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the study of the phase transition of frustrated spin systems on two - dimensional ( 2d ) lattices is a central problem in modern condensed mater physics . a competition of exchange interaction can lead to frustration , where spatial arrangement of magnetic ions in a crystal for which a simultaneous antiparallel ordering of all interacting spin is impossible . in particular , one of the frustrated 2d models most discussed is the quantum spin-@xmath0 heisenberg antiferromagnet on a square lattice with competing nearest - neighbor ( nn ) and next - nearest - neighbor ( nnn ) antiferromagnetic exchange interactions ( known as @xmath12 model ) @xcite . the criticality of this @xmath12 heisenberg model on a square lattice are relatively well known at @xmath13 . there are two magnetically long - range ordered phases at small and at large values of @xmath6 separated by an intermediate quantum paramagnetic phase without magnetic long - range order in the region between @xmath14 and @xmath15 , where the properties of these disordered phase are still under intensive debate . for @xmath16 @xmath17 , the system possesses antiferromagnetic ( af ) long - range order with wave vector @xmath18 , with a staggered magnetization smaller than the saturated value ( quantum fluctuations ) , which vanished continuously when @xmath19 . for @xmath20 we have two degenerate collinear states which are the helical states with pitch vectors @xmath21 and @xmath22 . these two collinear states are characterized by a parallel spin orientation of nearest neighbors in vertical ( or horizontal ) direction and an antiparallel spin orientation of nearest neighbors in horizontal ( or vertical ) direction , and therefore exhibit nel order within the initial sublattice a and b. at @xmath23 , the magnetization jumps from a nonzero to a zero value . the phase transition from nel to the quantum paramagnetic state is second order , whereas the transition from the collinear to the quantum paramagnetic state is first order@xcite . et al_.@xcite have shown that the intermediate quantum paramagnetic is a ( _ singlet _ ) plaquette crystal , and the ground and first excited states are separated by a finite gap . the interest to study the two - dimensional @xmath12 heisenberg antiferromagnet have been greatly stimulated by its experimental realization in vanadium phosphates compounds@xcite , such as li@xmath24vosio@xmath25 , li@xmath24vogeo@xmath25 , and  vomoo@xmath25 , which might be described by this frustrated model in the case of @xmath26 ( @xmath27 ) . these isostructural compounds are characterized by a layered structure containing v@xmath28 ( @xmath29 ) ions . the structure of v@xmath28 layer suggest that the superexchange is similar . in these compounds a second order phase transition to a long - range ordered magnetic phase has been observed . nmr spin - lattice relaxation measurements@xcite below @xmath30 shows that the order is collinear . due to the two - fold degeneracy of the ground - state for @xmath31 it is not possible to say _ a priori _ which will be the magnetic wave vector ( i.e. , @xmath21 and @xmath22 ) below @xmath30 . on the other hand , such a scenario can change by considering spin - lattice coupling which will lift the degeneracy of the ground - state and will lower its energy@xcite . then , any structural distortion should inevitably reduce this competing interactions and thus reduces the frustration . in the case of  this frustrated magnetic materials , the competing interactions are inequivalent but their topology and magnitudes can be tuned so that the strong quantum fluctuations destroy the long - range ordering . experimentally the ground state phase diagram of frustrated compounds , described by the @xmath12 model , can be explored continuously from high to the low @xmath6 regime by applying high pressures ( p ) , which modify the bonding lengths and angles . recent results from x - ray diffraction measurements@xcite on the li@xmath24vosio@xmath25 compound has shown that the ratio @xmath8 decreases by about @xmath32 when the pressure increases from @xmath33 to @xmath34gpa . a generalization of the @xmath12 heisenberg antiferromagnetic model on a square lattice was introduced by nersesyan and tsvelik@xcite and studied by other groups@xcite , the so - called @xmath35 model . in the @xmath36 model is considered inequivalence nn couplings @xmath1 and @xmath37 in the two orthogonal spatial lattice dimensions with all the nnn bonds across the diagonals to have the same strength @xmath3 . study of extensive band structure calculations@xcite for the vanadium phosphates abvo(po@xmath38)@xmath24 ( ab = pb@xmath24 , srzn , bazn , and bacd ) have indicated four inequivalent exchange couplings : @xmath1 and @xmath2 between nn and @xmath3 and @xmath39 between nnn . for example , in srznvo(po@xmath25)@xmath24 was estimated @xmath40 and @xmath41 causing a distortion of the spin lattice . this spatial anisotropy tends to narrow the critical region and destroys it completely at a certain value of the interchain parameter @xmath7 . on the other hand , by using the continuum limit of the @xmath36 spin-@xmath0 model starykh and balents@xcite have shown that this transition splits into two , with the presence of an intermediate quantum paramagnetic ( columnar dimer ) phase for @xmath42 . bishop , _ _ et al__@xcite , by using coupled cluster treatment found the surprising and novel result that there exists a quantum triple point ( * qtp * ) with coordinates at ( @xmath43 ) , below which there is a second - order phase transition between the * af * and * caf * phases while above this * qtp * are these two ordered phases separated by the intermediate magnetically disordered phase ( vbs or rvb ) . the order parameters of both the * af * and * caf * phases vanish continuously both below and above the * qtp * , which is typical of second - order phase transition . there is some evidence that the transition between the * caf * and intermediate phases is of first - order . using exact diagonalization@xcite with small lattice of @xmath44 ( @xmath45 ) size , the intermediate * qp * phase for all interval of @xmath46 $ ] has been obtained for the pure spin-@xmath0 @xmath12 model on a square lattice . these results are in accordance with results obtained by starykh and balentes@xcite , that predicted not the * qtp * in the ground - state phase diagram recently observed by bishop , _ _ et al.__@xcite . the ground state ( gs ) properties of the two - dimensional frustrated heisenberg antiferromagnet have been investigated by various methods . the exact diagonalization starts from singlet states on pairs of sites , which cover the whole 2d lattice . however , the manifold of these states which can be constructed is nonorthogonal and overcomplete . this numerical methods are limited to small clusters @xmath47 due to storage problems . the computation on the largest cluster @xmath45 has been performed by schulz and co - workes@xcite @xmath48 years ago . in spite of the great improvements achieved during this time , it is not possible so far to repeat this calculation for the next interesting cluster @xmath49 . this is only possible with other technique , as the quantum monte carlo simulation . due to the progress in computer hardware and the increased efficiency in programing , very recently@xcite the gs of the quantum spin-1/2 @xmath12 model have been calculated by the lanczos algoritm for a square lattice with @xmath50  sites . the theoretical treatment of the frustrated quantum models is far from being trivial . many of the standard many - body methods , such as quantum monte carlo techniques , may fail or become computationally infeasible to implement if frustration is present due to the minus - sign problem . hence , there is considerable interest in any method that can deal with frustrated spin systems . this considerable qualitative difference in the ground state phase diagram in the @xmath51 plane of the quantum spin-@xmath0 @xmath52 model further motivates us to study this issue by alternative methods .    using a variational approximation , in which plaquettes of four spins are treated exactly , oliveira@xcite has studied the ground state phase diagram of the pure @xmath12 heisenberg antiferromagnet on a square lattice , where the quantitative results are in good accordance with a more sophisticated method ( exact diagonalization ) . in this work , we generalize this variational method to treat the anisotropic square lattice ( @xmath52 model ) . the rest of this paper is organized as follows : in sec . ii , the model is presented and a brief discussion of results . in sec . iii , the method is applied for the case of one plaquette with four spins interacting with other plaquette type mean field approximation . main results will be presented in sec . iv , as well as some discussions . finally , in sec . v we will give a brief summary . the critical behavior of the quantum spin-@xmath0 @xmath12 heisenberg model has been studied for many years , but very little has been done in the anisotropic square lattice case , which is described by following hamiltonian : @xmath53 where @xmath54 is the spin-@xmath0 pauli spin operators , the index @xmath55 labels the @xmath56 ( row ) and @xmath57 ( column ) components of the lattice sites . the first sum runs over all nn and the second sum runs over all nnn pairs . we denote the