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. 2, 202 (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 $T\ensuremath{\approx}0.7$. A similar restriction to $T\ensuremath{\gtrsim}0.7$ 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 $T\ensuremath{\approx}0.25$. For the susceptibility $\ensuremath{\chi}(T)$ we find indications of a monotonous increase of $\ensuremath{\chi}$ upon decreasing of $T$ reaching ${\ensuremath{\chi}}_{0}\ensuremath{\approx}0.1$ at $T=0$. Moreover, the EM allows us to estimate the ground-state energy to ${e}_{0}\ensuremath{\approx}\ensuremath{-}0.52$.
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. 2, 202 (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\ensuremath≈0.7. A similar restriction to T\ensuremath≳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\ensuremath≈0.25. For the susceptibility \ensuremathχ(T) we find indications of a monotonous increase of \ensuremathχ upon decreasing of T reaching \ensuremathχ0\ensuremath≈0.1 at T=0. Moreover, the EM allows us to estimate the ground-state energy to e0\ensuremath≈\ensuremath−0.52.
We use a combination of three computational methods to investigate the notoriously difficult frustrated three-dimensional pyrochlore S=21 quantum antiferromagnet, at finite temperature T: canonical typicality for a finite cluster of $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2$ unit cells (i.e., 32 sites), a finite-T matrix product state method on a larger cluster with 48 sites, and the numerical linked cluster expansion (NLCE) using clusters up to 25 lattice sites, including nontrivial hexagonal and octagonal loops. We calculate thermodynamic properties (energy, specific heat capacity, entropy, susceptibility, magnetization) and the static structure factor. We find a pronounced maximum in the specific heat at T=0.57J, which is stable across finite size clusters and converged in the series expansion. At $T\ensuremath{\approx}0.25J$ (the limit of convergence of our method), the residual entropy per spin is 0.47kBln2, which is relatively large compared to other frustrated models at this temperature. We also observe a nonmonotonic dependence on T of the magnetization at low magnetic fields, reflecting the dominantly nonmagnetic character of the low-energy states. A detailed comparison of our results to measurements for the S=1 material NaCaNi2F7 yields a rough agreement of the functional form of the specific heat maximum, which in turn differs from the sharper maximum of the heat capacity of the spin ice material Dy2Ti2O7.
Over the last decade, the interest in the spin-1/2 Heisenberg antiferromagnet (HAF) on the square kagome (also called shuriken) lattice has been growing as a model system of quantum magnetism with a quantum paramagnetic ground state, flat-band physics near the saturation field, and quantum scars. A further motivation to study this model comes from the recent discovery of a gapless spin liquid in the square kagome magnet KCu6AlBiO4(SO4)5Cl [M. Fujihala et al., Nat. Commun. 11, 3429 (2020)]. Here, we present large-scale numerical investigations of the specific heat C(T), the entropy S(T), as well as the susceptibility $\ensuremath{\chi}(T)$ by means of the finite-temperature Lanczos method for system sizes of N=18,24,30,36,42,48, and N=54. We find that the specific heat exhibits a low-temperature shoulder below the major maximum which can be attributed to low-lying singlet excitations filling the singlet-triplet gap, which is significantly larger than the singlet-singlet gap. This observation is further supported by the behavior of the entropy S(T), where a change in curvature is present just at about T/J=0.2, the same temperature where the shoulder in C sets in. For the susceptibility the low-lying singlet excitations are irrelevant, and the singlet-triplet gap leads to an exponentially activated low-temperature behavior. The maximum in $\ensuremath{\chi}(T)$ is found at a pretty low temperature Tmax/J=0.146 (for N=42) compared to Tmax/J=0.935 for the unfrustrated square-lattice HAF signaling the crucial role of frustration also for the susceptibility. We find a striking similarity of our square kagome data with the corresponding ones for the kagome HAF down to very low T. The magnetization process featuring plateaus and jumps and the field dependence of the specific heat that exhibits characteristic peculiarities attributed to the existence of a flat one-magnon band are discussed as well.
We apply the contractor renormalization (CORE) method to the spin half Heisenberg antiferromagnet on the frustrated checkerboard and pyrochlore lattices. Their ground states are spin-gapped singlets which break lattice symmetry. Their effective Hamiltonians describe fluctuations of orthogonal singlet pairs on tetrahedral blocks, at an emergent low energy scale. We discuss low temperature thermodynamics and new interpretations of finite size numerical data. We argue that our results are common to many models of quantum frustration.
