Energy levels and their degeneracies for two-ring Ising chains of spins-
21 with NN and NNN couplings: spin frustration of ferromagnetic and antiferromagnetic orders
Abstract An even-site spins- linear chain of cyclic in the nearest-neighbour and next-nearest-neighbour interactions has necessarily a three-ring structure. In this study, the interactions are considered to be short-range through the coupling constants J and K , respectively. For an arbitrary even number of spins- , the exact classical solution of the partition function and the pair correlations between the nearest- and next-nearest-neighbours are calculated in terms of the constants J and K , without using the energy levels and their degeneracy factors. Then, this partition function is used to verify the validity of an ansatz on the allowed energies of the chain and their corresponding degeneracy factors. The influence of next-nearest-neighbour interactions K on the specific heat capacity per spin at the low-temperatures is analysed for the ferromagnetic and antiferromagnetic chains. We especially show that specific heat capacity exhibits the spin frustration when the ferromagnetic and antiferromagnetic orders are demanded via the coupling constants J and K , respectively. Also, the maximal frustration is obtained analytically by two distinct ground levels for even number of spins- .
Motivated by frustrated magnets and quasi-one-dimensional magnetic materials, we study the magnetic properties of one-dimensional (1D) Ising chains with nearest-neighbor (NN) and weaker next-to-nearest-neighbor (NNN) interactions in the presence of vacancy defects. The effect of a vacancy on the magnetic susceptibility of a spin chain is twofold: it reduces the length of the chain by an effective ``vacancy size'' and may also act as a free spin, a ``quasispin,'' with a Curie-type ${\ensuremath{\chi}}_{\text{quasi}}=\ensuremath{\langle}{S}^{2}\ensuremath{\rangle}/T$ contribution to the susceptibility. In chains with antiferromagnetic short-range order, the susceptibility of vacancy-free chains is exponentially suppressed at low temperatures, and quasispins dominate the effect of impurities on the chains' magnetic properties. For chains with antiferromagnetic NN interactions, the quasispin matches the value $\ensuremath{\langle}{S}^{2}\ensuremath{\rangle}=1$ of the Ising spins in the chain for ferromagnetic NNN interactions and vanishes for antiferromagnetic NNN interactions. For chains with ferromagnetic short-range order, quasispin effects are insignificant due to exponentially large low-temperature susceptibilities, and the dominant effect of a vacancy is effectively changing the length of the chain.
We study the ground-state phase diagram of a one-dimensional spin-
21 easy-plane XXZ model with a ferromagnetic nearest-neighbor (NN) coupling
J1 and a competing next-nearest-neighbor (NNN) antiferromagnetic coupling
J2 in the parameter range
0<J2/∣J1∣<0.4. When ${J}_{2}/|{J}_{1}|\ensuremath{\lesssim}1/4$, the model is in a Tomonaga-Luttinger liquid phase which is adiabatically connected to the critical phase of the XXZ model of
J2=0. On the basis of the effective (sine-Gordon) theory and numerical analyses of low-lying energy levels of finite-size systems, we show that the NNN coupling induces phase transitions from the Tomonaga-Luttinger liquid to gapped phases with either N\'eel or dimer order. Interestingly, these two types of ordered phases appear alternately as the easy-plane anisotropy is changed toward the isotropic limit. The appearance of the antiferromagnetic (N\'eel) order in this model is remarkable, as it is strongly unfavored by both the easy-plane ferromagnetic NN coupling and the antiferromagnetic NNN coupling in the classical-spin picture. We argue that emergent trimer degrees of freedom play a crucial role in the formation of the N\'eel order.