Ground-state phase diagram and magnetic properties of a tetramerized spin-\frac{1}{2} J_1-J_2 model: BEC of bound magnons and absence of the transverse magnetization

Ground-state phase diagram and magnetic properties of a tetramerized spin- model: BEC of bound magnons and absence of the transverse magnetization

Abstract

We study the ground state and the magnetization process of a spin-1/2 - model with a plaquette structure by using various methods. For small inter-plaquette interaction, this model is expected to have a spin-gap and we computed the first- and the second excitation energies. If the gap of the lowest excitation closes, the corresponding particle condenses to form magnetic orders. By analyzing the quintet gap and magnetic interactions among the quintet excitations, we find a spin-nematic phase around due to the strong frustration and the quantum effect. When high magnetic moment is applied, not the spin-1 excitations but the spin-2 ones soften and dictate the magnetization process. We apply a mean-field approximation to the effective Hamiltonian to find three different types of phases (a conventional BEC phase, “striped” supersolid phases and a 1/2-plateau). Unlike the BEC in spin-dimer systems, this BEC phase is not accompanied by transverse magnetization. Possible connection to the recently discovered spin-gap compound is discussed.

1Introduction

Magnetic frustration have provided us with many intriguing topics e.g. the phenomena of order-by-disorder, the residual entropy at absolute zero temperature, disordered spin liquids, etc[1]. There are a variety of models which are known to exhibit the so-called frustration effects. Among them, the - model on a square lattice has been extensively investigated over the last two decades as one of the simplest models to study how frustration destroys magnetic orders and stabilizes paramagnetic phases. The model is defined by adding antiferromagnetic interactions on diagonal bonds to the ordinary Heisenberg antiferromagnet on a square lattice (see FIG. ?):

where the summations (n.n.) and (n.n.n.) are taken for the nearest-neighbor- and the second-neighbor (diagonal) pairs, respectively. In the classical () limit, the ground state is readily obtained by computing Fourier transform of the exchange interactions and minimize it in the -space:

  • , : the ground state has Néel antiferromagnetic order (NAF).

  • : the ground state consists of two interpenetrating Néel-ordered square lattices. First quantum correction fixes the relative angle between the two ordering directions and selects the so-called collinear antiferromagnetic order (CAF).

  • otherwise: the ferromagnetic (FM) ground state is stabilized.

For , the next-nearest-neighbor (diagonal) interaction gives rise to no frustration and only the case with is non-trivial. The case has been extensively studied in the context of spin-gap phases stabilized by the frustrating interactions. Chandra and Doucot[2] investigated the model in the large- limit and concluded that a non-magnetic (neither NAF nor CAF) phase appeared around the classical phase boundary . The most quantum case has been studied later both by numerical[3] and by analytical methods [5] (for other literatures, see, for instance, Refs. and references cited therein). By now it is fairly well established that we have spin gapped phase(s) in the window although the nature of the spin-gap phase(s) is still in controversy.

The case with , has been less investigated and recent analyses[9] suggested that there is another non-magnetic (probably spin-nematic) phase around the classical boundary between CAF and FM. From an experimental viewpoint, most compounds[11] found so far correspond to the ordered phase (CAF) of the - model.

Recently, Kageyama et al. reported[13] a new two-dimensional Cu-based compound (CuCl)La. In this compound, two-dimensional sheets consisting of and are separated from each other by non-magnetic [La] layers and within each sheet the ions form a square lattice. The ions are located at the center of plaquettes and from a naive Goodenough-Kanamori argument the - model with and is suggested as the model Hamiltonian for (CuCl)La.

What is remarkable with this compound is that inelastic neutron scattering experiments [13] observed a finite spin gap 2.3meV(=26.7K) above the spin-singlet ground state. Subsequently, high-field magnetization measurements[14] were carried out to show that magnetization monotonically increased between two critical fields T and T. The data for (i) the Weiss temperature and (ii) the saturation field in principle determine the coupling constants and . Unfortunately, none of the solutions obtained in this way reproduced the spin-gap behavior[14]. Therefore, the usual - model does not seem to work.

The second intriguing point concerns the magnetization process. From the standard scenario [15], the onset of magnetization at in spin-gapped systems is understood as Bose-Einstein condensation (BEC, or superfluid onset, more precisely) of the lowest-lying triplet excitation (magnon) and the lower critical field at is given by the spin gap as . This BEC scenario has been confirmed in various spin gap compounds[16].

Recent specific-heat- and magnetization measurements [19] for (CuCl)La exhibited behavior typical of spin-BEC transitions and suggested that the magnetization-onset transition at may be described by BEC of a certain kind of magnetic excitations. However, we immediately find a serious difficulty when we try to understand this within the standard BEC scenario; the lower critical field T expected from the observed spin gap meV at the zero field (where the experimental value is used) in the standard scenario is much larger than the observed value[14] T. One possible explanation for this discrepancy may be that a lower-lying triplet excitation which is responsible for the BEC was not observed in the neutron-scattering experiments because of selection rules. However, this seems unlikely since powder samples were used and usually one can hardly expect a perfect extinction of a certain triplet excitation in such powder samples. Neither susceptibility measurements [13] nor NMR data [20] indicate such a hidden triplet excitation.

