In this article we provide a review of geometrical methods employed in the analysis of quantum phase transitions and non-equilibrium dissipative phase transitions. After a pedagogical introduction to geometric phases and geometric information in the characterisation of quantum phase transitions, we describe recent developments of geometrical approaches based on mixed-state generalisation of the Berry-phase, i.e. the Uhlmann geometric phase, for the investigation of non-equilibrium steady-state quantum phase transitions (NESS-QPTs ). Equilibrium phase transitions fall invariably into two markedly non-overlapping categories: classical phase transitions and quantum phase transitions, whereas in NESS-QPTs this distinction may fade off. The approach described in this review, among other things, can quantitatively assess the quantum character of such critical phenomena. This framework is applied to a paradigmatic class of lattice Fermion systems with local reservoirs, characterised by Gaussian non-equilibrium steady states. The relations between the behaviour of the geometric phase curvature, the divergence of the correlation length, the character of the criticality and the gap - either Hamiltonian or dissipative - are reviewed.

中文翻译：

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量子相变的几何学

在本文中，我们回顾了用于分析量子相变和非平衡耗散相变的几何方法。在对量子相变表征中的几何相和几何信息进行教学性介绍之后，我们描述了基于贝里相（即乌尔曼几何相）的混合态概括的几何方法的最新发展，用于研究非平衡稳态量子相变 (NESS-QPTs)。平衡相变总是分为两个明显不重叠的类别：经典相变和量子相变，而在 NESS-QPT 中，这种区别可能会消失。本次审查中描述的方法，除其他外，可以定量评估这种临界现象的量子特性。该框架应用于具有局部储层的格子费米子系统的范式类，其特征在于高斯非平衡稳态。回顾了几何相位曲率的行为、相关长度的散度、临界特性和间隙（哈密顿量或耗散量）之间的关系。