Superconducting correlations induced by charge ordering in cuprate superconductors and Fermi arc formation

TL;DR

This paper presents a comprehensive model linking charge order, pseudogap phenomena, and superconductivity in cuprates, reproducing key experimental temperature dependencies and elucidating the role of charge density waves in Fermi arc formation.

Contribution

It introduces a generalized phase separation model that connects charge order, pseudogap, and superconductivity, explaining Fermi arc formation and reproducing experimental doping dependencies.

Findings

Charge order modulates superconductivity via localized charge regions.

The model reproduces doping dependence of pseudogap and superconducting transition temperatures.

Fermi arcs arise from local phase fluctuations in the model.

Abstract

We have developed a generalized electronic phase separation model of high-temperature cuprate superconductors that links the two distinct energy scales of the superconducting (SC) and pseudogap (PG) phases via a charge-density-wave (CDW) state. We show that simulated electronic-density modulations resembling the charge order (CO) modulations detected in cuprates intertwine the SC and charge orders by localizing charge and providing the energy scale for a spatially periodic SC attractive potential. Bulk superconductivity is achieved with the inclusion of Josephson coupling between nanoscale domains of intertwined fluctuating CDW and SC orders, and local SC phase fluctuations give rise to the Fermi arcs along the nodal directions of the SC gap. We demonstrate the validity of the model by reproducing the hole-doping dependence of the PG onset temperature $T^*$, and the SC transition temperature $T_c$ of ${\rm YBa_2Cu_3O_y}$ and ${\rm Bi_{2-y}Pb_ySr_{2-z}La_zCuO_{6+\delta}}$. The results show that the periodicity of the CDW order is controlled by the PG energy scale, and the hole-doping dependence of the SC energy gap is controlled by the charge ordering free energy.

Tables

Table 1. Table 1: Experimental data for Δ SC subscript Δ SC \Delta_{\rm SC} and Δ PG subscript Δ PG \Delta_{\rm PG} determined from universal curves that describe Y123 and a number of other high- T c subscript 𝑇 𝑐 T_{c} cuprate superconductors (Ref. Huefner  et al. , 2007 ). An experimental value for Δ SC subscript Δ SC \Delta_{\rm SC} at p = 0.12 𝑝 0.12 p=0.12 is omitted, since deviations from the universal curve are expected for Y123 near 1/8 hole doping, where T c subscript 𝑇 𝑐 T_{c} plateaus. The value of T ∗ ≈ 278 superscript 𝑇 278 T^{*}\approx 278 K at p = 0.09 𝑝 0.09 p=0.09 is estimated from a linear extrapolation of data in (Ref. Vig  et al. , 2016 ). We tune the scaling factors explained in the text to yield the blue and green values at optimum doping. Red values are calculated with the same parameters.

	$p=0.16$	$p=0.12$	$p=0.09$
${\lambda }_{\mathrm{CO}}\left(a_0\right)$Blanco-Canosa et al. (2014); Hücker et al. (2014); Comin et al. (2015a)	$3.49\pm 0.16$	$3.15\pm 0.16$	$3.00\pm 0.16$
$⟨V_{\mathrm{GL}}\left(𝐫\right)⟩$	-0.156	-0.110	-0.105
${\rho }_b\left(1.05T_c\right)(\mu \Omega$ cm)Segawa and Ando (2001)	$\approx$ 40	$\approx$ 50	$\approx$ 70
${\Delta }_{\mathrm{SC}}$Huefner et al. (2007) (meV)	$42\pm 2$	-	$24\pm 1$
$⟨{\Delta }_d\left(𝐫\right)⟩$ (meV	43.8	25.5	23.3
$T_c$ (K)Segawa and Ando (2001); Daou et al. (2010)	93.4	66	55
	92.0	68.1	55.6
${\Delta }_{\mathrm{PG}}$Huefner et al. (2007)	76	104	124
	76	93.4 $\pm$ 9	103.3 $\pm$ 9
$T^{\ast }$Daou et al. (2010)	170	232	278
	170	209 $\pm$ 19	231 $\pm$ 21

Table 2. Table 2: Experimental data for Δ SC subscript Δ SC \Delta_{\rm SC} and Δ PG subscript Δ PG \Delta_{\rm PG} from STM with the indicated T c subscript 𝑇 𝑐 T_{c} values Wise  et al. ( 2008 ) . The hole doping p 𝑝 p was determined from the T c subscript 𝑇 𝑐 T_{c} versus p 𝑝 p relationship obtained by X-ray absorption experiments on (Pb, La)-Bi2201 and La-doped Bi2201 Schneider  et al. ( 2005 ) . The in-plane resistivity ρ ab subscript 𝜌 ab \rho_{\rm ab} data area for La-doped Bi2201 Ono and Ando ( 2003 ) and T ∗ superscript 𝑇 T^{*} data are from intrinsic tunneling measurements Yurgens  et al. ( 2003 ) . We tune the scaling factor explained in the text to yield the blue and green values at optimum doping. Red values are calculated without any extra parameters.

