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arXiv:1906.11576 · Physical review. B./Physical review. B

Similarity of thermodynamic properties of the Heisenberg model on triangular and kagome lattices

Peter Prelovšek, J. Kokalj

Published Feb 5, 2020• 8 citations
arXivHTMLPDF

Abstract

Derivation of a reduced effective model allows for a unified treatment and discussion of the ${J}_{1}\text{\ensuremath{-}}{J}_{2}\phantom{\rule{4pt}{0ex}}S=1/2$ Heisenberg model on triangular and kagome lattices. Calculating thermodynamic quantities, i.e., the entropy $s(T)$ and uniform susceptibility ${\ensuremath{\chi}}_{0}(T)$, numerically on systems up to effectively $N=42$ sites, we show by comparing to full-model results that low-$T$ properties are qualitatively well represented within the effective model. Moreover, we find in the spin-liquid regime similar variation of $s(T)$ and ${\ensuremath{\chi}}_{0}(T)$ in both models down to $T\ensuremath{\ll}{J}_{1}$. In particular, studied spin liquids appear to be characterized by the Wilson ratio vanishing at low $T$, indicating that the low-lying singlets are dominating over the triplet excitations.

DOI
10.1103/physrevb.101.075105

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Derivation of a reduced effective model allows for a unified treatment and discussion of the J1\ensuremath-J2S=1/2{J}_{1}\text{\ensuremath{-}}{J}_{2}\phantom{\rule{4pt}{0ex}}S=1/2J1​\ensuremath-J2​S=1/2 Heisenberg model on triangular and kagome lattices. Calculating thermodynamic quantities, i.e., the entropy s(T)s(T)s(T) and uniform susceptibility \ensuremathχ0(T){\ensuremath{\chi}}_{0}(T)\ensuremathχ0​(T), numerically on systems up to effectively N=42N=42N=42 sites, we show by comparing to full-model results that low-TTT properties are qualitatively well represented within the effective model. Moreover, we find in the spin-liquid regime similar variation of s(T)s(T)s(T) and \ensuremathχ0(T){\ensuremath{\chi}}_{0}(T)\ensuremathχ0​(T) in both models down to T\ensuremath≪J1T\ensuremath{\ll}{J}_{1}T\ensuremath≪J1​. In particular, studied spin liquids appear to be characterized by the Wilson ratio vanishing at low TTT, indicating that the low-lying singlets are dominating over the triplet excitations.
Over the last decade, the interest in the spin-1/21/21/2 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\mathrm{K}{\mathrm{Cu}}_{6}\mathrm{Al}\mathrm{Bi}{\mathrm{O}}_{4}{(\mathrm{S}{\mathrm{O}}_{4})}_{5}\mathrm{Cl}KCu6​AlBiO4​(SO4​)5​Cl [M. Fujihala et al., Nat. Commun. 11, 3429 (2020)]. Here, we present large-scale numerical investigations of the specific heat C(T)C(T)C(T), the entropy S(T),S(T),S(T), as well as the susceptibility $\ensuremath{\chi}(T)$ by means of the finite-temperature Lanczos method for system sizes of N=18,24,30,36,42,48N=18,24,30,36,42,48N=18,24,30,36,42,48, and N=54N=54N=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)S(T)S(T), where a change in curvature is present just at about T/J=0.2T/J=0.2T/J=0.2, the same temperature where the shoulder in CCC 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{T}_{\mathrm{max}}/J=0.146Tmax​/J=0.146 (for N=42N=42N=42) compared to Tmax/J=0.935{T}_{\mathrm{max}}/J=0.935Tmax​/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 TTT. 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 present a comprehensive analytical theory of quantum spin liquids (QSLs) on the kagome lattice with Heisenberg exchange J1, next-nearest-neighbor exchange J2, Dzyaloshinskii-Moriya interaction D, and scalar spin chirality interaction λ. The work contains three principal contributions. First, we derive new mean-field analytic relations connecting the chiral interaction λ to the spinon gap, low-temperature thermodynamics, and thermal Hall response, with explicit spin-S dependence and generalization across kagome, triangular, and honeycomb lattices. The central result is Δspinon ∝ S³λ/J12 with lattice-dependent prefactors derived from self-consistent mean-field equations. Second, we derive a controlled effective model by projecting the kagome Hamiltonian onto the triangle-doublet subspace, obtaining an effective spin-chirality model on the triangular lattice with complex-valued interactions arising from the C3 symmetry of the chirality eigenstates. We compute all matrix elements of the effective spin operator in the chirality basis and characterize the inter-chirality coupling mediated by shared lattice sites. Third, we provide a rigorous map of obstructions to exact analytical solutions, systematically examining eight classes of proof techniques and identifying the precise mathematical reason each fails for the kagome Hamiltonian. The Lieb-Schultz-Mattis theorem provides the single rigorous exact result: any gapped ground state on the kagome lattice with spin-1/2 per site must exhibit topological order. All other results are classified by derivation category with explicit uncertainty quantification. AI-Generated Content Statement: This manuscript was generated primarily with the assistance of artificial intelligence language models, including ChatGPT. The work represents an exploratory demonstration of AI-assisted knowledge generation and prompt engineering methods rather than traditional experimental research.
We derive exact results for close-packed dimers on the triangular kagome lattice (TKL), formed by inserting triangles into the triangles of the kagome lattice. Because the TKL is a nonbipartite lattice, dimer-dimer correlations are short ranged so that the ground state at the Rokhsar-Kivelson (RK) point of the corresponding quantum dimer model on the same lattice is a short-ranged spin liquid. Using the Pfaffian method, we derive an exact form for the free energy, and we find that the entropy is 13ln 2\frac{1}{3}\text{ln}\text{ }231​ln 2 per site, regardless of the weights of the bonds. The occupation probability of every bond is 14\frac{1}{4}41​ in the case of equal weights on every bond. Similar to the case of lattices formed by corner-sharing triangles (such as the kagome and squagome lattices), we find that the dimer-dimer correlation function is identically zero beyond a certain (short) distance. We find in addition that monomers are deconfined on the TKL, indicating that there is a short-ranged spin liquid phase at the RK point. We also find exact results for the ground-state energy of the classical Heisenberg model. The ground state can be ferromagnetic, ferrimagnetic, locally coplanar, or locally canted, depending on the couplings. From the dimer model and the classical spin model, we derive upper bounds on the ground-state energy of the quantum Heisenberg model on the TKL.
We study low-energy properties of the S=1/2 kagome-triangular-strip Heisenberg model. Using exact diagonalization, we calculate magnetization curves of the model in the ferromagnetic kagome coupling regime, where the singlet ground state and 1/3 plateau emerge. In order to analyze formation of the singlet ground state, we further derive a low-energy effective ladder model, focusing on the parity degrees of freedom for the two degenerating doublets of the unit triangle. We then discuss the entanglement between the spin and parity degrees of freedom.
In this paper we report the latest results of exact diagonalizations of SU (2) invariant models on various lattices (square, triangular, hexagonal, checkerboard and kagomé lattices) and revisit critically in this light some of our previous analysis. We focus on the low-lying levels in each S sector. The differences in behavior between gapless systems and gapped ones are exhibited. The plausibility of a gapless spin liquid in the Heisenberg model on the kagomé lattice is discussed. A rough estimate of the spin susceptibility in such an hypothesis is given. The evolution of the intra- S -channel spectra under the effect of a small perturbation is consistent with the proximity of a quantum critical point. We reemphasize that the very small intra- S -channel energy scale observed in exact spectra is a very interesting information to understand the low- T dynamics of this model.