Unied energy law for uctuating density wave orders in cuprate pseudogap phase

The origin of the pseudogap and its relationship to symmetry-broken orders in cuprates have 5 been extensively debated. Here, we report a uniﬁed energy law underlying the pseudogap, which 6 determines the scattering rate, pseudogap energy, and its onset temperature, with a quadratic scaling 7 of the wavevector of density wave order (DWO). The law is validated by data from over one hundred 8 samples, and a further prediction that the master order of pseudogap transforms from ﬂuctuating 9 spin to charge DWO is also conﬁrmed by independent measurements. Furthermore, the energy law 10 enables our derivation of the well-known linear scalings for the resistivity of the strange metal phase 11 and the transition temperature of the superconducting phase, shedding light on the universal origin 12 of various phases. Finally, it is concluded that ﬂuctuating orders provide a critical bridge linking 13 microscopic spectra to macroscopic transport in cuprates, showing promise for the quantiﬁcation of 14 other strongly correlated materials.


27
From a mean-field viewpoint, the DWO breaks trans-28 lational symmetry to produce an anisotropic gap, and 29 small Fermi surface (FS) pockets [1,6,7]. However, 30 experimentally observed static (charge and spin) DWOs 31 exhibit an onset temperature significantly lower than T * 32 and no further gap opening [1,2], apparently ruling out 33 the static DWO as the origin of the pseudogap [3,8]. 34 On the other hand, the nematic and loop current orders 35 are widely observed to emerge coincidentally during the 36 pseudogap opening at T * [4,5]. However, within the 37 mean-field theory, these intra-unit-cell orders are known 38 to be unable to break the lattice translation symmetry 39 to open a pseudogap [9,10]. Thus, the controversy of 40 whether the pseudogap opening can be related to the 41 conventional form of these two classes [1] remains to be 2 gap [1,23,24]. To answer these questions, we naturally 1 introduce the wavevector and amplitude as the two fun-2 damental quantities to characterize DWO and to inves-3 tigate their relationships with two critical energy scales 4 in the single-particle self-energy (see section 1 in the 5 "Methods"), namely, the gap energy and the scattering 6 rate. 7 To derive these relationships, we use an innovative 8 symmetry-breaking analysis inspired by a recent suc-9 cessful wall turbulence theory [25]. Similar to eddies 10 exposed to strongly wall-constrained shear turbulence, 11 the fluctuating DWO near T * has a complex spatio-12 temporal pattern, which is generally difficult to model 13 by an order parameter. However, the structural ensem-14 ble dynamics (SED) theory of wall turbulence [25,26] 15 states that eddy motions can be effectively self-organized 16 into several statistical ensembles. This self-organization 17 comes from the constraint of symmetry (e.g., dilation 18 associated with wall turbulence) and is expressed by 19 the fact that eddy lengths satisfy generalized power 20 laws, which determine the crucial energy distribution 21 [26] and transport coefficient [25]. Therefore, we as- 22 sume that the pseudogap energy scales should satisfy a 23 power-law relation with the wavevector (or wavelength) 24 of the fluctuating DWO. Specifically, as ubiquitously ob-25 served in cuprates [3,8,23,27], the mesoscopic (fluctu-   ing rate Γ k , we consider only the mean scattering rate Γ, 76 which is independent of k, k ′ , and q but closely related 77 to Q o in an Umklapp scattering process. This yields an 78 energy law for the mean scattering rate as follows: tions. Thus, we obtain an energy law similar to Eq. (1): Here, is a dimensionless parameter proportional to the ampli-10 tude of the mean-field.

11
From a pairing perspective, the DWO is equivalent to 12 pairing in the particle-hole channel [31]. In this context, 13 we define the onset temperature T * of the pseudogap 14 opening with the emergence of this particle-hole pairing.

15
This implies that the thermal-fluctuation energy k B T * is 16 linearly proportional to the pseudogap energy, as widely 17 observed in spectroscopic measurements [15,32]. This 18 naturally yields a relationship for determining T * , as 19 follows: where γ T ∝ γ ∆ is a dimensionless coefficient. In Gap energy scales associated with the CDW. The

25
CDW has been identified as a leading competitor of su-26 perconductivity in cuprates [3]. Therefore, it would be 27 intriguing to examine Eq. (3)    The red and black symbols represent the experimental data reported in Refs. [46,48]. The solid blue line represents the Eq. (5) prediction with γ * Γ = 0.11 and the CDW period lCDW, which is compared to the prediction (solid black line) with the antiferromagnetism (AF) period length 2a0, as well as the predictions (purple squares) from Planckian dissipation theory [49] with m * = 2.7me.
processes [45]. Thus, we assume that the (macroscopic) 1 transport scattering rate is proportional to but smaller 2 than the (microscopic) single-particle scattering rate, 3 /2τ = C τ Γ, where τ is the relaxation time and C τ 4 is a dimensionless coefficient less than 1. By substitut-5 ing τ and Eq. (1) into the Drude model, we obtain the 6 in-plane resistivity: where γ * Γ = 4πC τ γ Γ , R Q = h/e 2 is the quantum resis-8 tance, n c = pK/a 0 b 0 c 0 is the carrier density, a 0 and b 0 9 are the in-plane lattice constants, c 0 is the c-axis lattice 10 constant, K is the number of Cu or Fe ions in one unit 11 cell, and p is the carrier concentration per ion.

12
Eq. (5)  During the extraction, we cautiously avoided the influ-22 ence of superconductivity (SC) by selecting the "knee" 23 data ρ(T sf ) at the onset temperature T sf of the SC fluc-24 tuations, as presented in Fig. 4 (a).

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(5), we predict the critical sheet resistance as follows: where γ * Γ is the corresponding scattering coefficient and 81 p c is the critical carrier concentration. If γ * Γ is univer-82 sal, then Eq. (6) predicts that R is inversely propor-   The present work reports two universal scattering 48 coefficients for the low-energy pseudogap and resistivity, 49 i.e., γ ∆ ≈ γ * Γ ≈ J 2 /t 2 ≈ 0.11. It is essential to discuss  Characteristic energy scales and the DWO or-30 der parameter. In mean-field theory, the character- 31 istic energies that determine the anomalous electronic 32 spectrum and charge transport in the pseudogap phase 33 are described by the single-particle self-energy, which, 34 following Ref.
[6], takes the following form: where k, ω, ǫ, and Q are the wavevector, frequency, and 36 dispersion of a single-particle excitation and the DWO This paper aims to uncover a universal energy law link-48 ing ∆ k and Γ k to the wavevector Q and the amplitude 49 A of the fluctuating DWO.

52
The energy law for the scattering rate (i.e., Eq. (1)) can 53 be derived from a Umklapp scattering theory for phason 54 modes of DWO [88], assuming a small momentum differ-55 ence and a large sound velocity. This scattering obeys 56 the conservation of momentum as k ′ = k + q + nQ o .

57
Following Lee and Rice [88], we obtain the characteris-58 tic energy for the mean scattering rate: where v and E are the velocity and energy of a carrier, respectively.

65
For a small momentum difference, it is reasonable to 66 assume a linear relation, i.e., where s is a dimensionless constant. Furthermore, we 68 assume that, compared to Q o , the q of low-lying excita- By substituting this expression into Eq. (9), we obtain 71 Eq.
(1) and a dimensionless coefficient describing the 72 mean strength of carrier-phason scattering: