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The matrix product state (MPS) is utilized to study the ground stateproperties and quantum phase transitions (QPTs) of the one-dimensional quantumcompass model (QCM). The MPS wavefunctions are argued to be very efficientdescriptions of QCM ground states, and are numerically determined by imaginarytime projections. The ground state energy, correlations, quantum entanglementand its spectrum, local and nonlocal order parameters, etc., are calculated andstudied in details. It is revealed that the bipartite and block entanglemententropies, as well as the nearest neighbor correlation functions can be used todetect the second-order QPTs, but not the first-order ones, while fidelitydetections can recognize both. The entanglement spectrum is extracted from theMPS wavefunction, and found to be doubly degenerate in disordered phases ofQCM, where non-local string order parameters exist. Moreover, with linearizedtensor renormalization group method, the specific heat curves are evaluated andtheir low temperature behaviors are investigated.

Matrix Product State and Quantum Phase Transitions in the One-Dimensional Extended Quantum Compass Model