Quantum field theory for the three-body constrained lattice Bose gas. II. Application to the many-body problem

We analyze the ground-state phase diagram of attractive lattice bosons, which are stabilized by a three-body onsite hardcore constraint. A salient feature of this model is an Ising-type transition from a conventional atomic superfluid to a dimer superfluid with vanishing atomic condensate. The study builds on an exact mapping of the constrained model to a theory of coupled bosons with polynomial interactions, proposed in a related paper [S. Diehl, M. Baranov, A. Daley, and P. Zoller, Phys. Rev. B 82, 064509 (2010).]. In this framework, we focus by analytical means on aspects of the phase diagram which are intimately connected to interactions, and are thus not accessible in a mean-field plus spin-wave approach. First, we determine shifts in the mean-field phase border, which are most pronounced in the low-density regime. Second, the investigation of the strong coupling limit reveals the existence of a “continuous supersolid,” which emerges as a consequence of enhanced symmetries in this regime. We discuss its experimental signatures. Third, we show that the Ising-type phase transition, driven first order via the competition of long-wavelength modes at generic fillings, terminates into a true Ising quantum critical point in the vicinity of half filling.

Figure 1

(Color online) Phase diagram for the attractive three-body hardcore constrained attractive Bose-Hubbard model. The lower curve (black online) is the mean-field result, while the middle and upper curves (blue and red online) correspond to $d=2,3$, respectively, with fluctuations taken into account. The bound state formation of dimers $(n=0)$ and di-holes $(n=2)$ indicated by crosses [the upper crosses (red online) for $d=2$ and the lower ones (blue online) for $d=3$], determines the endpoints of the critical lines (Sec. 3). A bicritical point, characterized by energetically degenerate but different orders (superfluid and charge-density wave) is reached asymptotically at half filling. It can be detected experimentally ramping a superlattice (Sec. 4). An Ising quantum critical point, connecting the two ordered phases, is predicted in the vicinity of half filling, while the correlation length is large but finite away from this point (Sec. 5).

Figure 2

(Color online) ASF-DSF phase border for the attractive three-body hardcore constrained Bose-Hubbard model. The lower curve (black online) is the mean field result, while the upper (red online) and the middle (blue online) curves correspond to $d=2,3$. The crosses denote the endpoints of the critical ines. For high densities, the $d=2,3$ results are very close to each other except for a tiny region close to maximal fillings.

Figure 3

Condensate depletion due to the atomic two-point function in two dimensions. The overall shape of the curve with zero crossing and sign change is determined by the function $c^2-{\left|s\right|}^2$ premultiplying the two-point function, cf. Eq. (25). The small overall size shows that the depletion effects produce only very tiny corrections to the phase border.

Figure 4

(Color online) Computations of the ground state on a 1D lattice with 60 sites for $U/J=-20$ using TEBD methods. (a) Correlation functions characterizing the CDW and DSF phases in a system with open boundary conditions plotted on a log-log scale as a function of distance $x$. Values are shown for $N=60$ particles, where the correlation functions are almost identical, and $N=50$ particles, where the decay of the CDW correlations are significantly more rapid than the DSF correlations. (b) Algebraic decay exponents $K_{\text{DSF}}$ and $K_{\text{CDW}}$ that are fitted to the envelope of the correlation functions for varying mean density $n$. Error bars show typical errors in the fitted decay.

Figure 5

(Color online) Numerical validation of symmetry enhancement $SO\left(2\right)\to SO\left(3\right)$. The plot shows the exponents $K_{\text{DSF}}$ and $K_{\text{CDW}}$ describing the algebraic decay of DSF and CDW order in the ground state as a function of the ratio $U/J$ at unit filling $n=1$. For sufficiently large interactions, the coincidence of the decay exponents signals the degeneracy of the two kinds of orders. These calculations were performed with TEBD methods for 60 particles on 60 sites. Open boundary conditions were used, but the decay exponents were fitted within the central 30 sites. The fitting errors are similar to those in Fig. 4b. Number of Schmidt coefficients retained in TEBD calculations $\chi =200$.

Figure 6

(Color online) Computations of the ground state with 30 particles on 60 lattice sites with $U/J=-20$, in the presence of a harmonic trap with onsite potential $V\left(x\right)$. (a) Shaded plot of the CDW correlation function $⟨n_x{n}_y⟩-⟨n_x⟩⟨n_y⟩$ (with interpolated colors, and diagonal elements not shown), showing substantial order near $x=y=20$ and $x=y=40$, where the mean filling factor $n\sim 1$ regions. $V\left(x\right)=V_{\text{tr}}{\left(x-30.5\right)}^2$, $V_{\text{tr}}=1.33\times {10}^{-3}$. (b) Shaded plot of the DSF correlation function $\text{DSF}\left(x-y\right)=⟨b_x^{†}{b}_x^{†}{b}_y{b}_y⟩$ (with interpolated colors), showing substantial order across the occupied region of the lattice, for the same trap parameters as part (a). (c) Density $n\left(x\right)$ for the same parameters as parts (a) and (b). (d) Plots of the DSF correlation function $\text{DSF}\left(x\right)$ for a trap $V\left(x\right)=V_{\text{tr}}{\left(x-30\right)}^2$, $V_{\text{tr}}=0.7/{30}^2\approx 7.78\times {10}^{-4}$ with and without an additional superlattice potential $V_{SL}/2{\sum }_i{(-)}^i$.