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Physical review. B, Condensed matter

Specific heat of the quantum Heisenberg antiferromagnet on the kagomé lattice

Ken’ichi Fukamachi, Hidetoshi Nishimori

Published Jan 1, 1994• 13 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.49.651

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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.
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.
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.
We develop a simple and unbiased numerical method to obtain the uniform susceptibility of quantum many-body systems. When a Hamiltonian is spatially deformed by multiplying it with a sine-square function that smoothly decreases from the system center toward the edges, the size-scaling law of the excitation energy is drastically transformed to a rapidly converging one. Then, the local magnetization at the system center becomes nearly size independent; the one obtained for the deformed Hamiltonian of a system length as small as $L\ensuremath{\sim}10$ provides the value obtained for the original uniform Hamiltonian of $L\ensuremath{\sim}100$. This allows us to evaluate a bulk magnetic susceptibility by using the magnetization at the center by existing numerical solvers without any approximation, parameter tuning, or size-scaling analysis. We demonstrate that the susceptibilities of the spin-1/2 antiferromagnetic Heisenberg chain and square lattice obtained by our scheme at $L\ensuremath{\sim}10$ agree within ${10}^{\ensuremath{-}3}$ with exact analytical and numerical solutions for $L=\ensuremath{\infty}$ down to temperatures of 0.1 times the coupling constant. We apply this method to the spin-1/2 kagome lattice Heisenberg antiferromagnet which is of prime interest in the search for spin liquids.
The Suzuki-Takano quantum decimation technique is applied to the antiferromagnetic, nearest-neighbor, frustrated Heisenberg spin-\textonehalf{} system attached to a lattice with dimension d=ln3ln2d=\frac{\mathrm{ln}3}{\mathrm{ln}2}d=ln2ln3​. Some thermodynamical functions are calculated. The temperature dependence of the specific heat is very similar to that obtained for the Heisenberg spin system on a kagom\'e lattice.
We consider the S = 1 / 2 antiferromagnetic Heisenberg model on a frustrated kagome-lattice bilayer with strong nearest-neighbor interlayer coupling and examine its low-temperature magnetothermodynamics using a mapping onto a rhombi gas on the kagome lattice. Besides, we use finite-size numerics to illustrate the validity of the classical lattice-gas description. Among our findings there are (i) the absence of an order-disorder phase transition and (ii) the sensitivity of the specific heat at low temperatures to the shape of the system just below the saturation magnetic field even in the thermodynamic limit.
In analogy to the quantum dimer model on the square lattice introduced by Rokhsar and Kivelson, we identify the low-energy Hilbert space of the spin-1/2 kagom\'e Heisenberg antiferromagnet with the set of dimerized states (nearest-neighbor singlets) and derive an effective Hamiltonian which describes resonance and includes fluctuations into next-nearest-neighbor singlets. Complete diagonalization of this effective Hamiltonian for systems as large as 36 spins makes possible the study of the low-energy thermodynamics of this quantum antiferromagnet and strongly supports the double peak feature in the heat capacity.