An alternative and a more appealing scenario would be that the BEC occurs not in a single-particle channel but in a multi-particle channel. That is, what condenses to support a spin-superfluid is a bound state of magnon excitations. The possibility of multi-magnon condensation has been proposed theoretically [21] in the context of a kinetic quintet bound state in the Shastry-Sutherland model (see Ref. and references cited therein). In fact, gapped quintet excitations which come down as the external field is increased were observed in the ESR experiments [24] carried out for Sr, whereas small Dzyaloshinskii-Moriya interactions hindered a quintet BEC from being observed in that compound (see also Ref.).

One of the simplest --like models which realize the above scenario and have a finite spin gap would be the - model with a plaquette structure (see FIG. ?). A similar model () has been investigated to develop a plaquette series expansion[5]. In this paper, we mainly focus on the region , where the quintet excitation is expected to play an important role in low-energy physics.

The organization of the present paper is as follows. In section II, we briefly recapitulate the problem of a single plaquette mainly to establish the notations. The coupling among plaquettes will be taken into account in Section 3 by two different methods: (i) a plaquette extension of the bond-operator mean-field theory[26] and (ii) a perturbation expansion with respect to the inter-plaquette couplings. We find gapped triplets and quintet for small enough inter-plaquette couplings in both methods.

For larger values of inter-plaquette couplings, one of the gapped excitations softens and the form of the effective interactions among the soft excitations determines the resulting magnetic phases. By using the gaps obtained in the perturbation expansion, we determine the semi-quantitative phase diagram in section Section 4 (see FIG. ? and FIG. ?).

The effect of high magnetic field will be considered in Section 5. For high enough field compared with the spin gaps, we can approximate the low-energy sector by using only the singlet and the lowest excited state. For , we may expect that the quintet touches the singlet ground state first and a multi-particle BEC occurs. On general grounds, a single-particle (magnon) BEC phase is expected to have finite transverse magnetization. Actually, in the BEC phase of TlCuCl, the transverse magnetization has been observed in the experiment[16]. In the case of a multi-particle BEC, however, the transverse magnetization does not appear. To investigate the magnetization process, we shall keep only the singlet and the quintet to derive a hardcore boson model as the effective Hamiltonian valid in high enough magnetic field. A mean-field approximation[32] will be applied to the resulting effective Hamiltonian to draw a full magnetization curve. Interesting phases (a 1/2-plateau and supersolids) will be discussed. According to the value of the parameters, we shall roughly classify the magnetization curve in FIG. ?.

A summary of the main results and the discussion on the connection to the spin-gap compound will be given in sections Section 6 and Section 7, respectively. The equations omitted in the text will be summarized in the appendices.

2plaquette structure

We consider a spin-1/2 - model with a plaquette structure where the interactions among spin-1/2s are explicitly tetramerized (see FIG. ?). The model is made up of four-spin units (plaquettes) and the four sites constituting a single plaquette are connected by the nearest-neighbor- () and the second-neighbor () interactions as is shown in FIG. ?). The inter-plaquette interactions (both the nearest-neighbor- and the diagonal) which connect those units are multiplied by a distortion constant . This parameter may be thought of as modeling the distortion of the underlying lattice in a simple way. In the case of , this model reduces to the homogeneous - model, while when , the plaquettes are decoupled from each other.

Two dimensional square lattice with a plaquette structure to be considered in this paper. Filled circles denote spin-1/2s connected by the usual exchange interactions. Thin lines (both solid and broken) imply that the interactions are multiplied by the distortion parameter \lambda on these bonds.
Two dimensional square lattice with a plaquette structure to be considered in this paper. Filled circles denote spin-1/2s connected by the usual exchange interactions. Thin lines (both solid and broken) imply that the interactions are multiplied by the distortion parameter on these bonds.
Single plaquette. Dots represent spin 1/2s, and the solid- and the dashed lines respectively represent Heisenberg interaction with the couplings J_1 and J_2 between S=1/2 spins.
Single plaquette. Dots represent spin 1/2s, and the solid- and the dashed lines respectively represent Heisenberg interaction with the couplings and between spins.

2.1single plaquette

Let us begin by analyzing a single isolated plaquette, which corresponds to the case . The eigenstates of a single plaquette can be easily obtained as follows. First we note that a plaquette Hamiltonian can be rewritten as

where , and . Therefore, all the eigenstates are classified by the three quantum numbers as . The eigenvalues are given by

Here a constant has been dropped just for simplicity. The energy of these states is shown in FIG. ?. For , the spin-singlet state is the ground state, the triplets , are the first excited states, and the quintet is the second excited state. For , the singlet is the ground state, quintet is the first excited state, and triplets , are the second excited state.

