In a real space perspective, the doping dependences of energy scales for the pseudogap [11, 61] correspond to varying spatial organizing structures of collective electron motions during the increase of hole spacing [62–64]. Therefore, the energy-length relation lies at the heart of the pseudogap origin and its relationship with symmetry-breaking orders [65]. It is well known that the strong electron-electron correlations stimulate multiple symmetry-breaking orders in cuprates, whose competitions result in unprecedented prominence of collective fluctuations [1]. These fluctuations are intimately related to and affect the pseudogap phase. For instance, the emergence of magnetic fluctuations [66, 67] and nematicity order coincide with the pseudogap opening temperature T * [53–56], while the superconducting phase fluctuations defines T c , the onset temperature of thermal vortices in the pseudogap phase [13, 68–70]. Therefore, we propose that the pseudogap phase can be treated as a “critical” phase (ansatz No. 1), which holds various quantum [63, 71–75] or thermal critical points and lines in the T − p phase diagram. Thus, many collective fluctuations in this phase are of critical natures, e.g., having correlation functions with power law scalings. Although each order and its fluctuations have unique symmetry-broken form and energy, the common criticality may constrain the energy-length relationship to be a universal form.

In the following, taking pairing orders as an example, we derive this relation from the spatial organization of critical fluctuations. DWO and superconductivity are particle-hole and particle-particle pairings, respectively, with order parameters expressed by two-point correlation functions. In conventional metals, these orders are usually long-range coherent, so nontrivial pairing mainly occurs in the momentum space [76, 77]. On the contrary, strong electron-electron correlations and chemical disorder in cuprates constrain pairings to be short-range with coherence or correlation lengths at only several nanometers [20, 21, 78–80], revealing nontrivial spatial organizations [81–85]. Thus, we propose that they should be described by two-point correlation functions in real space. Moreover, for critical fluctuations, the universal criticality constrains the correlation function to be a power law versus length (ansatz No. 2), i.e., g ( l ) ∝ l − η , where η is a scaling exponent [60], l is the characteristic distance of pairing particle or hole.

In a classical theory of critical behavior, length scales are continuous coordinates. However, the Mott physics and strong correlations in cuprates result in a local tendency to phase separation [1], forcing the spatial organization of mesoscopic order and fluctuations occurring on specific nanoscales. Specifically, there are two kinds of length scales in the order parameter, i.e., the spatial period of the DWO and the characteristic length of the (charge, spin, or superfluid) density, presenting in phase and amplitude, respectively. Like Emery and Kivelson [13], we assume the critical fluctuations have the particle-wave duality constrained by the Heisenberg uncertainty principle (ansatz No. 3). Therefore, the energy scale corresponding to the two-point correlation function should satisfy an inverse square relation with its appropriate length as follows [65]. E o = γ o h 2 m * l o 2 , (1) where E o and l o are the characteristic energy (e.g., gap and its onset temperature) and length (e.g., CDW period or Cooper-pair spacing) of a mesoscopic order, respectively, and m * is the effective mass, h is the Planck constant, γ o is a dimensionless parameter.

The scaling in Eq. 1 is universal for various phase-fluctuating orders. It can be derived from a series of physical mechanisms, including the phase-disordering transition (e.g., Berezinskii-Kosterlitz-Thouless (BKT) [86–88] and Bose-Einstein phase transition [89]), the umklapp scattering [65], and quantum kinetic energy of a density modulation ¹ . For these mechanisms, γ o is determined by a geometric factor, the amplitude and coupling strength, as well as the ratio between density fluctuations and mean-field, respectively. Different γ o and l o for various symmetry-breaking forms would yield different magnitude and doping dependence, providing quantitative criteria to determine dominant orders of the pseudogap, as shown in the following.

It is also intriguing to mention that the power law form of the energy-length scaling is universal for both quantum and classical systems. However, the scaling index may vary for different systems owing to different physical mechanisms. For instance, the gravity and electric potentials of a point source are inversely proportional to the distance to this point (−1 index), while the fluctuation energy called Reynolds stress of a turbulent flow is proportional to the square of the stress length (+2 index) [90]. In contrast, for simple quantum phenomena with only one related length scale, the inverse square scaling (−2 index) must be universal due to the dimensional constraints with Planck constant h (i.e., constant angular momentum).

