Tunable anisotropic superfluidity in an optical kagome superlattice
We study the phase diagram of the Bose-Hubbard model on the kagome lattice with a broken sublattice symmetry. Such a superlattice structure can naturally be created and tuned by changing the potential offset of one sublattice in the optical generation of the frustrated lattice. The superstructure gives rise to a rich quantum phase diagram, which is analyzed by combining Quantum Monte Carlo simulations with the Generalized Effective Potential Landau Theory. Mott phases with non-integer filling and a characteristic order along stripes are found, which show a transition to a superfluid phase with an anisotropic superfluid density. Surprisingly, the direction of the superfluid anisotropy is changing between different symmetry directions as a function of the particle number or the hopping strength. Finally, we discuss characteristic signatures of anisotropic phases in time-of-flight absorption measurements.
Ultracold atoms in optical lattices are prominently used to simulate many-body systems in condensed matter physics qop1 (); qop2 (); qop3 (); qop4 (); qop5 (). One of the most striking experiments is the Mott insulator–superfluid quantum phase transition of ultracold bosons in optical lattice built with counter-propagating lasers bh1 (). It can be described by the seminal Bose-Hubbard model bh0 (); bh2 (), where each parameter is precisely adjustable in the experiment. With the rapid advances in experimental techniques toolbox () the many-body physics can now be analyzed on more complex lattice geometries. On the one hand, the lattice symmetry can be reduced by adding additional lasers or tuning their relative strength, leading to a superlattice structure sp1 (); sp2 (); sp3 (); sp4 (), which can give rise to insulator phases with fractional fillings sp5 (); sp6 (); sp7 (); sp8 (); tao (). On the other hand, it is also possible to enhance the residual entropy of the many-body system by using frustrated lattices, which have recently been realized using sophisticated optical techniques windpassinger (); sengstock (); klattice ().
Theoretically, many interesting phases have been predicted in frustrated lattices such as spin-liquids wen (); sl1 (); sl2 (); sl3 (); sl4 (); sl5 (); sl6 (), valence bond solids kl1 (), string excitations kl0 (), ordered metals tocchio (), chiral fractional edge states kl2 (), and supersolids tri_sc1 (); tri_sc2 (); tri_sc3 (); sellmann (). Unfortunately, however, in all these scenarios longer range interactions beyond the on-site Bose-Hubbard model are assumed, which require dipolar interactions and are experimentally much harder to handle. On the other hand, the intriguing interplay between a superlattice and the kagome lattice has never been explored before for the on-site Bose-Hubbard model. This is surprising since a superlattice structure can be created naturally in optically generated kagome lattices, and insulating phases with fractional filling occur without the need of longer range interaction as discussed below. We now analyze the detailed phase diagram of the Bose-Hubbard model on the kagome lattice with a tunable superlattice structure. In addition to the fractionally filled ordered phases in the quantum phase diagram, the most striking features are found in the unusual properties of the Bose condensed phase, where the anisotropic superfluid density spontaneously picks a preferred direction depending on filling and interaction. This leads to a characteristic signature in time-of-flight experiments.
Let us first consider the optical generation of a kagome lattice, which recently has been achieved in experiment by using standing waves from a long wavelength 1064 nm (LW) laser and from a short wavelength 532 nm (SW) laser, which are counterpropagating from three directions klattice (). The superposition of the corresponding two triangular lattices results in a kagome lattice if the laser strengths are exactly equal from all directions. Any slight variation of this setup results in a superlattice structure, which of course can in turn be used as an additional tunable parameter. For example, enhancing the potential from the LW laser in the -direction by a factor results in a combined optical potential
where in units of the longer wavelength . As depicted in Fig. 1 this potential leads to an offset of on one of the sublattices A. Note, however, that this offset preserves the parity symmetry along the - and the -direction and does not increase the unit cell of the kagome lattice, which contains three sites.
Interacting bosons on this lattice can be represented by Wannier states, which leads to the well-known Bose-Hubbard model for the description of the lowest band in second quantized language
where the nearest-neighbor hopping amplitude and the onsite interaction are tunable parameters, which depend on the scattering cross-section and the potential depth qop4 (). In principle, the potential shift also affects the Wannier states and hence other parameters in Eq. (2), but for reasonably small values of these higher order corrections can be neglected since they preserve the symmetry of the problem. The chemical potential is used to tune the particle number in the grand-canonical ensemble. In the following we will use the stochastic cluster series expansion algorithm sse1 (); sse2 (); scse () for unbiased quantum Monte Carlo (QMC) simulations of this model. In addition the Generalized Effective Potential Landau Theory (GEPLT) provides an analytic method to estimate the phase boundaries in an expansion of the hopping parameter tao (); santos ().
