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physica status solidi (b)

Ground‐State Properties of Finite Antiferromagnetic Chains

H. P. van de Braak, W.J. Caspers

Published Jan 1, 1969• 1 citation
Publisher

Abstract

Abstract Exact calculations are presented for the ground state of linear antiferromagnetic Heisenberg chains with open ends. The wave function and the energy are given for chains with N = 4, 6, and 8, and quantum number S = 1/2. It is shown that in the limit for N → ∞ the ground state is probably nondegenerate and the long‐range order tends to zero. The short‐range order shows a strongly oscillating character due to end effects persisting over relatively long distances. A comparison with closed rings is made and the connection with the observation of a reduction of the zero‐point deviation in the neighbourhood of non‐magnetic impurities in more‐dimensional systems is pointed out.

DOI
10.1002/pssb.19690350244

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The one-dimensional Heisenberg antiferromagnet with 8=1/2 is treated with the use of the Fermi operators and an additional condition that the long-range order is absent.Our spin wave spectrum of a one-dimensional Heisenberg antiferromagnet without longrange order is expressed by andThis is in good agreement with the exact spin wave spectrum role =1!'/2lsin kl which is obtained by des Cloizeaux and Pearson at T=O.Correlation functions <S1S2> and <SlSs> are in good agreement with the exact value at T=O, also.Thermodynamic properties of this system agree with those of Bulaevskii and those of Katsura and Inawashiro. § I. IntroductionThe one-dimensional Heisenberg antiferromagnet has been investigated by many authors.Most of previous theories have been for either zero or infinite temperatures.Recently various experiment on (CD3) 4NMnCl 3 (TMMC)ll requires further theoretical work of a Heisenberg linear-chain magnet for finite temperatures.The experimental results of linear antiferromagnet show that at low temperatures there are long-lived spin waves in spite of the absence of long-range order.At T= 0, the spin wave spectrum, wqoc I sin ql, is gained approximately by Anderson 2 l and exactly by des Cloizeaux and Pearson.8 l Recently Richards, 4 l Scales and Gersh, 5 l Kondo and Yamaji 6 l obtained the spin wave spectrum of linear antiferromagnet at finite temperatures with the use of the second-order Green's function theory.The spin wave spectrum, however, does not show exactly wqoc I sin ql at finite temperatures.