The energy of the triplets | 1,0;1 \rangle, | 0,1;1 \rangle and the quintet | 1,1;2 \rangle. We take the units of energy as J_2, and the energy is plotted as a function of J_1/J_2
The energy of the triplets , and the quintet . We take the units of energy as , and the energy is plotted as a function of

The singlet is written as

In what follows, the single-spin states in ket will be shown in the order of 1,3,2,4, i.e. in FIG. ?. For later convenience, we name the two triplets and as and () respectively. The explicit expressions of the two triplets are given as:

To label the quintet states, we use the eigenvalues of , i.e. whose expressions are given explicitly as:

3Effect of inter-plaquette interaction

For and , all plaquettes are in the singlet state . Finite inter-plaquette interactions induce various tunneling processes among plaquettes to change both the ground state and the excitations over it. For finite , we calculate the excitation energy by two different approaches. One is the bond-operator mean-field theory (MFT)[26], which gives the excitation energy of the triplets , . Another is the second-order perturbation theory in , and it gives the energy of the quintet as well as that of and . For sufficiently small , both approximations yield finite energy gaps for these excitations and when one of these gaps closes, the corresponding (bosonic) excitation condenses to form a magnetically ordered state. The energy of triplet excitations can be observed by inelastic neutron scattering experiments. Both approximations may not be reliable for large and small .

3.1bond-operator MFT

Let us begin with the bond-operator MFT[26]. For and , is the ground state and the degenerate triplets , are the first- or the second excited state (see FIG. ?). Therefore, we may truncate the Hilbert space and consider a subspace spanned by the singlet and the triplets , . This approximation is reliable to estimate the excitation energy of the triplets, unless condenses. In this subspace, nonzero matrix elements of is

where . Using boson operators satisfying the standard commutation relations, , , , , etc, the local spin operator can be written as

where the summation over repeated indices is implied. Since the restriction that each plaquette has exactly one particle leads to the local constraint , we introduce the Lagrange multiplier and add a term

to each plaquette Hamiltonian. We may assume that for each plaquette takes the same value for all plaquettes because of the translation invariance.

Next, we replace by its expectation value , since the boson condenses in the ground state. Moreover, since the triplet is dilute when the energy gap is positive, we may ignore the terms consisting of three or four triplet operators. In this way, we obtain the mean-field Hamiltonian consists only of bilinear terms in and . The mean-field parameters are determined by requiring the expectation values of the derivatives of with respect to the mean-field (MF) ground state vanish:

or equivalently by finding the extrema of the mean-field ground-state energy :

In particular, must be minimum for .

In this approximation, the inter-plaquette interactions associated with the site reads

where the site labels and are defined in FIG. ?.

Inter-plaquette interactions associated with the plaquette n (shown by a thick line).
Inter-plaquette interactions associated with the plaquette (shown by a thick line).

Summing up all four interactions and doing Fourier transformation, the total Hamiltonian reads

where we have defined

If we introduce a vector , the MF Hamiltonian can be written compactly as

where denotes the total number of plaquettes and the kernel is given as

Using a real matrix (see Appendix A for the detail), we can diagonalize by the Bogoliubov transformation:

As is shown in Appendix A, then reduces to

where the mean-field ground state energy is given as:

In eq. ( ?), the order of signs coincides on both sides. Since , condensation of the triplets and occurs when the equality holds at some . Otherwise, there is no condensation, and . Therefore, there exist rotational symmetry and no magnetic order. In this case, are the excitation energy of triplets.

We looked for the solutions to the set of equations (Equation 3) numerically. For example, we found for the set of parameters . The dispersion relation of the excitation energy is shown in Figure 1.

Figure 1: The dispersion relation of the excitation energy of the triplet p^\prime in (), which has the lower energy of the two triplets, for the parameters \lambda=0.3,\ J_1=-0.8,\ J_2=1
Figure 1: The dispersion relation of the excitation energy of the triplet in (), which has the lower energy of the two triplets, for the parameters

If the excitation becomes soft at some , the system is in a magnetically ordered phase. From the known results[9], we expect that ordered phase appears for sufficiently close to 1. To determine the phase boundary between the paramagnetic phase and magnetically ordered ones, we searched the plane for the points where the mean-field gap vanishes. Unfortunately we found that the gap did not close in the relevant parameter region , and that the disordered singlet phase persisted; the gap vanished only for larger . This unacceptable result may be attributed to the fact that the bond-operator mean-field theory probably overestimates the stability of the plaquette phase.

3.2Second order perturbation

In this section, we compute the energy gap of triplets , and the quintet by the second order perturbation theory in the distortion parameter . The naive expansion in is ill-behaved in the vicinity of the point and we have to use another perturbation scheme for that region.

