, and the strong coupling expansion simply tells us that the system behaves like a coupled-spin-$1/2$ chain [@Read90].

Fortunately, the antiferromagnetic spin-$1/2$ chain can be solved using the exact Bethe ansatz technique [@Giamarchi03]. The exact Bethe ansatz solution tells us that the antiferromagnetic spin-$1/2$ Heisenberg chain is critical, namely, the ground state has no long-range magnetic order but has a gapless excitation spectrum. Unlike integer spin chains, where the lowest-energy excitations carry spin $S=1$, the elementary excitation of this system has spin $S=1/2$. Combining this with the continuum theory leads to the Haldane conjecture for half-odd integer spin chains, namely, that they are all critical with elementary $S=1/2$ excitations.

### Open spin chains and end states

The Haldane conjecture has been checked numerically for quantum spin chains with different spin magnitudes and has been found to be correct in all cases that have been studied thus far. One may wonder whether the difference in spin magnitudes may manifest at all in some low-energy properties of quantum spin chains. The answer is yes, when we consider open spin chains.

Recall that we have always assumed periodic boundary conditions in deriving $S_T$. In fact, a periodic boundary condition is needed to define 