Static Phase Wave in Quantum and CDW Systems

• Static phase waves are spatially modulated oscillatory states where the phase of complex order parameters or wavefunctions is pinned, resulting in time-independent modulations.
• In charge density wave systems, they manifest as persistent speckle patterns via resonant x-ray photon correlation spectroscopy, evidencing frozen charge order even amidst superconductivity.
• The phase–amplitude approach in quantum mechanics enables computationally efficient and high-precision evaluation of large-scale wavefunctions by separating rapid oscillations from slow spatial variations.

A static phase wave is a spatially modulated oscillatory state in which the phase of a complex order parameter or wavefunction remains locked (pinned) across some spatial extent, yielding a coherent, time-independent modulation. The concept emerges in different contexts, notably in condensed matter systems exhibiting charge or pair-density waves, and in quantum calculations employing phase–amplitude representations of the solution to the Schrödinger equation. Static phase waves contrast with fluctuating or sliding scenarios, where the phase undergoes temporal evolution. Their realization and detection provide crucial insight into the interplay between order, disorder, and quantum coherence in many-body systems, as well as offering computational advantages in numerical quantum mechanics.

1. Static Phase Waves in Charge Density Wave Systems

In strongly correlated materials such as La$_{2-x}$Ba$_x$CuO$_4$, the charge density wave (CDW) order parameter is conventionally described as

$$\psi(\mathbf{r}) = \psi_0 \, e^{i[\mathbf{Q} \cdot \mathbf{r} + \phi]},$$

where $\psi_0$ is a real amplitude, $\mathbf{Q}$ is the incommensurate wavevector of period modulation, and $\phi$ is a global phase (Thampy et al., 2017). The physical electron density is modulated according to

$$\rho(\mathbf{r}) = \rho_0 + \psi_0 \cos(\mathbf{Q} \cdot \mathbf{r} + \phi).$$

A static phase wave is realized when the phase $\phi$ is spatially pinned by quenched disorder or lattice defects, resulting in a frozen-in modulation of the charge density. The characteristic feature is the lack of temporal evolution or domain wall motion: the phase does not slide or fluctuate dynamically but remains locked, establishing temporally persistent mesoscale charge patterns that can endure even in the presence of competing ground states such as superconductivity. This scenario has been directly observed in La$_{2-x}$Ba$_x$CuO$_4$ ($x=0.11$) via Cu $L$-edge resonant x-ray photon correlation spectroscopy (XPCS), where the speckle patterns from CDW Bragg peaks remain unchanged up to at least 100 minutes, indicating a truly static CDW as opposed to a slowly fluctuating one (Thampy et al., 2017).

2. Phase–Amplitude Representation and the Static Phase Wave

Within single-particle quantum mechanics, static phase waves denote wavefunction decompositions of the form

$\psi(r) = A(r) \sin \phi(r),$

where the amplitude $A(r)$ and phase $\phi(r)$ increase monotonically and vary slowly with respect to the rapidly oscillatory nature of $\psi(r)$. This representation, pioneered by Milne and further developed by Seaton and Peach, separates amplitude and phase evolution within solutions to the one-dimensional time-independent Schrödinger equation:

$$\psi''(r) + [k^2 - V_T(r)]\psi(r) = 0,\quad V_T(r) = L(L+1)/r^2 + V(r).$$

Staticity here refers to the fact that $A(r)$ and $\phi(r)$ are determined solely by spatial position—not time—as $\psi(r)$ itself is a stationary-state solution (Rawitscher, 2014).

The construction of a static phase wave in this formalism involves solving

$A^2(r) \phi'(r) = k,$

yielding $\phi'(r) = k/A^2(r)$. Substitution into the equation for $A(r)$ produces the non-linear Milne equation:

$$A''(r) + \left[k^2 - V_T(r)\right]A(r) - \frac{k^2}{A^3(r)} = 0.$$

This approach provides an economical pathway to accurate, spatially resolved solutions for quantum wavefunctions over large spatial domains, requiring significantly fewer degrees of freedom than direct finite-difference representations (Rawitscher, 2014).

3. Experimental Detection and Quantification

The manifestation of static phase waves in CDW materials has been decisively corroborated by XPCS techniques that leverage the coherent scattering of resonant x-ray beams. Incident x-rays at the Cu $L_3$-edge generate high-contrast speckle patterns at positions corresponding to CDW Bragg reflection. These speckles arise from coherent interference among different spatial regions of phase-pinned CDW order. The speckle's temporal stability is quantified using the normalized intensity–intensity correlation function:

$$g_2(\mathbf{q}, \tau) = \frac{\langle I(\mathbf{q}, t) I(\mathbf{q}, t+\tau)\rangle}{\langle I(\mathbf{q}, t)\rangle^2} = 1 + \beta |F(\mathbf{q}, \tau)|^2,$$

where $F(\mathbf{q}, \tau)$, the intermediate scattering function, characterizes the system's dynamics. A non-decaying $g_2 - 1$ plateau out to the maximum measured lag time, as observed in La$_{2-x}$Ba$_x$CuO$_4$ to $\sim100$ minutes and temperatures up to $0.85 T_{\mathrm{CDW}}$, is unambiguous evidence of static phase wave order (Thampy et al., 2017).

