Triplon mean-field analysis of an antiferromagnet with degenerate Shastry-Sutherland ground states

We look into the quantum phase diagram of a spin-$\frac{1}{2}$ antiferromagnet on the square lattice with degenerate Shastry-Sutherland ground states, for which only a schematic phase diagram is known so far. Many exotic phases were proposed in the schematic phase diagram by the use of exact diagonalization on very small system sizes. In our present work, an important extension of this antiferromagnet is introduced and investigated in the thermodynamic limit using triplon mean-field theory. Remarkably, this antiferromagnet shows a stable plaquette spin-gapped phase like the original Shastry-Sutherland antiferromagnet, although both of these antiferromagnets differ in the Hamiltonian construction and ground state degeneracy. We propose a sublattice columnar dimer phase which is stabilized by the second and third neighbor antiferromagnetic Heisenberg exchange interactions. There are also some commensurate and incommensurate magnetically ordered phases, and other spin-gapped phases which find their places in the quantum phase diagram. Mean-field results suggest that there is always a level-crossing phase transition between two spin gapped phases, whereas in other situations, either a level-crossing or a continuous phase transition happens.


I. INTRODUCTION
The emergence of exotic physical properties in frustrated quantum magnets is of great current interest. [1][2][3] In particular, the spin-gapped systems with dimerized singlet ground states have been given a lot of attention. There are quite a few of such actively investigated materials, of which the CuGeO 3 and SrCu 2 (BO 3 ) 2 serve as two nice and simple examples in 1D and 2D, respectively. Here, Cu 2+ acts as a spin-1/2. Various experimental investigations have shown that Cu-germanate is a good realization of the famous Majumdar-Ghosh (MG) quantum spin chain model [4][5][6] , and SrCu 2 (BO 3 ) 2 is a natural realization of the Shastry-Sutherland (SS) model. 7,8 As it happens, the MG and the SS models are historically important prototypical cases of how frustration and quantum mechanics lead to non-magnetic dimer (spin-singlet) ground states. The competing exchange interactions in the Majumdar-Ghosh spin chain give rise to a doubly degenerate dimer ground state at J 1 = 2J 2 . The Shastry-Sutherland model is an extension of the MG model on square lattice with three-quarters of the second-neighbor exchange interactions deleted out. The ground state of the SS model is a certain exact direct product of the pairwise singlets (or simply, dimers; see Fig. 2). A plaquette spin-gapped phase has also been shown to appear in a small range between the Néel and the SS phases. 9,10 There are cases when the dimer ground state is not exact, but arises (or expected to arise) nevertheless because of strong frustration. The spin-1/2 antiferromagnetic J 1 -J 2 model on square lattice is a very good example of this. It is known to have two antiferromagnetically ordered phases: Néel and collinear. Classically, the phase transition between these occurs at J 2 /J 1 = 0.5. But quantum fluctuations reduce the sublattice magnetization, and a "quantum" paramagnetic region appears in the range 0.4 J 2 /J 1 0.6. [11][12][13][14][15][16] Although there are many different views on the nature of this quantum paramagnetic ground state, the majority opinion favors the existence of a columnar dimer phase. 12,[17][18][19][20][21][22] An extended version of this model, the J 1 − J 2 − J 3 model, has also been studied recently on square lattice around (J 2 + J 3 )/J 1 = 1/2. 23 A difficult question that arises in the studies of such problems is the nature of quantum phase transition between the magnetically ordered (say, Néel) and the dimer (say, columnar) phases. For example, in the J 1 -J 2 model, it is predicted by some to be a weak first order transition. [24][25][26] However, within a field-theoretic framework, Senthil et al. proposed some years ago a generic scenario that goes beyond the Landau-Ginzburg-Wilson paradigm. 27,28 They proposed the possibility of a continuous quantum phase transition between a dimer and a magnetically ordered phase without using order parameter notions. Instead, the concept of "deconfined" quantum criticality in which free spinons at the quantum critical point act as the essential degrees of freedom is put forward. Motivated by this remarkable suggestion, many quantum spin models have been constructed and investigated. 19,[29][30][31][32] The signatures of a deconfined quantum critical transition seem to have been observed in a SU(2)invariant spin-1/2 multiple-spin exchange model on the square lattice. 30 However there is no finality yet. There are, for example, counter suggestions wherein a ringexchange model, which breaks the Landau-Ginzburg-Wilson framework, shows a first order phase transition. 29 With similar motivations, Gélle et al. constructed and studied a quantum spin-1/2 model on square lattice a few years ago. 31 The important part of this model is a certain projector interaction term (leading to four-spin exchange interactions; described in the following section) which helps in realizing the SS dimer state as exact ground state. Interestingly, this SS ground state is fourfold degenerate (unlike in the original SS model), as it spontaneously breaks the lattice symmetry. Adding the nearest or further neighbor Heisenberg interactions to this exactly solvable model generates a competition to the exact SS ground states, and making it more interesting as well as difficult to study. They investigated it by exact diagonalization and presented a schematic quantum phase diagram. In this paper, we study this model using triplon mean-field theory with respect to various dimer and plaquette phases, and present a rich quantum phase diagram.

