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
g 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.
We present a new general and much simpler scheme to construct various quantum phase transitions (QPTs) in spin chain systems with matrix product ground states. By use of the scheme we take into account one kind of matrix product state (MPS) QPT and provide a concrete model. We also study the properties of the concrete example and show that a kind of QPT appears, accompanied by the appearance of the discontinuity of the parity absent block physical observable, diverging correlation length only for the parity absent block operator, and other properties which are that the fixed point of the transition point is an isolated intermediate-coupling fixed point of renormalization flow and the entanglement entropy of a half-infinite chain is discontinuous.
We investigate quantum phase transitions (QPTs) in spin chain systems characterized by local Hamiltonians with matrix product ground states. We show how to theoretically engineer such QPT points between states with predetermined properties. While some of the characteristics of these transitions are familiar, like the appearance of singularities in the thermodynamic limit, diverging correlation length, and vanishing energy gap, others differ from the standard paradigm: In particular, the ground state energy remains analytic, and the entanglement entropy of a half-chain stays finite. Examples demonstrate that these kinds of transitions can occur at the triple point of "conventional" QPTs.
According to our scheme to construct quantum phase transitions (QPTs) in spin chain systems with matrix product ground states, we first successfully combine matrix product state (MPS) QPTs with spontaneous symmetry breaking. For a concrete model, we take into account a kind of MPS QPTs accompanied by spontaneous parity breaking, though for either side of the critical point the GS is typically unique, and show that the kind of MPS QPTs occur only in the thermodynamic limit and are accompanied by the appearance of singularities, diverging correlation length, vanishing energy gap and the entanglement entropy of a half-infinite chain not only staying finite but also whose first derivative discontinuous.
For the matrix product system of a one-dimensional spin-½ chain, we present a new model of quantum phase transitions and find that in the thermodynamic limit, both sides of the critical point are respectively described by phases |Ψ a 〉 = |1...1〉 representing all particles spin up and |Ψ b 〉 = |0...0〉 representing all particles spin down, while the phase transition point is an isolated intermediate-coupling point where the two phases coexist equally, which is described by the so-called N -qubit maximally entangled GHZ state . At the critical point, the physical quantities including the entanglement are not discontinuous and the matrix product system has long-range correlation and N -qubit maximal entanglement. We believe that our work is helpful for having a comprehensive understanding of quantum phase transitions in matrix product states of one-dimensional spin chains and of potential directive significance to the preparation and control of one-dimensional spin lattice models with stable coherence and N -qubit maximal entanglement.
We consider a spin ladder model which is known to have matrix product states as exact ground states with spin liquid characteristics. The model has two critical-point transitions at the parameter values
u=0 and $\ensuremath{\infty}$. We study the variation of entanglement and fidelity measures in the ground states as a function of
u and specially look for signatures of quantum phase transitions at
u=0 and $\ensuremath{\infty}$. The two different entanglement measures used are
S(i) (the single-site von Neumann entropy) and
S(i,j) (the two-body entanglement). At the quantum critical point (QCP) $u=\ensuremath{\infty}$, the entanglement measure
E [=S(i),S(i,j)] vanishes but remains nonzero at the other QCP
u=0. The first and second derivatives of
E with respect to the parameter
u and the entanglement length associated with
S(i,j) are further calculated to identify special features, if any, near the QCPs. We further determine the GS fidelity
F and a quantity
ln∣D∣ related to the second derivative of
F and show that these quantities calculated for finite-sized systems are good indicators of QPTs occurring in the infinite system.