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arXiv:cond-mat/0410229 · Physical Review B

Doping and temperature dependence of superfluid weight and spectral function: Universal scaling behavior of the pseudogap

Sung-Sik Lee, Sung‐Ho Suck Salk

Published Apr 26, 2005• 7 citations
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Abstract

Using the improved U(1) and SU(2) slave-boson approaches of the $t\ensuremath{-}J$ Hamiltonian [S.-S. Lee and Sung-Ho Suck Salk, Phys. Rev. B 64, 052501 (2001)] that we developed recently, we study the doping and temperature dependence of superfluid weight and spectral function and discuss our finding of the universal scaling behavior of the pseudogap. It is shown that at low hole doping concentrations $x$ and at low temperatures $T$, there exists a propensity of linear decrease in the superfluid weight ${n}_{s}∕{m}^{*}$ with temperature $T$, and a tendency of doping independence in the linearly decreasing slopes of $({n}_{s}∕{m}^{*})(x,T)$ with $T$ in qualitative agreement with the experimentally observed relation; that is, $({n}_{s}∕{m}^{*})(x,T)=({n}_{s}∕{m}^{*})(x,0)\ensuremath{-}\ensuremath{\alpha}T$, where $\ensuremath{\alpha}$ is a constant. It is also demonstrated that there exists a boomerang behavior; that is, both ${T}_{c}$ and ${n}_{s}∕{m}^{*}$ increase with hole doping concentration $x$ in the underdoped region, reach a saturation (maximum) at a hole doping above optimal doping, and decrease beyond the saturation point in the overdoped region, in agreement with $\ensuremath{\mu}\mathrm{SR}$ measurements. Based on our improved SU(2) slave-boson approach to the $t\ensuremath{-}J$ Hamiltonian, we further investigate the doping and temperature dependence of the spectral function and discuss our theoretical finding of a scaling behavior of the pseudogap. In addition, we discuss the cause of the hump and quasiparticle peak in the observed spectral functions of high ${T}_{c}$ cuprates. It is demonstrated that the sharpening of the observed quasiparticle peak below ${T}_{c}$ is attributed to the bose condensation of holon pair, in agreement with observations. From the predicted ratios of the pseudogap ${\ensuremath{\Delta}}_{0}$ to both the superconducting temperature ${T}_{c}$ and the pseudogap temperature ${T}^{*}$, i.e., $2{\ensuremath{\Delta}}_{0}∕{k}_{B}{T}_{c}$ and $2{\ensuremath{\Delta}}_{0}∕{k}_{B}{T}^{*}$, respectively, as a function of hole doping concentration $x$, we find that there exists a universal scaling behavior (sample independence) of these ratios with doping, by showing a nonlinearly decreasing behavior of the former; that is, $2{\ensuremath{\Delta}}_{0}∕{k}_{B}{T}_{c}\ensuremath{\sim}{x}^{\ensuremath{-}\ensuremath{\alpha}}$ with $\ensuremath{\alpha}\ensuremath{\sim}2$ and a near doping independence of the latter; that is, $2{\ensuremath{\Delta}}_{0}∕{k}_{B}{T}^{*}\ensuremath{\approx}4--6$.

