1.1 Antiferromagnetic spin systems without Néel order

  It has been believed that a macroscopic spin system with an antiferromagnetic interaction would freeze at a Néel temperature, which is comparable to the magnitude of the interaction (J). But recently, several theoretical situations have been proposed, in which an antiferromagnetic spin system prefers a many-body singlet ground state, rather than the Néel state. These situations include low dimensionality and/or geometrical frustration of the spins, so that the conventional Néel order is suppressed. Amazingly, several materials have been discovered which may realize the theoretical situations. In this thesis, I will report experimental results of three such spin systems, namely, (1) the spin-ladder system Sr_n-1Cu_n+1O_2n, (2) Haldane compound Y₂BaNiO₅ and (3) the spin-Peierls system CuGeO₃. Although the detailed structures of the ground states differ among these spin systems, they share one important concept for a general understanding of the many-body singlet ground states; it is the singlet pair formation of two spins.

  

Figure 1: (a, b) The Néel state and (c) the singlet pair state for a two S=1/2 spin system.

Suppose two S=1/2 spins interact with an antiferromagnetic coupling (J):

In classical mechanics, the ground state of this two-spin system is the Néel state, in which the two spins point in opposite directions (Fig.1a, b). In quantum mechanics, spin-flips caused by the xy terms of the Hamiltonian (eq.1) prevent the Néel state from serving as an eigenstate of the Hamiltonian. After a simple calculation, one finds that the spin singlet pair $(\vert\!\uparrow\downarrow\rangle-\vert\!\downarrow\uparrow\rangle)/\sqrt{2}$ is the quantum mechanical ground state of this system (Fig.1c).

The singlet pair state is a mixed state of the two Néel states and does not have a classical counterpart. Still, this state is often realized in localized two electron systems, such as valence bonds in molecules. One characteristic feature of the singlet pair is that the magnetic dipolar field from each spin is exactly canceled; in other words, there is no magnetic field induced around a singlet pair. This is the main reason why most molecules do not show magnetism. In macroscopic localized spin systems, singlet pair formation becomes difficult, because a spin on a lattice has at least two nearest neighbors. Since the singlet pair is a state for just two S=1/2 spins, one spin on the lattice must select one specific partner from the equivalent neighboring spins, which is difficult in translationally symmetric systems. Consequently, most macroscopic spin systems with an antiferromagnetic interaction exhibit Néel order.

  

Figure 2: (a) An example of a trivial spin-gap system: local singlet pairs without any correlations. (b) The first excited state.

As shown in the next section, some macroscopic spin systems still prefer a ground state based on the singlet pair formations of the spins. Some of these unconventional spin systems are characterized by an energy gap between the ground state and the magnetic excited states. Since this energy gap originates from the spin degrees of freedom, it is often called a `spin gap'. The existence/absence of the spin gap is probably related to how well the singlet pairs are localized. For example, a crystal made up of many uncorrelated singlet pairs (Fig.2a) is a trivial example of a spin-gap system; the energy excitation spectrum of this system will have a gap, which corresponds to the singlet-triplet excitation of a singlet pair (Fig.2b). Another example, the S=1/2 spin-chain with an antiferromagnetic Heisenberg interaction is a non-trivial gap-less system. The ground state of this system is a many-body singlet [11], which is approximately expressed as the superposition of every possible singlet pairing on the chain [12]. This ground state, which is known as the Resonating Valence Bond (RVB) state, has completely delocalized singlet pairs, and the excitation to the triplet state becomes gap-less [13]. In this thesis, materials with a spin gap, namely, the spin systems with relatively well localized singlet pairs, are investigated.