Thermodynamics of spin S = 1 / 2 antiferromagnetic uniform and alternating-exchange Heisenberg chains

The magnetic susceptibility ${\ensuremath{\chi}}^{*}(t)$ and specific heat $C(t)$ versus temperature t of the spin $S=1/2$ antiferromagnetic (AF) alternating-exchange ${(J}_{1}$ and ${J}_{2})$ Heisenberg chain are studied for the entire range $0l~\ensuremath{\alpha}l~1$ of the alternation parameter $\ensuremath{\alpha}\ensuremath{\equiv}{J}_{2}{/J}_{1}{(J}_{1},$ ${J}_{2}g~0,$ ${J}_{2}l~{J}_{1},$ ${t=k}_{\mathrm{B}}{T/J}_{1},$ ${\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the high-accuracy ${\ensuremath{\chi}}^{*}(t)$ and $C(t)$ Bethe ansatz data of Kl\"umper and Johnston (unpublished) are shown to agree very well at low t with the respective exact theoretical low-$t$ logarithmic correction predictions of Lukyanov [Nucl.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, $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}25}l~tl~5,$ 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 ${10}^{8}$ of the exact value $\mathrm{ln}2. $ Quantum Monte Carlo (QMC) simulations and transfer-matrix density-matrix renormalization group (TMRG) calculations of ${\ensuremath{\chi}}^{*}(\ensuremath{\alpha},t)$ are presented for $0.002l~tl~10$ and $0.05l~\ensuremath{\alpha}l~1,$ 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 $0.01l~tl~10$ and $0l~\ensuremath{\alpha}l~1. $ From the low-$t$ TMRG data, the spin gap $\ensuremath{\Delta}(\ensuremath{\alpha})$ is extracted for $0.8l~\ensuremath{\alpha}l~0.995$ and compared with previous results, and a fit function is formulated for $0l~\ensuremath{\alpha}l~1$ 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 ${C}_{\mathrm{p}}(T)$ data for ${\mathrm{NaV}}_{2}{\mathrm{O}}_{5}$ 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)/k}_{\mathrm{B}}=640(80)\mathrm{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 ${T}_{\mathrm{c}},$ which decreases with increasing temperature. From our ${C}_{\mathrm{p}}(T)$ 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 ${T}_{\mathrm{c}}$ is too small to account for the entropy of the transition. A quantitative analysis indicates that at ${T}_{\mathrm{c}},$ at least 77% of the entropy change due to the transition at ${T}_{\mathrm{c}}$ 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.