The antiferromagnetic ${J}_{1}\ensuremath{-}{J}_{2}$ model is a spin-1/2 chain with isotropic exchange J1>0 between first neighbors and ${J}_{2}=\ensuremath{\alpha}{J}_{1}$ between second neighbors. The model supports both gapless quantum phases with nondegenerate ground states and gapped phases with $\mathrm{\ensuremath{\Delta}}(\ensuremath{\alpha})>0$ and doubly degenerate ground states. Exact thermodynamics is limited to $\ensuremath{\alpha}=0$, the linear Heisenberg antiferromagnet (HAF). Exact diagonalization of small systems at frustration $\ensuremath{\alpha}$ followed by density matrix renormalization group calculations returns the entropy density $S(T,\ensuremath{\alpha},N)$ and magnetic susceptibility $\ensuremath{\chi}(T,\ensuremath{\alpha},N)$ of progressively larger systems up to N=96 or 152 spins. Convergence to the thermodynamic limit, $S(T,\ensuremath{\alpha})$ or $\ensuremath{\chi}(T,\ensuremath{\alpha})$, is demonstrated down to $T/J\ensuremath{\sim}0.01$ in the sectors $\ensuremath{\alpha}<1$ and $\ensuremath{\alpha}>1$. $S(T,\ensuremath{\alpha})$ yields the critical points between gapless phases with ${S}^{\ensuremath{'}}(0,\ensuremath{\alpha})>0$ and gapped phases with ${S}^{\ensuremath{'}}(0,\ensuremath{\alpha})=0$. The ${S}^{\ensuremath{'}}(T,\ensuremath{\alpha})$ maximum at ${T}^{*}(\ensuremath{\alpha})$ is obtained directly in chains with large $\mathrm{\ensuremath{\Delta}}(\ensuremath{\alpha})$ and by extrapolation for small gaps. A phenomenological approximation for $S(T,\ensuremath{\alpha})$ down to T=0 indicates power-law deviations ${T}^{\ensuremath{-}\ensuremath{\gamma}(\ensuremath{\alpha})}$ from $exp[\ensuremath{-}\mathrm{\ensuremath{\Delta}}(\ensuremath{\alpha})/T]$ with exponent $\ensuremath{\gamma}(\ensuremath{\alpha})$ that increases with $\ensuremath{\alpha}$. The $\ensuremath{\chi}(T,\ensuremath{\alpha})$ analysis also yields power-law deviations, but with exponent $\ensuremath{\eta}(\ensuremath{\alpha})$ that decreases with $\ensuremath{\alpha}$. Spin correlation functions account for $S(T,\ensuremath{\alpha})$ differences between frustration $\ensuremath{\alpha}<1$ within a chain and $\ensuremath{\alpha}>1$ between HAFs on sublattices. $S(T,\ensuremath{\alpha})$ and the spin density $\ensuremath{\rho}(T,\ensuremath{\alpha})=4T\ensuremath{\chi}(T,\ensuremath{\alpha})$ probe the thermal and magnetic fluctuations, respectively, of strongly correlated spin states. Gapless chains have constant $S(T,\ensuremath{\alpha})/\ensuremath{\rho}(T,\ensuremath{\alpha})$ for T<0.10. Remarkably, the ratio decreases (increases) with T in chains with large (small) $\mathrm{\ensuremath{\Delta}}(\ensuremath{\alpha})$.
We calculate the thermodynamic quantities (correlation functions ⟨S0Sn⟩, correlation length $\ensuremath{\xi}$, spin susceptibility $\ensuremath{\chi}$, and specific heat CV) of the frustrated one-dimensional spin-half ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg ferromagnet, i.e., for J2<0.25∣J1∣, using a rotation-invariant Green's-function formalism and full diagonalization of finite lattices. We find that the critical indices are not changed by J2, i.e., $\ensuremath{\chi}={y}_{0}{T}^{\ensuremath{-}2}$ and $\ensuremath{\xi}={x}_{0}{T}^{\ensuremath{-}1}$ at $T\ensuremath{\rightarrow}0$. However, the coefficients y0 and x0 linearly decrease with increasing J2 according to the relations ${y}_{0}=(1\ensuremath{-}4{J}_{2}/|{J}_{1}|)/24$ and ${x}_{0}=(1\ensuremath{-}4{J}_{2}/|{J}_{1}|)/4$, i.e., both coefficients vanish at J2=0.25∣J1∣ indicating the zero-temperature phase transition that is accompanied by a change in the low-temperature behavior of $\ensuremath{\chi}$ $(\ensuremath{\xi})$ from $\ensuremath{\chi}\ensuremath{\propto}{T}^{\ensuremath{-}2}$ $(\ensuremath{\xi}\ensuremath{\propto}{T}^{\ensuremath{-}1})$ at J2<0.25∣J1∣ to $\ensuremath{\chi}\ensuremath{\propto}{T}^{\ensuremath{-}3/2}$ $(\ensuremath{\xi}\ensuremath{\propto}{T}^{\ensuremath{-}1/2})$ at J2=0.25∣J1∣. In addition, we detect the existence of an additional low-temperature maximum in the specific heat when approaching the critical point at J2=0.25∣J1∣.
Phase diagram of the antiferromagnetic J1-J2 spin-1 pyrochlore Heisenberg model
We study the phase diagram of the antiferromagnetic J1-J2 Heisenberg model on the pyrochlore lattice with S=1 spins at zero and finite temperatures. We use a combination of complementary state-of-the-art quantum many-body approaches such as density matrix renormalization group (DMRG), density-matrix purification and pseudo-Majorana functional renormalization group (PMFRG). We present an efficient approach to preserve the applicability of the PMFRG for spin-1 systems at finite temperatures despite the inevitable presence of unphysical spin states. The good performance of our methods is first demonstrated for the nearest-neighbor pyrochlore Heisenberg model where the finite temperature behavior of the specific heat and uniform susceptibility show excellent agreement within PMFRG and density-matrix purification. Including an antiferromagnetic second neighbor coupling we find that the non-magnetic ground-state phase of the nearest neighbor model extents up to $J_2/J_1 \sim 0.02 $ within DMRG, beyond which magnetic k=0 long-range order sets in. Our PMFRG calculations find the phase transition in a similar regime J2/J1∼0.035(8) which, together with the DMRG result, provides a strong argument for the existence of a small but finite non-magnetic ground-state phase in the spin-1 pyrochlore Heisenberg model. We also discuss the origin of discrepancies between different versions of the functional renormalization group concerning the location of this phase transition.