Supersolid phases in the magnetic fields

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Supersolid phases in the magnetic fields

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Ji-Guo Wang, Shi-Jie Yang ∗

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Department of Physics, Beijing Normal University, Beijing 100875, China

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Article history: Received 22 September 2016 Received in revised form 12 December 2016 Accepted 17 December 2016 Available online xxxx Communicated by F. Porcelli

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We study the ground state phases of the ultracold atomic condensates loaded in a two-dimensional optical lattice with the magnetic fields. Apart from uniform superfluid (SF) phase, four types of supersolid (SS) phases in the presence of the uniform magnetic fluxes and two types of SS phases in the presence of the staggered magnetic fluxes are found. For the system without magnetic flux, except for a certain unit phase factor φx( y ) = π , the magnetic field has no effect on the system. © 2016 Published by Elsevier B.V.

Keywords: ???

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1. Introduction

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The applications of the strong magnetic fields with charged particles have led to many discoveries, such as the integer and the fractional quantum Hall effects [1–3]. Meanwhile, many great developments, i.e., topological insulators [4,5] and fractional Chern insulators [6–8] are also found in condensed matter physics. Inspired by the realization of artificial gauge fields [9–12], the manipulation of the neutral atoms in the gauge fields has been revitalized in the ultracold atom community [13–20]. The effective magnetic fields of the optical lattices with atomic gases are realized by using Raman lasers and a potential energy gradient to imprint a phase on the motion of the atoms as they hop from site to site [16,17,21–30]. The neutral rubidium atoms that are loaded into the lasers induced periodic potentials provide the evidence for the realization of the Hofstadter–Harper (HH) Hamiltonian [30–33] by the two research groups [18–20]. The solid phase was predicted in many studies on the lattice boson systems in a weak magnetic field [34–37]. However, the structure of the ground state, in particular, the SS state is not known in the system of interacting bosons in the strong magnetic field. The SS has caused much attractions since it was first reported in 4 He [38]. However, the existence of the supersolid phase is still controversial. The report [39] by measuring the rotational inertia of a solid sample in torsional oscillator experiment raised the possibility of a SS phase in 4 He. The SS phase also exists in the hard-core bosonic Hubbard model with dipole–dipole interaction (DDI) [40–43] and the ultracold atoms with the spin orbit coupling

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(SOC) [44–46]. Anderson [47] argued that the ground state of the bosonic Hubbard model exhibits a SS phase at integer fillings with short-range interaction. In this paper, we consider the interacting ultracold atomic condensates in a two-dimensional optical lattice with the magnetic fields. The system shows the interesting physics far beyond the usual systems [25,48–50]. The magnetic flux breaks the translational invariant and time reversal invariant of the Hamiltonian [26, 51–53], the ground states of the system can be considered as the SS phases. Apart from the uniform SF phase, four types of SS phases, i.e., the stripe supersolid (SSS), triangular supersolid (TSS), rectangular supersolid (RSS) and checkerboard supersolid (CSS) phases are found in the presence of the uniform magnetic fluxes and two types of supersolid phases, i.e., the SSS and RSS phases are found in the presence of the staggered magnetic fluxes, where superfluidity and density-wave order coexist. For the system without magnetic flux, the density distribution of the unit phase factors φx = φ y = π is the same with the case if uniform or stagger flux per plaquette φ = π . It is remarkable that the SS phase occurs in our short-range model, with purely on-site interactions and nearest-neighbor hopping in the effective magnetic fields. The paper is organized as follows. In Sec. 2 we introduce a two-dimensional Bose–Hubbard model with an effective magnetic field. We present our main results and some discussions of the density distribution of the ultracold atomic condensates with effective uniform and staggered magnetic fluxes in Sec. 3.1 and without magnetic flux in Sec. 3.2. A brief summary is included in Sec. 4.

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2. Model and Hamiltonian

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Corresponding author. E-mail address: [email protected] (S.-J. Yang).

http://dx.doi.org/10.1016/j.physleta.2016.12.035 0375-9601/© 2016 Published by Elsevier B.V.

We first consider the ultracold atomic condensates in a twodimensional Bose–Hubbard model with an effective magnetic field.

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J.-G. Wang, S.-J. Yang / Physics Letters A ••• (••••) •••–•••

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The system can be described by an effective Hamiltonian [18,25, 30–33,49]

ˆ = −t H



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account the effects of the magnetic field that is applied perpendicular to the lattice, where A 0 (a) (a = x, y) is the vector potential and φ0 = 2π h¯ /e is the magnetic flux quantum. The integral is to be taken along the straight line path between the positions xi ,n and x j ,n ( ym,i and ym, j ) of the two sites. We can adjust the phase factors by combining with the experiments that adjust the Raman lasers strength and the potential energy gradients in Refs. [18,30]. In the tight-binding approximation limit, the wave function can

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In the paper, the coordinates of lattice sites are denoted by x = ml, y = nl (l is the lattice spacing) at the lattice site (m, n).