	$p=0.16$	$p=0.12$	$p=0.09$
${\lambda }_{\mathrm{CO}}\left(a_0\right)$Wise et al. (2008)	$6.2\pm 0.2$	$5.1\pm 0.2$	$4.5\pm 0.2$
$⟨V_{\mathrm{GL}}\left(𝐫\right)⟩$	-0.1022	-0.1018	-0.1021
${\rho }_b\left(1.05T_c\right)(\mu \Omega$ cm)Ono and Ando (2003)	18.3	24.8	28.3
${\Delta }_{\mathrm{SC}}$Wise et al. (2008) (meV)	15	-	-
$⟨{\Delta }_d\left(𝐫\right)⟩$ (meV	15.8	13.2	9.2
$T_c$ (K)Wise et al. (2008)	35	32	25
	35.2	32.5	24.7
${\Delta }_{\mathrm{PG}}$Wise et al. (2009)	30$\pm$12	45$\pm$15	68$\pm$20
	30	44.3	54.2
$T^{\ast }$Yurgens et al. (2003)	241	355	446
	241	327.9	427.4

Equations

$$f(u)=\frac{1}{2}\epsilon |\nabla u{|}^2+V_{GL}(u,T),$$

$$V(T,p)=V(0,p)[1-T/T_{\mathrm{PS}}(p){]}^2,$$

$$⟨E_{\mathrm{J}}(p,T)⟩=\frac{\pi \hslash ⟨{\Delta }_d(\mathbf{r},T)⟩}{2e^2{R}_{\mathrm{n}}(p)}\tanh \left[\frac{⟨{\Delta }_d(\mathbf{r},T)⟩}{2k_{\mathrm{B}}T}\right]$$

$$⟨{\Delta }_d(p,T,\varphi )⟩=$$

$$\frac{|\cos (2\varphi )|}{\Delta {\Phi }_j(p,T,\varphi )}{\int }_0^{\Delta {\Phi }_j(p,T,\varphi )}⟨\Delta (p,T)⟩\cos (\Phi )d\Phi$$

$$=⟨\Delta (p,T)⟩|\cos (2\varphi )|\frac{1}{\Delta {\Phi }_j(p,T,\varphi )}\sin [\Delta {\Phi }_j(p,T,\varphi )].$$

$$R=\frac{|\cos (2\varphi )|}{\Delta \Phi (p,T,\varphi )}\times \sin [\Delta {\Phi }_j(p,T,\varphi )]$$

$$\frac{\partial u}{\partial t}=-\nabla .J=-M{\nabla }^2\left[{\epsilon }^2{\nabla }^{2}u+\frac{\partial V_{\mathrm{GL}}}{\partial u}\right],$$

$$H$$

$$\left(\begin{array}{cc}K& {\Delta }_d(\mathbf{r})\\ & \\ {\Delta }_d^{\ast }(\mathbf{r})& -K^{\ast }\end{array}\right)\left(\begin{array}{c}{u}_n(\mathbf{r})\\ \\ v_n(\mathbf{r})\end{array}\right)=E_n\left(\begin{array}{c}{u}_n(\mathbf{r})\\ \\ v_n(\mathbf{r})\end{array}\right)$$

$$K{u}_n(\mathbf{r})$$

$${\Delta }_d{u}_n(\mathbf{r})$$

$${\Delta }_d(\mathbf{r})=$$

$$p(\mathbf{r})=1-2\sum _n[|u_n(\mathbf{r}){|}^2{f}_n+|v_n(\mathbf{r}){|}^2(1-f_n)],$$

Full text

I Introduction

Experiments using different methods have established the occurrence of short-range, incommensurate static CDW correlations in a variety of high-temperature SC cuprates Chang et al. (2012); Ghiringhelli et al. (2012); Comin et al. (2014); Blanco-Canosa et al. (2014); Hücker et al. (2014); da Silva Neto et al. (2015); Comin et al. (2015a, b); da Silva Neto et al. (2014); Fujita et al. (2014); Wise et al. (2008, 2009); Hoffman et al. (2002); Howald et al. (2003); Vershinin et al. (2004); Hanaguri et al. (2004); McElroy et al. (2005). With the exception of La-based cuprates in which CDW order is accompanied by spin order, the charge order (CO) observed in different cuprate families appears to be similar. In zero magnetic field, the CDW order is essentially two-dimensional. The wave vector of the CDW order is parallel to the Cu-O bond directions along the a and b axes, and decreases in magnitude with increased charge doping. While much of the experimental data cannot distinguish between checkerboard (bidirectional) or alternating stripe (unidirectional) CO, recent resonant X-ray scattering (RXS) experiments on underdoped ${\rm YBa_{2}Cu_{3}O_{7}}$ (Y123)Comin et al. (2015a) and an analysis of scanning tunneling microscopy (STM) data for ${\rm Bi_{2}Sr_{2}CaCu_{2}O_{8+\delta}}$Hamidian et al. (2016) indicate that the inter-unit-cell character is one of segregated or overlapping unidirectional charge-ordered stripes. Furthermore, it has been found that the CDW order possesses a $d$-wave intra-unit-cell symmetry with the modulated charge primarily on the O-$2p$ orbitals linking the Cu atoms Comin et al. (2015b); da Silva Neto et al. (2014); Fujita et al. (2014). Since the SC order parameter also has $d$-wave symmetry, this local charge or bond order symmetry supports theoretical proposals that suggest the charge and SC order parameters are intimately intertwined. Some attribute the $d$- wave CO symmetry to quasiparticle scattering by antiferromagnetic (AF) fluctuations near a metallic quantum critical point, which also gives rise to the d-wave superconductivitySachdev and La Placa (2013); Efetov K. B. et al. (2013); Hayward et al. (2014). Alternatively, it has been proposed that CDW order in cuprates is a consequence of a pair-density wave (PDW) phase, in which the SC order parameter is periodically modulated in space due to the Cooper pairs having finite momentumPépin et al. (2014); Fradkin et al. (2015); Wang et al. (2015a, b).

