Unified energy law for fluctuating density wave orders in cuprate pseudogap phase

The quantum origin of the cuprate pseudogap is a central conundrum of condensed matter physics. Although many symmetry-broken scenarios were previously proposed, universal quantitative relationships have been rarely studied. Here, we report a unified energy law underlying the pseudogap, which determines the scattering rate, pseudogap energy, and its onset temperature, with a quadratic scaling of the wavevector of density wave order (DWO). The law is validated by data from over one hundred samples, and a further prediction that the master order of pseudogap transforms from fluctuating spin to charge DWO is also confirmed. Furthermore, the energy law enables our derivation of the well-known linear scalings for the resistivity of the strange metal phase and the transition temperature of the superconducting phase. Finally, it is concluded that fluctuating orders provide a critical bridge linking microscopic spectra to macroscopic transport, showing promise for the quantification of other strongly correlated materials. The origin of the pseudogap in the cuprates and how it is related to the mechanism of their superconductivity has been a topic of long standing debate. Here, the authors suggest that the pseudogap is governed by a unified energy law the basis for which lies in the theory of wall turbulence.

A central puzzle of high-temperature cuprate superconductors is the pseudogap Δ* that occurs below a characteristic temperature T*, manifested by suppressing the electronic density of states around the Fermi level 1,2 . The critical question that remains unanswered is, what is the underlying quantum order that determines the pseudogap? Experimentally, in the pseudogap phase, two distinct classes of orders are widely reported 1,3-5 , namely, zero wavevector Q = 0 states (e.g., nematicity, loop current and Cooper pair) and the finite wavevector Q ≠ 0 density wave order (DWO). From a mean-field viewpoint, the DWO breaks translational symmetry to produce an anisotropic gap, and small Fermi surface (FS) pockets 1,6,7 . However, experimentally observed static (mainly short-range) DWOs exhibit an onset temperature significantly lower than T* and no further gap opening 1,2 , apparently ruling out the static DWO as the origin of the pseudogap 3,8 . On the other hand, the nematic and loop current orders are observed to emerge coincidentally during the pseudogap opening at T* 4,5 . However, within the mean-field theory, these intra-unit-cell orders are known to be unable to break the lattice translation symmetry to open a pseudogap 9,10 . Thus, the controversy of whether the pseudogap opening can be related to the conventional form of these two classes 1 remains to be solved.
Recent theoretical advances suggest that the pseudogap phase can be understood with intertwined orders, including the Q ≠ 0 DWO and the Q = 0 nematic, loop-current or superconducting states 8,9,[11][12][13] . For instance, it was demonstrated that a partially melted unidirectional DWO (either spin or charge) could generate a vestigial nematic phase 13 . In this context, using elaborate data analysis, various experiments were performed to achieve precise measurements of energy scales and wavevectors associated with a charge density wave (CDW), revealing an intimate link among CDWs, the nematic order, and the pseudogap. Specifically, recent Raman measurements of the spectral gap associated with CDWs (Δ CDW ) in Bi 2 Sr 2 CaCu 2 O 8+δ (Bi-2212), HgBa 2 -CuO 4+δ (Hg-1201), HgBa 2 Ca 2 Cu 3 O 8+δ (Hg-1223) and YBa 2 -Cu 3 O 6+δ (Y-123), display the same doping dependence as the pseudogap energy 14,15 , indicating that the pseudogap and the CDW energy scale may have a common microscopic origin. Furthermore, by analysing tunnelling conductance from distinct regions of momentum space, Mukhopadhyay et al. identified energies characterizing the CDW, nematicity, and pseudogap in Bi-2212 and found that they are identical 16 , which suggested that the pseudogap may originate from highly disordered unidirectional DWO. This viewpoint was further supported by more recent observations from scanning tunnelling microscopy (STM) and nuclear magnetic resonance (NMR): the CDW phase is locally unidirectional (with significant phase fluctuations globally) for underdoped Bi-2212 [17][18][19] and Y-123 20 , and the local spectral gap in (Bi,Pb) 2 (Sr,La) 2 CuO 6+δ (Pb-Bi2201) is positively correlated with fluctuating CDW wavevector 21 .
While the above-mentioned theoretical and experimental progress indicates a mainly qualitative link between the CDW and the pseudogap and highlights the crucial role of phase fluctuations (either thermal fluctuations or spatial disorders) in this link, we are devoted to investigating whether there exists a universal quantitative relationship between energy scales of the pseudogap and the fluctuating DWO. In this context, three critical challenges remain to be addressed 15,22 . First, one needs to define the energy scales linked to the intrinsic nature of DWO to determine whether the measured gap is from the CDW rather than other orders. Second, one needs to have a clear explanation for the origin of the universal doping dependence of Δ * and T*. Finally, one needs to explain the transition from a CDW to a spin density wave (SDW) and its influence on the pseudogap 1,23,24 . To answer these questions, we naturally introduce the wavevector and amplitude as the two fundamental quantities to characterize DWO and to investigate their relationships with two critical energy scales in the single-particle self-energy (see the "Methods" subsection "Characteristic energy scales and the DWO order parameter"), namely, the gap energy and the scattering rate.
