Quantitative theory of $T_c$ and the spin-gap-to-superconductivity conversion in the cuprates within the spin-charge-separated (RVB) kinetic-energy-recovery framework

Anderson's 2000 synthesis of the resonating-valence-bond (RVB) theory of the cuprate superconductors proposed that doping the Mott-insulating parent compounds yields spin-charge separation with vanishing quasiparticle residue ($Z=0$), a doping-dependent spin-gap (pseudogap) crossover $T^{*}(x)$ driven by a Cooper-like instability of the spinons, and superconductivity below $T^{*}$ caused by the recovery of kinetic energy frustrated by the opening of the spin gap. The paper itself stated that this picture 'is far from a quantitative theory of $T_c$': its doping scales rested on ad hoc interpolations $p_Fv_s\simeq J+x(t-J)$ and $J_{eff}(x)=J-tx$; the mechanism converting the spin-gap RVB state into a charge-coherent superconducting condensate was sketched, not derived; the role of interlayer (c-axis) kinetic energy in bilayer and multilayer materials relative to the one-layer mechanism was left unsettled; and the claim that the Landau Fermi liquid is a $Z_2$-symmetric quantum critical point was asserted without formal derivation. The central objective - a quantitative, material-specific theory of $T^{*}(x)$ and $T_c(x)$ derived rather than fitted from the kinetic-energy-recovery mechanism, together with a settled interlayer kinetic-energy budget - remains open.

Background

The high-$T_c$ superconductors discovered in 1986 are layered copper oxides ('cuprates') such as La$_{2-x}$Sr$_x$CuO$_4$, YBa$_2$Cu$_3$O$_{6+\delta}$, and Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$. Their undoped parent compounds are antiferromagnetic Mott insulators - insulators in which the on-site Coulomb repulsion $U$, rather than band filling, localizes one electron per Cu site - and superconductivity appears when holes are introduced at a concentration $x$ per Cu site. Because $U$ is the largest energy scale, the low-energy description is the $t$-$J$ Hamiltonian: holes hop between nearest-neighbor Cu sites with amplitude $t$, while the localized spins interact through the antiferromagnetic superexchange $J\sim t^2/U$.

The resonating-valence-bond (RVB) idea (Anderson, 1987) holds that doping does not immediately restore a conventional metal: the singlet 'valence bonds' of the insulator persist as a fluctuating background while doped holes move through it, so that the electron fractionalizes into spinons (neutral, fermionic spin-$\tfrac{1}{2}$ excitations) and holons (charged, spinless excitations). This spin-charge separation implies that the quasiparticle residue $Z$ - the weight of the sharp electron-like pole of the single-particle Green's function whose existence defines the Landau Fermi liquid, the standard theory of interacting metals - is zero ($Z=0$). Anderson's 2000 synthesis argued that such a 'quantum protectorate' (a term the source borrows from Laughlin for a state whose many-body correlations are so strong that perturbations scattering individual particles, such as phonons or impurities, are ineffective) extends across the cuprate phase diagram, and that this, rather than an adjustable failure of perturbation theory, explains the observed absence of phonon self-energy and impurity-scattering signatures.

Upon doping, the spin sector is proposed to pass into a $d$-wave RVB spin-gap state below a crossover temperature $T^{*}$ (the pseudogap temperature observed in NMR, optical, photoemission, and neutron experiments), with antiferromagnetism surviving only as a soft collective mode - identified with the well-known neutron-scattering resonance. The crossover is modeled as a Cooper-like instability of the spinons,

$kT^{*}\sim p_Fv_s\,\exp\!\left(-\frac{p_Fv_s}{J_{eff}(x)}\right),$

where $p_F$ is the Fermi momentum of the doped holes, $v_s$ the spinon velocity, and $J_{eff}(x)$ an effective antiferromagnetic interaction that weakens with doping. To reproduce the rapid decrease of $T^{*}$ with doping and its disappearance near $x\simeq 0.3$, the 2000 paper fixed these scales by rough interpolation: $p_Fv_s\simeq J+x(t-J)$ (connecting the $x\to 0$ insulating limit $v_s=J/p_F$ to the $x\to 1$ limit $v_s=t/p_F$) and $J_{eff}(x)=J-tx$.