The excitation energies of triplets

Let us consider the states where there exists only one triplet and all the other plaquettes are in the singlet state. If the coupling constant of inter-plaquette interaction =0, these states are -fold degenerate, where is the number of plaquettes. For finite , the second order perturbation induces hopping of the triplet to nearest or next nearest neighbors and lifts the degeneracy.

Rotational symmetry forbids the hopping which changes the spin label or the magnetic quantum number. On the other hands, the transitions between two different triplets and of the same label occur. For example, the hopping amplitude of to the nearest-neighbor plaquette is given by

The degeneracy is partially resolved by the hopping of . The transition between and will be considered later. In the second-order perturbation, the processes that the triplet returns to the original site is also allowed. Including this effect, the energy change of -particle is given by in ( ?). Similarly, that of is given by in ( ?).

Next, we consider the transition between and . The transition amplitude is given by

Therefore, for each , eigenstates satisfies

The expressions of are given in Appendix ?. After this procedure, the degeneracy with respect both to the position and to the species and is resolved. There also exists an energy shift in the ground state. Taking all these into account, we obtain the energy of the triplets:

where denotes the energy shift of the bare ground state where all plaquettes are occupied by the singlet and is given by eq.( ?). The dispersion relation of the lower branch is shown in FIG. ?

The dispersion relation of the excitation energy E_t^{-}({\bf k}) of triplet at \lambda=0.3,\ J_1=-0.8,\ J_2=1.
The dispersion relation of the excitation energy of triplet at .

The lower branch takes its minimum at the -point , and gives spin gap . The second order expression of is given in ( ?). The expression tells us that has a pole at and that the standard perturbation breaks down near the pole. To remedy this, we introduce another perturbation parameter and carry out a double expansion in both and . Then, we obtain the energy gap in a modified method given in eq.( ?). This improved energy gap is expressed to give a better approximation around

excitation energy of quintet

Next, we consider states containing only one quintet in a background of the singlet plaquettes. As before, the degeneracy with respect to the position of the quintet plaquette is resolved by hopping. Up to the second order in , the hopping to nearest neighbor is given by

and the hopping to next nearest neighbor does not occur. Taking into account the processes that the quintet returns to the original site and the energy shift of the ground state, the excitation energy of quintet is given by in ( ?). The dispersion relation is shown in FIG. ?

The dispersion relation of the excitation energy E_q({\bf k}) of quintet at \lambda=0.3,\ J_1=-0.8,\ J_2=1.
The dispersion relation of the excitation energy of quintet at .

Since the quintet dispersion takes its minimum at , the quintet gap is given by

We note that there is the pole at and the approximation becomes poor for .

4Ground State Phases

If the inter-plaquette coupling is increased, one of the energy gaps of the triplets (( ?) and ( ?)) and the quintet ( ?) becomes at a certain critical value of . When it happens, the corresponding particle condenses and a phase transition occurs from the gapped spin-singlet phase to superfluid phases with magnetic long-range order. Therefore, we can classify the phases according to what kind of particles condense and what kind of magnetic orders is stabilized by a given set of interactions among them. In FIGs. ? and ?, we plot the value of at which the smallest energy gap becomes 0.

The value of \lambda at which the energy gaps \Delta_{t} and \Delta_{q} close. The energy gap of the triplets \Delta_{t} is given in (), which is not reliable near J_1/J_2=-2 because of the pole there, and that of the quintet \Delta_{q} is in (), which is not reliable near J_1/J_2=0.
The value of at which the energy gaps and close. The energy gap of the triplets is given in (), which is not reliable near because of the pole there, and that of the quintet is in (), which is not reliable near .
Plot of the value of \lambda when the energy gaps are equal to 0. We use \Delta_{t,\text{mod}} (eq.()) for the triplets, which is reliable even in the vicinity of J_1/J_2=-2. The curve for the quintet is the same as in FIG. . We only plot the region -2<J_1/J_2<-1.5.
Plot of the value of when the energy gaps are equal to 0. We use (eq.()) for the triplets, which is reliable even in the vicinity of . The curve for the quintet is the same as in FIG. . We only plot the region .

If we assume that no further condensation occurs in the other kinds of particles once the triplets or the quintet condenses, the phase diagram FIG. ? is obtained. When we mapped out the phase diagram FIG. ?, we have used two different expressions ( ?) and ( ?) for the energy gap of the lowest triplet in the vicinity of and away from it (), respectively. We have also neglected the quintet around since the collapse of the quintet gap there (see FIG. ?) can be attributed to the existence of a pole and is just an artifact of the perturbative approximation. Note that the phase boundary between the two regions covered by eq.( ?) and eq.( ?) is only schematic.