To derive these relationships, we use an innovative symmetrybreaking analysis inspired by a recent successful wall turbulence theory 25 . This structural ensemble dynamics (SED) theory 25,26 demonstrated that, although wall turbulence encompasses fully spatio-temporal chaotic motions (or eddies), due to shear stress, its time-averaged behaviour, constrained by wall-induced dilation symmetry, is spatially self-organized into several simple ensembles. Within each ensemble, turbulent eddies have their characteristic length size as the crucial similarity parameter, on which momentum transport coefficient has a universal power-law dependence. This theory predicts the energy distribution 26 and transport coefficient 25 over the entire domain. Here, we regard fluctuating DWO below T* as eddies in wall turbulence, whose statistical behaviours must obey similar self-organized structures dictated by underlying symmetry-braking characterizing the electron-electron correlations in pseudogap phase through an appropriate length. Specifically, as ubiquitously observed in cuprates 3,8,23,27 , the mesoscopic (fluctuating) DWO emerges as a consequence of translational symmetry-breaking. Therefore, following SED, we assume that the pseudogap energy scales should satisfy a power-law relation with the wavevector (or wavelength) of the fluctuating DWO. In a microscopic scattering view, its wavelength l o = 2π/Q o corresponds to a small wavevector Q o (smaller than the reciprocal lattice vector, where o represents the order type), which connects momentum states on the FS, favouring the Umklapp scattering associated with Q o to determine the pseudogap energy scales in the single-particle selfenergy.
Here, we regard T* as the onset temperature of the particle-hole pairing that breaks translational symmetry locally and instantaneously, and the transition temperature T DW of quasi-static DWO 27 as the onset temperature of phase coherence, and the dynamic DWO above T* 28 as amplitude fluctuations of this pairing. Especially, we assume that the highly disordered DWO presented below and near T* is a particular nematic order that breaks the two-dimensional rotational symmetry on a large scale but maintains the unidirectional DWO locally. Therefore, in this work, we focus on the locally fluctuating unidirectional DWO below T* (for both the nematic and the quasi-static states), which, we assume, can be described by one-dimensional order parameters with a small-wave vector or low-frequency phase fluctuations.
In the following, we first introduce an Umklapp phason scattering mechanism to derive a quadratic scaling with the DWO wavevector for the scattering rate and further extend it to the pseudogap energy Δ * and the onset temperature T*. The scalings are then validated by the spectral gap and the onset temperature data obtained for over one hundred samples, indicating that the universal monotonic decrease with increasing doping of Δ * and T* in the intermediate doping regime originates from the variation in the CDW wavevector Q CDW . Furthermore, by using resistivity data of high-quality single crystals, we demonstrate the validity of the quadratic scaling, with a universal scattering coefficient for both the CDW and SDW. Besides, by using the energy law, we can derive a length, from the resistivity and pseudogap energy data (with the so-called length mapping method), of the master order, which displays a universal transition in the pseudogap phase from the SDW (at light doping) to the CDW (at intermediate doping), as confirmed by independent measurements. Finally, the new energy law offers a straightforward explanation for both the strange metal resistivity and the linear scaling between the superconducting transition temperature and superfluid density, indicating that the law is universal for all three phases of hole-doped cuprates. These findings let us conclude that mesoscopic (fluctuating) orders provide a crucial bridge (in universal energy scaling) linking microscopic spectra to macroscopic transport in cuprates. This discovery offers a potential breakthrough towards a comprehensive theoretical description of strongly correlated materials.

Results
Universal energy law for fluctuating DWO. The fluctuating DWO generates phason modes to induce carrier scattering, which requires conservation of momentum as k 0 ¼ k þ q þ nQ o , where k and k 0 are the initial and final states, q is the phason wavevector, and n is a nonzero integer. Ignoring the anisotropy of the scattering rate Γ k , we consider only the mean scattering rate Γ, which is independent of k, k 0 , and q but closely related to Q o in an Umklapp scattering process. This yields an energy law for the mean scattering rate as follows: where γ Γ is a dimensionless coefficient describing the mean scattering strength and m Ã is the effective mass of a carrier. Eq. (1) is not only the most straightforward explicit function of Γ(Q o ) with correct dimensionality and inversion symmetry but also a natural result of the Umklapp scattering theory under the small momentum difference and a long-wavelength approximation (see the "Methods" subsection "Umklapp scattering rate associated with DWO"). The scattering theory also predicts that γ Γ is proportional to the module square of the carrier-DWO coupling. Based on a simple energy correspondence to the t − J model 29 , we assume that this coupling is proportional to the ratio between the superexchange energy J and the hopping energy t, which yields: where the dimensionless coefficient F Γ is proportional to the square of the fluctuating DWO amplitude A, i.e., F Γ ∝ A 2 . The second equality originates from the approximation J/t ≈ 1/3, supported by ab initio calculations and Raman scattering data 30,31 . Generally, F Γ may depend on temperature and doping. However, recent spectroscopy experiments reveal that the dynamical charge and spin excitations pervading the phase diagram of cuprates and their scattering energy (e.g., peak locations and intrinsic width) are little affected by temperature and doping in the pseudogap phase 28,32,33 . These findings reveal that in the intermediate temperature and doping regime discussed below (see also the "Discussion" section), the temperature and doping contributions may be renormalized. Based on these experimental observations, we assume F Γ and γ Γ away from quantum critical points to be doping and temperatureindependent and let the experimental data determine their magnitudes. Furthermore, as for the scattering rate, we propose that the characteristic gap amplitude Δ originates from the Umklapp scattering by the fluctuating DWO. Therefore, Δ can be predicted from Eq. (1) by extending the scattering source from order fluctuations to its mean-field, which involves only the substitution of the amplitude of phason modes at q ≠ 0 in Eq. (2) for the amplitude of modes at q = 0 without further calculations. Thus, we obtain an energy law similar to Eq. (1): Here, γ Δ = F Δ J 2 /t 2 ≈ 0.11F Δ is the dimensionless coefficient describing the scattering strength, where F Δ is a dimensionless parameter proportional to the square of the mean-field amplitude. Similarly, we assume F Δ and γ Δ below and away from T* to be doping and temperature-independent and let the experimental data determine their magnitudes. From a pairing perspective, the DWO is equivalent to pairing in the particle-hole channel 34 . In this context, we define the onset temperature T* of the pseudogap opening with the emergence of this particle-hole pairing. This implies that the thermalfluctuation energy k B T* is linearly proportional to the pseudogap energy, as widely observed in spectroscopic measurements 15,35 . This naturally yields a relationship for determining T*, as follows: where γ T ∝ γ Δ is a dimensionless coefficient. In addition, in the following, we express Q o in units of 2π/a 0 , where a 0 is the inplane lattice constant.
