Dimensional Crossover of Charge-Density Wave Correlations in the Cuprates

TL;DR

This paper investigates how magnetic fields influence charge-density wave correlations in cuprates, revealing a transition from short-range, out-of-phase order to long-range, in-phase order driven by disorder and Coulomb interactions.

Contribution

It introduces a theoretical framework explaining the magnetic field-induced crossover in charge-density wave order in cuprates using large-N analysis and Monte-Carlo simulations.

Findings

Long-range charge-density wave order develops above a critical magnetic field.

The transition involves a change from out-of-phase to in-phase correlations along the c-axis.

Results align with experimental observations in YBCO and suggest broader implications for cuprates.

Abstract

Short-range charge-density wave correlations are ubiquitous in underdoped cuprates. They are largely confined to the copper-oxygen planes and typically oscillate out of phase from one unit cell to the next in the c-direction. Recently, it was found that a considerably longer-range charge-density wave order develops in YBCO above a sharply defined crossover magnetic field. This order is more three-dimensional and is in-phase along the c-axis. Here, we show that such behavior is a consequence of the conflicting ordering tendencies induced by the disorder potential and the Coulomb interaction, where the magnetic field acts to tip the scales from the former to the latter. We base our conclusion on analytic large-N analysis and Monte-Carlo simulations of a non-linear sigma model of competing superconducting and charge-density wave orders. Our results are in agreement with the observed phenomenology in the cuprates, and we discuss their implications to other members of this family, which have not been measured yet at high magnetic fields.

Equations

$$|{\psi }_{\mu j}{|}^2+|{\mathbf{\Phi }}_{\mu j}{|}^2=1,$$

$$H$$

$$H_{\mu j}$$

$$G_{\mu {\mu }^{\prime }}^{\alpha }(\mathbf{q},l)$$

$$G_{00}^{\alpha }(0,l)$$

$$G_{01}^{\alpha }(0,l)$$

$$\frac{T_{CDW}}{{\rho }_s}=\kappa r_0^2\sqrt{tU}-\frac{V^2}{2U},$$

$$\frac{B_{CDW}{r}_0^2}{{\varphi }_0}\approx {\ln }^{-2}\left[{\kappa }^2{r}_0^{2}U{\left(\frac{2U}{V^2}\right)}^2\right].$$

$$H$$

$$|{\psi }_{\mu j}{|}^2+\sum _{\alpha =1}^N(n_{\mu j}^{\alpha }{)}^2=1.$$

$$Z=e^{-\beta F}=\int \mathcal{D}\psi \mathcal{D}{n}^{\alpha }\prod _{\mu j}\delta \left[|{\psi }_{\mu j}{|}^2+\sum _{\alpha }(n_{\mu j}^{\alpha }{)}^2-1\right]e^{-S},$$

$$S=\beta H-\int d^{2}r{d}^2{r}^{\prime }\sum _{\mu {\mu }^{\prime }j{j}^{\prime }}\sum _{\alpha \beta }{K}_{\mu {\mu }^{\prime }j{j}^{\prime }}^{\alpha \beta }(\mathbf{r},{\mathbf{r}}^{\prime })n_{\mu j}^{\alpha }(\mathbf{r})n_{{\mu }^{\prime }{j}^{\prime }}^{\beta }({\mathbf{r}}^{\prime }),$$

$$G_{\mu {\mu }^{\prime }j{j}^{\prime }}^{\alpha \beta }(\mathbf{r},{\mathbf{r}}^{\prime })\equiv \overline{⟨n_{\mu j}^{\alpha }(\mathbf{r})n_{{\mu }^{\prime }{j}^{\prime }}^{\beta }({\mathbf{r}}^{\prime })⟩}={-\frac{\delta \beta \overline{F}}{\delta K_{\mu {\mu }^{\prime }j{j}^{\prime }}^{\alpha \beta }(\mathbf{r},{\mathbf{r}}^{\prime })}|}_{K=0}.$$

$$-\beta \overline{F}=\overline{\ln Z}=\underset{m\to 0}{\lim }\frac{\overline{Z^m}-1}{m}\equiv \underset{m\to 0}{\lim }\frac{e^{-\beta F(m)}-1}{m}.$$

$$\overline{F}=\underset{m\to 0}{\lim }\frac{F(m)}{m}$$

$$e^{-\beta F(m)}$$

$$\tilde{S}(m)$$

$${\hat{L}}_{j{j}^{\prime }}^a$$

$${\hat{M}}_{j{j}^{\prime }}$$

$$S(m)$$

$$({\hat{G}}^{-1}{)}_{j{j}^{\prime }}^{\alpha \beta ab}(\mathbf{r},{\mathbf{r}}^{\prime })=\beta {\rho }_s\left[{\delta }_{ab}{\hat{L}}_{j{j}^{\prime }}^a-{\hat{M}}_{j{j}^{\prime }}\right]{\delta }_{\alpha \beta }\delta (\mathbf{r}-{\mathbf{r}}^{\prime }),$$

$$K_{\mu {\mu }^{\prime }j{j}^{\prime }}^{\alpha \beta ab}(\mathbf{r},{\mathbf{r}}^{\prime })={\delta }_{ab}{K}_{\mu {\mu }^{\prime }j{j}^{\prime }}^{\alpha \beta }(\mathbf{r},{\mathbf{r}}^{\prime }).$$

$$\frac{\delta S(m)}{\delta {\sigma }_{\mu j}^a(\mathbf{r})}$$

$$\frac{\delta S(m)}{\delta {\psi }_{\mu j}^{a\ast }(\mathbf{r})}$$

$${\hat{G}}_{j{j}^{\prime }}^{\alpha \beta }(\mathbf{r},{\mathbf{r}}^{\prime })\equiv {\delta }_{\alpha \beta }{\hat{G}}_{j{j}^{\prime }}(\mathbf{r},{\mathbf{r}}^{\prime })={\delta }_{\alpha \beta }\underset{m\to 0}{\lim }\frac{1}{m}\sum _a{\hat{G}}_{j{j}^{\prime }}^{aa}(\mathbf{r},{\mathbf{r}}^{\prime }).$$

