Supersolid in a one dimensional model of hardcore bosons

Supersolid in a one dimensional model of hardcore bosons

Tapan Mishra, Ramesh V. Pai and Subroto Mukerjee International Center for Theoretical Sciences, TIFR, Bangalore, 560 012, India Department of Physics, Goa University, Taleigao Plateau, Goa 403 206, India Department of Physics, Indian Institute of Science, Bangalore, 560 012, India Centre for Quantum Information and Quantum Computing (CQIQC), Indian Institute of Science, Bangalore, 560 012, India
July 4, 2019

We study a system of hardcore boson on a one-dimensional lattice with frustrated next-nearest neighbor hopping and nearest neighbor interaction. At half filling, for equal magnitude of nearest and next-nearest neighbor hopping, the ground state of this system exhibits a first order phase transition from a Bond-Ordered (BO) solid to a Charge-Density-Wave(CDW) solid as a function of the nearest neighbor interaction. Moving away from half filling we investigate the system at incommensurate densities, where we find a SuperSolid (SS) phase which has concurrent off-diagonal long range order and density wave order which is unusual in a system of hardcore bosons in one dimension. Using the finite-size Density-Matrix Renormalization Group (DMRG) method, we obtain the complete phase diagram for this model.

75.40.Gb, 67.85.-d, 71.27.+a

I Introduction

Supersolid phases of matter which feature both off diagonal superfluid order and long range crystalline order have been a subject of intense research in the last decade. While there is still no clear evidence for the occurrence of this phase in solid Helium kim_2012 (), there are proposals for creating such a state in optical lattices of cold atoms kellmann_2009 (). Model Hamiltonians which describe supersolid phases exist, of which one of the most rigourously studied is one of hardcore bosons on a lattice with further neighbor interactions herbert_2001 (); batrouni_ss (); boninsegni (); wessel (); damle (); melko (). Supersolid phases have also been predicted in models of softcore bosons and binary mixturesbatrouni_prl_2005 (); mishrass (); mishra_mixture (); batrouni_bose_fermi () and quantum spin systemslee (); mila (). It has been proposed that a system of polar gases in optical lattices is a suitable test bed to observe this exotic phase of matter. Pioneering experiments on chromium Bose-Einstein condensates (BECs) Griesmaier2005 () have been recently followed by the realization of quantum gases in other highly-magnetic species, including dysprosium Bose and Fermi gases Lu2011 () and erbium condensates Aikawa2012 (). Significantly more dipolar gases may be realized by means of polar molecules, which have large electric dipole moments of the order of the Debye or larger. Seminal experiments on KRb molecules at JILA Ni2008 () has opened the door towards achieving a quantum degenerate gas of polar molecules, and various experimental groups worldwide are currently involved in this enterprise Wu2012 (); Takekoshi2012 (). Rydberg gases constitute yet another possible realization of highly dipolar gases Gallagher2008 (). The successful manipulation of polar lattice gases in optical lattice experiments could lead to the observation of supersolid phases.

On the other hand, the ability to produce frustration in optical lattices of cold atoms has opened up possibilities to realize interesting superfluid and Mott states which have additional kinds of order arising from the kinetic frustration dhar1 (); dhar2 (); santos_cmi (). Kinetic frustration in these systems is produced by the competition of two different hopping processes from a site to different sites with different signs of the hopping amplitude. The two different hopping processes could be to a nearest neighbor site and a next-nearest neighbor site. It is thus interesting to study the interplay between nearest neighbour interaction and kinetic frustration away from commensurate densities, which can potentialy stabilize supersolid phase.

Figure 1: (Color online) (a) lattice with nearest and next-nearest neighbour hopping. (b) Equivalent triangular ladder geometry. The arrows are the representation of the hopping directions.

