 Award ID(s):
 1849751
 NSFPAR ID:
 10323610
 Date Published:
 Journal Name:
 Nature Communications
 Volume:
 12
 Issue:
 1
 ISSN:
 20411723
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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null (Ed.)The defining characteristic of holedoped 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 strongcoupling meanfield theory of cuprates, to model the atomicscale electronic structure of an eightunitcell 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 highprecision electronic visualization experiments using spectroscopic imaging scanning tunneling microscopy (STM). The PDW + DSC model predictions include the intraunitcell 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 scatteringwavevector space q  space . Consistency between all these predictions and the corresponding experiments indicates that lightly holedoped 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 symmetrybreaking 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.more » « less

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 electronpair density wave (PDW). To search for evidence of a fieldinduced 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 fieldinduced PDW, including two sets of particlehole 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 fieldinduced state in underdoped superconducting cuprates is a PDW, with approximately eight CuO 2 unitcell periodicity and coexisting with its secondary CDWs.more » « less

The CuO 2 antiferromagnetic insulator is transformed by holedoping into an exotic quantum fluid usually referred to as the pseudogap (PG) phase. Its defining characteristic is a strong suppression of the electronic densityofstates D ( E ) for energies  E  < Δ * , where Δ * is the PG energy. Unanticipated brokensymmetry phases have been detected by a wide variety of techniques in the PG regime, most significantly a finite Q densitywave (DW) state and a Q = 0 nematic (NE) state. Sublatticephaseresolved imaging of electronic structure allows the doping and energy dependence of these distinct brokensymmetry 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 energygap 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 Fermisurface gap). We demonstrate how this perplexing phenomenology of thermal transitions and energygap 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 .more » « less

Abstract 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 nonrigorous replica method. We perform a higherorder analysis for LQLS in the smallerror 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.

Abstract Covariance matrices are fundamental to the analysis and forecast of economic, physical and biological systems. Although the eigenvalues $\{\lambda _i\}$ and eigenvectors $\{\boldsymbol{u}_i\}$ of a covariance matrix are central to such endeavours, in practice one must inevitably approximate the covariance matrix based on data with finite sample size $n$ to obtain empirical eigenvalues $\{\tilde{\lambda }_i\}$ and eigenvectors $\{\tilde{\boldsymbol{u}}_i\}$, and therefore understanding the error so introduced is of central importance. We analyse eigenvector error $\\boldsymbol{u}_i  \tilde{\boldsymbol{u}}_i \^2$ while leveraging the assumption that the true covariance matrix having size $p$ is drawn from a matrix ensemble with known spectral properties—particularly, we assume the distribution of population eigenvalues weakly converges as $p\to \infty $ to a spectral density $\rho (\lambda )$ and that the spacing between population eigenvalues is similar to that for the Gaussian orthogonal ensemble. Our approach complements previous analyses of eigenvector error that require the full set of eigenvalues to be known, which can be computationally infeasible when $p$ is large. To provide a scalable approach for uncertainty quantification of eigenvector error, we consider a fixed eigenvalue $\lambda $ and approximate the distribution of the expected square error $r= \mathbb{E}\left [\ \boldsymbol{u}_i  \tilde{\boldsymbol{u}}_i \^2\right ]$ across the matrix ensemble for all $\boldsymbol{u}_i$ associated with $\lambda _i=\lambda $. We find, for example, that for sufficiently large matrix size $p$ and sample size $n> p$, the probability density of $r$ scales as $1/nr^2$. This powerlaw scaling implies that the eigenvector error is extremely heterogeneous—even if $r$ is very small for most eigenvectors, it can be large for others with nonnegligible probability. We support this and further results with numerical experiments.more » « less