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Journal of the Physical Society of Japan

Fractional Magnetization Plateau in the Alternating Spin Heisenberg Chain

Tetsuji Kuramoto

Published Jun 15, 1999• 12 citations
Publisher

Abstract

We study the S =1/2 and S =1 alternating spin Heisenberg chain with the next-nearest neighbor (NNN) interaction J 2 between neighboring S =1/2 spins, by using the numerical diagonalization method of a finite size system. It is found that a fractional plateau appears at magnetization m =1/2 in the magnetization curve for J 2 > J 2 c =0.106±0.001. The phase transition at J 2 c is characterized by the exponent σ=1, which controls the critical behavior of the spin gap Δ as Δ∼ exp(- a /( J 2 - J 2 c ) 1+σ ). For J 2 < J 2 c , the model at m =1/2 is described by the conformal field theory with central charge c =1 and exponent η for the spin correlation function of the transverse component to a magnetic field is calculated.

DOI
10.1143/jpsj.68.1813

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We study the S =1/2 and S =1 alternating spin Heisenberg chain with the next-nearest neighbor (NNN) interaction J 2 between neighboring S =1/2 spins, by using the numerical diagonalization method of a finite size system. It is found that a fractional plateau appears at magnetization m =1/2 in the magnetization curve for J 2 > J 2 c =0.106±0.001. The phase transition at J 2 c is characterized by the exponent σ=1, which controls the critical behavior of the spin gap Δ as Δ∼ exp(- a /( J 2 - J 2 c ) 1+σ ). For J 2 < J 2 c , the model at m =1/2 is described by the conformal field theory with central charge c =1 and exponent η for the spin correlation function of the transverse component to a magnetic field is calculated.
We study the magnetization plateau at m =1/2 in the S =1/2 and S =1 alternating spin Heisenberg chain with two next-nearest-neighbor (NNN) interactions J a between neighboring S =1/2 spins and J b between neighboring S =1 spins, using the numerical diagonalization and the density-matrix renormalization group (DMRG) method. We present the m =1/2 plateau phase diagram in the J a – J b plane obtained by the level crossing method useful for the Berezinskii–Kosterlitz–Thouless (BKT) transition. In this phase diagram, we find that there are two plateau phases distinguished by a parity of the alternating spin chain.
The one dimensional S=1/2S=1/2S=1/2 Heisenberg model with dimerization (1−j1-j1−j) and quadrumerization (δδδ) in the magnetic field is studied by means of the numerical exact diagonalization of finite size systems and the conformal field theory. It is found that the magnetization plateau at half of the saturation value exists for δ≠0δ\neq 0δ=0. For δ=0δ= 0δ=0, this model is described by the conformal field theory with central charge c=1c=1c=1 at this value of magnetization. The critical exponent ννν which characterizes the δδδ-dependence of the width of the plateau is calculated using the level spectroscopy method. The jjj-dependence of the critical exponent ννν is found to be non-monotonic and discontinuous at j=0j = 0j=0. The effective theory of the magnetization plateau is also presented for various limiting cases.
The S =1/2 bond alternating Heisenberg spin chain in a magnetic field is studied by the numerical diagonalization and the finite size scaling based on the conformal field theory. It is found that the magnetic state (0< m <1) is gapless (the central charge c =1) and described by the Luttinger liquid, which corresponds to a spinless Fermion system with repulsive interaction for antiferromagnetic-antiferromagnetic alternating chains, but attractive one for ferromagnetic-antiferromagnetic chains. The critical exponents δ and η are determined at the critical fields. The results suggest that the phase transitions belong to the same universality class as the S =1 antiferromagnetic chain for both alternating cases.
The magnetization process of the S=1/2S=1/2S=1/2 Heisenberg chain with next-nearest-neighbor (NNN) and alternating nearest-neighbor interactions is investigated using the bosonization technique. The existence of the NNN interaction is shown to stabilize a ``diluted'' dimer order at magnetization mz=1/4,{m}^{z}=1/4,mz=1/4, leading to a plateau, where two degenerate ground states appear. Effective Hamiltonians describing the low-energy physics in the vicinity of the mz=1/4{m}^{z}=1/4mz=1/4 plateau are also presented. The analogy to metal-insulator transitions is discussed briefly.
The S =1 antiferromagnetic Heisenberg chain in a magnetic field along the z -axis at T =0 is studied by numerical diagonalizations up to N =16 and an analysis of size corrections. We check that the system obeys the conformal field theory with the central charge c =1 in a magnetic state (0< m <1). We investigate the asymptotic form of the transverse and parallel spin correlations and give the exponents η and η z , defined by < S 0 x S _r^x>≃(-1)^rr^-η a n d <S_0^zS_r^z>-m^2≃cos (2k_Fr)r^-η^z. W e d e t e r m i n e η=1/2 a n d η^z=2 a t m=0 a n d 1. I n a d d i t i o n , w e c h e c k t h e r e l a t i o n ηη^z=1, w h i c h i s c o n s i s t e n t w i t h t h e L u t t i n g e r l i q u i d c o n c e p t . I n t e r m s o f t h e c o n c e p t , t h e a n o m a l i e s o f m a g n e t i z a t i o n a t m=0 a n d 1 a r e d i s c u s s e d .