2 2 Δ δ 2 Δ d 2 Δ s Δ s / Δ d δ 2 Δ δ 2 Δ p 2 Δ * FIG. 1: (color online) The doping dependence of the order pa- rameters for a model system describing doped Mott-Hubbard in- sulators on a two-dimensional square lattice, whose ground-state wave function is a superconducting state with mixed spin-singlet wave and spin-triplet wave symmetries in the presence of an anti-ferromagnetic background, with the order parameters for the -wave, -wave, and )-wave superconducting components, together with the anti-ferromagnetic order parameter . Left panel: The superconducting gaps and for the spin-singlet -wave and -wave components as a function of doping , with their ratio shown in the inset. Right panel: The energy scales 2 and res for the anti-ferromagnetic order and the spin-triplet wave superconducting order as a function of doping , with being the anti-ferromagnetic order, and the Boltzmann constant. A crossover from the Bose-Einstein condensation (BEC) regime to the Bardeen-Cooper-Schrie er (BCS) regime occurs, when the two energy scales merge into one single energy scale in the (heavily) overdoped regime. Note that, in general, res . Indeed, res 25 , as predicted from the two-dimensional model. In addition, res scales with the superconducting transition temperature res d + s − p$_{x}$ ( p$_{y}$ ) − s d p$_{x}$ ( p$_{y}$ N Δ$_{d}$ Δ$_{s}$ d s δ Δ$_{s}$ / Δ$_{d}$ Δ ∗ ∼ k$_{B}$T ∗ ∼ N E$_{res}$ = 2 Δ$_{p}$ p$_{x}$ ( p$_{y}$ ) − δ N k$_{B}$ the Bardeen-Cooper-Schrie ff er (BCS) regime occurs, when the two E$_{res}$ < 2 Δ$_{d}$ . E$_{res}$ ≈ 1 . Δ$_{d}$ t − J E$_{res}$ T$_{c}$ : E$_{res}$ ∼ k$_{B}$T$_{c}$ mining the transition temperature T$_{c}$ . Actually, two distinct energy scales 2 Δ ∗ and E$_{res}$ are involved, in a marked contrast with the conventional superconductors: 2 Δ ∗ arises from the anti-ferromagnetic N´eel order parameter N , which is respon- sible for pairing, with its coupling strength decreasing almost linearly with doping, whereas E$_{res}$ = 2 Δ$_{p}$ , which is respon- sible for condensation. Therefore, E$_{res}$ must scale with the superconducting transition temperature T$_{c}$ , i.e., E$_{res}$ ∼ k$_{B}$T$_{c}$ , with k$_{B}$ being the Boltzmann constant [see Fig. 1, right panel]. Similarly, 2 Δ ∗ scales as 2 Δ ∗ ∼ k$_{B}$T $^{∗}$, with T ∗ being the so- called pseudogap temperature [13, 14]. In addition, one may expect that E$_{res}$ < 2 Δ$_{d}$ , simply due to the fact that the pre- dominant d -wave superconducting component survives ther- mal fluctuations. Considering that both the superconductin g gap Δ$_{d}$ and the transition temperature T$_{c}$ characterize the su- perconductivity, they should track each other in the entire dop- ing range, implying Δ$_{d}$ ∼ k$_{B}$T$_{c}$ . In fact, for the t − J model, our simulation indicates that E$_{res}$ ≈ 1 . 25 Δ$_{d}$ [9]. This in turn allows us to estimate a universal coe ffi cient κ ≈ 5 . 37 in the scaling relation: E$_{res}$ = κ k$_{B}$T$_{c}$ . and res anti-ferromagnetic Neel order parameter res res res with being the Boltzmann constant [see Fig. , right panel]. , with called pseudogap temperature [13 14]. In addition, one may res mal fluctuations. Considering that both the superconducting gap res 25 ]. This in turn 37 res Note that the two distinct energy scales in the underdoped regime are split o ff from one single energy scale in the (heav- ily) overdoped regime. This naturally results in a crossover from the Bose-Einstein condensation (BEC) regime to the Bardeen-Cooper-Schrie ff er (BCS) regime, as conjectured in Ref. [15], which in turn is essentially equivalent to the phase fluctuation picture proposed by Emery and Kivelson [16]. However, there is an important di ff erence: the superconduc- tivity weakens in the heavily underdoped regime, not only be- cause of the loss of phase coherence, but also because of the decrease of the superconducting gap Δ$_{d}$ with underdoping. Bardeen-Cooper-Schrie er (BCS) regime, as conjectured in However, there is an important di erence: the superconduc- Now a fundamental question is whether or not such a sce- nario is really relevant to the high T$_{c}$ problem. This brings us to the phenomenology of the high temperature cuprate super- conductors. underdoped regime merge into one single energy scale in the (heavily) overdoped regime as a consequence of the evolution of the Fermi arcs in the underdoped regime to a large Fermi surface in the (heavily) overdoped regime [30 31]. We em- phasize that the pseudogap near the antinodal region does not characterize a precursor to the superconducting state, in the sense that the pseudogap smoothly evolves into the supercon- ducting gap at 13 30 31]. Instead, it coexists with the superconducting gap of the -wave symmetry in the su- perconducting state. More likely, a precursor pairing occurs