The thermodynamic quantities (spin-spin correlation functions $\ensuremath{\langle}{\mathbf{S}}_{0}{\mathbf{S}}_{n}\ensuremath{\rangle}$, correlation length $\ensuremath{\xi}$, spin susceptibility $\ensuremath{\chi}$, and specific heat
CV) of the frustrated one-dimensional
J1-
J2 Heisenberg ferromagnet with arbitrary spin quantum number
S below the quantum critical point, i.e., for
J2<∣J1∣/4, are calculated using a rotation-invariant Green-function formalism and full diagonalization as well as a finite-temperature Lanczos technique for finite chains of up to
N=20 sites. The low-temperature behavior of the susceptibility $\ensuremath{\chi}$ and the correlation length $\ensuremath{\xi}$ are well described by $\ensuremath{\chi}=\frac{2}{3}{S}^{4}\left(|{J}_{1}|\ensuremath{-}4{J}_{2}\right){T}^{\ensuremath{-}2}+A{S}^{5/2}{\left(|{J}_{1}|\ensuremath{-}4{J}_{2}\right)}^{1/2}{T}^{\ensuremath{-}3/2}$ and $\ensuremath{\xi}={S}^{2}\left(|{J}_{1}|\ensuremath{-}4{J}_{2}\right){T}^{\ensuremath{-}1}+B{S}^{1/2}{\left(|{J}_{1}|\ensuremath{-}4{J}_{2}\right)}^{1/2}{T}^{\ensuremath{-}1/2}$ with $A\ensuremath{\approx}1.1\ensuremath{\cdots}1.2$ and $B\ensuremath{\approx}0.84\ensuremath{\cdots}0.89$. The vanishing of the factors in front of the temperature at
J2=∣J1∣/4 indicates a change of the critical behavior of $\ensuremath{\chi}$ and $\ensuremath{\xi}$ at $T\ensuremath{\rightarrow}0$. The specific heat may exhibit an additional frustration-induced low-temperature maximum when approaching the quantum critical point. Such a maximum appears for
S=1/2 and
S=1, but was not found for
S>1.
Based on the ballistic macroscopic fluctuation theory, the integration of the spin correlation function (spin conductivity) is analyzed for the spin-1/2 XXZ chain in the critical regime. In the time when the magnetization of an infinite spin chain fluctuates from an initial state with a wavelength as long as the infinite length
N, the equal-time two-point spin correlation function is scaled up to
O(1/N). In the state where the ballistic spin transport decays at high temperature
T, the diffusive transport remains on a large scale. We show that the spin conductivity is proportional to
1/T in the limit
T→∞ and its high temperature proportionality constant diverges in the case where one-quasiparticle magnetization is infinitely large. This analysis informs that the superdiffusive spin transport is driven by the
1/N-scaled long-range spin correlation and sheds a light on the dynamic scaling in spin transport at the isotropic point.
Qualitative changes in thermodynamic and single-particle properties characterize the transition between the pseudogapped electronic liquid and the Fermi liquid. Recent cold-atom experiments on a simulator of the Hubbard model with nearest-neighbor hoppings [L. H. Kendrick , .] showed that the isothermal compressibility κ ( δ ) has a maximum as a function of doping δ . Here, we use the two-particle self-consistent plus approach to explain these experiments and connect the maximum in κ ( δ ) to the transformation of the single-particle spectrum from the pseudogapped to the metallic regime. This elucidates the nature of the pseudogap. Specifically, the maximum in κ ( δ ) practically coincides with the doping at which the precursor of the lower ( π , π ) spin-density-wave (SDW) band at the antinodal point crosses the zero-frequency ω = 0 . The Knight shift χ sp ( 0 , 0 ) ( δ ) , as a function of doping, should also have a maximum. In addition, we demonstrate that two-particle self-consistent plus correctly predicts a maximum in the temperature dependence of the Knight shift χ sp ( 0 , 0 ) ( T ) , consistent with recent ultracold-atom experiments [T. Chalopin , .]. The maxima in both quantities should exist, at sufficiently low temperatures ( T ), in both the intermediate U ≈ U Mott and weak U < U Mott interaction limits. In both limits, the mechanism is critical thermal SDW fluctuations. At the antinodal pseudogap, the correlation length at δ max ( T ) can be small, controlled not by static but by dynamic critical thermal fluctuations. We also find that the SDW fluctuations are incommensurate at δ = δ max . We predict that, at low T , the multiple peaks in the spin susceptibility in the incommensurate case lead to more than two SDW precursor peaks in the spectral function and density of states. By allowing access to parameter regimes relevant to cuprates—including further-neighbor hopping ( t ′ , t ′′ ) and low temperatures, our work provides a high-impact tool for further studies by the broader community.