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Equilibrium Phases of Dipolar Lattice Bosons in the Presence of Random Diagonal Disorder

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Equilibrium Phases of Dipolar Lattice Bosons in the Presence of Random Diagonal Disorder Equilibrium phases of dipolar lattice bosons in the presence of random diagonal disorder C. Zhang,1 A. Safavi-Naini,2 and B. Capogrosso-Sansone3 1Homer L. Dodge Department of Physics and Astronomy, The University of Oklahoma, Norman, Oklahoma ,73019, USA 2JILA , NIST and Department of Physics, University of Colorado, 440 UCB, Boulder, CO 80309, USA 3Department of Physics, Clark University, Worcester, Massachusetts, 01610, USA Ultracold gases offer an unprecedented opportunity to engineer disorder and interactions in a controlled manner. In an effort to understand the interplay between disorder, dipolar interaction and quantum degeneracy, we study two-dimensional hard-core dipolar lattice bosons in the presence of on-site bound disorder. Our results are based on large-scale path-integral quantum Monte Carlo simulations by the Worm algorithm. We study the ground state phase diagram at fixed half-integer filling factor for which the clean system is either a superfluid at lower dipolar interaction strength or a checkerboard solid at larger dipolar interaction strength. We find that, even for weak dipolar interaction, superfluidity is destroyed in favor of a Bose glass at relatively low disorder strength. Interestingly, in the presence of disorder, superfluidity persists for values of dipolar interaction strength for which the clean system is a checkerboard solid. At fixed disorder strength, as the dipolar interaction is increased, superfluidity is destroyed in favor of a Bose glass. As the interaction is further increased, the system eventually develops extended checkerboard patterns in the density distribution. Due to the presence of disorder, though, grain boundaries and defects, responsible for a finite residual compressibility, are present in the density distribution. Finally, we study the robustness of the superfluid phase against thermal fluctuations. I. INTRODUCTION large enough disorder always destroys superfluidity in fa- vor of the BG. In the weakly-interacting limit, interac- Since their first theoretical investigation three decades tions compete with Anderson localization thus enhanc- ago [1{4], many-body bosonic systems in the presence ing superfluidity [26, 32]. This results in sizable disorder of disorder have attracted a great deal of attention both strength needed in order to destroy superfluidity. On the experimentally and theoretically [5{26]. While Ander- other hand, in the strongly-interacting limit, interactions son localization [5,6, 11] and classical trapping of non- suppress superfluidity and, at integer filling, eventually interacting bosons [27] are well understood, the inter- completely destroy it, leaving the system either in the play between disorder, interaction and quantum degen- Mott insulator phase at lower disorder strength or in a eracy in strongly-correlated bosonic systems may give BG phase at larger disorder strength [21, 26]. rise to new fascinating phenomena. A certain degree In this paper, we consider dipolar lattice bosons in two- of disorder is ubiquitous in condensed matter systems, dimensions, as described by the extended Bose-Hubbard but a thorough understanding of these systems is hin- model, in the presence of on-site bound disorder. Dipolar dered by poor control over the nature of disorder and lattice systems are now accessible experimentally. They competing interactions. Ultracold gases, on the other can be realized with polar molecules [33, 34], atoms with hand, offer an unprecedented level of control over inter- large magnetc moments [35, 36], and Rydberg atoms [37{ actions and disorder. Specifically, interactions and dis- 39]. Unlike single-component atomic systems purely in- order can be tuned independently. Experimentally, the teracting via Van-der-Waals interactions, dipolar systems most common way to generate disorder is via optical interacting via the long-ranged and anisotropic dipolar speckle fields [13, 28, 29]. Other possibilities to engi- interaction can realize novel superfluid, solid, and topo- neer disorder include quasi-periodic potentials generated logical phases [40{46]. While in-depth theoretical studies by non-commensurate bichromatic optical lattices [30], of dipolar lattice bosons in presence of disorder are still the introduction of impurity atoms to the system [15], lacking, as suggested in [47], many of these phases may and holographic techniques which produce point-like dis- not be robust in presence of disorder. arXiv:1711.04879v1 [cond-mat.dis-nn] 13 Nov 2017 order [31]. In this paper we study two-dimensional hard-core dipo- A paradigmatic example is that of lattice bosons de- lar bosons trapped in a square optical lattice and in the scribed by the Bose-Hubbard model in the presence of presence