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arXiv:1211.4357 · Physical Review B

Even-odd effect in short antiferromagnetic Heisenberg chains

Anna Machens, Nikolaos P. Konstantinidis, O. Waldmann, Imke Schneider, Sebastian Eggert

Published Apr 10, 2013• 42 citations
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Abstract

Motivated by recent experiments on chemically synthesized magnetic molecular chains, we investigate the lowest-lying energy band of short spin-$s$ antiferromagnetic Heisenberg chains focusing on effects of open boundaries. By numerical diagonalization we find that the Land\'e pattern in the energy levels, i.e., $E(S)\ensuremath{\propto}S(S+1)$ for total spin $S$, known from, e.g., ring-shaped nanomagnets, can be recovered in odd-membered chains, while strong deviations are found for the lowest excitations in chains with an even number of sites. This particular even-odd effect in the short Heisenberg chains cannot be explained by simple effective Hamiltonians and symmetry arguments. We go beyond these approaches, taking into account quantum fluctuations by means of a path-integral description and the valence bond basis, but the resulting quantum edge-spin picture which is known to work well for long chains does not agree with the numerical results for short chains and cannot explain the even-odd effect. Instead, by analyzing also the classical chain model, we show that spatial fluctuations dominate the physical behavior in short chains, with length $N\ensuremath{\lesssim}{e}^{\ensuremath{\pi}s}$, for any spin $s$. Such short chains are found to display a unique behavior, which is not related to the thermodynamic limit and cannot be described well by theories developed for this regime.

