Formation of Incommensurate Charge Density Waves in Cuprates

TL;DR

This study reveals that charge density waves in cuprates develop in two stages, with a doping-independent precursor phase that seeds the low-temperature, doping-dependent CDW, shedding light on their complex formation mechanism.

Contribution

It uncovers a two-stage development of CDWs in cuprates, highlighting a precursor phase and its phase mode as fundamental to understanding their electronic ground state.

Findings

Precursor CDW appears at high temperature with a quasi-commensurate wavevector.

The precursor CDW is doping-independent and originates from a phase mode coupled with a phonon.

The low-temperature CDW is strongly doping-dependent and seeded by the precursor phase.

Abstract

Although charge density waves (CDWs) are omnipresent in cuprate high-temperature superconductors, they occur at significantly different wavevectors, confounding efforts to understand their formation mechanism. Here, we use resonant inelastic x-ray scattering to investigate the doping- and temperature-dependent CDW evolution in La2-xBaxCuO4 (x=0.115-0.155). We discovered that the CDW develops in two stages with decreasing temperature. A precursor CDW with quasi-commensurate wavevector emerges first at high-temperature. This doping-independent precursor CDW correlation originates from the CDW phase mode coupled with a phonon and "seeds" the low-temperature CDW with strongly doping dependent wavevector. Our observation reveals the precursor CDW and its phase mode as the building blocks of the highly intertwined electronic ground state in the cuprates.

Equations

$$|Q|=\frac{{ϵ}_0}{1-e^{-E_{\text{g}}/2k_{\text{B}}T}}$$

$$|Q|={ϵ}_0(1-e^{-E_{\text{g}}/2k_{\text{B}}T}).$$

$$H=\sum _{i,j,\sigma }{t}_{ij}{c}_{i,\sigma }^{†}{c}_{j,\sigma }+U\sum _i{n}_{i,\uparrow }{n}_{i,\downarrow }$$

$$\underline{\underline{\chi}}_{\alpha,\beta}({\bf q})=\left(\begin{array}[]{cc}\chi^{\rho,\rho}_{\alpha,\beta}({\bf q})&\chi^{\rho,m}_{\alpha,\beta}({\bf q})\\ \chi^{m,\rho}_{\alpha,\beta}({\bf q})&\chi^{m,m}_{\alpha,\beta}({\bf q})\\ \end{array}\right)\,.$$

$${\underline{\underline{\chi }}}^{ee}(\mathbf{q})=\underline{\underline{{\chi }^0}}(\mathbf{q})+\underline{\underline{{\chi }^0}}(\mathbf{q})\underline{\underline{V^{ee}}}(\mathbf{q})\underline{\underline{\chi }}(\mathbf{q})$$

$$\underline{\underline{V}}_{\alpha,\beta}({\bf q})=\left(\begin{array}[]{cc}\frac{U}{2}&0\\ 0&-\frac{U}{2}\end{array}\right)\delta_{\alpha,\beta}\,.$$

$$\underline{\underline{D}}(\mathbf{q},\omega )={\left[\underline{\underline{1}}+\underline{\underline{\Lambda }}(\mathbf{q})\underline{\underline{D^0}}(\mathbf{q},\omega )\underline{\underline{{\chi }_{ee}}}(q,\omega )\underline{\underline{\Lambda }}(\mathbf{q})\right]}^{-1}\underline{\underline{D^0}}(\mathbf{q},\omega )$$

$${\underline{\underline{\Lambda }}}_{\alpha ,\beta }(\mathbf{q})$$

$${\underline{\underline{D}}}_{\alpha ,\beta }^0(\mathbf{q})$$

$$D_0(\mathbf{q},\omega )=\frac{2{\Omega }_{\mathbf{q}}}{{\omega }^2-{\Omega }_{\mathbf{q}}^2}.$$

$${\Omega }_{\mathbf{q}}$$

$$g_{\mathbf{q}}$$

$$|4a_0-\frac{1}{Q_{\text{CDW}}}|\le \frac{2{\pi }^2}{{\lambda }_{\text{epc}}^{1/2}}\frac{E_{\text{cond}}}{D}$$

$$S$$

$$=k_{B}N[ϵ\text{ln}ϵ-\delta \text{ln}\delta -(ϵ-\delta )\text{ln}(ϵ-\delta )].$$

$$S=k_{B}N\left[\frac{ϵ}{2}\text{ln}\frac{ϵ}{2}-|\delta |\text{ln}|\delta |-\left(\frac{ϵ}{2}-|\delta |\right)\text{ln}\left(\frac{ϵ}{2}-|\delta |\right)\right].$$