The schematic phase diagram of the ground state determined by the particle whose excitation gap closes first. In the green region, the quintet and the singlet condense, in the red do the triplet and the singlet, and in the blue does the singlet. In the region marked by blue, the energy gap exists. The phase shown by red may be considered as collinear antiferromagnetic (CAF) state. The nature of the green phase, where the quintet condensation occurs, is closely investigated by using an effective Hamiltonian H_{\text{qu}} (eq.()).
The schematic phase diagram of the ground state determined by the particle whose excitation gap closes first. In the green region, the quintet and the singlet condense, in the red do the triplet and the singlet, and in the blue does the singlet. In the region marked by blue, the energy gap exists. The phase shown by red may be considered as collinear antiferromagnetic (CAF) state. The nature of the green phase, where the quintet condensation occurs, is closely investigated by using an effective Hamiltonian (eq.()).

Now let us discuss the nature of the ordered phases appearing after the condensation. In the region shown as “CAF” (highlighted in red) in FIG. ?, condensation occurs to the singlet and the triplets. Then, we may expect:

which, combined with ( ?), implies

provided that and . Note that all the plaquettes are in the same state, since the energy of the triplet takes its minimum at the -point (see FIG. ?). When the combination of the two bosons condenses, the relation holds and the ground state has the transversely aligned (i.e. ) collinear antiferromagnetic order. In the case where condenses, on the other hand, we have instead and the system is in the collinear antiferromagnetic ground state in the longitudinal () direction. This is consistent with the known results[9].

Now we move on to a more interesting case. In the green region in FIG. ?, frustration is strong () and nontrivial order may be expected. In fact, Shannon et al.[9] analyzed the uniform model by numerical exact diagonalizations up to clusters of 36 spins and found a spin-nematic phase with -wave (or ) symmetry for . In the state with the nematic order, the expectation value of the rank-1 tensor vanishes , while we have a finite expectation value of the following traceless rank-2 tensor:

where and label the lattice sites.

As is shown in FIG. ?, the singlet and the quintet condense in the region of interest. This is analogous to the spinor Bose-Einstein condensation of spin-2 particles (here particles are defined not on the lattice sites but on the plaquettes). We consider a single plaquette (see FIG. ?) and, as before, denote the singlet and the quintet respectively by and . To investigate what kind of magnetic order is stabilized in the condensate, let us introduce the following mean-field ansatz for the ground state:

where the product is over all plaquettes and the complex numbers satisfy . Then, since the rank-1 tensor can not give rise to transitions between the spin-0 states and the spin-2 ones by the Wigner-Eckart theorem[27], we have , and consequently . If we introduce the cyclic operator which translates the state as , we obtain from (Equation 2) and ( ?). Therefore, the spin-nematic tensor defined on the bond satisfies . This implies that the spinor condensate of our quintet boson has the same (-wave) symmetry as the spin-nematic state discussed in Ref..

However, this is not the end of the story. Since the local spin operator with assumes several different states (e.g. polarized, nematic, etc.) and it is not obvious if or not for our - model. To determine the actual value of , we need the explicit mean-field solution for a given set of . Since we are considering the situation where the gap between the singlet ground state and the quintet excitation is vanishingly small, it would be legitimate to keep only the singlet and the quintet for each plaquette to write down the effective Hamiltonian.

The form of the effective Hamiltonian is determined by using the second-order perturbation theory and it contains the kinetic part describing the hopping of the quintet particles and the magnetic part which concerns the interactions among them. Since within a mean-field treatment the spinor part is determined by the magnetic interactions, it suffices to consider only the magnetic part of the effective Hamiltonian:

where denotes the spin operator, and the symbols and mean the nearest-neighbor- and the next-nearest-neighbor pairs, respectively. For different types of three-plaquette clusters , we assign different three-body (i.e. three-plaquette) interactions () in ( ?). The correspondence between six types of clusters and the strength of the three-plaquette interaction is shown in FIG. ?. The full expressions of , and are given in Appendix. ?. Note that our effective Hamiltonian in its full form contains the kinetic term and charge interactions as well as magnetic ones . In this sense, our effective model is a generalization of the Bose Hubbard Hamiltonian for cold atoms in optical lattices[29] and the determination of the full phase diagram and the identification of various phases found in systems of cold atoms in our magnetic system would be interesting in its own right.

Clusters involved in the 3-points interaction in (). The plaquette corresponding to i^{\prime\prime} is always located on the center of the clusters. We identify all clusters obtained from a given one by rotation by \pi /2,\pi,3\pi /2 and reflection.
Clusters involved in the 3-points interaction in (). The plaquette corresponding to is always located on the center of the clusters. We identify all clusters obtained from a given one by rotation by and reflection.

We investigate this Hamiltonian by means of a mean-field theory by assuming an -independent uniform , for simplicity. Since the parametrization of the spin-2 states is cumbersome, we adopt the method used by Bacry[28] and Barnett et al. [29]. First we note that arbitrary (normalized) spin- states are parametrized by a set of unit vectors except for obvious gauge redundancy. Using rotational symmetry, we can further reduce the number of free parameters needed to express arbitrary spin-2 states to (see Appendix. Appendix B). We numerically minimized the mean-field energy with respect to these five parameters. The result is shown in FIG. ?.