The RVB spin-gap state is itself not superconducting; superconductivity is attributed to frustrated kinetic energy. Opening a gap near the Fermi level prevents the momentum distribution $n(k)$ from approaching its optimal step-function form, raising the kinetic energy; by the Ferrell-Glover-Tinkham (FGT) optical sum rule,

$\int_0^{\infty}\sigma(\omega)\,d\omega=e^2\langle K.E.\rangle,$

(the integral of the real part of the optical conductivity measures the electronic kinetic energy), the optical gap observed in underdoped cuprates implies a kinetic-energy cost that scales with the hole concentration $x$, setting an upper bound $T_c<T_{KE}\propto x$. Making the charge fluctuations coherent below $T^{*}$ recovers that energy, and this recovery is proposed as the immediate cause of superconductivity, with an $s$-wave-like condensation in the charge channel accompanying the $d$-wave spin gap. A second, earlier proposal (Anderson, 1995) attributed the pairing energy specifically to interlayer (c-axis) tunneling in bilayer and multilayer cuprates; by 2000 Anderson conceded that a one-layer mechanism is 'quite effective in some cases' while retaining a role for interlayer kinetic energy - leaving their relative importance unsettled.

Finally, the framework rests on a structural claim about the Fermi liquid itself: that the Landau quasiparticle spectrum carries an extra discrete $Z_2$ symmetry - local in momentum space, mixing spin and charge (roughly, a hole-particle transformation applied to one spin species only) - which is not a symmetry of the interacting Hamiltonian. If so, the Fermi liquid is not a generic stable phase but a fine-tuned quantum critical point, and no summation of Feynman diagrams about it can reach the spin-charge-separated states. In one dimension the claim is vindicated: the Fermi liquid gives way to the Luttinger liquid, a state with separate spin and charge velocities. In two and three dimensions the claim was asserted in the source paper and elaborated in a companion note by Anderson and Haldane (2000), but never formalized.

Subsequent work filled parts of this picture without closing it. Gutzwiller-projected (renormalized) mean-field theory - a variational treatment in which doubly occupied sites are projected out of the trial wave function to enforce the Mott no-double-occupancy constraint - and slave-boson approaches - mean-field schemes that implement the spinon-holon fractionalization explicitly - reproduce $d$-wave pairing trends and pseudogap phenomenology qualitatively; phase-fluctuation arguments (Emery and Kivelson, 1995) explain the empirical proportionality between $T_c$ and the superfluid density; optical sum-rule analyses of the c-axis kinetic-energy change below $T_c$ led different groups to opposite conclusions about interlayer tunneling; and Anderson's own later program (the projected and 'hidden' Fermi liquid, 2005-2011, and his 2016 'Last Words') developed the normal-state theory and qualitative scale arguments but not a derivable, material-specific $T_c(x)$. Standard reviews of the doped-Mott-insulator problem accordingly still lack a quantitative microscopic theory of $T_c$. The unresolved target is stated below.

Problem Statement

Within the spin-charge-separated (RVB) description of the doped cuprate planes - modeled by the two-dimensional $t$-$J$ Hamiltonian (hole hopping amplitude $t$, antiferromagnetic superexchange $J$, hole density $x$ per Cu site), supplemented where relevant by interlayer hopping for bilayer and multilayer materials - construct the quantitative theory of the superconducting transition that Anderson's 2000 synthesis explicitly left unfinished. In that synthesis the pseudogap crossover is estimated from a Cooper-like spinon instability,

$kT^{*}\sim p_Fv_s\,\exp\!\left(-\frac{p_Fv_s}{J_{eff}(x)}\right),$

with the rough interpolations $p_Fv_s\simeq J+x(t-J)$ and $J_{eff}(x)=J-tx$, and superconductivity below $T^{*}$ is attributed to the recovery of kinetic energy frustrated by the opening of the spin gap, bounded by $T_c<T_{KE}\propto x$ through the Ferrell-Glover-Tinkham sum rule. The problem is to deliver, from the microscopic Hamiltonian rather than from fitted or interpolated inputs, the following jointly supporting components of one common objective:

1. (a) Derived expressions for the spinon scale $p_Fv_s(x)$ and the effective pairing interaction $J_{eff}(x)$ (or their justifiably corrected replacements); (b) from these scales, quantitative, material-specific predictions of $T^{*}(x)$ and $T_c(x)$; (c) quantitative, material-specific predictions of the superconducting condensation energy; and (d) quantitative, material-specific predictions of the optical spectral-weight transfer - the predictions of (b)-(d) being for the cuprate families YBa$_2$Cu$_3$O$_{6+\delta}$, Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$, and La$_{2-x}$Sr$_x$CuO$_4$, including the effects of multilayer structure.