4. Interplay with Competing Orders and Microscopic Coexistence

In La$_{2-x}$Ba$_x$CuO$_4$, static CDW order with pinned phase coexists with superconductivity over a temperature interval $10\,\mathrm{K} < T < 20\,\mathrm{K}$ for $x=0.11$. This coexistence challenges scenarios in which fluctuating CDW order is a prerequisite for unconventional pairing mechanisms, as the direct observation of static order over significant timescales ($\lesssim100$ minutes) and spatial ranges ($\xi \approx 150\,\text{\AA}$ at base temperature) demonstrates that superconductivity and static charge order can intertwine without direct temporal competition (Thampy et al., 2017). The relevant Ginzburg–Landau-type free energy expansion,

$$F[\psi, \Delta] = \alpha(T)|\psi|^2 + \beta|\psi|^4 + a(T)|\Delta|^2 + b|\Delta|^4 + \gamma |\psi|^2 |\Delta|^2 + \ldots,$$

encompasses the possibility of either weak competition or the emergence of more complex intertwined states such as pair-density waves (PDW), wherein superconducting order parameters modulate with the same spatial periodicity as the static CDW, inherently relating to the concept of a static phase wave at the superconducting order parameter level.

5. Computational Construction and Efficiency in Quantum Problems

The phase–amplitude approach, particularly when employing spectral Chebyshev expansions as implemented by (Rawitscher, 2014), allows the construction of static phase wave solutions to the Schrödinger equation over large spatial intervals with high numerical efficiency. For instance, with a long-range $V_3(r)\sim-1/r^3$ potential, the entire range $[0,2000]$ can be represented with only $N=301$ Chebyshev support points. The amplitude $A(r)$ and phase $\phi(r)$ vary slowly, facilitating high-precision evaluation of $\psi(r)$ without resolving every oscillation. A single first-order Seaton–Peach iteration reduces numerical error in the tail of the solution by an order of magnitude compared to standard WKB: at $r\approx2000$, errors are $0.12\%$ for the phase–amplitude approach versus $1.2\%$ for WKB at $k=0.01$ (Rawitscher, 2014).

This static phase wave representation also enables efficient computation of overlap integrals by leveraging the separation of fast and slow oscillatory components inherent in decomposing $\sin \phi_1 \sin \phi_2$ into $\cos(\phi_1 - \phi_2)$ and $\cos(\phi_1 + \phi_2)$. The slowly oscillating part approximates the full integral to within sub-percent error, further reducing computational cost.

6. Physical and Theoretical Implications

The existence of static phase waves, specifically through phase pinning, defines non-trivial mesoscale textures—regions of alternating charge density or pairing strength—which can persist in the presence of macroscopic quantum coherence. In CDW superconductors, pinned phase waves necessitate spatial adaptation of the superconducting state, producing either spatially inhomogeneous or modulated (PDW-like) superconductivity. This texture offers a natural mechanism for reconciling robust, static charge order and global superconducting coherence, accommodating states that are neither simply competing nor trivially coexisting (Thampy et al., 2017).

In quantum computations, static phase wave decompositions not only enable significant computational savings but also yield direct access to physically meaningful quantities such as phase shifts, resonance conditions, and asymptotic behaviors. The monotonic and slow variation of the amplitude and phase compared to the rapidly oscillating wavefunction profoundly aids in analysis and numerical integration over large domains (Rawitscher, 2014).

7. Summary Table: Key Aspects of Static Phase Waves

Context	Formalism/Order Parameter	Defining Criterion
CDW in La$_{2-x}$Ba$_x$CuO$_4$ (Thampy et al., 2017)	$$\psi(\mathbf{r}) = \psi_0 e^{i[\mathbf{Q} \cdot \mathbf{r} + \phi]}$$	$\phi$ pinned by disorder, yielding static modulation
Quantum scattering (Rawitscher, 2014)	$\psi(r) = A(r) \sin \phi(r)$	$A(r),\,\phi(r)$ monotonic, spatially determined only; no explicit time evolution

Static phase waves play a pivotal role in understanding static modulations in correlated electron materials and provide computational efficacy in bulk quantum calculations, presenting a paradigm in which spatial coherence, pinning, and order coexist with robust physical consequences across many-body systems and quantum dynamics alike.