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arXiv:0803.2940 · Physical Review A

Operator fidelity susceptibility: An indicator of quantum criticality

Xiaoguang Wang, Zhe Sun, Z. D. Wang

Published Jan 9, 2009• 60 citations
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Abstract

We introduce an operator fidelity and propose to use its susceptibility for characterizing the sensitivity of quantum systems to perturbations. Two typical models are addressed: One is the transverse Ising model exhibiting a quantum phase transition, and the other is the one-dimensional Heisenberg spin chain with next-nearest-neighbor interactions, which has the degeneracy. It is revealed that the operator fidelity susceptibility is a good indicator of quantum criticality regardless of the system degeneracy.

DOI
10.1103/physreva.79.012105

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We introduce an operator fidelity and propose to use its susceptibility for characterizing the sensitivity of quantum systems to perturbations. Two typical models are addressed: One is the transverse Ising model exhibiting a quantum phase transition, and the other is the one-dimensional Heisenberg spin chain with next-nearest-neighbor interactions, which has the degeneracy. It is revealed that the operator fidelity susceptibility is a good indicator of quantum criticality regardless of the system degeneracy.
We show a general connection between the reduced-fidelity susceptibility and quantum phase transitions, and derive an explicit expression of the reduced-fidelity susceptibility for the one-dimensional spin-1/2 dimerized Heisenberg chain, which has both SU(2) and translational symmetries. We find that the reduced-fidelity susceptibility is directly related to the square of the second derivative of ground-state energy, which means that it is an effective indicator of the second-order quantum phase transitions. In terms of this indicator, we explicitly examine the critical behavior of the spin-1/2 dimerized Heisenberg chain. Moreover, we give another two exemplifications to show that the results may also be extended to high-spin systems.
In this article, we study the high order term of the fidelity of the Heisenberg chain with next-nearest-neighbor interaction and analyze its connection with quantum phase transition of Beresinskii-Kosterlitz-Thouless type happened in the system. We calculate the fidelity susceptibility of the system and find that although the phase transition point can't be well characterized by the fidelity susceptibility, it can be effectively picked out by the higher order of the ground-state fidelity for finite-size systems.
We study slightly generalized quantum fidelity susceptibilities where the differential change in the fidelity is measured with respect to a different term than the one used for driving the system toward a quantum phase transition. As a model system we use the spin-1/21/21/2 J1{J}_{1}J1​-J2{J}_{2}J2​ antiferromagnetic Heisenberg chain. For this model, we study three fidelity susceptibilities, ${\ensuremath{\chi}}_{\ensuremath{\rho}}$, ${\ensuremath{\chi}}_{D}$, and ${\ensuremath{\chi}}_{\mathrm{AF}}$, which are related to the spin stiffness, the dimer order, and antiferromagnetic order, respectively. All these ground-state fidelity susceptibilities are sensitive to the phase diagram of the J1{J}_{1}J1​-J2{J}_{2}J2​ model. We show that they all can accurately identify a quantum critical point in this model occurring at ${J}_{2}^{c}\ensuremath{\sim}0.241{J}_{1}$ between a gapless Heisenberg phase for J2<J2c{J}_{2}<{J}_{2}^{c}J2​<J2c​ and a dimerized phase for J2>J2c{J}_{2}>{J}_{2}^{c}J2​>J2c​. This phase transition, in the Berezinskii-Kosterlitz-Thouless universality class, is controlled by a marginal operator and is therefore particularly difficult to observe.
We study the quantum phase transitions in the two-dimensional spin-orbit models in terms of fidelity susceptibility and reduced fidelity susceptibility. An order-to-order phase transition is identified by fidelity susceptibility in the two-dimensional Heisenberg XXZ model with Dzyaloshinsky-Moriya interaction on a square lattice. The finite size scaling of fidelity susceptibility shows a power-law divergence at criticality, which indicates the quantum phase transition is of second order. Two distinct types of quantum phase transitions are witnessed by fidelity susceptibility in Kitaev-Heisenberg model on a hexagonal lattice. We exploit the symmetry of two-dimensional quantum compass model, and obtain a simple analytic expression of reduced fidelity susceptibility. Compared with the derivative of ground-state energy, the fidelity susceptibility is a bit more sensitive to phase transition. The violation of power-law behavior for the scaling of reduced fidelity susceptibility at criticality suggests that the quantum phase transition belongs to a first-order transition. We conclude that fidelity susceptibility and reduced fidelity susceptibility show great advantage to characterize diverse quantum phase transitions in spin-orbit models.
We investigate the ability of quantum fidelity in detecting the classical phase transitions (CPTs) in a two-dimensional Heisenberg—Ising mixed spin model, which has a very rich phase diagram and is exactly soluble. For a two-site subsystem of the model, the reduced fidelity (including the operator fidelity and the fidelity susceptibility) at finite temperatures is calculated, and it is found that an extreme value presents at the critical temperature, thus shows a signal for the CPTs. In some parameter region, the signal becomes blurred. We propose to use the "normalized fidelity susceptibility" to solve this problem.