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arXiv:1108.2597 · Physical Review B

Quantum fidelity for one-dimensional Dirac fermions and two-dimensional Kitaev model in the thermodynamic limit

Victor Mukherjee, Amit Dutta, Diptiman Sen

Published Jan 3, 2012• 25 citations
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Abstract

We study the scaling behavior of the fidelity ($F$) in the thermodynamic limit using the examples of a system of Dirac fermions in one dimension and the Kitaev model on a honeycomb lattice. We show that the thermodynamic fidelity inside the gapless as well as gapped phases follow power-law scalings, with the power given by some of the critical exponents of the system. The generic scaling forms of $F$ for an anisotropic quantum critical point for both the thermodynamic and nonthermodynamic limits have been derived and verified for the Kitaev model. The interesting scaling behavior of $F$ inside the gapless phase of the Kitaev model is also discussed. Finally, we consider a rotation of each spin in the Kitaev model around the $z$ axis and calculate $F$ through the overlap between the ground states for the angle of rotation $\ensuremath{\eta}$ and $\ensuremath{\eta}+d\ensuremath{\eta}$, respectively. We thereby show that the associated geometric phase vanishes. We have supplemented our analytical calculations with numerical simulations wherever necessary.

DOI
10.1103/physrevb.85.024301

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We study the scaling behavior of the fidelity (F ) in the thermodynamic limit using the examples of a Tomonaga-Luttinger liquid in one dimension and the Kitaev model on a honeycomb lattice. We show that the thermodynamic fidelity inside the gapless as well as gapped phases follow powerlaw scalings, with the power given by some of the critical exponents of the system. The generic scaling forms of F for an anisotropic quantum critical point for both thermodynamic and nonthermodynamic limits have been derived and verified for the Kitaev model. The behavior of F inside the gapless phase of the Kitaev model is found to depend strongly on its distance from the anisotropic quantum critical point. Finally, we consider a rotation of each spin in the Kitaev model around the z axis and calculate F through the overlap between the ground states for angle of rotation η and η+ dη, respectively. We thereby show that the associated geometric phase vanishes. We have supplemented our analytical calculations with numerical simulations wherever necessary.
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.
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We study the behavior of the defect and heat densities under sudden quenching near the quantum critical points in the two-dimensional Kitaev honeycomb model both in the thermodynamic and nonthermodynamic limits. We consider quenches starting from a quantum critical point into the gapped as well as the gapless phases. We choose points on the lines of anisotropic quantum critical points as well as different points of intersection of these lines as the initial points from where the quenching starts. We find that the defect and heat densities display the expected power-law scalings along with logarithmic corrections to scaling (or cusp singularities) in certain cases. In the vicinity of some of the intersection points, the scaling behaviors change, indicating an effective dimensional reduction; the scaling behavior near these points depends on the number of critical lines crossed in the process of quenching. All the analytical predictions are also verified by numerical integration.