Fidelity And Quantum Critical Phenomena

Quantum phase transitions (QPTs) occur at absolute zero, which is driven by some non-thermal parameter like pressure, chemical composition or magnetic field. QPTs are caused by quantum fluctuations which result from Heisenberg's uncertainty principle. The characteristic length scale diverges at a quantum critical point (QCP). Although QPTs occurs at absolute zero as a function of a non-thermal parameter, they can leave fingerprints at finite temperature. For this reason, QPTs have become prime suspects in the ongoing experimental investigation of strongly correlated systems like heavy fermions and cuprate superconductors.In section 1 we collect a few basic facts about phase transitions and critical behavior and then briefly discuss newly-developed fidelity approach to QPTs based on tensor network algorithms. Fidelity, a basic notion of quantum information science, may be used to characterize QPTs, regardless of what type of internal order is present in quantum many-body states. There are two cases: (1) they are in the same phase if the distinguish ability results from irrelevant local information; or (2) they are in different phases if the distinguish ability results from relevant long-distance information. To quantify irrelevant and relevant information, the scaling parameter extracted from the fidelity, i.e. the fidelity per lattice site, was introduced to characterize QPTs. In section 2 we use fidelity per lattice site to characterize QPTs for one dimensional quantum Ising model in a transverse field. It is found that, in the thermodynamic limit, the fidelity per lattice site is singular, and the derivative of its logarithmic function with respect to the transverse field strength is logarithmically divergent at the critical point. The scaling behavior is confirmed numerically by performing a finite-size scaling analysis for systems of different sizes, consistent with the conformal invariance at the critical point. This allows us to extract the correlation length critical exponent, which turns out to be universal in the sense that the correlation length critical exponent does not depend on either the anisotropic parameter or the transverse field strength. In section 3 we use newly-developed tensor network algorithms on infinite-size lattices to simulate quantum Ising model in a transverse magnetic field, the spin 1/2 XYX model in an external magnetic field and spin 1/2 XXZ model on an infinite-size lattice in one spatial dimension. The newly-developed tensor network algorithms on infinite-size lattices can be used to compute fidelity per lattice site, which characterizes spontaneous symmetry breaking occurs by a bifurcation in a system, when some control parameter crosses its critical value. Traditionally spontaneous symmetry breaking occurs in a system when its Hamiltonian possesses certain symmetry, whereas the ground state wave functions do not preserve it. In last chapter, we discuss multi-scale entanglement renormalization ansatz (MERA) algorithm and then use this algorithm to simulate spin 1/2 XYX model in an external magnetic field. From the reduce density matrix to compute fidelity and entanglement, which characterize QPTs.Fidelity per lattice site may be used to detect QPTs in condensed matter systems. Antother feature worth to be mentioned is that the fidelity per lattice site is simple to be evaluated in the tensor network representation. We can compute fidelity per lattice site by tensor network algorithms for systems on infinite-size lattices to detect QPTs as bifurcation. Practically, bifurcation is also important since it makes possible to locate transition points without the need to compute the derivatives of the ground state fidelity per lattice site with respect to the control parameter, which is usually a formidable task.