of random on-site disorder and use large-scale on-site, bound disorder, where it was proven analyti- quantum Monte Carlo simulations by the Worm algo- cally and confirmed numerically that the gapless Bose rithm [48] to study the robustness of the equilibrium glass (BG) phase, characterized by finite compressibility phases to disorder. In particular we show that the in- and absence of off-diagonal long-range order, always in- terplay of the long-range interactions and the on-site dis- tervenes between the Mott-insulator and superfluid (SF) order leads to the suppression of the checkerboard (CB) state [25, 26]. Moreover, numerical studies at commensu- order, while enhancing the SF order, and stabilizing a rate and incommensurate filling factor have shown that BG phase. 2 II. SYSTEM HAMILTONIAN 10 BG The system is comprised of hard-core, dipolar lattice ● ● ● bosons in a 2d square lattice, with the dipole moments 8 ● aligned perpendicular to the lattice by an external static ● ● electric field so that the dipolar interaction is purely re- pulsive. At half-integer filling and in the absence of dis- 6 ● J order, the system is either in a SF state at lower dipo- / SF /J ● lar interaction strength or in a checkerboard (CB) solid Δ ∆ phase at larger dipolar interaction strength [41]. In the 4 0.60 ● 0.55 ▲▼ ■ presence of on-site disorder, the system is described by 0.50 L L * s ρ ⇢ 0.45 the Hamiltonian: ▲■ 0.40 ▼ 2 0.35 ● X X ninj X H = −J aya + V − (" − µ)n ; 0.30 i j r3 i i 3.4 3.5 3.6 3.7 3.8 3.9 hi ji i<j ij i V/JV/J 0 ● where the first term is the kinetic energy characterized by 0 1 2 3 4 5 6 hopping amplitude J. Here h· · · i denotes nearest neigh- V/J y boring sites, ai (ai) are the bosonic creation (annihila- V/J tion) operators satisfying the usual commutation rela- FIG. 1. (Color online) Phase diagram of the system described y y tions and the hard-core constraint ai ai = 0. The second by Eq.1 at filling factor n = 0:5. The horizontal and vertical term is the purely repulsive dipolar interaction charac- axes are the dipole-dipole repulsive interaction strength V=J terized by strength V = d2=a3, d is the induced dipole and the disorder strength ∆=J, respectively. Solid red circles moment and a is the lattice spacing, rij is the relative represent superfluid to insulator transition points as deter- distance (measured in units of a) between site i and site mined by standard finite size scaling. The solid line is a guide y to the eye. The black dotted line is the interaction strength j, ni = ai ai is the particle number operator. The third at which superfluidity disappears in favor of a checkerboard term is the chemical potential term with chemical poten- solid in the clean system. The inset shows finite size scal- tial µ shifted by the on site random disorder potential ing of the superfluid stiffness ρs. We plot ρsL vs. V=J at "i, where "i is uniformly distributed within the range ∆=J = 7 for system sizes L = 12, 16, 20 and 24 (red circles, [∆; −∆]. We set our unit of energy and length to be the blue squares, green up triangles and purple down triangles, hopping amplitude J and the lattice spacing a, respec- respectively). The transition point corresponds to the value tively. of V=J where curves referring to different system sizes cross. Here Vc=J = 3:68 ± 0:25. Error bars are within the symbols if not visible in the plots. III. GROUND STATE PHASE DIAGRAM In this section, we present our numerical results for the sizable disorder strength ∆=J ∼ 75 is needed in order to ground state phase diagram of model1 at fixed filling fac- destroy superfluidity [21]. This can be easily understood tor n = 0:5, as shown in Fig.1. The horizontal and ver- as follows. In the weak interaction limit, interactions tical axes are the dipole-dipole interaction strength V=J compete with Anderson localization resulting in an en- and the disorder strength ∆=J, respectively. Red circles hancement of superfluidity. Here, instead, the hard-core represent SF-insulator transition points. The red solid nature of bosons suppresses superfluidity even at weak line is a guide to the eye. At zero disorder, the system is dipolar interaction. As a matter of fact, for V=J < 3:5 either in the SF phase for V=J < 3:5, or the CB phase superfluidity is destroyed in favor of the BG at a nearly (dotted line) [41]. The SF phase is characterized by fi- constant value of disorder strength ∆=J ∼ 7 ÷ 8, that is, nite superfluid stiffness ρs which can be determined from the critical disorder strength is nearly independent of the the statistics of winding numbers in space. Specifically, dipolar interaction strength. Furthermore, at finite dis- 2 d−2 2 2 2 ρs = hW i=dL β, where W = Wx + Wy , Wx;y being order strength ∆=J . 7, the SF phase persists for dipolar the winding number in spatial directions x and y, d = 2 interaction V=J > 3:5, whereas the clean system featured is the spatial dimension, and β is the inverse tempera- a CB solid at this point.
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