Significance Although compelling evidence now exists that a pair density wave (PDW) state occurs in hole-doped CuO2 materials, its microscopic phenomenology and mechanism remain a mystery. A prevalent theoretical concept has been that the two-dimensional strongly correlated electrons in these materials can spontaneously generate a unidirectional, lattice-commensurate (striped) PDW state that is not induced by any preexistent charge density wave order. Here, in samples with virtually no charge density wave phenomenology, we show by direct scanned Josephson tunneling microscopy visualization of the modulating electron pair density that the PDW state in Bi2Sr2CaCu2O8+x is a vestigial nematic phase, with impurity-atom pinned Ising domains of strongly unidirectional and commensurate pair density modulations.

pdf/identification-of-a-nematic-pair-density-wave-state-in-36bu0q9v.pdf

Identification of a nematic pair density wave state in Bi2Sr2CaCu2O8+x

Condensation of bosons in Bose-Einstein condensates or Cooper pairs in superconductors refers to a macroscopic occupation of a few single- or two-particle states. A condensate is called "fragmented" if not a single, but multiple states are macroscopically occupied. While fragmentation is known to occur in particular Bose-Einstein condensates, we propose that fragmentation naturally takes place in striped superconductors. To this end, we investigate the nature of the superconducting ground state realized in the two-dimensional t-t^{'}-J model. In the presence of charge density modulations, the condensate is shown to be fragmented and composed of partial condensates located on the stripes. The fragments of the condensates hybridize to form an extended macroscopic wave function across the system. The results are obtained from evaluating the singlet-pairing two-particle density matrix of the ground state on finite cylinders computed via the density matrix renormalization group method. Our results shed light on the intricate relation between stripe order and superconductivity in systems of strongly correlated electrons.

Fragmented Cooper Pair Condensation in Striped Superconductors.

We investigate the nematic and loop-current type orders that may arise as vestigial precursor phases in a model with an underlying pair-density wave (PDW) instability. We discuss how such a vestigial phase gives rise to a highly anisotropic stiffness for a coexisting single-component superconductor with low intrinsic stiffness, as is the case for the underdoped cuprate superconductors. Next, focusing on a regime with a mean-field PDW ground state with loop-current and nematic $xy$ (B$_{2g}$) order, we find a preemptive transition into a low and high-temperature vestigial phase with loop-current and nematic order corresponding to $xy$ (B$_{2g}$) and $x^2-y^2$ (B$_{1g}$) symmetry respectively. Near the transition between the two phases, a state of soft nematic order emerges for which we expect that the nematic director is readily pinned away from the high-symmetry directions in the presence of an external field. Results are discussed in relation to findings in the cuprates, especially to the recently inferred highly anisotropic superconducting fluctuations [W{\aa}rdh {\em et al.}, ``Colossal transverse magnetoresistance due to nematic superconducting phase fluctuations in a copper oxide'', arXiv:2203.06769], giving additional evidence for an underlying ubiquitous PDW instability in these materials.

Nematic single-component superconductivity and loop-current order from pair-density wave instability

Focused ion beam (FIB) milling is a mask-free lithography technique that allows the precise shaping of 3D materials on the micron and sub-micron scale. The recent discovery of electronic nematicity in La2−xSrxCuO4 (LSCO) thin films triggered the search for the same phenomenon in bulk LSCO crystals. With this motivation, we have systematically explored FIB patterning of bulk LSCO crystals into micro-devices suitable for longitudinal and transverse resistivity measurements. We found that several detrimental factors can affect the result, ultimately compromising the possibility of effectively using FIB milling to fabricate sub-micrometer LSCO devices, especially in the underdoped regime.

/pdf/effects-of-focused-ion-beam-lithography-on-la2-xsrxcuo4-1elmxc3a.pdf

Effects of Focused Ion Beam Lithography on La2−xSrxCuO4 Single Crystals

The structural phase and polymorphs play important roles in determining the symmetry and properties of materials, providing an important route to engineering exotic electronic phases and expanding the functionalities for applications. Here, we report the observation of a two-fold symmetry of transport anisotropy associated with the crystallographic directions in 2D ultrathin Mo2C superconducting crystals. Through angle-resolved transport measurements, we demonstrate the modulation of the longitudinal and transverse resistance on an in-plane magnetic field. The onset of anisotropy coincides with an enhanced superconducting fluctuation near the superconducting transition. Moreover, large nonreciprocal charge transport is observed, indicating spatial inversion symmetry breaking. These results can be understood from the spontaneous symmetry breaking due to the emergence of mixed polymorphs and metastable phases in ultrathin Mo2C crystals, which is further supported by atomic-scale characterizations. Our work reveals the potential of engineering phases and polymorphs to manipulate superconducting properties and functionalities in ultrathin transition metal carbides.

Observation of a Two-fold Anisotropy in Ultrathin Mo 2 C Superconducting Crystals

Electronic anisotropy (or ‘nematicity’) has been detected in all main families of cuprate superconductors by a range of experimental techniques — electronic Raman scattering, THz dichroism, thermal conductivity, torque magnetometry, second-harmonic generation — and was directly visualized by scanning tunneling microscope (STM) spectroscopy. Using angle-resolved transverse resistance (ARTR) measurements, a very sensitive and background-free technique that can detect 0.5% anisotropy in transport, we have observed it also in La2-xSrxCuO4 (LSCO) for 0.02 ≤ x ≤ 0.25. Arguably the key enigma in LSCO is the rotation of the nematic director with temperature; this has not been seen before in any material. Here, we address this puzzle by measuring the angle-resolved transverse magnetoresistance (ARTMR) in LSCO. We report a discovery of colossal transverse magnetoresistance (CTMR) — an order-of-magnitude drop in the transverse resistivity in the magnetic field of 6T, while none is seen in the longitudinal resistivity. We show that the apparent rotation of the nematic director is caused by superconducting phase fluctuations, which are much more anisotropic than the normal-electron fluid, and their respective directors are not parallel. This qualitative conclusion is robust and follows straight from the raw experimental data. We quantify this by modelling the measured (magneto-)conductivity by a sum of two conducting channels that correspond to distinct anisotropic Drude and Cooper-pair effective mass tensors. Strikingly, the anisotropy of Cooper-pair stiffness is significantly larger than that of the normal electrons, and it grows dramatically on the underdoped side, where the fluctuations become effectively quasi-one dimensional. Our analysis is deliberately general rather than model-dependent, but we also discuss some candidate microscopic models including coupled strongly-correlated ladders where the transverse (inter-ladder) phase stiffness is low compared to the longitudinal intra-ladder stiffness, as well as the anisotropic superconducting fluctuations expected close to the transition to a pair-density wave state. The results provide important clues about the pseudogap state in the cuprate superconductors.

Colossal transverse magnetoresistance due to nematic superconducting phase fluctuations in a copper oxide

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Colossal transverse magnetoresistance due to nematic superconducting phase fluctuations in a copper oxide