$$\beta {\rho }_s\sum _{cl}\int d^2\tilde{r}\left[{\delta }_{ac}{\hat{L}}_{jl}-{\hat{M}}_{jl}\right]\delta (\mathbf{r}-\tilde{\mathbf{r}}){\hat{G}}_{l{j}^{\prime }}^{cb}(\tilde{\mathbf{r}},{\mathbf{r}}^{\prime })=\hat{I}{\delta }_{j{j}^{\prime }}{\delta }_{ab}\delta (\mathbf{r}-{\mathbf{r}}^{\prime }).$$

$${\hat{G}}_{j{j}^{\prime }}^{ab}(\mathbf{r},{\mathbf{r}}^{\prime })=\frac{1}{N_c}\sum _{k_z}\sum _{st}{\hat{G}}_{st}^{ab}{e}^{i{k}_z(j-j^{\prime })c}{\varphi }_s(\mathbf{r}){\varphi }_t^{\ast }({\mathbf{r}}^{\prime }),$$

$$\frac{\beta {\rho }_s}{N_c}\sum _c\sum _{k_z}\sum _{st}\left[{\delta }_{ac}{\hat{\tilde{L}}}_s(k_z)-\hat{\tilde{M}}(k_z)\right]{\hat{G}}_{st}^{cb}{e}^{i{k}_z(j-j^{\prime })c}{\varphi }_s(\mathbf{r}){\varphi }_t^{\ast }({\mathbf{r}}^{\prime })=\hat{I}{\delta }_{j{j}^{\prime }}{\delta }_{ab}\delta (\mathbf{r}-{\mathbf{r}}^{\prime }),$$

$${\hat{\tilde{L}}}_s(k_z)$$

$$\hat{\tilde{M}}(k_z)$$

$${\hat{G}}_{st}^{ab}={\delta }_{st}\left[{\hat{A}}_s(k_z){\delta }_{ab}+{\hat{B}}_s(k_z)\right],$$

$${\hat{G}}_{j{j}^{\prime }}(\mathbf{r},{\mathbf{r}}^{\prime })=\frac{1}{N_c}\sum _{k_z}\sum _s\left[{\hat{A}}_s(k_z)+{\hat{B}}_s(k_z)\right]e^{i{k}_z(j-j^{\prime })c}{\varphi }_s(\mathbf{r}){\varphi }_s^{\ast }({\mathbf{r}}^{\prime }),$$

Full text

The cuprate high-temperature superconductors are in a delicate state of balance between various electronic orders intertwined . In particular, experiments have revealed a subtle interplay between the superconducting (SC) and charge-density wave (CDW) orders. Much of the evidence for the latter, coming from x-ray scattering Ghiringhelli ; Chang ; Achkar1 ; Blackburn13 ; Achkar2 ; Comin1 ; daSilva ; Le-Tacon ; Croft ; Christensen ; Tabis ; Campi ; SilvaNeto ; Comin3 ; Comin-symmetry ; Forgan-structure ; Peng-Bi2201 and nuclear magnetic resonance (NMR) measurements NMR-shortcor , points at short-range, in-plane, CDW order which is in competition with superconductivity. Concretely, the intensity of the CDW scattering peak grows as the system is cooled towards the SC transition temperature, $T_{c}$, and then decreases or saturates upon entering the SC phase. Furthermore, the CDW signal is enhanced when a magnetic field is used to partially quench superconductivity.

However, recent x-ray scattering measurements of YBa2Cu3O6+x (YBCO) Gerber3D ; Chang3D ; Jang3D have detected additional Bragg peaks that are different in several respects from the signal described above. First, the new peaks are much sharper, thus corresponding to considerably longer-ranged CDW correlations. Secondly, whereas both the short-range and long-range CDW peaks share the same incommensurate in-plane wave-vector, the latter appear only along the $b$-direction and at integer $c$-axis wave-vectors (measured in reciprocal lattice units), $l$. This stands in contrast to the bidirectional nature and the half-integer $l$ of the former. Thirdly, as also found by NMR NMR-nature ; NMR-nature-comm and ultrasound measurements ultrasound , the longer-range CDW order sets in only above a magnetic field $H_{3D}\approx 15$T, and at temperatures below $T_{3D}\approx 50$K. The short-range correlations, however, appear already at zero field and survive up to about $T_{ch}=150$K.

Previously, aspects of competing SC and CDW orders were studied via Ginzburg-Landau and non-linear sigma models (NLSM) Zachar ; Demler ; Efetov ; Hayward1 ; Nie ; NLSM-layers ; Meier ; Einenkel-vortex . Those directly related to recent experiments include the CDW temperature dependence Hayward1 , the effects of disorder Nie ; NLSM-layers and of a magnetic field NLSM-layers ; Meier ; Einenkel-vortex . However, a framework in which to understand the complete phenomenology, especially the relation between short and long-range CDW order in YBCO, is lacking. Here, we offer such a scheme by including in our recent NLSM NLSM-layers the structure and couplings of YBCO, by elucidating the different effects of the disorder on the chain layers and on the CuO2 planes, and by going beyond the inter-plane mean-field approximation. The gained insights are then applied to other cuprates.

We use analytical large-$N$ and replica techniques alongside Monte-Carlo simulations to show that the physics is driven by opposing forces. While the Coulomb interaction causes the CDW order to change sign from one plane to the next within a CuO2 bilayer, its relative phase between consecutive unit cells in the $c$-direction is frustrated. On the one hand, the disordered dopant potential on the chain layer tends to induce the same CDW configuration on the two adjacent CuO2 bilayers. On the other hand, such an arrangement is costly from the point of view of their mutual capacitive energy, which is minimized by having them host out-of-phase CDWs.