The aim of our work is to establish a quantitative link between the inter-unit-cell dependence of the CO resolved by RXS and imaged in real space by STM, and the energy gaps of the PG and SC phases. Our model is based on an intrinsic propensity for mesoscale electronic phase separation below an onset temperature $T_{\rm PS}$ that follows the hole-doping dependence of the PG temperature $T^{*}$. This picture is similar to that previously advocated by Fradkin and KivelsonFradkin and Kivelson (2012). We presume the onset of fluctuating CDW order domains at $T^{*}$ , where STM measurements on ${\rm Bi_{2}Sr_{2}CaCu_{2}O_{8+\delta}}$ have detected the emergence of charge stripes that extend into the overdoped regimeParker et al. (2010). The short-range static CO that has been observed by X-rays at a lower temperature $T_{\rm CO}\leq T^{*}$ is assumed to be confined to local regions where fluctuating CDW order has become pinned by disorder. Contrary to this assumption, we note that in ${\rm HgBa_{2}CuO_{4+\delta}}$ (Hg1201) CDW order observed by X-ray scattering vanishes already well below optimal dopingTabis W. et al. (2017). This seems to be due to the presence of pairs of interstitial oxygens within the same unit cell specific to Hg1201. Although not captured by our model, it is also important to recognize that the pseudogap region marks the onset of an intra-unit-cell magnetic orderFauqué et al. (2006); Li Y. et al. (2008), a true phase transition that modifies ultrasonic wavesShekhter Arkady et al. (2013), an increase in antiferromagnetic correlationsChan M. K. et al. (2016) and global inversion-symmetry breakingZhao L. et al. (2017).

Another important ingredient of our model is the experimental observation that the CDW periodicity is independent of temperature, leading us to surmise that the CDW periodicity is set by the onset of the PG at $T^{*}$ . This infers that the CDW order is a consequence of the PG formation. At low doping ($p\leq 0.12$) where $T_{\rm CO}$ decreases with decreasing doping, CDW order is potentially suppressed by a slowing down of spin fluctuations and a tendency toward static SDW order. Compatible with experimental signatures of pairing or SC correlations persisting above $T_{c}$Corson et al. (1999); Xu Z. A. et al. (2000); Gomes et al. (2007); Yang H.-B. et al. (2008); Dubroka et al. (2011); Li et al. (2005); Wang et al. (2005), our model shows that CDW order in the PG regime may induce SC domains that grow and connect to establish bulk superconductivity at $T_{c}$.

II Simulation of the charge-ordered state

Our approach is to first simulate spatial modulations of the electronic structure resembling experimentally resolved inter-unit-cell CO modulations, using the time-dependent Cahn-Hilliard (CH) differential equationCahn and Hilliard (1958). Besides generating the desired CDW order, the CH approach yields the associated free-energy modulations, which we assume scales with a periodic attractive potential in the subsequent SC calculations. The starting point is the introduction of a time-dependent conserved order parameter associated with the local electronic density, $u({\bf r},t)=[p({\bf r},t)-p]/p$, where $p$ is the average charge density and $p({\bf r},t)$ is the charge density at a position ${\bf r}$ in the plane. The Ginzburg-Landau (GL) free energy density of the system is of the form

[TABLE]

where ${V_{\rm GL}}(u,T)=-\alpha[T_{\rm PS}-T]u^{2}/2+B^{2}u^{4}/4+...$ is a double-well potential that characterizes the electronic phase separation below $T_{\rm PS}$. The parameters $\alpha$ and $B$ are constants, and $\varepsilon$ controls the spatial separation of the charge-segregated patches. The CH equation obtained from the continuity equation for the local free energy current density ${\bf{\rm J}}=-{\rm M}\nabla\mu$ (where M is the charge mobility and $\mu=\partial f/\partial\mu$ is the chemical potential). Its solution is described in the Apendix I on charge ordering simulations. For each time step the CH equation is solved for $u({\bf r},t)$, and $p({\bf r},t)$ is obtained. We adjust the parameters of the free energy such that when the periodicity of $p({\bf r},t)$ matches that of the experimentally observed CDW order, the calculation is stopped and the solution is taken to be the spatially-dependent static electronic density $p({\bf r})$. Since the method described here does not generate an intra-unit-cell CO symmetry, it is applicable to systems that have SC pairing and CO symmetries other than d-wave.

Figure 1(a) shows a simulation of alternating planar domains of $90^{0}$-rotated charge stripes with an intra-domain periodicity compatible with RXS data for detwinned Y123 at $p=0.12$[Comin et al., 2015b]. Within each domain the CO wavelength is $\lambda_{\rm CO}=3.15a_{0}$, where the in-plane lattice constant is $a_{0}=3.85\AA$. The CO wavelength in Y123 measured by various X-ray methods increases with increased hole dopingBlanco-Canosa et al. (2014); Hücker et al. (2014); Comin and Damascelli (2016) . We have used such data to generate similar CO striped patterns for Y123 at $p=0.16$ (Fig. 1(d)) and $p=0.09$.

STM differential conductance maps for optimally-doped ($T_{c}=35$ K) and underdoped ($T_{c}=32$ K and $T_{c}=25$ K) ${\rm Bi_{2-y}Pb_{y}Sr_{2-z}La_{z}CuO_{6+\delta}}$ [(Pb, La)-Bi2201] samplesWise et al. (2008, 2009) exhibit checkerboard patterns (indicative of the simultaneous presence of both CDW domains) with $6.20.2a_{0}$ , $5.10.2a_{0}$ , and $4.50.2a_{0}$ unit cell ($a_{0}=3.83\AA$) periodicity, respectively. The increase in the CO wavelength with increased hole doping agrees well with X-ray scattering and STM measurements on Bi2201 without cation substitutionsBlanco-Canosa et al. (2014) . We have used the following formula from Ref. Schneider et al., 2005 to calculate the average number of holes per Cu for the (Pb, La)-Bi2201 samples in Ref. Wise et al., 2008: $T_{c}/T^{\rm max}_{c}=1-250(p-0.16)^{2}$ , where $T^{\rm max}_{c}=35$ K. This calculation yields $p=0.126$ and $p=0.141$ for the two underdoped samples. Figs. 2(a) and 2(d) display CO checkerboard patterns simulated by the CH equation that resemble the STM differential conductance maps for (Pb, La)-Bi2201 at $p=0.126$ and $p=0.16$.