For vanishing hopping amplitude in the atomic limit, the competition between and can induce several incompressible insulating phases. When is less than , no site is occupied, and the Mott- phase is the energetically favored state. For a larger chemical potential , only sublattice A will be occupied with one boson per site while the other sites remain empty. This phase is therefore filled with an order in the form of occupied horizontal stripes. Such a striped density phase (SD) can also occur spontaneously in the extended Hubbard model when nearest and next-nearest interactions are included kl2 (). However, longer range interactions are notoriously difficult in optical lattices, so that the proposed superlattice is a convenient tool to study this phase. For positive values of , the system enters the familiar uniform Mott-1 insulator with filling factor one. Continuing this analysis for larger , we deduce that SD- phases with fractional filling factor occur for , which are separated by Mott- insulators with integer filling for .
Both the integer filled Mott- phases and the fractional SD- phases remain stable for small finite hopping . As shown in Fig. 2 for there are plateaus of the average density as a function of chemical potential, which are characteristic of those incompressible phases. In the fractionally filled SD- phases the density difference between the sublattices shows plateaus with a value that is slightly reduced from unity due to virtual quantum excitations. The plateau states are separated by compressible phases, which are characterized by a finite superfluid density, i.e. an off-diagonal order with a spontaneously broken U(1) gauge symmetry which will be analyzed in more detail below.
The corresponding phase diagram is mapped out in Fig. 3 using large scale QMC simulations. The second order GEPLT approximation is much less demanding and agrees quite well with the QMC data, except near the tips of the Mott lobes. With increasing offset the fractionally filled SD phases extend over a larger range not only in the chemical potential but also in the hopping . In fact, the SD-1 phase for is remarkably stable up to larger values of hopping than the uniform Mott-1 phase. The transitions to the superfluid phase are always of second order and can be understood in terms of additional condensed particles (holes) on top of the Mott states as the chemical potential is increased (decreased).
One interesting detail in the phase diagram in Fig. 3 is the drastic dependence on of the shape of the Mott-0 phase transition line in the limit of small hopping, which changes from linear behavior for to quadratic behavior for large . The linear dependence for can be understood from a competition of chemical potential with the kinetic energy, analogously to the quantum melting on the triangular lattice tri_sc1 (). For finite , on the other hand, the melting of the Mott-0 phase takes place by additional particles on the sublattice A only, which is not connected by any first-order hopping processes. In the limit of small , the kinetic energy of those particles is therefore determined by the second order hopping coefficient , which explains the quadratic behavior of the phase boundary. The exact shape of the Mott-0 transition can be determined from the single particle energy on the superlattice.
We now turn to the analysis of the order parameter in the superfluid phase. In the QMC simulations we determine the superfluid density along the lattice vector direction using the winding number and analogous for along the lattice direction winding (). We use a system with sites and periodic boundary conditions with unit cells in both the and directions, which ensures that for the perfect kagome lattice. Note, that in general the superfluid density is a response tensor with four elements in the --coordinate system ueda (). Due to reflection symmetry the off-diagonal elements must vanish. The relation to the superfluid densities along the lattice vectors is given by and .
In order to analyze a possible anisotropy we consider the average superfluid density and the difference between the two lattice vector directions in Fig. 4 as a function of filling . For finite offsets and the superfluid density is indeed anisotropic, but surprisingly also changes the preferred direction with increasing filling . For low densities just above the Mott-0 phase the superfluid density is dominated by virtual hopping processes between the A sublattice. As illustrated in the left inset of Fig. 4 this virtual hopping process is not possible along the lattice vector , which leads to an anisotropic superfluid density with .
When the filling reaches the superfluid density drops to zero in the SD-1 phase as expected, but then shows the opposite anisotropy for , which signals a different mechanism: At the A sublattice is completely filled, so that for slightly larger densities excess particles on the B and C sublattices are now responsible for the superfluid density. As shown in the right inset of Fig. 4, the B and C sublattices correspond to connected chains along the direction, which are disconnected by occupied A sites. This immediately explains why in this case.