Gap energy scales associated with the CDW. The CDW has been identified as a leading competitor of superconductivity in cuprates 3 . Therefore, it would be intriguing to examine Eqs. (3) and (4) to evaluate whether the CDW is indeed related to the pseudogap and satisfies these simple energy laws. Recently, using STM, Webb et al. carried out simultaneous measurements of the spectral gap and Q CDW for local charge modulations in (Bi, Pb) 2 (Sr, La) 2 CuO 6+δ (Pb-Bi2201) at a superconducting state (at 6 K) 21 , providing an appropriate sample for this evaluation. Taking m Ã = 2.7m e (m e is the electron mass) for Bi-2201 based on the optical conductivity measurement 36 and γ Δ = 0.135, Eq. (3) were used to predict a quadratic scaling, Δ ¼ _ 2 Q 2 CDW =20m e , that is quantitatively consistent with the majority of the reported data shown in Fig. 1. This agreement confirms the quadratic scaling between Δ and Q CDW .
It is interesting to compare the present prediction with previous theories. In the past, the connection between the pseudogap and Q CDW was attributed to fermiology, which explains the decrease in Q CDW with increasing doping as a result of a growing hole pocket 37,38 . The predictions of two models based on this fermiology 21 are presented in Fig. 1. In the antinodal (AN) case, Q CDW connects the nested AN segments of the FS, and in the antiferromagnetic zone boundary (AFZB) case, Q CDW connects the points at which the FS crosses the AFZB. Both models show apparent overestimation for the optimal and underdoped regimes. Therefore, the present proposal that the spectral gap associated with CDW originates from scattering by CDW rather than the simple FS instability is well confirmed. As discussed in previous studies 21,37 , the failure of these simple Fermiology may be the neglect of the renormalization of strong interactions, which is captured in the present theory. In addition, one may wonder whether the global incommensurability is compatible with recent STM and NMR observations that the CDW is locally commensurate for underdoped Bi-2212 and Y-123 [17][18][19][20] . Assuming phase fluctuations to be of Gaussian type, we derive from Eq. (3) a gap distribution that is quite similar to the experimental observations (see Supplementary Note 2), implying that the present energy law applies to the locally commensurate CDW and the accompanying phase fluctuations.
Note that, the spectral gap in Fig. 1 is defined as the peak location of the differential conductance spectra of the lowtemperature (6 K) superconducting phase. Therefore, it may represent a composite gap synthesized contributions of both the CDW and the superconducting gaps. Thus, it is important to find a way to distinguish these gaps. The recent Raman response measurements 14,15 defined the CDW gap (Δ CDW ) as the nodal hump location persisting above the superconducting transition temperature T c , the high-energy scale pseudogap Δ Ã H as the AN depletion location above T c , as shown in Fig. 2. Interestingly, with simple normalizations (i.e., divide energy by constant), the experimental data for both gaps in Hg-1201 (hollow symbols) and Bi-2212 (solid symbols) collapse to the solid blue line predicted by Eq. (3). The fitting parameters are γ Δ = 0.1, m * = 2.45m e (determined from the quantum oscillation experiment of Hg-1201 at p = 0.09 39 ) and Q CDW = 0.343-0.699p (linearly fitted from the experimental data for Hg-1201 40,41 ; see Supplementary Note 1). On the other hand, the angle-resolved photoemission spectroscopy (ARPES) measurement in the AN direction usually suggests two energy scales for the pseudogap, namely, the lowenergy scale (Δ Ã L , e.g., the peak location) and the high-energy scale (Δ Ã H , e.g., the hump location) 42 . It is remarkable that both energy scales for Bi-2212 determined from ARPES measurement 35,42 follow the doping dependence of our prediction as well.
Figures 1 and 2 contain three types (STM, Raman, and ARPES) of data for over 50 samples of three compounds; thus, our theory is firmly validated. In addition, since the pseudogap onset temperature is the gap opening temperature, T* should also satisfy the same quadratic scaling. As shown in Fig. 3, the predictions from Eq. (4) are indeed consistent with experimental data over a wide doping range of Bi-2201, Bi-2212 and Hg-1201 15,43 , confirming the universality of the energy law. Note that in the fitting of Bi-2201, a power-law model Q CDW = 0.269[1 − (p/0.261) 3.79 ] fitted from the previously reported experimental data 24,44 has been used (see Supplementary Note 1).