At zero magnetic field the disorder prevails. CDW puddles that nucleate at locally favorable potential regions on the nearest CuO2 planes to the chain layer, tend to be in phase to each other and opposite to the CDW order that develops on the other CuO2 plane within their bilayer. This leads to a CDW structure factor that is centered near half-integer $l$ with a $c$-axis correlation length, $\xi_{c}$, of about one lattice constant. The in-plane correlation lengths, $\xi_{a,b}$, are longer but still extend over only few wave periods. For disorder fluctuations that are larger than the slight anisotropy induced by the chains, which benefits $b$-axis CDW, the nucleated CDW regions are distributed evenly between the $a$ and $b$ directions, thus leading to scattering peaks in both directions DelMaestro ; Robertson .

A magnetic field introduces vortices into the system at which superconductivity is suppressed and the CDW amplitude is significantly larger than its typical value without the field Hamidian-nematic . This in turn implies that the inter-layer Coulomb interaction and the chain-induced anisotropy play a more important role in the energetic balance governing these regions. As an outcome, the CDW halos formed around a vortex line tend to order with integer $l$ in the $c$-direction and orient preferentially, although not exclusively, along the $b$-axis. The disorder, on its part, interferes with the establishment of inter-halo coherence, both along the vortex and more importantly between different vortices. However, as the field is made stronger vortices move closer together until correlations between the $b$-oriented halos start rapidly increasing. Our calculations indicate that this growth would eventually turn into true long-range order at a critical field. Such a transition is possible since the chain disorder couples to the gradient of the integer-$l$ CDW order, thereby reducing the lower critical dimension to $d_{L}=2$. In contrast, any disorder on the CuO2 layers, for which $d_{L}=4$ Imry-Ma , would smear the transition into a crossover. Nevertheless, as long as this disorder is not too strong the high-field state will still exhibit unidirectional integer-$l$ CDW correlations persisting over long distances in all three dimensions.

The model.– Our NLSM of YBCO consists of $N_{c}$ bilayers, see Fig. 1, hosting complex SC and CDW order parameters, $\psi_{\mu j}({\bf r})$ and $\Phi_{\mu j}^{a,b}({\bf r})$. The latter describe density variations $\delta\rho_{\mu j}=e^{i{\bf Q}_{a}\cdot{\bf r}}\Phi_{\mu j}^{a}({\bf r})+e^{i{\bf Q}_{b}\cdot{\bf r}}\Phi_{\mu j}^{b}({\bf r})+{\rm c.c.}$, along the $a$ and $b$ directions with incommensurate wave-vectors ${\bf Q}_{a,b}$. Here, $j$ is the bilayer index and $\mu=0,1$ corresponds to the bottom (top) layer within a bilayer. Focusing on $T<T_{ch}$ we assume the existence of some type of local order and the competition between its components, as encapsulated by the constraints Hayward1 ; Nie ; NLSM-layers

[TABLE]

where ${\bm{\Phi}}_{\mu j}=(\Phi_{\mu j}^{a},\Phi_{\mu j}^{b})^{T}$. The Hamiltonian reads

[TABLE]

with the SC stiffness, $\rho_{s}$, setting the overall energy scale. We model the Coulomb interaction between CDW fields within a bilayer by a local coupling $\tilde{U}$, and denote the intra-bilayer Josephson tunneling amplitude by $\tilde{J}$. The (weaker) Coulomb interaction and Josephson coupling between nearest-neighbor CuO2 layers belonging to adjacent unit cells are denoted by $U$ and $J$, respectively. The disorder due to the doped oxygens on the chain layers couples via Coulomb interaction to the CDW fields. We include its interaction with the neighboring bilayers, assuming that the coupling to the outer CuO2 planes is reduced by a factor $\gamma$ compared to the coupling to the inner CuO2 planes. The disorder is described by independent random Gaussian fields ${\bm{V}}_{j}=(V^{1}_{j}+iV^{2}_{j},V^{3}_{j}+iV^{4}_{j})^{T}$, satisfying $\overline{V_{j}^{\alpha}}({\bf r})=0$ and $\overline{V_{j}^{\alpha}({\bf r})V_{j^{\prime}}^{\beta}({\bf r}^{\prime})}=V^{2}\delta_{\alpha\beta}\delta_{jj^{\prime}}\delta({\bf r}-{\bf r}^{\prime})$, with the overline signifying disorder averaging. Within a layer the physics is governed by

[TABLE]

where $\lambda\rho_{s}$ is the CDW stiffness and $g\rho_{s}$ is the energy density penalty for CDW ordering. The $\Delta g$ term reflects our assumption that the chain potential favors ordering along the $b$-axis, either directly or via amplification by nematic interactions between the CDW components Nie-nematic ; Hayward1 ; Nie . We consider the extreme type-II limit where the magnetic field, $B$, is uniform and points in the $c$-direction. Therefore, we include only its orbital coupling to the SC order. Finally, $\tilde{\bm{V}}_{\mu j}$ is the disorder potential on the CuO2 layers, which we model by Gaussian random fields with zero mean and $\overline{\tilde{V}_{\mu j}^{\alpha}({\bf r})\tilde{V}_{\mu^{\prime}j^{\prime}}^{\beta}({\bf r}^{\prime})}=\tilde{V}^{2}\delta_{\alpha\beta}\delta_{\mu\mu^{\prime}}\delta_{jj^{\prime}}\delta({\bf r}-{\bf r}^{\prime})$.