Full text

A remarkable phenomenon of the cuprates is the coexistence of multiple nearly-degenerate electronic orders or instabilities that intertwine at low temperature to form the novel electronic liquid which precipitates high-$T_{c}$ superconductivity Fradkin et al. (2015); Lee (2014). While unidirectional charge density waves (CDWs), also known as stripes, have been theoretically predicted for doped Mott insulators Zaanen and Gunnarsson (1989); Poilblanc and Rice (1989); Emery et al. (1990); Castellani et al. (1995) and experimentally discovered in La-based cuprates over two decades ago Tranquada et al. (1995), a full CDW phase diagram for different cuprate systems, as shown schematically in Fig. 1(a), was only established very recently Hücker et al. (2011); Ghiringhelli et al. (2012); Blanco-Canosa et al. (2014); Tabis et al. (2017); Comin et al. (2014); Abbamonte et al. (2005); Croft et al. (2014); Thampy et al. (2014); Miao et al. (2017, 2018); Arpaia et al. . Consistent results are also found in state-of-the-art numerical calculations of realistic 2D $t-J$ and Hubbard models near $1/8$ doping, where stripe ordering or fluctuations are found to be one of the leading electronic instabilities of the ground state Corboz et al. (2014); Huang et al. (2017); Zheng et al. (2017). While this progress indicates a universal CDW mechanism, consensus about the nature of this mechanism has not been reached due to the opposite evolution of the CDW wavevectors with doping in different cuprates families. Figure 1(b) summarizes the doping dependence of the CDW wavevector as determined by diffraction measurements Hücker et al. (2011); Blanco-Canosa et al. (2014); Tabis et al. (2017); Comin et al. (2014); Chaix et al. (2017); Miao et al. (2017, 2018). In the La-based cuprates, such as La2-xBaxCuO4 (LBCO), CDW wavevectors increase with doping and saturate at doping levels beyond $x=0.125$. In the Bi-, Y- and Hg-based cuprates, however, CDW wavevectors monotonically decrease with doping. These observations have motivated different pictures for CDW formation mechanisms based on either real-space local interactions or weak coupling Fermi surface (FS) driven mechanisms Zaanen and Gunnarsson (1989); Poilblanc and Rice (1989); Tranquada et al. (1995); Comin et al. (2015); Shen et al. (2005). However, as is evident from the incommensurate-commensurate crossover in transition metal chalcogenides Grüner (2018), low-temperature ordering wavevectors are not necessarily representative of the CDW formation mechanism, which is instead encoded in the inelastic spectrum above the transition temperature. As we show in Fig. 1c, resonant-inelastic x-ray scattering (RIXS) can probe electronic degrees of freedom via its resonant process Ament et al. (2011); Dean (2015). Together with the improvement of energy resolution, RIXS can thus reveal CDW order and its fluctuations in great detail.

To understand the nature and formation mechanism of the CDW, we use RIXS to study the doping and temperature dependent CDW evolution in LBCOn (n=115, 125 and 155, corresponding to x=0.115, 0.125 and 0.155 in La2-xBaxCuO4, respectively). By carefully tracing the doping and temperature dependent elastic and inelastic CDW signals in the RIXS spectra, we discovered that a doping-independent precursor CDW with quasi-commensurate wavevector is developed first at high temperature. This short-ranged CDW correlation originates from the phase mode of the CDW and “seeds” the long-ranged CDW with strong doping-dependent incommensurate wavevectors at lower temperature. This two-stage CDW evolution uncovers the locally commensurate CDW together with its inelastic excitation as the building block of the charge correlations in the underdoped cuprates and suggests that the doping-dependent incommensurate CDW wavevectors are driven by the subtle balance of intertwined spin, charge and lattice correlations.

