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Modern Physics Letters B

THERMODYNAMIC PROPERTIES OF SPIN-1/2 ISING-LIKE HEISENBERG MODEL ON TRIANGLE-BASED LATTICES

Makoto Isoda, Hiroki Nakano, Tôru Sakai

Published May 23, 2011• 3 citations
Publisher

Abstract

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.

DOI
10.1142/s0217984911026619

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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.
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 investigate thermodynamic properties like specific heat c V and susceptibility χ in anisotropic J 1 – J 2 triangular quantum spin systems ( ). As a universal tool we apply the finite temperature Lanczos method (FTLM) based on exact diagonalization of finite clusters with periodic boundary conditions. We use clusters up to N = 28 sites where the thermodynamic limit behavior is already stably reproduced. As a reference we also present the full diagonalization of a small eight-site cluster. After introducing model and method we discuss our main results on c V ( T ) and . We show the variation of peak position and peak height of these quantities as function of control parameter . We demonstrate that maximum peak positions and heights in Néel phase and spiral phases are strongly asymmetric, much more than in the square lattice J 1 – J 2 model. Our results also suggest a tendency to a second side maximum or shoulder formation at lower temperature for certain ranges of the control parameter. We finally explicitly determine the exchange model of the prominent triangular magnets Cs 2 CuCl 4 and Cs 2 CuBr 4 from our FTLM results.
Thermodynamic properties of S=1 antiferromagnetic Heisenberg chains with free and periodic boundaries are investigated by a quantum Monte Carlo method. In particular, temperature dependences of the specific heat, the magnetic susceptibility, and the hidden order parameter, which are inherent to the Haldane phase, are investigated. The specific heat turns out to have a peak at a temperature Tpeak{\mathit{T}}_{\mathrm{peak}}Tpeak​\ensuremath{\sim}2\ensuremath{\Delta}, where \ensuremath{\Delta} is the energy gap of the present model, although the temperature dependence of the specific heat is very similar to the Schottky type. In open chains, due to the fourfold degeneracy of the ground state, the magnetic susceptibility shows a Curie-like divergence regardless of the number of spins in the chain. The amplitude of the Curie-like divergence is consistent with the edge moments (S=1/2). On the other hand, it is confirmed that the long-range hidden order exists only in the ground state. How the hidden order grows as the temperature goes to zero (T\ensuremath{\rightarrow}0) is investigated by introducing an intrinsic correlation length ${\ensuremath{\xi}}_{\mathit{D}}$ of the ordering, which diverges as T\ensuremath{\rightarrow}0. A qualitative difference in fluctuations between the ground state and the finite-temperature states is discussed making use of snapshots of the transformed two-dimensional Ising system on which a Monte Carlo simulation is being performed.
We study the spin-1/2 Ising-XXZ model on a decorated honeycomb lattice composed of five spins per unit cell, one Ising spin, and four Heisenberg spins. This model involving the Heisenberg exchange interaction is one of the few models that can be exactly solvable through the generalized star-triangle transformation. The significance of this model is its close relationship to the fully decorated quantum Heisenberg honeycomb lattice since 4/5 of the particles are Heisenberg spins. We investigate the phase diagram at zero temperature and identify a relevant quantum spin frustrated phase resulting from the contribution of quantum Heisenberg exchange interaction. We obtain an exact residual entropy for the quantum spin frustrated phase, which coincides with the residual entropy of the antiferromagnetic spin-1/2 Ising model on a triangular lattice. We also thoroughly explore its thermodynamic properties, focusing mainly on the frustrated region such as entropy, specific heat, spontaneous magnetization, and critical temperature under several conditions.
The spin-12\frac{1}{2}21​ Ising-Heisenberg model on two geometrically related triangles-in-triangles lattices is exactly solved through the generalized star-triangle transformation, which establishes a rigorous mapping correspondence with the effective spin-12\frac{1}{2}21​ Ising model on a triangular lattice. Basic thermodynamic quantities were exactly calculated within this rigorous mapping method along with the ground-state and finite-temperature phase diagrams. Apart from the classical ferromagnetic phase, both investigated models exhibit several unconventional quantum ordered and disordered ground states. A mutual competition between two ferromagnetic interactions of basically different character generically leads to the emergence of a quantum ferromagnetic phase, in which a symmetric quantum superposition of three up-up-down states of the Heisenberg trimers accompanies a perfect alignment of all Ising spins. In the highly frustrated regime, we have either found the disordered quantum paramagnetic phase with a substantial residual entropy or a similar but spontaneously ordered phase with a nontrivial criticality at finite temperatures. The latter quantum ground state exhibits a striking coexistence of imperfect spontaneous order with partial disorder, which is evidenced by a quantum reduction of the spontaneous magnetization of Heisenberg spins that indirectly causes a small reduction of the spontaneous magnetization of otherwise classical Ising spins.
The specific heat of the S = 1 transverse-crystal-field Ising model with an antiferro-magnetic exchange interaction J on a ∆-chain is studied by using the exact-diagonalization, quantum Monte Carlo and finite-temperature Lanczos methods. We find that when the strength of the transverse crystal field Γ is chosen to be sufficiently small, the specific heat exhibits a three-peak structure as a function of temperature, with peaks at T ∼ 0.05Γ, 0.2Γ and 0.7J. In addition to the two energy scales Γ and J, a new energy scale ∼ 0.1 × Γ appears due to the geometrical frustration and the existence of the Sz = 0 states in the present system. Geometrically frustrated spin systems have attracted considerable attention for a long time. It is expected that the introduction of quantum fluctuations into a clas-sically degenerate manifold leads to exotic properties of ground states and thermo-dynamics. From this point of view, S = 1/2 transverse-field Ising antiferromagnets