The CO periodicity is manifest in the spatial dependence of the free-energy potential $V_{\rm GL}({\bf r})$, shown in Figs. 1(b) and 1(e), and Figs. 2(b) and 2(e). The central assumption in our model is that by confining charge, a fluctuating CDW periodic potential that scales with $V_{\rm GL}({\bf r})$ mediates the attractive two-body SC interaction. In particular, we assume the fluctuating periodic potential has the same periodicity as the static CO detected experimentally and has a time-averaged potential well depth that is proportional to the depth of the static periodic potential. In what follows, we make the approximation $~{}\langle V_{\rm GL}({\bf r})\rangle$, where $\langle V_{\rm GL}({\bf r})\rangle$ is the spatial average of $V_{\rm GL}({\bf r})$ over a $100\times 100$ unit cell area.

III Superconducting calculations in the charge-ordered state

Next we use the free-energy simulations and experimentally determined input parameters for the optimally-doped compounds to deduce the SC energy gap $\Delta_{\rm SC}$, the pseudogap $\Delta_{\rm PG}$, $T_{c}$ and $T^{*}$ for the underdoped samples. To derive the local SC gap we solved the Bogoliubov-deGennes (BdG) equations via self-consistent calculations based on a Hubbard Hamiltonian [Eq. 7]. The calculations were performed for a sub-lattice about the center of the simulated charge density maps, using periodic boundary conditions and governed by self-consistent conditions for a spatially-varying d-wave pairing potential $\Delta_{d}({\bf r})$ and hole density $p({\bf r})$ [Eq. 10) and 11]. We find that the spatial-average $\langle\Delta_{d}({\bf r})\rangle$ decreases with a reduction of $p$ (below $p=0.16$), but increases with decreasing $\lambda_{\rm CO}$. The latter behavior is because as the two holes are forced closer together by the narrower confining potential the binding energy of the two-body interaction increases. The results on the CuO2 plane shown in Figs. 1(c), 1(f), 2(c) and 2(f) indicate that in our approach the PDW is a consequence of the CDW.

The values of $~{}\langle V_{\rm GL}({\bf r})\rangle$ at optimal doping were multiplied by a scaling factor, such that the calculations (Figs. 1(f) and 2(f)) generate an average value of the pairing potential $\langle\Delta_{d}({\bf r})\rangle$ for $p=0.16$ that is close to the experimentally estimated value of the low-temperature SC gap $\Delta_{\rm SC}$. To calculate $\langle\Delta_{d}({\bf r})\rangle$ for the underdoped samples (Tables 1 and 2), this same scaling factor was subsequently applied to the respective values of $~{}\langle V_{\rm GL}({\bf r})\rangle$. For (Pb, La)-Bi2201, the value of $~{}\langle V_{\rm GL}({\bf r})\rangle$ varies little with doping, and hence $p$ and $\lambda_{\rm CO}$ are responsible for the hole-doping dependence of $\langle\Delta_{d}({\bf r})\rangle$. Experimental estimates of the SC gap for the $p=0.126$ and 0.141 samples are not reported, but the calculated values of $\langle\Delta_{d}({\bf r})\rangle$ for the underdoped samples (Table 2) roughly follow the trend expected if the ratio $\Delta_{\rm SC}/k_{\rm B}T_{c}$ is independent of $p$. In contrast to (Pb, La)-Bi2201, the doping dependence of $\lambda_{\rm CO}$ in Y123 is weaker, and the CH simulations of charge stripes are characterized by a significant change in $\langle V_{\rm GL}({\bf r})\rangle$ with doping (Fig.4). Consequently, the depth of the periodic potential plays an important role in the calculation of the doping dependence of $\langle\Delta_{d}({\bf r})\rangle$ for Y123. The calculated values of $\langle\Delta_{d}({\bf r})\rangle$ at p = 0.09 and 0.16 agree well with an empirical relation for $\Delta_{\rm SC}(p)$ that describes a number of high-$T_{c}$ cuprate superconductorsHuefner et al. (2007). The calculated result at $p=0.12$ falls below this universal curve, which is consistent with the well-known plateau of $T_{c}(p)$ for Y123 near 1/8 hole doping.

An estimate of $T_{c}$ is obtained by self-consistently solving the BdG equations with a temperature-dependent GL potential

[TABLE]

where we take $V(0,p)\approx\langle V_{\rm GL}({\bf r},p)\rangle$ and $T_{\rm PS}(p)$, the onset of phase separation transition, is taken to be equan to $T^{*}(p)$ in the calculations. Because of the BdG approach and the above equation, the value of $\langle\Delta_{d}({\bf r},T)\rangle$ decreases with increasing temperature, but the it remains finite in many regions of the system for a significant range of temperature above Tc. This is consistent with the body of experimental results on cuprates mentioned earlier that are suggestive of persisting SC correlations above $T_{c}$Corson et al. (1999); Xu Z. A. et al. (2000); Gomes et al. (2007); Yang H.-B. et al. (2008); Dubroka et al. (2011); Li et al. (2005); Wang et al. (2005). Typical $\langle\Delta_{d}({\bf r},T)\rangle$ plots of three Y123 compounds are shown in Fig. A2. Next we assume that bulk superconductivity is achieved via Josephson coupling between different closely spaced patches of intertwined CO and SC pairing. We assume there is SC phase coherence within the patches, and that there are many such closely spaced SC domains slightly above $T_{c}(p)$ forming junctions with an average tunnel y proportional to the normal-state resistance immediately above $T_{c}(p)$. As explained previouslyde Mello and Sonier (2014), for a $d$-wave superconductor in single-crystal form it is sufficient to use the following relation for the average Josephson coupling energy