Zero field.– Our main interest lies in $k$-space (measured from ${\bf Q}_{a,b}$) CDW correlations encapsulated by the matrix

[TABLE]

with $A$ the layer area. To make analytical progress we increase the number of independent components of $\Phi^{a,b}$ from two to large $N/2$, assume $T\ll\rho_{s}$, and use a saddle-point approximation suppmat . For $B=0$, and $\gamma=\tilde{V}^{2}=0$, $U\ll\tilde{U}$ we find (see Ref. suppmat for the general result)

[TABLE]

where $\epsilon_{\alpha}=g+\Delta g\delta_{\alpha a}+J+\tilde{J}$, and $\epsilon_{\perp}(l)=[U^{2}+\tilde{U}^{2}+2U\tilde{U}\cos 2\pi l]^{1/2}$. Two ordering tendencies are apparent in Eqs. (5),(6). While the temperature terms reach a maximum at integer $l$, the disorder terms peak at half-integer $l$ as long as $\epsilon_{\alpha}>3U+\tilde{U}$, and dominate the correlation matrix if $V^{2}>2UT/\rho_{s}$, which is our case of interest. A small $\Delta g/g$ makes $\epsilon_{a}>\epsilon_{b}$ and introduces a slight tendency towards ordering along the $b$-axis.

To go beyond the limitations of the saddle-point approximation we used Monte-Carlo (MC) simulations of the lattice version of Eqs. (1)-(3). The results are for a system of size $64\times 64\times 32$ (16 bilayers), which is open in the $a$-direction, periodic in the $b$ and $c$-directions and whose parameters are $ga_{0}^{2}=1.1$, $\Delta ga_{0}^{2}=0.1$, $\tilde{J}a_{0}^{2}=0.15$, $Ja_{0}^{2}=0.015$, $\tilde{U}a_{0}^{2}=0.85$, $Ua_{0}^{2}=0.12$, $V^{2}a_{0}^{2}=0.1$, $\gamma=0.15$, $\lambda=1$. Here, $a_{0}$ is the in-plane lattice constant of the coarse grained model, which we assume is roughly the observed CDW wavelength, i.e., about 3 Cu-Cu spacings. Each data point was averaged over 50-70 disorder realizations suppmat . In order to establish contact with the x-ray scattering experiments we obtained the CDW structure factor $S_{\alpha}({\bf q},l)$ by convolving $G$ with the measured CDW form factors Forgan-structure ; suppmat . Fig. 1 depicts the $l$ dependence of the structure factor at the in-plane peaks $S_{a,b}(l)=S_{a,b}(0,l)$. We find that for $B=0$ both $S_{a}$ and $S_{b}$ exhibit a broad maximum centered around $l=0.6-0.7$, whose asymmetry is largely due to the $l$ dependence of the form factors. The correlation lengths $\xi_{b,c}=1/\sigma_{b,c}$, extracted from fits to $e^{-q^{2}/2\sigma^{2}}$, are both of order one lattice constant, see Fig. 2, and the peak height reaches a maximum slightly above $T_{c}\simeq 0.42\rho_{s}$, see Fig. 3(a), all qualitatively consistent with experiments.

The effects of $B$.– The suppression of superconductivity inside magnetically induced vortices facilitates CDW nucleation there. At low $B$ these localized modes form a narrow band due to their small overlap and appear in tandem to the more extended CDW states, which are already present at $B=0$ and produce the $l=1/2$ correlations. Their contribution to $G$ is similar to Eqs. (5),(6) apart of two modifications that determine its $B$ dependence suppmat . First is an overall $B$-linear factor reflecting the number of vortices. Secondly, $\epsilon_{\alpha}$ is given by the bottom of the vortex band, which drifts down with $B$ as CDW halos move closer together. Consequently, the maximum of the disorder terms shifts towards integer $l$ and the entire $G$ increases in magnitude. The effect, however, is very sensitive to the small anisotropy, $\Delta g$. Our MC results, shown in Figs. 1 and 2, demonstrate that the CDW core regions orient predominantly along the $b$ axis, causing $S_{b}$ to form a rapidly growing peak near $l=1$ for fields beyond $B_{3D}=0.06-0.07B_{0}$. Since $B_{0}=\phi_{0}/2\pi a_{0}^{2}\simeq 250{\rm T}$, where $\phi_{0}=\pi/e$ is the flux quantum, this crossover scale corresponds to 15-18T for the set of parameters used by us. At the same time Fig. 1 shows that $S_{a}$ is only weakly modified by the presence of the field.

The agreement of the calculated high-field signal with the observed x-ray phenomenology Gerber3D ; Chang3D ; Jang3D extends beyond its unidirectionality and sharp $B$-dependence. Fig. 2 shows that the increase of $S_{b}(l=1)$ is accompanied by a substantial growth of the $l=1$ correlation lengths. At $B=0.11B_{0}$ - the highest field we could handle without significant finite size effects, $\xi_{c}$ extends over 5 $c$-axis lattice constants and $\xi_{b}=40a_{0}$ is found within the planes. In contrast, the $l=1/2$ correlation lengths change very little with $B$ and remain short. The dichotomy between the two types of correlations is also reflected by their $T$-dependence, depicted in Fig. 3. While the $T=0.5\rho_{s}$ peak of the $l=1/2$ signal turns into a low-temperature saturation in the presence of high magnetic fields Chang , $S_{b}(l=1)$ and its associated correlation lengths exhibit a rapid upturn below $T_{3D}=0.5\rho_{s}$ at large $B$.

