The identity of the fundamental broken symmetry (if any) in the underdoped cuprates is unresolved. However, evidence has been accumulating that this state may be an unconventional density wave. Here we carry out site-specific measurements within each CuO 2 unit cell, segregating the results into three separate electronic structure images containing only the Cu sites [Cu(r)] and only the x/y axis O sites [O x (r) and O y (r)]. Phase-resolved Fourier analysis reveals directly that the modulations in the O x (r) and O y (r) sublattice images consistently exhibit a relative phase of π. We confirm this discovery on two highly distinct cuprate compounds, ruling out tunnel matrix-element and materials-specific systematics. These observations demonstrate by direct sublattice phaseresolved visualization that the density wave found in underdoped cuprates consists of modulations of the intraunit-cell states that exhibit a predominantly d-symmetry form factor.CuO 2 pseudogap | broken symmetry | density-wave form factor U nderstanding the microscopic electronic structure of the CuO 2 plane represents the essential challenge of cuprate studies. As the density of doped holes, p, increases from zero in this plane, the pseudogap state (1, 2) first emerges, followed by the high-temperature superconductivity. Within the elementary CuO 2 unit cell, the Cu atom resides at the symmetry point with an O atom adjacent along the x axis and the y axis (Fig. 1A, Inset). Intraunit-cell (IUC) degrees of freedom associated with these two O sites (3, 4), although often disregarded, may actually represent the key to understanding CuO 2 electronic structure. Among the proposals in this regard are valence-bond ordered phases having localized spin singlets whose wavefunctions are centered on O x or O y sites (5, 6), electronic nematic phases having a distinct spectrum of eigenstates at O x and O y sites (7,8), and orbital-current phases in which orbitals at O x and O y are distinguishable due to time-reversal symmetry breaking (9). A common element to these proposals is that, in the pseudogap state of lightly hole-doped cuprates, some form of electronic symmetry breaking renders the O x and O y sites of each CuO 2 unit cell electronically inequivalent.Electronic Inequivalence at the Oxygen Sites of the CuO 2 Plane in Pseudogap State Experimental electronic structure studies that discriminate the O x from O y sites do find a rich phenomenology in underdoped cuprates. Direct oxygen site-specific visualization of electronic structure reveals that even very light hole doping of the insulator produces local IUC symmetry breaking, rendering O x and O y inequivalent (10), that both Q ≠ 0 density wave (11) and Q = 0 C 4 -symmetry breaking (11, 12, 13) involve electronic inequivalence of the O x and O y sites, and that the Q ≠ 0 and Q = 0 broken symmetries weaken simultaneously with increasing p and disappear jointly near p c = 0.19 (13). For multiple cuprate compounds, neutron scattering reveals clear intraunit-cell breaking of rotational symmetry (14,15...

A sensitive probe of unconventional order is its response to a symmetry-breaking field. To probe the proposed p(x) ± ip(y) topological superconducting state of Sr2RuO4, we have constructed an apparatus capable of applying both compressive and tensile strains of up to 0.23%. Strains applied along ⟨100⟩ crystallographic directions yield a strong, strain-symmetric increase in the superconducting transition temperature T(c). ⟨110⟩ strains give a much weaker, mostly antisymmetric response. As well as advancing the understanding of the superconductivity of Sr2RuO4, our technique has potential applicability to a wide range of problems in solid-state physics.

The quantum condensate of Cooper pairs forming a superconductor was originally conceived as being translationally invariant. In theory, however, pairs can exist with finite momentum Q, thus generating a state with a spatially modulated Cooper-pair density. Such a state has been created in ultracold (6)Li gas but never observed directly in any superconductor. It is now widely hypothesized that the pseudogap phase of the copper oxide superconductors contains such a 'pair density wave' state. Here we report the use of nanometre-resolution scanned Josephson tunnelling microscopy to image Cooper pair tunnelling from a d-wave superconducting microscope tip to the condensate of the superconductor Bi2Sr2CaCu2O8+x. We demonstrate condensate visualization capabilities directly by using the Cooper-pair density variations surrounding zinc impurity atoms and at the Bi2Sr2CaCu2O8+x crystal supermodulation. Then, by using Fourier analysis of scanned Josephson tunnelling images, we discover the direct signature of a Cooper-pair density modulation at wavevectors QP ≈ (0.25, 0)2π/a0 and (0, 0.25)2π/a0 in Bi2Sr2CaCu2O8+x. The amplitude of these modulations is about five per cent of the background condensate density and their form factor exhibits primarily s or s' symmetry. This phenomenology is consistent with Ginzburg-Landau theory when a charge density wave with d-symmetry form factor and wavevector QC = QP coexists with a d-symmetry superconductor; it is also predicted by several contemporary microscopic theories for the pseudogap phase.

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.