II. MODEL
We consider the following spin-1/2 model on the square lattice.

+K
[i,j,k;l,m,n] P 3/2 (i, j, k) P 3/2 (l, m, n). (1) The couplings J 1 , J 2 , and J 3 are respectively the first, second, and third neighbor antiferromagnetic Heisenberg exchange interactions. Originally in Ref. 31, only J 1 and J 2 were considered. But we have also included J 3 for the reasons that will become clear shortly. The last term of strength K is the projector interaction. Here, K ≥ 0, and the summation [i, j, k; l, m, n] denotes the sum over all horizontal and vertical (six-spin) plaquettes (see Fig. 1). On each plaquette, we designate two oppositely oriented triangles with vertices i, j, k and l, m, n. The projector interaction is thus given by the product of the spin-projection operators, P 3/2 , defined on the two triangles of a plaquette. The projection operator P 3/2 on a triangle is defined below.
The operator P 3/2 annihilates those states for which the total spin of the triangle is 1/2. Otherwise, it is one. For this particular choice of the projector interaction, the Hamiltonian (in the absence of the Heisenberg interactions) was shown to have an exact spontaneously dimerized singlet ground state, consisting of four degenerate Shastry-Sutherland configurations (see Fig. 2). For the details of the exact solution, see Ref. 31. The projector interaction, when expanded, would have the two-spin (Heisenberg) terms, and certain four-spin exchange interactions. Since the two triangles of a (vertical or horizontal) plaquette belong to two different sublattices of the square lattice, only J 2 and J 3 type Heisenberg exchange would arise out of the K interaction. Therefore, it seems natural to also consider the competition from J 3 , in addition to J 1 and J 2 as done in the original model.
Using the definition of projection operator in Eq. (2), the Hamiltonian Eq. (1) can be rewritten as where L is the number of lattice sites,J 2 = J 2 + 4 3 K, In the following sections, we will discuss this model using triplon mean-field theory.

III. TRIPLON MEAN-FIELD THEORY
As shown in previous section, the H K Hamiltonian gives four symmetry broken SS dimer states. This means that the model is exactly solvable at the limit J 1 = J 2 = J 3 = 0. The triplon mean-field theory 17 (also known as bond-operator mean-field theory) is a very good analytical approach on the situations where spontaneous dimerization of singlet dimers happen as in the present case. We formulate triplon mean-field theory on a chosen SS dimer state. Moreover, The previous research of J 1 -J 2 model on square lattice predict columnar-dimer 12,17,18 , plaquette-RVB 20-22 states in the paramagnetic region between Néel and collinear phases. So, we also devised the triplon mean-field theory on columnar dimer and plaquette-RVB states.
In this section, we describe two kind of triplon meanfield theories: (1) Dimer triplon mean-field theory for SS and columnar states, and (2) Plaquette triplon mean-field theory for plaquette-RVB state.

A. Dimer triplon mean-field theory
The Hilbert space of a spin dimer is four dimensional that is one singlet |s and three triplet |t x , |t y , |t z states. We transform these states into Fock space. In the Fock space, the singlet and triplet states are forming by the application of creation operators on the vacuum |0 . These operators are known as bond-operators 17 and defined as By calculating matrix elements of S 1 and S 2 , we can express the spin operators in terms of bond operators as where αβγ is the totally antisymmetric tensor. The bond operators are canonical bosons and satisfy commutation algebra. The number of bosons on a dimer is restricted by the constraint: We choose a reference SS dimer state shown in Fig. 3. For the sake of generality, we assume different singlet condensation on each dimer in unit cell. The singlet condensation amplitudes on the dimers (1) and (2) are some c-numberss 1 ands 2 respectively. We also used distinct type of triplet operators on these dimers, say t (1) α and t (2) α . The constraints are imposed by introducing global chemical potentials µ (1) and µ (2) on the dimers of unit cell: where R refers to position of unit cell. The chemical potentials fix the number of bosons on the dimers. We ignored the triplet-triplet interactions as they change the results very minutely. Using these approximations, the mean-filed Hamiltonian can be written as where, The renormalized effective chemical potentials λ (i) and ξ (i) k are give by following equations A close inspection of Eqs. (9) tells us that the meanfield Hamiltonian decoupled into independent triplet operators that is no mixing of triplet operators t In the above equation, E k = λ (λ − 2s 2 ξ k ) ≥ 0 is triplon quasiparticle dispersion. The energy of the meanfield Hamiltonian Eq. (12) gives ground energy in the absence of quasi particles. The saddle point equations of the ground state energy with respect to unknown mean-field parameterss, λ lead to the self-consistent equations. The solution of these equations give mean-field results. The detailed procedure of finding solution of self-consistent equations have been discussed in Ref. 19.

