Using high temperature expansions for the equal time correlator S(q) and static susceptibility \ensuremath{\chi}(q) for the t-J model, we present evidence for quantum critical (QC), z=1, behavior at intermediate temperature in a broad range of t/J ratio, doping, and temperatures. We find that the dynamical susceptibility is very close to the universal scaling function computable for the asymptotic QC regime, and that the dominant energy scale is temperature. Our results are in excellent agreement with measurements of the spin-echo decay rate, 1/${\mathit{T}}_{2\mathrm{G}}$, in ${\mathrm{La}}_{2}$${\mathrm{CuO}}_{4}$, and provide qualitative understanding of both 1/${\mathit{T}}_{1}$ and 1/${\mathit{T}}_{2\mathrm{G}}$ nuclear relaxation rates in doped cuprates.
Using high temperature expansions for the equal time correlator S(q) and static susceptibility \ensuremath{\chi}(q) for the t-J model, we present evidence for quantum critical (QC), z=1, behavior at intermediate temperature in a broad range of t/J ratio, doping, and temperatures. We find that the dynamical susceptibility is very close to the universal scaling function computable for the asymptotic QC regime, and that the dominant energy scale is temperature. Our results are in excellent agreement with measurements of the spin-echo decay rate, 1/T2G, in {\mathrm{La}}_{2}$${\mathrm{CuO}}_{4}, and provide qualitative understanding of both 1/T1 and 1/T2G nuclear relaxation rates in doped cuprates.
The antiferromagnetic spin correlation function SQ, the staggered spin susceptibility ${\ensuremath{\chi}}_{\mathbf{Q}}$, and the energy scale ${\ensuremath{\omega}}_{FL}={S}_{\mathbf{Q}}∕{\ensuremath{\chi}}_{\mathbf{Q}}$ are studied numerically within the $t\text{\ensuremath{-}}J$ model and the Hubbard model, as relevant to cuprates. It is shown that ${\ensuremath{\omega}}_{FL}$, related to the onset of the non-Fermi-liquid spin response at $T>{\ensuremath{\omega}}_{FL}$, is very low in the regime below the ``optimum'' hole doping ${c}_{h}<{c}_{h}^{*}\ensuremath{\sim}0.16$, while it shows a steep increase in the overdoped regime. A quantitative analysis of the NMR spin-spin relaxation rate 1∕T2G for various cuprates reveals a similar behavior, indicating a pronounced crossover between a Fermi-liquid and a non-Fermi-liquid behavior as a function of doping.
Finite-temperature spin dynamics in the planar $t\ensuremath{-}J$ model are studied using a method based on the Lanczos diagonalization of small systems. The dynamical spin structure factor at moderate dopings shows the coexistence of free-fermion-like and spin-fluctuation time scales. At T<J, low-frequency and static susceptibilities show a pronounced T dependence, supporting a scenario related to the marginal Fermi liquid one, for the explanation of neutron-scattering and NMR relaxation experiments in cuprates. Calculated NMR relaxation rates reasonably reproduce experimental ones.
Recent results for the finite-temperature static and dynamical properties of the planar t−J model, obtained by a novel numerical method for small correlated systems, are reviewed. Of particular interest are T>0 charge and spin response functions: optical conductivity σ(ω) and spin susceptibility χ(q,ω), which show very universal features at intermediate doping. In spin dynamics the universality is best seen in the local spin correlation function, which appears to be T-independent and only weakly doping dependent. Results apply directly to the inelastic neutron scattering and to the NMR relaxation experiments in cuprates. Analysing σ(ω) we find that the current correlation function C(ω) is nearly T- and moreover ω-independent in the regime $T
A relaxation-function theory for the dynamic spin susceptibility in the $t\text{\ensuremath{-}}J$ model is presented. By a sum-rule-conserving generalized mean-field approximation (GMFA), the two-spin correlation functions of arbitrary range, the staggered magnetization, the uniform static susceptibility, and the antiferromagnetic correlation length are calculated in a wide region of hole doping and temperatures. A good agreement with available exact diagonalization (ED) data is found. The correlation length is in reasonable agreement with neutron-scattering experiments on ${\text{La}}_{2\ensuremath{-}\ensuremath{\delta}}{\text{Sr}}_{\ensuremath{\delta}}{\text{CuO}}_{4}$. Going beyond the GMFA, the self-energy is calculated in the mode-coupling approximation. The spin dynamics at arbitrary frequencies and wave vectors is studied for various temperatures and hole doping. At low doping a spin-wave-type behavior is found as in the Heisenberg model, while at higher doping a strong damping caused by hole hopping occurs, and a relaxation-type spin dynamics is observed in agreement with the ED results. The local spin susceptibility and its $\ensuremath{\omega}/T$ scaling behavior are calculated in a reasonable agreement with experimental and ED data.