[TABLE]

where $\langle\Delta_{d}({\bf r},T)\rangle=\sum_{i}^{N}\langle\Delta_{d}(r_{i},p,T)\rangle/N$ is the pairing potential. The quantity $R_{\rm n}(p)$ is proportional to the normal state in-plane resistivity $\rho_{ab}(p,T\geq T_{c})$. In what follows we assume $R_{\rm n}(p)$ for the optimally-doped compounds, and in the case of (Pb, La)-Bi2201 use experimental values of the in-plane resistivity ratio $\rho_{ab}(p)/\rho_{ab}(p=0.16)$ to calculate $R_{\rm n}(p)$ for the underdoped samples. Since the relationship between $T_{c}/T_{c}^{\rm max}$ and the hole concentration $p$ for (Pb, La)-Bi2201 is the same as for La-doped Bi2201Schneider et al. (2005), we have used available resistivity data for La-doped Bi2201Ono and Ando (2003) in our calculations shown in Table 2. For orthorhombic Y123, we instead used experimental values of the $b$-axis resistivity ratio $\rho_{b}(p)/\rho_{b}(p=0.16)$ from Ref. Segawa and Ando, 2001 to estimate $R_{\rm n}(p)/R_{\rm n}(p=0.16)$.

As the temperature is lowered below $T^{*}$, thermal fluctuations diminish and long-range phase coherence between the individual SC domains is established when $k_{\rm B}T\approx\langle E_{\rm J}(p,T)\rangle$. The temperature $T$ at which this occurs defines the bulk critical temperature $T_{c}$. Figs. 3(a) and 3(b) show the temperature dependence of $\langle E_{\rm J}(p,T)\rangle$ for both compounds at the different dopings. The intersection of the $\langle E_{\rm J}(p,T)\rangle$ curves with the $k_{\rm B}T$ line yields values of $T_{c}$ in good agreement with the actual values for Y123 and (Pb, La)-Bi2201 (Fig. 4 and Tables 1 and 2).

IV The pseudogap

Next, we use the free-energy simulations to make a simple estimate of the PG, under the assumption that the PG appears due to the mesoscale phase separationFradkin and Kivelson (2012) that creates small domains of CO wavelength below $T_{\rm PS}\approx T^{*}$. For Bi2201 we consider a single-particle state bound to a two-dimensional (2-D) square box of depth $\langle V_{\rm GL}({\bf r})\rangle$ and sides of length $\lambda_{\rm CO}$. For Y123 we consider a single-particle state bound to a stripe-like 2-D rectangular box of depth $\langle V_{\rm GL}({\bf r})\rangle$, width $\lambda_{\rm CO}$, and length equivalent to the CDW correlation length, which is much longer than $\lambda_{\rm CO}$Blanco-Canosa et al. (2014); Hücker et al. (2014); Comin et al. (2015a). We assume in both cases that the PG is proportional to the numerical solution of the corresponding 2-D Schrödinger equation for the ground state binding energy. The proportionality factor is estimated using experimental values of the pseudogap $\Delta_{\rm PG}$ for Y123 and (Pb, La)-Bi2201 at $p=0.16$ (Tables 1 and 2), and the values of $\Delta_{\rm PG}$ are calculated for the underdoped samples using the respective values of $\langle V_{\rm GL}({\bf r})\rangle$ and $\lambda_{\rm CO}$. To further assess the accuracy of the results for the underdoped samples, we convert $\Delta_{\rm PG}$ to $T^{*}$ using the experimental ratios $T^{*}/\Delta_{\rm PG}=170$ K/76 meV and $T^{*}/\Delta_{\rm PG}=241$ K/ 30 meV for optimally-doped Y123 and (Pb, La)-Bi2201, respectively. As shown in Fig. 4 and Table 2, the calculated values of $T^{*}$ for (Pb, La)-Bi2201 at p = 0.126 and 0.141 agree well with measurements of the PG onset temperature for La-doped Bi2201Yurgens et al. (2003). Reasonable agreement is also obtained between the calculated and experimentalDaou et al. (2010) values of $T^{*}$ for Y123 at $p=0.09$ and 0.12 (Fig. 4 and Table 1). As mentioned in the introduction we presume the onset of CO domains at $T^{*}$ above the long range temperature $T_{\rm CO}(p)<T^{*}(p)$, a behavior that has been detected by some different experimentsComin et al. (2014, 2015a); da Silva Neto et al. (2014); Parker et al. (2010).

V Comparison of Calculated and Experimental Parameters

Tables 1 and 2 contain values of experimental parameters (denoted by black text) used in the calculations for each compound, and the calculated parameters (denoted by red text). For each cuprate family the calculated values at $p=0.16$ (which are denoted by blue and green text) were multiplied by a scaling factor to match experimental values as follows:

i) The proportionality constant between $\langle V_{\rm GL}({\bf r})\rangle$ and the attractive pairing potential $V$ of Eq. (10) was adjusted to yield a calculated value of $\langle\Delta_{d}({\bf r})\rangle$ that approximately equals the experimental value of the SC gap $\Delta_{\rm SC}$ at $p=0.16$. This proportionality constant, once determined, was subsequently used for all other values of $p$.

ii) The scaling factor between the normal resistance $R_{n}$ in Eq. (3) and the resistivity $\rho_{\rm b}(1.05Tc)$ just above $T_{c}$ was adjusted until the calculated value of $T_{c}$ at $p=0.16$ approximately equaled the experimental value. This same scaling factor was used for all other values of $p$.

iii) The ground state binding energy of a single-particle in a 2-D square (rectangular) box was multiplied by a proportionality factor so as to equal the PG of (Pb, La)-Bi2201 (Y123) at $p=0.16$. Again, this same proportionality constant was used for the calculations at other dopings.

