Strong-coupling phases of the spin-orbit-coupled spin-1 Bose-Hubbard chain: Odd-integer Mott lobes and helical magnetic phases

We study the odd-integer filled Mott phases of a spin-1 Bose-Hubbard chain and determine their fate in the presence of a Raman induced spin-orbit coupling which has been achieved in ultracold atomic gases; this system is described by a quantum spin-1 chain with a spiral magnetic field. The spiral magnetic field initially induces helical order with either ferromagnetic or dimer order parameters, giving rise to a spiral paramagnet at large field. The spiral ferromagnet-to-paramagnet phase transition is in a universality class with critical exponents associated with the divergence of the correlation length $\nu \approx 2/3$ and the order-parameter susceptibility $\gamma \approx 1/2$. We solve the effective spin model exactly using the density-matrix renormalization group, and compare with both a large-$S$ classical solution and a phenomenological Landau theory. We discuss how these exotic bosonic magnetic phases can be produced and probed in ultracold atomic experiments in optical lattices.

Figure 1

Comparison of the low-energy eigenvalues of $H_{\mathrm{BH}}$ at unit filling with the spectrum of $H_{\mathrm{spin}}$ for two lattice sites, shown as a function of $t/U_0$ for (a) uniform field and (b) SOC with pitch angle $\eta =\pi /5$. In both cases we take $h=0.001U_0$ and $U_2=-0.1U_0$. The spectra are indistinguishable for sufficiently small $t/U_0$ and the SOC causes avoided level crossings in the spectrum in the regime $t^2/U_0\sim h$. At larger $t$, as the spectra begin to differ quantitatively, the level ordering nonetheless remains identical.

Figure 2

Average spin along the chain in the SFM (a), (b), and (c), as well as the SPM (d) phase computed using DMRG with chain length $L=80$, bond dimension $M=50$, and $\theta =1.1\pi$. For a pitch $\eta =\pi /10$ the spiral is commensurate with $L$ and there are large finite-size effects for small $h$ near the chain boundaries; we have also confirmed this behavior in the classical model [see Figs. 6 and 6]. Away from the chain boundaries we find $\langle S^x\left(r\right)\rangle \propto cos\left(\eta r\right)$ and $\langle S^y\left(r\right)\rangle \propto sin\left(\eta r\right)$ consistent with the Landau theory (see Sec. 3c). For moderate fields the boundary effects are weak and $\langle S^z\left(r\right)\rangle$ is roughly constant in the center of the chain. Crossing the critical field $h_c(\theta =1.1\pi )/J=0.0798\left(3\right)$ we find $M_z=0$.

Figure 3

Spin susceptibility computed from DMRG as a function of momentum $q$ in the longitudinal (a) and transverse (b) directions with $L=80, M=50, \theta =1.1\pi$, and the legend is shared across both figures. In the SFM phase ${\chi }^z\left(q\right)$ is strongly peaked at $q=0$ with a peak height monotonically decreasing with increasing $h$ going to zero at $h_c$. The nonzero SOC produces peaks in ${\chi }^{\perp }\left(q\right)$ at $q=\pm \eta (=\pm 0.1\pi )$ and the peak continuously increases as a function of $h$.

Figure 4

Longitudinal correlation length ${\xi }^z$ defined in Eq. (17) for various system sizes as a function of $h$ for $\theta =0.6\pi$ (a) and (b) as well as $\theta =1.21\pi$ (c) and (d). We find a clear crossing in the data for ${\xi }^z/L$ vs $h$ for several $L$ and take this location as an unbiased estimate of $h_c$; we find $h_c(\theta =0.6\pi )=0.0266\left(2\right)$ and $h_c(\theta =1.21\pi )=0.0625\left(3\right)$. In the vicinity of the transition we perform finite-size scaling from Eq. (18) and collapse the data in terms of $L^{1/\nu }\delta$ and estimate the quality of collapse using a ${\chi }^2$ analysis [62]. We find $\nu =0.68\left(8\right)$ for $\theta =0.6\pi$ and $\nu =0.66\left(6\right)$ for $\theta =1.21\pi$.

Figure 5

Spin susceptibility as a function of system size $L$ and magnetic field $h$ for $\theta =0.6\pi$ (a) and $\theta =1.21\pi$ (b). In the SFM phase ${\chi }^z\left(0\right)\sim L$ (red) and in the SPM ${\chi }^z\left(0\right)\sim \mathrm{const}$ (black). At the critical point the spin susceptibility diverges with $L$ like ${\chi }^z\left(0\right)\sim L^{\gamma /\nu }$ (gray) and, using the value of $\nu$ found in Fig. 4, we find $\gamma =0.52\left(4\right)$ and $0.50\left(4\right)$ for $\theta =0.6\pi$ and $1.21\pi$, respectively.