We start by revealing the two-stage CDW evolution in LBCO115. Figures 1(d) and (e) show a typical RIXS intensity plot and the integrated RIXS intensity ($\pm 100$ meV with respect to zero energy loss) at 20 K. The strong intensity centered at zero-energy and $|\mathbf{Q}|=0.225$ in reciprocal lattice units (r.l.u.) corresponds to static CDW order in LBCO115. RIXS intensity plots of LBCO115 below 200 meV at 45 and 35 K are shown in Figs. 2(a) and (b), respectively. Representative constant momentum spectra in a wider energy range are shown in Figs. 2(c) and (d), where the well-established dispersionless $dd$ excitations ($\sim 1.7$ eV) and dispersive paramagnon ($200\sim 350$ meV) are observed Miao et al. (2017); Dean et al. (2012); Ghiringhelli et al. (2012); Le Tacon et al. (2011). At 45 K, the RIXS spectra below 100 meV are dominated by dispersive charge excitations (identified by red arrows in Fig. 2a and c) whose intensity quickly fades away below $|\mathbf{Q}|\sim 0.2$ r.l.u. This new feature was not observed in previous RIXS studies of this system, due to poorer energy resolution Dean et al. (2013a); Miao et al. (2017). A zero-energy (0 meV) cut of the intensity plot shows a broad quasielastic peak along the $H$ direction (yellow curve at the bottom of Fig. 2(e)) hereafter referred to as the precursor-CDW peak (pCDW). At higher energy, the peak position of the constant energy cut shifts to higher $\mathbf{Q}$ which may affect the $\mathbf{Q}_{\text{pCDW}}$ in energy integrated diffraction study. This broad peak intensity is completely suppressed when changing the incident photon energy 1.5 eV below the Cu $L_{3}$ edge (gray curve in Fig. 2(e)), thus proving that the signal is dominated by the resonant process. The large inelastic contribution and broad peak width of the pCDW suggest dynamic charge fluctuations as discussed extensively in a different cuprate family recently Arpaia et al. . Intriguingly, Fig. 2(e) shows that $H$ cuts at 50 and 100 meV show stronger spectral weight at larger values of $|H|$ indicating that dynamic charge correlations may tend to exist at higher $|H|$. It is these higher-energy dynamic correlations that drive the motion of the total energy-integrated CDW peak, and the associated phonon softening, to $H=0.272$ r.l.u. at higher temperatures of 90 K, although the worse energy resolution of the previous RIXS measurements was insufficient to separate out this effect Miao et al. (2017, 2018). As we cool down to 35 K an elastic peak emerges on top of the broad dispersive feature and eventually evolves to the intense CDW peak shown in Fig. 1(d). To distinguish these two CDW peaks, we refer to the low temperature peak as the low temperature-CDW (lCDW).

To understand the origin of the inelastic excitation and its connection with the CDW, we calculated the dynamic charge susceptibility, $\chi^{ee}_{\mathbf{q}}(\omega)$, of a phenomenological model that reproduces our observations. This assumes the presence of metallic stripes within a correlated Hubbard model at low temperatures, since phonons are known to have large contributions in the energy range of interest Chaix et al. (2017); Devereaux et al. (2016); Reznik et al. (2006), we also include a phonon mode of energy $\Omega_{\mathbf{q}}$, which couples to the electrons with interaction vertex, $g_{\mathbf{q}}$. Figure 2(f) and (g) show the calculated spectra and the experimental resolution convoluted spectra, respectively. We choose parameters so that the phase mode of the CDW yields an acoustic mode dispersing out from $\mathbf{Q}_{\text{CDW}}=0.25$ r.l.u. and interacts strongly with the phonon mode at low $Q$. This regime of soft phasons was invoked before to explain the optical conductivity Lorenzana and Seibold (2003). Here, the phonon-phason coupling yields the large momentum dependence of the inelastic intensity observed in Fig. 2(a) and (b).

Figure 2(g) shows that the model reproduces quite well the features observed at low temperatures even though disorder is neglected. The sensitivity of a CDW to disorder is dictated by its stiffness to local phase changes i.e. the energy cost to distorting the CDW phase locally, so that it can pin to a point defect Grüner (2018). A stiff CDW will tend to preserve its local phase and will therefore be inefficiently pinned by disorder; whereas a flexible CDW will distort such that it efficiently pinned. The high-temperature signal is consistent with a flexible pCDW that is strongly pinned by disorder while the low-temperature features can be assigned to a small fraction of the CDW which becomes stiff and is therefore inefficiently pinned. This can be seen by noting that the total $q$-integrated scattering from the pCDW is 7 times larger than that from the lCDW Miao et al. (2017). Such a phenomenology explains the concomitant presence of long-range charge order and a well-defined phason mode due to poor pinning. It is worth emphasizing that in the pCDW state the phason mode is still clearly present but yields a broad structure at low energy.

