# Pair Superfluidity of Three-Body Constrained Bosons in Two Dimensions

###### Abstract

We examine the equilibrium properties of lattice bosons with attractive on-site interactions in the presence of a three-body hard-core constraint that stabilizes the system against collapse and gives rise to a dimer superfluid phase formed by virtual hopping processes of boson pairs. Employing quantum Monte Carlo simulations, the ground state phase diagram of this system on the square lattice is analyzed. In particular, we study the quantum phase transition between the atomic and dimer superfluid regime and analyze the nature of the superfluid-insulator transitions. Evidence is provided for the existence of a tricritical point along the saturation transition line, where the transition changes from being first-order to a continuous transition of the dilute bose gas of holes. The Berzinskii-Kosterlitz-Thouless transition from the dimer superfluid to the normal fluid is found to be consistent with an anomalous stiffness jump, as expected from the unbinding of half-vortices.

###### pacs:

67.85.Hj,64.70.Tg,75.40.MgDue to their remarkable versatility, cold-atom systems are ideally suited to realize quantum simulators for the physics of strong correlations lewenstein07 (). A fascinating approach towards enhancing correlations in atomic systems steams from the fact that they can emerge via dissipative processes. In fact, dissipation-induced correlation effects were observed in an experiment with Feshbach molecules subject to strong inelastic collisions syassen08 (). More recently, it was found that three-body loss processes for bosons in an optical lattice give rise to an effective Bose-Hubbard model description with a local three-body hardcore constraint daley09 (). Such a constraint stabilizes the system in the presence of strong attractive interactions, where dimer bound states proliferate. The effective lattice model describing this situation is the Bose-Hubbard Hamiltonian

(1) |

where denotes the tunneling matrix element for nearest neighbor sites and an attractive on-site interaction daley09 (). The filling is controlled by varying the chemical potential , () denote bosonic annihilation (creation) operators and the number operator for bosons on lattice site . In contrast to the usual Bose-Hubbard model, the Hilbert space is now restricted by the contraint to a maximum of two bosons on each lattice site. In another recent proposal mazza10 (), a similar effective lattice model of bosons with a three-body constraint was derived for spin-1 atoms, which in addition includes an explicit correlated hopping term of the dimer bound states.

The model in Eq. (1) exhibits an intriguing phase diagram, shown in Fig. 1 for the case of a two-dimensional square lattice. Recent analytical calculations daley09 (); diehl09a (); diehl10 (); lee10 () and numerical works for the one-dimension case daley09 (); diehl09a (); diehl10 () exhibited that besides the trivially insulating phases at and , the system stabilizes two kinds of superfluid phases. The atomic superfluid (ASF) is characterized by a finite atomic condensate with and a finite superfluid response . For strong interactions however, a dimer (pair) superfluid phase (DSF) valatin58 () is stabilized, which is characterized by a vanishing atomic condensate and , but a finite dimer condensate density with an order parameter . Such single-component DSF phases have been observed before in models with explicit correlated or pair hopping processes schmidt06 (); zhou09 (); mazza10 (). In the DSF phase, the symmetry of the Hamiltonian is partially broken down to , and at the DSF to ASF transition, this remaining symmetry gets broken. With respect to this partial internal symmetry breaking, DSF are thus related to spin nematic states andreev84 (). Within Ginzburg-Landau theory, a Feshbach resonance term couples the ASF and DSF order parameter fields, which implies an effective symmetry similar as in the boson Feshbach resonance problem radzihovsky04 (); radzihovsky08 (); diehl09a (); lee10 (). From such analysis, the ASF-DSF transition was found to be Ising-like at unit filling, , and driven first-order by fluctuations via the Coleman-Weinberg mechanism coleman73 () for diehl09a (); diehl10 ().

Here, we study the ground state and thermal phase diagram of the three-body constrained Bose-Hubbard model in Eq. (1) using quantum Monte Carlo (QMC) simulations. We focus on a two-dimensional square lattice geometry, on which true off-diagonal long-range order (ODLRO) within the ASF and DSF regimes emerges in the ground state. Besides establishing the presence of ODLRO, we assess the above mentioned theory for the ASF-DSF transition diehl09a (); diehl10 () as well as a recent effective potential approach to the insulator-ASF quantum phase transitions lee10 (), which finds that they are driven first-order close to . We indeed obtain evidence for a tricritical point along one of these transition lines. The thermal Berzinskii-Kosterlitz-Thouless (BKT) transition out of the DSF phase is found to be consistent with an anomalous jump in the superfluid density, driven by the unbinding of half-vortices mukerjee06 ().