Progress of Theoretical Physics•Jan 1, 1973•7 citations
Abstract Exact calculations are presented for the ground state of linear antiferromagnetic Heisenberg chains with open ends. The wave function and the energy are given for chains with N = 4, 6, and 8, and quantum number S = 1/2. It is shown that in the limit for N → ∞ the ground state is probably nondegenerate and the long‐range order tends to zero. The short‐range order shows a strongly oscillating character due to end effects persisting over relatively long distances. A comparison with closed rings is made and the connection with the observation of a reduction of the zero‐point deviation in the neighbourhood of non‐magnetic impurities in more‐dimensional systems is pointed out.
The ground state properties of the Heisenberg antiferromagnetic spin- 1 / 2 chain with interactions varying as (distance) -p are investigated from finite cell calculations. The gap between the ground state and the first excited state as well as the end-to-end spin correlation function have been calculated for finite open chains up to 14 spins and 12 spins respectively. Aided by first-order perturbation calculations near p=0, the authors can extrapolate the gap in the thermodynamic limit, and they find that the gap vanishes for every p not=0. From the analysis of the spin correlation function they find that long-range order exists for p<1 and does not exist for p>2. The existence of long-range order in the range 1<p<2 remains an open question.
Ground state properties of finite anisotropic antiferromagnetic Heisenberg chains are studied for odd half-integer spins S= 1 / 2 , 3 / 2 ... as well as integer spins S=1, 2 .... Finite size scaling analysis of the results clearly distinguishes the half-integer (S= 1 / 2 , 3 / 2 ) from the integer (S=1) spin situation. It gives strong support to a recent conjecture which postulates that the T=0 phase structure is very different in the two cases. According to this idea there exists a new phase between the planar and the antiferromagnetic region, for integer spins only. This phase, which includes the isotropic point, has a finite energy gap and no long range order.
Full magnetization curves of S=1 antiferromagnetic Heisenberg chains with periodic and open boundaries are investigated by a quantum Monte Carlo method. The boundary condition has a significant effect on the magnetization process especially in the field range where the Zeeman energy is less than the Haldane gap. The magnetization of the periodic chain in the range is strongly suppressed at low temperatures because the ground state is nonmagnetic. On the other hand, the magnetic ground of the open chain bring about finite magnetization. The difference between both the magnetization curves is also investigated in detail by subtracting the magnetization of the periodic chain with L spins from one of the open chains with L+1 spins. It agrees well with the double of the Brillouin function for S=1/2, rather than the Brillouun function for S=1.
The ground state energy, the correlation functions and the excitation energy of a Heisenberg antiferromagnetic chain of spin S are investigated by the real-space renormalization-group method. The asymptotic form ( S →∞) of the ground state energy is compared with the result obtained by the spin wave theory. The calculated correlation function < S ( i ) · S ( i + l ) > shows no long range order for a system of finite S . The excitation spectrum is found to have singular dips at q =(odd integer) ×π/3 n a . This is associated with the shortcoming of the three-site renormalization.
We have calculated the two-point correlation functions ${\ensuremath{\omega}}_{\mathrm{il}}$(N)=(4/3)〈S${\ensuremath{\rightarrow}}_{i}$\ensuremath{\cdot}S${\ensuremath{\rightarrow}}_{i+l}$ 〉 and their averages over i,${\ensuremath{\omega}}_{l}$ (N), in the ground state of the one-dimensional antiferromagnetic Heisenberg model for N=4,6,8,...,16 spins. Both periodic (rings) and free-end (chains) boundary conditions are considered. Surprisingly tight lower and upper bounds have been obtained for ${\ensuremath{\omega}}_{l}$(\ensuremath{\infty}) under reasonable assumptions. In addition to showing the rather strong even-l--odd-l alternation in \ensuremath{\Vert}${\ensuremath{\omega}}_{l}$(N)\ensuremath{\Vert}, known from earlier results of Bonner and Fisher for rings with N up to 10, our bounds indicate a smooth behavior in l\ensuremath{\Vert}${\ensuremath{\omega}}_{l}$(\ensuremath{\infty})\ensuremath{\Vert} for l odd and l even, with, surprisingly, a broad maximum attained within the odd-l values. The bounds obtained from the chain results were essential to seeing this maximum (because of the larger l values available for given N). The quantity l\ensuremath{\Vert}${\ensuremath{\omega}}_{l}$(N)\ensuremath{\Vert} for chains with fixed N also shows such a maximum, and in addition shows a similar maximum for even l's. If the indicated trends for large l and N continue in ${\ensuremath{\omega}}_{l}$(\ensuremath{\infty}) and in SN{S}_{N}SN​, the structure factor at wave vector \ensuremath{\pi}, then finite-size contributions to ${\ensuremath{\omega}}_{l}$(N) will have to contribute to the (seemingly) logarithmic divergence of SN{S}_{N}SN​ as N\ensuremath{\rightarrow}\ensuremath{\infty}. We are not aware of any models where a similarly weak divergence shows such a finite-size contribution.
The one-dimensional Heisenberg antiferromagnet with 8=1/2 is treated with the use of the Fermi operators and an additional condition that the long-range order is absent.Our spin wave spectrum of a one-dimensional Heisenberg antiferromagnet without longrange order is expressed by andThis is in good agreement with the exact spin wave spectrum role =1!'/2lsin kl which is obtained by des Cloizeaux and Pearson at T=O.Correlation functions <S1S2> and <SlSs> are in good agreement with the exact value at T=O, also.Thermodynamic properties of this system agree with those of Bulaevskii and those of Katsura and Inawashiro. § I. IntroductionThe one-dimensional Heisenberg antiferromagnet has been investigated by many authors.Most of previous theories have been for either zero or infinite temperatures.Recently various experiment on (CD3) 4NMnCl 3 (TMMC)ll requires further theoretical work of a Heisenberg linear-chain magnet for finite temperatures.The experimental results of linear antiferromagnet show that at low temperatures there are long-lived spin waves in spite of the absence of long-range order.At T= 0, the spin wave spectrum, wqoc I sin ql, is gained approximately by Anderson 2 l and exactly by des Cloizeaux and Pearson.8 l Recently Richards, 4 l Scales and Gersh, 5 l Kondo and Yamaji 6 l obtained the spin wave spectrum of linear antiferromagnet at finite temperatures with the use of the second-order Green's function theory.The spin wave spectrum, however, does not show exactly wqoc I sin ql at finite temperatures.