Adapting Planck's route to investigate the thermodynamics of the spin-half pyrochlore Heisenberg antiferromagnet

The spin-half pyrochlore Heisenberg antiferromagnet (PHAF) is one of the most challenging problems in the field of highly frustrated quantum magnetism. Stimulated by the seminal paper of M.~Planck [M.~Planck, Verhandl. Dtsch. phys. Ges. {\bf 2}, 202-204 (1900)] we calculate thermodynamic properties of this model by interpolating between the low- and high-temperature behavior. For that we follow ideas developed in detail by B.~Bernu and G.~Misguich and use for the interpolation the entropy exploiting sum rules [the ``entropy method'' (EM)]. We complement the EM results for the specific heat, the entropy, and the susceptibility by corresponding results obtained by the finite-temperature Lanczos method (FTLM) for a finite lattice of $N=32$ sites as well as by the high-temperature expansion (HTE) data. We find that due to pronounced finite-size effects the FTLM data for $N=32$ are not representative for the infinite system below $T \approx 0.7$. A similar restriction to $T \gtrsim 0.7$ holds for the HTE designed for the infinite PHAF. By contrast, the EM provides reliable data for the whole temperature region for the infinite PHAF. We find evidence for a gapless spectrum leading to a power-law behavior of the specific heat at low $T$ and for a single maximum in $c(T)$ at $T\approx 0.25$. For the susceptibility $\chi(T)$ we find indications of a monotonous increase of $\chi$ upon decreasing of $T$ reaching $\chi_0 \approx 0.1$ at $T=0$. Moreover, the EM allows to estimate the ground-state energy to $e_0\approx -0.52$.Comment: 17 pages, 24 figure

Spin-half Heisenberg antiferromagnet on a symmetric sawtooth chain: Rotation-invariant Green's functions and high-temperature series

We apply the rotation-invariant Green's function method to study the finite-temperature properties of a $S{=}1/2$ sawtooth-chain (also called $\Delta$-chain) antiferromagnetic Heisenberg model at the fully frustrated point when the exchange couplings along the straight-line and zig-zag paths are equal. We also use 13 terms of high-temperature expansion series and interpolation methods to get thermodynamic quantities for this model. We check the obtained predictions for observable quantities by comparison with numerics for finite systems. Although our work refers to a one-dimensional case, the utilized methods work in higher dimensions too and are applicable for examining other frustrated quantum spin lattice systems at finite temperatures.Comment: 13 pages, 8 figure

Thermodynamics of the S=1/2S=1/2 hyperkagome-lattice Heisenberg antiferromagnet

The $S=1/2$ hyperkagome-lattice Heisenberg antiferromagnet, which for instance is related to the experimentally accessible spinel oxide Na$_4$Ir$_3$O$_8$, allows to study the interplay of geometrical frustration and quantum as well as thermal fluctuations in three dimensions. We use 16 terms of a high-temperature series expansion complemented by the entropy-method interpolation to examine the specific heat and the uniform susceptibility of the $S=1/2$ hyperkagome-lattice Heisenberg antiferromagnet. We obtain thermodynamic quantities for the two possible scenarios of either a gapless or a gapped energy spectrum. We have found that the specific heat $c$ exhibits, besides the high-temperature peak around $T\approx 0.669$, a low-temperature one at $T\approx 0.021\ldots0.033$. The functional form of the uniform susceptibility $\chi$ below about $T=0.5$ depends strongly on whether the energy spectrum is gapless or gapped. The value of the ground-state energy can be estimated to $e_{0}\approx-0.440\ldots-0.435$. In addition to the entropy-method interpolation we use the finite-temperature Lanczos method to calculate $c$ and $\chi$ for finite lattices of $N=24$ and $36$ sites. A combined view on both methods leads us to favour a gapless scenario since then the maximum of the susceptibility agrees better between both methods.Comment: 10 pages, 7 figure

Quantum Heisenberg model on a sawtooth-chain lattice: Rotation-invariant Green's function method

We apply the rotation-invariant Green's function method (RGM) to study the spin $S=1/2$ Heisenberg model on a one-dimensional sawtooth lattice, which has two nonequivalent sites in the unit cell. We check the RGM predictions for observable quantities by comparison with the exact-diagonalization and finite-temperature-Lanczos calculations. We discuss the thermodynamic and dynamic properties of this model in relation to the mineral atacamite Cu$_2$Cl(OH)$_3$ complementing the RGM outcomes by results of other approaches.Comment: 14 pages, 11 figure