Method.– We employ a generalized directed loop algorithm in the stochastic series expansion (SSE) representation sandvik99b (); alet05 () at finite temperatures . The simulations are performed on finite square lattices of linear extend (and sites), with periodic boundary conditions, such that the superfluid density is obtained from the winding number fluctuations, pollock87 (); bernardet02 (). Since ODLRO in our system is forbidden at finite mermin66 (), we need to perform the simulations at sufficiently low to probe ground state properties of the finite system, as detailed below. Using two kinds of directed loops, in which the worm heads carry either a single creation (annihilation) operator () or a pair operator (), provides direct access to the equal-time Green’s function of the atoms as well as the dimers dorneich01 (); alet05 (). The atomic and dimer condensate densities are obtained as and respectively after extrapolations to the thermodynamic limit. A similar scheme with pair-worms was shown recently to be efficient for simulating two-component boson systems ohgoe10 (). Here, we find that accessing the dimer condensate density based on still becomes problematic at the relevant low temperatures: Histograms of individual measurements for (related to the lengths of the pair operator loops) exhibit fat-tailed distributions, i.e. the estimator of this quantity is dominated by rare events that make its sampling inefficient. A typical histogram within the DSF region is shown in Fig. 2. This behavior results form the fact that pairs of bosons proliferate at large near . A worm head carrying a bosonic pair operator performs an off-diagonal move (corresponding to the hopping of a dimer) only if it encounters a bond that shares a dimer (or an empty site, if the worm either carries annihilation or creation operators) and a single atom. Such processes are thus strongly suppressed in the relevant parameter regime.

Moreover, the fat tail of the histogram fits well to a Fréchet distribution. The exponent of the power-law decay in the tail is ( for the histogram in Fig. 2), so that the variance does not exist, and the central limit theorem for the mean value does not hold.

Hence, we resign to alternative estimates of the dimer condensate density in the DSF region. We employ two different means of adding correlated hopping terms to the original Hamiltonian that allow for efficient updates of the boson pairs. The results from both approaches agree within error bars, and with results based on in cases, where the distribution of the loop lengths leads to a finite variance. In the first approach, we add to the correlated hopping term , such that . The other approach couples directly to the Hamiltonian, such that features correlated hopping terms between all sites of the system (the diagonal term allows the insertion of long-range vertices into the SSE operator string). The coupling ensures an extensive energy. Besides the diagonal update of the short-range terms in , we insert/remove long-range vertices using heat-bath probabilities sandvik03 (). We can now measure based on the estimator , where denotes all correlated hopping terms in and the atomic kinetic energy term on bond . This method remains robust down to very low values of , so that we extract the condensate density of the model by fitting to a low-degree polynomial (the data is found to be essentially linear up to ), performing a bootstrap analysis for the error estimation. Such a scaling is shown in the right panel of Fig. 2, and extrapolations to the thermodynamic limit in the left panel.

ASF and DSF phases.– We concentrate on the fixed line (cf. Fig. 1) and vary in order to reach both the ASF and the DSF phases. The finite-temperature phase diagram along this line (cf. Fig. 1) exhibits the temperature scale at which the system undergoes a BKT transition, below which superfluidity emerges. The transition temperature was determined from the finite size scaling of with system sizes up to , following Ref. weber87 (). As seen from Fig. 1, the slope of the BKT-line changes near . For smaller , the system enters the DSF regime and the superflow is driven by virtual pair hopping processes. Correspondingly, we observe in the DSF region a strong even-odd effect in the winding number schmidt06 () (i.e. only even values are measured), which is absent in the ASF regime, cf. the left inset in Fig. 3. This allows to locate the phase boundary separating the ASF and DSF phases (diamonds in Fig. 1). Moreover, due to the nematic nature of the DSF, the BKT transition to this paired phase is driven by unbinding of half-vortices instead of the usual integer vortices as for e.g. the BKT transition to the ASF phase mukerjee06 (). This leads to an anomalous universal stiffness jump at instead of as for the ASF phase mukerjee06 (). This anomaly is consistent with the finite size behavior of shown in the inset of Fig. 1, and was accounted for in the determination of based on the scheme in Ref. weber87 (). To assess the extend of the ASF phase within the ground state, we measured for different system sizes at , which we found necessary in order to access the finite system’s ground state properties. The finite size data is shown in the main panel of Fig. 3. From extrapolations of to the thermodynamic limit such as shown in Fig. 2 (left panel), we find that below the atomic condensate density vanishes in the thermodynamic limit. Since remains finite below this value, we still expect the emergence of ODLRO from a dimer condensate beyond the ASF regime. In fact, the data of shown in Fig. 2 extrapolates to finite values both within the ASF and the DSF region.

