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Structural Superfluid-Mott Insulator Transition for a Bose Gas in Multi Structural Superfluid-Mott Insulator Transition for a Bose Gas in Multi-Rods Omar Abel Rodríguez-López,1, ∗ M. A. Solís,1, y and J. Boronat2, z 1Instituto de Física, Universidad Nacional Autónoma de México, Apdo. Postal 20-364, 01000 Ciudad de México, México 2Departament de Física, Universitat Politècnica de Catalunya, Campus Nord B4-B5, E-08034 Barcelona, España (Dated: Modified: January 12, 2021/ Compiled: January 12, 2021) We report on a novel structural Superfluid-Mott Insulator (SF-MI) quantum phase transition for an interacting one-dimensional Bose gas within permeable multi-rod lattices, where the rod lengths are varied from zero to the lattice period length. We use the ab-initio diffusion Monte Carlo method to calculate the static structure factor, the insulation gap, and the Luttinger parameter, which we use to determine if the gas is a superfluid or a Mott insulator. For the Bose gas within a square Kronig-Penney (KP) potential, where barrier and well widths are equal, the SF-MI coexistence curve shows the same qualitative and quantitative behavior as that of a typical optical lattice with equal periodicity but slightly larger height. When we vary the width of the barriers from zero to the length of the potential period, keeping the height of the KP barriers, we observe a new way to induce the SF-MI phase transition. Our results are of significant interest, given the recent progress on the realization of optical lattices with a subwavelength structure that would facilitate their experimental observation. I. INTRODUCTION close experimental realization of a sequence of Dirac-δ functions, forming the well-known Dirac comb poten- Phase transitions are ubiquitous in condensed matter tial [13], and could be useful to test many mean-field physics. In particular, at very low temperatures, quan- calculations on the weakly-interacting Bose gas in this tum phase transitions are the onset to distinguish new kind of potentials [14–18]. In the near future, we might phenomena in quantum many-body systems. Following see the realization of complex subwavelength optical lat- the realization of a Bose-Einstein condensate (BEC) in tices, including multiscale design, as a niche to observe 1995 [1,2], a new and successful research line has been new quantum phenomena [11, 19]. From the point of to study BEC within periodic optical lattices. It has view of theoretical simplicity, adaptable periodic struc- been possible to study the properties and behavior of tures ranging from the subwavelength to typical optical atomic gases trapped in three-dimensional (3D) multilay- lattices could be simulated and engineered by applying ers, multi-tubes, or a simple cubic array of dots through an external Kronig-Penney (KP) potential, for example, the superposition of two opposing lasers in one, two, or to a quantum gas. three mutually perpendicular directions [3,4], respec- In this paper, we analyze the physical properties of a tively. One of the most exciting achievements has been degenerate interacting 1D Bose gas in a lattice formed the observation of the superfluid (SF) to Mott insula- by a succession of permeable rods at zero temperature. tor (MI) phase transition in a BEC with repulsive in- To create this multi-rod lattice, we apply the KP poten- teractions, held both in a 3D [3,5] and one-dimensional tial [20] (1D) [6] optical lattices. 1 Although there have been great advances in the cre- X V (z) = V Θ(z (jl+a)) Θ(z (j+1)l) ; (1) ation of many types of optical lattices, efforts are still KP 0 − − − being made to overcome their limited spatial resolution, j=−∞ which is of the order of one-half the laser wavelength to the gas, where V0 is the barrier height and Θ(z) the λOL=2, to manipulate atoms. Recently, there has been Heaviside step function. The rods, distributed along a notorious interest and advances in developing tools to the z direction, have width b, and are separated by surpass the diffraction limit, not only in the field of cold empty regions of length a, such that the lattice period atoms but also in nanotechnology [7,8]. Nowadays, the is l a + b. Here, we use the KP potential to study physical realization of optical lattices formed by sub- quantum≡ gases due to its relatively simple shape, robust- arXiv:2101.03668v1 [cond-mat.quant-gas] 11 Jan 2021 wavelength (ultranarrow) optical barriers of width be- ness, and versatility while preserving the system essence low λOL=50 [9–12], is a reality. These so-called