Quantum phases of hardcore bosons with long-range interactions on a square lattice
We study the ground-state phase diagrams of hardcore bosons with long-range interactions on a square lattice using the linear spin-wave theory and a cluster mean-field method. Specifically, we consider the two types of long-range interaction: One consists only of the nearest- and next-nearest-neighbor interactions, and the other is the dipole-dipole interaction that decays with the interparticle distance as . It is known from previous analyses by quantum Monte Carlo methods that a checkerboard supersolid (CSS) is absent in the ground-state phase diagram of the former case while it is present in the latter. In the former, we find that quantum fluctuations around mean-field solutions are enhanced by the direct competition between the checkerboard and striped solid orders and that they destabilize the CSS phase. On the other hand, the emergence of the CSS phase in the latter case can be attributed to the absence of such a competition with other solid orders. We also show that the cluster mean-field method allows for the determination of phase boundaries in a precise quantitative manner when scaling with respect to the cluster size is taken into account. It is found that the phase transition between the superfluid and the solid (or CSS) is of the first order in the vicinity of the particle-hole symmetric line.
pacs:03.75.-b, 05.30.Jp, 67.80.kb
Can a solid exhibit superfluidity in lattice systems? This question was first investigated theoretically by Matsuda and Tsuneto matsuda-70 (); liu-72 () in the context of the quantum lattice-gas model for He, which assumes that atoms move only on fixed lattice points even in the liquid phase. matsubara-56 () Using the lattice representation, they discussed the possibility of supersolidity, which is characterized by the coexistence of solid (diagonal) and superfluid (off-diagonal) long-range orders, in bulk and thin film of He. In the lattice system, the continuous translational invariance of the system is broken by the presence of the background discrete structure, and the “solid” means a state in which a discrete translational invariance is broken spontaneously. Recently, this issue has attracted renewed interest in connection with ultracold Bose gases in optical lattices. The creation of gases with strong dipole-dipole interactions griesmaier-05 (); lu-11 (); aikawa-12 (); ni-08 (); aikawa-10 (); deiglmayr-08 () has provided an ingredient essential for the emergence of supersolid phases, namely long-range interactions. Moreover, the precise controllability of optical-lattice systems has inspired theoretical explorations of supersolid phases in various types of lattice structure, such as chain, batrouni-06 (); burnell-09 () square, goral-02 (); kovrizhin-05 (); sengupta-05 (); scarola-05 (); yi-07 (); danshita-09 (); sansone-10 (); danshita-10 () triangular, wessel-05 (); heidarian-05 (); melko-05 (); boninsegni-05 (); hassan-07 (); sen-08 (); pollet-10 (); bonnes-11 (); zhang-11 (); yamamoto-12 () honeycomb, wessel-07 (); gan-07 () kagome, isakov-06 () and cubic yi-07 (); kyamamoto-09 (); xi-11 (); ohgoe-12 () lattices. We also note that the formation of checkerboard density-wave order has been experimentally observed in Bose-Einstein condensates coupled with an optical cavity. baumann-10 ()
For understanding lattice supersolids, it is important to address the following questions: in what situations the coexistent state can emerge and why it can be stable in such situations. Extensive studies over the past few decades have provided answers to these questions. For example, previous researches demonstrated that no supersolid phases can exist in the ground-state phase diagram of the hardcore Bose-Hubbard model with the nearest-neighbor (NN) interaction for bipartite lattices such as square batrouni-00 () and honeycomb wessel-07 (); gan-07 () lattices. In these cases, uniform supersolid states are unstable towards the formation of domain walls, zhang-03 (); sengupta-05 () and the system undergoes phase separation into superfluid and solid phases. In order for supersolid phases to be present, one has to modify the model by, e.g., introducing dipole-dipole interactions sansone-10 (); ohgoe-11 () or treating softcore bosons. kovrizhin-05 (); sengupta-05 () In contrast, the triangular-lattice system of hardcore bosons with only the NN interaction has stable supersolid phases for the fillings . murthy-97 (); wessel-05 (); bonnes-11 (); zhang-11 (); yamamoto-12 () As for the case of the kagome lattice, although the mean-field (MF) analysis predicts the existence of supersolid states, murthy-97 () they are destabilized by the effects of strong quantum fluctuations. isakov-06 ()
In this paper, focusing on the supersolid phase with checkerboard solid order, we analyze ground-state properties of hardcore bosons with long-range interactions on a square lattice by means of the linear spin-wave (LSW) theory and a cluster mean-field (CMF) method. In this system, the range of the interactions makes a qualitative difference in the emergence of checkerboard supersolid (CSS) states. The previous quantum Monte Carlo (QMC) calculations batrouni-00 () have shown that no CSS phase is present between the superfluid (SF) and checkerboard solid (CS) phases in the system with only the NN interaction and the next-nearest-neighbor (NNN) interaction . On the other hand, it is known that the infinite-range dipole-dipole interaction, which decays as the inverse cube of the distance, can stabilize the CSS states. sansone-10 () We will clarify the reasons why the dipole-dipole interaction can stabilize the CSS states unlike the case of only the NN and NNN interactions.