Our findings reveal that, in the CDW-dominated regime, the well-known monotonic decreases in Δ and T* for three compounds with short-range DWOs mainly result from the reduction of the CDW wavevector with increasing doping. Furthermore, our theory enables us to propose a preliminary explanation for the origin of the high-energy pseudogap, i.e., Δ Ã H . Specifically, we assume that Δ Ã H originates from the multi-fold scattering rather than the one-fold scattering process of the lowenergy pseudogap. For instance, the coupling of x-directional and y-directional one-fold scattering by unidirectional CDWs (with wave vectors |Q x | = |Q y | = Q CDW ) results in two-fold pseudogap energy, i.e., γ Δ ℏ 2 |Q x + Q y | 2 /m * = 2Δ CDW , which then provides a natural explanation for the high-energy pseudogap of Raman data. Besides, the early-stage ARPES measurements 42 determined higher pseudogap energy than the Raman response, for it defined the gap with raw data without subtracting the high-temperature signal, which we suspect may also be explained by higher-fold scattering processes. A thorough theoretical description of the high-energy process with a multi-fold scattering scenario will soon be communicated in the future. Note also that the experimental data begin to deviate from the prediction at p~0.2, likely because the CDW amplitude begins to decrease noticeably near the pseudogap quantum critical point (QCP), which might be described by the defect power law of γ Δ (the dashed blue lines in Figs. 2 and 3). Furthermore, near p~1/8, T* for Bi-2212 and Hg-1201 is lower than our prediction, which may reveal an anomaly in the CDW gap opening. Its relevance to the temperature and doping variations of the CDW amplitude and the effective mass (assumed to be doping independent here) should be investigated in the future.
Characteristic resistivity associated with a CDW. Despite considerable discussion on the contribution of CDWs to the electronic spectrum 6,9,34,45,46 , the connection between CDWs and charge transport is rarely mentioned because of the difficulty in associating the scattering rate with the CDW. However, our phenomenology provides a simple way to make this connection. Generally, transport dissipation stems from a substantial momentum transfer (e.g., backward scattering), which is a subset of all-microscopic scattering processes 47 . Thus, we assume that the (macroscopic) transport scattering rate is proportional to but smaller than the (microscopic) single-particle scattering rate, ℏ/ 2τ = C τ Γ, where τ is the relaxation time and C τ is a dimensionless coefficient <1. By substituting τ and Eq. (1) into the Drude model, we obtain the in-plane resistivity: where γ Ã Γ ¼ 4πC τ γ Γ , R Q = h/e 2 is the quantum resistance, n c = pK/a 0 b 0 c 0 is the carrier density, a 0 and b 0 are the in-plane lattice constants, c 0 is the c-axis lattice constant, K is the number of Cu or Fe ions in one unit cell, and p is the carrier concentration per ion.
Equation (5) quantifies the characteristic resistivity determined by DWO fluctuations. A good candidate for the verification is the underdoped Bi-2201 because its short CDW correlation length fluctuates between 0.75 and 1.5 times the CDW period l CDW 44 , which may efficiently induce scattering. We extract the characteristic resistivity of CDW scattering from previously reported data obtained from high-quality single crystals 48,49 , which have very small residual resistivity at optimum doping. During the extraction, we cautiously avoided the influence of superconductivity (SC) by selecting the "knee" data ρ(T sf ) at the onset temperature T sf of the SC fluctuations, as presented in Fig. 4a.
In Fig. 4b, we fit ρ(T sf ) with Eq. (5) and the CDW period length l CDW = 2π/Q CDW , where Q CDW is estimated with the same power law, 0.269[1 − (p/0.261) 3.79 ], used in Fig. 3a. We find that a constant scattering coefficient γ Ã Γ ¼ 0:11 % J 2 =t 2 makes the predictions agree well with the data from 10 samples between p = 0.12 and 0.18 48,49 . This outcome is consistent with the presence of CDW ordering in the p = 0.11 − 0.16 region observed by resonant inelastic X-ray scattering 44 . Quantitatively, when the doping increases from 0.12 to 0.18, l 2 CDW increases by 80%, which is higher than the 50% increase in the carrier density, revealing that the decrease in ρ(T sf ) is dominated by the variation in the CDW period length.
It would be helpful to compare the present analysis with other theoretical approaches. In Planckian dissipation theory 50,51 , the   48 . The pairs of vertical lines indicate the T sf error bars determined from the specific-heat measurement 95 . b Doping (p) dependence of ρ(T sf ). The red and black symbols represent the experimental values determined from previously reported measurements 48,49 . The vertical errors are estimated from the resistivity variation associated with the T sf error bars. The solid blue line represents Eq. (5) prediction with the scattering parameter γ Ã Γ ¼ 0:11 and the charge density wave (CDW) period l CDW , which is compared to the prediction (solid black line) with the antiferromagnetism period length 2a 0 , as well as the predictions (purple squares) from Planckian dissipation theory 50 with the effective mass m Ã = 2.7m e (m e is the electron mass) and T sf . The root mean squared error of the solid blue line fit to data above p = 0.12 is 20 μΩ cm. scattering rate solely determined by temperature is ℏ/τ = k B T, which predicts the resistivity at T sf to be ρ(T sf ) = (m Ã /n c e 2 )(k B T sf /ℏ). Taking m Ã = 2.7m e from optical conductivity measurements 36 , this prediction (Fig. 4b, purple squares) underestimates the data by nearly 50%. One way to remedy this discrepancy is to attribute it to the "residual resistivity" induced by impurities. However, the stochastic nature of impurities makes the minimum residual resistivity of optimally doped cuprates difficult to explain 52,53 . Therefore, the above results reveal that the "knee" resistivity at T sf in the pseudogap phase has a simple scaling of ρðT sf Þn c / l À2 CDW / Q 2 CDW and thus satisfies the energy law of Eq. (1).
