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Physica Scripta

Hidden topological order in integer quantum spin chains

S. M. Girvin, Daniel P. Arovas

Published Jan 1, 1989• 135 citations
Publisher

Abstract

Using an idea of Rommelse and den Nijs, we have investigated a direct analogy between the topological order in integer quantum antiferromagnets and the topological order in the fractional quantum Hall effect (FQHE). Numerical results are presented for the S = 1 generalized Heisenberg chain. The analogy between the Laughlin gap in the FQHE and the Haldane gap in integer spin chains is discussed.

DOI
10.1088/0031-8949/1989/t27/027

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Using an idea of Rommelse and den Nijs, we have investigated a direct analogy between the topological order in integer quantum antiferromagnets and the topological order in the fractional quantum Hall effect (FQHE). Numerical results are presented for the S = 1 generalized Heisenberg chain. The analogy between the Laughlin gap in the FQHE and the Haldane gap in integer spin chains is discussed.
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.
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 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.
We discuss the relationship between fractional quantum Hall (FQH) states at filling factor $\ensuremath{\nu}=p/(2p+1)$ and quantum spin chains. This series corresponds to the Jain series $\ensuremath{\nu}=p/(2mp+1)$ with m=1m=1m=1 where the composite fermion picture is realized. We show that the FQH states with toroidal boundary conditions beyond the thin-torus limit can be mapped to effective quantum spin S=1S=1S=1 chains with ppp spins in each unit cell. We calculate energy gaps and the correlation functions for both the FQH systems and the corresponding effective spin chains, using exact diagonalization and the infinite time-evolving block decimation (iTEBD) algorithm. We confirm that the mass gaps of these effective spin chains are decreased as ppp is increased which is similar to S=pS=pS=p integer Heisenberg chains. These results shed new light on a link between the hierarchy of FQH states and the Haldane conjecture for quantum spin chains.
Since the Heisenberg spin chain can be considered the simplest realistic model of magnetism, surprise and some degree of controversy have resulted from recent work of Haldane. The prediction is that quantum spin chains with half-integer spin should all display T=0 phase behavior equivalent to that of the Bethe Ansatz integrable (solvable) spin-1/2 quantum chain. More remarkably, the class of integer spin chains is predicted to show very different phase behavior. In particular, a gap should be present in the spectrum of a Heisenberg antiferromagnetic chain. This remarkable feature is counterintuitive in terms of accepted wisdom in magnetism (spin-wave theory, spin-Peierls theory) and critical phenomena. Consequently the vertification of the prediction is of great interest. A considerable amount of numerical work has been done, involving finite-chain, finite-size scaling, variational, Monte Carlo and other calculations, which will be reviewed here. The present consensus is that the weight of numerical evidence supports the prediction, although puzzling features still remain. Adding additional interactions to the basic Heisenberg Hamiltonian such as spin (XXZ) anisotropy, single-ion anisotropy, biquadratic exchange, and an applied magnetic field, generates a rich and complicated phase diagram for chains with spin >1/2, particularly for the case of integer spin. The s=1 phase diagram seems to display critical behavior of a type not previously encountered. A theoretical appraisal of the Haldane phenomenon will include a discussion of the possible role of nonintegrability. Mention will also be made of current progress in experimental investigation of the phenomenon, including problems that might be encountered. More recent work of Affleck has greatly generalized the field-theoretic mappings which underlay the original work of Haldane. A number of interesting problems have been mapped into quantum spin chains of various types, including field theoretic phenomena and the localization problem of the quantum Hall effect.
We have investigated Haldane's conjecture for the S = 2 antiferromagnetic quantum spin chain with nearest-neighbour exchange J . Using a density matrix renormalization group algorithm for chains up to L = 350 spins, we show that the ground state has hidden topological order: the physics of the S = 2 chain can be captured by a valence bond solid description. We also observe effective free spin-1 states at the ends of an open S = 2 chain. We find in the thermodynamic limit a gap Δ = 0.085(5) J and a spin correlation length ξ = 49(1) lattice spacings in the isotropic case. We give the phase diagram in the presence of single-ion and exchange anisotropy, showing how the Haldane phase is squeezed out when the spin S increases.