VI Fermi arc formation

Next we show that the phase separation approach considered above is able to reproduce the ungapped portion of the Fermi surface (Fermi arcs) that is known to occur near the nodal region just above $T_{c}$Lee et al. (2007); Yoshida et al. (2012); Keimer B. et al. (2015). We start by recalling that Figs. 1(c), 1(f), 2(c) and 2(f) show domains of SC order parameter modulations. To each domain we assign a label $j$ and a complex SC order parameter $\Delta_{d}(k,T)\exp(i\Phi_{j})$, where $\Delta_{d}(k,T)=\Delta_{0}(T)[\cos(k_{x}a_{0})-\cos(k_{y}a_{0})]=\Delta_{0}(T)\cos(2\phi)$, $\phi$ is the azimuthal angle measured from the $x$-direction in the CuO2 plane and $\Delta_{0}(T)$ is the wave function amplitude in the jth domain at temperature $T$. The $d$-wave symmetry implies larger supercurrents flowing in the CuO2 plane along the antinodal directions parallel to the Cu-O bonds, and vanishing values along the nodal directions $\phi=\pm\pi/4$ and $\pm\pi/4$. The local intrinsic SC phase $\Phi_{j}$ and the superfluid density $n_{j}\propto\Delta_{d}(r_{j},T)^{2}$ are canonically conjugate variablesSpivak and Kivelson (1991), leading to large fluctuations of the phase $\Phi_{j}$ along the nodal directions, where $n_{j}$ and $\Delta n_{j}$ vanish. This is due to the quantum uncertainty principle and we may write $\Delta\Phi_{j}(\phi)\propto 1/\cos(2\phi)$ to indicate the azimuthal dependence of the phase uncertainty, which has its maximum and minimum values along the nodal and antinodal directions, respectively. Furthermore, $\Delta\Phi_{j}$ has a clear dependence on the Josephson coupling. In particular, as shown in the previous section, at $T<T_{c}$ all $\Phi_{j}$ are locked together leading to long range SC order, but at $T>T_{c}$ phase decoupling occurs because $\langle E_{\rm J}(p,T)\rangle<k_{\rm B}T$ and concomitantly $\Delta\Phi_{j}$ increases with $T$ up to the temperature at which $\langle E_{\rm J}(p,T)\rangle$ vanishes. In particular, $\Delta\Phi_{j}$ increases monotonically from near zero at $T_{c}$ to very large values near $T^{*}$. Furthermore, $\Delta\Phi_{j}$ has a large anisotropy when combined with the quantum effects discussed above. The two distinct contributions are separable, such that $\Delta\Phi_{j}(p,T,\phi)=\Delta\Phi_{j}(p,T)\Delta\Phi_{j}(\phi)$. We drop the index $j$ because $\Delta\Phi_{j}(\phi)$ is the same for all domains, and assume $\langle E_{\rm J}(p,T)\rangle$ is the same for all $j$. These considerations imply that just above $T_{c}$ the electrons ejected by ARPES from different domains come from regions where the SC order parameter has essentially the same $\Phi_{j}$ along the antinodal directions, and $\Delta\Phi_{j}(\phi)\approx 0$. On the other hand, such phase coherence is lost near the nodal directions where $\Delta\Phi_{j}(\phi)\approx\pi$ is maximum. Consequently, the average SC amplitude measured by ARPES may be written as followsde Mello (2012),

[TABLE]

This expression contains the two distinct contributions that weaken phase coherence, one from quantum oscillations that depends only on the azimuthal angle $\phi$ and one from thermal oscillations that competes with the average Josephson coupling, leading to $\Delta\Phi_{j}(p,T,\phi)=\Delta\Phi_{j}(p,T)\Delta\Phi_{j}(\phi)$ for all “j“. We may take $\Delta\Phi(\phi)\sim 1/\cos^{2}(2\phi)$, which satisfies the expected inverse cosine dependence and the square makes it symmetric and always positive around the nodal directions ($\phi=\pm\pi/4$ and $\pm\pi/4$). Now we can infer the functional form of $\Delta\Phi(p,T)$ noting that for $T<Tc$, all $\Phi_{j}$ are locked together leading to long range order and $\Delta\Phi_{j}\sim 0$. On the other hand, for $T\leq T_{c}$ all $\Phi$ decouple because $\langle E_{\rm J}(p,T)\rangle<k_{\rm B}T$ and $\Delta\Phi>0$. Above $T_{c}$, $\langle E_{\rm J}(p,T)\rangle$ decreases with increasing $T$ and vanishes near $T^{*}$. Concomitantly $\Delta\Phi$ increases. Thus, there are three distinct temperature dependent regimes:

(i) $T\leq T_{c}$: Since $\Delta\Phi\sim 0$, Eq. (4) is easy to solve and we obtain the “bare“ expression $\langle\Delta_{d}({\bf r},T,\phi)\rangle=\langle\Delta_{d}(p,T)\rangle|\cos(2\phi)$.

(ii) $T>T^{*}$: $\langle\Delta(p,T)\rangle\sim 0$ and it is clear that there is no gap.

(iii) $T_{c}<T$: Taking into account the effect of $\langle E_{\rm J}(p,T)\rangle$ we assume $\Delta\Phi(p,T)=A[1-\langle E_{\rm J}(p,T)\rangle/k_{\rm B}T_{c}(p)$ where $A$ is a constant. This expression vanishes at $T_{c}$ and increases monotonically with $p$, as expected from ARPES experimentsLee et al. (2007); Yoshida et al. (2012). Thus, putting all together, $\Delta\Phi(p,T,\phi)=[A/\cos^{2}(2\phi)][1-\langle E_{\rm J}(p,T)\rangle/k_{\rm B}T_{c}(p)$ and we obtain the value of $A$ by comparing with the onset of the measuredLee et al. (2007) gapless region for a given sample. To reproduce the measured gapless regions we also assume in Eq. (4) that $\langle\Delta_{d}(p,T,\phi)\rangle\sim 0$ whenever $\Delta\Phi(p,T,\phi)\geq\pi$, due to destructive phase interference from electrons ejected from different domains.

