Abstract:

The recent work on fidelity approach to quantum phase transitions in the context of tensor network algorithms is summarised.
First, the fidelity, a basic notion of quantum information science, may be used to characterize quantum phase transitions, regardless of what type of internal order is present in quantum many-body states. If the fidelity of two given states vanishes, then there are two cases: (1) they are in the same phase if the distinguishability results from irrelevant local information; or (2) they are in different phases if the distinguishability results from relevant long-distance information. The irrelevant and relevant information are quantified, which allows us to identify unstable and stable fixed points (in the sense of renormalization group theory).
Second, for any D-dimensional quantum lattice system, the fidelity between two ground state many-body wave functions is mapped onto the partition function of a D-dimensional classical statistical vertex lattice model with the same lattice geometry. The fidelity per lattice site, analogous to the free energy per site, is well defined in the thermodynamic limit and can be used to characterize the phase diagram of the model. Remarkably, the fidelity per lattice site may be computed in the context of tensor network algorithms,.
Third, the tensor network algorithms are exploited to takle a few condensed matter systems, especially the two-dimensional quantum Ising model with transverse and parallel magnetic fields, the quantum two-dimensional anisotropic spin 1/2 antiferromagnetic XYX model in an external magnetic field and the two-dimensional quantum anisotropic compass model.
 
 
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