For the pseudogap origin, charge orders were identified as an important candidate in moderately doped cuprates. Recently, the pseudogap opening at T * was observed to coincide with the emergence of an electron-nematic order breaking the rotational symmetry [53–56], which is proposed to be generated by a partially melted unidirectional DWO with significant global phase fluctuations [8]. To test this assertion, it is crucial to quantify the spectral gap originating from a locally unidirectional CDW with space-time phase fluctuations. Taking the x-axis CDW in Figure 2A, for instance, this is considered as follows [65]: Ψ ( x , t ) = A exp [ i ( Q o x + ϕ ( x ,   t ) ) ] . (2)

Here, A is the amplitude of charge modulation, Q o = 2 π / l o is a phase-averaged wavevector, and ϕ ( x ,   t ) is the residual phase fluctuations, whose simplest expression is, for phason mode, ϕ ( x ,   t ) = q x − ω q t . This fluctuating DWO generates an effective potential V proportional to the density wave modulations to induce the umklapp scattering (i.e., k → k ′ = k + q + n Q o , where k and k ′ are the initial and final states, n is a nonzero integer). Following Lee and Rice [95], the characteristic energy of the mean scattering rate is Γ = 4 π m * ∑ | v k − v k ′ | 2 | 〈 k ′ | V | k 〉 | 2 F k , k ′ , where v k and v k ′ are velocity, F k , k ′ is a complex function composed of Dirac and distribution functions. Under the assumption of small momentum difference ( v k − v k ′ ≈ s ℏ ( k − k ′ ) / m * ) and long-wavelength approximation ( q ≪ n Q o ), we obtain | v k ′ − v k | 2 ≈ n 2 s 2 ℏ 2 Q o 2 / ( m * ) 2 [65]. Therefore, an energy gap related to umklapp scattering in the form of energy-period scaling is obtained as: Δ = γ Δ ℏ 2 Q o 2 m * = γ Δ h 2 m * l o 2 , (3)

which is a specific expression of Eq. 1.

Taking γ Δ / m * as constant for a moderate doping regime, Eq. 3 predicts an inverse square of energy-period scaling for cuprates, which is quantitatively consistent with the reported data of (Bi,Pb)₂(Sr,La)₂CuO_6+δ (Pb-Bi2201) [42], as shown in Figure 2B. In contrast, the previous theories [44, 96] attribute the connection between the pseudogap and Q C D W to fermiology in momentum space, whose predictions show apparent overestimation for the optimal and underdoped regimes (see Figure 2B) due to the neglect of the renormalization of strong correlations [42, 96]. In contrast, the mean scattering strength γ Δ = 0.135 in our theory is found to be close to the square ratio between the superexchange energy and the hopping energy ( J 2 / t 2 ≈ 0.11 [42, 96]), revealing it is closely related to the antiferromagnetic correlation [22].

Another advantage of the energy-length scaling is its ability to characterize the global phase fluctuations, which was observed as spatial homogeneity of gap amplitude and period between different local CDW plaques. Assuming this phase fluctuations as a Gaussian type, we derive from Eq. 2 a gap distribution as [65]: Δ = f 0 exp [ − ( Δ − Δ ¯ ) 2 ( Δ m − Δ ¯ ) 2 ] , (4) where Δ ¯ = γ Δ h 2 / m * l ¯ 2 , Δ m = γ Δ h 2 / m * ( l ¯ − δ l ) 2 , l ¯ is the globally averaged value of l C D W , δ l is the standard deviation, and f 0 is the peak intensity. As shown in Figure 2C, the experimental observations are consistent with Eq. 4, indicating that both the local CDW and its global phase fluctuations satisfy the energy-length scaling.