DOI
10.1103/physrevb.87.144409

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Motivated by recent experiments on chemically synthesized magnetic molecular chains, we investigate the lowest-lying energy band of short spin-sss antiferromagnetic Heisenberg chains focusing on effects of open boundaries. By numerical diagonalization we find that the Land'e pattern in the energy levels, i.e., E(S)\ensuremath∝S(S+1)E(S)\ensuremath{\propto}S(S+1)E(S)\ensuremath∝S(S+1) for total spin SSS, known from, e.g., ring-shaped nanomagnets, can be recovered in odd-membered chains, while strong deviations are found for the lowest excitations in chains with an even number of sites. This particular even-odd effect in the short Heisenberg chains cannot be explained by simple effective Hamiltonians and symmetry arguments. We go beyond these approaches, taking into account quantum fluctuations by means of a path-integral description and the valence bond basis, but the resulting quantum edge-spin picture which is known to work well for long chains does not agree with the numerical results for short chains and cannot explain the even-odd effect. Instead, by analyzing also the classical chain model, we show that spatial fluctuations dominate the physical behavior in short chains, with length N\ensuremath≲e\ensuremathπsN\ensuremath{\lesssim}{e}^{\ensuremath{\pi}s}N\ensuremath≲e\ensuremathπs, for any spin sss. Such short chains are found to display a unique behavior, which is not related to the thermodynamic limit and cannot be described well by theories developed for this regime.
Linear Heisenberg antiferromagnets (HAFs) are chains of spin-SSS sites with isotropic exchange JJJ between neighbors. Open and periodic boundary conditions return the same ground-state energy per site in the thermodynamic limit, but not the same spin SG{S}_{G}SG​ when $S\ensuremath{\ge}1$. The ground state of open chains of NNN spins has SG=0{S}_{G}=0SG​=0 or SSS, respectively, for even or odd NNN. Density-matrix renormalization-group calculations with different algorithms for even and odd NNN are presented up to N=500N=500N=500 for the energy and spin densities $\ensuremath{\rho}(r,N)$ of edge states in HAFs with S=1S=1S=1, 3/2, and 2. The edge states are boundary-induced spin density waves (BI-SDWs) with $\ensuremath{\rho}(r,N)\ensuremath{\propto}{(\ensuremath{-}1)}^{r\ensuremath{-}1}$ for r=1,2,...,Nr=1,2,...,Nr=1,2,...,N. The SDWs are in phase when NNN is odd, are out of phase when NNN is even, and have finite excitation energy $\mathrm{\ensuremath{\Gamma}}(N)$ that decreases exponentially with NNN for integer SSS and faster than 1/N1/N1/N for half integer SSS. The spin densities and excitation energy are quantitatively modeled for integer SSS chains longer than $5\ensuremath{\xi}$ spins by two parameters, the correlation length $\ensuremath{\xi}$ and the SDW amplitude, with $\ensuremath{\xi}=6.048$ for S=1S=1S=1 and 49.0 for S=2S=2S=2. The BI-SDWs of S=3/2S=3/2S=3/2 chains are not localized and are qualitatively different for even and odd NNN. Exchange between the ends for odd NNN is mediated by a delocalized effective spin in the middle that increases $|\mathrm{\ensuremath{\Gamma}}(N)|$ and weakens the size dependence. The nonlinear sigma model ($\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$) has been applied to the HAFs, primarily to S=1S=1S=1 with even NNN, to discuss spin densities and exchange between localized states at the ends as $\mathrm{\ensuremath{\Gamma}}(N)\ensuremath{\propto}{(\ensuremath{-}1)}^{N}exp(\ensuremath{-}N/\ensuremath{\xi})$. S=1S=1S=1 chains with odd NNN are fully consistent with the $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$; S=2S=2S=2 chains have two gaps $\mathrm{\ensuremath{\Gamma}}(N)$ with the same $\ensuremath{\xi}$ as predicted whose ratio is 3.45 rather than 3; the $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$ is more approximate for S=3/2S=3/2S=3/2 chains with even NNN and is modified for exchange between ends for odd NNN.
The qualitative difference in low-energy properties of spin S quantum antiferromagnetic chains with integer S and half-odd-integer S discovered by Haldane [F. D. M. Haldane, arXiv:1612.00076 (1981); Phys. Lett. A 93, 464–468 (1983); Phys. Rev. Lett. 50, 1153–1156 (1983)] and Tasaki [Tasaki, Graduate Texts in Physics (Springer, 2020)] can be intuitively understood in terms of the valence-bond picture proposed by Affleck et al. [I. Affleck, Phys. Rev. Lett. 59, 799 (1987); Commun. Math. Phys. 115, 477–528 (1988)]. Here, we develop a similarly intuitive diagrammatic explanation of the qualitative difference between chains with odd S and even S, which is at the heart of the theory of symmetry-protected topological (SPT) phases. (There is a 24 min video in which the essence of the present work is discussed: https://youtu.be/URsf9e_PLlc.) More precisely, we define one-parameter families of