The MF ground-state (classical) properties of the hardcore Bose-Hubbard models with dipole-dipole interaction and with only the NN and NNN interactions have already been discussed separately in previous works. danshita-10 (); bruder-93 (); scalettar-95 (); pich-98 () We first review those results from the standpoint of comparing the two types of interactions. When assuming that the system is in the phases with checkerboard (two-sublattice) order, the MF energy of the dipolar model can be naturally written in the same form as that of the model with effective NN and NNN interactions, and . We find that the value of is very large, and it leads to a large region of CSS phase in the ground-state phase diagram at the MF level. Second, from the LSW analysis, we show that quantum fluctuations around the MF solutions are not so strong compared to the case of only the NN and NNN interactions, which is attributed to the absence of the direct competition between the checkerboard and other solid orders. These two factors lead to the emergence of the stable CSS state in the dipolar system unlike the case of the shorter-range interactions.
Moreover, including the effects of quantum fluctuations, we derive the ground-state phase diagrams. Although some of the results have already been known from previous QMC works, we reconsider the issue in detail in terms of another numerical approach based on a large-size CMF method. yamamoto-12 () From a comparison with the QMC result sansone-10 () for the model with dipole-dipole interaction, it is shown that the CMF method combined with cluster-size scaling can locate the phase boundaries quantitatively. We also derive the phase diagram of the model with only the NN and NNN interactions and confirm that the region of stable CSS phase almost completely disappears due to the strong quantum fluctuations. Moreover, we find the first-order phase transition between the SF and the CS (or the CSS) in the close vicinity of the particle-hole symmetry line for the both models. It is worth stressing that our CMF procedure is free from the minus-sign problem even when applying to frustrated systems. Moreover, it is useful to study metastability phenomena such as hysteresis, yamamoto-12 () since one can get all stationary points of the free energy including metastable and saddle-point solutions.
The remainder of the paper is organized as follows. In Sec. II, we introduce our models describing hardcore bosons with two types of long-range interactions in a square lattice. In Sec. III, we show the ground-state phase diagrams of the two models within the mean-field theory. In Sec. IV, we perform the LSW analyses to discuss the strength of quantum fluctuations around the MF solutions. In Sec. V, applying a CMF method and the cluster-size scaling, we obtain the phase diagrams including the effects of the quantum fluctuations. Moreover, we summarize the reasons why the dipole-dipole interaction stabilizes the CSS states, based on the results obtained in Secs. III-V. The conclusion is given in Sec. VI.
Ii Hardcore Bose-Hubbard Models
We consider interacting hardcore bosons on a square lattice given by the following Hamiltonian:
where and are the creation and number operators of the hardcore bosons at site , denotes the hopping amplitude between NN pairs, and the chemical potential. The hardcore boson limit means the situation where two or more bosons are not allowed to occupy the same site due to the strong on-site interaction . We assume the existence of a long-range interaction between the hardcore bosons and consider two different forms of such that we study the effect of long-range interactions on the stability of supersolid phases through the comparison of the two models.
The first one is given by
where is the lattice spacing and with integers and is a lattice vector at site . The parameters and represent the strength of the NN and NNN interactions, respectively. The NN interaction tends to induce the checkerboard density-wave order depicted in Fig. 1(I), while the strong NNN interaction favors the stripe pattern in Fig. 1(II). pich-98 (); batrouni-00 ()
Thus, the Hamiltonian in Eq. (1) with Eq. (2), which we refer to as the “- model,” is a minimal model for studying the competition among two different solid orders and superfluidity induced by the hopping . bruder-93 (); batrouni-95 (); scalettar-95 (); pich-98 (); batrouni-00 (); ng-0810 () We will focus on the regime of checkerboard ordering, , batrouni-95 (); scalettar-95 (); pich-98 ().