In this paper, we study such a model of hard core bosons hopping on a one-dimensional lattice with nearest neighbor hopping and interaction and next-nearest neighbor hopping that induces kinetic frustration as shown in Fig. 1(a). This model is equivalent to a system of triangular ladder as shown in Fig. 1(b). The model describing such a system can be described by the Hamiltonian


where and are creation and annihilation operators for hard core bosons at site , and is the boson number operator at site . and are the nearest and next-nearest neighbor hopping amplitudes. represents the nearest neighbor repulsion. Frustration in this model is introduced by choosing and . The model described by Eq. 1 can also be thought of as a triangular ladder where the nearest neighbor hopping and interaction are along the rungs and the next-nearest neighbor hopping is along the legs. In this work we scale the energies with respect to by considering , therefore, all the parameters considered are dimensionless. As discussed in Ref. mishra_ttpv (), this model does not have a simple representation in terms of spinless fermions due to the presence of the next-nearest-neighbour hopping term. At half filling and for , apart from the trivial point , there exists one more point corresponding to , where the exact ground state can be obtained mishra_ttpv ().

The model of Eq. 1 has been studied recently by us at half filling mishra_ttpv (). The ground state phase diagram has three different phases, a uniform superfluid (SF), an insulating charge density wave (CDW) crystal and a bond ordered insulator (BO). When , only the insulating (gapped) phases occur and there is a first order transition between them as a function of . In this paper we study the system by doping it away from half filling to see what types of gapless phases might arise. By performing a detail analysis using the FS-DMRG method, we obtain a complete ground state phase diagram for this model. Our main result is that in addition to the gapped CDW and BO phases, the phase diagram contains a gapless supersolid(SS) phase, which is a phase with concurrent superfluid and charge density wave order. This is summarized in the phase diagram of Fig. 2, which is plotted as a function of the chemical potential and interaction . In the following sections, we present details of our calculations

Figure 2: (Color online)Phase diagram of the model at commensurate and incommensurate with in the chemical potential and interaction plane. Both and are in units of and corresponds to the case of half filling where only the CDW and BO phases occur. For other values of , we obtain the additional supersolid phase SS. The phase boundaries marked with green circles are those between gapped and gapless phases and are obtained by calculating the charge gap. The phase boundaries marked with red triangles and blue squares with error bars for some values of are between the gapless superfluid and the supersolid phases and are determined by looking at the diverging compressibility and the structure factor respectively as explained in the text.

Ii Details of the DMRG method

We study the model described by Eq. 1 using the finite-size DMRG method with open boundary conditions white_92 (); schollwock_review_05 (). This method is best suited for (quasi-)one-dimensional problems schollwock_review_05 (). For most of our calculations we study system sizes up to 200 sites and retain up to density matrix eigenstates with the weight of the discarded states in the density matrix less than . We compute various physical quantities to characterize the different phases. Some of these quantities have been calculated by us using the DMRG method to study related models mishra_ttpv (); tapan_tvvp (). We describe below the quantities which are most important for the characterization of the different phases.

In order to distinguish between gapped and gapless phases we calculate the chemical potentials


where and . In Eq. (2), is the ground-state energy of the system with sites and bosons.

The CDW order in the system can be quantified by calculating the structure factor, which is the Fourier transform of the density-density correlation function


The BO phase is characterized by a non-zero value of the bond-order parameter




Iii Results and discussion

We first discuss how to obtain the signature of the gapless and gapped phases at incommensurate and commensurate densities respectively. This is done by computing the chemical potential defined in Eq. 2 for various densities . We start at a value of far away from half filling and then dope the system to increase gradually. Since the model considered is particle-hole symmetric, the signatures at densities above half filling are mirror reflections of those below half filling. The gapless to gapped transition can be seen in the plot as shown in Fig. (3). It can be seen that there exists a jump in as a function of at for different values of . The corresponding length of the plateau in decreases as where the gap exists only very close to the transition between CDW and BO at and again increases. The end points of the plateaux trace out the BO nd CDW phases which are shown in Fig. (2). The BO and CDW phases are characterised by the finite bond order parameter and density wave structure factor as defined in Eq. 4 and Eq. 3 respectively.

Figure 3: (Color online)(a),(b),(c),(d) show the vs. plotted for different values of . The appearance of a plateau indicates an incompressible (gapped) phase. The length of the plateau measured along the direction gives the size of the gapped region in the phase diagram at a given . At , which is where the direct transition from CDW to BO occurs at , the gap exists only very close to the transition point and hence there is no plateau.

