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1. Atomic-scale electronic structure of the cuprate pair density wave state coexisting with superconductivity

   https://doi.org/10.1073/pnas.2002429117  

   Choubey, Peayush ; Joo, Sang Hyun ; Fujita, K. ; Du, Zengyi ; Edkins, S. D. ; Hamidian, M. H. ; Eisaki, H. ; Uchida, S. ; Mackenzie, A. P. ; Lee, Jinho ; et al ( June 2020 , Proceedings of the National Academy of Sciences)

   null (Ed.)

   The defining characteristic of hole-doped cuprates is d -wave high temperature superconductivity. However, intense theoretical interest is now focused on whether a pair density wave state (PDW) could coexist with cuprate superconductivity [D. F. Agterberg et al., Annu. Rev. Condens. Matter Phys. 11, 231 (2020)]. Here, we use a strong-coupling mean-field theory of cuprates, to model the atomic-scale electronic structure of an eight-unit-cell periodic, d -symmetry form factor, pair density wave (PDW) state coexisting with d -wave superconductivity (DSC). From this PDW + DSC model, the atomically resolved density of Bogoliubov quasiparticle states N r , E is predicted at the terminal BiO surface of Bi 2 Sr 2 CaCu 2 O 8 and compared with high-precision electronic visualization experiments using spectroscopic imaging scanning tunneling microscopy (STM). The PDW + DSC model predictions include the intraunit-cell structure and periodic modulations of N r , E , the modulations of the coherence peak energy Δ p r , and the characteristics of Bogoliubov quasiparticle interference in scattering-wavevector space q - space . Consistency between all these predictions and the corresponding experiments indicates that lightly hole-doped Bi 2 Sr 2 CaCu 2 O 8 does contain a PDW + DSC state. Moreover, in the model the PDW + DSC state becomes unstable to a pure DSC state at a critical hole density p *, with empirically equivalent phenomena occurring in the experiments. All these results are consistent with a picture in which the cuprate translational symmetry-breaking state is a PDW, the observed charge modulations are its consequence, the antinodal pseudogap is that of the PDW state, and the cuprate critical point at p * ≈ 19% occurs due to disappearance of this PDW. 

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2. Magnetic field–induced pair density wave state in the cuprate vortex halo

   https://doi.org/10.1126/science.aat1773  

   Edkins, S. D. ; Kostin, A. ; Fujita, K. ; Mackenzie, A. P. ; Eisaki, H. ; Uchida, S. ; Sachdev, Subir ; Lawler, Michael J. ; Kim, E.-A. ; Séamus Davis, J. C. ; et al ( June 2019 , Science)

   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 Bi 2 Sr 2 CaCu 2 O 8 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 Q P and 2 Q P , 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 CuO 2 unit-cell periodicity and coexisting with its secondary CDWs. 

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3. Evidence for a vestigial nematic state in the cuprate pseudogap phase

   https://doi.org/10.1073/pnas.1821454116  

   Mukhopadhyay, Sourin ; Sharma, Rahul ; Kim, Chung Koo ; Edkins, Stephen D. ; Hamidian, Mohammad H. ; Eisaki, Hiroshi ; Uchida, Shin-ichi ; Kim, Eun-Ah ; Lawler, Michael J. ; Mackenzie, Andrew P. ; et al ( July 2019 , Proceedings of the National Academy of Sciences)

   The CuO 2 antiferromagnetic insulator is transformed by hole-doping into an exotic quantum fluid usually referred to as the pseudogap (PG) phase. Its defining characteristic is a strong suppression of the electronic density-of-states D ( E ) for energies | E | < Δ * , where Δ * is the PG energy. Unanticipated broken-symmetry phases have been detected by a wide variety of techniques in the PG regime, most significantly a finite- Q density-wave (DW) state and a Q = 0 nematic (NE) state. Sublattice-phase-resolved imaging of electronic structure allows the doping and energy dependence of these distinct broken-symmetry states to be visualized simultaneously. Using this approach, we show that even though their reported ordering temperatures T DW and T NE are unrelated to each other, both the DW and NE states always exhibit their maximum spectral intensity at the same energy, and using independent measurements that this is the PG energy Δ * . Moreover, no new energy-gap opening coincides with the appearance of the DW state (which should theoretically open an energy gap on the Fermi surface), while the observed PG opening coincides with the appearance of the NE state (which should theoretically be incapable of opening a Fermi-surface gap). We demonstrate how this perplexing phenomenology of thermal transitions and energy-gap opening at the breaking of two highly distinct symmetries may be understood as the natural consequence of a vestigial nematic state within the pseudogap phase of Bi 2 Sr 2 CaCu 2 O 8 . 