We now explore the doping-dependent evolution of the two-stage CDW, as enabled by higher RIXS throughput Brookes et al. (2018). In Fig. 3, we show the quasielastic RIXS intensity of LBCO115, LBCO125 and LBCO155 along the $H$ and $K$ directions. This is obtained by integrating $\pm 100$ meV respect to the elastic line in order to achieve higher sensitivity than cuts at 0 meV. At 20 K (blue symbols), the lCDW peaks are strongly doping-dependent. The peak intensity is largest in LBCO125 and significantly weaker in LBCO155, consistent with the lCDW dome centered at $1/8$ doping Hücker et al. (2011); Blanco-Canosa et al. (2013); Tabis et al. (2017). A similar trend is shown in the correlation length, $\xi$, defined as the inverse peak half-width-at-half-maximum (1/HWHM), that is largest in LBCO125 and shortest in LBCO155. The peak position, $\mathbf{Q}_{\text{lCDW}}$, increases with doping and saturates for $x>0.125$. Here $x$ is the hole doping. As we warm up, the intensity of the lCDW decreases and disappears at $41\pm$2 K, $55\pm 1$ K, $43\pm 3$ K in LBCO115, LBCO125 and LBCO155, respectively. Near these critical temperatures, the two-stage CDW structure is evident along both the $H$ and $K$ directions.

To quantify the doping and temperature dependence of the two-stage CDWs, we summarize the fitted CDW peak intensity, HWHM and the wavevectors in Figs. 4(a) and (b). Most remarkably, as we show in Fig. 4(b), we discovered that while the wavevectors of the lCDW is strongly doping-dependent, the wavevectors of the pCDW are doping independent and broadly peaked at $|\mathbf{Q}_{\text{pCDW}}|=0.240$ r.l.u. The corresponding real space CDW period, $\lambda_{pCDW}=1/\mathbf{Q}_{\text{pCDW}}\sim 16\AA$, is similar to the extracted correlation length of pCDW, $\xi_{\text{pCDW}}=$18(2), 13(2) and 21(3)Å for LBCO115, LBCO125 and LBCO155, respectively, and suggests the existence of locally commensurate correlations without extended phase coherence. This picture is also in agreement with our theoretical considerations pointing to a “soft” pCDW that is pinned by disorder. As we go on to discuss, these observations have important implications for the CDW phenomena observed in underdoped cuprates.

Following Fig. 1(b), the wavevector of the CDW appears to fall in two categories with distinct doping-dependent trends. Figure 4(c) illustrates the real space stripe CDW mechanism, where by locating holes at the anti-phase SDW domain boundaries, the kinetic energy of the strongly correlated electrons is reduced. In this picture, when both CDW and SDW are static, the CDW wavevector is expected to follow the SDW with a simple $|\mathbf{Q}_{\text{CDW}}|=2\delta_{\text{SDW}}$ relation that is observed in La-based cuprates at low temperature and reconfirmed in our RIXS study. When the SDW is dynamic with a spin gap and no magnetic Bragg peak, the CDW wavevector is expected to unlock from the spin correlations with nearly degenerate wavevectors Miao et al. (2017, 2018); Nie et al. (2017); Huang et al. (2017); Zheng et al. (2017). In the FS-based mechanism, the CDW is determined by FS portions with large density-of-states (DOS), and the free energy is minimized by reducing the DOS near the FS. Since hole doping shifts the chemical potential down in Fig. 4(d), the CDW wavevectors are expected to decrease with doping as has been observed in Bi-, Y- and Hg-based cuprates. Our observations of the pCDW and its phase mode demonstrate that the intrinsic CDW correlations emerge first with doping independent quasi-commensurate periods. This strongly points towards models in which CDW order is driven by local real-space correlations. Similar short-ranged CDW correlations that persist even to room temperature were recently observed in YBa2Cu3O6+δ (YBCO) Wu et al. (2015); Arpaia et al. , Bi2Sr2CaCu2O8+δ (Bi2212) Chaix et al. (2017), La2-xSrxCuO4 (LSCO) Croft et al. (2014); Thampy et al. (2013) and electron doped Nd2-xCexCuO4 da Silva Neto et al. (2018), strongly indicating an ubiquitous pCDW phase in underdoped cuprates. Our results are also compatible with previous STM studies of various cuprate families without magnetic stripe order at low temperature, such as Bi2212 and Ca2-xNaxCuO2Cl2, where CDWs are found to be locally commensurate with large phase slips Hanaguri et al. (2004); Howald et al. (2003); Mesaros et al. (2016). The CDW phase mode and the pCDW thus serve as the “seed” of the lCDW that couples strongly to different types of correlations at low temperature and is dragged to distinct wavevectors. An important prerequisite of this picture is that CDW states with different period are close in free energy. This is indeed supported by early computations Lorenzana and Seibold (2002) and recent state-of-the-art numerical studies of realistic $t-J$ model and 2D Hubbard model near 1/8 doping Corboz et al. (2014); Huang et al. (2017); Zheng et al. (2017), where multiple CDW periods are nearly degenerate in energy. It would be interesting and important for future studies to explore the pCDW and its phase mode in heavily underdoped and overdoped cuprates (e.g. LSCO) and built its connections with the puzzling pseudogap and strange metal phase.