On columnar dimer state
In the case of J 1 -J 2 model on square lattice, the triplon mean-field theory predicts columnar dimer state in the range 0.19 J 2 /J 1 0.67 (see Ref. 17 ). The present Hamiltonian also has these two interactions, J 1 and J 2 . In this regard, we formulate triplon mean-field theory on a reference columnar dimer state shown in Fig. 4. This dimer state has single dimer per unit cell. We assume singlet Bose condensation on the dimers (shown thick line segments in Fig. 4). Let say it is some c-numbers.
Here again, we ignore triplon interactions and the constraint,s 2 + α t † α t α = 1, is applied globally by choosing chemical potential µ on each bonded spins uniformly: where R denotes position of a dimer. The triplon mean-field Hamiltonian of the model on the CDS state is now can written as where In the above equations, λ = 1 4 J 1 − µ is effective chemical potential and ξ k is given by ξ k = 1 2 J 1 (cos 2k x − 2 cos k y ) +J 2 (cos 2k x cos k y + cos k y ) −J 3 (cos 2k x + cos 2k y ).
The mean-field Hamiltonian is diagonalized by unitary Bogoliubov canonical transformation. The diagonalized mean-field Hamiltonian can now be written as The dispersion relation is given by E k = λ(λ − 2s 2 ξ k ).

B. Plaquette triplon mean-field theory
Consider the labelled plaquette shown in Fig. 5. The antiferromagnetic Heisenberg Hamiltonian on plaquette can be written as Let the total spin on bonds (13) and (24) are S 13 and S 24 respectively. Then, the total spin of the plaquette will be S = S 13 + S 24 ; (S 13 = S 1 + S 3 and S 24 = S 2 + S 4 ) .
The obtained energy spectrum of the Hamiltonian H p is given in Table (I). If J 1 is greater than J 2 , we can restrict ourselves in the restricted subspace of the singlet state with energy −2J 1 + 1 2 J 2 and the three degenerate triplet states having energy −J 1 + 1 2 J 2 . These states can viewed as the states emerging out of vacuum by the boson creation operators as |s := s † |0 , |t + := t † + |0 , |t z := t † z |0 and |t − := t † − |0 , where The matrix elements of the spin-operators,Ŝ x ,Ŝ y , and S z , in the subspace {|s , |t + , |t z , |t − } give the following definition of the spin-operatorŝ where the vertices of the plaquette are labelled by m. To fix the boson numbers on each plaquette, the constraint Let introduce the operators t x and t y as following Now, the constraint equation becomes s † s + t † x t x + t † y t y + t † z t z = 1, and the spin-operators shown in Eq. (22) can be written aŝ where α, β, γ = x, y, z and αβγ is Levi-Civita antisymmetric tensor.
In the simplified triplon mean-field theory, we ignore the mixing of different triplet bosons and higher order triplet interactions. Then, the exchange interactions within plaquette and inter-plaquette can be written as following where R and R are the position vectors of different plaquettes. Let the singlet Bose condensation amplitude on the plaquettes (see the shaded plaquettes in Fig. 5) is some c-numbers, i.e. s † = s =s. The global constraint is imposed on each plaquette: µ R s 2 + α t † α (R)t α (R) − 1 . The mean-field Hamiltonian now becomes where (cos 2k x + cos 2k y ) +J 2 cos 2k x cos 2k y . (32) In the above equations, we have used the definition of renormalized effective potential, that is, λ = We diagonalize the mean-field Hamiltonian Eq. (28) by Bogoliubov transformation. Thus, we get the following diagonalized form of the meanfield Hamiltonian

