechanism: such transitions have to be of the Griffiths type when the vanishing of the gap at the critical point is due to a zero concentration of rare regions where extreme fluctuations of disorder mimic a regular gapless system. An exactly solvable random transverse field Ising model in one dimension is used to illustrate the point. A highly nontrivial overall shape of the phase diagram is revealed with the worm algorithm. The phase diagram features a long superfluid finger at strong disorder and on-site interaction. Moreover, bosonic superfluidity is extremely robust against disorder in a broad range of interaction parameters; it persists in random potentials nearly 50 (!) times larger than the particle half-bandwidth. Finally, we comment on the feasibility of obtaining this phase diagram in cold-atom experiments, which work with trapped systems at finite temperature.
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