Finally we discuss the temperature-dependent commensurability effect observed in Fig. 4(b). Similar effects have been observed in prototypical stripe ordered La2-xSrxNiO4 (LSNO, $x\sim 1/3$). In these materials, the CDW wavevectors also follow a simple $|\mathbf{Q}_{\text{CDW}}|=\delta_{\text{SDW}}\sim x$ relation at low-temperature and move to $\mathbf{Q}_{\text{CDW}}=$1/3 at high temperature. An entropy-driven self-doping mechanism has been proposed to explain the commensurability effect in LSNO Ishizaka et al. (2004). This model considers the entropy of doped holes as $S=k_{B}\text{ln}(N_{c}$) where $N_{c}$ is the number of configurations for a given concentration of holes. The number of configurations is computed as the number of ways to accommodate indistinguishable particles in boxes representing equivalent sites along the core of the domain wall that can accommodate holes. We expanded the entropy model to our case as described in Appendix D. This requires the commonly applied assumption of ordered holes along the stripe so that half-filled stripes are insulating with zero entropy and satisfy $\mathbf{|Q|}=2x$. At finite temperatures it is convenient either to increase or decrease the incommensurability with respect to the $2x$ value to gain entropy (see Fig. 5). For doping levels below $x=0.125$ the solution with larger incommensurability has lower free energy and the CDW is predicted to be at

[TABLE]

where $\epsilon_{0}\sim 2x$ is the low-temperature incommensurability and $E_{\text{g}}$ is the energy gap due to the secondary order along the stripe. For doping levels higher than $x=0.125$ the computation is more complicated because stripes overlap and inter-stripe interactions become important producing a saturation of the low temperature incommensurability Lorenzana and Seibold (2002). Qualitatively we expect that the solution in which the incommensurability decreases with temperature prevails yielding at low temperatures,

[TABLE]

This simple computation predicts an activated increase (decrease) of the incommensurablity for $x\lessapprox 1/8$ ($x\gtrapprox 1/8$) as indeed found (see Fig. 6).

In summary, we report detailed measurements of the doping and temperature dependent CDW correlations in LBCO. We discovered that CDW order forms from a doping-independent pCDW with quasi-commensurate period and a soft phase mode. Our observation thus uncovers the basic foundation underpinning the emergence of CDW order in the cuprates.

Appendix A Methods

La2-xBaxCuO4 ($x$=0.115-0.155) single crystals were grown using the floating zone method and cleaved in-situ to reveal a face with a $[001]$ surface normal. The wavevectors used here are described using the high temperature tetragonal ($I4/mmm$) space group. The orientation matrix is determined by (002), (101) and (-101) fundamental peaks at 1700 eV.