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4. Phase transition and higher order analysis of Lq regularization under dependence

   https://doi.org/10.1093/imaiai/iaae005  

   Huang, Hanwen ; Zeng, Peng ; Yang, Qinglong ( February 2024 , Information and Inference: A Journal of the IMA)

   We study the problem of estimating a $k$-sparse signal ${\boldsymbol \beta }_{0}\in{\mathbb{R}}^{p}$ from a set of noisy observations $\mathbf{y}\in{\mathbb{R}}^{n}$ under the model $\mathbf{y}=\mathbf{X}{\boldsymbol \beta }+w$, where $\mathbf{X}\in{\mathbb{R}}^{n\times p}$ is the measurement matrix the row of which is drawn from distribution $N(0,{\boldsymbol \varSigma })$. We consider the class of $L_{q}$-regularized least squares (LQLS) given by the formulation $\hat{{\boldsymbol \beta }}(\lambda )=\text{argmin}_{{\boldsymbol \beta }\in{\mathbb{R}}^{p}}\frac{1}{2}\|\mathbf{y}-\mathbf{X}{\boldsymbol \beta }\|^{2}_{2}+\lambda \|{\boldsymbol \beta }\|_{q}^{q}$, where $\|\cdot \|_{q}$  $(0\le q\le 2)$ denotes the $L_{q}$-norm. In the setting $p,n,k\rightarrow \infty $ with fixed $k/p=\epsilon $ and $n/p=\delta $, we derive the asymptotic risk of $\hat{{\boldsymbol \beta }}(\lambda )$ for arbitrary covariance matrix ${\boldsymbol \varSigma }$ that generalizes the existing results for standard Gaussian design, i.e. $X_{ij}\overset{i.i.d}{\sim }N(0,1)$. The results were derived from the non-rigorous replica method. We perform a higher-order analysis for LQLS in the small-error regime in which the first dominant term can be used to determine the phase transition behavior of LQLS. Our results show that the first dominant term does not depend on the covariance structure of ${\boldsymbol \varSigma }$ in the cases $0\le q\lt 1$ and $1\lt q\le 2,$ which indicates that the correlations among predictors only affect the phase transition curve in the case $q=1$ a.k.a. LASSO. To study the influence of the covariance structure of ${\boldsymbol \varSigma }$ on the performance of LQLS in the cases $0\le q\lt 1$ and $1\lt q\le 2$, we derive the explicit formulas for the second dominant term in the expansion of the asymptotic risk in terms of small error. Extensive computational experiments confirm that our analytical predictions are consistent with numerical results.

    

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5. Multiple intertwined pairing states and temperature-sensitive gap anisotropy for superconductivity at a nematic quantum-critical point

   https://doi.org/10.1038/s41535-019-0192-x  

   Klein, Avraham ; Wu, Yi-Ming ; Chubukov, Andrey V. ( November 2019 , npj Quantum Materials)

   The proximity of many strongly correlated superconductors to density-wave or nematic order has led to an extensive search for fingerprints of pairing mediated by dynamical quantum-critical (QC) fluctuations of the corresponding order parameter. Here we study anisotropics-wave superconductivity induced by anisotropic QC dynamical nematic fluctuations. We solve the non-linear gap equation for the pairing gap$$\Delta (\theta ,{\omega }_{m})$$$\Delta \left(\theta ,{\omega }_m\right)$and show that its angular dependence strongly varies below$${T}_{{\rm{c}}}$$$T_c$. We show that this variation is a signature of QC pairing and comes about because there are multiples-wave pairing instabilities with closely spaced transition temperatures$${T}_{{\rm{c}},n}$$$T_{c,n}$. Taken alone, each instability would produce a gap$$\Delta (\theta ,{\omega }_{m})$$$\Delta \left(\theta ,{\omega }_m\right)$that changes sign$$8n$$$8n$times along the Fermi surface. We show that the equilibrium gap$$\Delta (\theta ,{\omega }_{m})$$$\Delta (\theta ,{\omega }_m)$is a superposition of multiple components that are nonlinearly induced below the actual$${T}_{{\rm{c}}}={T}_{{\rm{c}},0}$$$T_c=T_{c,0}$, and get resonantly enhanced at$$T={T}_{{\rm{c}},n}\ <\ {T}_{{\rm{c}}}$$$T=T_{c,n}<T_c$. This gives rise to strong temperature variation of the angular dependence of$$\Delta (\theta ,{\omega }_{m})$$$\Delta \left(\theta ,{\omega }_m\right)$. This variation progressively disappears away from a QC point.

    

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