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Progress of Theoretical Physics

Thermodynamic Properties of the Transverse-Crystal-Field Ising Model on a Chain: -- Three Peak Structure in the Specific Heat --

Y. Fukumoto, Akihide Oguchi

Published May 1, 2006• 2 citations
Publisher

Abstract

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

DOI
10.1143/ptp.115.847

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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
We study effects of applying a transverse crystal field with strength Γ to the fully frustrated spin‐1 Ising Δ chain with an antiferromagnetic interaction J. Using Oittma and Brasch’s theorem for the ground state wave function of the transverse crystal field Ising models, we show that a ground state with no spontaneous symmetry breaking is stabilized by the transverse crystal field in our system for all Γ/J. The thermodynamic properties are studied numerically, and we found that this system shows a three‐staged entropy release as a function of temperature if J ≫ Γ.
We have numerically studied the thermodynamic properties of the spin 12\frac{1}{2}21​ XXZXXZXXZ chain in the presence of a transverse (noncommuting) magnetic field. The thermal, field dependence of specific heat and correlation functions for chains up to 20 sites has been calculated. The area where the specific heat decays exponentially is considered as a measure of the energy gap. We have also obtained the exchange interaction between chains in a bulk material using the random phase approximation and derived the phase diagram of the three-dimensional material with this approximation. The behavior of the structure factor at different momenta verifies the antiferromagnetic long-range order in the yyy direction for the three-dimensional case. Moreover, we have concluded that the low temperature Lanczos results [M. Aichhorn et al., Phys. Rev. B 67, 161103(R) (2003)] are more accurate for low temperatures and closer to the full diagonalization ones than the results of finite temperature Lanczos method [J. Jaklic and P. Prelovsek, Phys. Rev. B 49, 5065 (1994)].
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.
Using the Jordan-Wigner transformation and continued fractions we calculate rigorously the thermodynamic quantities for the spin-1/2 transverse Ising chain with periodically varying intersite interactions and/or on-site fields. We consider in detail the properties of the chains having a period of the transverse field modulation equal to 3. The regularly alternating transverse Ising chain exhibits several quantum phase transition points, where the number of transition points for a given period of alternation strongly depends on the specific set of the Hamiltonian parameters. The critical behavior in most cases is the same as for the uniform chain. However, for certain sets of the Hamiltonian parameters the critical behavior may be changed and weak singularities in the ground-state quantities appear. Due to the regular alternation of the Hamiltonian parameters the transverse Ising chain may exhibit plateaulike steps in the zero-temperature dependence of the transverse magnetization vs transverse field and many-peak temperature profiles of the specific heat. We compare the ground-state properties of regularly alternating transverse Ising and transverse XX chains and of regularly alternating quantum and classical chains. Making use of the corresponding unitary transformations we extend the elaborated approach to the study of thermodynamics of regularly alternating spin-1/2 anisotropic XY chains without field. We use the exact expression for the ground-state energy of such a chain of period 2 to discuss how the exchange interaction anisotropy destroys the spin-Peierls dimerized phase.
The exactly solvable spin-1/2 Ising-Heisenberg distorted diamond chain in the presence of the external magnetic field is investigated for the case of antiferromagnetic Ising and ferromagnetic XXZ Heisenberg interactions. The influence of quantum fluctuations and the distortion on the ground state, magnetic and thermal properties of the model are studied in detail.In particular, it is established that the zero-temperature magnetization curve may involve intermediate plateaus just at zero and 1/3 of the saturation magnetization. It is demonstrated that the temperature dependence of the specific heat reveals up to four distinct peaks at zero magnetic field and up to five distinct peaks at a weak magnetic field. The physical origin of all observed additional peaks of the specific heat has been clarified on the grounds of dominating thermal excitations. We have shown that the quantum fluctuations give rise to an effective geometrical frustration in this chain.
An antiferromagnetic Ising model on the Δ chain, with a macroscopic ground state degeneracy and exponentially decaying spin correlation functions, is studied. By introducing transverse magnetic fields Γ on spins at tips of the triangles and λΓ on spins on the bottom chain, we study the ground state as a function of λ and Γ. The λ=0 problem is exactly solved and the ground state has a staggered magnetization on the bottom chain for any Γ>0. Because the transverse field on the bottom chain with finite λ destroys the staggered order, the order–disorder transition occurs as λ is increased. We calculate the transition point λ c (Γ/ J ) as a function of Γ/ J , where J denotes the magnitude of the exchange interaction, by using the series expansion method around the λ=0 limit and conclude that λ c (Γ/ J =+0)≃0.9459. It is shown that states in the degenerate ground state manifold at Γ=0 are crucial in getting a finite value of λ c (+0).