Pair density waves (PDWs) are highly exotic superconductors that are thought to occur in quantum materials. Ordinary superconductors, those that occur in simple metals like Aluminum, `look' very uniform, featureless and ordered throughout a sample material -- like the calm water in a tranquil lake. PDWs, on the other hand, are superconductors that are also well-ordered throughout a material but periodically modulate in space, just like waves in the sea. In certain scenarios, however, PDWs can lack ordering in the sense that one part of the material is independent of another despite being spatially modulated -- like chaotic waves in the turbulent ocean!
In more technical terms, PDWs are superconducting states that spontaneously break translational symmetry while preserving time reversal symmetry. These states contain long-range order and are characterized by a non-uniform order parameter for the Cooper pairs with a non-zero modulation wave vector Q. They can be thought of as analogous to charge and spin ordered phases occurring in the pairing sector. FPDWs, on the other hand, also contain finite Q Cooper pairs as in a PDW, but lack coherent long range order. They are thus analogues of the widely studied “preformed Cooper pair” scenario but with a finite center-of-mass momentum pair.
There is now substantial evidence for both FPDWs and PDWs in several material families like cuprates, kagome, heavy fermion systems etc. The first clear observation of vortex-induced PDW in BSCCO at low temperature was seen in scanning tunneling microscopy (STM) measurements . More recent STM experiments over the past few years [2, 3] provide further evidence in favor of a short- range PDW coexisting with the d-wave superconductivity and evolving into a PDW state. This phase is characterized by a gap at finite temperatures but lacks long-range order. Very recently, such finite momentum pair correlations were also found in the kagome lattice materials like CsV3Sb5  and the Uranium based superconductor UTe2 
However, currently there is no clear-cut understanding for why FPDWs form, and the exact nature of their relationship to PDWs and uniform d-wave superconductivity is an outstanding problem in the physics of quantum matter. This is mainly because there is no analytically tractable model that clarifies the microscopic ingredients responsible for such phenomenology starting from the well studied Fermi liquid (FL).
In this paper, we write an analytically solvable model that demonstrates that a FL subject to a finite, anisotropic attractive interaction spontaneously generates a modulated (non-uniform) pairing state. Whether this phase is an FPDW or PDW is determined by temperature as well as the interaction strength defined by the ratio of the Fermi and pairing energy. An appealing aspect of our work is that the FPDW phase is already stable for moderate/intermediate pairing strengths, very much applicable to real materials including cuprates, and where BCS theory is qualitatively reliable. At weak pairing, we recover the well-known uniform d-wave pairing solution. Our calculations therefore provide a unifying framework that subsumes the FPDW, PDW and uniform d-wave superconducting phases under a single paradigm by providing a concrete description of their microscopic origin (see phase diagram below).
PDWs and FPDWs have been subject to intense theoretical and numerical debate lately. This is further compounded by the recent strong evidence of FPDW and PDW phases in the cuprates. Our work not only provides a mechanism for the FPDW phase, but clarifies the relationship between PDW, FPDW and uniform d-wave pairing in a very simple model. Hence our work opens new avenues to explore modulated phases in other material systems. The two microscopic ingredients outlined in our work — pair anisotropy and intermediate coupling — thus form a suitable starting point for further theoretical and numerical exploration of these exotic phases of matter.
 Hamidian, M.H., Edkins, S.D., Joo, S.H., Kostin, A., Eisaki, H., Uchida, S., Lawler, M.J., Kim, E.A., Mackenzie, A.P., Fujita, K. and Lee, J., 2016. Detection of a Cooper-pair density wave in Bi2Sr2CaCu2O8+ x. Nature, 532(7599), pp.343-347.
 Du, Z., Li, H., Joo, S.H., Donoway, E.P., Lee, J., Davis, J.S., Gu, G., Johnson, P.D. and Fujita, K., 2020. Imaging the energy gap modulations of the cuprate pair-density-wave state. Nature, 580(7801), pp.65-70.
 Edkins, S.D., Kostin, A., Fujita, K., Mackenzie, A.P., Eisaki, H., Uchida, S., Sachdev, S., Lawler, M.J., Kim, E.A., Séamus Davis, J.C. and Hamidian, M.H., 2019. Magnetic field–induced pair density wave state in the cuprate vortex halo. Science, 364(6444), pp.976-980.
 Chen, H., Yang, H., Hu, B., Zhao, Z., Yuan, J., Xing, Y., Qian, G., Huang, Z., Li, G., Ye, Y. and Ma, S., 2021. Roton pair density wave in a strong-coupling kagome superconductor. Nature, 599(7884), pp.222-228.
 Gu, Q., Carroll, J.P., Wang, S., Ran, S., Broyles, C., Siddiquee, H., Butch, N.P., Saha, S.R., Paglione, J., Davis, J.C. and Liu, X., 2022. Detection of a Pair Density Wave State in UTe $ _2$. arXiv preprint arXiv:2209.10859.