For the one-dimensional chain system, on the other hand, quite a few
things are known exactly. For example, the antiferromagnetic
Heisenberg model of the *S*=1/2 spin-chain was exactly solved by
H. A. Bethe in 1931 [11], and using a similar technique,
the one dimensional Hubbard model, as well as the model in
special cases (*J*=0 and 2*t*) have also been solved
[42,43]. Considering these successes in
one-dimensional systems, one approach to tackle the superconducting
cuprates is to investigate the quasi one-dimensional lattices
known as `spin ladder' structures (Fig.25a), which are
strips of square lattice with a finite width and infinite length.

Early theoretical works of the spin ladder system were numerical studies on isolated 2-leg spin ladders. For the Heisenberg model of a 2-leg spin ladder, a many-body singlet ground state was suggested, with a finite energy gap (spin-gap) to the excited states [20,44]. The ground state structure is mainly composed of spin singlet pairs on the rungs, as shown in Fig.25b [20]. The hole doped 2-leg ladders have also been investigated theoretically, in relation to the superconductivity [45,46,47]. These works suggest that two doped holes will form a bound state so that the surrounding spins can remain spin singlet. The paired holes, which behave as hard-core bosons, were suggested to take a superconducting ground state [45,46].

Behavior of wider spin-ladder systems was first emphasized by
T. M. Rice *et al.* in 1993 [19]. In Ref. [19],
the authors first pointed out that the homologous series of cuprate Sr_{n-1}Cu_{n+1}O_{2n} should realize the *N*-leg ladder structure, and pointed out
contrasting magnetic behavior between the even-number-leg ladders and the
odd-number-leg ladders. The even-number-leg ladders were to manifest
a spin-gap, while the odd-number-leg system should be gapless. This
idea of alternating ground states has been supported by numerical
simulations of the spin ladders up to 4-legs [48] and a mean-field
calculation of the 2-leg and the 4-leg ladder systems [21].