Transition to long-range order.– The following Imry-Ma argument Imry-Ma shows that in the absence of in-plane disorder the integer-$l$ CDW can become long-ranged. Consider a domain of linear size $L$ of such a CDW. If the order is constant the interaction of the chain disorder with its neighboring CuO2 planes cancels out. However, if the order varies as $1/L$ along the $c$-axis the averaged squared interaction scales, in $d$ dimensions, as $L^{d-2}$ and can lead to a typical energy gain of $L^{d/2-1}$. Since the elastic energy to create the domains scales as $L^{d-2}$ their proliferation become favorable only in $d\leq 2$. Our large-$N$ analysis reflects this physics suppmat . For $\tilde{V}^{2}=0$ and $V^{2}\ll(r_{0}U)^{2}<U\ll\tilde{U}$ we find that $\xi_{c}$ diverges at

[TABLE]

where $r_{0}$ is the vortex core radius, $t\sim r_{0}^{-2}e^{-b\sqrt{\phi_{0}/Br_{0}^{2}}}$ and $\kappa$,$b$ constants. Hence, the clean system orders for any small magnetic field at low enough temperatures. In the presence of chain disorder a transition occurs only above a critical field, which for $T=0$ is approximately

[TABLE]

Similar expressions for the case of stronger disorder can be found in Ref. suppmat .

The effects of in-plane disorder.– Contrary to the chain disorder, the in-plane disorder couples to each layer separately, leads to a typical energy gain which scales as $L^{d/2}$, and thus prevents long-range order at $d\leq 4$ Imry-Ma . Indeed, our saddle-point equations suppmat do not admit a diverging $\xi_{c}$ when $\tilde{V}^{2}>0$. Fig. 4 shows that as $\tilde{V}^{2}/V^{2}$ approaches 1 the rapid increase of the $l=1$ correlations is averted. We therefore infer that in the physical systems $\tilde{V}^{2}\ll V^{2}$.

The sensitivity to $\Delta g$.– It is difficult to ascertain the magnitude of the anisotropy in $g$. A proxy might be the resistivity anisotropy which is roughly $\rho_{a}/\rho_{b}\approx 1.5$ in the relevant YBCO samples Ando02 . Much larger values have been measured for the ratio of the Nernst coefficients Taileffer-rotational . The presented MC results are for $\Delta g/g=9\%$ and we have checked that deviations from a unidirectional $l=1$ signal commence only around $\Delta g/g=3\%$.

Discussion.– Let us conclude by pointing out few consequences of our model. First, since the enhancement of CDW by a magnetic field is driven by the suppression of superconductivity, one is led to infer the existence of local SC order as long as CDW correlations continue to increase with $B$. Surprisingly, in ortho-VIII YBCO Jang3D the $l=1$ scattering intensity and correlation volume grow up to $H=32$T, well in excess of the resistive critical field $H_{c2}=24$T Hc2 . Hence, an interesting possibility arises that in this system local SC order continues to exist long after global superconductivity is lost.

Secondly, like in YBCO the disorder due to doped oxygens in HgBa2CuO4+δ resides on planes (HgO) Campi shared by consecutive unit cells along the $c$-axis. The arguments presented above would then imply that in this single-layer compound low-field CDW correlations should broadly peak near integer $l$. Zero-field measurements Tabis found CDW peaks at $l=1.12$ and $l=1.25$, but experimental constraints make it currently impossible to determine whether these are the true maxima. In a magnetic field the interaction between CDW halos on neighboring planes is expected to move the scattering peaks towards half-integer $l$. Since HgBa2CuO4+δ is tetragonal, with no dopant order or signs of nematicity, the signal would likely remain bidirectional. On the other hand, in the La2-xSrxCuO4 unit cell each of the two CuO2 planes is separately affected, at least to first approximation, by the Sr disorder on its adjacent LaO layers. Furthermore, consecutive CuO2 planes are offset by half a lattice constant and Coulomb interactions between next-nearest-neighbor planes dominate and lead to half-integer-$l$ peaks at low fields Croft ; Christensen . We then expect a high field to strengthen and sharpen the peaks without shifting their $l$.

Supplementary Material for ”Dimensional Crossover of Charge-Density Wave Correlations in the Cuprates”

.1 A. The model in the large-$N$ approximation

With the aim of applying a saddle-point approximation to the model defined by Eqs. (1)-(3) of the main text, we enlarge the number of components of $\Phi^{a,b}$ from 2 to $N/2$. The CDW order parameters are then described by the real fields $n_{\mu j}^{\alpha}({\bf r})$, with $\alpha=1,\cdots,N/2$ corresponding to $\Phi^{a}$ and $\alpha=N/2+1,\cdots,N$ to $\Phi^{b}$. The Hamiltonian becomes

[TABLE]

where we have set $\Delta g=0$ in order to slightly simplify the following analysis. We will adapt the results to the general case where $g$ depends on $\alpha$ at the end of the calculation. The constraints read

[TABLE]

The partition function

[TABLE]

is defined by the action

[TABLE]

to which we have added a source term that yields the correlation function via

[TABLE]

To calculate the free energy averaged over realizations of disorder, $\overline{F}$, we employ the replica method where we consider $m$ replicas of the original model, and use

[TABLE]

This implies that

[TABLE]

with $F(m)$ defined by

[TABLE]

Integrating over $V_{\alpha}$, $\tilde{V}_{\alpha}$ and analytically continuing to $\bar{\sigma}_{\mu j}^{a}=i\frac{\beta\rho_{s}}{2}\sigma_{\mu j}^{a}$ we find $e^{-\beta F(m)}=\int\mathcal{D}\psi^{a}\mathcal{D}n^{\alpha a}\mathcal{D}\sigma^{a}e^{-\tilde{S}(m)}$ given in terms of the effective action

[TABLE]

where a hat denotes a $2\times 2$ matrix whose indices are $\mu,\mu^{\prime}$, and

[TABLE]

with $\upsilon^{2}={\tilde{V}}^{2}/V^{2}$. Next, we integrate over the CDW fields to obtain $e^{-\beta F(m)}=\int\mathcal{D}\psi^{a}\mathcal{D}\sigma^{a}e^{-S(m)}$, where

