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International Journal of Modern Physics B

Quantum phase transitions in composite matrix product states of one-dimensional spin-1/2 chains

Jing-Min Zhu

Published Feb 16, 2015• 5 citations
Publisher

Abstract

For matrix product states of one-dimensional spin-1/2 chains, we investigate the properties of quantum phase transition of the proposed composite system. We find that the system has three different ferromagnetic phases, one line of the two ferromagnetic phases coexisting equally describes the paramagnetic state, and the other two lines of two ferromagnetic phases coexisting equally describe the ferrimagnetic states, while the three phases coexisting equally point describes the ferromagnetic state. Whether on phase transition lines or at the phase transition point, the system is always in an isolated mediate-coupling state, the physical quantities are discontinuous and the system has long-range correlation and has long-range classical correlation and long-range quantum correlation. We believe that our work is helpful for comprehensively and profoundly understanding the quantum phase transitions, and of some certain guidance and enlightening on the classification and measure of quantum correlation of quantum many-body systems.

DOI
10.1142/s021797921550071x

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Abstract For our proposed composite parity-conserved matrix product state (MPS), if only a spin block length is larger than 1, any two such spin blocks have correlation including classical correlation and quantum correlation. Both the total correlation and the classical correlation become larger than that in any subcomponent; while the quantum correlations of the two nearest-neighbor spin blocks and the two next-nearest-neighbor spin blocks become smaller and for other conditions the quantum correlation becomes larger, i.e., the increase or the production of the long-range quantum correlation is at the cost of reducing the short-range quantum correlation, which deserves to be investigated in the future; and the ration of the quantum correlation to the total correlation monotonically decreases to a steady value as the spacing spin length increasing.
We consider a one-parameter family of matrix-product states of spin-1 particles on a periodic chain and study in detail the entanglement properties of such a state. In particular, we calculate exactly the entanglement of one site with the rest of the chain, and the entanglement of two distant sites with each other, and show that the derivative of both these properties diverge when the parameter ggg of the states passes through a critical point. Such a point can be called a point of quantum phase transition, since at this point the character of the matrix-product state, which is the ground state of a Hamiltonian, changes discontinuously. We also study the finite size effects and show how the entanglement depends on the size of the chain. This later part is relevant to the field of quantum computation where the problem of initial state preparation in finite arrays of qubits or qutrits is important. It is also shown that the entanglement of two sites have scaling behavior near the critical point.