The second one is the isotropic dipole-dipole interaction that is more realistic from an experimental point of view. In experiments of ultracold gases, one of the most promising way to prepare long-range interacting systems is the use of the so-cold “dipolar” atoms, such as chromium, griesmaier-05 () dysprosium, lu-11 () and erbium, aikawa-12 () or molecules, such as KRb ni-08 (); aikawa-10 () and LiCs. deiglmayr-08 () These atoms and molecules have a large (magnetic or electric) dipole moment, which leads to strong long-range forces among the dipolar particles. We assume that the dipole moments are fully polarized along the direction perpendicular to the lattice plane. In this case, the interaction between the dipoles works isotropically and its long-range part can be well approximated by
We refer to the model given by the Hamiltonian in Eq. (1) with Eq. (3) as the “ model,” hereafter. The dipole-dipole interaction falls off as the inverse cube of the distance as (for the NN, NNN, third, fourth neighbors). Therefore, it appears that most of the essential physics can be captured with just the first two terms, namely within the - model. In fact, as will be shown in the next section, the MF phase diagrams of the two models are very similar; there are regions of the standard SF phase, solid phases, and supersolid phases, including the CSS phase.
However, previous numerical analyses based on the QMC method demonstrated that the correct ground-state phase diagrams, which include quantum fluctuations, have a crucial difference between the finite-range - model and the infinite-range model. For the - model, the authors of Ref. batrouni-00, concluded that the CSS phase predicted by the MF theory is completely destabilized by strong quantum fluctuations and does not appear in the QMC calculations for any value of , although they checked it only for . In contrast, as for the model, the main features of the MF phase diagram can survive, sansone-10 () including the existence of the CSS phase. This indicates that the long-range part of the dipole-dipole interactions plays a crucial role in stabilizing the CSS phase. In the following sections, we shall analyze the ground states of the two models and discuss the role of the long-range interactions in the emergence of the CSS state in order to clarify the reasons why the two models have the qualitative difference.
The instability of supersolid phases against phase separation has often been discussed from a perturbative point of view assuming that is small; the total energy gains from the lowest-order hopping process of doped bosons (or holes) and from the surface energy are compared on classical solid states with/without a domain-wall. zhang-03 (); sengupta-05 (); wessel-05 (); ohgoe-12 () However, in the model, the CSS states can appear even for relatively large values of , and the structure called the with many different types of solid states emerges in the region of small values of . sansone-10 () Hence, we will present a more careful discussion from a different angle by using the LSW and CMF methods.
Iii Classical Ground States
To begin with, we show the ground-state properties within the MF theory. From the equivalence of the hardcore-boson and spin-1/2 operators, matsubara-56 () Eq. (1) can be mapped onto the spin- model with long-range Ising-type interactions:
where is the pseudospin operator which satisfies the commutation relations
In the pseudospin language, the occupied and unoccupied states of bosons correspond to the spin-up and spin-down states, respectively. Thus the filling factor, which is the average density per site, of hardcore bosons can be calculated through the relation . Here, is the number of lattice sites. The pseudospin raising and lowering operators play the role of the creation and annihilation of the hardcore bosons; . The effective magnetic field acting on the pseudospins is given by with
where is the coordination number of the square lattice. As can be obviously seen from the definition, the zero magnetic field corresponds to the particle-hole symmetric point () of the hardcore-boson model. Moreover, the density and the condensate wave function of bosons are expressed by the longitudinal components and the transverse components as and . matsuda-70 (); liu-72 () From these correspondences, we can use the calculation methods which have been developed in the field of quantum spins for studying hardcore-boson systems.