[TABLE]

Here

[TABLE]

and

[TABLE]

The integrals over $\psi^{a}_{\mu j}$ and $\sigma^{a}_{\mu j}$ are to be calculated using a saddle-point approximation, which is justified in the limits $N\to\infty$ and $\beta\to\infty$, provided that the disorder is weak and satisfies $V^{2}a^{2}={\cal O}(1/N)$, see Eq. (50) below. Within this approximation, $\beta F(m)=S(m)$, where $S(m)$ is evaluated using the saddle-point configurations, satisfying

[TABLE]

with $\bar{\mu}=1-\mu$. We have neglected ${\cal O}(K)$ terms in the above equations since they lead to an ${\cal O}(K^{2})$ contribution to $S(m)$, which is irrelevant for the purpose of calculating $G$, as follows from Eq. (5). Furthermore, since $G^{-1}$ is diagonal in $\alpha,\beta$ and symmetric in $a,b$, ${\bf r},{\bf r}^{\prime}$, and under exchange of both $j,j^{\prime}$ and $\mu,\mu^{\prime}$ so is $G^{\alpha\beta ab}_{\mu\mu^{\prime}jj^{\prime}}=\delta_{\alpha\beta}G_{\mu\mu^{\prime}jj^{\prime}}^{ab}$. Thus, from Eqs. (5) and (20) one finds

[TABLE]

We will calculate the $2\times 2$ correlation matrix $\hat{G}_{jj^{\prime}}^{ab}$ by assuming a replica-symmetric solution of the saddle-point equations, which is also independent of $j$ and $\mu$, i.e., $\psi_{\mu j}^{a}({\bf r})=\psi({\bf r})$ and $\sigma_{\mu j}^{a}({\bf r})=\sigma({\bf r})$. Under this assumption the operator $\hat{L}_{jj^{\prime}}^{a}=\hat{L}_{jj^{\prime}}$ is also replica symmetric, and $\hat{G}_{jj^{\prime}}^{ab}$ is determined from

[TABLE]

Expanding $\hat{G}_{jj^{\prime}}^{ab}$ in the eigen-basis of $L_{0}=-\lambda\nabla^{2}+g+\sigma({\bf r})$, defined by $L_{0}\phi_{s}({\bf r})=\epsilon_{s}\phi_{s}({\bf r})$,

[TABLE]

and plugging it into Eq. (26) gives

[TABLE]

where $k_{z}$ is quantized in units of $2\pi/N_{c}c$, with $c$ the $c$-axis lattice constant, and

[TABLE]

Eq. (28) is solved by

[TABLE]

leading to the correlation matrix

[TABLE]

with

[TABLE]

where

[TABLE]

and

[TABLE]

whose components are

[TABLE]

Let us comment on the changes incurred in the preceding analysis as a result of an $\alpha$ dependent $g$. In such a case the operator $L_{0}$, appearing on the diagonal of Eqs. (16), (21), and (26), turns into $L_{0}^{\alpha}=-\lambda\nabla^{2}+g_{\alpha}+\sigma({\bf r})$. Its eigenfunctions $\phi_{s}$ are unchanged but the spectrum, $\epsilon_{s}^{\alpha}$, is shifted and acquires $\alpha$ dependence. Consequently, so does $\hat{G}$, which now reads

[TABLE]

where $\hat{A}^{\alpha}_{s}(k_{z})$, and $\hat{B}^{\alpha}_{s}(k_{z})$ are obtained from Eqs. (37) and (39) via the substitution $\epsilon_{s}\rightarrow\epsilon_{s}^{\alpha}$.

.2 B. The zero-field case

In the absence of a magnetic field the saddle-point equations (23,24) possess a constant solution $\psi_{\mu j}^{a}=\psi_{0}$, and $\sigma_{\mu j}^{a}=\tilde{J}+J$. Consequently, the eigenfunctions of $L_{0}$ are plane waves $\phi_{s}({\bf r})=\phi_{{\bf k}}({\bf r})=\frac{1}{\sqrt{A}}e^{i{{\bf k}}\cdot{\bf r}}$, with eigenvalues $\epsilon^{\alpha}_{s}=\epsilon^{\alpha}_{{\bf k}}=\lambda k^{2}+g_{\alpha}+\tilde{J}+J$. Hence the correlation matrix

[TABLE]

takes the form

[TABLE]

Neglecting the intra-cell form factor, with which we deal in Sec. D, and using the fact that the CuO2 planes within a bilayer are separated by approximately $c/3$ the contribution of the $\alpha$ component to the structure factor is given by

[TABLE]

The intensity at the incommensurate peak, for $\gamma=\upsilon^{2}=0$, is

[TABLE]

where $\epsilon^{\alpha}_{0}=g_{\alpha}+\tilde{J}+J$. For $T/\rho_{s}\ll V^{2}/(\epsilon^{\alpha}_{0}-\tilde{U})$, the intensity $S^{\alpha}(0,q_{z})$ is dominated by the $V^{2}$ term and has the form of an asymmetric peak. For $\epsilon^{\alpha}_{0}\gg\epsilon^{\alpha}_{0}-\tilde{U}\gg U$ it becomes $4V^{2}\sin^{2}(q_{z}c/2)\sin^{2}(q_{z}c/6)/(\epsilon^{\alpha}_{0}-\tilde{U})^{2}$, which reaches a maximum at $q_{z}=0.63(2\pi/c)$, with FWHM=$0.42(2\pi/c)$, see Fig. 5.