Universal sheet resistance for an antiferromagnetic SDW. An antiferromagnetic (AF) SDW is another widespread DWO in the underdoped regime of HTSCs. Thus, it would be intriguing to examine Eq. (5) to determine whether the AF SDW satisfies the energy law of Eq. (1). In both cuprate-based and iron-based HTSCs, the phase transition between AF and SC is characterized by a low-temperature plateau for the resistivity 49,54 , as shown in Fig. 5a. In a superconductor-insulator (SI) transition scenario, the corresponding sheet resistance is predicted to be a universal value, namely, h/4e 2 = 6450 Ω in the Boson localization theory 55 . However, experimental observations revealed that the critical resistance is sample-dependent within a factor of 0.5-2 of the predicted values, which motivated Goldman's question: "What different physical models govern the various SI transitions which have different critical resistances?" 55 . We now provide a quantitative explanation for this sample-dependent resistance, using Eq. (5). In our theory, the plateau is attributed to the Umklapp scattering by characteristic fluctuations of the AF SDW with l o ≈ 2a 0 . By substituting this l o into Eq. (5), we predict the critical sheet resistance as follows: where γ Ã Γ is the corresponding scattering coefficient and p c is the critical carrier concentration. If γ Ã Γ is universal, then Eq. (6) predicts that R □ is inversely proportional to p c . As shown in Fig. 5b, the prediction with a constant γ Ã Γ ¼ 0:11 is quantitatively consistent with the reported data for both hole-doped cuprates and iron pnictides within a wide doping range.
The validation of Eq. (6) yields a surprising prediction that the critical sheet resistance per carrier is a universal value, R & p c ¼ γ Ã Γ h=4e 2 % 710 Ω, which represents the true universal feature behind observed critical resistances. This universal sheet resistance of 710 Ω per carrier is not present in the total resistance description (R □ ) by the localization theory 55 , and we have achieved a good answer to Goldman's question. Furthermore, since the parents of iron pnictides are metals but not insulators, the validity of Eq. (6) above implies that the energy law is applicable for AF fluctuations in both AF insulator and metal states.
Transition from a CDW to an SDW. It is well known that decreasing doping in cuprates induces an order transition from charge to spin sectors 1,56 . Remarkably, the present theory fully confirms this transition from a fluctuating CDW at intermediate doping to an SDW at light doping in the validations of Eqs. (5) to (6) in Figs. 4 and 5. Specifically, Eq. (5) enables us to quantify DWO's period length from resistivity data: where ρ(T sf ) is observational data. As shown in Fig. 4b for Bi-2201, as the doping decreases, ρ(T sf ) increases sharply from 236 μΩ cm at p = 0.12 to 871 μΩ cm at p = 0.10. Taking γ Ã Γ ¼ 0:11, the sharp increases in ρ(T sf ) is explained by an order's period change from l CDW ≈ 4a 0 to l SDW ≈ 2a 0 , which indicates a possible transition from a CDW-dominated regime to an AF SDW-dominated regime. This assertion is remarkably consistent with an observation from NMR measurements 57 that suggests that the CDW supersedes AF near p = 0.11 in Bi-2201.
In addition to the consistency for the above-predicted order transition from a CDW to an AF SDW, let us further investigate whether it can also describe the transition in the pseudogap energy scale. Recently, STM measurements of Bi-2201 have exhibited a monotonic increase in the gap energy from p = 0.12 to p = 0.03, and the absence of a CDW at p = 0.03 24 . The latter suggests that there must be another order dominating the gap in the lightly doped AF regime. According to our theory, SDWs have a larger wavevector (near 0.5) than CDWs (near 0.25), resulting in a larger energy gap, and this finding is qualitatively consistent with the STM observation. Quantitatively, taking l o = 2a 0 (the AF period), γ Δ = 0.135 (the same value as Pb-Bi2201) and m Ã = 2.2 ± 1m e (from optical conductivity measurements at p = 0.03 36 ), we predict that Δ * = 170-470 meV, which is close to the gap energy scale (i.e., 400-600 meV) measured by STM at p = 0.03. Note that this estimation is much higher than that measured by STM for the CDW (≲100 meV), which concludes that the pseudogap in the lightly doped regime is indeed determined by the AF SDW.