IV. RESULTS AND DISCUSSION
In this section, we discuss the results for three different cases of the model. In these cases, parametrization of the exchange couplings is as follows self-consistent equations obtained by dimer triplon meanfield theory on SS dimer state. It gives the phase boundary between collinear phase and SS dimer state. Importantly, the SS dimer state does not compete with Néel state as the SS dimer state is also an eigenstate for the nearest-neighbor antiferromagnetic Heisenberg interaction, J 1 i,j S i · S j . Thus, we get an overestimated phase boundary of the collinear state. We can evade from this overestimation by formulating the triplon mean-field theory on the columnar dimer state (CDS). This is physically correct adoption as, for J 1 − J 2 model, the triplon mean-field results show columnar singlet dimerization in the range 0.19 J 2 /J 1 0.67 17,19 . The triplon meanfield analyses on the CDS show two commensurate magnetically ordered phases: Néel and collinear, and an incommensurate phase with ordering wave vector (0, ϕ), The triplon mean-field theory on the plaquette crystal state gives an energetically more favorable PRVB phase inside the columnar dimer phase. The resultant quantum phase diagram is shown in Fig. 6. We explored plaquette valence-bond crystal phase in the quantum phase diagram because some earlier studies show the existence of plaquette phase in the paramagnetic region of the J 1 − J 2 model on square lattice [20][21][22] . The phase diagram contains continuous as well as firstorder phase transitions. The Néel phase shows continuos phase transition with columnar dimer phase. The collinear phase also has continuous quantum phase transitions with columnar and SS dimer states. The incommensurate magnetic phase shows first order phase transition (i.e., level crossing) with SS dimer phase and continuos phase transition with columnar dimer phase. The paramagnetic phases, CDS and SS, show first order quantum phase transition with each other. The plaquette valence bond crystal phase shows a level crossing with CDS. The exact diagonalization study 31 shows the phase boundary of the collinear phase upto κ = 0.87 at J 2 = 1 axis whereas our study underestimating and giving 0.52. At J 1 = 1 axis, the Néel phase is bounded by κ 0.2 (exact diagonalization), 0.08 (triplon mean-field theory). Again, triplon mean-field theory underestimating. However, the SS dimer phase exists for κ 0.58 at J 1 = 1 axis in triplon mean-field theory which is very close to numerical value, 0.6, obtained by exact diagonalization on 32 site cluster. The numerical study predicts many kind of phases in between the Néel phase and the SS dimer phase. This is happing because of the presence of different competing dimer correlations as revealed by triplon mean-field analyses.
When we are close to a magnetically ordered phase, the triplon mean-field results vary from exact numerical results. For example, in the case of J 1 − J 2 model, the exact diagonalization predicts the paramagnetic region in the range 0.4 J 2 /J 1 0.6 [11][12][13][14][15][16] whereas the triplon mean-field theory gives 0.19 J 2 /J 1 0.67 17,19 . This is because we always assume singlet condensation in triplon mean-field theory calculation and triplons are created on the top of the frozen singlets. This means that there is always phase coexistence in ordered phase, that is, s = 0 and t α = 0. In this phase diagram, there are two magnetically ordered phases: collinear phase and an incommensurate phase with ordering wave vectors (0, 0) and (π/2, π/2) respectively. The ordering wave vectors are with respect to SS dimer state.

Case (b):
In the absence of J 1 interaction, the model consists only second and third nearest-neighbor interactions. This leads to competition between a antiferromagnetic phase and SS dimer phase. The quantum phase diagram for this case shown in Fig. 7. This is obtained by formulating the triplon mean-field theory on the SS dimer state. Here, we have two magnetically ordered phase with ordering wave vectors (π/2, π/2) and (0, 0) (i.e., collinear . All the wave-vectors defined with respect to CDS. phase). The SS dimer state gives continuous quantum phase transitions with collinear and (π/2, π/2) magnetically ordered phases.
Case (c): For this case, the quantum phase diagram is shown in Fig. 8. We do the triplon mean-field analyses on the columnar and SS dimer states. The phase diagram contains Néel phase, an incommensurate phase with respective ordering wave vectors (0, π) and (π/2, π − ϕ), where ϕ = cos −1 J1 4J3 . The SS dimer state shows energy level crossings with the incommensurate phase and columnar dimer phase. The columnar dimer phase shows continuous quantum phase transitions with Néel and incommensurate phase.

V. SUMMARY
We studied the model proposed by Gellé et al. 31 using triplon mean-field theory with respect to various dimer and plaquette phases. We presented the quantum phase diagrams for the three different cases of the extended model: (a)J 1 − J 2 − K model (b)J 2 − J 3 − K model, and (c)J 1 − J 3 − K model. In our analyses, we used dimer and plaquette triplon mean-field theories. In J 1 − J 2 − K model, we get three valence bond crystal phases namely SS, CDS, and PRVB. There are three magnetic ordered phases emerge out with ordering wave vectors (0, π), (0, 0), and (0, ϕ), where ϕ = cos −1 J 2 2J3 . The phase transition among these magnetic ordered and valence bond crystal phases are either continuous or first-order. In the case of J 2 − J 3 − K model, the commensurate ordered phases (0, 0) and (π, π) show continuous phase transitions with SS dimer state. In the last case, that is for J 1 − J 3 − K model, the CDS phase shows continuos phase transition with Néel and an incommensurate