Thermodynamics of spin $S=1/2\mathrm{}$ antiferromagnetic uniform and alternating-exchange Heisenberg chains

The magnetic susceptibility ${\chi }^{*}\mathrm{}\left(t\right)\mathrm{}$ and specific heat $C\left(t\right)\mathrm{}$ versus temperature t of the spin $S=1/2\mathrm{}$ antiferromagnetic (AF) alternating-exchange ${(J}_1$ and $J_2)$ Heisenberg chain are studied for the entire range $0<~\alpha <~1$ of the alternation parameter $\alpha \equiv J_2{/J}_1{(J}_1,$ $J_2>~0,$ $J_2<~J_1,$ ${t=k}_{\mathrm{B}}{T/J}_1,$ ${\chi }^{*}=\chi J_1{/Ng}^2{\mu }_B^2).\mathrm{}$ For the uniform chain $(\alpha =1),$ the high-accuracy ${\chi }^{*}\mathrm{}\left(t\right)\mathrm{}$ and $C\left(t\right)\mathrm{}$ Bethe ansatz data of Klümper and Johnston (unpublished) are shown to agree very well at low t with the respective exact theoretical low-$t$ logarithmic correction predictions of Lukyanov [Nucl.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 $\left(\sim {10}^{-7}\right)$ independent empirical fits to the respective data are obtained over t ranges spanning 25 orders of magnitude, $5\times {10}^{-25}<~t<~5,$ which contain extrapolations to the respective exact $t=0\mathrm{}$ limits. The infinite temperature entropy calculated using our $C\left(t\right)\mathrm{}$ fit function is within 8 parts in ${10}^8$ of the exact value $\ln 2.$ Quantum Monte Carlo (QMC) simulations and transfer-matrix density-matrix renormalization group (TMRG) calculations of ${\chi }^{*}(\alpha ,t)$ are presented for $0.002<~t<~10$ and $0.05<~\alpha <~1,$ and an accurate $(2\mathrm{}\times {10}^{-4})$ two-dimensional $(\alpha ,t)$ fit to the combined data is obtained for $0.01<~t<~10$ and $0<~\alpha <~1.$ From the low-$t$ TMRG data, the spin gap $\Delta \left(\alpha \right)$ is extracted for $0.8<~\alpha <~0.995$ and compared with previous results, and a fit function is formulated for $0<~\alpha <~1$ 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 $\alpha \gtrsim \mathrm{0.99.}$ We examine in detail the theoretical predictions of Bulaevskii [Sov. Phys. Solid State $11,$ 921 (1969)], for ${\chi }^{*}(\alpha ,t)$ and compare them with our results. To illustrate the application and utility of our theoretical results, we model our experimental $\chi \mathrm{}\left(T\right)\mathrm{}$ and specific heat $C_{\mathrm{p}}\mathrm{}\left(T\right)\mathrm{}$ data for ${\mathrm{NaV}}_2{\mathrm{O}}_5$ single crystals in detail. The $\chi \mathrm{}\left(T\right)\mathrm{}$ data above the spin dimerization temperature $T_{\mathrm{c}}\approx 34\mathrm{K}$ are not in quantitative agreement with the prediction for the $S=1/2\mathrm{}$ uniform Heisenberg chain, but can be explained if there is a moderate ferromagnetic interchain coupling and/or if J changes with T. Fitting the $\chi \mathrm{}\left(T\right)\mathrm{}$ data using our ${\chi }^{*}(\alpha ,t)$ fit function, we obtain the sample-dependent spin gap and range $\Delta {\mathrm{}(T=0\mathrm{})/k}_{\mathrm{B}}=103\mathrm{}\left(2\right)\mathrm{}\mathrm{K},$ alternation parameter $\delta \mathrm{}\left(0\right)\mathrm{}\equiv \left(1\mathrm{}-\alpha \right)/(1+\alpha )=0.034\mathrm{}\left(6\right)\mathrm{}$ and average exchange constant ${J\left(0\right)/k}_{\mathrm{B}}=640\mathrm{}\left(80\right)\mathrm{}\mathrm{K}.$ The $\delta \mathrm{}\left(T\right)\mathrm{}$ and $\Delta \mathrm{}\left(T\right)\mathrm{}$ are derived from the data. A spin pseudogap with magnitude $\approx 0.4\Delta \mathrm{}\left(0\right)\mathrm{}$ is consistently found just above $T_{\mathrm{c}},$ which decreases with increasing temperature. From our $C_{\mathrm{p}}\mathrm{}\left(T\right)\mathrm{}$ measurements on two crystals, we infer that the magnetic specific heat at low temperatures $T\lesssim 15\mathrm{K}$ 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.