subwave- coming from the periodicity of its structure. We use it length optical lattices (SWOLs) can be seen as a very 2 to model a typical optical lattice V (z) = V0 sin (kOLz), with kOL = 2π/λOL, in the symmetric case b = a. Also, we model the subwavelength optical barriers in a SWOL through a Kronig-Penney lattice with . Further- ∗ [email protected]; https://orcid.org/0000-0002-3635- b a 9248 more, we take advantage of the KP potential versatil- y masolis@fisica.unam.mx ity by defining a new parameter b=a, and analyze how z [email protected] its variation affects the ground-state properties of the 2 Bose gas within a fixed-period multi-rod lattice. Then, (a) we show a new structural mechanism to induce a reen- trant, commensurate SF-MI-SF quantum phase transi- tion, by varying the ratio b=a while keeping the interac- tion strength and lattice period fixed. We propose that → ) z this transition, infeasible in experiments with typical op- ( tical lattices under similar conditions, could be observed KP in experiments with cold atoms in SWOLs. V 2l l 0 l 2l II. MODEL AND THEORY − − z (b) → (c) Our analysis relies on the ab-initio diffusion Monte Carlo (DMC) method [21], adapted for the calculation of pure estimators through the forward-walking tech- → → nique. The DMC method solves stochastically the many- ) ) z z body Schrödinger equation, providing exact results for ( ( the ground-state of the system within some statistical KP KP noise. In Fig.1, we show three faces of the KP potential V V depending on the ratio b=a while keeping V0 and l fixed: 2l l 0 l 2l 2l l 0 l 2l (a) the symmetric case where b = a, which is our refer- − − − − ence potential; (b) when the barriers become very thin, z z → → i.e. b a; and (c) when b a. The limits b=a 0 and b=a convert the KP potential in constant potentials! ! 1 Figure 1. Multi-rod lattices: (a) square lattice with b=a = 1; VKP = 0 and VKP = V0, respectively. The bosonic par- (b) very thin lattice with b=a = 1=10; (c) lattice with very ticles interact through a contact-like, repulsive potential broad barriers, b=a = 10. of arbitrary magnitude. Consequently, our system corre- sponds to the Lieb-Liniger (LL) Bose gas [22] within the multi-rod lattice Eq. (1). The Hamiltonian of the system and the Luttinger parameter K = ~πn1=mcs, play a cen- with N bosons is tral role in the description of the ground-state. In par- ticular, for any commensurate filling n l = j=p, with j 2 N 2 N 1 ~ X @ X and p integers, the system can undergo a SF-MI transi- H^ = + VKP(zi) +g1D δ(zi zj); (2) −2m @z2 − tion when takes the critical value 2 [23, 28], i=1 i i<j K Kc = 2=p where p is the commensurability order. For p = 1, i.e., for 2 an integer number of bosons per lattice site, . The where m is the mass of the particles, g1D 2~ =ma1D is Kc = 2 ≡ 1D Bose gas remains superfluid while , and vari- the interaction strength, and a1D is the one-dimensional K > Kc scattering length. In the absence of the multi-rod lattice, ations in the interaction strength and the lattice height we recover the LL Bose gas, which is exactly solvable for can push K towards Kc. The excitation energy spec- any magnitude of the dimensionless interaction param- trum is non-gapped and increases linearly with the quasi- eter 2 , with the momentum as E(k) = cs~ k in the SF state [27, 30], γ mg1D=~ n1 = 2=n1a1D n1 = N=L j j linear density.≡ whereas it develops an excitation gap ∆ in the MI phase, q The presence of a lattice can produce a quantum phase 2 2 E(k) = (cs~ k ) + ∆ [23]. Remarkably, in 1D gases, j j transition in the system from a superfluid state to a an arbitrarily weak periodic potential is enough to drive Mott-insulator state or vice-versa, commonly known as a Mott transition provided that the interactions are suf- Mott transition [5,6, 23]. In deep optical lattices, that ficiently strong [6, 29]. is, when the lattice height is way larger than the recoil A fingerprint of the Mott transition can be obtained energy 2 2 , the well-known Bose-Hubbard ER = ~ kOL=2m from the static structure factor S(k) [31, 32], (BH) model [3,5,6, 24–26] captures the essence of the transition. According to this model, the competition be- 1 2 S(k) n^ n^ − n^ ; (3) tween the on-site interaction U and the hopping energy J ≡ N h 1;k 1; ki − jh 1;kij between adjacent lattice sites drives the transition from the SF to the MI state. In the Mott insulator state, each where the operator n^1;k is the Fourier transform of the PN lattice site contains the same number of particles. On density operator n^1(z) δ(z zj).
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