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3.1. Effective uniform and staggered magnetic fluxes

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Fig. 1. (Color online.) The density distribution of the ultracold atomic condensates with various uniform magnetic fluxes per plaquette, the type of phase factor is x = nφx ,  y = −mφ y from the site (m, n) to the site (m + 1, n + 1). From upper to lower the magnetic fluxes per plaquette φ are: π /2, π and 2π . From left to right column the phase factor along: the x direction, the y direction and both the x and y directions.

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m ,i

be written as |ψ = m,n ψm,n bˆ m,n |0, one arrives at the following coupled discrete nonlinear schrödinger (DNLS) equation (h¯ = 1).

∂ψm,n = −(ψm+1,n e i x + ψm−1,n e −i x ) ∂t

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is the bosonic annihilation (creation) opera-

ˆ describes the hoptor at lattice site (m, n). The first term of H ping amplitude t of bosons with the phase factor x( y ) along the x( y ) direction. In this paper, we set t = 1. The phase facx y tors x = φ1 x j,n A 0 (x)dx and  y = φ1 y m, j A 0 ( y )dy take into

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According to the definition of the phase factor in the section 2, we can obtain x = nφx ,  y = −mφ y from the site (m, n) to the site (m + 1, n + 1) along both the x and y directions, where φx( y ) is the magnetic flux per plaquette of the x( y ) direction. We restrict the magnetic flux per plaquette to be 0 ≤ φx( y ) < 2π throughout this paper. Similarly, we also can obtain x = nφx ,  y = 0 from the site (m, n) to site (m + 1, n) along the x direction and x = 0,  y = −mφ y from the site (m, n) to site (m, n + 1) along the y direction, respectively. The magnetic flux per plaquette is φ = φx +φ y in this system. A type of the effective uniform magnetic flux is realized in this model. Here, we set U = 0 for the single-particle problem. The single-particle excitation spectrum is determined by ˆ ψ = E ψ for all k values in the solving the Schrödinger equation H first magnetic Brillouin zone. The minimal single-particle excitation spectrum  = mink ( E ) as a function of φ is the celebrated Hofstadter butterfly [31,49]. The translational invariance of the lattice is broken by the magnetic flux per plaquette φ and the interaction U in our model, the system can form the SS phase. Fig. 1 displays the density distribution of the ultracold atomic condensates with uniform magnetic flux. When only considering the phase factor along one direction, i.e., the x direction φx = 0, φ y = 0 or the y direction φx = 0, φ y = 0, the ground state shows density wave order, in particular, unidirectional stripe order. A SSS phase with periodic density modulation along the corresponding direction is stabilized. The SSS phase is induced by the magnetic flux, and the density modulation period is h = 2π /φx( y ) , where h

is the lattice sites number between the nearest stripes, one can see in Figs. 1(a1–a2), (b1–b2) and (c1–c2). Figs. 1(a3), (b3) and (c3) show the magnetic flux of the both directions, i.e. φx = 0, φ y = 0, the ground states also show the density wave order, and are analogous to a SS. The CSS phases with the periodic density modulations along both directions are found when the magnetic flux per plaquette φ = 2π . For the magnetic flux per plaquette φ = 2π , the magnetic flux has no effect on the system and the CSS phase disappears. When the phase factor along the both directions, the magnetic flux per plaquette φ = 2φx( y ) leads to the density modulation period reduced by half compared with one direction. We also discuss another type of the effective uniform magnetic flux. The tunneling term of the x direction with a phase factor x = (m + n)φ from the site (m, n) to the site (m + 1, n), while the bare tunneling along the y direction  y = 0, where φ is the magnetic flux per plaquette. Fig. 2 shows the density distribution of the ultracold atomic condensates with this type of the uniform magnetic flux. The system is a uniform SF phase at φ = 0, as shown in Fig. 2(a). When the magnetic flux per plaquette is weak (φ = π /2), the system is an ordinary RSS phase. Further increasing φ drives a transition from the RSS phase to the TSS phase. Fig. 2(c) shows a TSS phase at φ = 4π /5. The system is a SSS phase at φ = π , as shown in Fig. 2(d). The ground states of the system also exhibit the RSS phases at φ = 5π /4, 7π /5, but the types of the RSS phases are different, one can see in Figs. 2(e) and (f). The system is dominated by the uniform SF phase, RSS phase, TSS phase, SSS phase and RSS phase upon increasing magnetic flux per plaquette. For this type of the uniform magnetic flux, when the flux per plaquette φ = p π /q, where p and q are integers with no common factor, the density modulation periods along the both directions contain q lattice sites at p = even number, and have 2q lattice sites at p = odd number. Fig. 2(g) illustrates the ground-state phase diagram versus the magnetic flux per plaquette φ . The system is a SF phase at φ = 0 (the solid blue square). With increasing the magnetic flux per plaquette strength, the system undergoes a phase transition from the SF phase to a phase with the periodic density modulations along the x and y directions, and hence can be considered as a SS state. Due to the magnetic flux per plaquette φ can adjust the tunneling amplitude, the SS phase features a triangular lattice and a rectangular lattice, respectively. The system favors a RSS phase at 0 < φ < π /2 and φ > π (the dash green lines) a TSS phase at