At , the system is in the ferromagnetic state for and is in the spin-nematic state for . This result slightly differs from the numerical results[9] . However, this is not surprising since our results are based on a mean-field treatment of the magnetic Hamiltonian obtained by perturbation expansion in . Our result may be improved by taking the number of sublattice larger, since and there are various 3-site interactions and .

The schematic phase diagram obtained in a similar manner to in FIG. . We zoom up the region around J_{1}/J_{2}=-2 in FIG. . In the two regions on the left (green and yellow), the quintet and the singlet condense and we determined the resulting magnetic orders by a mean-field approximation to the magnetic Hamiltonian H_{\text{qu}}. In the green region, the quintet and the singlet condense, and the spin-nematic phase appears. In the red, on the other hand, conventional ferromagnetic order is stabilized. The red and the blue region are the same as FIG. .
The schematic phase diagram obtained in a similar manner to in FIG. . We zoom up the region around in FIG. . In the two regions on the left (green and yellow), the quintet and the singlet condense and we determined the resulting magnetic orders by a mean-field approximation to the magnetic Hamiltonian . In the green region, the quintet and the singlet condense, and the spin-nematic phase appears. In the red, on the other hand, conventional ferromagnetic order is stabilized. The red and the blue region are the same as FIG. .

5Magnetization process

Having mapped out the phase diagram in the absence of external magnetic field, we consider next the magnetization process of the plaquette model by mapping the original model onto a hardcore boson model or an equivalent pseudo-spin model. Tachiki and Yamada[32] applied this method to obtain the magnetization curve of the spin-dimer model, which consists of pairs of spins. The coupling to the external magnetic field is incorporated into the Hamiltonian by adding the Zeeman term . For convenience, we set and assume that is pointing the -direction: .

Although the original treatment in Ref. is for a coupled dimer systems, we can readily generalize the method to our plaquette system as follows. We denote the plaquette states by , where are defined in (). From ( ?), the energies of a single plaquette satisfy

for . As is shown in Figure 2, with increasing the magnetic field, the quintet level comes down to faster than the lowest triplet levels and .

Figure 2: The energy of eigenstates of a single plaquette as a function of magnetic field h.
Figure 2: The energy of eigenstates of a single plaquette as a function of magnetic field .

Therefore, in order to describe the low-energy physics in the presence of strong magnetic field (), we may keep only the two lowest-lying states and for each plaquette and restrict ourselves to the subspace spanned by them. In what follows, we regard the singlet and the quintet respectively as the up- and the down state of a pseudo spin-1/2. That is,

Then, the resulting effective Hamiltonian is written in terms of the Pauli matrices ( spins) defined on each strongly-coupled plaquette.

Note that the approximation to treat only the subspace spanned by and probably breaks down for where all the components () of the quintet come into play. Also the validity of the approximation may be questionable for sufficiently large where the singlet-triplet gap may be much smaller than the singlet-quintet gap, since the triplet states and are important there (see FIG. ?).

If we simply project the original Hamiltonian to the restricted subspace as in ( ?), no spin-flipping term (or, hopping term, in terms of hardcore bosons) appears. This is because the projection is equivalent to the ordinary first-order perturbation theory and no transition between the singlet and the quintet occurs in the first-order processes. Therefore, we need take into account the second-order processes to obtain the meaningful effective Hamiltonian. The amplitude that a quintet state (spin ‘down’) hops to the adjacent plaquette is given by

The hopping to the next nearest-neighbor does not occur at this order of approximation. The energy gap between the state where there exists only one static ‘down’ spin () in a background of the ‘up’ spins (singlet plaquettes) and the one where all plaquettes are ‘up’ is given by in ( ?). The interaction between the two adjacent ‘up’ spins () is given by in ( ?) and that between the next-nearest-neighbor pair is given by in ( ?). We note that this approximation becomes poor near the pole of and at . On top of them, we have several three-‘site’ processes and putting them all together, we obtain the effective Hamiltonian:

where s denote the Pauli matrices and . The symbols and mean that the summation is taken over the nearest-neighbor- and the next-nearest-neighbor plaquettes, respectively. As in Section 4, there are six types of for different bond configurations (see FIG. ?). We label the different three-plaquette interactions by and the corresponding bond-configurations are shown in FIG. ?. The concrete expressions of and are given in Appendix. ?. We note that the transverse components and can be translated to the creation- and the annihilation operator of a hardcore boson, respectively.

We analyze the Hamiltonian ( ?) within a mean-field approximation. Since , which have the first order contributions in , are dominant for small , we may assume two different two-sublattice structures: (i) “checkerboard” and (ii) “stripe” shown in FIG. ? in the calculation .