in the nodal region [32], with its onset temperature lower than , but above , which may be identified with the Nernst regime [33 34]. As observed, the superconducting gap near the nodes scales as 29]. This makes a strong case for our argument, if one takes into account the smallness of the -wave superconducting gap. On the other hand, ample evi- dence has been accumulated, over the years, for the universal scaling relation res , valid for both soft modes, i.e., the spin-triplet resonance mode in inelastic neutron scattering experiments [35 36 37 38 39 40 41] and the spin-singlet mode observed as a peak in electronic Raman scatter- ing [21 42 43 44 45 46], respectively. The experimentally determined is around 6, quite close to our theoretical es- timate. This presents a possible resolution to the mysterious problem [42]: the mode is a charge collective mode as a bound state of (quasiparticle) singlet pairs originating from (heavily) overdoped regime as a consequence of the evolution of the Fermi arcs in the underdoped regime to a large Fermi surface in the (heavily) overdoped regime [30, 31]. We em- phasize that the pseudogap near the antinodal region does not characterize a precursor to the superconducting state, in the sense that the pseudogap smoothly evolves into the supercon- ducting gap at T$_{c}$ [13, 30, 31]. Instead, it coexists with the superconducting gap of the d + s -wave symmetry in the su- perconducting state. More likely, a precursor pairing occurs in the nodal region [32], with its onset temperature lower than T ∗ , but above T$_{c}$ , which may be identified with the Nernst regime [33, 34]. As observed, the superconducting gap near the nodes scales as k$_{B}$T$_{c}$ [29]. This makes a strong case for our argument, if one takes into account the smallness of the s -wave superconducting gap. On the other hand, ample evi- dence has been accumulated, over the years, for the universal scaling relation E$_{res}$ = κ k$_{B}$T$_{c}$ , valid for both soft modes, i.e., the spin-triplet resonance mode in inelastic neutron scattering experiments [35–41] and the spin-singlet mode observed as a A$_{1}$$_{g}$ peak in electronic Raman scattering [21, 42–46], respec- tively. The experimentally determined κ is around 6, quite close to our theoretical estimate. This presents a possible res- olution to the mysterious A$_{1}$$_{g}$ problem [42]: the A$_{1}$$_{g}$ mode is a charge collective mode as a bound state of (quasiparticle) singlet pairs originating from the fluctuating p$_{x}$ ( p$_{y}$ )-wave su- perconducting order. Second, is the pairing symmetry really of d + s -wave na- First, let us focus on the two distinct energy scales 2 Δ ∗ and E$_{res}$ . Physically, the two distinct energy scales mea- sure, respectively, the pairing strength and the coherence of the superfluid condensate. This naturally leads to two di ff er- ent phases: one is characterized by incoherent pairing, which may be identified with the pseudogap phase; the other is as- sociated with the emergence of a coherent condensate of su- perconducting pairs, which may be identified with the super- conducting phase of d + s -wave symmetry [see Fig. 2]. Evi- dence for the two distinct energy scales was reported in angle- resolved photoemission spectra [17–20], electronic Raman spectra [21–24], scanning tunneling microscopy [25], c -axis conductivity [26], Andreev reflection [27], magnetic penet ra- tion depth [28], and other probes (for a review, see, Ref. [29]. Actually, these studies indicate that the gap near the antinodal region, which is identified as the pseudogap, does not scale with T$_{c}$ in the underdoped regime, whereas the gap near the nodal region may be identified as the superconducting order parameter in the cuprates [17–23]. This identification o ff ers a natural explanation why the two distinct energy scales in the underdoped regime merge into one single energy scale in the and $_{res}$. the superfluid condensate. This naturally leads to two di er- perconducting pairs, which may be identified with the su- perconducting phase of -wave symmetry [see Fig. ]. Evidence for the two distinct energy scales was reported in angle-resolved photoemission spectra [17 18 19 20], elec- tronic Raman spectra [21 22 23 24], scanning tunneling microscopy [25], -axis conductivity [26], Andreev reflec- tion [27], magnetic penetration depth [28], and other probes (for a review, see, Ref. [29]. Actually, these studies indi- cate that the gap near the antinodal region, which is identi- fied as the pseudogap, does not scale with in the under- doped regime, whereas the gap near the nodal region may be identified as the superconducting order parameter in the cuprates [17 18 19 20 21 22 23]. This identification o ers a natural explanation why the two distinct energy scales in the