A further question that our theory should address is whether the CDW extends into the overdoped regime or the energy law is valid across the whole doping range of cuprates 3 from the pseudogap energy scale using spectroscopic data. While the pseudogaps in Bi-2201, Bi-2212, and LSCO are experimentally shown to persist up to a high doping level of p = 0.20-0.22 2,15,21,58 , our length mapping formula predicts the existence of CDWs in overdoped cuprates accompanying the pseudogap, consistent with recent STM and RXS measurements of Bi-2201 (up to p = 0.23) 44 and LSCO (up to p = 0.21) 59 . On the other hand, Eq. (5) enables us to confirm CDW's dominance from resistivity data for overdoped cuprates, such as Y-123 60 , Tl 2 Ba 2 CuO 6+δ 61 and La 2−x Ce x CuO 4 62 , for which the predicted l o in p = 0.16-0.22 increases from 3a 0 to 5.5a 0 . The latter coincides with the characteristic CDW period l CDW ≈ 4a 0 , indicating the possible presence of a dynamic CDW in the overdoped regime. Furthermore, the transport scattering rate of the typical CDW is _=2τ ¼ πγ Ã Γ _ 2 =m Ã l 2 CDW % 4:6 meV with l CDW = 4a 0 and m Ã = 2.5m e . These findings are highly consistent with Arpaia et al.'s observations that the short-range dynamical charge density fluctuations are characterized by energies of a few meV, and pervade a large area of the phase diagram 28 .
Then, it is of interest to verify these predictions in other strongly correlated materials using the length mapping formula. A preliminary examination has already yielded consistent outcomes. For instance, from Eq. (3) with γ Δ = 0.135 and m Ã = 4m e 63 , we predict the charge order length for iridate from the pseudogap amplitude data (i.e., 70-300 meV) to be l o = 1.7-3.6a 0 , consistent with the STM measurement 64 .

Conclusion
In summary, complementary to current theories focusing on intertwined mechanisms of various orders 8,9,[11][12][13] , we here uncover a universal energy law linking the pseudogap and the DWO, namely, all three pseudogap energy scales (the scattering rate, the pseudogap energy, and its onset temperature) have a quadratic scaling with the DWO wavevector. All (more than one hundred) single-crystal sample data fully support the present energy law, revealing that the pseudogap originates from the fluctuating DWO, i.e., an SDW at light doping and a CDW at intermediate doping. In our opinion, the universal energy law (and scattering coefficients) represents the zeroth-order relationship between pseudogap energy scales and the DWO wavevector for doping dependence, which provides an important reference point to resolve several conundrums associated with the pseudogap origin.
First, the onset temperature of the pseudogap, if defined as the emergence for the particle-hole pairing of fluctuating SDWs or CDWs, would be naturally higher than the onset temperature of static (coherent) SDWs or CDWs. Second, the universal monotonic decrease in Δ * and T* with increasing doping stems from the variation in the amplitude and the wavevector of the SDW and CDW. At light doping, the decreases in Δ * and T* are due to a reduction in the amplitude of the SDW, supported by the observation that the antiferromagnetic spectral weight decreases with doping 23 . On the other hand, at intermediate doping, the decreases are due to reductions in the Q CDW and the CDW amplitude, which is supported by spectroscopic measurements 27,44 , as shown in Figs. 2 and 3 for Bi-2201, Bi-2212 and Hg-1201.

Discussion
In contrast, the La-based cuprates exhibit a weak increase in Q CDW with doping (observed in p = 0.115-0.21) 27,56,59,65,66 as a consequence of mutual locking of long-range spin and charge orders at low temperatures 65,67,68 , a situation that never occurs in other cuprates. Based on a simple analysis of the resistivity (Fig. 5b) and gap data (Supplementary Table 3), we find that the main predictions (e.g., Eqs. (3) and (6)) in this work holds for independent SDW and CDW in the La-based cuprate, but γ Δ and γ Ã Γ have different values and doping dependencies (see also Supplementary Note 6). It reveals that the decreases in Δ * and T* with increasing doping in the CDW-dominated regime for this exceptional class might be due to the reduction of the CDW amplitude, which is preliminarily observed by the decreasing peak height intensity of X-ray diffraction with doping 59 . However, the intertwining effects of spin and charge DW in striped order are neglected in this work and should be further studied in the future.
Furthermore, one may wonder why the CDW transition temperature (T CDW ) does not follow the CDW gap (Δ CDW ) and onset temperature (T*) as shown in previously reported measurements 14,15 . Although the clarification of the specific physical mechanism underlying this difference goes beyond the scope of this paper, the present work inspires a preliminary understanding. Specifically, we conjecture that in contrast to Δ CDW and T* determined by the particle-hole pairing energy, T CDW represents the phase coherence (perhaps local for the short-range charge order) energy of the pairing. If this is correct, there may be a positive correlation between T CDW and the CDW's correlation length characterizing the phase coherence strength. Indeed, this positive correlation is consistent with RXS experiments of Hg-1201 40 , Bi-2201 37 , YBa 2 Cu 3 O x 69 and La 2−x Sr x CuO 4 59,66 . Therefore, we propose that the T CDW dome has an intimate relationship with the phase coherence energy of CDW, which is an intriguing conjecture worth further study.
The present work reports two universal scattering coefficients for the low-energy pseudogap and resistivity, i.e., γ Δ % γ Ã Γ % J 2 =t 2 % 0:11. It is essential to discuss the doping range for the universality of these two scattering coefficients. Our preliminary understanding is that the universality is preserved for compounds with short-range DWO in the intermediate doping regime away from both the AF insulating phase and the pseudogap QCP. The reason is that the AF insulator contains long-range AF correlations, leading to stronger scattering behaviour than short-range fluctuations. In contrast, the pseudogap QCP yields a significant decrease in the DWO amplitude, resulting in a weakening of the scattering. As explained in the "Methods" section, we presently use data from high-quality single crystals with the least degree of impurity among all reported data associated with the same compound. However, our theory may be extended to discuss data obtained for less pure samples 70,71 to quantify the additional impurity effect (see Supplementary Note 3). Note that long-range ordering effects in Y-123 and La-based cuprates may significantly affect the magnitudes of the mean-field and fluctuation intensity of DWO (e.g., γ Δ , see Supplementary Note 6), which is an intriguing issue to be explored in the future.