Specifically we use the ARPES measurementsLee et al. (2007) at $T=102$ K for the Bi2212 compound with $T_{c}=92$ K, which shows a gapless region between $28^{0}\leq\phi\leq 62^{0}$ to derive the constant $A$. Equating $\Delta\Phi(p\sim 0.15,T=102{\rm K},\phi=28^{0},62^{0})=\pi$, yields $\langle\Delta_{d}(p,T,\phi)\rangle\sim 0$ and this is possible if we take $A=2.84\pi$. Note that $\phi$ is measured from the $(\pi,\pi)$ to $(0,\pi)$ direction of the Brillouin zone according to Refs. Lee et al., 2007 and Yoshida et al., 2012. With $\Delta\Phi(p,T,\phi)$ determined, we may apply the derived equation to any sample. In particular, we apply this expression to the other two Bi2212 compounds measured in Ref. Lee et al., 2007. Some above $T_{c}$ values of $\langle E_{\rm J}(p,T)\rangle$ used in the calculations are plotted in Fig.(5) for illustration purpose. Accordingly we obtain for the $T_{c}=86$ K compound a gapless region at $23.8^{0}\leq\phi\leq 66.2^{0}$, which compares well with the experimental $25^{0}\leq\phi\leq 65^{0}$. For the underdoped $T_{c}=75$ K Bi2212 sample at $T=85$ K, we obtain $36.5^{0}\leq\phi\leq 53.5^{0}$, which is also in good agreement with the experimental resultLee et al. (2007). We summarize the Fermi arc calculations for the three samples in Fig. 6, where the results of the ”envelope” phase factor of $\langle\Delta_{d}(p,T,\phi)\rangle$

[TABLE]

from Eq. (4) are in good agreement with the experimentsLee et al. (2007). The arrows mark the experimentally determined onset of the gapless regions for each sample, as described above.

VII Summary and conclusions

Our theory infers a fundamental link between the periodicity of the CDW and the PG and SC energy scales of high-temperature cuprate superconductors, and shows that within this framework it is possible to account for the onset temperatures $T_{c}$ and $T^{*}$ of two different cuprate families. We stress that the only quantitative assumptions made in our calculations for underdoped Y123 and (Pb, La)-Bi2201 pertain to a natural scaling factor, which we have determined by scaling calculated free energy parameters to achieve values of the PG and SC gap that agree with experimental values at one particular doping. Our model is general in the sense that it can be applied to other cuprate families, provided the doping dependence of the CDW order is known.

Our approach generates a local free energy potential having a spatial periodicity that matches that of the experimentally observed short-range static CDW order. Our calculations in the framework of BdG theory yield different SC amplitudes in distinct charge-ordered domains that generally vanish only above $T_{c}$. Our approach is consistent with experimentsda Silva Neto et al. (2014); Wise et al. (2008); Gomes et al. (2007); Dubroka et al. (2011) that measure a finite SC amplitude above $T_{c}$, and promotes the scenario whereby the SC resistive transition marks the onset of global phase coherence between SC domains. In our model Fermi arcs appear above $T_{c}$ because there are large phase fluctuations along the nodal directions where the superfluid density vanishes. The increase of the arcs size with $p$ is reproduced because the dependence of $\langle E_{\rm J}(p,T)\rangle$ on the temperature changes rapidly with doping.

Finally we address the experimental observations indicating a competition between superconductivity and CO. While X-ray experiments show a decrease of the CDW diffraction intensity and correlation length below $T_{c}$Chang et al. (2012); Ghiringhelli et al. (2012); Blanco-Canosa et al. (2014); Hücker et al. (2014); Comin et al. (2015b), these measurements seem to be detecting static charge correlations. Static CDW order competes with superconductivity by reducing the number of charge carriers available for pairing. On the other hand, our theory requires that dynamic CO is also present to induce a fluctuating hole-pair potential that scales with $V_{\rm GL}$. While there is some evidence for CDW fluctuations from optical pump-probeTorchinsky et al. (2013) and low-energy, momentum-resolved electron energy-loss spectroscopyVig et al. (2016) experiments, there is currently insufficient experimental information to assess the pervasiveness or significance of fluctuating CO in cuprates.

Acknowledgements.

We thank S. Ono and Y. Ando for providing us with their resistivity data. We also thank Andrea Damascelli, André-Marie Tremblay and John Tranquada for informative discussions. Supported by the Brazilian agencies FAPERJ and CNPq (E.V.L.M.); the Canadian Institute for Advanced Research (CIFAR) and the Natural Sciences and Engineering Research Council of Canada (J.E.S.).

Appendix A CHARGE ORDER SIMULATIONS

To describe the growth and development of spatial charge inhomogeneity in the CuO2 planes we applied a theory of phase-ordering dynamics, whereby the system evolves through domain coarsening when quenched from a homogeneous into a broken-symmetry phase. The time-dependent CH approach provides a simple way to determine the time evolution of the CO processCahn and Hilliard (1958). The CH equation can be written in the form of the following continuity equation for the local free energy current density ${\bf J}=-M\nabla^{2}((\partial f/\partial u)$Bray (1994)

[TABLE]

where $M$ is the charge mobility that sets the phase separation time scale. The order parameter varies between $u({\bf r},t)\sim 0$ for the homogeneous system above the phase-separation onset temperature $T_{\rm PS}$, and $u({\bf r},t\rightarrow\infty)=\pm 1$ for the extreme case of complete phase separation. We solved the CH equation by a stable and fast finite difference scheme with free boundary conditionsde Mello (2012). The spatial dependence of the charge density obtained by numerically solving the CH equation evolves with time $t=n\delta t$, where n is the number of time steps $\delta t$. When the order parameter is conserved, as in phase separation, the charges can only exchange locally rather than over large distances. This leads to diffusive transport of the order parameter. Consequently, at early times or small n, we obtain charge modulations with periodicities of only a few lattice constants. Using different parameters and initial conditions we are able to reproduce the experimentally determined CO patterns in cuprates. Although these simulations are not the stable solutions of the CH equation (as is clear from the time evolution of the simulations shown in Fig. 7), the aim here is to generate periodic charge modulations with experimentally determined wavelengths that can be subsequently used to calculate the SC gap and PG in our phase- separation model. For convergence the time step $\delta t$ and spatial step $h\approx 1/N$ for a square lattice of $N^{2}$ sites must be such that $\delta t\leq h^{2}/9$ (Ref. de Mello, 2012). For the calculations here we used $\delta t\leq h^{2}/10$ and $h=1/100$.

