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arXiv:0902.3848 · Physical Review A

Reduced fidelity in topological quantum phase transitions

Erik Eriksson, Henrik Johannesson

Published Jun 3, 2009• 43 citations
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Abstract

We study the reduced fidelity between local states of lattice systems exhibiting topological order. By exploiting mappings to spin models with classical order, we are able to analytically extract the scaling behavior of the reduced fidelity at the corresponding quantum phase transitions out of the topologically ordered phases. Our results suggest that the reduced fidelity, albeit being a local measure, generically serves as an accurate marker of a topological quantum phase transition.

DOI
10.1103/physreva.79.060301

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We study the reduced fidelity between local states of lattice systems exhibiting topological order. By exploiting mappings to spin models with classical order, we are able to analytically extract the scaling behavior of the reduced fidelity at the corresponding quantum phase transitions out of the topologically ordered phases. Our results suggest that the reduced fidelity, albeit being a local measure, generically serves as an accurate marker of a topological quantum phase transition.
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We study reduced fidelity and reduced fidelity susceptibility in the Kitaev honeycomb model. It is shown that the nearest-two-site reduced fidelity susceptibility manifests itself as a peak at the quantum phase transition point, although the one-site reduced fidelity susceptibility vanishes. Our results directly reveal that the reduced fidelity susceptibility can be used to characterize the quantum phase transition in the Kitaev honeycomb model, which suggests that, despite its local nature, the reduced fidelity susceptibility is an accurate marker of the topological phase transition when it is properly chosen.
We investigate the relation between Bell function values (BFV) of the reduced density matrix and the topological quantum phase transitions in the Kitaev-Castelnovo-Chamon model. % [Phys. Rev. B \textbf{77}, %054433 (2008)]. We find that the first order derivative of BFV exhibits singular behavior at the critical point and we propose that it can serve as a good and convenient marker for the transition point. More interestingly, the value of the critical point can be analytically obtained in this approach. Since the BFV serves as a measure of nonlocality when it is greater than the classical bound of the correlation functions, our work has established a link between quantum nonlocality and phase transitions.
This dissertation puts forward the concept of fidelity as a way to understand and characterize quantum phase transitions involving topological phases. It lies at the interphase between Condensed Matter and Quantum Phase Transitions. ❧ We find that the fidelity metric, which characterizes the degree of distinghuishability between two quantum states, can serve as a useful tool in the understanding of transitions involving topological phases. ❧ The methodology does not require a complete knowledge of the physical mechanism involved in the occurrence of the critical point, such as the identification of a local order parameter and a symmetry breaking mechanism, the fundamental pillars of critical phenomena. Topological phases are unique in that they cannot be classified under these methodologies, due to their lack of local order parameters. It is one of the purposes of this thesis to understand how well we can characterize the phase boundaries between these phases using fidelity concepts. ❧ In the first part, we introduce two quantum models that exhibit a phase transition between an ordered and a topological phase, as quantified by the topological entanglement entropy. In the thermodynamic limit, the different components of the fidelity metric exhibit a divergence, signalling the occurrence of the critical point. ❧ In the second part, we introduce models that have phase boundaries between two distinct topological phases, and we show that the fidelity is successful at identifying the occurrence of a critical region. Moreover, a finite size scaling analysis is carried out, from which we determine the value of critical exponents.
The ground-state fidelity per lattice site is shown to be able to detect quantum phase transitions for the Kitaev model on the honeycomb lattice, a prototypical example of quantum lattice systems with topological order. It is found that, in the thermodynamic limit, the ground-state fidelity per lattice site is nonanalytic at the phase boundaries; the second-order derivative of its logarithmic function with respect to a control parameter describing the interaction between neighboring spins is divergent. A finite-size scaling analysis is performed, which allows us to extract the correlation length critical exponent from the scaling behaviors of a part of the ground-state fidelity per lattice site.
We establish an intriguing connection between quantum phase transitions and bifurcations in the reduced fidelity between two different reduced density matrices for quantum lattice many-body systems with symmetry-breaking order. Our finding is based on the observation that in the conventional Landau–Ginzburg–Wilson paradigm a quantum system undergoing a phase transition is characterized in terms of spontaneous symmetry breaking that is captured by a local-order parameter, which in turn results in an essential change of the reduced density matrix in the symmetry-broken phase. Two quantum systems on an infinite lattice in one spatial dimension, i.e. a quantum Ising model in a transverse magnetic field and a quantum spin-1/2 XYX model in an external magnetic field, are considered in the context of the tensor network algorithm based on the matrix product state representation.