Fractional magnetization plateaux of the spin1/2 Heisenberg orthogonaldimer chain revisited: strongcoupling approach developed from the exactly solved IsingHeisenberg model
Abstract
The spin1/2 Heisenberg orthogonaldimer chain is considered within the perturbative strongcoupling approach, which is developed from the exactly solved spin1/2 IsingHeisenberg orthogonaldimer chain with the Heisenberg intradimer and the Ising interdimer couplings. Although the spin1/2 IsingHeisenberg orthogonaldimer chain exhibits just intermediate plateaux at zero, onequarter and onehalf of the saturation magnetization, the perturbative treatment up to second order stemming from this exactly solvable model additionally corroborates the fractional onethird plateau as well as the gapless Luttinger spinliquid phase. It is evidenced that the approximate results obtained from the strongcoupling approach are in an excellent agreement with the stateoftheart numerical data obtained for the spin1/2 Heisenberg orthogonaldimer chain within the exact diagonalization and densitymatrix renormalization group method. The nature of individual quantum ground states is comprehensively studied within the developed perturbation theory.
pacs:
04.25.Nx, 05.30.Rt, 75.10.Jm, 75.10.KtI Introduction
Fractional magnetization plateaux in lowdimensional quantum Heisenberg spin systems are one of the most fascinating and most targeted topics in the modern condensed matter physics, because they often resemble intriguing quantum ground states with extremely subtle spin order (1); (2); (3). From the experimental point of view, the fractional plateaux have been detected in magnetization curves of a variety of insulating magnetic materials, which mostly provide realworld representatives of zerodimensional Heisenberg spin clusters (4); (5); (6); (7); (8); (9); (10), onedimensional Heisenberg spin chains (11); (12); (13); (14); (15); (16); (17); (18); (19); (20); (21); (22); (23); (24); (25); (26); (27) or twodimensional Heisenberg spin lattices (28); (29); (30); (31); (32); (33); (34); (35); (36).
The fractional magnetization plateaux of onedimensional quantum Heisenberg chains should satisfy the quantization condition ( is a period of the ground state, and are the total spin and total magnetization per elementary unit, is a set of the integer numbers), which has been derived by Oshikawa, Yamanaka, Affleck (OYA) by extending the LiebSchultzMattis theorem (37); (38); (39). It is worthwhile to remark that the OYA criterion provides for a given period of the ground state a necessary (but not a sufficient) condition for a presence of fractional magnetization plateaux. To the best of our knowledge, all intermediate plateaux of the quantum Heisenberg chains observed to date experimentally are in agreement with the OYA rule when assuming either simple period or just the period doubling . For instance, the experimental representatives of the spin1/2 Heisenberg diamond chain (11); (12); (13), the trimerized spin1/2 Heisenberg chain (14); (15); (16) and the mixed spin(1/2,1) Heisenberg chain (17) display onethird plateau, the experimental realizations of the tetramerized spin1/2 Heisenberg chain (18); (19); (20), the spin1/2 Heisenberg bond alternating chain (21) as well as the spin1 Heisenberg bond alternating chain (22) exhibit onehalf plateau, the experimental realization of the spin1 Heisenberg ladder (23); (24) shows onequarter plateau, etc.
From this perspective, it is quite curious that the spin1/2 Heisenberg orthogonaldimer (or equivalently dimerplaquette) chain seems at first sight to contradict the OYA rule, which predicts just its three most pronounced fractional plateaux at zero, onequarter and onehalf of the saturation magnetization when the period of ground state does not exceed doubling of unit cell (i.e. ). Contrary to this, it has been argued by Schulenburg and Richter on the basis of exact numerical diagonalization data (40); (41) that the spin1/2 Heisenberg orthogonaldimer chain exhibits in between onequarter and onehalf plateaux an infinite series of smaller fractional plateaux at of the saturation magnetization corresponding to the ground state with the period of unit cell. It could be thus concluded that the overall magnetization curve of the spin1/2 Heisenberg orthogonaldimer chain is not consistent with any finite period of the ground state.