DOI
10.1103/physrevb.71.134518

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Using the improved U(1) and SU(2) slave-boson approaches of the t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J Hamiltonian [S.-S. Lee and Sung-Ho Suck Salk, Phys. Rev. B 64, 052501 (2001)] that we developed recently, we study the doping and temperature dependence of superfluid weight and spectral function and discuss our finding of the universal scaling behavior of the pseudogap. It is shown that at low hole doping concentrations xxx and at low temperatures TTT, there exists a propensity of linear decrease in the superfluid weight ns∕m∗{n}_{s}∕{m}^{*}ns​∕m∗ with temperature TTT, and a tendency of doping independence in the linearly decreasing slopes of (ns∕m∗)(x,T)({n}_{s}∕{m}^{*})(x,T)(ns​∕m∗)(x,T) with TTT in qualitative agreement with the experimentally observed relation; that is, (ns∕m∗)(x,T)=(ns∕m∗)(x,0)\ensuremath−\ensuremathαT({n}_{s}∕{m}^{*})(x,T)=({n}_{s}∕{m}^{*})(x,0)\ensuremath{-}\ensuremath{\alpha}T(ns​∕m∗)(x,T)=(ns​∕m∗)(x,0)\ensuremath−\ensuremathαT, where \ensuremathα\ensuremath{\alpha}\ensuremathα is a constant. It is also demonstrated that there exists a boomerang behavior; that is, both Tc{T}_{c}Tc​ and ns∕m∗{n}_{s}∕{m}^{*}ns​∕m∗ increase with hole doping concentration xxx in the underdoped region, reach a saturation (maximum) at a hole doping above optimal doping, and decrease beyond the saturation point in the overdoped region, in agreement with \ensuremathμSR\ensuremath{\mu}\mathrm{SR}\ensuremathμSR measurements. Based on our improved SU(2) slave-boson approach to the t\ensuremath−Jt\ensuremath{-}Jt\ensuremath−J Hamiltonian, we further investigate the doping and temperature dependence of the spectral function and discuss our theoretical finding of a scaling behavior of the pseudogap. In addition, we discuss the cause of the hump and quasiparticle peak in the observed spectral functions of high Tc{T}_{c}Tc​ cuprates. It is demonstrated that the sharpening of the observed quasiparticle peak below Tc{T}_{c}Tc​ is attributed to the bose condensation of holon pair, in agreement with observations. From the predicted ratios of the pseudogap \ensuremathΔ0{\ensuremath{\Delta}}_{0}\ensuremathΔ0​ to both the superconducting temperature Tc{T}_{c}Tc​ and the pseudogap temperature T∗{T}^{*}T∗, i.e., 2\ensuremathΔ0∕kBTc2{\ensuremath{\Delta}}_{0}∕{k}_{B}{T}_{c}2\ensuremathΔ0​∕kB​Tc​ and 2\ensuremathΔ0∕kBT∗2{\ensuremath{\Delta}}_{0}∕{k}_{B}{T}^{*}2\ensuremathΔ0​∕kB​T∗, respectively, as a function of hole doping concentration xxx, we find that there exists a universal scaling behavior (sample independence) of these ratios with doping, by showing a nonlinearly decreasing behavior of the former; that is, 2\ensuremathΔ0∕kBTc\ensuremath∼x\ensuremath−\ensuremathα2{\ensuremath{\Delta}}_{0}∕{k}_{B}{T}_{c}\ensuremath{\sim}{x}^{\ensuremath{-}\ensuremath{\alpha}}2\ensuremathΔ0​∕kB​Tc​\ensuremath∼x\ensuremath−\ensuremathα with \ensuremathα\ensuremath∼2\ensuremath{\alpha}\ensuremath{\sim}2\ensuremathα\ensuremath∼2 and a near doping independence of the latter; that is, 2\ensuremathΔ0∕kBT∗\ensuremath≈4−−62{\ensuremath{\Delta}}_{0}∕{k}_{B}{T}^{*}\ensuremath{\approx}4--62\ensuremathΔ0​∕kB​T∗\ensuremath≈4−−6.
Based on our improved SU(2) slave-boson approach (Phys. Rev. B 64, 052501(2001)) to the t-J Hamiltonian, we report a scaling behavior of pseudogap with doping and the temperature and doping dependence of spectral functions. In addition we discuss the cause of hump and quasi-particle peak in the observed spectral functions of high Tc cuprates. It is demonstrated that the sharpening of the observed quasi-particle peak below Tc is attributed to the bose condensation of holon pair. From the computed ratios of pseudogap Δ0Δ_0Δ0​ to both the superconducting temperature Tc and the pseudogap temperature T∗T^*T∗ as a function of hole doping concentration x, we find that there exists a universal scaling of these ratios with doping, that is, the hyperbolic scaling behavior of the former, 2Δ0kBTc∼x−α\frac{2 Δ_0}{k_B T_c} \sim x^{-α}kB​Tc​2Δ0​​∼x−α with α∼2α\sim 2α∼2 and near doping independence of the latter, 2Δ0kBT∗≈4∼6\frac{2 Δ_0}{k_B T^*} \approx 4 \sim 6kB​T∗2Δ0​​≈4∼6 are found.
