Antiferromagnetic ladder model

Now we should consider in particular the special case [11] x = 1/2. In this case the product gi S gi+1 contains only one singlet and one triplet and does not contain any quintet. Therefore, in this case the cell Hamiltonian can be written in the form [31] 

It follows from Eq. (42) that the ground state has ultrashort-range correlations with rc 1. For example, rc(y = 0) = 2 log-13, which coincides with the correlation length of the AKLT model. But at y = all correlations are zero except (S2i S2i+i) = — It implies that at y = the model (46-48) has a dimer ground state. 

The value u — 1) changes sign at y =, and as follows from Eqs.(42), the correlators show the antiferromagnetic structure of the ground state at y , while at 0 y there are ferromagnetic correlations inside pairs (1,2), (3,4). and antiferromagnetic correlations between the pairs. 

The Hamiltonian (46-48) of the cyclic ladder has a singlet-triplet gap A for finite N. It is evident that for y = the gap exists for N — oo and A( ) = 4. The existence of the finite gap at the thermodynamic limit in the range 0 y follows from the continuity of the function A(y). It is also clear that A (y) at N — oo vanishes at the boundary points y = 0 and y = when the ground state is degenerate and there are low-lying spin-wave excitations. 

Unfortunately, a method for the exact calculation of A(y) in the thermodynamic limit is unknown. For the approximate calculation A(y) we use the trial function of the triplet state in the form 

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One of these models is the spin- ladder with competing interactions of the ferro- and antiferromagnetic types at the F-AF transition line. The exact singlet ground-state wave function on this line is found in the special form expressed in terms of auxiliary Bose-operators. The spin correlators in the singlet state show double-spiral ordering with the period of spirals equal to the system size. 

In the general case the proposed form of the wave function corresponds to the MP form but with matrices of infinite size. However, for special values of parameters of the model it can be reduced to the standard MP form. In particular, we consider a spin-1 ladder with nondegenerate antiferromagnetic ground state for which the ground state wave function is the MP one with 2x2 matrices. This model has some properties of ID AKLT model and reduces to it in definite limiting case. 

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