In this regard, it appears worthwhile to revisit the zerotemperature magnetization curve of the spin1/2 Heisenberg orthogonaldimer chain by some another rigorous method, which may capture a formation of the fractional magnetization plateaux of quantum origin. To this end, we will develop in the present work a strongcoupling approach starting from the exactly solved spin1/2 IsingHeisenberg orthogonaldimer chain with the Heisenberg intradimer and Ising interdimer interactions (42); (43); (44). It will be demonstrated that the developed strongcoupling approach actually brings insight into character of individual quantum ground states realized at particular fractional magnetization plateaux. The validity of the method will be also examined by the comparison with the results of the combined numerical approach described in Appendix A.
It should be also mentioned that the strongcoupling approach and its modification, the localizedmagnon approach, has been recently applied to the asymmetric orthogonaldimer chain (47). However, this study was merely restricted to high magnetic fields and the effect of the asymmetry.
The organization of this paper is as follows. In Sec. II we will introduce the model and suggest its approximate perturbative treatment. The exact solution for the spin1/2 IsingHeisenberg orthogonaldimer chain is formulated within the projection operator technique in Sec. III. In Sec. IV we will develop the strongcoupling approach for the spin1/2 Heisenberg orthogonaldimer chain from the exactly solved IsingHeisenberg model. The main results are summarized in Sec. V.
Ii Heisenberg orthogonaldimer chain and perturbation method
Let us consider the spin1/2 quantum Heisenberg orthogonaldimer chain given by the Hamiltonian:
(1)  
which involves the coupling constants and accounting for the Heisenberg intradimer and interdimer interactions, respectively, in addition to the usual Zeeman’s term (see Fig. 1 for a schematic illustration of the considered magnetic lattice). It has been found that the model defined through the Hamiltonian (1) exhibits a singletdimer ground state for (48) at zero magnetic field and reveals the peculiar infinite series of the fractional magnetization plateaux in between 1/4 and 1/2 of the saturation magnetization (40); (41).
Recently, we have exactly solved the simplified version of this frustrated quantum spin model, the socalled spin1/2 IsingHeisenberg orthogonaldimer chain defined by the Hamiltonian:
(2)  
which takes into account the Heisenberg intradimer interaction and the Ising interdimer interaction (42); (43). The only difference between two models lies in replacing the Heisenberg interdimer coupling in the Hamiltonian (1) through the Ising interdimer coupling in the Hamiltonian (2). The simplified spin1/2 IsingHeisenberg orthogonaldimer chain (2) can be rigorously solved either by the transfermatrix method (42); (43) or the mapping transformation technique (44), whereas this model still exhibits some common features with its full Heisenberg counterpart like intermediate magnetization plateaux at onequarter and onehalf of the saturation magnetization. However, the exactly solved IsingHeisenberg model given by the Hamiltonian (2) does not reproduce neither an infinite series of the fractional magnetization plateaux in between onequarter and onehalf of the saturation magnetization nor an existence of the TomonagaLuttinger spinliquid phase above the intermediate onehalf plateau. Instead it shows the macroscopically degenerate groundstate manifold at each critical field accompanied with the magnetization jump (43). This fact enables us to develop an approximate theory for the spin1/2 Heisenberg orthogonaldimer chain based on the exactly solved spin1/2 IsingHeisenberg orthogonaldimer chain when treating the part of the interdimer coupling perturbatively.
To this end, let us decompose the total Hamiltonian (1) of the spin1/2 Heisenberg orthogonaldimer chain into two parts
(3) 
where the former unperturbed (ideal) part corresponds to the exactly solved spin1/2 IsingHeisenberg orthogonaldimer chain (42) rewritten as
(4)  
while the latter perturbed part contains all remaining terms from the total Hamiltonian (1) of the spin1/2 Heisenberg orthogonaldimer chain
(5) 
It is noteworthy that the perturbed Hamiltonian includes except the part of the interdimer coupling also difference between the true magnetic field and its respective critical value , around each of which one should separately perform the perturbative expansion due to a macroscopic degeneracy of the groundstate manifold of the spin1/2 IsingHeisenberg orthogonaldimer chain (42); (43). The macroscopic degeneracies at the critical fields and their values will be given and discussed in the next section. Though we have singled out the part of the interdimer interaction explicitly, the isotropic limit of the quantum Heisenberg model will be later recovered by putting in all final expressions. Besides, our further consideration will be limited only to the most interesting case with the antiferromagnetic interactions under the simultaneous constraint , which favors the singletdimer phase as the zerofield ground state of the spin1/2 Heisenberg orthogonaldimer chain (48).