We study scaling behavior in two-dimensional, square lattice, S=1/2 and S=1 Heisenberg antiferromagnets using the data on full q dependences of the equal-time structure factor and the static susceptibility, calculated through high-temperature series expansions. We also carry out comparisons with a model of two coupled S=1/2 Heisenberg planes with the interlayer exchange coupling tuned to the T=0 critical point (two-plane model hereafter). For both S=1/2 and S=1 models, we separately determine the spin-wave velocity c and mass m=c/\ensuremath{\xi}, in addition to the correlation length, \ensuremath{\xi}, and find that c is temperature dependent; only for temperatures below T\ensuremath{\lesssim}JS, where J is the exchange coupling, c approaches its known T=0 values, c0. This nonuniversal lattice effect is caused by the quantum nature of spin, and is therefore not captured by the quantum nonlinear \ensuremath{\sigma} model.Despite this temperature dependence of the spin-wave velocity, full q and \ensuremath{\omega} dependences of the dynamical susceptibility \ensuremath{\chi}(q,\ensuremath{\omega}) agree with those of the universal scaling function, computable for the \ensuremath{\sigma} model, for temperatures up to T0\ensuremath{\sim}0.6c0/a, and their further analysis leads us to the inference that below T0 the S=1 model is in the renormalized classical (RC) regime, the two-plane model is in the quantum critical (QC) regime in agreement with earlier work, and the S=1/2 model exhibits a RC-QC crossover, centered around crossover and for the two-plane model at all T=0.55J. In particular, for the S=1/2 model above the RC-QC temperatures where calculated, the obtained spin-wave mass m=c/\ensuremath{\xi} is in excellent agreement with the known universal QC prediction, m\ensuremath{\simeq}1.04T. In contrast, for the S=1/2 model below the RC-QC crossover, and for the S=1 model at all temperatures, the behavior agrees with the exact RC expression. for all three models, nonuniversal behavior occurs above T\ensuremath{\sim}0.6c0/a. Our results strongly support the conjecture by Chubukov and Sachdev that the S=1/2 model is close enough to the T=0 critical point to exhibit QC behavior at intermediate temperatures.
We demonstrate quantum critical scaling for an S=1/2 Heisenberg antiferromagnetic chain compound Cu(C4H4N2)(NO3)2 in a magnetic field around saturation, by analyzing previously reported magnetization [Y. Kono et al., Phys. Rev. Lett. 114, 037202 (2015)], thermal expansion [J. Rohrkamp et al., J. Phys.: Conf. Ser. 200, 012169 (2010)], and NMR relaxation data [H. K\"uhne et al., Phys. Rev. B 80, 045110 (2009)]. The scaling of magnetization is demonstrated through collapsing the data for a range of both temperature and field onto a single curve without making any assumption for a theoretical form. The data collapse is subsequently shown to closely follow the theoretically predicted scaling function without any adjustable parameters. Experimental boundaries for the quantum critical region could be drawn from the variable range beyond which the scaled data deviate from the theoretical function. Similarly to the magnetization, quantum critical scaling of the thermal expansion is also demonstrated. Further, the spin dynamics probed via NMR relaxation rate 1/T1 close to the saturation is shown to follow the theoretically predicted quantum critical behavior as $1/{T}_{1}\ensuremath{\propto}{T}^{\ensuremath{-}0.5}$ persisting up to temperatures as high as ${k}_{\mathrm{B}}T\ensuremath{\simeq}J$, where J is the exchange coupling constant.