A further interesting outcome is that the present energy law extends to the strange metal and superconductivity phases. The former comprises fluctuating vortices as an emergent dynamic order, characterized by the thermal de Broglie wavelength (l T ∝ T −1/2 ) or magnetic length (l B ∝ B −1/2 ), which scatters carriers with a scattering rate inversely proportional to the square of these lengths 72 . The scattering rate also satisfies the energy law Eq. (1) with a linear dependence on temperature or magnetic field, consistent with recent experimental observations 53,[72][73][74] . On the other hand, comprehensive measurements have demonstrated a universal linear relation between T c and the superfluid density ρ s in most doping regimes of holedoped cuprates [75][76][77] . This indicates that the phase coherence energy k B T c is inversely proportional to the square of the Cooperpair distance (l p ¼ ffiffiffiffi ρ s p ); thus, Eq. (4) is satisfied. Furthermore, Raman response measurements show that the nodal superconducting gap Δ N SC has the same dome-like doping dependence on T c 15 , indicating Δ N SC / T c / l À2 p , again satisfying Eq. (3). Therefore, although originating from different microscopic mechanisms (e.g., the Umklapp scattering by DWO for pseudogap and strong phase fluctuations for superconductivity and strange metal), the characteristic energies (i.e., gap, transition temperature, and scattering rate) of the strange metal, pseudogap, and superconducting phases all satisfy an inverse square scaling on the characteristic lengths of mesoscopic orders (e.g., vortex, DWO, and Cooper pairs). This unified energy law reveals that, regardless of how complex the symmetry-broken forms are, a common invariance constrains the mesoscopic collective electronic motions in cuprates, providing a unified cross-scale link between the microscopic spectrum and macroscopic transport. We speculate that this invariance exhibits an intrinsic quantum nature of strongly correlated electrons and is worthy of further experimental verification and theoretical research in three directions: First, it is interesting to verify the energy law for other strongly correlated materials, such as iron-based HTSC, iridate, organic, and heavy Fermion superconductors. Second, we suggest further explorations of the physical origin of the energy law to determine whether there is a local quantum wave state whose single-particle excitation is constrained by the period of a mesoscopic order and thus has quantum kinetic energy determined by the order's period, i.e., E / h 2 =m Ã l 2 o . It is highly plausible that the local quantum wave state and the unified energy law can be derived from an action of mesoscopic ordering (e.g., quantum XY model 5 ) or the microscopic Hamiltonian (e.g., t − J or Hubbard model) of correlated electrons through some renormalization calculations.
Finally, the present universality provides a clue for quantifying the intertwined behaviours of various forms of collective orders and fluctuations by considering the quantum coupling of multiple pairing or scattering channels. First of all, a thorough understanding of the pseudogap origin requires clarifying the intertwining relationship between the CDW and superconductivity. For instance, the comprehensive description of ARPES data at low and high-T regimes involves an intertwining of particle-particle and particle-hole pairing channels 78 . More interestingly, the recent Raman response measurements 14,15 found that the AN superconducting gap Δ AN SC is close to the CDW gap over a substantial doping range. Based on the present theory, it means that both gaps have a universal quadratic scaling with the CDW wavevector, i.e., Δ AN SC $ Δ CDW / Q 2 CDW . These universal energy laws provide an intriguing clue for advancing microscopic researches of the intertwining effects of particle-particle and particle-hole pairing channels 11,79 . Furthermore, many recent experiments observed that high magnetic fields suppress superconductivity to enhance or even induce CDW for cuprate compounds 57,[80][81][82][83][84][85][86][87] . While the clarification of a specific microscopic mechanism for the origin of this field-induced CDW is beyond the scope of the present work, we can offer a preliminary description for the field dependence of the magnetoresistance, consistent with the primary revelation of experimental observations, see Supplementary Note 7. On the other hand, the quantum coupling of multiple scattering channels will yield a comprehensive explanation for the preliminary observations that the total scattering rate can be expressed as a coupling formalism of multiple energy laws in recent experiments, e.g., ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðk B TÞ 2 þ ðμ B BÞ 2 q 74,88,89 , where μ B is the Bohr magneton. The progress in these directions will significantly advance the understanding of non-Fermi liquids in strongly correlated electronic systems 90 .

Methods
Characteristic energy scales and the DWO order parameter. In mean-field theory, the characteristic energies that determine the anomalous electronic spectrum and charge transport in the pseudogap phase are described by the singleparticle self-energy, which, following Norman et al.'s theory 6 , takes the following form: where k, ω, ϵ, and Q o are the wavevector, frequency, and dispersion of a singleparticle excitation and the DWO wavevector, respectively. Here, the anisotropic gap Δ k and the anisotropic scattering rate Γ k are two critical energy scales describing the electron scattering by the mean-field effective potential and bosonic excitations of the fluctuating DWO (or impurities), respectively. This paper aims to uncover universal energy laws linking Δ k and Γ k to the wavevector Q o and the amplitude of the fluctuating DWO.