At zero temperature, replacing the local pseudospin operators in Eq. (4) with the classical vectors of length ,
we obtain the MF (classical) energy as a function of the orientation of the local pseudospins :
Minimizing the MF energy with respect to , we derive the classical pseudospin configurations in the usual manner, bruder-93 (); scalettar-95 (); pich-98 () and translate the results into the hardcore-boson language. Without loss of generality, we can take , which means that the canted spins are assumed to lie in the plane. The procedure described here gives the same results as those obtained by the standard decoupling technique for the intersite spin-exchange interaction terms (i.e., the Weiss molecular-field theory) at .
iii.1 The MF results for the - model
In this subsection, let us briefly review the MF results for the - model. bruder-93 (); scalettar-95 (); pich-98 () We mainly focus on the phases with the two-sublattice structure described in Fig. 1(I). Within this checkerboard structure, we can describe the CS and CSS states in addition to the uniform SF state. The CS state, which is an insulating state appearing at half filling, has the checkerboard density-wave order characterized by with , while the SF state has the off-diagonal long-range order characterized by . The CSS state has both of the two (diagonal and off-diagonal) orders. In addition, completely empty () and fully occupied () states also appear. These trivial incompressible states can be regarded as a kind of Mott insulator (MI) states.
In the classical limit, these states can be expressed in terms of pseudospin angles as follows:
and the filling factor is given by . The ground-state phases are determined so as to minimize the MF energy with respect to and . Figures 2(a) and 2(b) show the phase diagrams in the (, )-plane for two different values of . The value of at the tip of the CS phase, , is given by within the MF theory. The phase boundaries between CS and CSS, between CSS and SF, and between SF and MI are given by , , and , where
In the figures, the quantities on the axes are also scaled by and to compare the results in the same scale. In addition to the CS and CSS phases, other solid (with , and ) and supersolid (SS2a) phases are formed due to the competition of the NN and NNN repulsions. bruder-93 (); pich-98 () These phases have the sublattice structures depicted in Fig. 3.
As seen in Eqs. (12), the CSS phase can emerge as long as the NNN interaction is finite, and the window gets wider as increases. Thus, it appears that we just have to prepare the system with a stronger NNN interaction in order to obtain the stable CSS phase in a wider range of the parameters. However, when the value of is large, the striped solid order shown in Fig. 1(II) is more favored than the checkerboard. The general expression of the MF energy for striped phases is given by
where and are the canting angles of the pseudospins on even and odd rows [see Fig. 1(II)]. For example, let us consider the solid orders emerging at the half-filling (). Putting in Eq. (11) and in Eq. (13), we obtain the MF energies of the CS and striped solid states:
From the comparison, one finds that the striped solid state has lower energy than the CS state when . Also for the supersolid phase, the striped one takes the place of the CSS phase in this regime. pich-98 () Because of the transitions to the striped phases, we cannot extend the CSS region by exceeding the limit of . Moreover, as shown in Fig. 2(b), when the value of approaches the boundary to the stripe regime, the SS2a phase is extended toward the large region due to the competition of the two density-wave orders. This competition also causes strong quantum fluctuations that destabilize the CSS states, as will be discussed in Sec. IV.
iii.2 The MF results for the model
Next, let us move onto the model. In Ref. danshita-09, , we have applied the MF theory to this model, and examined the stability of superflow in the CSS state. Here, we present more detailed information on the MF ground states, and discuss the comparison with the results for the - model.
The MF energy per site of the model for the checkerboard pattern can be written as
Here, () is just the summation of the long-range interactions between the pseudospins on the same (different) sublattice sites:
The index () means the th site on sublattice (). Only by replacing and with and in Eq. (11), we can immediately obtain the expression of Eq. (15). This means that the MF properties of the checkerboard phases of the model can be described exactly by the - model with the effective NN and NNN interactions and . For example, the phase boundaries between the CS, CSS, SF, and MI phases are obtained by replacing and in Eqs. (12) with and . Moreover, the tip of the CS lobe is given by . It is worth noting that the MF energy of Eq. (15) is valid not only for the model, but generally for systems with checkerboard sublattice structure regardless of the form of .
However, many additional phases emerge for the region of smaller due to the long-ranged character of the dipole-dipole interaction. Within our analysis (see Appendix A), we found the supersolid, named SS2b, and the solid phases with and in addition to the phases appearing in the - model. Unlike the - model, the two possible structures of the () solid state shown in Fig.3(II) can be distinguished even within the MF theory; the b-type structure has lower energy. The emergence of these solid phases is consistent with the QMC results. sansone-10 () Although many other phases can emerge for smaller , we do not extend the calculations to more complex sublattice structures, since our main focus is the stability of the CSS phase.