 Two-sublattice structures assumed in the calculation: (i) striped- (left) and (ii) checkerboard (right) case. Circles (whether filled or open) denote the strongly-coupled plaquettes shown by thick lines in FIG. .
Two-sublattice structures assumed in the calculation: (i) striped- (left) and (ii) checkerboard (right) case. Circles (whether filled or open) denote the strongly-coupled plaquettes shown by thick lines in FIG. .

By using the relations

we can rewrite ( ?) in terms of . Since we are interested in the ground state energy at , we can simply replace operators in ( ?) by their expectation values on each site, e.g. for the “checkerboard” case. For convenience, we introduce the following two-component vector:

Since there is rotational symmetry in the - plane, the mean-field energy is parametrized by , , , and the angle between and . The Hamiltonian ( ?) reduces to

where denotes the total number of plaquettes and are given in Appendix ? both for the case of “checkerboard” and for the “striped” case. Correspondingly, the total magnetization is given simply as

In both cases, and is minimized for . Since any spin-1/2 states satisfy the following relation among the expectation values (see eq. (Equation 28))

the transverse magnetization can be expressed in terms of the longitudinal one . Hence, there remain two variational parameters and in . From the definition (Equation 10), the expectation values and respectively correspond to the singlet state and the fully polarized (or, saturated) state.

The critical field which marks the onset of magnetization is given by after substituting , i.e.

The right-hand side is exactly the same as (Equation 9).

Once spin-gap closes at , the quintet particle condenses, i.e. , . If , and there exists a finite expectation value of . In the hardcore boson language discussed below ( ?), can be viewed as the boson annihilation operator and its finite expectation value implies that Bose-Einstein condensation of the quintet particle occurs. In particular, if and in BEC phase, the state is in the so-called “supersolid” phase[34]. For convenience, we shall call the BEC phase satisfying a normal BEC.

It should be noted that even when , the transverse magnetization vanishes unlike the BEC in the spin-dimer model[16]. In fact, since the creation operator of the quintet particle can be written in terms of the original spin operators as

the existence of the condensate (or ) implies that we have a finite expectation value of the following spin-nematic operator:

The form (Equation 12) of the quintet creation operator suggests that we should think of the plaquette quintet as a tightly-bound magnon pair (or magnon molecule).

The critical field where the saturation occurs is given by after substituting , i.e.

We minimized numerically and we found that the energy in the “stripe” case was always equal to or smaller than that in the “checkerboard” case. We show various types of magnetization curves obtained in this way in FIG. ?. In FIG. ?, we also classified the parameter regions (in the -plane) according to the qualitative behavior of the magnetization curve. There appears (i) the normal BEC phase, (ii) the “striped” supersolid phase and (iii) the “striped” 1/2-plateau. At the 1/2-plateau, the pseudo-spins are ordered in a collinear manner and (see FIG. ?).

Schematic classification of the magnetization curve. (i) In the green region, the curve is smooth and the system is always in the normal BEC phase. (ii) In the red region, the magnetization curve has a 1/2-plateau. Except at the plateau, the system is in the normal BEC phase. (iii) The region where we have additional supersolid phases around the 1/2-plateau is highlighted in blue. (iv) In the region colored by yellow magnetization jumps to saturation and the magnetization process is step-like. The concrete expression of curves is shown in FIG. . The phase boundary is only schematic.
Schematic classification of the magnetization curve. (i) In the green region, the curve is smooth and the system is always in the normal BEC phase. (ii) In the red region, the magnetization curve has a 1/2-plateau. Except at the plateau, the system is in the normal BEC phase. (iii) The region where we have additional supersolid phases around the 1/2-plateau is highlighted in blue. (iv) In the region colored by yellow magnetization jumps to saturation and the magnetization process is step-like. The concrete expression of curves is shown in FIG. . The phase boundary is only schematic.
Magnetization curve for various values of the distortion parameter \lambda. The frustration parameters is fixed to J_{1}/J_{2}=-1.4. The colors of the curves correspond to those used in FIG.  (except for yellow). The blue curve has supersolid phase around the 1/2-plateau and the phase transition between the normal BEC and the supersolid phase is of second-order. All curves in BEC and supersolid phase is convex down because of the 3-point interaction \gamma in ().
Magnetization curve for various values of the distortion parameter . The frustration parameters is fixed to . The colors of the curves correspond to those used in FIG. (except for yellow). The blue curve has supersolid phase around the 1/2-plateau and the phase transition between the normal BEC and the supersolid phase is of second-order. All curves in BEC and supersolid phase is convex down because of the 3-point interaction in ().

The magnetization curve in the BEC and the supersolid phase is convex down because of 3-point interaction in ( ?) which breaks the particle-hole symmetry. The “striped” supersolid phase always appears around the 1/2-plateau and the width of the supersolid phase appearing on the left of the 1/2 plateau is broader than that on the right due to the convex down character. The equivalent Hamiltonian ( ?) without the 3-point interactions has been investigated by using the mean field theory[35] and Monte-Carlo simulations[35]. They found that the “striped” supersolid phase around the 1/2-plateau is stable[36]. Therefore, our result that the supersolid phase exists may be correct beyond the mean-field approximation, since the 3-point interaction in ( ?) is weak. There are other models accompanied by the supersolid phase, e.g. spin dimer XXZ model[37], spin-1/2 XXZ model on the triangular lattice[38], etc.