RIXS measurements were performed at the ID32 beamline of the European Synchrotron Radiation Facility (ESRF). The resonant condition was achieved by tuning the incident x-ray energy to the maximum of the Cu $L_{3}$ absorption peak around 931.5 eV. The scattering geometry is shown in Fig. 1c. $\sigma$ and $\pi$ x-ray polarizations are defined as perpendicular and parallel to the scattering plane, respectively. $H$ and $K$ scans are achieved by rotating the sample around the $\theta$ and $\chi$ axes, without changing $2\theta$, thus changing the in-plane component of the momentum transfer $\mathbf{Q}$ = $\mathbf{k}_{f}$-$\mathbf{k}_{i}$. By doing this, we are assuming that the scattering is independent of $L$, which is reasonable as the inter-layer coupling in the cuprates is known to be weak Wilkins et al. (2011); Hücker et al. (2011); Dean et al. (2013b). All intensities are normalized to beam current and counting time. In this study, we used $\sigma$-polarized incident x-rays and negative $H$ values to enhance charge excitations Ghiringhelli et al. (2012); Miao et al. (2017). In principle one can use the polarization analyzer at ID32 to ensure that the excitation is a pure charge mode ($\Delta S=0$) da Silva Neto et al. (2018). However, the efficiency of this setup is an order of magnitude lower than the standard setup which makes its use very time consuming for the present problem. As a consequence we cannot completely exclude a spin flip component of the dispersing mode, although we consider it very unlikely because of the association of the mode with the charge quasi-elastic scattering. The spectrometer scattering angle (2$\theta$) was fixed at $118^{\circ}$ such that $L\approx 1.5$ and the total instrumental energy resolution (full-width at half maximum) was set to 70 meV to increase the counting rate. The quasi-elastic intensity was obtained by integrating the RIXS spectrum in an energy window of $\pm 100$ meV around 0 meV.

Appendix B Charge excitations of stripes coupled to phonons

Our calculations are based on the single-band Hubbard model

[TABLE]

where we include nearest ($\sim t$) and next-nearest neighbor ($\sim t^{\prime}$) hopping. Stripe solutions are evaluated within Hartree-Fock (HF) and we calculate binding energies with respect to the homogeneous antiferromagnet (AF) for a configuration where the domain wall of the AF order parameter is bond-centered. Within the HF approximation site-centered stripes involve paramagnetic sites with charge density $n$ and the associated energy cost $\sim Un^{2}/4$ makes them energetically unfavorable with respect to bond-centered configurations for large $U$. This is not anymore the case if correlations beyond the HF approximation are taken into accountSeibold and Lorenzana (2004). Here, for simplicity, we keep the HF approximation but choose parameters $U/t=4$ and $t^{\prime}/t=-0.25$ which reproduce the “Yamada-plot” Yamada et al. (1998), i.e. the low-temperature relation between spin incommensurability $\delta_{\text{SDW}}$ and doping, $x=\delta_{\text{SDW}}$ (see Fig. 4(c)).

Excitations on top of the mean-field stripes are computed with the random phase approximation (RPA). The striped ground state couples charge fluctuations $\delta\rho_{\bf q}$ which differ by multiples $\alpha$ of the stripe modulation wave-vector ${\bf Q}_{s}$ with $\alpha$ an integer. Moreover charge fluctuations are coupled with fluctuations of the magnetization $\delta m_{{\bf q}+\alpha{\bf Q}_{s}}$ so that for each $\alpha$, $\beta$ the susceptibility is a $2\times 2$ matrix

[TABLE]

The total susceptibility matrix is then of dimension $2\lambda_{mag}\times 2\lambda_{mag}$ where $\lambda_{mag}$ is the magnetic periodicity (in units of the lattice spacing). The corresponding RPA equation reads

[TABLE]

with the interaction given by

[TABLE]

Upon including also the coupling to lattice fluctuations (vertex $g_{\bf q}$, frequency $\Omega_{\bf q}$) the renormalized phonon propagator can be obtained from

[TABLE]

where $\underline{\underline{\chi_{ee}}}({\bf q},\omega)$ is evaluated from Eq. (4). The vertex and phonon Greens function matrices are given by

[TABLE]

with the bare phonon Green function

[TABLE]

The phonon propagator can then be used to compute the phonon contribution to RIXS following the approach of Ref. Devereaux et al., 2016

In the main part of the paper we show results for a longitudinal acoustic phonon with frequency and coupling given by

[TABLE]

The stripe phason mixes phonons which differ by a reciprocal lattice vector of the stripe lattice. This coupling is particularly strong at the stripe momentum ${\bf Q}_{\text{CDW}}=(0.25,0)$ where it can induce a quasicritical mode due to a change of the respective stability of bond- and site centered stripes similar to Ref. Lorenzana and Seibold (2003). This mode is shown in Fig. 1 (g,f) for $\Omega_{0}/t=0.42$ and $g_{0}/t=0.73$.