It is obvious from the Fig. (3) that the compressibility is zero along the plateau and is finite on the shoulders around the plateau. However, it should be noted that there is a kink in the vs. plot for , where the chemical potential tends to saturate with respect to and therefore the compressibility diverges. These kinks appear for all the values of considered in our calculation. The divergent compressibility can be regarded as the signature of a phase transition which can be located from the kink position. This phase transition corresponds to the transition among the gapless phases, the SS and the SF. Once, is increased beyond the position of the kink, increases monotonically with indicating a finite compressibility in the gapless SS phase. The kink positions which give us the phase boundaries between the gapless phases are shown in Fig. (2). Note that we cannot characterize the nature of these gapless phases (i.e. say whether they are SS or SF) from the above analysis. For that we require to calculate the correct order parameters as well, which we discuss in the following sections.

Figure 4: (Color online)On-site number density plotted as a function of site index for different values of with (a),(b), (c), (d) and (e) for . It can be seen that while there is perfect CDW order at , there is a modulation over at a wave vector that appears as one moves away from .
Figure 5: (Color online) vs for different densities for . It can be seen that the location of the peak shifts away from as the filling decreases for . The shift is found to be linear in such that the modulation vector .

The CDW and BO phases which occur at are gapped and thus one would expect them to remain robust to small changes in . Thus, we would expect the CDW and BO phases to appear as lobes (as seen in Fig. (2)). To understand what happens as we move away from half filling, we calculate the density-density structure factor as defined in Eq. 3 and also look at the local density as a function of lattice site . It can be seen from Fig. (4) that there is a modulation of the density with wavevector superimposed on the CDW order as the filling is changed from . We can quantify the dependence of on by plotting the density-density structure factor as in Fig. (5). With a modulation in the density, has a peak at , which can be seen as the peak shifts away from as changes. Tracking the positions of the peaks yields . Similar feature has been studied before in a system of hardcore bosons in zig-zag ladder rossini (). From Fig. (3), it can be seen that the state, one obtains for , where the density modulation occurs is a compressible (gapless) state and thus corresponds to a supersolid (SS). However, it is a supersolid, where the charge ordering wavevector is dependent on the filling .

Figure 6: (Color online) is plotted as a function of for different . The peak position drifts linearly with the filling fraction . The modulation wavevector .
Figure 7: (Color online) vs for for different system sizes shows that there is a fairly steep jump at a particular value of which does not appear to drift appreciably with increasing system size. However, the jump seems to be come more gradual with increasing system size.

The SS phase shares phase boundaries with the CDW, BO phase and SF phases as can be seen from Fig. (2). The phase boundary between the SS and gapped CDW and BO phases is obtained by measuring the charge gap which is zero in the SS phase but finite in the gapped phases. The other phase boundary between the SS and the SF phases cannot be obtained in this way since they are all gapless. To obtain this boundary we plot the peak value of the structure factor as a function of for different values of as shown in Fig. (6). The value of at each density as a function of shows a fairly steep increase at a particular value of . This value of at which this happens does not drift appreciably with increasing system as shown in Fig. (7) although it appears steeper for smaller system sizes. The phase boundary obtained this way coincides with the one obtained from the positions of the kinks in the plots of Fig. (3) as discussed earlier. This validates this particular way of obtaining the phase boundary. The lines of constant density in the SS and SF phases of the phase diagram can be seen in Fig. (8).

Figure 8: (Color online) An enlarged view of the SS lobe and regions around it of the phase diagram of Fig. (2), with the lines of constant density marked.

The SF phase might be stabilized over a larger part of the phase diagram and be easier to detect if we choose a different set of parameters, say . As we have seen in our previous work, for such a choice of parameters, it is possible to obtain a regular superfluid phase even exactly at half-filling mishra_ttpv () and it is quite likely that this phase will remain over a fairly large part of the phase diagram even when we move away from half-filling. However, for these parameters, it is likely that the SS phase will occupy a smaller region of the phase diagram.

Iv Conclusions

We have studied a system of hardcore bosons in a one dimensional optical lattice with frustrated next-nearest-neighbour hopping and nearest neighbour interaction. Using the finite-size DMRG method we have obtained the ground state phase diagram of this model and shown that in addition to gapped CDW and BO phases, it also displays the regular SS phase, which has concurrent superfluid and CDW order.

V Acknowledgments

We would like to thank Arun Paramekanti and B. P. Das, Thierry Giamarchi and Abhishek Dhar for many useful discussions. SM thanks the Department of Science and Technology, Govt. of India for support. RVP thanks the UGC, Govt. of India for support.


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