It should be noted that the ratio of the effective NNN interaction strength to the NN one is fixed in the model as. This value obviously exceeds the limit , above which the striped phases emerge in place of the checkerboard ones in the case of the - model. Nevertheless, we have to keep in mind that the effective interactions are made by the summation of the long-range interactions between various pairs with different distances. Therefore, the limit predicted for the - model cannot be directly applied to the model.
In the model, the MF energy per site for the striped phases is written as
This expression is formally equivalent to that of the - model [Eq. (13)] with the effective interactions
However, the effective NN and NNN interactions have different values for the checkerboard () and striped () phases [compare Eqs. (16) and (18)]. Hence, the large value of in the checkerboard phases does not mean that the striped phases are energetically preferred, and the checkerboard order is always favored over the striped one in the model. As an example, we show the comparison of the MF energies of the CS and striped solid states:
According to Eqs. (12), the CSS region, , gets wider for a larger value of (). In the model, the ratio is larger than of the - model with the checkerboard order, the CSS region is also larger in the MF level. This is one of the two main reasons why the CSS phase is stable in the case of the dipole-dipole interactions. Comparing the width in units of , for example, at in Figs. 2(a), 2(b), and 4, we indeed see that the model has a wider region of the CSS phase than the - model. Moreover, despite the large value of , the SS2a region in Fig. 4 is relatively suppressed compared with that in Fig. 2(b). This means that the direct competition of the checkerboard and striped density-wave orders is much weaker than the case of the - model. The suppression of the competition can be also seen in the excitation spectra, which will be discussed in the next section.
Iv Linear spin-wave analysis
In this section, we discuss the strength of quantum fluctuations around the MF ground states within the linear spin-wave (LSW) theory. pich-98 (); scalettar-95 (); coletta-12 () First, we perform local rotations of the spin reference frame in Eq. (4), so that the new spin quantization axis is oriented along the direction of the classical pseudospin vector:
Furthermore, we introduce new bosonic variables via the Holstein-Primakoff transformation,
to describe quantum fluctuations around the classical spin angles. Within the LSW approximation, we keep the terms up to the second order in the boson operators:
where is identical to the MF energy given by Eq. (8). The linear term in boson operators disappears by substituting the MF solutions into . Diagonalizing , we calculate the LSW excitation spectra and the number of “spin waves” to estimate the strength of the quantum fluctuations (see Appendix B for details). By calculating the number of spin waves, one can roughly estimate the strength of quantum fluctuations around the MF solutions obtained in Sec. III. In the spin language, the value of corresponds to the spin reduction from its classical value due to the zero-point fluctuations.
We will show the results of the excitation spectra in Sec. IV.1 and of the the number of spin waves in Sec. IV.2. For the - model, the LSW excitation spectra of SF, CS, and CSS states have already been discussed in detail in Ref. scalettar-95, , and it was confirmed that the softening of roton excitations causes the phase transition from the SF to CSS state. As for the model, although we used in Ref. danshita-10, the LSW theory to discuss the critical velocity of flowing CSS states, detailed results of the spectra have not been presented yet. Moreover, to date, no studies estimating the strength of quantum fluctuations from the values of have been demonstrated for the comparison of the two models.
iv.1 The excitation spectra
As mentioned above, the LSW excitation spectra for the - model have already been analyzed in Ref. scalettar-95, . Hence, we show here only the case of the dipole-dipole interaction given in Eq. (3). In the calculations, we have to take an infinite summation in Eq. (44) due to the long-range nature. To avoid the practical difficulty, we introduce a cutoff distance on the dipole-dipole interaction as for only in this section (namely, in Secs. IV.1 and IV.2). For this truncated dipole-dipole interaction, the values of the effective NN and NNN interactions are and and the ratio is .
Solving Eq. (46), we plot in Figs. 5(a)-5(c) the excitation spectra for the SF, CSS, and SS2a phases along the line marked in Fig. 4. The excitation spectra have a Nambu-Goldstone mode reflecting the spontaneous breaking of the U(1) symmetry. The spectrum of the CSS phase consists of two branches due to the two-sublattice structure, and the lower branch has gapless, linear dispersions around and , which is the ordering vector of the checkerboard phases. In the SS2a phase, in which the checkerboard and stripe orders coexist, the lowest mode is gapless at in addition to at and .