Before moving on to include the effects of a magnetic field let us consider the saddle-point equation for $\sigma$, Eq. (23), which for the case $g_{\alpha}=g$ reads

[TABLE]

For $\gamma=0$ this can be written as

[TABLE]

where we assume $\epsilon_{0}>U+\tilde{U}$ to avoid divergence of the $k_{z}$ integral. The ${\bf k}$ integral requires ultraviolet regularization, which we achieve by restricting it to a disk of radius $\pi/a_{0}$. Defining the clean mean-field transition temperature

[TABLE]

we obtain

[TABLE]

where it can be checked that the last term is negative. Therefore, disorder reduces $|\psi_{0}|^{2}$ and $T_{c}$. It cannot be too strong otherwise $\psi=0$, and one needs to take into account fluctuations in $\psi$.

.3 C. The system in a magnetic field

In the presence of a magnetic field an Abrikosov lattice of vortices develops. Within each vortex core $\psi$ vanishes linearly with the distance to the vortex center and $\sigma$ is reduced from its value $J+\tilde{J}$ far from the vortex. From Bloch’s theorem we know that

[TABLE]

where ${\bf k}$ is in the magnetic Brillouin zone (MBZ) and $u_{n{\bf k}}({\bf r}+{\bf R})=u_{n{\bf k}}({\bf r})$ for any vector ${\bf R}$ in the Abrikosov lattice. Writing ${\bf r}=\tilde{{\bf r}}+{\bf R}$, where $\tilde{{\bf r}}$ is in the magnetic unit cell (m.u.c) and using the periodicity of $u_{n{\bf k}}$ we obtain from Eqs. (27), (35), and (43) that

[TABLE]

where in the reminder we specify to the case $g_{\alpha}=g$. Let us decompose ${\bf q}={\bf Q}+{\bf q}^{\prime}$, with ${\bf Q}$ a reciprocal magnetic vector and ${\bf q}^{\prime}$ lying in the MBZ. Using $\sum_{{\bf R}}e^{i({\bf k}-{\bf q})\cdot{\bf R}}=\sum_{{\bf R}}e^{i({\bf k}-{\bf q}^{\prime})\cdot{\bf R}}=N_{v}\delta_{{\bf k},{\bf q}^{\prime}}$, where $N_{v}$ is the number of unit cells (vortices) in the Abrikosov lattice we obtain

[TABLE]

For $R\gg r_{0}$, where $r_{0}$ is the core radius, we expect the scattering states of $\hat{L}$ to be close to plane waves (somewhat reduced inside the cores), with a spectrum that is close to the free dispersion. The index $n$ for these states is the band index originating from folding the free dispersion onto the MBZ, and is thus given by a reciprocal wavevector ${\bf K}$, i.e.,

[TABLE]

As a result, $\int_{m.u.c}d^{2}re^{-i{\bf Q}\cdot{\bf r}}u_{n{\bf q}^{\prime}}({\bf r})\simeq\frac{R^{2}}{\sqrt{A}}\delta_{{\bf K},{\bf Q}}$ leading to

[TABLE]

where $A_{1}<1$ is a numerical factor expressing the suppression of the scattered waves inside the cores. Thus, the contribution of the scattering state to the correlation matrix is given approximately by

[TABLE]

where we have used the fact that $\hat{A}_{{\bf Q}{\bf q}^{\prime}}(q_{z})=\hat{A}_{\bf q}(q_{z})$, and similarly for $\hat{B}$.

The reduction in $\sigma$ within the vortex core acts as an attractive potential for the CDW fields. Consequently, one expects that in the presence of a vortex the spectrum of $\hat{L}$ contains in addition to the scattering states also a discrete set of bound states within the core. Our numerical solution of the saddle-point equations confirms that there is a single (normalized) bound state, $\varphi_{v}({\bf r})$, that decays at large distances as $\varphi_{v}({\bf r})\sim\exp(-r/r_{0})$. In the presence of a dilute Abrikosov lattice of vortices, i.e., $R\gg r_{0}$, the small overlap between bound states in neighboring cores leads to the formation of a tight-binding band $\phi_{v{\bf k}}({\bf r})$. For a square Abrikosov lattice (assuming a triangular lattice yields similar results with modified numerical constants), $R=\sqrt{\phi_{0}/B}$ and

[TABLE]

with

[TABLE]

where the bound state eigenvalue, $\epsilon_{v}$, includes the shift due to the change in the effective core potential induced by the other vortices, and where the overlap integral scales as $t\sim r_{0}^{-2}e^{-b\sqrt{\phi_{0}/Br_{0}^{2}}}$ with a constant $b$. Assuming an exponential bound state $\varphi_{v}(r)=(2\pi r_{0}^{2})^{-1/2}e^{-r/2r_{0}}$ we find for $qr_{0}\ll 1$

[TABLE]

which leads, together with $\epsilon_{v}({\bf q}^{\prime})=\epsilon_{v}({\bf q}^{\prime}+{\bf Q})=\epsilon_{v}({\bf q})$ and Eq. (53), to the vortex part of the correlation matrix

[TABLE]

Here, $\hat{A}_{v\bf q}(q_{z})$ and $\hat{A}_{v\bf q}(q_{z})$ are obtained by substituting $\epsilon_{s}=\epsilon_{v{\bf q}}$ in Eqs. (37) and (39).