Figure 6

Classical approximation results in the large-$S$ limit. In (a) we show a typical spin configuration corresponding to the SFM ($\theta =1.1\pi , h=0.1J$) In (b) we plot the order parameter of the SFM phase with increasing $h$, demonstrating the existence of a classical SFM to SPM transition and the effect of changing $\theta$. In (c) we show a 3D plot of a segment of the staggered phase, demonstrating that neighboring spins are nearly perpendicular (with an overall spiral component), resulting from a purely classical frustration due to a positive $K$ ($\theta =0.7\pi , h=0.1J$). This phase does not appear in the DMRG results, suggesting that quantum effects preserve the stability of the ferromagnet for positive $K$.

Figure 7

Critical field $h_c$ as a function of the pitch of the spiral $\eta$ for $\theta =0$, for DMRG (black squares) and the classical solution (red circles); the dashed lines are a fit to the mean-field result $h_c\propto {\eta }^2$. We find the deviation between the quantum and classical solutions grow with $\eta$ and the fit to the mean-field result only works well for the DMRG and classical data at small $\eta$. Note that $h_c(\eta =0.1\pi )$ from DMRG and the classical solution do deviate (see Fig. 13).

Figure 8

Dispersion from Landau theory truncating the effective action in Eq. (25) to quadratic order. As a prototypical example, we take $r_0/K=-0.5, u_0/K=0.2$, and $\eta =0.4\pi$, which has a critical field $h_c/K=3.53$. Evolution of the spin-wave dispersion as a function of increasing magnetic field $h/K=0.5$ (a), $h/K=1.0$ (b), $h/K=2.5$ (c), and $h/K=3.0$ (d).

Figure 9

Average nearest-neighbor spin product along the chain for $\theta =1.3\pi$ and $L=80$ in the SD phase (a), (b), (c), and in the SPM phase for $\theta >1.25\pi$ (d). In contrast, for $\theta <1.25\pi$ (e) is in the SFM phase and (f) is in the SPM phase. We find the effect of the magnetic field in the SD phase is to break the spin symmetry between the $x$ (red), $y$ (blue), and $z$ (black) components of the spin. For large fields in the SPM phase the translational symmetry is restored and the value of $\theta$ dictates the sign of the spin product.

Figure 10

Average spin along the chain for $\theta =1.3\pi$ in the SD phase (a) and in the SPM phase (b) for $L=80$ and $M=200$ and $x$ (red), $y$ (blue), and $z$ (black) spin components. The dimerization is visible in the average spin in the SD phase, whereas for large fields the translational symmetry is restored in the SPM phase.

Figure 11

Dimer order parameter defined in Eq. (26) for $\theta =1.3\pi$ as a function of $h$ and $L$. $D$ vs $h$ for various $L$ (a) and $D$ vs $L$ for various $h$ (b). In the SD phase we find the dimer order parameter is saturating to a constant for large $L$ and weak $h$. At large $h, D$ goes to zero at large $L$, and we take an estimate for transition from the smallest value of $h$ where $D$ still seems to vanish at large $L$.

Figure 12

Dimer correlations for $\theta =1.3\pi$ and $M=225$. (a) Dimer correlation function measured from the center of the chain ($r^{\prime }=L/2$) for various values of the spiral magnetic field and a system size $L=120$. For small fields in the SD phase $C_D(r,L/2)$ saturates to a constant at large $r$, whereas in the SPM phase it goes to zero. The dimer correlation length [from $C_D(r,L/2)$ in Eq. (17)] as a function of the magnetic field for various system sizes for $\theta =1.3\pi$ (b) and $\theta =1.4\pi$ (c). The crossing of ${\xi }_D/L$ estimates the location of the quantum phase transition into the SPM phase; here we find $h_c(\theta =1.3\pi )/J=0.033\left(8\right)$ and $h_c(\theta =1.4\pi )/J=0.25\left(2\right)$. The drift in the crossing with $L$ implies large corrections to finite-size scaling.

Figure 13

Phase diagram as a function of $h$ and $\theta$ for a fixed pitch $\eta =0.1\pi$ extracted from the DMRG calculations. The dashed line marks the classical SFM phase boundary, which completely misses the dimerized phase and underestimates the stability of the SFM for positive $\tilde{K}$. Error bars are determined based on the spread of the crossing in the corresponding correlation length (the error bars for SFM-to-SPM are smaller than the symbol). (Inset) Zoomed in on the region near $\theta =1.25\pi$, here we have added a schematic dashed red line, where we expect the critical field is exponentially small.

Figure 14

(a) Spin correlation length as a function of $h$ for $\theta =1.245\pi$. In close proximity to $\theta =1.25\pi$ we find the SFM to SPM transition remains continuous. (b) Quadrupolar correlation length for $\theta =1.28\pi$, as a function of $h$ for various systems sizes. We find ${\xi }_Q$ is going to zero with $L$ and monotonically decreases with increasing $h$. Thus the spiral magnetic field suppresses the quadrupolar correlations.