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Chinese Physics C

Critical behavior of the XY spin chain with three-site interaction studied in terms of the Loschmidt echo

Han‐li Lian

Published Jun 1, 2012• 2 citations
Publisher

Abstract

The effect of the three-site interaction (α) on the critical behaviors of the XY spin chain is studied in terms of the Loschmidt echo ( LE ). The critical lines can be shifted by α, and the anisotropy parameter of the XY chain has no effect on the critical lines. The scaling behaviors of the LE reveal that the dynamical behaviors of the LE are reliable for characterizing quantum phase transition (QPT).

DOI
10.1088/1674-1137/36/6/001

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The effect of the three-site interaction (α) on the critical behaviors of the XY spin chain is studied in terms of the Loschmidt echo ( LE ). The critical lines can be shifted by α, and the anisotropy parameter of the XY chain has no effect on the critical lines. The scaling behaviors of the LE reveal that the dynamical behaviors of the LE are reliable for characterizing quantum phase transition (QPT).
We study the Loschmidt echo (LE) of a coupled system consisting of a central spin and its surrounding environment described by a general XYXYXY spin-chain model. The quantum dynamics of the LE is shown to be remarkably influenced by the quantum criticality of the spin chain. In particular, the decaying behavior of the LE is found to be controlled by the anisotropy parameter of the spin chain. Furthermore, we show that due to the coupling to the spin chain, the ground-state Berry phase for the central spin becomes nonanalytical and its derivative with respect to the magnetic parameter $\ensuremath{\lambda}$ in spin chain diverges along the critical line $\ensuremath{\lambda}=1$, which suggests an alternative measurement of the quantum criticality of the spin chain.
By introducing the Dzyaloshinsky–Moriya (DM) interaction, the Loschmidt Echo (LE) of a quantum system consisting of a central spin and its surrounding environment characterized by an XY spin chain was investigated. The analytical expression of the LE was obtained. The effects of magnetic field, DM interaction, spin chain size, and anisotropy parameter on the evolution behavior of the LE were discussed numerically. At the critical points of magnetic field, the LE presents an obvious decay following with local oscillation. The decaying and oscillation behavior of the LE can be tuned by the intensity of the DM interaction. With increasing the spin chain size, the decaying amplitude is increased and the following oscillation behavior vanishes for the spin chain with larger size. The effect of the anisotropy parameter on the LE presents different properties at both sides of the knee point. The effect of the DM interaction on the decay of the LE is also related to the value of anisotropy parameter.
By introducing the Dzyaloshinsky-Moriya (DM) interaction, the Loschmidt Echo ( LE ) of a quantum system consisting of a central spin and its surrounding environment characterized by an XY spin chain was investigated analytically and numerically. At the critical points of the magnetic field, the LE presents an obvious decay. The decay amplitude can be tuned by the DM interaction. In some specific intervals the DM interaction can remarkably delay the decay of the LE . On the other hand, the DM interaction can change the effects of the anisotropy parameter on the LE.
Chiral phase and quantum phase transitions of anisotropic XY chains with the XZY − YZX type of three-site interaction are studied. The phase diagram is obtained by the study of the Lee–Yang zeros, order parameters, and ground-state energy. Besides the paramagnetic and ferromagnetic phases, a new chiral phase is found after the introduction of three-site interactions. In the new phase, we find that the fidelity is zero and the von Neumann entropy SL is proportional to Alog2L with the coefficient A≈1/3 the same as that in the anisotropy transition line of XY chain without three-site interactions. From the ferromagnetic phase to the paramagnetic phase, the quantum critical behavior is same as that identified for the XY spin model without three-site interactions. In this critical line, SL is proportional to Alog2L with A≈1/6, the same as that in the Ising transition line of the XY chain without three-site interactions. From the ferromagnetic phase to the chiral phase and from the chiral phase to the paramagnetic phase, the quantum phase transitions are similar to the anisotropy transitions.
The Loschmidt echo (LE) of a central two-level qubit coupled to nonuniform anisotropic XYXYXY chains in a transverse field is studied. A general formula for LE is derived, which we use to discuss the influence of the criticality of the environment on LE. It is found that for the periodic XYXYXY chain the behaviors of LE in the vicinity of the critical points are similar to those of the uniform case. It is different for the disordered transverse Ising chains. For the aperiodic chains, if the surrounding systems are bounded chains, the behaviors of LE are similar to those of the uniform case, while if the surrounding systems are unbounded chains, they are similar to those of the disordered case.