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Physical Review Letters

Nonlocal Topological Order in Antiferromagnetic Heisenberg Chains

Shaojin Qin, Jizhong Lou, Liqun Sun, Changfeng Chen

Published Feb 12, 2003• 28 citations
Publisher

Abstract

We demonstrate the existence of nonlocal topological (string) order in half-integer-spin antiferromagnetic Heisenberg chains on macroscopic scale on the basis of analytical scaling analysis and density matrix renormalization group calculations. Strong numerical evidence leads to a conjecture that chains with S = (2m-1)/2 and m (m = integers) belong to the same topological class defined by the topological angle theta/pi = 1/m that plays a role similar to the fictitious gauge field in the fractional quantum Hall effect.

DOI
10.1103/physrevlett.90.067202

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We demonstrate the existence of nonlocal topological (string) order in half-integer-spin antiferromagnetic Heisenberg chains on macroscopic scale on the basis of analytical scaling analysis and density matrix renormalization group calculations. Strong numerical evidence leads to a conjecture that chains with S = (2m-1)/2 and m (m = integers) belong to the same topological class defined by the topological angle theta/pi = 1/m that plays a role similar to the fictitious gauge field in the fractional quantum Hall effect.
The manifestations of topological effects in finite antiferromagnetic Heisenberg chains are examined by density-matrix renormalization-group techniques. We find that the difference between integer and half-integer spin chains shows up in ground-state energy per site when the length of the spin chain is longer than \ensuremath{\sim}\ensuremath{\xi}, where $\ensuremath{\xi}\ensuremath{\sim}\mathrm{exp}(\ensuremath{\pi}S)$ is a spin-spin correlation length, for spin magnitudes S up to 5/2.5/2.5/2. For open chains with spin magnitudes S=5/2S=5/2S=5/2 to S=5,S=5,S=5, we verify that end states with fractional spin quantum numbers ${S}^{\ensuremath{'}}$ exist and are visible in the structure of the low-energy excitation spectrum even when the chain length is much smaller than the correlation length \ensuremath{\xi}. The robustness of end states points to the possibility of the existence of general hidden topological order in quantum spin chains with spin magnitudes S> 1.S>~1.S> 1.
We derive the extended string order parameter O(S) for antiferromagnetic Heisenberg chains with half-integer spin S in the valence-bond-solid picture. We obtain the analytic power-law scaling of O(S) versus the chain length L and show that O(S) scales at an extremely slow pace that decreases rapidly with growing spin magnitude. Furthermore, accurate numerical calculations show that the power-law scaling sets in only when L exceeds a characteristic length scale l(S) which increases very fast with growing S. Consequently, a pseudo-long-range string order exists in half-integer-spin Heisenberg chains. The implications of this result and its relationship to other topological features such as the end-chain states are discussed.
Antiferromagnetic Heisenberg spin chains for S =1 and S =2 are treated numerically. We investigate a generalized θ -string correlation function O θ ( j )=< S z 1 exp ( i θ ∑ j -1 k =1 S z k ) S j z > which is a generalization of the string correlation function of den Nijs-Rommelse. When the value of θ lies in an appropriate region, the phase of O θ ( j ) is fixed and O θ ( j ) behaves smoothly as a function of j. This means alternative sign changes of the usual spin-spin correlation O θ =0 ( j ) are well canceled in O θ ( j ) for such values of θ . The cancellation is best at θ =π and π /2 for the S=1 and 2 cases, respectively. The generalized string which appears in the θ -string correlation function plays a role similar to that of a fictitious gauge potential in the definition of the hidden order parameter of the fractional quantized Hall state by Girvin and MacDonald.
We study the ground-state properties of antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ chains with half-integer spins ranging from S=32S=\frac{3}{2}S=23​ to S=112S=\frac{11}{2}S=211​ using the density-matrix renormalization group method. We map out the ground-state phase diagrams as a function of J2J1\frac{{J}_{2}}{{J}_{1}}J1​J2​​ containing topological phases with alternating $\frac{2S\ensuremath{-}1}{2}$ and 2S+12\frac{2S+1}{2}22S+1​ valence bonds. We identify these topological phases and their boundaries by calculating the string order parameter, the dimer order parameter, and the spin gap for those $\mathrm{high}\text{\ensuremath{-}}S$ systems in the thermodynamic limit (finite-size scaling). We find that these topological regions narrow down inversely with SSS and converge to a single point at J2J1=14\frac{{J}_{2}}{{J}_{1}}=\frac{1}{4}J1​J2​​=41​ in the classical limit---a critical threshold between commensurate and incommensurate orders. In addition, we extend the discussion of the Majumder-Ghosh state, previously noted only for S=12S=\frac{1}{2}S=21​, and speculate about its possible presence as a ground state in half-integer high-spin systems over a substantial range of J2J1\frac{{J}_{2}}{{J}_{1}}J1​J2​​ values.
The antiferromagnetic Heisenberg spin chain with integer spin has a short-range magnetic order and an excitation energy gap above the ground state. This so-called Haldane gap is proportional to the exchange coupling constant J of the Heisenberg chain. In this study, we discuss recent results about the spin dependence of the Haldane gap and conjecture an analytical formula valid asymptotically for large spin values. We then discuss the robustness of the edge states of the spin-1 chain by studying a spin-1 ladder using the density matrix renormalization group (DMRG) algorithm. We show that the peculiar hidden topological order of the spin-1 chain disappears smoothly by increasing the ladder rung coupling without any intervening phase transition. This is evidence for the fragile character of the topological order of the spin-1 chain.