Homogeneous and inhomogeneous magnetic phases of constrained dipolar bosons

We study the emergence of several magnetic phases in dipolar bosonic gases subject to the three-body loss mechanism employing numerical simulations based on the density matrix renormalization group (DMRG) algorithm. After mapping the original Hamiltonian in spin language, we find a strong parallelism between the bosonic theory and the spin-1 Heisenberg model with single-ion anisotropy and long-range interactions. A rich phase diagram, including ferromagnetic, antiferromagnetic, and nonlocal ordered phases, emerges in the one-dimensional case and is preserved even in the presence of a trapping potential.

Figure 1

Homogeneous phase diagram for dipolar bosons on an optical lattice with three-body hard core constraint at filling $\mathrm{n̄}=1$ (see text). Triangles, squares, diamonds, and black and red points denote numerical results, while the black dashed line describes an approximate strong coupling description for $\left|U\right|\gg J$.

Figure 2

Cartoon of magnetic phases related to the model in Eqs. (1), (3). From top to bottom, ferromagnetic, Haldane, and Néel phase in bosonic and spin language (see text).

Figure 3

Superfluid correlations in double logarithmic scale as a function of the distance from the middle in a $L=120$ chain. Red (dashed) and black (thick) lines represent DSF and SF phase, respectively, with $U=-6.5,\Lambda =0$ and $U=-3,\Lambda =0.05$.

Figure 4

Magnetic order parameters in double logarithmic scale as a function of the distance from the middle in a $L=120$ chain. Black (dashed), red (thick), and green (dot-dashed) represent $(U=-3,\Lambda =0.05),(-3,0.7),(-3,1.3)$, respectively.

Figure 5

${\mathcal{C}}_z(x)$ correlation function at the HI-NP boundary for chains of different lengths $L=60,80,100,120$, in panels a, b, c, and $d$, respectively; correlations are taken with respect to the center of the chain. Here, $U=-3$, and from top to bottom, $\Lambda =1.1$ (orange, dotted), $\Lambda =1.05$ (blue, dashed), $\Lambda =1$ (green, dot-dashed), $\Lambda =0.95$ (red, thick), and $\Lambda =0.9$ (black, dot-dot-dashed).

Figure 6

String correlator ${\mathcal{O}}_x(x)$ at the HI-NP boundary for chains of different lengths $L=60,80,100,120$, in panels a, b, c, and $d$, respectively; correlations are taken with respect to the center of the chain. Here, $U=-3$, and from top to bottom, $\Lambda =0.9$ (black, dot-dot-dashed), $\Lambda =0.95$ (red, thick), $\Lambda =1$ (green, dot-dashed), $\Lambda =1.05$ (blue, dashed), and $\Lambda =1.1$ (orange, dotted).

Figure 7

Left panel: Dependence of $\Delta ={\Delta }_1-{\Delta }_2$ on $U$ for $\Lambda =-0.1$ and different chain lengths: $L=8$ (black circles), $L=12$ (red squares), $L=16$ (green stars), $L=20$ (blue diamonds), and $L=24$ (orange triangles). Dashed lines are guides for the eye. Right panel: Critical value of the SF-DSF transition as a function of $1/L$ for $\Lambda =-0.1$; circles represent numerical data, line is a best fit of the type $a_1+a_2/L+a_3/L^2$.

Figure 8

Finite-size scaling of ${\mathcal{O}}_z(L/2)$ for $U=-0.8$ by using Eq. (7) (see text). From top to bottom: $\Lambda =0.5,0.45,0.4,0.35,0.3,0.25,0.2,0.15$, and $0.1$.

Figure 9

Upper panel: String order parameter $C_1$ for $U=-0.8,-2$ (black circles and red squares, respectively) as a function of $\Lambda$. Lower panel: $C_1$ for $\Lambda =0.1,0.2,0.3$ (black squares, red triangles, and blue circles, respectively) as a function of $U$. In both panels, lines are guide for the eye, and the size of each point denotes the maximum error on the extrapolated value $C_1$.

Figure 10

Density distribution as a function of the distance from the trap center for $U=-2.5,\Lambda =0.9$. Top panel: Fixed population $N=40$ and different trap strength $k=7.5k^{*}$ (black, dotted), $8.5k^{*}$ (red, thick), $9.5k^{*}$ (green, dashed), and $10.5k^{*}$ (blue, dot-dashed). Bottom panel: Fixed trap strength $k=9k^{*}$ and different populations: $N=36$ (black, dot-dashed), 38 (red, dashed), 40 (green, thick), 42 (blue, dot-dot-dashed), and 44 (orange, dotted).

Figure 11

Magnetic order parameters ${\mathcal{C}}_z(L/2,j)$ (top panel) and ${\mathcal{O}}_z(L/2,j)$ (bottom panel) as a function of the distance from the trap center for $U=-2.5,\Lambda =0.9,N=40$, and different values of $k$: $k=7.5k^{*}$ (black, dotted), $8.5k^{*}$ (red, thick), $9.5k^{*}$ (green, dashed), and $10.5k^{*}$ (blue, dot-dashed).

Figure 12

Magnetic order parameters ${\mathcal{C}}_z(L/2,j)$ (top panel) and ${\mathcal{O}}_z(L/2,j)$ (bottom panel) as a function of the distance from the trap center for $U=-2.5,\Lambda =0.9,k=9k^{*}$, and different number of particles $N=36$ (black, dot-dashed), 38 (red, dashed), 40 (green, thick), 42 (blue, dot-dot-dashed), and 44 (orange, dotted).

Figure 13

Density distribution as a function of the distance from the trap center for $U=-4,\Lambda =1.5$. Top panel: Fixed population $N=40$ and different trap strength $k=5k^{*}$ (black, dotted), $7k^{*}$ (red, thick), $9k^{*}$ (green, dashed), $11k^{*}$ (blue, dot-dashed), and $20k^{*}$ (orange, dot-dot-dashed). Bottom panel: Fixed trap strength $k=12k^{*}$ and different populations: $N=36$ (black, dot-dashed), 38 (red, dashed), 40 (green, thick), and 42 (blue, dot-dot-dashed).

Figure 14

Magnetic order parameters ${\mathcal{C}}_z(L/2,j)$ (top panel) and ${\mathcal{O}}_z(L/2,j)$ (bottom panel) as a function of the distance from the trap center for $U=-4,\Lambda =1.5,N=40$, and different values of $k$: $k=5k^{*}$ (black, dotted), $7k^{*}$ (red, thick), $9k^{*}$ (green, dashed), $11k^{*}$ (blue, dot-dashed), and $20k^{*}$ (orange, dot-dot-dashed).

Figure 15

Magnetic order parameters ${\mathcal{C}}_z(L/2,j)$ (top panel) and ${\mathcal{O}}_z(L/2,j)$ (bottom panel) as a function of the distance from the trap center for $U=-4,\Lambda =1.5,k=12k^{*}$, and different number of particles: $N=36$ (black, dot-dashed), 38 (red, dashed), 40 (green, thick), and 42 (blue, dot-dot-dashed).