Iii Exact solution of the IsingHeisenberg orthogonaldimer chain in terms of the projection operators
Although the exact solution of the spin1/2 IsingHeisenberg orthogonaldimer chain given by the Hamiltonian (2) [or equivalently by the Hamiltonians (4)] have been already reported by two independent methods, i.e. the transfermatrix method (42); (43) and the mapping transformation technique (44), it appears worthwhile to rederive it by making use of the projection operators in view of a subsequent development of the perturbative strongcoupling approach. For this purpose, let us introduce first the dimerstate basis
(6) 
and the corresponding projection operators (49); (50); (51)
(7) 
One can find the representation of spin operators through the introduced projection operators (the explicit correspondence is given in Appendix B) and rewrite in terms of these operators the local Hamiltonians (4) pertinent to the vertical and horizontal Heisenberg dimers (see Fig. 1):
(8)  
(9)  
Here, denotes the component of the total spin on th vertical dimer, whereas an explicit form of the total spin and on two neighboring vertical dimers has been retained in Eq. (9) for the sake of compactness. It is quite evident that the Hamiltonians of the vertical dimers (8) are already diagonal in the dimer representation, while the Hamiltonians of the horizontal dimers (9) can be diagonalized by a unitary transformation:
(10) 
It should be stressed that and depend on eigenvalues of the operators , , and they can be reduced to an algebraic form using the van der Waerden identity (see e.g. Refs. (45); (46)). The explicit expressions for and is given in Appendix C. Apparently, two polarized triplet states and are invariant under the unitary transformation (10), while the singlet and the zerocomponent of the triplet state are mutually entangled to a more complex quantum state:
(11)  
After performing the local unitary transformation (10) one consequently obtains the diagonal form of the Hamiltonian of the ()st horizontal dimer
(12) 
Here is a symbolic notation of Kronecker delta. Its algebraic representation through the spin and projection operators can be found in Appendix C (see Eq. (40)).
Using this procedure, the total Hamiltonian (2) of the spin1/2 IsingHeisenberg orthogonaldimer chain has been put into a fully diagonal form and the ground state of the model can be easily found by minimizing a sum of its local diagonal parts (8) and (12) (see also Ref. (42)). By inspection, one finds just four different ground states in the investigated parameter space and , namely,

singletdimer (SD) phase: ,

modulated ferrimagnetic (MFI) phase:

staggered bond (SB) phase:
, 
saturated (SAT) phase: .
It is worthwhile to recall that the ground state is macroscopically degenerate at the critical fields, where the magnetization discontinuously jumps due to successive fieldinduced (firstorder) phase transitions SDMFISBSAT upon strengthening of the magnetic field. The explicit form of the critical fields corresponding to the relevant groundstate phase boundaries were found in Ref. (42):

SDMFI: ,

MFISB: ,

SBSAT: .
The groundstate manifold along with its macroscopic degeneracy at a given critical field can be obtained from the condition of the phase coexistence of both individual ground states. For instance, all horizontal dimers have to be in the singletlike state at SDMFI boundary, while the polarized triplet states can be randomly distributed on the vertical dimers on assumption that the hardcore repulsion between the nearestneighboring polarized states on the vertical dimers is fulfilled (the remaining vertical dimers have to be in the singletlike state ). Thus, the groundstate manifold at SDMFI phase boundary can be defined through the following projection operator:
(13) 
Similarly, the groundstate manifold at SBSAT boundary can be built from any random configuration of the singletlike states and on the horizontal and vertical dimers, which satisfies the hardcore repulsion between the singletlike states on the nearestneighbor dimers (the remaining dimers should occupy the polarized triplet states and ). The groundstate manifold at SBSAT phase boundary is thus given by the following projection operator:
(14) 
The situation at MFISB phase boundary is much more intricate and it does not allow such a transparent representation. However, the groundstate manifold at MFISB phase boundary can be defined through the projection operator as follows:
(15) 
Iv Strongcoupling approach developed from the exactly solved IsingHeisenberg model
The strongcoupling approach is based on the manybody perturbation theory (see e.g. Ref. (52); (53)), where the unperturbed and perturbed parts of the Hamiltonian can be singled out:
(16) 
and the eigenvalue problem for the ideal part becomes exactly tractable. If is the projection operator on a ground state of the unperturbed model subspace and , the perturbative expansion can be formally found out for the effective Hamiltonian acting in the projected subspace :
(17)  
where . Note that the perturbative expansion (17) is still exact, but one usually has to truncate it due to computational difficulties arising out from higherorder contributions of the effective Hamiltonians. In the present work we will restrict ourselves to the secondorder perturbative expansion, which will take into account the zeroth, first and the secondorder contributions to the effective Hamiltonian: , and , respectively. In what follows we will develop the perturbation theory for the spin1/2 Heisenberg orthogonaldimer chain from the exactly solved spin1/2 IsingHeisenberg orthogonaldimer chain by considering separately the macroscopically degenerate groundstate manifold at each its phase boundary.