Furthermore, the pseudogap opening temperature can be defined with the emergence of the particle-hole pairing of CDW [39], implying k B T * is proportional to the pseudogap energy [10, 11]. In other words, k B T * also satisfies the energy-length scaling as follows, k B T * = γ T h 2 m * l o 2 . (5)

As m * are nearly constant in a moderate doping regime [97, 98], Eqs 3, 5 predict that the previously known monotonic decreases in T * and Δ * with increasing doping actually originates from the reduction of the CDW wavevector [20] and its amplitude [99]. As shown in Figure 2D and Ref. [65], this prediction is verified by experimental data of T * and Δ * over a wide doping range of various cuprate compounds [11, 100]. Therefore, the energy-length scaling provides solid proof that the pseudogap mainly originates from phase-fluctuating CDW in the moderate doping regime ( p = 0.1 ∼ 0.2 ). It is worth mentioning that the energy-length scaling may also be approximatively reflected in momentum-space structures, which is worth further exploration. For instance, we suspect that the antinodal pseudogap (3.5 or 2 times of local gap [65]) measured by ARPES or Raman response may have a similar inverse square relationship with the wave vector connecting the Fermi arc tips (close to the CDW wave vector [96]).

Besides charge orders, superconducting phase fluctuations have been accepted as an important participator in part of the pseudogap regime [101]. It is well known that the two-dimensionality (2D) and low carrier density result in significant phase fluctuations in underdoped cuprates [13]. In the Berezinskii-Kosterlitz-Thouless (BKT) phase transition scenario [86, 88], the 2D superconducting phase fluctuations are generated by the real-space unbinding of vortex-antivortex pairs at a critical temperature, (see Figure 2E). The balance between the vortex excitation energy and the entropy (i.e., E ϕ = T c S ϕ ) determines this BKT phase transition temperature as [87] k B T c = π ℏ 2 n s 2 m * = π ℏ 2 2 m * l c p 2 , (6) where l c p = n s is the Cooper pair distance. Eq. 6 indicates that the phase coherence energy k B T c has an inverse square proportion to the Cooper-pair distance, satisfying the energy-length scaling Eq. 1. It provides another typical example of the energy-length scaling with length from the amplitude of a uniform phase but not the period of DWO.

Although there are other specific contexts to derive the scaling in Eq. 6 with different prefactors [13, 14], the linear scaling between T c and n s is now widely observed and accepted [68, 70]; see Figure 2F. Moreover, the recent observations of Δ S C N ∝ T c ∝ n s in both underdoped and overdoped cuprates [11, 68, 69] demonstrate that the superconducting dome of the nodal superconducting gap Δ S C N and T c of thin cuprate films both originate from the dome-like doping dependence of the n s , see Figures 2G,H. On the other hand, the relation between T c and n s in overdoped bulk samples shows more complexities [102] and need further clarifications. Furthermore, above the T c dome, the BKT phase transition generates free thermal vortices in the low- T part ( T c < T ≪ T * ) of the pseudogap phase. The best circumstantial evidence for this scenario comes from diamagnetism and Nernst signals [94, 103], which were accurately described by the time-dependent Ginzburg-Landau equation [104, 105] and vortex transport theory [106]. These evidences confirm that a spatial organization of superconducting phase-fluctuations (i.e., the unbinding of vortex-antivortex pair) determines the cuprate superconducting transition. However, contrary to Loktev et al.’s work [107] attributed the pseudogap formation mainly to superconducting phase fluctuations, we think the phase fluctuations of DWO dominate the upper pseudogap phase, and the superconducting phase fluctuations play a role in the pseudogap origin only in the lower part of the pseudogap phase.

It is intriguing that the present energy-length scaling (Eq. 1) also applies to the topological excitations of loop current in the strange metal phase at higher temperatures or beyond the pseudogap critical point [108]. These excitations are fluctuating vortices of unpaired electrons, whose characteristic lengths are the thermal de Broglie wavelength ( l T ∝ T − 1 / 2 ) or magnetic length ( l B ∝ B − 1 / 2 ). These vortices scatter carriers with rates inversely proportional to the square of these lengths, which satisfies Eq. 1 and determines the well-known linear resistivity [108]. In summary, the above results in this section demonstrate the importance and universality of the spatial organization mechanism and the energy-length scaling for phase fluctuations of density wave, superconductivity, and loop current orders.