Research on high-temperature superconducting cuprates is at present focused on identifying the relationship between the classic 'pseudogap' phenomenon 1,2 and the more recently investigated density wave state 3-13 . This state is generally characterized by a wavevector Q parallel to the planar Cu-O-Cu bonds 4-13 along with a predominantly d-symmetry form factor 14-16 (dFF-DW). To identify the microscopic mechanism giving rise to this state 17-29 , one must identify the momentum-space states contributing to the dFF-DW spectral weight, determine their particle-hole phase relationship about the Fermi energy, establish whether they exhibit a characteristic energy gap, and understand the evolution of all these phenomena throughout the phase diagram. Here we use energy-resolved sublattice visualization 14 of electronic structure and reveal that the characteristic energy of the dFF-DW modulations is actually the 'pseudogap' energy ∆ 1 . Moreover, we demonstrate that the dFF-DW modulations at E = −∆ 1 (filled states) occur with relative phase π compared to those at E = ∆ 1 (empty states). Finally, we show that the conventionally defined dFF-DW Q corresponds to scattering between the 'hot frontier' regions of momentum-space beyond which Bogoliubov quasiparticles cease to exist [30][31][32] . These data indicate that the cuprate dFF-DW state involves particle-hole interactions focused at the pseudogap energy scale and between the four pairs of 'hot frontier' regions in momentum space where the pseudogap opens.A conventional 'Peierls' charge density wave (CDW) in a metal results from particle-hole interactions which open an energy gap at specific regions of k-space that are connected by a common wavevector Q. This generates a modulation in the density of free charge at Q along with an associated modulation of the crystal lattice parameters. Such CDW states are now very well known 33 . In principle, a density wave modulating at Q can also exhibit a 'form factor' (FF) with different possible symmetries 34,35 (see Supplementary Section 1). This is relevant to the high-temperature superconducting cuprates because numerous researchers have recently proposed that the 'pseudogap' regime 1,2 (PG in Fig. 1a) contains an unconventional density wave with a d-symmetry form factor [17][18][19][20][21][22][23][24][25][26][27][28][29] . The basic phenomenology of such a state is that intraunit-cell (IUC) symmetry breaking renders the O x and O y sites within each CuO 2 unit-cell electronically inequivalent, and that this inequivalence is then modulated periodically at wavevector Q parallel to (1,0);(0,1). The real-space (r-space) schematic of such a d-symmetry FF density wave (dFF-DW) at Q x , as shown in Fig. 1b, exemplifies periodic modulations at the O x sites that are π out of phase with those at the O y sites. Such a state is then described by A(r) = D(r) cos(φ(r) + φ 0 (r)), where A(r) represents whatever is the modulating electronic degree of freedom, φ(r) = Q x · r is the DW spatial phase at location r, φ 0 (r) represents disorde...

We report the design and construction of piezoelectric-based apparatus for applying continuously tuneable compressive and tensile strains to test samples. It can be used across a wide temperature range, including cryogenic temperatures. The achievable strain is large, so far up to 0.23% at cryogenic temperatures. The apparatus is compact and compatible with a wide variety of experimental probes. In addition, we present a method for mounting high-aspect-ratio samples in order to achieve high strain homogeneity. (C) 2014 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License

High magnetic fields suppress cuprate superconductivity to reveal an unusual density wave (DW) state coexisting with unexplained quantum oscillations. Although routinely labeled a charge density wave (CDW), this DW state could actually be an electron-pair density wave (PDW). To search for evidence of a field-induced PDW, we visualized modulations in the density of electronic states N(r) within the halo surrounding Bi2Sr2CaCu2O8 vortex cores. We detected numerous phenomena predicted for a field-induced PDW, including two sets of particle-hole symmetric N(r) modulations with wave vectors QP and 2QP, with the latter decaying twice as rapidly from the core as the former. These data imply that the primary field-induced state in underdoped superconducting cuprates is a PDW, with approximately eight CuO2 unit-cell periodicity and coexisting with its secondary CDWs.

Essentials of the scientific discovery process have remained largely unchanged for centuries 1 : systematic human observation of natural phenomena is used to form hypotheses that, when validated through experimentation, are generalized into established scientific theory. Today, however, we face major challenges because automated instrumentation and large-scale data acquisition are generating data sets of such volume and complexity as to defy human analysis. Radically different scientific approaches are needed, with machine learning (ML) showing great promise, not least for materials science research 2-5 . Hence, given recent advances in ML analysis of synthetic data representing electronic quantum matter (EQM) 6-16 , the next challenge is for ML to engage equivalently with experimental data. For example, atomic-scale visualization of EQM yields arrays of complex electronic structure images 17 , that frequently elude effective analyses. Here we report development and training of an array of artificial neural networks (ANN) designed to recognize different types of hypothesized order hidden in EQM imagearrays. These ANNs are used to analyze an experimentally-derived EQM image archive from carrier-doped cuprate Mott insulators. Throughout these noisy and complex data, the ANNs discover the existence of a lattice-commensurate, four-unitcell periodic, translational-symmetry-breaking EQM state. Further, the ANNs find these phenomena to be unidirectional, revealing a coincident nematic EQM state. Strong-coupling theories of electronic liquid crystals 18,19 are congruent with all these observations. 1Frontier research in EQM concentrates on exotic electronic phases that emerge when electrons interact so strongly that they lack a definite momentum. These electrons often self-organize into complex new states of EQM including, for example, electronic liquid crystals 18,19 , high temperature superconductors 20,21 , fractionalized electronic fluids and quantum spin liquids. In this field, vast experimental data sets have emerged, for example from real space (r-space) visualization of EQM using spectroscopic imaging scanning tunneling microscopy 17 (SISTM), from momentum space (k-space) visualization of EQM using angle resolved photoemission (ARPES), or from modern X-ray 22 and neutron scattering. The challenge is to develop ML strategies capable of scientific discovery using such large and complex experimental data structures from EQM experiments. 2An excellent example is the electronic structure of the CuO2 plane in the cuprate compounds supporting high temperature superconductivity 20 (Fig. 1a). With one electron per Cu site, strong Coulomb interactions produce charge localization in an antiferromagnetic Mott insulator (MI) state. Removing p electrons (adding p 'holes') perCuO2 plaquette generates the 'pseudogap' (PG) phase 20 . It exhibits strongly depleted density-of-electronic states ( ) for energies |E| < Δ ! , where Δ ! is the characteristic pseudogap energy scale that emerges for < * ( ) (Fig. 1a). Although the PG pha...