iv.1 SDMFI boundary
The phase boundary between SD and MFI ground states of the spin1/2 IsingHeisenberg orthogonaldimer chain is defined by the critical field and the projection operator to the macroscopically degenerate groundstate manifold is given by Eq. (13). The straightforward application of Eq. (17) results in the firstorder term:
(18)  
where is the unitarytransformed perturbation operator. The secondorder term requires the calculation of the matrix elements and is much more involved (see the details of the calculations in Appendix D):
(19)  
(20) 
Summing up all contributions up to second order we get the following effective Hamiltonian:
(21) 
Obviously, the effective Hamiltonian (21) is essentially onedimensional classical model with a simple mapping correspondence to the latticegas model, which can be established by considering the singletlike (polarized triplet) states on the vertical dimers as being empty (filled) sites: , , , . The effective Hamiltonian in the latticegas representation is extraordinarily simple and it satisfies the hardcore constraint for the polarized triplet states on the nearestneighbor vertical dimers as dictated by the projection operator (13):
(22) 
The ground state corresponds either to the latticegas model with all empty sites for or the halffilled case for . The former condition with all empty sites ( for all ) is consistent with SD ground state of the original spin model (1), while the latter condition with a regular alternation of empty and filled sites apparently corresponds to MFI phase. It could be thus concluded that the secondorder perturbative expansion around SDMFI phase boundary does not create any novel ground state, but it only renormalizes the critical field of a discontinuous phase transition between SD and MFI ground states accompanied with the magnetization jump from zero to onequarter of the saturation magnetization. It is quite evident from Eq. (19) that the secondorder correction to the first critical field is negative (), which is consequently shifted to lower values of the magnetic field in an excellent accordance with the stateoftheart numerical data obtained from the densitymatrix renormalization group (DMRG) and exact diagonalization (ED) calculations described in Appendix A (c.f. Figs. 2 and 3).
iv.2 MFISB boundary
The phase boundary between MFI and SB ground states of the spin1/2 IsingHeisenberg orthogonaldimer chain occurs at the second critical field , at which the projector (III) determines the macroscopically degenerate groundstate manifold. One may use the same procedure as before in order to get the effective Hamiltonian. The firstorder contribution to the effective Hamiltonian is determined by the diagonal elements of the perturbed part of the Hamiltonian:
(23) 
After cumbersome calculations one gets of the following result for the secondorder contribution to the effective Hamiltonian (see Appendix E for further details):
(24) 
Since all three expansion coefficients are negative () one generally has and if horizontal dimers are in the singletlike states. It is quite straightforward to show that the ground state corresponds to the state with , i.e. . Therefore, the states with the polarized horizontal triplets can be excluded from the consideration if we are seeking only for the ground state. Let us introduce the notation , in order to rewrite the Hamiltonian (24) in the latticegas representation:
(25)  
Similarly to the previous case one obtains the classical effective Hamiltonian with the hardcore potential, but there also appears some additional nextnearestneighbor interaction. When looking for the lowestenergy states of the effective latticegas model given by the Hamiltonian (25), one finds three different ground states either with empty, onethirdfilled or halffilled states upon varying the external magnetic field. These lowestenergy states correspond to the fractional plateaux at the onehalf, onethird or onequarter of the saturation magnetization, whereas two conditions of a phase coexistence determine the critical fields associated with the respective magnetization jumps:
(26) 
The perturbation expansion around the MFISB phase boundary thus surprisingly verifies an existence of the fractional onethird magnetization plateau, which is totally absent in a zerotemperature magnetization curve of the spin1/2 IsingHeisenberg orthogonaldimer chain (see Fig. 3(a)). Besides, the method also brings insight into a microscopic nature of the spin arrangement realized within the 1/3plateau, in which singletlike states are spread over all horizontal dimers and each third vertical dimer. It can be seen from Fig. 2(b) that the developed perturbation theory predicts the critical field between 1/4 and 1/3plateaux in a perfect agreement with the numerical results (see Appendix A), while the other critical field between 1/3 and 1/2plateaux lies in a middle of the tiny region involving an infinite sequence of the fractional magnetization plateaux . There are strong indications that the other tiny fractional magnetization plateaux could be also recovered if the perturbation expansion would be performed up to higher orders. In this case, the repulsion between further neighbors in the latticegas representation appears leading to the onequarterfilled, onefifthfilled, …, states. These states correspond to the 3/8, 2/5plateaux in the original spin model using the relation .