2. The microscopic mechanism and symmetry content by which the spin-gap RVB state (a $d$-wave spin gap with a claimed $s$-wave-like charge-channel condensation) converts into a charge-coherent superconducting state - a derivation of the conversion rather than a sketch.

3. A quantitative apportionment of the interlayer (c-axis, bilayer and multilayer) kinetic-energy recovery versus the single-layer mechanism in determining material-specific $T_c$.

These deliverables serve one common objective: a quantitative, derivable theory of $T_c$ in the cuprates from spin-charge separation and kinetic-energy recovery. Such a theory derives its energy scales rather than fitting them and reproduces - within the scatter of the measured data - the measured $T^{*}(x)$ and $T_c(x)$ phase diagrams, the superconducting condensation energies, and the optical spectral-weight transfer of the cuprate families named in component 1, together with the conversion mechanism and symmetry content of component 2 and the interlayer kinetic-energy budget of component 3.

References

1. P. W. Anderson, 'Spin-charge separation is the key to the high Tc cuprates', Physica C 341-348, 9-10 (2000). DOI: 10.1016/S0921-4534(00)00378-6; arXiv:cond-mat/0007185. https://doi.org/10.1016/s0921-4534(00)00378-6
2. P. W. Anderson, 'The Resonating Valence Bond State in La2CuO4 and Superconductivity', Science 235, 1196-1198 (1987). DOI: 10.1126/science.235.4793.1196. https://doi.org/10.1126/science.235.4793.1196
3. P. W. Anderson and F. D. M. Haldane, 'The Symmetries of Fermion Fluids at Low Dimensions', arXiv:cond-mat/0009309 (2000). https://arxiv.org/abs/cond-mat/0009309
4. P. W. Anderson, 'Physics of the Pseudogap Phase of High Tc Cuprates, or, RVB Meets Umklapp', J. Phys. Chem. Solids 63, 2145-2148 (2002). DOI: 10.1016/S0022-3697(02)00247-0; arXiv:cond-mat/0108522. https://doi.org/10.1016/S0022-3697(02)00247-0
5. P. W. Anderson, 'Physics of the Resonating Valence Bond (Pseudogap) State of the Doped Mott Insulator: Spin-Charge Locking', Phys. Rev. Lett. 96, 017001 (2006). DOI: 10.1103/PhysRevLett.96.017001. https://doi.org/10.1103/PhysRevLett.96.017001
6. P. W. Anderson, 'The "Strange Metal" is a Projected Fermi Liquid with Edge Singularities', arXiv:cond-mat/0512471 (2005); published as Nature Physics 2, 626-630 (2006). DOI: 10.1038/nphys388. https://doi.org/10.1038/nphys388
7. P. A. Casey and P. W. Anderson, 'Hidden Fermi Liquid: Self-Consistent Theory for the Normal State of High-Tc Superconductors', Phys. Rev. Lett. 106, 097002 (2011). DOI: 10.1103/PhysRevLett.106.097002; arXiv:1101.3609. https://doi.org/10.1103/PhysRevLett.106.097002
8. P. W. Anderson, 'Last Words on the Cuprates', arXiv:1612.03919 (2016). https://arxiv.org/abs/1612.03919
9. P. W. Anderson, 'Interlayer Tunneling Mechanism for High-Tc Superconductivity: Comparison with c Axis Infrared Experiments', Science 268, 1154-1155 (1995). DOI: 10.1126/science.268.5214.1154. https://doi.org/10.1126/science.268.5214.1154
10. A. J. Leggett, 'Interlayer Tunneling Models of Cuprate Superconductivity: Implications of a Recent Experiment', Science 274, 587-589 (1996). DOI: 10.1126/science.274.5287.587. https://doi.org/10.1126/science.274.5287.587
11. S. Chakravarty, 'Do electrons change their c-axis kinetic energy upon entering the superconducting state?', Eur. Phys. J. B 5, 337-343 (1998). DOI: 10.1007/s100510050451; arXiv:cond-mat/9801025. https://doi.org/10.1007/s100510050451
