A recent scanning tunneling microscopy (STM) experiment reports the observation of charge density wave (CDW) with period of approximately 8a in the halo region surrounding the vortex core, in striking contrast to the approximately period 4a CDW that are commonly observed in the cuprates. Inspired by this work, we study a model where a bi-directional pair density wave (PDW) with period 8 is at play. This further divides into two classes, (1) where the PDW is a competing state of the d wave superconductor and can exist only near the vortex core where the d wave order is suppressed, and (2) where the PDW is the primary order, the so called "mother state" that persists with strong phase fluctuations to high temperature and high magnetic field and lies behind the pseudogap phenomenology. We study the charge density wave structures near the vortex core in these models. We emphasize the importance of the phase winding of the d-wave order parameter. The PDW can be pinned by the vortex core due to this winding and become static. Furthermore, the period 8 CDW inherits the properties of this winding, which gives rise to a special feature of the Fourier transform peak, namely, it is split in certain directions. There are also a line of zeros in the inverse Fourier transform of filtered data. We propose that these are key experimental signatures that can distinguish between the PDW-driven scenario from the more mundane option that the period 8 CDW is primary. We discuss the pros and cons of the options considered above. Finally we attempt to place the STM experiment in the broader context of pseudogap physics of underdoped cuprates and relate this observation to the unusual properties of X-ray scattering data on CDW carried out to very high magnetic field.

We present a theory of optical conductivity in systems with finite-momentum Cooper pairs. In contrast to the BCS pairing where AC conductivity is purely imaginary in the clean limit, there is nonzero AC absorption across the superconducting gap for finite-momentum pairing if we break the Galilean symmetry explicitly in the electronic Hamiltonian. Vertex correction is crucial for maintaining the gauge invariance in the mean-field formalism and dramatically changes the optical conductivity in the direction of the pairing momentum. We carried out a self-consistent calculation and gave an explicit formula for optical conductivity in a simple case. This result applies to the Fulde-Ferrell-Larkin-Ovchinnikov state and candidates with pair density waves proposed for high-Tc cuprates. It may help detect PDW and determine the pairing gap as well as the direction of the pairing momentum in experiments.

We present a way to quantum-disorder a pair density wave, and propose it to be a candidate of the effective low-energy description of the pseudogap metal which may reveal itself in a sufficiently high magnetic field to suppress the d-wave pairing. The ground state we construct is a small-pocket Fermi liquid with a hidden bosonic Mott insulator in the density-wave-enlarged unit cell. At low energy, the charge density is mainly carried by charge 2e bosons, which develop a small insulating gap. We discuss a number of experimental consequences. Optical excitation across the boson gap can explain the onset of mid infra-red absorption reported long ago. The interplay between the electron and the small-gap boson results in a characteristic electron spectral function consistent with ARPES. We also postulate a secondary magnetization density wave along (1,1) direction which breaks timereversal symmetry, mirror symmetry, and C4 rotation. Our treatment of fluctuating pair density wave provides a non-perturbative mechanism of gap generation on part of the Fermi surface, which can be generalized to other fluctuating orders. I. INTRODUCTIONPseudogap occupies a large region in the phase diagram of underdoped cuprates, up to a high temperature wellseparated from the superconductivity. Theoretically, it is desirable to kill the superconductivity and probe the pseudogap phenomena at zero temperature, where different quantum phases are sharply distinct. This line of thinking brings us to the study of high-field ground state of underdoped cuprates. At around 12% hole doping, the magnetic field needed to destroy long-range superconductivity corresponds to an energy scale well below the anti-nodal gap [1][2][3]; it is reasonable to expect the 'pseudogap' to be present in the high-field ground state. Surprisingly, as the superconductivity is destroyed, quantum oscillation of a small Fermi pocket was detected [4][5][6], and charge density wave (CDW) with correlation length hundreds of lattice spacing was discovered [7][8][9][10][11]. Harrison and Sebastian postulate that the CDW momentum connects the tips of 'Fermi arcs' and makes the closed pocket seen in quantum oscillations (for a review, see [12]). Another hint of the high-field ground state comes from the recent STM discovery of a new period 8 short-range but static CDW below T c in the 'vortex halo' [13][14][15], pointing to its possible origin in a pair density wave (PDW) with the same period. This experiment suggests the relevance of PDW to the high field pseudo-gap ground state in addition to the CDW. The possible relevance of PDW to the Cuprates have been under discussion for some time [16][17][18][19][20][21] and recently reviewed by Agterberg et al. [22] The large antinodal gap in the normal state and the decrease of low-energy fermionic carrier density, which are the most prominent phenomena of the pseudogap region, was initially attributed to fluctuating d-wave superconductivity. However, careful ARPES study [23,24] reported a momentum-dependence different from a d-wave su...

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The level s(A) of a (commutative) ring A is the smallest natural number s such that -1 is a sum of s squares in A. [l,p. 184]).In this note, we announce the following. While the above results are of an algebraic nature, their proofs (at least as so far discovered) are purely topological. One uses ideas from homotopy and