Physical property of the zigzag spin chain: quantum critical surface, spiral spin state, and multiferroics

Multiferroic LiCu2O2 is an interesting material. When we apply magnetic field, its polarization changes. In type II multiferroic material, it has a spiral spin state as the ground state. These spiral states can be found in many complex transition metal compounds, where competing exchange interactions of the neighboring spins can cause such periodically modulated spin state. The system with competing exchange interactions can be modeled by zigzag spin chain. Therefore, here we consider the zigzag spin chain system with competing nearest and next nearest superexchange interactions. Most importantly, we also considered anisotropic exchange interaction and Dzyaloshinskii-Moriya(DM) interaction. In this thesis, we divide our work into two part; the first part is to model the system and the second part is to analyze the ESR data. In the first part, we had planed two work. The first one is to find the quantum critical surface of the XXZ zigzag spin chain, and the second one is to find the spiral spin state as the ground state in a zigzag spin chain system. In the first work, we analyze the ground state of a zigzag spin chain with applied magnetic field. Starting from a local Hamiltonian Hi,i+1,i+2 for the i−th, i+1−th, i+2−th spins, a parameter x is introduced to give the applied magnetic field on respective spins as B/(2 + x), xB/(2 + x) and B/(2 + x). We are able to identify the ground state as the fully polarized state and one-magnon states in the region b = (4 + f)^2/8 where b = B/J2, f = J1/J2; J1 and J2 are the nearest neighbor and next nearest neighbor exchange interaction, respectively. With the theorem of positive semi-definite matrix, we showed that b = (4 + f)^2/8 is a quantum critical line for f ≥ 0. For f < 0, we are able to show with the Bethe ansatz that the fully polarized state and one-magnon states have energy lower than those states with |S| ≥ N/6 for b ≥ (4 + f)^2/8. Hence, the line b = (4 + f)^2/8 is likely a quantum critical line for both positive and negative J1. We can generalize our result to xxz zigzag spin chains. In the second work, we intend to identify the conditions for spiral spin state as the ground state in this spin chain system. We start from building the connection between the spiral state and the fully polarized (FP) state with a unitary transform. Under this transformation, anisotropic exchange interaction and the Dzyaloshinskii-Moriya (DM) interaction can be transformed to each other. Then we use positive semi-definite theorem to identify the region of FP state being the GS for the transformed Hamiltonian, and it is the region of spiral spin state as GS of the original Hamiltonian. We showed that, to have the spiral GS, the effect of DM interaction is important, and its strength is related to the pitch angle of spiral spins. Our system indicates the connection between spiral spins and magnetic frustration, which is a feature of the multiferroics. This method not only can be applied to spin-1/2 system, but also to any other spin systems. In the second part of our work, we intend to explain the electron spin resonance (ESR) of LiCu2O2. Based on the icture of classical spins and the spin wave theory, we calculate the low lying excitations. Our result shows that the resonance ν1 ∼ 30GHz corresponds to the spin wave state of wave vector q = Q (where Q = (0.5, 0.174, 0) is the wave vector of spiral spins). The mechanism for the spin gap and hence, the resonance can be either the DM interaction or an anisotropic superexchange interaction. Thus, when the applied magnetic field is parallel to the spiral axis (R), there are two branches; when it is perpendicular to the spiral axis, there is only one branch. By comparing the result of our theoretical calculation with that of experiment, we discuss the system properties, such as anisotropy and DM interaction. We also predict that the spiral axis will lies on a-b plane, and close to a +b, which is determined by the direction of the DM interaction. As a result of this work, we revealed the condition for spiral spin state to be the ground state in the zigzag spin chain, in which the DM interaction is very important. Furthermore, we understand the origin of ν1 in LiCu2O2 is from spin wave with wave vector q = Q. Then we obtain the system parameter, such as J, delta, D , spiral axis R and predict how the spiral ground state changes under applied magnetic field