The magnetic susceptibility ${\ensuremath{\chi}}^{*}(t)$ and specific heat
C(t) versus temperature t of the spin
S=1/2 antiferromagnetic (AF) alternating-exchange
(J1 and
J2) Heisenberg chain are studied for the entire range $0<~\ensuremath{\alpha}<~1$ of the alternation parameter $\ensuremath{\alpha}\ensuremath{\equiv}{J}_{2}{/J}_{1}{(J}_{1},$
J2> 0, J2< J1, t=kBT/J1, ${\ensuremath{\chi}}^{*}=\ensuremath{\chi}{J}_{1}{/Ng}^{2}{\ensuremath{\mu}}_{B}^{2}).$ For the uniform chain $(\ensuremath{\alpha}=1),$ the high-accuracy ${\ensuremath{\chi}}^{*}(t)$ and
C(t) Bethe ansatz data of Kl\"umper and Johnston (unpublished) are shown to agree very well at low t with the respective exact theoretical low-
t logarithmic correction predictions of Lukyanov [Nucl. Phys. B
522, 533 (1998)]. Accurate $(\ensuremath{\sim}{10}^{\ensuremath{-}7})$ independent empirical fits to the respective data are obtained over t ranges spanning 25 orders of magnitude, $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}25}<~t<~5,$ which contain extrapolations to the respective exact
t=0 limits. The infinite temperature entropy calculated using our
C(t) fit function is within 8 parts in
108 of the exact value
ln2. Quantum Monte Carlo (QMC) simulations and transfer-matrix density-matrix renormalization group (TMRG) calculations of ${\ensuremath{\chi}}^{*}(\ensuremath{\alpha},t)$ are presented for
0.002< t< 10 and $0.05<~\ensuremath{\alpha}<~1,$ and an accurate $(2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4})$ two-dimensional $(\ensuremath{\alpha},t)$ fit to the combined data is obtained for
0.01< t< 10 and $0<~\ensuremath{\alpha}<~1.$ From the low-
t TMRG data, the spin gap $\ensuremath{\Delta}(\ensuremath{\alpha})$ is extracted for $0.8<~\ensuremath{\alpha}<~0.995$ and compared with previous results, and a fit function is formulated for $0<~\ensuremath{\alpha}<~1$ by combining these data with literature data. We infer from our data that the asymptotic critical regime near the uniform chain limit is only entered for $\ensuremath{\alpha}\ensuremath{\gtrsim}0.99.$ We examine in detail the theoretical predictions of Bulaevskii [Sov. Phys. Solid State
11, 921 (1969)], for ${\ensuremath{\chi}}^{*}(\ensuremath{\alpha},t)$ and compare them with our results. To illustrate the application and utility of our theoretical results, we model our experimental $\ensuremath{\chi}(T)$ and specific heat
Cp(T) data for
NaV2O5 single crystals in detail. The $\ensuremath{\chi}(T)$ data above the spin dimerization temperature ${T}_{\mathrm{c}}\ensuremath{\approx}34\mathrm{K}$ are not in quantitative agreement with the prediction for the
S=1/2 uniform Heisenberg chain, but can be explained if there is a moderate ferromagnetic interchain coupling and/or if J changes with T. Fitting the $\ensuremath{\chi}(T)$ data using our ${\ensuremath{\chi}}^{*}(\ensuremath{\alpha},t)$ fit function, we obtain the sample-dependent spin gap and range $\ensuremath{\Delta}{(T=0)/k}_{\mathrm{B}}=103(2)\mathrm{K},$ alternation parameter $\ensuremath{\delta}(0)\ensuremath{\equiv}(1\ensuremath{-}\ensuremath{\alpha})/(1+\ensuremath{\alpha})=0.034(6)$ and average exchange constant
J(0)/kB=640(80)K. The $\ensuremath{\delta}(T)$ and $\ensuremath{\Delta}(T)$ are derived from the data. A spin pseudogap with magnitude $\ensuremath{\approx}0.4\ensuremath{\Delta}(0)$ is consistently found just above
Tc, which decreases with increasing temperature. From our
Cp(T) measurements on two crystals, we infer that the magnetic specific heat at low temperatures $T\ensuremath{\lesssim}15\mathrm{K}$ is too small to be resolved experimentally, and that the spin entropy at
Tc is too small to account for the entropy of the transition. A quantitative analysis indicates that at
Tc, at least 77% of the entropy change due to the transition at
Tc and associated order parameter fluctuations arise from the lattice and/or charge degrees of freedom and less than 23% from the spin degrees of freedom.
We have extracted the leading low-temperature contributions to the specific heat and the magnetic susceptibility from the thermodynamic Bethe-Ansatz equations of the SU(3) invariant ferromagnetic Heisenberg chain of spin 1. The critical exponents for the specific heat, the susceptibility, and the correlation length are $\ensuremath{\alpha}=\ensuremath{-}\frac{1}{2}$, $\ensuremath{\gamma}=2$, and $\ensuremath{\nu}=1$, respectively. The susceptibility and the correlation length exhibit logarithmic corrections, which are quenched by relatively small magnetic fields. In large fields, the energy required to flip a spin gives rise to an exponential activation.
We have extracted the leading low-temperature contributions to the specific heat and the magnetic susceptibility from the thermodynamic Bethe-ansatz equations of the SU(2) invariant ferromagnetic Heisenberg chain of arbitrary spin S. This extends previous analytic results for S=1/2 to other spin values. The critical exponent for the specific heat is \ensuremath{\alpha}=-1/2, that for the susceptibility is \ensuremath{\gamma}=2, and that for the correlation length is \ensuremath{\nu}=1. The susceptibility and the correlation length exhibit logarithmic corrections, reflecting the analogy to the Kondo problem. These logarithmic corrections are quenched by relatively small magnetic fields. In large fields the energy required to flip a spin gives rise to an exponential activation.
A detailed account of the results of the numerical solution of the thermodynamic Bethe-ansatz equations for the isotropic ferromagnetic S=(1/2) Heisenberg chain is presented. The extrapolation procedure used in approximating the infinite set of coupled nonlinear integral equations is discussed. The data for the truncated integral equations are analyzed in terms of a finite-string-size scaling. Analytic expressions for the free energy and susceptibility for T\ensuremath{\rightarrow}0 are obtained. All the results are consistent with entropy S\ensuremath{\sim}(T/\ensuremath{\Vert}J\ensuremath{\Vert}
)1/2 and \ensuremath{\chi}\ensuremath{\sim} \ensuremath{\Vert}J\ensuremath{\Vert} /
T2 [${\mathrm{scrL}}^{\mathrm{\ensuremath{-}}1}$+(lnscrL)/
scrL2+...]+O (${T}^{\mathrm{\ensuremath{-}}3/2}$) , where scrL=ln(\ensuremath{\Vert}J\ensuremath{\Vert}/T), suggesting the existence of a marginal variable. The logarithmic corrections reflect the analogy to the Kondo problem.