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sometimes , when carrying out a preliminary computation in order to be sure that well understood llimits of a problem are handled correctly , results which are interesting in their own right emerge .
i will now discuss one such result which i have obtained as part of such a preliminary study of the hubbard model .    the hubbard model at half - filling is equivalent to a heisenberg anti - ferromagnet ( haf ) and it is important to show that core treats this limiit correctly .
this was shown to be the case for the spin-1/2 haf in ref.@xcite .
what that paper did not address was the subtle question of how the physics of the one - dimensional anti - ferromagnet changes when the spin-1/2 on each lattice site is replaced by spin @xmath1 . in 1983
it was argued by f.d.m .
haldane , ref.@xcite that when @xmath1 is a half - integer , then the spectrum has no mass - gap , but when @xmath1 is an integer , a mass - gap develops . in this talk
i describe a core computation for the spin-1/2 and spin-1 anti - ferromagnet which not only supports haldane s conjecture , but shows how the spin-1 case leads to an understanding of a more general class of theories defined by a hamiltonian of the form @xmath2 about which , until now , very little was known . .
the core method consists of two parts .
first , a theorem which defines the hamiltonian analogue of wilson s exact renormalization group transformation ; second , a set of approximation procedures which render the calculation of the _ renormalized hamiltonian _ doable .
core replaces the lagrangian notion of integrating out degrees of freedom by that of throwing away hilbert space states ; i.e. , defining a ojection operator , @xmath3 , which acts on the original hilbert space , @xmath4 to create the space of _ retained states _ @xmath5 .
the central feature of this approach is a formula which relates the original hamiltonian , @xmath6 , to a _
renormalized hamiltonian _ , @xmath7 , which acts on @xmath8 and which has , in a sense which was made precise in ref.@xcite , exactly the same low energy physics as @xmath6 .
this formula is @xmath9 } [ [ t(t ) h t(t ) ] ] \sqrt{[[t(t)^2 ] ] }   \label{basicform}\ ] ] where @xmath10 and where we have defined @xmath11= p o p$ ] for any operator @xmath12 which acts on @xmath4 .
it was explained in ref.@xcite that @xmath13 is an extensive operator and can be written as a sum of finite - range connected operators ; i.e. , @xmath14 where each @xmath15 can be computed on a finite sublattice of the original infinite lattice .
although this formula involves an infinite number of terms examples have shown that a highly accurate approximation to @xmath13 only requires computing a few terms in the cluster expansion , typically range-2 and/or range-3 terms are adequate .
to obtain the hamiltonian for a general heisenberg anti - ferromagnet set @xmath16 in eq.[betaham ] .
the complete discusion of a range-2 calculation for the spin-1/2 haf appears in ref.@xcite and only a brief review of the results will be discussed here .
all core computations begin with a choice of a thinning procedure and the procedure used in ref.@xcite was based upon dividing the lattice into three site blocks @xmath17 , and solving the corresponding block - hamiltonian ( obtained by restricting the full hamiltonian to those terms which are completely contained in block @xmath17 ) , keeping a small set of low - lying eigenstates . since the haf hamiltonian has an @xmath18 symmetry its eigenstates fall into degenerate multiplets and to guarantee that the renormalized hamiltonian has the same symmetry as the original it is necessary to keep full multiplets .
three - site blocks were chosen for the spin-1/2 haf because that is the smallest size block for which a non - trivial core computation can be done .
this is because the eigenstates of the two - site block - hamiltonian fall into spin-0 and spin-1 multiplets , with the spin-0 state having the lowest energy and keeping only the lowest lying multiplet would amount to keeping a single state , a trivial renormalization group transformation .
going to three sites avoids this problem since the eigenstates of the block hamiltonian fall into two spin-1/2 multiplets and one spin-3/2 multiplet . in this case one of the spin-1/2 multiplets has the lowest energy and so truncating to the space generated by keeping only these two - states per block leads to a renormalized hamiltonian for which the operator @xmath19 is a multiple of the unit matrix and @xmath20 is , up to a multiplicative constant which is less than unity , exactly the same starting hamiltonian ; i.e. , the spin-1/2 haf hamiltonian is at a fixed point of the core transformation .
iterating this calculation shows the spin-1/2 haf is massless , in agreement with what is known about this system .