iv.3 SBSAT boundary
The phase boundary between SB and SAT ground states of the spin1/2 IsingHeisenberg orthogonaldimer chain represents quite exceptional case, because the perturbative strongcoupling approach will lead in this specific case to the effective Hamiltonian of a quantum nature. The critical field relevant to this phase boundary is given by , while the macroscopically degenerate groundstate manifold is defined by the projection operator (14). Applying the perturbation theory one obtains the following firstorder contribution to the effective Hamiltonian:
(27) 
After tedious calculations (see Appendix F) one may also find the secondorder perturbation term:
(28) 
In above, we have introduced the following notation for the coefficients:
(29) 
which enable to write the secondorder contribution (28) to the effective Hamiltonian in a more compact form. Next, let us proceed to a notion of the quantum lattice gas achieved through the following transformation: , , , ( and are Pauli operators which anticommute on one site and commute on different sites). The overall effective Hamiltonian can be subsequently rewritten in the particle representation as:
(30)  
The effective quantum latticegas model (30) contains two types of particles. The particles on odd sites are mobile and they hop in between nearestneighbor odd sites, while the particles on even sites are localized. The projection operator (14) additionally leads to the hardcore repulsion, which blocks the occupation of nearestneighbor sites. To get the ground state, one has to find such a configuration of the localized particles on the even sites given by the set of occupation numbers , which corresponds to the lowestenergy eigenstate of the quantum subsystem on the odd sites. It is worthy to note that the corresponding quantum part is split into two open chains at each even site occupied by the particle, whereas occupation of the neighboring odd sites and should be then excluded (). In the following we will show that the energy of the system increases whenever the empty even site changes to the filled one (i.e. changes from 0 to 1). This fact should be proven separately for two cases: and . Let us denote by the energy for the empty (filled) even site . It is quite clear from previous arguments that , where and are the lowest energies of the left and right parts of the system split by (see Fig. 4).
