Charge-density waves: observations, competition and reconstruction

Scope

A charge-density wave (CDW) modulates the electronic density at a finite wave vector, often together with atomic displacements. Cuprates and kagome compounds exhibit several distinct charge-order patterns. Their coexistence or competition with superconductivity depends on composition, temperature, field, pressure and the probe's spatial and temporal resolution. Neither a density modulation nor a nearby superconducting dome identifies the pairing interaction. R2 R3 R8 R11

This page addresses observations and their interpretation. Pair-density waves instead modulate the superconducting pair field; spin-density waves modulate magnetization. Charge order, structural chirality, broken time-reversal symmetry and superconducting chirality are different claims. The retained source chapter's citation of Stahl for the conjoined-CDW paper is corrected: that paper is by Li and colleagues; Stahl's temperature-dependent stacking study has a different DOI. R7 R8 R9 R10

Mathematical assumptions

A single-component modulation can be written as $\rho(\mathbf r)=\rho_0+\operatorname{Re}[\rho_{\mathbf Q}e^{i\mathbf Q\cdot\mathbf r}]$. Peak position identifies periodicity, while width constrains correlations under a stated line-shape model. For the illustrative lattice correlation $C(n)=r^{|n|}$, with $r=e^{-1/\xi}$ and $\xi$ in lattice spacings, the Fourier sum is $S(q)=(1-r^2)/(1+r^2-2r\cos q)$. Here $q$ is dimensionless radians per spacing, measured from the ordering wave vector. Finite $\xi$ produces a finite-width peak; observing that peak does not establish infinite-range order. This is an independent correlation model, not a fit to the cited specimens. R4 R5

Correlation lengths also depend on resolution and conventions. If both intrinsic and instrumental profiles are Gaussian, $\sigma_{\rm obs}^2=\sigma_{\rm int}^2+\sigma_R^2$. The convention $\xi=1/\sigma_{\rm int}$ then requires momentum widths in inverse-length units. Reciprocal-lattice widths need the appropriate lattice conversion. Uncorrected $1/\sigma_{\rm obs}$ is a lower bound within this model. Gaussian standard deviations, Lorentzian half-widths and full widths are not interchangeable; a resolution-limited peak yields a bound rather than a precisely measured divergence. R4 R5

A phenomenological free energy $F=a_s|\Delta|^2+b_s|\Delta|^4+a_c|\rho_{\mathbf Q}|^2+b_c|\rho_{\mathbf Q}|^4+g|\Delta|^2|\rho_{\mathbf Q}|^2$ illustrates static competition when $g>0$. With positive quartic coefficients, a stable interior coexistence minimum additionally requires $4b_sb_c-g^2>0$ and positive equilibrium squared amplitudes. Competition alone neither guarantees coexistence nor determines the frequency-dependent pairing kernel. Charge fluctuations could contribute to an effective interaction in a specified microscopic model without making every observed CDW a pairing mechanism. R3 R11

Experimental criteria

Tranquada's historical neutron result concerns Nd-substituted La1.6−xNd0.4SrxCuO4 at $x=0.12$. Its static stripe evidence and suppressed superconductivity should not be assigned indiscriminately to every one-eighth-doped cuprate. Ghiringhelli's resonant X-ray study adds an element-sensitive charge response in underdoped YBCO. The selected author text uses 130 meV instrumental resolution and argues for a much smaller energy bound; that bound is not a separately resolved sharp collective mode. R1 R2

Chang's 2012 experiment on detwinned ortho-VIII YBa2Cu3O6.67 finds charge-order intensity suppressed on entering the superconducting regime and enhanced by magnetic field below it. Those controlled changes support competition in this specimen. They do not isolate every field-induced change in vortices or electronic structure, and do not identify a unique microscopic interaction. R3

Gerber's high-field diffraction shows additional interlayer-correlated order developing above about 15 T at 10 K. The author Figure 3 separates peaks near integer and half-integer out-of-plane coordinates and extracts Gaussian widths. Its reported lengths are explicitly uncorrected for resolution. Chang's 2016 study instead reports resolution-corrected widths: for YBCO6.67, the c-axis scale reaches about four lattice periods within the measured field range. Finite three-dimensional correlations must remain distinct from unlimited long-range coherence. These analyses cannot be combined by silently treating corrected and uncorrected widths as equivalent. R4 R5

Kagome studies probe different structures and surfaces. Liang's STM step-edge phase shift suggests $2\times2\times2$ order. Stahl's bulk diffraction reports a low-temperature $2\times2\times2$ structure changing to $2\times2\times4$ near 60 K on warming; its null response to fields up to 10 T does not rule out broken time-reversal symmetry. Li's separate CsV3Sb5 study resolves conjoined $2\times2\times1$ and $2\times2\times2$ charge-order responses, with distinct resonant behavior and a roughly 4 K onset separation at 1 GPa. That separation concerns charge order, not two superconducting transitions. R6 R7 R8

Limits and disputes

A scattering superlattice peak needs electronic, structural and instrumental interpretation; it is not by itself proof of an exclusively electronic instability. STM conductance modulations must be compared across bias, surface termination and bulk probes. Li's surface modulation and bulk-null comparison illustrates why a surface pattern cannot automatically be promoted to a universal bulk phase. The present review does not independently reproduce those sensitivity limits or resolve every reported stacking structure. R6 R8

Mielke's accepted-manuscript abstract reports enhanced internal-field widths in KV3Sb5 below charge ordering, interpreted as time-reversal symmetry breaking. This is a normal-state charge-order claim, not a direct determination of chiral superconducting pairing. Chen's spatially modulated tunnelling signatures motivate a kagome PDW interpretation, but interpretation of the pair field requires additional evidence beyond ordinary charge contrast. Material identity, probe coupling and the superconducting transition must remain explicit. R9 R10

Complete scientific acceptance remains open. Selected methods and figures, historical abstracts and independent line-shape checks do not replace full source-version comparisons, raw diffraction reproduction, all supplementary calibrations or competing-structure assessment. A Hall anomaly, a Fermi-surface pocket or a charge onset below a pseudogap scale cannot alone specify which charge structure causes reconstruction or whether it mediates pairing. R3 R4 R5

Discovery implications

Preserve the measured wave vector and reciprocal units, sample composition, oxygen order, field direction, pressure medium, line shape and resolution treatment. Compare competing-order amplitudes on characterized specimens and distinguish a static order parameter from dynamic spectral weight. Useful predictions should connect a particular charge structure to a testable electronic reconstruction and superconducting response, including results that could falsify the proposed connection. R2 R3 R5 R11

Related topics

Pseudogap, Charge-fluctuation pairing, and Pair-density waves.

Material cases

Values describe individual reports, not a ranking. Unknown means the checked source did not supply the field at this review scope.