the same calculation produces a ground - state energy density which is within one - percent of the exact answer .      applying the same three site blocking to the spin-1 haf yields a different result . in this case
the eigenstates of the block hamiltonian fall into a spin-0 , spin-1 and spin-2 multiplet , with the spin-1 multiplet having the lowest energy .
truncating to the space generated by keeping only the spin-1 multiplet per block gives , as in the spin-1/2 case , @xmath19 proportional to the unit matrix , but now @xmath20 takes the more general form @xmath21 thus , after one three site blocking the core analysis of the spn-1 haf turns into the study of renormalization group flows for the theory defined by @xmath22 where i have ignored the term proportional to the identiy operator . except for the special values @xmath23
very little is known about the hamiltonian eq.[genham ] . the case @xmath24 is special , in that it can be solved for the groundstate , which is a valence - bond solid with a non - vanishing mass gap . moreover , when @xmath25 the theory can be solved by means of the bethe - ansatz and is massless .
finally , when @xmath26 the theory becomes @xmath27 symmetric and it is conjectured that the mass gap vanishes at this point too , although no exact result is known for this case .    unlike the spin-1/2 case sensible core computations can be done for the spin-1 system defined by eq.[genham ] using either two or three - site blocking .
if two - site blocks are used , then the states of the block hamiltonian fall into a spin-0 , spin-1 and spin-2 representation and keeping the spin-0 and spin-1 states defines a non - trivial renormalization group transformation .
the determination that the spin-0 and spin-1 multiplets are the ones to keep is made by studying the @xmath28 dependence of their energies .
this shows that one of these representations always lies lowest in energy and that they become degenerate and then cros at @xmath24 .
similarly , the eigenstates of the three - site hamiltonian divide into one spin-3 , two spin-2 , three spin-1 and one spin-0 representations of @xmath18 and a study of the @xmath28 dependence of the eigenenergies shows that the lowest lying spin-0 and spin-1 states are the ones to keep because one or the other of these multiplets always lies lowest in energy and they become degenerate and cross at @xmath24 . since either two or three site blocking leads to the same truncation algorithm i study the two - site blocking procedure because it is easier to carry out .
the approximate , range-2 , two - site blocking core computation of the mass gap for @xmath29 is shown in the figure .
several points should be made about this plot .
first , the core computation solves the @xmath24 case exactly .
analysis of the flow at this point shows that at each renormalized hamiltonian is isomorphic to a dimerized spin-1/2 anti - ferromagnet and the fixed point is the place where one of the two nearest neighbor couplings vanish .
furthermore , all of the hamiltonians defined by eq.[genham ] flow to the same massive fixed point for the range of @xmath28 shown in the graph .
the general picture which emerges from this calculation is that there are three fixed points to the flow in this region , an attractive , massive fixed point at @xmath24 ( the valence - bond solid ) and two repulsive massles fixed points near @xmath30 .
the fact that the mass gap does nt vanish for @xmath31 but for a value of @xmath28 which is 10% further from the valence bond solid point is typical of what occurs in range-2 approximations .
going to range-3 or better should reduce this error by an order of magnitude .
the reason i cut off the plot at @xmath32 , before the massgap goes to zero , is that at this point the theory becomes @xmath27 symmetric ( a fact which is obvious from the core computation ) and so the spin-0 and spin-2 representations become degenerate . because the program i wrote to do this calculation just grabbed the four lowest lying states the calculation destroys the manifest @xmath18 symmetry.a clearly a more careful programming job has to be done to go beyond this pont keeping the symmetry intact .
note , the calculated mass gap is shown by the solid curve , the dotted curve is just included to heuristically remind the reader of what might be expected in an exact calculation .
note , the spin-1 haf at @xmath16 lies quite close to the @xmath24 point where the core computation is exact and is well within in the basin of attraction of the massive theory , thus providing the advertised support for the haldane conjecture .
the contractor renormalization group method ( core ) , originally developed for application to lattice gauge theories , is very well adapted the study of spin systems and systems with fermions . as an warmup exercise for studying hubbard models this method 
 is applied to spin-1/2 and spin-1 anti - ferromagnets in one space dimension in order to see if it is able to explain the physics of the haldane conjecture . 
 the method not only provides support for haldane s conjecture but provides insight into the physics of a more general class of spin-1 systems with hamiltonians of the form @xmath0 about which , until now , little was known .