According to this analysis, positive anisotropies are therefore a hallmark of an off-diagonal U(1) order parameter coexisting with a striped density order of a filled sublattice A. This situation is reminiscent of a supersolid where a stable density order exists on one filled sublattice and excess particles contribute to the superfluidity tri_sc1 (), with the main difference that in supersolids both the U(1) symmetry and the translational symmetry are spontaneously broken. Normally supersolid phases require longer range interactions beyond on-site, which are experimentally difficult to achieve. The creation of supersolid-like regions by introducing a superlattice is experimentally straight-forward, however. Similar to the ordinary supersolid, the supersolid-like regions considered here are also only stable for relatively small hopping, while for larger hopping the ”ordinary” superfluid behavior dominates as we will see below.
As long as the hopping is sufficiently small the alternation of anisotropies between Mott and SD phases continues as the density is increased due to the same reasoning as above. However, this is not the full story since for larger hopping or larger filling the Mott and SD phases are not stable, so it is not clear where the different regions of positive and negative are separated. Indeed as shown in Fig. 4, the superfluid density does not drop to zero for and , since the corresponding line is just outside the lobe of the SD-2 phase as shown in Fig. 3(b). Also the anisotropy no longer changes sign. We find that in the limit of large hopping the overall density becomes irrelevant. The sublattice A remains slightly more occupied for all values of and . Since particles on the A sublattice hardly hop in the direction, this leads to in the weak coupling limit . We call this behavior the ”ordinary” superfluid, in contrast to the supersolid-like regions of the positive anisotropy , which are basically confined between the lobes of the SD-n and Mott-n phases.
To analyze the crossover between different anisotropy regions we show the normalized anisotropy parameter as a function of for different values and in Fig. 5. For small hopping the anisotropy parameter is positive in supersolid-like regions () and negative between the Mott- and SD- phase () as discussed above. For larger the anisotropy parameters approach small negative values in all cases, corresponding to the ordinary superfluid. According to the analysis above, the sign-change of as a function of coincides with the delocalization of the particles on the A sublattice, which start to contribute to the superfluid density in the direction. This behavior can be interpreted as a continuous ”melting” of the supersolid-like phase to the ordinary superfluid, reminiscent of the melting of the sublattice order in an interaction driven supersolid tri_sc1 ().
Anisotropic superfluid densities appear in a variety of different systems such as dipolar Bose-Einstein condensates with disorder krumnow (); nikolic (); ghabour (), spin-orbit coupled Fermi gases devreese (), coupled spin dimer systems strassel (), and systems with rectangular shape sf (). However, an anisotropic superfluidity which is tunable by the isotropic hopping and changes sign when the order on one sublattice melts has not been discussed before to our knowledge.
The observation of the superfluid-Mott transition by time-of-flight (TOF) experiments has been pioneered many years ago bh1 (). The TOF absorption picture measures the momentum distribution and turns out to also show a clear signature of the anisotropy parameter. To demonstrate this effect, we used a QMC technique for calculating the off-diagonal long-range correlation function during the loop update off (), which allows a direct simulation of the TOF absorption signal. As shown in Fig. 5 for (upper inset) and for (lower inset) the TOF images display a clear signature of the anisotropy, which can be used for straight-forward measurements of the melting from supersolid-like to ordinary superfluid states.
In conclusion, we analyzed ultracold bosons in a kagome superlattice, which can be created and tuned by enhancing the long wavelength laser in one direction based on recent progress for creating highly frustrated lattices klattice (). By using numerical QMC simulations and the Generalized Effective Potential Landau Theory, we obtained the entire quantum phase diagram including Mott phases and fractionally filled charge density phases. In the superfluid phase an anisotropic superfluid density is found, which changes direction as the overall density or the hopping is changed. By tuning the hopping it is possible to induce a continuous melting from a supersolid-like state with a filled sublattice A and positive anisotropy parameter to an ordinary superfluid phase, which generically is characterized by a negative anisotropy parameter . Both the fractionally filled insulating phases and supersolid phases have received much attention by using models with longer-range interactions kl1 (); kl0 (); tocchio (); kl2 (); tri_sc1 (); tri_sc2 (); tri_sc3 (). Using the superlattice structure proposed in this work these phases become experimentally much more accessible by a simple laser setup instead of using dipolar interactions. Moreover, the characteristic signature of those effects can be measured in straight-forward TOF absorption experiments, without the need of single site resolution. In particular, by implementing off-diagonal measurements in QMC loop updates, it was possible to simulate TOF flight images which show a clear signature of the anisotropic superfluid density and the change of its direction, when the melting takes place.
Acknowledgements.X.-F. Zhang thanks for discussions with D. Morath about TOF calculations and with Y.C. Wen about superlattices. This work was supported by the Allianz für Hochleistungsrechnen Rheinland-Pfalz and by the German Research Foundation (DFG) via the Collaborative Research Center SFB/TR49.
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