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Journal of the Physical Society of Japan

Order–Disorder Transition in a Fully Frustrated Transverse-Field Ising Chain

Yoshiyuki Fukumoto, Akihide Oguchi

Published Sep 15, 2003• 0 citations
Publisher

Abstract

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).

DOI
10.1143/jpsj.72.2317

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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).
We study a one-dimensional chain of corner-sharing triangles with antiferromagnetic Ising interactions along its bonds. Classically, this system is highly frustrated with an extensive entropy at T=0T=0T=0 and exponentially decaying spin correlations. We show that the introduction of a quantum dynamics via a transverse magnetic field removes the entropy and opens a gap, but leaves the ground state disordered at all values of the transverse field, thereby providing a particularly simple realization of the ``disorder by disorder'' scenario first proposed by Fazekas and Anderson in their search for resonating valence bond states. Our conclusion relies on exact diagonalization calculations as well as on the analysis of a 14th-order perturbation expansion about the large transverse field limit. This test of the perturbation expansion method suggests that it could be applied to studies of frustrated transverse-field magnets in higher dimensions.
We study the transverse susceptibility of the fully frustrated antiferromagnetic Ising Δ-chain, extending Minami's transfer-matrix method to the transverse susceptibility of general-type Ising linear-chains [J. Phys. Soc. Jpn. 67 (1998), 2255]. For the transverse fields Γ1 in tip spin sites and Γ2 in bottom spin sites, we calculate zero-field transverse susceptibilities χxtip = lim Γ1 →0Mxtip/Γ1 and χxbottom = lim Γ2 →0Mxbottom/Γ2 while keeping the ratio Γ1/Γ2 fixed, where Mxtip (bottom) denotes the magnetization for tip (bottom) spin sites. Both the transverse susceptibilities follow Curie's law at low temperatures. We also calculate χxbottom(Γ1 > 0), the transverse susceptibility of the bottom spin chain under a finite tip-spin transverse field, to understand the Curie-type behavior in zero-field susceptibility. Using the second-order perturbation theory, we also discuss the Γ1 dependence of χxbottom (Γ1) at zero temperature.
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.
The ground-state properties of the triangular antiferromagnetic Ising model with Ising constant J in a transverse field h are investigated by a real-space renormalisation group method. For h=0, the known properties (energy=-J, finite entropy, no order) are recovered. For h not=0, the ground state remains highly degenerate, no order exists and the correlation function exhibits power-law behaviour up to (h/J)c approximately=1.41, where a phase transition takes place to a non-degenerate ground state. At h/J=(h/J)c, power-law behaviour is still observed for the correlation function but with a different exponent. Just above this region a gap Delta opens ( Delta approximately ((h/J)-(h/J)c)s with s=0.8) and the coherence length of the correlation function xi diverges as xi varies as ((h/J)-(h/J)c)-v with v=0.8. This corresponds to a three-dimensional classical model where h is replaced by the temperature. In this type of system, where the ground state has a degeneracy proportional to N2/3 it is predicted that a transition will occur at a finite T c to a low-temperature phase without order but with power-law behaviour for the correlation function, as for the classical XY model in two dimensions.
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
The ground state properties of an Ising chain with nearest (J1{J}_{1}J1​) and next-nearest neighbor (J2{J}_{2}J2​) interactions in a transverse field are investigated using the density matrix renormalization group and cluster mean-field theory methods. Its quantum phase diagram has two ordered regions (separated by a quantum disordered region), one of which has simple ferromagnetic or N\'eel order depending upon the sign of J1{J}_{1}J1​, and the other is the double-staggered antiferromagnetic phase. The presence of the edge modes in these ordered phases is inferred by calculating the end-to-end spin-spin correlations on open chains. It is found that there occur four edge modes in the double-staggered phase, while the ferromagnetic and N\'eel phases support two edge modes, except very near J1=0{J}_{1}=0J1​=0, where it seems they have four edge modes like the exact case at J1=0{J}_{1}=0J1​=0 itself.