6Comparison with the experimental data of

In this section, we compare our results with the experimental data obtained for . Since we have three parameters , and , three experimental inputs in principle determine the set of coupling constants. Then, we use those values of coupling constants to compare the magnetization curve of our model with the experimental one[14].

We use the triplet gap observed in inelastic neutron scattering[13], the lower critical field (or if is used), which marks the onset of magnetization, and the saturation field[14] as the experimental input.

The triplet gap has been calculated in Section 3 and are given by eq.( ?) or ( ?). In Section 5, we have obtained the critical field (eq.(Equation 11)) and (eq.(Equation 13)). We compare these results with the experimental ones to determine two exchange couplings , and the distortion parameter . The result is:

where we have used ( ?) for the excitation energy of the triplet. The magnetization curve for the ratio and the distortion obtained above is shown in FIG. ? (see FIG. ?). This curve is similar to that obtained in the high-field magnetization measurement[14] except for the little convex down character.

Magnetization curve obtained from () by using the parameter set ().
Magnetization curve obtained from () by using the parameter set ().

However, a remark is in order here. Recent NMR experiments[20] suggest the displacement patterns of which yield different magnetic interactions from what have been assumed here. In particular, the system does not have any explicitly tetramerized structure (see FIG. ?), although has period 2 both in the - and the direction. Therefore, our results should not be taken literally. Instead, our plaquette model should be thought of as one of the simplest Hamiltonians realizing the BEC of magnon bound states which is applicable to much wider class of systems including our simple - model.

7Conclusion

Motivated by the recent discovery of a new two-dimensional spin-gap compound , we have studied spin-1/2 - model with a plaquette structure. For the small inter-plaquette interactions, i.e., for small , there exists a finite spin gap over the spin-singlet ground state.

We have computed the excitation energy of the triplets and the quintet in Section 3 in two different methods. If the gap of the lowest excitation closes, the corresponding particle condenses and a phase transition occurs from a paramagnetic phase to magnetically ordered phases. For the case of ferromagnetic considered here, we have two possibilities. For relatively small , the triplet particles ( and ) condense and generically we may expect CAF appears after the condensation (see FIG. ?).

For larger values of , however, the quintet excitation matters and we may have various phases. In the situation of relevance, we have either a usual ferromagnetic phase or a less conventional spin-nematic phase. One of these phases is selected by magnetic interaction among the quintet particles. We have derived an effective Hamiltonian governing the magnetic part by using the second-order perturbation and mapped out the magnetic phase diagram (see FIG. ?). A mean-field calculation predicted a finite window of the spin-nematic phase (green region in FIG. ?) in agreement with recent numerical results[10] obtained for . From the properties of the condensing particle, we found the nematic order for in the homogeneous () - model in Section 4. We remark that our effective Hamiltonian is closely related to that for cold atoms in optical lattices [29].

We have studied the magnetization process in Section 5. In the region of interest, magnetization is carried by spin-2 particles, which should be identified with a tightly-bound magnon pair (magnon molecule),and we have constructed an effective hardcore boson (or, pseudo spin-1/2) model for these spin-2 particles. By using a mean-field ansatz, we have determined the ground state of the above effective Hamiltonian as a function of the external field . We have found three different phases: (i) the normal BEC phase, (ii) the ‘striped’ supersolid phase and (iii) the ‘striped’ 1/2-plateau. In the normal BEC phase, the transverse magnetization vanishes unlike the conventional BEC in the spin-dimer model[16].

We have compared the results obtained for our - model with the experimental data of in Section 6. Although we have found that our model could qualitatively explain the magnon gap in the inelastic neutron scattering experiments[13] and the magnetization curve[14], the structure suggested by NMR measurements[20] is inconsistent with our tetramerized - model and this agreement should not be taken literally. Nevertheless, we hope that our scenario ‘molecular spin-BEC’ based on a simple - model will capture the basic physics which underlies the magnetism of the compound .

Acknowledgments

We would like to thank H. Kageyama for sharing his unpublished results, many helpful discussions, and comments on the manuscript. We are also grateful to A. Kitada and T. Miki for many useful discussions, and Mike Zhitomirsky for careful reading of the manuscript. This work is supported by the Grant-in-Aid for the 21st Century COE of Education, Culture, Sports, Science and Technology (MEXT) of Japan.


ADiagonalization of Hamiltonian by Bogoliubov transformation

For convenience, we briefly summarize the method of Bogoliubov transformation. We want to diagonalize

where

Now are boson operators, and . We introduce Bogoliubov transformation

where is real matrix, and