We review the physics of pair density wave (PDW) superconductors. We begin with a macroscopic description that emphasizes order induced by PDW states, such as charge density wave, and discuss related vestigial states that emerge as a consequence of partial meting of the PDW order. We review and critically discuss the mounting experimental evidence for such PDW order in the cuprate superconductors, the status of the theoretical microscopic description of such order, and the current debate on whether the PDW is a "mother order" or another competing order in the cuprates. In addition, we give an overview of the weak coupling version of PDW order, Fulde-Ferrell-Larkin-Ovchinnikov states, in the context of cold atom systems, unconventional superconductors, and non-centrosymmetric and Weyl materials.The GLW energy density consistent with time-reversal, parity, space group, and gauge symmetries is (5, 15)The parameters βi depend upon the specific microscopic model. Depending on which values are found for these, one of five possible ground states can be realized. These five phases include the following: the FF-type phase with only one momentum component, the FF * phase which is a bidirectional version of the FF-type phase, the LO type which include pairing with opposite momentum components: these include the unidirectional phase, and the bidirectional-I (II) phases which have a phase factor of 0 (π/2) between the two unidirectional components. These five states give rise to different patterns of induced order, providing a means to distinguish them. We now turn to these induced orders.Induced Order Parameters www.annualreviews.org • The Physics of Pair Density Waves 3 6 Agterberg et al.

Summary. Neutron scattering studies have provided important information about the momentum and energy dependence of magnetic excitations in cuprate superconductors. Of particular interest are the recent indications of a universal magnetic excitation spectrum in hole-doped cuprates. That starting point provides motivation for reviewing the antiferromagnetic state of the parent insulators, and the destruction of the ordered state by hole doping. The nature of spin correlations in stripeordered phases is discussed, followed by a description of the doping and temperature dependence of magnetic correlations in superconducting cuprates. After describing the impact on the magnetic correlations of perturbations such as an applied magnetic field or impurity substitution, a brief summary of work on electron-doped cuprates is given. The chapter concludes with a summary of experimental trends and a discussion of theoretical perspectives. IntroductionNeutron scattering has played a major role in characterizing the nature and strength of antiferromagnetic interactions and correlations in the cuprates. Following Anderson's observation [1] that La 2 CuO 4 , the parent compound of the first high-temperature superconductor, should be a correlated insulator, with moments of neighboring Cu 2+ ions anti-aligned due to the superexchange interaction, antiferromagnetic order was discovered in a neutron diffraction study of a polycrystalline sample [2]. When single-crystal samples became available, inelastic studies of the spin waves determined the strength of the superexchange, J, as well as weaker interactions, such as the coupling between CuO 2 layers. The existence of strong antiferromagnetic spin correlations above the Néel temperature, T N , has been demonstrated and explained. Over time, the quality of such characterizations has improved considerably with gradual evolution in the size and quality of samples and in experimental techniques.Of course, what we are really interested in understanding are the optimallydoped cuprate superconductors. It took much longer to get a clear picture of the magnetic excitations in these compounds, which should not be surprising

The unconventional normal-state properties of the cuprates are often discussed in terms of emergent electronic order that onsets below a putative critical doping of xc ≈ 0.19. Charge density wave (CDW) correlations represent one such order; however, experimental evidence for such order generally spans a limited range of doping that falls short of the critical value xc, leading to questions regarding its essential relevance. Here, we use X-ray diffraction to demonstrate that CDW correlations in La2−xSrxCuO4 persist up to a doping of at least x = 0.21. The correlations show strong changes through the superconducting transition, but no obvious discontinuity through xc ≈ 0.19, despite changes in Fermi surface topology and electronic transport at this doping. These results demonstrate the interaction between CDWs and superconductivity even in overdoped cuprates and prompt a reconsideration of the role of CDW correlations in the high-temperature cuprate phase diagram.

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We report muon spin rotation and magnetic susceptibility experiments on in-plane stress effects on the static spin-stripe order and superconductivity in the cuprate system La 2−x Ba x CuO 4 with x ¼ 0.115. An extremely low uniaxial stress of ∼0.1 GPa induces a substantial decrease in the magnetic volume fraction and a dramatic rise in the onset of 3D superconductivity, from ∼10 to 32 K; however, the onset of at-least-2D superconductivity is much less sensitive to stress. These results show not only that largevolume-fraction spin-stripe order is anticorrelated with 3D superconducting coherence but also that these states are energetically very finely balanced. Moreover, the onset temperatures of 3D superconductivity and spin-stripe order are very similar in the large stress regime. These results strongly suggest a similar pairing mechanism for spin-stripe order and the spatially modulated 2D and uniform 3D superconducting orders, imposing an important constraint on theoretical models.