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arXiv:2003.04898 · Physical review. B./Physical review. B

Pyrochlore S=12 Heisenberg antiferromagnet at finite temperature

Robin Schäfer, Imre Hagymási, Roderich Moessner, David J. Luitz

Published Aug 4, 2020• 60 citations
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Abstract

We use a combination of three computational methods to investigate the notoriously difficult frustrated three-dimensional pyrochlore $S=\frac{1}{2}$ 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 $0.47{k}_{B}ln2$, 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 ${\mathrm{NaCaNi}}_{2}{\mathrm{F}}_{7}$ 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 ${\mathrm{Dy}}_{2}{\mathrm{Ti}}_{2}{\mathrm{O}}_{7}$.

DOI
10.1103/physrevb.102.054408

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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=32N=32N=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=32N=32N=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 TTT and for a single maximum in c(T)c(T)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 TTT reaching ${\ensuremath{\chi}}_{0}\ensuremath{\approx}0.1$ at T=0T=0T=0. Moreover, the EM allows us to estimate the ground-state energy to ${e}_{0}\ensuremath{\approx}\ensuremath{-}0.52$.
We address the ground-state properties of the long-standing and much-studied three-dimensional quantum spin liquid candidate, the S=1/2 pyrochlore Heisenberg antiferromagnet. By using SU(2) density-matrix renormalization group (DMRG), we are able to access cluster sizes of up to 128 spins. Our most striking finding is a robust spontaneous inversion symmetry breaking, reflected in an energy density difference between the two sublattices of tetrahedra, familiar as a starting point of earlier perturbative treatments. We also determine the ground-state energy, E_{0}/N_{sites}=-0.490(6)J, by combining extrapolations of DMRG with those of a numerical linked cluster expansion. These findings suggest a scenario in which a finite-temperature spin liquid regime gives way to a symmetry-broken state at low temperatures.
We employ a recently developed variant of the functional renormalization group method for spin systems, the so-called pseudo Majorana functional renormalization group, to investigate three-dimensional spin-1/2 Heisenberg models at finite temperatures. We study unfrustrated and frustrated Heisenberg systems on the simple cubic and pyrochlore lattices. Comparing our results with other quantum many-body techniques, we demonstrate a high quantitative accuracy of our method. Particularly, for the unfrustrated simple cubic lattice antiferromagnet ordering temperatures obtained from finite-size scaling of one-loop data deviate from error controlled quantum Monte Carlo results by ∼5%\sim5\%∼5% and we further confirm the established values for the critical exponent ν\nuν and the anomalous dimension η\etaη. As the PMFRG yields results in good agreement with QMC, but remains applicable when the system is frustrated, we next treat the pyrochlore Heisenberg antiferromagnet as a paradigmatic magnetically disordered system and find nearly perfect agreement of our two-loop static homogeneous susceptibility with other methods. We further investigate the broadening of pinch points in the spin structure factor as a result of quantum and thermal fluctuations and confirm a finite width in the extrapolated limit T→0T\rightarrow0T→0. While extensions towards higher loop orders ℓ\ellℓ seem to systematically improve our approach for magnetically disordered systems we also discuss subtleties when increasing ℓ\ellℓ in the presence of magnetic order. Overall, the pseudo Majorana functional renormalization group is established as a powerful many-body technique in quantum magnetism with a wealth of possible future applications.
We study the phase diagram of the antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the pyrochlore lattice with S=1S=1S=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 nonmagnetic ground-state phase of the nearest-neighbor model extents up to ${J}_{2}/{J}_{1}\ensuremath{\sim}0.02$ within DMRG, beyond which magnetic $\mathbit{k}=0$ long-range order sets in. Our PMFRG calculations find the phase transition in a similar regime ${J}_{2}/{J}_{1}\ensuremath{\sim}0.035(8)$ which, together with the DMRG result, provides a strong argument for the existence of a small but finite nonmagnetic 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.
Many frustrated spin models on three-dimensional (3D) lattices are currently being investigated, both experimentally and theoretically, and develop new types of long-range orders in their respective phase diagrams. They present finite-temperature phase transitions, most likely in the Heisenberg 3D universality class. However, the combination between the 3D character and frustration makes them hard to study. We present here several methods derived from high-temperature series expansions (HTSEs), which give exact coefficients directly in the thermodynamic limit up to a certain order; for several 3D lattices, supplementary orders than in previous literature are reported for the HTSEs. We introduce an interpolation method able to describe thermodynamic quantities at T>TcT>{T}_{c}T>Tc​, which we use here to reconstruct the magnetic susceptibility and the specific heat and to extract universal and nonuniversal quantities (for example, critical exponents, temperature, energy, entropy, and other parameters related to the phase transition). While the susceptibility associated with the order parameter is not usually known for more exotic long-range orders, the specific heat is indicative of a phase transition for any kind of symmetry breaking. We present examples of applications on ferromagnetic and antiferromagnetic models on various 3D lattices and benchmark our results whenever possible.
We employ a recently developed variant of the functional renormalization group method for spin systems, the so-called pseudo Majorana functional renormalization group, to investigate three-dimensional spin-1/2 Heisenberg models at finite temperatures. We study unfrustrated and frustrated Heisenberg systems on the simple cubic and pyrochlore lattices. Comparing our results with other quantum many-body techniques, we demonstrate a high quantitative accuracy of our method. Particularly, for the unfrustrated simple cubic lattice antiferromagnet ordering temperatures obtained from finite-size scaling of one-loop data deviate from error controlled quantum Monte Carlo results by \sim5\% ∼ 5 % and we confirm consistency of our data with established critical exponents \nu ν and \eta η of the three-dimensional Heisenberg universality class. As the PMFRG yields results in good agreement with QMC, but remains applicable when the system is frustrated, we next treat the pyrochlore Heisenberg antiferromagnet as a paradigmatic magnetically disordered system and find nearly perfect agreement of our two-loop static homogeneous susceptibility with other methods. We further investigate the broadening of pinch points in the spin structure factor as a result of quantum and thermal fluctuations and confirm a finite width in the extrapolated limit T\rightarrow0 T → 0 . While extensions towards higher loop orders \ell ℓ seem to systematically improve our approach for magnetically disordered systems we also discuss subtleties when increasing \ell ℓ in the presence of magnetic order. Overall, the pseudo Majorana functional renormalization group is established as a powerful many-body technique in quantum magnetism with a wealth of possible future applications.