Figures 6(a)-6(d) show the excitation spectra at the phase transitions between the different phases. When one approaches the CSS phase from the SF region, a roton-like minimum at develops, and it touches zero at the boundary as shown in Fig. 6(a), causing the second-order phase transition to the CSS state. In a similar way, a roton-like mode at causes the second-order transition from CSS to SS2a [see, Fig. 6(b)]. In contrast, as shown in Fig. 6(c), such a signal does not appear in the spectra at the first-order phase transitions. The above-mentioned properties of the excitations qualitatively agree with the case of the - model.
iv.2 The number of spin waves
For all the cases of Figs. 2(a), 2(b), and 4, the system exhibits the phase transition from SF to CSS, and then it reaches the CS phase if the value of increases from a negative value to zero along the line of . We will plot the number of spin waves along this line as a function of the filling factor . Within the MF analysis, the filling factor shows a linear increase with the chemical potential both in the SF and CSS phases, and the slope of the line, which is proportional to the compressibility, is always larger in the CSS phase than in the SF phase:
where the upper (lower) signs are for positive (negative) values of . Recall that has to be replaced with for the model. It should be noted that the critical filling factor at the SF-CSS transition point, which is obtained by substituting into Eq. (23), can be expressed as a function only of :
It takes for in the low-density (negative ) side.
Figures 7(a)-7(c) show the results for the - model with and , and for the model. In all the cases, we can see that has a peak at the SF-CSS phase transition, which means that quantum fluctuations are particularly strong at the phase boundary. The number of spin waves for in Fig. 7(b) is much larger than the case of in Fig. 7(a). This is attributed to the strong competition of the NN and NNN interactions. In fact, as shown in the right panel of Fig. 7(b), the excitation spectrum exhibits a remarkable drop at , which indicates the existence of strong striped density-wave fluctuations. The maximum value of for reaches about 77 percent of the classical value of the spin length , which means that the predictions for the ground states within the MF theory are unreliable. Actually we will show in Sec. V that the CSS phase predicted by the MF theory almost completely disappears due to the strong quantum fluctuations.
On the other hand, the excitation spectrum for the model in the right panel of Fig. 7(c) does not exhibit a significant drop at unlike the case of . This comes from the fact that is not just the NNN interaction but the summation of various long-range interactions that weakens the competition with the stripe order. Therefore, as shown in Fig. 7(c), the quantum fluctuations in the model are relatively weak for its large value of . This is the second reason for the stability of the CSS state in the model. The CSS region predicted by the MF theory is significantly reduced by the quantum fluctuations but still remains sufficiently large (see Ref. sansone-10, and Sec. V of this paper).
V Large-size cluster mean-field method and scaling analysis
We drew the ground-state phase diagram of the two models within the MF theory in Figs. 2(a), 2(b), and 4. However, according to the previous section, the fluctuations around the classical ground states are too large to completely ignore in any case. In this section, considering the MF results obtained in Sec. III as a starting point, we discuss how the quantum fluctuations change the features of the ground-state phase diagrams by employing a large-size CMF method. yamamoto-12 () We perform the calculations based on rectangular-shaped clusters, and then extrapolate the results with respect to the cluster size. The obtained results will be compared with the QMC data in Ref. sansone-10, for the model. As for the - model, although the authors of Ref. batrouni-00, concluded that the CSS state is thermodynamically unstable from the QMC calculations for , the entire phase diagram including the effects of quantum fluctuation has not been produced yet. We will confirm, in the -plane, that the CSS phase is almost completely destroyed by the strong quantum fluctuations in the - model. Hereafter, we will use the full (untruncated) dipole-dipole interaction again in the model.
v.1 The CMF method
First, we describe the details of our CMF approach. yamamoto-12 () The standard MF theory approximates the system by single-site problems in effective fields. A natural extension of the single-site approximation is the use of “clusters” of multiple sites as an approximate system. bethe-35 (); peierls-36 (); weiss-48 (); campbell-72 (); du-03 (); etxebarria-04 (); neto-06 (); oguchi-55 (); buonsante-04 (); mcintosh-12 () For example, the Bethe-Peierls-Weiss (BPW) method bethe-35 (); peierls-36 (); weiss-48 () employs a cluster consisting of one central site and its directly connected sites, e.g., a cluster of sites for a triangular-lattice system with NN and NNN interactions. campbell-72 () Treating exactly the interactions within the cluster, one can partially take into account the effects of correlations between particles (or spins). However, the BPW method and its extensions du-03 (); etxebarria-04 () cannot be applied to an infinite-range interaction model like the model since all sites are “directly connected” by the long-range interactions.