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Fig. 2. (Color online.) The density distribution of the ultracold atomic condensates with various uniform magnetic fluxes per plaquette, the type of phase factor is x = (m + n)φ ,  y = 0 from the site (m, n) to the site (m + 1, n). The magnetic fluxes per plaquette φ are: 0, π /2, 4π /5, π , 5π /4 and 7π /5. The insets are the amplified part of the density distribution. (g) The phase diagram with respect to the magnetic flux per plaquette φ .

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Fig. 4. (Color online.) The density distribution of the ultracold atomic condensates without magnetic flux, the type of phase factor is x = mφx ,  y = nφ y from the site (m, n) to the site (m + 1, n + 1). From upper to lower φ = φx + φ y are: π , 2π and 3π . From left to right column the phase factors along: the x direction, the y direction and both the x and y directions.

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Fig. 3. (Color online.) The density distribution of the ultracold atomic condensates with various staggered magnetic fluxes per plaquette, the type of the phase factor is x = (−1)m (m + n)φ ,  y = 0 from the site (m, n) to the site (m + 1, n). The magnetic fluxes per plaquette φ are: 0, π /5, π /2, 4π /5, π and 8π /5.

π /2 ≤ φ < π (the dash red line). The tunneling strength consists of alternating t and −t along the x direction at φ = π , the opposite signs of the tunneling give rise to a SSS phase at φ = π (the solid black square). The bare tunneling along the y direction  y = 0 and the tunneling term of the x direction with a phase factor x = (−1)m (m + n)φ from the site (m, n) to the site (m + 1, n) realize the effective staggered magnetic flux. Fig. 3 shows the density distribution of the ultracold atomic condensates with various staggered magnetic fluxes per plaquette φ . The staggered magnetic flux weaken the effect of the field compared with the uniform magnetic flux. Two types of the supersolid phases, i.e., the SSS and the RSS phases are found when the system with the staggered magnetic flux.

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adjusting the tunneling strength. From the site (m, n) to the site (m + 1, n + 1), the phase factor is x = mφx ,  y = nφ y . Similarly, we also obtain x = mφx ,  y = 0 from the site (m, n) to the site (m + 1, n) along the x direction and x = 0,  y = nφ y from the site (m, n) to the site (m, n + 1) along the y direction, respectively. Where φx( y ) is the unit phase factor along the x( y ) direction. The system is a uniform SF phase at φx = φ y = π , which shows the phase factor has no effect on the density distribution, one can see in Figs. 4(a1–a3) and (c1–c3). When φx = φ y = π , the tunneling strength consists of alternating t and −t along both the x and y directions, this is similar the effect of the magnetic flux per plaquette at φ = π . The density distribution in Fig. 4(b3) is the same with the Figs. 2(d) and 3(e).

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We discuss the density distribution of the atomic condensates with a position dependent artificial magnetic field, i.e. the system without magnetic flux. The phase factor x( y ) only as a role of

In this paper, we have investigated the ultracold atoms in a two-dimensional optical lattice with the magnetic fields. Apart from the uniform SF phase, we have demonstrated several types of the supersolid phases are induced by different types of the magnetic fluxes, where superfluidity and density-wave order coexist. There are four types of the supersolid phases in the uniform magnetic flux system and two types of the supersolid phases in the staggered magnetic flux system. For the same type of uniform magnetic flux, different RSS phases are induced by the different strengths of the magnetic fluxes per plaquette. For the system without magnetic flux, the density distribution of the unit phase factors φx = φ y = π is the same with the case of uniform or stagger flux per plaquette φ = π .

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Acknowledgements

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This work is supported by the NSF of China under grant No. 11374036, the National Basic Research Program of China under grant No. 2012CB821403. References [1] K.v. Klitzing, G. Dorda, M. Pepper, Phys. Rev. Lett. 45 (1980) 494. [2] D.C. Tsui, H.L. Stormer, A.C. Gossard, Phys. Rev. Lett. 48 (1982) 1559. [3] R.B. Laughlin, Phys. Rev. Lett. 50 (1983) 1395.

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