12. S. Chakravarty, H. Kee, and E. Abrahams, 'Frustrated Kinetic Energy, the Optical Sum Rule, and the Mechanism of Superconductivity', Phys. Rev. Lett. 82, 2366-2369 (1999). DOI: 10.1103/PhysRevLett.82.2366. https://doi.org/10.1103/PhysRevLett.82.2366
13. D. Munzar, T. Holden, and C. Bernhard, 'Approximate tight-binding sum rule for the superconductivity-related change of c-axis kinetic energy in multilayer cuprate superconductors', Phys. Rev. B 67, 020501 (2003). DOI: 10.1103/PhysRevB.67.020501; arXiv:cond-mat/0207511 (2002). https://doi.org/10.1103/PhysRevB.67.020501
14. D. N. Basov and T. Timusk, 'Electrodynamics of high-Tc superconductors', Rev. Mod. Phys. 77, 721-779 (2005). DOI: 10.1103/RevModPhys.77.721. https://doi.org/10.1103/RevModPhys.77.721
15. J. P. Carbotte and E. Schachinger, 'c-axis optical sum in underdoped superconducting cuprates', Phys. Rev. B 86, 224512 (2012). DOI: 10.1103/PhysRevB.86.224512. https://doi.org/10.1103/PhysRevB.86.224512
16. E. Gull and A. J. Millis, 'Superconducting and Pseudogap effects on the interplane conductivity and Raman scattering cross section in the two dimensional Hubbard Model', Phys. Rev. B 88, 075127 (2013). DOI: 10.1103/PhysRevB.88.075127; arXiv:1304.6406. https://doi.org/10.1103/PhysRevB.88.075127
17. Y. Yanase and M. Ogata, 'Kinetic Energy, Condensation Energy, Optical Sum Rule and Pairing Mechanism in High-Tc Cuprates', J. Phys. Soc. Jpn. 74, 1534-1543 (2005). DOI: 10.1143/JPSJ.74.1534; arXiv:cond-mat/0412508 (2004). https://doi.org/10.1143/JPSJ.74.1534
18. M. Ogata, H. Yokoyama, Y. Yanase, Y. Tanaka, and H. Tsuchiura, 'Kinetic-energy pairing and condensation energy in cuprates', J. Phys. Chem. Solids 67, 37-40 (2006). DOI: 10.1016/j.jpcs.2005.10.017. https://doi.org/10.1016/j.jpcs.2005.10.017
19. P. Wróbel, R. Eder, and P. Fulde, 'Pairing-induced kinetic energy lowering in doped antiferromagnets', J. Phys.: Condens. Matter 15, 6599-6616 (2003). DOI: 10.1088/0953-8984/15/38/025. https://doi.org/10.1088/0953-8984/15/38/025
20. C. Giannetti, F. Cilento, S. Dal Conte, G. Coslovich, G. Ferrini, H. Molegraaf, M. Raichle, R. Liang, H. Eisaki, M. Greven, A. Damascelli, D. van der Marel, and F. Parmigiani, 'Revealing the high-energy electronic excitations underlying the onset of high-temperature superconductivity in cuprates', Nature Communications 2, 353 (2011). DOI: 10.1038/ncomms1354. https://doi.org/10.1038/ncomms1354
21. V. J. Emery and S. A. Kivelson, 'Superconductivity in Bad Metals', Phys. Rev. Lett. 74, 3253-3256 (1995). DOI: 10.1103/PhysRevLett.74.3253. https://doi.org/10.1103/PhysRevLett.74.3253
22. P. A. Lee, N. Nagaosa, and X.-G. Wen, 'Doping a Mott insulator: Physics of high-temperature superconductivity', Rev. Mod. Phys. 78, 17-85 (2006). DOI: 10.1103/RevModPhys.78.17. https://doi.org/10.1103/RevModPhys.78.17
23. M. Guidry, Y. Sun, and L.-A. Wu, 'The Superconducting Critical Temperature', arXiv:2004.04066 (2020). https://arxiv.org/abs/2004.04066
24. L. Yang, I. Hamad, L. O. Manuel, and A. Feiguin, 'The fate of pairing and spin-charge separation in the presence of long-range antiferromagnetism', Phys. Rev. B 105, 195104 (2022). DOI: 10.1103/PhysRevB.105.195104; arXiv:2110.15856 (2021). https://doi.org/10.1103/PhysRevB.105.195104