The term “particle–hole symmetry” is beset with conflicting meanings in contemporary physics. Conceived and written from a condensed-matter standpoint, the present paper aims to clarify and sharpen the terminology. In that vein, we propose to define the operation of “particle–hole conjugation” as the tautological algebra automorphism that simply swaps single-fermion creation and annihilation operators, and we construct its invariant lift to the Fock space. Particle–hole symmetries then arise for gapful or gapless free-fermion systems at half filling, as the concatenation of particle–hole conjugation with one or another involution that reverses the sign of the first-quantized Hamiltonian. We illustrate that construction principle with a series of examples including the Su–Schrieffer–Heeger model and the Kitaev–Majorana chain. For an enhanced perspective, we contrast particle–hole symmetries with the charge-conjugation symmetry of relativistic Dirac fermions. We go on to present two major applications in the realm of interacting electrons. For one, we offer a heuristic argument that the celebrated Haldane phase of antiferromagnetic quantum spin chains is adiabatically connected to a free-fermion topological phase protected by a particle–hole symmetry. For another, we review the recent proposal by Son [Phys. Rev. X 5, 031027 (2015)] for a particle–hole conjugation symmetric effective field theory of the half-filled lowest Landau level, and we comment on the emerging microscopic picture of the composite fermion.

中文翻译：

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凝聚态中的孔洞对称性

在现代物理学中，“粒子-空穴对称性”一词具有相互矛盾的含义。本文是从简洁的角度构思和编写的，旨在阐明和强化术语。在这种情况下，我们建议将“粒子-空穴共轭”的运算定义为重言式代数自同构，该同构仅交换单费米子生成和an灭算符，然后构造其对Fock空间的不变提升。然后，在半填充时，无隙或无隙自由费米子系统出现了粒子-孔的对称性，因为粒子-孔的共轭与一次或多次对合的级联使第一个量化的哈密顿量的符号反转。我们通过一系列示例（包括Su–Schrieffer–Heeger模型和Kitaev–Majorana链）来说明该构造原理。为了增强视角，我们将粒子-孔的对称性与相对论狄拉克费米子的电荷-共轭对称性进行了对比。我们继续介绍相互作用电子领域中的两个主要应用。首先，我们提供一种启发式的论点，即著名的反铁磁量子自旋链的Haldane相绝热地连接到受粒子-孔对称性保护的自由费米子拓扑相。另外，我们回顾了Son [Phys。第十版 我们提供了一种启发式的论据，即著名的反铁磁量子自旋链的Haldane相绝热地连接到受粒子-孔对称性保护的自由费米子拓扑相。另外，我们回顾了Son [Phys。第十版 我们提供了一种启发式的论据，即著名的反铁磁量子自旋链的Haldane相绝热地连接到受粒子-孔对称性保护的自由费米子拓扑相。另外，我们回顾了Son [Phys。第十版[ 5，031027（2015）]给出了半填充最低朗道能级的粒子-孔共轭对称有效场理论，我们对复合费米子的新兴微观图片进行了评论。