.4 The transition to long-range CDW order

The expressions for the elements of $\hat{G}_{v}(0,q_{z})$, which determine the vortex contribution to the scattering peak intensity, all share the denominator $(\epsilon_{v}-4t)^{2}-U^{2}-\tilde{U}^{2}-2U\tilde{U}\cos(q_{z}c)$, or its square. For small $q_{z}c$, it can be written as $U\tilde{U}c^{2}(\xi_{c}^{-2}+q_{z}^{2})$, where the inverse correlation length is

[TABLE]

We are interested in calculating the conditions under which $\xi_{c}$ diverges, signaling long-range CDW order with $q_{z}=0$. To this end, we return to the saddle-point equation for $\sigma$,

[TABLE]

The contribution of the scattering states to the left hand side equals $1-|\psi_{0}|^{2}-|\delta\psi({\bf r})|^{2}$, where $\delta\psi$ is appreciable only within the vortex cores. Thus, we conclude that the contribution of the vortex band is

[TABLE]

Next, we integrate the equation over the plane using $\int d^{2}r|\phi_{v{\bf k}}({\bf r})|^{2}=1$ and $\int d^{2}r[|\psi_{0}|^{2}+|\delta\psi({\bf r})|^{2}-\left|\psi({\bf r})\right|^{2}]=N_{v}{\cal C}r_{0}^{2}|\psi_{0}|^{2}$, with ${\cal C}$ a constant. This leads to

[TABLE]

For $\gamma=0$, this can be expressed as

[TABLE]

We evaluate the remaining integral in the limit $\epsilon\equiv(\epsilon_{v}-4t)/(U+\tilde{U})\rightarrow 1$, corresponding to a transition to long-range CDW order with integer $q_{z}c/2\pi$, see Eq. (62). In this limit the integral is dominated by small momenta and we expand $\epsilon_{v{\bf k}}$ to order $k^{2}$. Furthermore, we approximate the MBZ by a disk of radius $\pi/R$. We have checked that these approximations lead only to an overall factor of order 1 relative to an exact numerical evaluation of the integral.

Eq. (66) admits a solution with $\epsilon=1$ provided $\upsilon^{2}=0$. The $\upsilon^{2}$ term, on the other hand, diverges as $1/\sqrt{\epsilon-1}$, precludes such a solution and smears the transition into a crossover. Concentrating on the case $\upsilon^{2}=0$, and assuming $U\ll\tilde{U}$ along with $t\ll\tilde{U}$, one finds from Eq. (66) that long-range order onsets when

[TABLE]

where

[TABLE]

For $t\ll U$ Eq. (67) implies an ordering temperature

[TABLE]

The critical field is given by

[TABLE]

or more explicitly,

[TABLE]

Hence, we conclude that the clean system orders for any small magnetic field at low enough temperatures. In the presence of disorder a transition occurs, i.e., $T_{CDW}\geq 0$, only if the field is sufficiently strong $r_{0}^{2}t\geq(V^{2}/2U)^{2}/\kappa^{2}r_{0}^{2}U$. This condition is compatible with the assumption $t\ll U$ only for weak disorder satisfying $V^{2}\ll(r_{0}U)^{2}$. To maintain the condition of well separated vortices, i.e., $r_{0}^{2}t\ll 1$, it suffices to require in addition $r_{0}^{2}U<1$.

For $U\ll t$ the ordering temperature is

[TABLE]

At low temperatures, such that $T/\rho_{s}\ll(V^{2}/U)/\ln(t/U)$ the critical field is given by

[TABLE]

or

[TABLE]

This result fulfills $U\ll t$ and $r_{0}^{2}t\ll 1$ provided the disorder satisfies $U\gg V^{2}\gg(r_{0}U)^{2}$. At higher temperatures satisfying $T/\rho_{s}\gg(V^{2}/U)/\ln(t/U)$

[TABLE]

where $\Delta t/t\rightarrow 0$ as $r_{0}^{2}U\rightarrow 0$.

.5 D. Making contact with the x-ray scattering experiments

To relate our model to x-ray measurements of the cuprates we assume that each lattice point in the model, ${\bf R}=({\bf r},j)=(m,n,j)*(3a,3b,c)$, corresponds to a supercell comprising of $3\times 3$ YBa2Cu3O6+x  unit cells, containing approximately one CDW oscillation in each direction, with $n_{\mu=0,1}$ corresponding to the amplitude of the CDW in the lower (upper) half of the supercell. The x-ray scattering intensity is proportional to the structure factor

[TABLE]

where ${\bf r}_{i}$ are the equilibrium positions of the ions in the supercell, ${\bf u}_{i}$ are the deviations from these positions and $f_{i}$ are ionic structure factors. Assuming small deviations we find

[TABLE]

Consider, for example, the ordered CDW state along the $b$-axis, whose in-plane wave-vector we approximate by $(0,2\pi/3b)$. The deviations in the lower ($\mu=0$) and upper ($\mu=1$) halves of the supercell [in which we include half of the central Y ion and half of the bottom (top) chain layer] take the form

[TABLE]

where $b_{i}$ is the position along the $b$-axis (in units of $b$) of the $i$th ion. The $u_{i}^{b,c}$ are the displacement amplitudes of the group of ions to which the $i$th ion belong, i.e., Y, Ba, etc. Next, we assume that in the thermally fluctuating state the deviations are proportional to the CDW amplitudes of the NLSM

[TABLE]

Here we chose the sign such that full anti-phase between $n_{0}$ and $n_{1}$ reproduces the ordered CDW state whose sign changes from one plane to the next within a CuO2 bilayer.

Averaging over thermal fluctuations and using $\langle n({\bf R})\rangle=0$ one obtains

[TABLE]

For ${\bf Q}=(0,k,l)*(2\pi/a,2\pi/b,2\pi/c)$ the part which gives a CDW scattering peak is

[TABLE]

Using the fact that $\langle n^{\alpha}n^{\beta}\rangle\propto\delta_{\alpha\beta}$ and independent of $\alpha$ we have

[TABLE]

where the form factor is given by

[TABLE]

.6 E. Disorder averaging of the Monte-Carlo data

We have found that for our relatively large system of 64$\times$64$\times$32 sites disorder averaging of the Monte-Carlo results converged rather quickly. Among the various quantities which we have calculated, the $l$ dependence of the CDW structure factor turned out to be the slowest to converge. Nevertheless, as shown in Fig. 6, apart from occasional isolated points it becomes practically independent of the disorder sample size, once more than 70 disorder realizations are included in the averaging.