hamiltonian ( 1 )  by @xmath35 model , with strength @xmath1 along the row direction , @xmath37 along the column direction , @xmath58 along the diagonals , and we assume all couplings to be positive with @xmath59 . the classical ( @xmath60 ) model ( 1 ) has only two ordered ground - states : * af * ( or nel ) for @xmath61 and columnar stripe ( * caf * ) for @xmath62 separated by a first - order line at @xmath63 . quantum fluctuations play a significant role in the magnetic phase diagram of the system at zero temperature . we will investigate the role of quantum fluctuations on the stability of the nel and collinear phases . in the @xmath29 case ( quantum limit ) , the line splits into two phase transitions , where the ordered states ( * af * and * caf * ) are separated by an intermediate quantum paramagnetic ( * qp * ) phase , both on a square lattice . exact diagonalization@xcite has estimated a critical line at @xmath64 , for the transition between the * caf * and * qp * states , and at @xmath65 between the * af * and * qp * states . the phase diagram in the @xmath51 plane obtained is in accordance with starykh and balents@xcite . however , the existence of * qtp * ( _ quantum triple point _ ) that was predicted by bishop , _ et al_.@xcite , is not present in their obtained phase diagram . moreover , they found only presence of second - order phase transitions in the phase diagram . this contradictory qualitative results ( existence or not of * qtp * ) is the primary motivation behind this present work .    on the other hand , a critical endpoint ( * ce * ) is a point in the phase diagram where a critical line meets and is truncated by a first - order line . this * ce * appear in the phase diagram of many physical systems such as binary fluid mixtures , superfluids , binary alloys , liquid crystals , certain ferromagnets , etc , and have been known for over a century@xcite . despite the * ce * long history , new singularities at the * ce * were predicted . fisher and upton@xcite argued that a new singularity in the curvature of the first - order phase transition line should arise at a * ce*. this prediction was confirmed by fisher and barbosas@xcite phenomenological studies for an exactly solvable spherical model . in conclusion of the analysis of the multicritical behavior observed in the ground - state phase diagram in the @xmath66 plane for the @xmath52 model , we have the presence of a _ quantum critical endpoint _ ( * qce * ) and not * qtp * as mentioned other works@xcite therefore , the objective of this work is to obtain the * qce * using the variational method , that was developed previously by oliveira@xcite in the pure limit ( @xmath67 ) case . we first express the fluctuations around the classical ground state ( * af * and * caf * phases ) , where consider a trial vector state @xmath68 for the ground state as a product of plaquette state @xmath69 . we denote the plaquettes by @xmath70 label , that is composed of four spins , where it do not overlap ( _ mean field _ ) on the square lattice as illustrated in figure 1 . each plaquette state is given by     and @xmath71 spin operators that are considered in eq . ( 2 ) . ] @xmath72    where \{@xmath73 , @xmath74 } is the vector basis with @xmath75 , \{@xmath76 } are real variational parameters obeying the normalization condition @xmath77 . with this choice of vector states , the mean value of the spin operator in each site of the plaquette is given by @xmath78 @xmath79 , where the components in the @xmath56 and @xmath57 directions are null . using the trial vector state defined in the eq . ( 2 ) , we obtain the magnetizations at each site that are given by @xmath80@xmath81@xmath82 and@xmath83 where we have used the same set of parameters ( _ canonical transformation _ ) of ref.@xcite , i.e. , @xmath84 , and @xmath85 , which obeys the normalization condition @xmath86 .    the ground state energy per spin and unit of @xmath1 , @xmath87 , is given by @xmath88 with @xmath89 and@xmath90   , \tag{9}%\end{aligned}\ ] ] where @xmath91 is the mean value of a given observable @xmath92 calculated in the vector state of the @xmath93 plaquette as illustrated in fig . the variational energy can be evaluated using the properties of the spin-@xmath0 pauli operator components , i.e. , @xmath94,@xmath95 and @xmath96 ,  that is expressed for@xmath97 + \nonumber\\ &   2\left (   1-\lambda\right )   \left (   y^{2}v^{2}-z^{2}\omega^{2}\right ) + 2x\left (   y\lambda+z\right )   + \alpha\left [   \dfrac{1}{2}-\left (   y - z\right ) ^{2}-v^{2}-\omega^{2}+6x^{2}u^{2}-6y^{2}v^{2}-6z^{2}\omega^{2}\right ]   . \tag{10}%\end{aligned}\ ] ]    to obtain the minimum energy with a boundary condition given by normalization @xmath98 , we