Generally, similar to electron-phonon scattering, the scattering energies in Eq. (7) is related to the real-space order parameters of the fluctuating DWO through the quasiparticle-phason scattering mechanism. To characterize the temporal fluctuations added to usual static DWO 17 , we express the order parameter of the unidirectional DWO to be a space-time variable associated with a phase-averaged wavevector Q o and the residual phase fluctuations ϕ(x, t) (ansatz no. 1): Generally, phase fluctuations ϕ(x, t) can have any form. However, there is a special simple case, i.e., the plane wave mode with ϕ(x, t) = qx − ω q t. For ω q ≠ 0, the longterm average of the order parameter is zero, revealing that the order is fluctuating.
Umklapp scattering rate associated with DWO. This density wave (Eq. (8)) introduces an effective potential to scatter quasiparticles. Specifically, the corresponding carrier-phason interaction is proportional to the density wave modulations, i.e., V ¼ g DW A cos½Q o x þ ϕðx; tÞ, where g DW is the carrier-DW coupling strength. Note that experimental evidence supported a strong coupling perspective of density wave instability in cuprates, which can be quantified by the t − J model 11,91 . In this context, we introduce the ansatz no. 2 that the cuprate DWO can be described by the t − J model, and the carrier-DW coupling strength is proportional to the superexchange energy, i.e., g DW ∝ J. The energy law for the scattering rate (i.e., Eq. (1)) can be derived from a Umklapp scattering theory for phason modes of DWO 92 , taking a small momentum difference and a long-wavelength approximation (ansatz no. 3). This scattering obeys the conservation of momentum as k 0 ¼ k þ q þ nQ o . For the characteristic energy of the mean scattering rate, Lee and Rice 92 predicted that: where v and E are the velocity and energy of a carrier, respectively; ω q is the frequency of the phason; and f and n q are the distribution functions of the carrier and phason modes, respectively. Besides, W k;k 0 ¼ jhk 0 jVjkij 2 / A 2 J 2 is the module square of the transition matrix element describing the carrier-phason scattering. For a small momentum difference, it is reasonable to assume a linear relation, i.e., ðv k 0 À v k Þ % s_ðk À k 0 Þ=m Ã , where s is a dimensionless constant. Furthermore, we assume that, compared to Q o , the q of low-lying excitations is small, yielding ðv k 0 À v k Þ 2 % n 2 s 2 _ 2 Q 2 o =ðm Ã Þ 2 . By substituting this expression into Eq. (9), we obtain Eq. (1) and a dimensionless coefficient describing the mean strength of carrier-phason scattering: Finally, considering γ Γ ∝ A 2 J 2 is dimensionless, we can express it as γ Γ = F Γ J 2 /t 2 (i.e., the Eq. (2)) in the t − J model, while the proportionality coefficient F Γ includes the residual integral associated with electronic dispersions. The microscopic calculations of these electronic dispersions and integral beyond the scope of this work. Therefore, we introduce the ansatz no. 4 based on recent experimental observations 28,32,33 , that is, we assume γ Γ (and γ Δ ) in the pseudogap phase away from quantum critical points (and T*) to be doping and temperature-independent and let the experimental data determine their magnitudes, see also discussions following Eq. (2).
Model the doping dependence of Q CDW . To make continuous predictions dependence of the gap energy, the scattering rate, and the resistivity on doping, we propose a least-squares fit for the doping-dependent phase-averaged Q CDW data. During the fitting procedure, we assume that the amplitude of the fluctuating CDW is nonzero in the doping range considered in this work. Under the constraints of simplicity and physical consistency, we found that the linear fit and the defect power law are the two most suitable fitting functions for Hg-1201, Bi-2212, and Bi-2201, as shown in Supplementary Fig. 1 (see Supplementary Note 1). For instance, although a polynomial provides a more accurate fit for Q CDW of Bi-2201, it is inconsistent with the observation that the CDW is nearly commensurate in the lightly doped Bi-2201 21,24 . In contrast, a defect power law, i.e., Q CDW = 0.269[1 − (p/0.261) 3.79 ], is the simplest function consistent with this scenario.
The choice of validating samples. To date, there have been thousands of reported experimental measurements devoted to the pseudogap, DWO, and resistivity, involving considerable diversity, compound series, doping regimes, and sample qualities. To identify a universal energy law, we restrict ourselves mainly to highquality single crystals with short-range DWO to remove various higher-order effects, e.g., excessive impurities and the long-range ordering effect. Therefore, in this paper, the samples selected for validation are high-quality single-crystal series of hole-doped Bi-2201, Bi-2212, and Hg-1201 prepared by highly respected experimental groups for their strong two-dimensional nature and systematic observations of short-range DWO 14,15,27,40,44 , as well as the lowest reported resistivity values among all reported data associated with the same compound (e.g., Bi-2201 prepared by Ando's group 48,52 ). Therefore, the choice of our validating samples is not arbitrary but consistent with our strict theoretical requirements. Please refer to Supplementary Notes 4 and 5 for the corresponding list of the data sources used in this work.

Data availability
All experimental data analysed and presented in the main text and supplementary materials can be found in an online repository https://github.com/luhaozhang/ Validation-for-unified-energy-law-in-pseudogap-phase.git. Further material is available from the corresponding author upon reasonable request.

Code availability
All code utilized in the simulations presented in the main text and supplementary text can be found in an online repository https://github.com/luhaozhang/Validation-forunified-energy-law-in-pseudogap-phase.git. Further material is available from the corresponding author upon reasonable request.