We shall note that (i) $\Omega_{0}$ is the “bare” phonon energy, which will be renormalized by the electron-phonon coupling to an energy of order 70-80 meV for $t\sim 0.3$ eV; (ii) the optical phonon is also active in the energy range of interest, in this case the prominent asymmetric intensity distribution (Fig. 2a and b) is possibly caused by the more complicated cross-section effect of the RIXS process Ament et al. (2011), which we did not take into account in our model calculations.

Appendix C Commensurate vs incommensurate CDW

A commensurate CDW with period $Ma_{0}$ is know to have a strong lattice effect. Charge modulations mix electronic states with momentum $k+nQ_{\text{CDW}}$ ($n$ is an integer) and yields an additional phase dependent condensation energy Lee et al. (1974). As temperature changes, this additional commensurate energy may thus drive an incommensurate to commensurate crossover. In mean field theory for a 1D CDW, the approximate crossover condition for $M=4$ is formulated as

[TABLE]

where $E_{\text{cond}}=\frac{1}{2}n(\epsilon_{\text{F}})\Delta_{\text{CDW}}^{2}$ is the phase independent CDW condensation energy, $\Delta_{\text{CDW}}$ is the CDW gap, $D$ is the cut-off energy close to the bandwidth or Fermi energy and $\lambda_{\text{epc}}$ is a dimensionless electron-phonon coupling constant Grüner (2018). Evidence for this effect has been observed in conventional CDW materials, such as K0.3MoO4, TaS3 and NdSe3 Grüner (2018), where the CDW wavevector is temperature dependent and becomes commensurate at base temperature. This CDW evolution differs from our observations, where the commensurate CDW forms at high-temperature and persists to low-temperature.

It is worth to note that the pCDW is short-ranged without long range phase coherence. Since the multiple CDW periods are nearly degenerate in energy Corboz et al. (2014); Huang et al. (2017); Zheng et al. (2017), it is reasonable to expect that multiple CDW periods coexist with $\lambda=4a_{0}$ being statistically dominated. This might be the reason of why $\mathbf{Q}_{\text{pCDW}}$ is slightly off 0.25 r.l.u. If possible, it would be interesting to directly check this in a future STM study.

Appendix D Entropy model for cuprate stripes

Ishizaka et al Ishizaka et al. (2004) considered a successful model to explain the shift of incommensurability with temperature in nickelates. Here we first briefly review their model. They consider the entropy of doped holes as $S=k_{B}\text{ln}(N_{c}$) where $N_{c}$ is the number of configurations for a given concentration of holes. The number of configurations is computed as the number of ways to accommodate indistinguishable particles in boxes representing equivalent sites along the core of the domain wall that can accommodate holes. Nickelates have insulating stripes at $T=0$. For filled stripes, there is only one configuration ($N_{c}=1$) and $S=0$. If the distance between the stripes is decreased at fixed $x$ there are not enough holes to fill completely all stripe core sites. Calling $\delta$ the concentration of electrons, the incommensurability, $\epsilon$, is now determined by the total number of “domain wall sites” or boxes being occupied by holes (concentration $x$) or electrons (concentration $\delta$), namely $\epsilon=x+\delta$. If there are $N$ total Ni sites in the system the entropy is

[TABLE]