use the lagrange multiplier method which correspond the minimization of the functional@xmath99    the stationary solutions ( @xmath100 ) are obtained by solving the set of nonlinear equations @xmath101{c}% -\left (   \lambda+1\right )   x-4\left (   \lambda+1\right )   xu^{2}+2\left ( y\lambda+z\right )   + 12\alpha xu^{2}=2\eta x\\ \left (   1-\lambda\right )   \left (   y+4yv^{2}\right )   + 2x\lambda-2\alpha\left ( y - z\right )   -12\alpha yv^{2}=2\eta y\\ -\left (   1-\lambda\right )   z-4\left (   1-\lambda\right )   z\omega^{2}% + 2x+2\alpha\left ( y - z\right )   -12\alpha z\omega^{2}=2\eta z\\ -\left (   \lambda+1\right )   u=2\eta u\\ \left (   1-\lambda\right )   v-2\alpha v=2\eta v\\ -\left (   1-\lambda\right )   \omega-2\alpha\omega=2\eta\omega \end{array } \right .   , \tag{12}%\ ] ] where @xmath102 is the lagrange multiplier . the variational parameters @xmath103 , and @xmath102 are determined simultaneously solving the system of equations ( 12 ) combined with the normalization condition @xmath98 for each phase . in the quantum paramagnetic ( * qp * ) phase we have @xmath104 . we note that in the isotropic limit ( @xmath67 ) , our results reduce the same expression obtained by oliveira@xcite . in this disordered phase , the ground state vector @xmath105 is an eigenvector of @xmath106 , where @xmath107 is the total spin of the @xmath70th plaquette of four spins , with zero eigenvalue ( singlet state ) . in the * af * ordered phase we have the boundary condition @xmath108 , and in the * caf * phase @xmath109 .    the order parameters @xmath110 and @xmath111 are numerically obtained as a function of frustration parameter @xmath8 for a given value of spatial anisotropy @xmath7 . we observe that the order parameter @xmath112 goes smoothly to zero when the frustration parameter ( @xmath8 ) increases from zero to @xmath113 with @xmath114 characterizing a second - order phase transition . a simple fitting of the form @xmath115 in the vicinity of the second - order transition gives the same classical value for the critical exponent @xmath116 . on the other hand , for @xmath114 and @xmath117 the staggered magnetization @xmath118 increases monotonically with the frustration parameter @xmath8 in the * caf * phase , with a discontinuity of @xmath118 at @xmath119 , which is a first - order phase transition . for @xmath120 , the * qp * intermediate phase between the two ordered states ( * af * and * caf * ) disappears , and a direct transition between the magnetically ordered * af * and * caf * located at the crossing point @xmath11 correspondent to the classical value . the ground state ( @xmath13 ) phase diagram in the @xmath66 plane is displayed in fig . the solid line indicate the critical points and the dashed lines represent first - order frontiers . we observe three different phases , namely : * af * ( antiferromagnetic ) ,  * caf * ( collinear antiferromagnetic ) and * qp * ( quantum paramagnetic ) . the * af * and * qp * phases are separated by a second - order transition line @xmath121 , while the * qp * and * caf * phases are separated by a first - order transition line @xmath122 . the presence of the interchain parameter @xmath7 has the general effect of suppressing the * qp * phase . the * qp * region decreases gradually with the decrease of the @xmath7 parameter , and it disappears completely at the _ quantum _ _ critical endpoint _ * * qce**@xmath123(@xmath124 ) where the boundaries between these phases emerge . below this * qce * , i.e. , for @xmath120 , there is a direct first - order phase transition between the * af * and * caf * phases , with a transition point @xmath125 ( classical value ) .    in order to illustrate the nature of the phase transition , we also show , in inset fig . 2 , the behavior of the staggered magnetization ( order parameter ) as a function of the frustration parameter ( @xmath8 ) for @xmath126 and @xmath127 @xmath128 . from curves such as those shown in fig . 2 we see that for @xmath129 there exists an intermediate region between the critical point @xmath130 at which @xmath131 for the * af * phase , characterizing a second - order transition , and the point @xmath132 at which the @xmath133 order parameter presents a discontinuity for the * caf * phase , characterizing a first - order transition . for @xmath134 , the order parameter of the * af * phase decreases monotonically with increase of the frustration parameter from @xmath135 , for @xmath136 , to zero for @xmath137 ( @xmath138 ) . in the * caf * phase @xmath133 decreases from @xmath139 for @xmath140 to @xmath141 for @xmath142 , characterizing a direct first - order transition between the magnetically ordered * af * and * caf * phases located at the crossing point . we note that the definition of the order parameter @xmath143 ( @xmath144 ) difer of @xmath0 factor when compared with calculations which use other methods ( i.e. , @xmath145 ) . therefore , in the limit of the not frustrated ( @xmath136 ) square lattice ( @xmath67 ) antiferromagnetic , solving the equations ( 12 ) and applying the corrections factor we found @xmath146 which is consistent with the numerical results obtained by various methods such as series expansion , quantum monte carlo simulation , and others@xcite , and can also be compared with experimental results for the k@xmath24nif@xmath25 , k@xmath24mnf@xmath25 , and rb@xmath147mnf@xmath25 compounds@xcite . plane for the quantum spin-@xmath0 @xmath35 model on a square lattice , where @xmath6 and @xmath148 . the dashed and solid lines corresponds the first- and second - order transitions lines , respectively . the black point represents the quantum critical endpoint ( * qce * ) . the notations indicated by * af * , * caf * and * qp * corresponds the antiferromagnetic , collinear antiferromagnetic and quantum paramagnetic phases , respectively . the dotted line correspond the classical solution @xmath63 . ] in summary , we have studied the effects of quantum fluctuations due to spatial ( @xmath7 ) and frustration ( @xmath8 ) parameter in the quantum spin-@xmath0 @xmath149 heisenberg model . using a variational method we calculated the sublattice magnetization for the * af * and * caf * phases . for values of @xmath150 the frustration contributes significantly to the existence of a disordered intermediate state ( * qp * ) between the two * af * and * caf * ordered phases , while for @xmath151 , we have a direct first - order transition between the * af * and * caf * phases . we have observed , by analyzing the order parameters of the * af * and * caf * phases , that the phase transitions are of second and first - order between the * af - qp * and * caf - qp * , respectively . the obtained phase diagram can be compared with recent results which used effective - field theory@xcite and coupled - cluster method@xcite , showing the same qualitative results predicting a paramagnetic region for small interlayer parameter ( i.e. , @xmath152 ) , and for @xmath120 this * qp * phase disappears by presenting a direct first - order transition between the * af * and * caf * phases . on the other hand , recent calculations of second order spin wave theory@xcite have indicated that the intermediate * qp * phase exists for all @xmath42 in accordance with results of exact diagonalization@xcite . we speculate that by using a more sophisticated method , for example , quantum monte carlo simulations@xcite and density matrix renormalization group ( dmrg ) method@xcite , this disordered region should disappear for certain values of @xmath120 . r. f. bishop , p. h. y. li , r. darradi , j. schulenburg , and j. richter , _ phys . b _ * 78 * , 054412 ( 2008 ) ; r. f. bishop , p. h. y. li , r. darradi , j. schulenburg , j. richter , and c. e. campbell , _ j. phys . : condens . matter _ * 20 * , 415213 ( 2008 ) . see also , r. darradi , j. richter , j. schulenburg , r. f. bishop , and p. h. y. li , _ j. phys . : conference series _ * 145 * , 012049 ( 2009 ) . p. carretta , r. melzi , n. papinutto , and p. millet , _ phys . * 88 * , 047601 ( 2002 ) ; p. carretta , n. papinutto , c. b. azzoni , m. c. mozzati , e. pavarini , s. gonthier , and o. millet , _ phys . b_. * 66 * , 094420 ( 2002 ) .
the phase transition of the quantum spin-@xmath0 frustrated heisenberg antiferroferromagnet on an anisotropic square lattice is studied by using a variational treatment . 
 the model is described by the heisenberg hamiltonian with two antiferromagnetic interactions : nearest - neighbor ( nn ) with different coupling strengths @xmath1 and @xmath2 along x and y directions competing with a next - nearest - neighbor coupling @xmath3 ( nnn ) . 
 the ground state phase diagram in the ( @xmath4 ) space , where @xmath5 and @xmath6 , is obtained . depending on the values of @xmath7 and @xmath8 
 , we obtain three different states : antiferromagnetic ( * af * ) , collinear antiferromagnetic ( * caf * ) and quantum paramagnetic ( * qp * ) . for an intermediate region 
 @xmath9 we observe a * qp * state between the ordered * af * and * caf * phases , which disappears for @xmath7 above some critical value @xmath10 . 
 the boundaries between these ordered phases merge at the _ quantum critical endpoint _ 
 ( * qce * ) . below this * qce * there is again a direct first - order transition between the * af * and * caf * phases , with a behavior approximately described by the classical line @xmath11 . 
 * pacs numbers * : 75.10.jm , 05.30.-d , 75.40.-s , 75.40.cx