Quantum Phases and Magnetization Plateaus of Skewed Spin Ladders
This thesis deals with theoretical studies of skewed spin ladders which are two legged ladder with periodic slanted rung bonds and can be viewed as periodic fused rings with two different number of vertices. The spin at the vertices (sites) considered in this thesis are either spin ½ or 1. The skewed ladder is named an n/m ladder when the fused adjacent rings are with n and m vertices. These are quasi one-dimensional systems. The systems are studied by varying the rung exchange $J_1$, keeping the nearest neighbor leg exchange $J_2$ fixed at unity. The first chapter gives a brief historical introduction and discusses the origin of exchange interactions in materials. The spin-1/2 and spin-1 chains and quantum phases of the frustrated and dimerized spin-1/2 chains as well as the qualitative distinction between spin-1/2 and spin-1 antiferromagnetic Heisenberg chains are discussed. For investigating the skewed ladder systems, computational methods such as exact diagonalization technique in valence bond (VB) and constant Ms basis is described in chapter 2. For studying the large systems closer to the thermodynamic limit, the density matrix renormalization group (DMRG) method is introduced. Both finite and infinite DMRG algorithms are discussed and their implementation are outlined. Chapter three presents the quantum phases in a 5/7 spin-1 skewed ladder system, studied numerically using the exact diagonalization technique up to 16 spins and the density matrix renormalization group method for larger system sizes. The study of diverse gs properties such as spin gap, spin-spin correlations, spin density, and bond order reveals that the system has four distinct phases, namely, the AF phase at small $J_1$; the ferrimagnetic phase with gs spin $S_G = n$ for $1.44 < J_1 < 4.74$ and with $S_G = 2n$ for $J_1 > 5.63$, where n is the number of unit cells; and a reentrant nonmagnetic phase at $4.74 < J_1 < 5.44$. The system also shows the presence of spin current at specific $J_1$ values due to simultaneous breaking of both reflection and spin parity symmetries. Quantum phase transitions of spin-1 3/4 and 3/5 skewed ladders are reported in chapter four. The ground state (gs) of the 3/4 ladder switches from a singlet to a magnetic state for $J_1 ≥ 1.82$. The gs of of the 3/5 skewed ladder is highly frustrated and has spiral spin arrangements and switches between singlet state and low spin magnetic states multiple times on tuning $J_1$ in a finite size system. The switching pattern is non-monotonic as a function of $J_1$, and depends on the system size. The 3/5 system also exhibits spontaneous spin parity and mirror symmetry breaking giving rise to spin current in the gs. Magnetization plateaus, which are some of the most striking manifestations of frustration in low-dimensional spin systems, are observed in the spin-1/2, 5/7 skewed ladder. This system exhibits three significant plateaus at m = 1/4, 1/2 and 3/4, consistent with the Oshikawa-Yamanaka-Affleck condition. Our numerical as well as perturbative analysis shows that the ground state can be approximated by three weakly coupled singlet dimers and two free spins, in the absence of a magnetic field. With increasing applied magnetic field, the dimers progressively become triplets with large energy gaps to excited states, giving rise to stable magnetization plateaus. The cusps at the ends of a plateau follow the algebraic square-root dependence on the magnetic field strength. Magnetization plateaus in the spin- 1/2 model are observed at m = 1/3, 2/3 and 1 for 3/4 skewed ladder, m = 1/4, 1/2, 3/4 and 1 for 3/5 skewed ladder, and m = 0, 1/3, 2/3 and 1 for 5/5 skewed ladder. The occurrence of the m = 0 plateau in the 5/5 ladder implies a finite gap between the lowest energy levels in the singlet and triplet sectors. In the case of 3/4 and 5/5 ladders, quadrupolar phases were observed below the 1/3rd plateau, due to the binding of two magnons. These studies form the contents of chapters five and six. The last chapter of the thesis deals with investigation of the quantum phase transitions of the skewed ladders using entanglement and fidelity analysis. All the calculations are based on the exact diagonalization technique for finite system size. At the transition points, the entanglement entropy exhibits a discontinuous change, while fidelity shows a sharp dip. The transition points are accurately determined from the characteristics changes in entanglement entropy and fidelity.