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Edge states in open antiferromagnetic Heisenberg chains

Shaojin Qin, Tai-Kai Ng, Zhao-Bin Su

1995
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Physical Review B (Condensed Matter)
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In this letter we report our results in investigating edge effects of open antiferromagnetic Heisenberg spin chains with spin magnitudes S = 1/2, 1, 3/2, 2 using the density-matrix renormalization group (DMRG) method initiated by White. For integer spin chains, we find that edge states with spin magnitude S edge = S/2 exist, in agreement with Valence-Bond-Solid model picture. For half-integer spin chains, we find that no edge states exist for S = 1/2 spin chain, but edge state exists in S = 3/2
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... spin chain with S edge = 1/2, in agreement with previous conjecture by Ng. Strong finite size effects associated with spin dimmerization in half-integer spin chains will also be discussed. PACS 75.10.Jm, 75.40.Mg The antiferromagnetic Heisenberg spin chains has been a subject of immense interest in the last decade since Haldane conjectured that the low energy physics of integer and half integer spin chains are fundamentally different [1]. It is now generally believed that half-integer Heisenberg spin chains have gapless excitation spectrum, whereas gaps exist in integer spin chains (Haldane gap). More recently, there has been increasing interests in studies of spin chains with defects [2] [3] [4] [5] . In particular, the properties of broken S = 1 quantum spin chains have received much attention because of the experimental observation of S = 1/2 excitations localized at the ends of broken S = 1 spin chains [2] . More generally, one may address the question of whether edge states are genuine properties of finite quantum spin chains as in Fractional Quantum Hall Effect [6, 7] . Recently, a theory of edge states based on the Non-linear-sigma model (N LσM ) plus topological θ-term has been developed by one of us [8] where it was conjectured that edge states are genuine properties of antiferromagnetic quantum spin chains with spin value S > 1/2. In this letter, we shall address this question for both integer and half integer spin chains numerically using the recently developed density matrix renormalization group (DMRG) method [9] . We shall present results for open spin chains with spin values S = 1/2, 1, 3/2, 2 up to chain length of 100 sites. The DMRG method has proved to be tremendously successful in studying S = 1 and S = 1/2 antiferro-magnetic Heisenberg spin chains [9, 11] . The method was found to be particularly suitable for studying spin chains with open boundary condition and is thus well suited for our purpose of studying edge states. We use the infinite chain algorithm [9] in our study. Two new sites are added to the spin chain in each optimal step of the calculation from length L = 4 to L = 100. That is, open spin chains with even number of sites are studied. The number of the kept optimized states in our calculation is m = 120. The largest truncation errors for S = 1/2, 1, 3/2, and 2 chains are found to be smaller than 10 −10 , 10 −8 , 3 × 10 −6 and 10 −5 respectively when the chain length reaches L = 100 in the final step. Properties of the ground state and a few lowest excited states are obtained by looking at the lowest energy state with fixed total z-component of spin angular momentum S tot z . In particular, the ground state corresponds to the lowest energy state in the sector S tot z = 0. We shall look at the excitation energies of various states with different S tot z and the corresponding average z-component of angular momentum on each site i =< S i z >. The dimmerization parameter q(i) =< S i .S i+1 − S i−1 .S i > for the ground state will also be examined. We start with the excitation energies for integer spin chains. Fig.1 shows the excitation energies E n − E 0 for n = 1 to 3 as a function of chain length L for the S = 1 spin chain. E n is the energy of the lowest energy state in the sector S tot z = n. Energy is measured in units of Heisenberg coupling J. According to the valence bond picture [10, 12] , for S = 1 spin chain, two S = 1/2 spins are left at two ends of the spin chain, and are coupled with effective coupling J ef f ∼ Je −L/ξ , where ξ is the correlation length. For even spin chains, the coupling is antiferromagnetic and the resulting ground state is a spin singlet(S tot z = 0). The lowest energy state with S tot z = 1 can be constructed by exciting the singlet formed by the two edge spins into a triplet, and the excitation energy is of order J ef f , which goes to zero exponentially as length of spin chain increases. Excited states with larger S tot z cannot be constructed by exciting only the edge spins any more and bulk excitations must be involved in constructing states with S tot z > 1, implying that the excitation energy will be of order E n − E 0 ∼ (n − 1) × E H as L → ∞, 1

doi:10.1103/physrevb.52.12844
pmid:9980452
fatcat:bbuqnmuly5hnbdradjk73ilfba