Abstract

We study fidelity out-of-time-order correlators (FOTOCs) in the Lipkin–Meshkov–Glick (LMG) model and one of its variants dubbed as the extended LMG model. We demonstrate that these exhibit distinctive behaviour at quantum phase transitions (QPTs) in both the ground and the excited states. We show that the dynamics of the FOTOC have different behaviour in the symmetric and broken phases and as one approaches QPT. Rescaling the FOTOC operator with time, we establish that for small times the rescaled operator is identical to the Loschmidt echo. We also compute the Nielsen complexity of the FOTOC operator for both models and apply this operator on the ground and excited states to obtain the quasi-scrambled states. The FOTOC operator introduces a small perturbation on the original ground and excited states. For this perturbed state, we compute the quantum information metric to first and second order in perturbation, in the thermodynamic limit. We find that the associated Ricci scalar does not show any divergence or discontinuity at QPTs in the LMG model. Instead, the amplitude of oscillations is relatively lower in the symmetric phase than in the broken phase. However, in the case of extended LMG model, the Ricci scalar diverges at the QPT from the broken phase side, in contrast to the zeroth order result. Finally, we comment upon the Fubini-Study complexity in the extended LMG model.

Graphical abstract

中文翻译：

---

Lipkin-Meshkov-Glick 模型及其变体中的 Fotoc 复杂度

摘要

我们研究 Lipkin-Meshkov-Glick (LMG) 模型及其变体之一（称为扩展 LMG 模型）中的保真度乱序相关器 (FOTOC)。我们证明它们在基态和激发态的量子相变（QPT）中表现出独特的行为。我们表明，FOTOC 的动力学在对称相和破碎相以及接近 QPT 时具有不同的行为。随着时间的推移重新缩放 FOTOC 算子，我们发现在很短的时间内，重新缩放的算子与洛施密特回波相同。我们还计算了两个模型的 FOTOC 算子的 Nielsen 复杂度，并将该算子应用于基态和激发态以获得准扰乱态。FOTOC 算子对原始基态和激发态引入了小扰动。对于这种扰动状态，我们在热力学极限下计算扰动中的一阶和二阶量子信息度量。我们发现相关的 Ricci 标量在 LMG 模型中的 QPT 处没有表现出任何分歧或不连续性。相反，对称相中的振荡幅度比破碎相中的振荡幅度相对较低。然而，在扩展 LMG 模型的情况下，Ricci 标量在 QPT 处从断相侧发散，与零阶结果相反。最后，我们评论了扩展 LMG 模型中的 Fubini-Study 复杂性。

图形概要