states, which we call the asymmetric valence-bond solid (VBS) states, that continuously interpolate between the Affleck–Kennedy–Lieb–Tasaki (AKLT) state and the trivial zero state in quantum spin chains with S = 1 and 2. The asymmetric VBS state is obtained by systematically modifying the AKLT state. It always has exponentially decaying truncated correlation functions and is a unique gapped ground state of a short-ranged Hamiltonian. We also observe that the asymmetric VBS state possesses the time-reversal, the Z2×Z2, and the bond-centered inversion symmetries for S = 2 but not for S = 1. This is consistent with the known fact that the AKLT model belongs to the trivial SPT phase if S = 2 and to a nontrivial SPT phase if S = 1. Although such interpolating families of disordered states were already known, our construction is unified and is based on a simple physical picture. It also extends to spin chains with general integer S and provides us with an intuitive explanation of the essential difference between models with odd and even spins.
The S = 1/2 Heisenberg chains with bond alternation and randomness on the strong bonds are studied by the density matrix renormalization group method. It is assumed that the odd-th bond is antiferromagnetic with strength J and the even-th bond can take the values JA and JF (JA > J > 0 > JF) randomly. The ground state of this model interpolates between the Haldane and dimer phases via a randomness dominated intermediate phase. On the basis of the scaling of the low energy spectrum and mean field treatment of the interchain coupling, we find that the magnetic long range order is induced by randomness in the intermediate regime. In the magnetization curves, there appears a plateau at the fractional value of the saturated magnetization. The fine structures of the magnetization curves and low energy spectrum are understood using the cluster picture. The relation to the recent experiment for (CH3)2CHNH3Cu(ClxBr1-x)3 is discussed.
The exact eigenvalue spectrum and thermodynamic properties of the spin Hamiltonian $\mathcal{H}=2|J|\ensuremath{\Sigma}\stackrel{\frac{N}{2}}{i=1}({\mathrm{S}}_{2i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{2i\ensuremath{-}1}+a{\mathrm{S}}_{2i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}_{2i+1})$ are calculated for short chains of N=4,6,8,and10N=4,6,8, \mathrm{and} 10N=4,6,8,and10 spins, each of spin \textonehalf{}, for alternation parameter a=0.2,0.4,0.6,and0.8a=0.2,0.4,0.6, \mathrm{and} 0.8a=0.2,0.4,0.6,and0.8, with ${\mathrm{S}}_{N+1}\ensuremath{\equiv}{\mathrm{S}}_{1}$. The behavior for $N=\ensuremath{\infty}$ is estimated by extrapolation. Comparison is made with the known results for a=0a=0a=0 and 1. The ground-state energy, ground-state short-range order, energy gap, and triplet excitation spectrum are compared with various approximate theories. Zero-temperature infinite-chain magnetization curves are inferred from the finite-chain results. The energy, entropy, specific heat, and magnetic susceptibility for N=10N=10N=10 are shown to approximate well the behavior for $N=\ensuremath{\infty}$ when kT∣J∣>a\frac{\mathrm{kT}}{|J|}>a∣J∣kT​>a. The magnetic-susceptibility data on the free radical 2,2-bis (ppp-nitrophenyl)-1-picrylhydrazyl are shown to agree well with the theoretical results for an alternating chain with $a\ensuremath{\approx}0.6$.
For the antiferromagnetic J1{J}_{1}J1​-J2{J}_{2}J2​ quantum spin chain with an even number of sites, the point J2d=J1/2{J}_{2}^{d}={J}_{1}/2J2d​=J1​/2 is a disorder point. It marks the onset of incommensurate real space correlations for J2>J2d{J}_{2}>{J}_{2}^{d}J2​>J2d​. At a distinct larger value of J2L=0.52036(6)J1{J}_{2}^{L}=0.520\phantom{\rule{0.16em}{0ex}}36(6){J}_{1}J2L​=0.52036(6)J1​, the Lifshitz point, the peak in the static structure factor begins to move away from $k=\ensuremath{\pi}$. Here, we focus on chains with an odd number of sites. In this case, the disorder point is also at J2d=J1/2{J}_{2}^{d}={J}_{1}/2J2d​=J1​/2 but the behavior close to the Lifshitz point, ${J}_{2}^{L}\ensuremath{\simeq}0.538{J}_{1}$, is quite different: starting at J2L{J}_{2}^{L}J2L​, the ground state goes through a sequence of level crossings as its momentum changes away from $k=\ensuremath{\pi}/2$. An even length chain, on the other hand, is gapped for any J2>0.24J1{J}_{2}>0.24{J}_{1}J2​>0.24J1​ and has the ground-state momentum k=0k=0k=0. This gradual change in the ground-state wave function for chains with an odd number of sites is reflected in a dramatic manner directly in the ground-state on-site magnetization as well as in the bipartite von Neumann entanglement entropy. Our results are based on DMRG calculations and variational calculations performed in a restricted Hilbert space defined in the valence bond picture. In the vicinity of the point J2=J1/2{J}_{2}={J}_{1}/2J2​=J1​/2, we expect the variational results to be very precise.