In the main paper we present detailed CO, $T_{c}$ and $T^{*}$ calculations for six compounds. Three of the Bi2201 and three of the Y123 families. In the next two paragraphs we give the values of some parameter used in the CH simulations.

(Pb, La)-Bi2201: Simulations with $\alpha=B=1$, time steps of $n=700,900,1300$ and $\varepsilon=0.012,0.014,0.0175$ yield checkerboard CO patterns with the desired wavelengths $\lambda_{\rm CO}$ for (Pb, La)-Bi2201 (at $p=0.126,0.141$ and $0.16$) near $4.5a_{0}$, $5.1a_{0}$, and $6.2a_{0}$, respectively. The fewer time steps required to simulate the CO patterns of the underdoped samples is indicative of a reduced charge mobility, and is consistent with an increase of the normal-state resistivity. At later times (i.e., greater n) the periodic electronic structure evolves into an irregular patch-like system of segregated low- and high-charge density regions. In addition, the length scale of the system increases with the phase separated regions forming larger domains. This latter situation was considered previouslyde Mello and Sonier (2014). Fig. 7 shows CH simulations of $u({\bf r},t)$ at times beyond where checkerboard CO with $\lambda_{\rm CO}=4.5a_{0}$ is observed in (Pb, La)-Bi2201 at $p=0.126$.

Y123: Simulations with $\alpha$ = 1, $B$ = 5, $\varepsilon$ = 0.0053, 0.0055, 0.0058 and time steps $n=35,38,42$ yield charge stripe patterns with the desired wavevectors $Q=0.333,0.317$ and 0.287 r.l.u. ($\lambda_{\rm CO}=1/Q$) estimated from (Refs. Blanco-Canosa et al., 2014; Hücker et al., 2014; Comin et al., 2015a) for Y123 at $p=0.09,0.12$ and 0.16, respectively. Note that the values of n are much shorter than needed to simulate the checkerboard CO patterns of (Pb, La)-Bi2201. Because of the fewer time steps, the simulations for Y123 are somewhat less sharp.

Appendix B Combined Bogoliubov-deGennes (BdG) and Cahn-Hilliard (CH) Calculations

We performed self-consistent calculations with the BdG theory (Refs.de Mello and Caixeiro, 2004; de Mello et al., 2009) for each of the CH simulated charge density maps (Figs. 1(a) and 1(d), and Figs. 2(a) and 2(d)). To calculate the SC pairing amplitude we assumed the attractive interaction potential $V$ scales with $V_{\rm GL}^{\rm min}\sim\langle V_{\rm GL}({\bf r})\rangle$. The SC calculations begin with the extended Hubbard Hamiltoniande Mello and Caixeiro (2004); de Mello et al. (2009). To describe the charge carriers dynamics in the CuO2 planes of the HTSC we consider this Hamiltonian in a square lattice

[TABLE]

where $c_{i\sigma}^{\dagger}(c_{i\sigma})$ is the usual fermionic creation (annihilation) operator at site $i$, the spin $\sigma$ is up $\uparrow$ or down $\downarrow$. $n_{i\sigma}=c_{i\sigma}^{\dagger}c_{i\sigma}$ is the number operator, and $t_{ij}$ is the hopping between sites $i$ and $j$. $U$ is the magnitude of the on-site repulsion, and $V$ is the magnitude of the nearest-neighbor attractive interaction. $\mu_{i}$ is the local chemical potential derived in the self-consistent process through which the local charge density is calculated by the CH equation and is maintained fixed. For (Pb, La)-Bi2201 we used nearest-neighbor hopping $t=0.15$ eV, next-nearest-neighbor hopping $t_{2}=-0.27t$, and third nearest-neighbor hopping $t_{3}=0.19t$ derived from angle-resolved photoemission spectroscopy (ARPES) dispersion relationsNorman et al. (1995). For Y123, we used the ARPES results $t=0.15$ meV, $t_{2}=-0.50t$ and $t_{3}=0.16t$ (Ref. Schabel et al., 1998). The BdG mean-field equations arede Mello et al. (2009)

[TABLE]

with

[TABLE]

and similar equations for $v_{n}({\bf r})$, where r+R is the position of the nearest-neighbor sites, and $\mu({\rm{\bf r}})\equiv\mu_{i}$ is the local chemical potential. These equations are solved numerically for eignevalues $E_{n}(\geq 0)$ self-consistently with the spatially-varying $d$-wave pairing potentialde Mello and Caixeiro (2004)

[TABLE]

where $V=V(T,p)$ was defined in Eq. 2. The results of $\langle\Delta_{d}({\bf r},T)\rangle$ are plotted in Fig. 8 for the three compounds of the Y123 system. Concomitantly, the spatially-varying hole density of charge carriers is given by

[TABLE]

where $f_{n}=[\exp(E_{n}/k_{\rm B}T+1]^{-1}$ is the Fermi occupation function. It is important to emphasize that the spatially inhomogeneous distribution of charge generated by the CH equation for different dopings was kept fixed while the local chemical potential $\mu({\bf r})$ was self-consistently determined in the convergence process. This procedure incorporates the charge inhomogeneity in the calculations in a natural way.