The following inequality can be also obtained , which furnishes the proof for :
(31) 
In the opposite case we have to use the property that a sum of the groundstate energies of two separate chains and clusters is larger than the groundstate energy of the whole system, which is obtained by joining the separate subsystems together. This property implies a validity of the following inequality
(32) 
After some algebra one can also show that the following inequality holds for
(33) 
Accordingly, the ground state should correspond to the particular case with all empty even sites ( for all ), whereas the effective Hamiltonian (30) of the quantum latticegas model then reduces to
(34) 
The SAT ground state corresponds to the empty state in the particle language ( for all ), while the SB ground state pertinent to the 1/2plateau emerges when all odd sites are filled by particles and all even sites are being empty (, for all ). To get the respective values of the critical fields, it is more convenient to convert the effective quantum latticegas model (34) into a pseudospin language. As a matter of fact, one gets the effective Hamiltonian of the spin1/2 Heisenberg chain using , , :
(35) 
The critical fields for the quantum antiferromagnetic Heisenberg chain are exactly known: . Bearing this in mind, the saturation field and the upper critical field for the 1/2plateau can be found from the relations
(36) 
It can be seen from Fig. 2(b) that both critical fields and obtained from the perturbative strongcoupling approach quantitatively agree with the numerical DMRG data up to a relative strength between the inter and intradimer couplings , while the critical field () is slightly underestimated (overestimated) for greater values of the interaction ratio . Most importantly, the perturbative expansion around SBSAT phase boundary predicts the gapless TomonagaLuttinger spinliquid (SL) ground state in a relatively wide range of the magnetic fields in spite of the fact that the simplified IsingHeisenberg model does not exhibit this ground state at all [c.f. Fig. 2(a) and (b)]. It should be also pointed out that the spin1/2 Heisenberg orthogonaldimer chain undergoes true continuous (secondorder) quantum phase transitions at the critical fields and delimiting a stability region of the SL ground state in contrast with discontinuous (firstorder) phase transitions associated with the magnetization jumps between the other fractional plateaux (see Fig. 3). Last but not least, the perturbative strongcoupling approach brings a deeper insight into the character of the SL phase, because the number of the odd filled sites within the effective quantum latticegas model continuously decreases with increasing of the magnetic field by keeping all even sites empty. When returning back to the spin language this result is taken to mean that the total number of (mobile) singlet states on the horizontal dimers gradually decreases within the SL ground state from its maximum value at the critical field down to zero at while keeping all vertical dimers in the polarized triplet state.
V Conclusions
The present work dealt with the perturbative strongcoupling calculation for the quantum spin1/2 Heisenberg orthogonaldimer chain in a magnetic field, which has been developed from the exactly solved spin1/2 IsingHeisenberg orthogonaldimer chain with the Heisenberg intradimer and Ising interdimer interactions up to the second order. Notably, the quantum spin1/2 Heisenberg orthogonaldimer chain represents a paradigmatic example of quantum spin chain with plethora of outstanding quantum ground states, which are manifested in a zerotemperature magnetization curve either as extensive zero, onequarter and onehalf magnetization plateaux, an infinite sequence of tiny fractional () magnetization plateaux or the TomonagaLuttinger spinliquid phase. Despite of this complexity, we have convincingly evidenced an impressive numerical accuracy of the strongcoupling approach stemming from the exactly solved IsingHeisenberg model through a direct comparison of the derived results with the stateoftheart numerical data obtained within DMRG and ED methods. It has been found that the strongcoupling approach not only substantially improves phase boundaries between the already existing ground states of the idealized IsingHeisenberg orthogonaldimer chain, but it also gives rise to completely novel quantum ground states such as the fractional onethird plateau or the TomonagaLuttinger spinliquid phase. Based on the effective latticegas model at MFISB boundary, we presumed that higherorder perturbation terms result in the repulsion interactions of a longer range. It is an indication that other tiny fractional plateaux in between the onequarter and onehalf of the saturation magnetization could be recovered within the higherorder perturbation theory.
It is also worth noticing that the perturbative strongcoupling approach could be alternatively developed from the limit of isolated dimers as it is shown in Appendix G. However, this simpler version of the perturbative treatment has serious deficiency in that it does not reproduce in the second order neither onequarter nor onethird magnetization plateaux. It could be thus concluded that the perturbative strongcoupling method developed from the exactly solved IsingHeisenberg orthogonaldimer chain is quite superior with respect to its simplified version derived from the limit of isolated dimers. It therefore appears worthwhile to remark that there exist several exact solutions for the hybrid IsingHeisenberg models, which could be used as useful starting ground for the perturbative analysis (see Ref. (54) and references cited therein). Quite recently, the similar perturbation procedure starting from the exactly solved spin1/2 IsingHeisenberg diamond chain has been applied to corroborate an existence of the TomonagaLuttinger spinliquid phase in between the intermediate onethird plateau and saturation magnetization of the quantum spin1/2 Heisenberg diamond chain (55). Our further goal is to apply the developed strongcoupling approach to the quantum spin1/2 Heisenberg model on the ShastrySutherland lattice to verify or disprove a presence of the questioned fractional magnetization plateaux by making use of the exact solution reported for the spin1/2 IsingHeisenberg model on the ShastrySutherla