Oguchi’s method oguchi-55 () is another simple way to extend the MF theory to clusters. In Ref. oguchi-55, , Oguchi studied ferromagnetism and antiferromagnetism of the low-dimensional Heisenberg model by using a cluster of up to three spins to include the short-range correlations between the spins. Since the influence from the spins outside of the cluster is also included as effective internal fields, we can treat even a system with infinite-range interactions. However, as Oguchi himself pointed out, the cluster of two or three sites is too small to sufficiently take into account the effects of the correlations (or quantum fluctuations).
Our CMF approach yamamoto-12 () is an extension of Oguchi’s method to larger-size clusters and to multiple-sublattice problems. Although we use here the pseudospin form of the Hamiltonian, Eq. (4), to explain the procedure of our method, the same manner can be applied straightforwardly to hardcore bosons and even to softcore-boson models. First, we assume a sublattice structure expected to emerge in the parameter range. Then, we embed a cluster of sites into the background sublattice structure. Figures 8(I) and 8(II) show, as examples, the cases of and , which we refer to as CMF- and CMF- under the assumption of the checkerboard sublattice structure.
As for the case of the CMF-, we have two inequivalent choices for embedding the cluster. We have to deal with both of them equally as in the BPW method for multiple-sublattice systems. neto-06 () Now, instead of treating the many-body problem in the whole system given by Eq. (4), we consider the effective cluster Hamiltonian written as the following general form:
in which the interactions within the cluster are treated exactly, while the interactions between the spins in the cluster and the rest of the system are approximately included via the effective fields
where is the part of the system outside the cluster and are the expectation values within the CMF method, which act as the mean fields from the spins in . Here, we chose again the plane as the plane in which the spins lie; i.e., . If we have two or more possibilities of choosing the cluster like in the CMF-, we should consider all the corresponding cluster Hamiltonians like , , .
Note that in our CMF method, we consider the -site problem in the cluster just as a reference system to estimate the values of the mean fields , which depend only on the background sublattice index of the site; i.e., . For example, the effective fields acting on the top-left site “” in the cluster of Fig. 8(I) can be written as the following explicit forms:
for the - model and
for the model (see Appendix C for more details). The values of the mean fields and are calculated self-consistently as the expectation values of the pseudospins inside the cluster as follows:
where (we take in this paper), is the number of the possible choices of the cluster, and is the summation of the number of sites belonging to sublattice in cluster . For example, in the CMF- for checkerboard phases in Fig. 8(II) and we have and for the two clusters and , respectively, which leads to .
This method reduces to the conventional MF (namely, Weiss’s molecular-field) theory for , and becomes exact in the limit . We have to diagonalize the cluster Hamiltonian to take the trace on the right-hand side of the self-consistent equation, Eq. (29). Thus the practical limit of the cluster size is determined by the largest number of sites which can be treated by exact diagonalization techniques. It should be noted, however, that some of the symmetries of the original Hamiltonian, Eq. (4), are broken in the effective cluster Hamiltonian due to the existence of mean fields.
v.2 The CMF results for the cluster
First, we show the results of the CMF- method for the two models and compare them with the MF results. We focus on the checkerboard phases and discuss the influence of quantum fluctuations on the locations of the phase boundaries between the CS, CSS, and SF phases, namely and in Eqs. (12). Note that there are no quantum fluctuations at the boundary between the SF and MI phases, , and thus the expression in Eq. (12c) does not change for any . Other phases with more complex symmetries, such as three-sublattice and four-sublattice phases, are affected more strongly by the quantum fluctuations, and the locations are shifted towards the region of much smaller values of than those of the MF theory. sansone-10 (); ng-10 () Thus, we leave them out of the scope of the rest of this paper, treating only a relatively large- region.
For the checkerboard phases, we can obtain the SF order parameter