Study On Matrix Product States And Quantum Discord In One-dimensional Systems

The ground states of one-dimensional spin-1/2 chain and ladder with some predetermined symmetries are exactly expressed as matrix product states (MPS). The singularities of observable quantities and quantum correlations in MPS are investigated to denote the quantum phase transition (QPT).In the one-dimensional spin-1/2 chain, the ground state with spin-flip symmetry and parity symmetry is investigated, the spin magnetization in the x direction and the correlation functions show their singularities, which implies the QPT. It is interesting that the correlation functions between two spins have no correlation with the distance between them. All the pairs of the spins have equal correlations with each other in this quantum system, which plays a guiding role in searching the spin pairs with equal correlation.Two types of ground states in one-dimensional spin-1/2 ladder are considered, one is the type which has the SO(2) symmetry (rotation around the z axis symmetry, spin flip symmetry, parity symmetry, and leg exchange symmetry), the other is the type which has full SO(3) symmetry (rotation in spin space symmetry and spin flip symmetry). In the SO(2) symmetrical state, the quantum discord (QD) and entanglement entropy (EE) show singularities, which implies the QPT in the ladder. The QD per site in SO(2) symmetrical state shows singularity outside the QPT point (this phenomenon is called sudden change), which means that QD is unreliable to denote QPT at zero temperature. It is known that EE disappears at high temperature (this phenomenon is called sudden death).To denote the QPT, the QD and EE are suitable for high and low temperature, respectively. In SO(3) symmetrical state, both the QD and the EE are continuous. Therefore, the ladder has no QPT.In addition, the QD of the site including two spins at the different rungs is zero. Therefore, in both two types of state, there is no quantum correlation between two spins at the different rungs.