Here we used the relation $\text{ln}N!=N\text{ln}N-N$ for $N\rightarrow\infty$. The computation is completed by postulating that the total energy is $E=\mu^{-}\delta N$ where $\mu^{-}$ is the energy to remove holes. Notice that this expression holds only for $\delta>0$. For hole addition a different energy is involved because the stripes are filled and the AF regions have to accommodate the holes. We call that energy $\mu^{+}$. The fact that $\mu^{-}$ and $\mu^{+}$ are different means simply that the filled stripe is an insulator and there is a jump in the chemical potential around $\delta=0$. To generalize this model to cuprates one should first identify the particles and the boxes. This is less trivial than in nickelates. For vertical stripes as in cuprates, if $d$ is the distance among domains (in units of the lattice constant $a$) the charge incommensurability is $\epsilon=1/d$. The first model of stripes Zaanen and Gunnarsson (1989) assumed insulating stripes as in nickelates and this leads to $\epsilon=x$. However, in the cuprates, $\epsilon=2x$ is observed for $x<1/8$ Tranquada et al. (1995); Hücker et al. (2011) which leads to half-filled stripes. From the theory side, a more accurate computation Lorenzana and Seibold (2002) indeed predicted half-filled stripes in accord with experiment. On the other hand, metallic half-filled stripes pose a problem for the entropy model since the half-filled system has maximum entropy. Therefore, the temperature will only stabilize more this configuration and the incommensurability would be independent of temperature in contradiction with experiment. However, as we show in Fig. 1(a) the CDW is enhanced around $x=1/8$. It was proposed by White and Scalapino White and Scalapino (2003) that this 1/8-anomaly is due to the tendency of stripes to develop addition hole ordering along the stripe Lorenzana and Seibold (2002). Indeed, assuming stripes in neighboring planes are perpendicular to each other, the Coulomb potential of one plane favors a half-filled CDW along the stripe in the next plane only at $x=1/8$ consistent with the increased stability of the CDW at that doping.

We can assume that decreasing the doping this configuration is still favorable as suggested by mean-field computations which picture the secondary CDW along the stripe as a lattice of Copper pair singletsBosch et al. (2001). In strong coupling, the pattern along the stripe is 00$\updownarrow\updownarrow$00$\updownarrow\updownarrow$, where 00 and $\updownarrow\updownarrow$ represent holes and disordered spins respectively. Notice that this has different periodicity than the pattern often assumed, $\updownarrow$0$\updownarrow$0$\updownarrow$. Since at $T=0$ this state would be nominally insulating we assume again that the energy to add or remove hole is different, i.e. the pattern of site diagonal energy is assumed to be $\mu^{-}\mu^{-}\mu^{+}\mu^{+}\mu^{-}\mu^{-}\mu^{+}\mu^{+}$. As for the nickelates, this configuration has zero entropy. We now compute the entropy associated with an increase of the incommensurability i.e. a decrease of $d$ at fixed $x$. The incommensurability in this case satisfies $x+\delta=\epsilon/2$ where $\delta$ is the concentration of extra electrons. The entropy reads:

[TABLE]

where $N$ is the number of Cu sites and for simplicity we neglected the entropy due to the spin degrees of freedom which does not change the physics.

If we consider a decrease of the incommensurability $x+\delta=\epsilon/2$ is still valid if we allow $\delta$ to be negative and interpret $-\delta$ as the concentration of holes added to $\mu^{+}$ sites. For fixed $\epsilon$, the entropy of the ordered half-filled stripe is symmetric with respect to adding or removing holes so Eq. (9) holds as written with the modulus and the energy can be written as $e=|\delta|E_{g}/2$ where we took $\mu^{\pm}=\pm E_{g}/2$ so the free energy is also symmetric. Notice however that the latter has to be minimized with respect to $\epsilon$ at fixed $x$ which is not anymore symmetric. Indeed, there are two solutions which minimize the free energy at finite temperature as shown in Fig. 5 having either $\epsilon<2x$ or $\epsilon>2x$ and deviating from the zero temperature solution $\epsilon=2x$. The solution in which the incommensurability increases with temperature has lower free energy and leads to Eq. 1. For $x>1/8$ the interaction between domain walls has to be taken into accountLorenzana and Seibold (2002). A detailed theoretical study is left for future work as it goes beyond our present scope. In particular it would require adding additional terms to the energy that makes the low-temperature incommensurability to saturate at $\epsilon\sim 1/4$ and include the effect of Fermi surface wrapping which frustrates the secondary order along the stripe Anisimov et al. (2004). For simplicity, here we neglect these effects and simply assume the solution in which the incommensurability decrease with temperature is favored due to the lower energetic cost. This leads to Eq. 2 for doping larger than 1/8.

We shall note our entropy model can qualitatively explain the incommensurate-commensurate crossover below or near the lCDW transition temperature, it, however, does not yield the saturation of $\epsilon_{0}$ as a function of doping or temperature due to the crude approximations. Figure 6 shows a fit of experimental data below 60 K by using Eq. 1 and Eq. 2. We also note that experimental studies of YBCO suggest that the local commensurate period is more consistent with $3a_{0}$ Wu et al. (2015), which might be due to the special chain structure that favors a different period. A more sophisticated model that incorporating the unidirectional field may be needed to explain the result in YBCO.