Based on an improved U(1) slave-boson approach to the t-J Hamiltonian, we invest igate a relationship between the superfluid weight ns/m∗n_s/m^*ns​/m∗(the superconducting charge carrier density/the effective mass of the charge carrier) and the superc onducting temperature TcT_cTc​. From the present study we find a linear increase of ns/m∗n_s/m^*ns​/m∗ with TcT_cTc​ with th e doping concentration in the underdoped region, a saturation around the optimal doping and a decrease in both ns/m∗n_s/m^*ns​/m∗ and TcT_cTc​ in the overdoped region. Such a trend of the `boomerang' shaped locus in ns/m∗n_s/m^*ns​/m∗ vs. TcT_cTc​ with increas ing doping concentration from the underdoped to the heavily overdoped region is predicted to be in complete agreement with muon-spin-relaxation measurements. The boomerang behavior is found to occur in correlation with reduction in the sp in singlet pairing(spinon pairing) order in the heavily overdoped region.
Based on our recent holon-pair boson theory of the t-J Hamiltonian (Phys. Rev. B 64, 052501 (2001)) we report the doping dependence of the bose condensation energy, superfluid weight and spectral peak intensity. We find a universality of doping dependence in these physical quantities, by equally showing an arch shape in the variations of their magnitudes with the hole doping concentration. We find that all of these physical quantities scale well with the positive charge carrier (hole) density x, but not with the electron density 1-x for the entire range of hole doping. It is shown that the doping dependence of the condensation energy U at T = 0 K is given by the relation, U(0)≈αx2∣Δ0∣2U(0) \approx αx^2 |Δ_0|^2U(0)≈αx2∣Δ0​∣2 with Δ0Δ_0Δ0​, the pairing gap at 0 K and ααα, a constant.
To supplement our recent brief report on the theory of holon-pair boson approach to the t-J Hamiltonian [S.-S. Lee and Sung-Ho Suck Salk, Phys. Rev. B {\bf 64}, 052501(2001)], in this paper we present a full exposure to the theory, detailed physical implications and predicted various physical properties of high TcT_cTc​ cuprates. We discuss the significance of coupling (interplay) between the spin and charge degrees of freedom in the Heisenberg interaction term of the t-J Hamiltonian. We discuss its importance in causing the arch-shaped superconducting transition temperature TcT_cTc​ and the pseudogap (spin gap) temperature T∗T^*T∗ tangential to TcT_cTc​ in the overdoped region in the observed phase diagram of high TcT_cTc​ cuprates. A universal parabolic scaling behavior of T∗/TcT^*/T_cT∗/Tc​ (or Tc/T∗T_c/T^*Tc​/T∗) with hole doping concentration is predicted in agreement with observations, indicating that there exists correlation between the pseudogap (spin gap) phase and the superconducting phase through antiferromagnetic fluctuations. Our proposed holon-pair boson theory is shown to be self-consistent in that it not only yields the arch (dome) shape structure of TcT_cTc​ but also reproduces various other physical properties such as superfluid weight, bose condensation energy, spectral function, optical conductivity and spin susceptibility, including their temperature and doping dependence.
One of the major theoretical challenges in high- T c superconductivity is to first reproduce the observed phase diagrams that display the monotonously decreasing pseudogap temperature T * and the dome-shaped superconducting phase transition temperature T c in the plane of temperature vs. hole concentration. Earlier Lee and Salk [J. Korean Phys. Soc. 37, 545 (2000); Phys. Rev. B 64, 052501 (2001)] reported a successful reproduction of the phase diagram by providing a realistic gauge theoretic [SU(2)/U(1)] slave-boson approach to the t - J Hamiltonian. Most recently, we [S.-H. S. Salk, Quantum Studies: Mathematics and Foundations 5, 149 (2018)] presented a comprehensive discussion on both the SU(2) and the U(1) approaches from which one can readily understand the intimate relationship between the two formalisms and discussed that both approaches can lead to room-temperature superconductivity with suitably high values of the antiferromagnetic coupling constant J , owing to the demonstration of identical physical propensities, i.e. , the higher the J is, the higher the superconducting phase transition temperature T c is. Here, we discuss hither-to-unreported detailed numerical computations of the phase diagrams by varying the values of J by using the U(1) gauge slave-boson approach to the t − J Hamiltonian. For the sake of testing convergence, we vary the unit-cell lattice sizes from 10 × 10 to a sufficiently large size of 50 × 50. We find that even a small square lattice size of 20 × 20 is seen to show reliable agreement with the results for higher lattice sizes while the 50 × 50 lattice size displays complete convergence in both T* and T c . In addition, we present a physical analysis of the structure of the high- T c phase diagram, focusing on the role of the spin pairing order in association with the interplay between the pseudo-gap (spin gap) temperature and the bose-condensation temperature/dome-shaped superconducting phase transition temperature.