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arXiv:1911.09266 · Physical Review Research

Finite-temperature properties of the Kitaev-Heisenberg models on kagome and triangular lattices studied by improved finite-temperature Lanczos methods

Katsuhiro Morita, Takami Tohyama

Published Feb 25, 2020• 43 citations
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Abstract

The authors investigate the finite-temper properties of the S=1/2 Kitaev-Heisenberg models on kagome and triangular lattices studied by improved finite-temperature Lanczos methods. In both lattice models, the specific heat exhibits multiple-peak structures.
DOI
10.1103/physrevresearch.2.013205

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We present an improvement of the finite temperature Lanczos method in order to apply this method to systems at very low temperature. One proposal is to introduce two steps in this method. In the first step, we use the Chebyshev polynomial expansion to calculate exp(-H/T1) random vector> at moderate temperature T1. In the second step, we apply the ordinary finite temperature Lanczos method using the calculated state as the initial state of the Lanczos method. Another proposal is to employ a sampling method for selecting a random vector. By this sampling, we can improve an efficiency of calculations. Using the improved finite temperature Lanczos method, we calculate the specific heat of the spin-1/2 Heisenberg model on the kagome lattices of 27 and 30 sites.
Thermodynamic properties, specific heat and magnetic susceptibility, of spin-1/2 Ising-like Heisenberg model are investigated by an exact diagonalization method for small finite size kagome- and triangular-lattices up to 18-spins. The enhancement of magnetic susceptibility from the Curie–Weiss law and the peaks-structure of specific heat, rather generally detected experimentally in triangle-based spin systems including herbertsmithite, are interpreted as an intrinsic property of triangle-based frustrated spin systems with some extent of exchange anisotropy and are inferred as owing to a quantum-classical crossover.
We propose a numerical technique, the forced-oscillator method, to investigate finite-temperature properties of quantum spin systems. The space and time complexity of this method are linear in the matrix dimension, which should be compared with the square or quadratic dependence of conventional methods. We apply this method to calculations of the specific heat of the spin-1/2 quantum Heisenberg antiferromagnet on the kagome$iaa--- lattice. We find only a single-peak in contrast to a double-peak structure claimed in the literature.
A two-dimensional Ising model on a particular topological structure, a ``triangles-in-triangles'' Kagom\'e structure, which is called a triangular-Kagom\'e lattice, is investigated. In this model, we consider two interaction parameters Jaa{J}_{aa}Jaa​ and Jab{J}_{ab}Jab​, corresponding to the interactions among spins on the inner sites and between spins on the inner and vertex sites of the triangular in the Kagom\'e lattice, respectively. By summing over all spins at inner sites in the partition function, we arrive at the partition function of the Kagom\'e lattice. The effective interaction of the corresponding Kagom\'e lattice is always ferromagnetic, even for antiferromagnetic Jaa{J}_{aa}Jaa​. The critical properties of the system depend only on ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}$. When ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}>\ensuremath{-}1$, the system has long range order at low temperature. However, when ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}<\ensuremath{-}1$, the partition function of the triangular-Kagom\'e lattice can be related to that of the Kagom\'e lattice with effective interaction at a temperature higher than its critical temperature. Therefore, the system will be in paramagnetic phase at all temperatures for ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}<\ensuremath{-}1$. The phase diagram for ${J}_{aa}∕\ensuremath{\mid}{J}_{ab}\ensuremath{\mid}$ is given exactly.
We investigate the spin-1 kagome-lattice Heisenberg antiferromagnet using large-scale Lanczos diagonalization and the finite-temperature Lanczos method. The zero-temperature magnetization process exhibits plateaus at m = 0 , 1 / 3 , 7 / 9 , and 8 / 9 , where m is the normalized magnetization. The m = 0 plateau is identified as a trimer valence-bond-crystal state, while the high-field plateaus at m = 7 / 9 and 8 / 9 are identified as magnon crystals. In particular, the m = 8 / 9 plateau corresponds to the exact localized-magnon crystal state. A smoothed zero-temperature magnetization curve constructed using the Gaussian-kernel smoothing method indicates magnetization jumps at the lower-field edge of the m = 1 / 3 plateau and at the upper-field edges of the m = 7 / 9 and 8 / 9 plateaus. At finite temperatures, the specific heat exhibits a double-peak structure with peaks around T / J ≃ 0.1 and T / J ≃ 1.1 , and the low-temperature peak may be related to trimer valence-bond-crystal ordering. The finite-temperature magnetization curves show that the m = 1 / 3 plateau remains visible at low temperatures, whereas the high-field plateaus are rapidly smeared out by thermal effects. These results provide benchmark data for thermodynamic and high-field magnetization measurements in candidate spin-1 kagome-lattice materials.
Abstract We study the temperature dependence of the thermodynamic properties of spin-1/2 antiferromagnets on two-dimensional lattices. Our analysis employs the sine-square deformation (SSD), in which a real-space envelope function is applied to the Hamiltonian so that the local energy scale is smoothly reduced to zero at the system boundaries. The quantum eigenstates of the SSD Hamiltonian exhibit bulk-like behavior near the system center, effectively mimicking the thermodynamic limit even in small finite-size calculations. Using these fictitious bulk states, we compute the energy density, specific heat, and magnetic susceptibility as functions of temperature. We find that both the triangular- and kagome-lattice antiferromagnets show either a shoulder or a pronounced double-peak structure in the low-temperature specific heat, whereas the kagome case particularly shows a strong enhancement of magnetic susceptibility down to the lowest temperature range. These direct comparisons, together with the square-lattice and one-dimensional cases, reveal that although both frustrated systems retain a substantial amount of entropy, the low-energy excitations below ∼ 0.5 J ∼0.5J\sim 0.5J∼0.5J of the kagome lattice are predominantly governed by the magnetic excited states, whereas not much for the triangular lattice.