ASF - DSF quantum phase transition.– Upon fixing , we cross the ASF-DSF quantum phase transition line slightly away from unit filling (the density for is about ), so it is expected that the transition at fixed is first-order diehl09a (); diehl10 (). However, as pointed out in Refs. diehl09a (), the correlation length in the vicinity of the Ising critical point at is expected to be large, diverging as . Close to , this severely exceeds the finite sizes accessible to our simulations. Based on this scenario, the accessible finite systems could be still controlled by the adjacent Ising critical point at . We indeed observe an approximate scaling in the data of close to the transition point, described by the standard finite-size scaling ansatz where , and denotes the transition point. In the inset of Fig. 3, we show a corresponding data collapse of the available finite-system data for from the main panel, using three-dimensional Ising critical exponents and hasenbusch99a () and (for a dynamical critical exponent ), giving . Unfortunately, we are not able to discern whether the quantum phase transition indeed is first-order or not (histograms of various observables did not exhibit any two-peak structures on the accessible system sizes at these low temperatures). Controlling as to fix for all finite systems in order to directly address the case also turned out unfeasible. Given the above estimate of , the mean-field value daley09 () is found to underestimate the stability of the DSF phase by about . The calculations of Ref. diehl09a (), while accounting for quantum fluctuations effects, still show a similar deviation.

ASF-insulator quantum phase transitions.– Next, we address the quantum phase transitions between the ASF and the (trivially) insulating phases at and . While SF to insulator transitions are often continuous, they can be driven first-order for attractive interactions kuklov04b (); radzihovsky08 (); lee10 (). We find from our QMC simulations that the transition from the empty system () to the ASF is indeed first-order over an extended parameter regime. This is evident from robust discontinuities in both the filling and superfluid density , such as those shown in the left panel of Fig. 4 at . From Ref. lee10 (), the transition is expected to turn continuous beyond a tricritical point near . Performing simulations down to , we did not obtain evidence for a continuous transition, indicating strong effects of the Feshbach resonance coupling for . The scenario of Ref. lee10 (), including a tricritical point, is found to be realized for the transition from the ASF to the full system (), which we find to be continuous beyond , cf. Fig. 1. The right panel of Fig. 4 shows the filling and the superfluid density along the line , where no discontinuities are observed. To assess if this transition corresponds to that of a dilute bose gas of holes sachdev99b (), we analyze the behavior close to the transition point in more detail. In that case and for two dimensions, the density of the holes should exhibit a linear increase with a logarithmic correction zhitomirsky98 () in the vicinity of the transition point at , while in the dilute limit bernardet02 (). As seen from Fig. 4 (right panel), both observables fit well to these functional forms; the dynamical critical exponent equals at this transition sachdev99b (), while our QMC temperature scales are sufficiently low to sample the finite system’s ground state.

Conclusion.– The DSF in the considered boson lattice model is restricted to a narrow region of parameter space; the anomalous jump in the stiffness at the BKT transition should however provide a direct experimental signature of this phase. The ASF-insulator transitions are found in overall agreement with the effective potential approach lee10 (), which however underestimates the Feshbach resonance coupling effects at low filling. We established the presence of a tricritical point along the ASF to insulator transition line; determining its precise location and critical properties remain challenges for future studies. It will also be interesting to explore finite temperature transitions between the DSF and the ASF.

###### Acknowledgements.

We thank H.P. Büchler, A. J. Daley, A.M. Läuchli, S. Manmana, L. Pollet, K.P. Schmidt, and B. Svistunov for discussions, and acknowledge the allocation of CPU time at HLRS Stuttgart and NIC Jülich. Support was also provided through the Studienstiftung des Deutschen Volkes (LB) and the DFG within SFB/TRR 21 (SW). Note added.– After the completion of our simulations, we became aware of recent results ng11 () on the thermal phase transitions in an extended three-body constrained boson lattice model, consistent with our findings.## References

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