Under the widely accepted but unproven assumption that the one-dimensional S=1 antiferromagnetic Heisenberg model has a unique gapped ground state, we prove that the model belongs to a nontrivial symmetry-protected topological phase. In other words, we rigorously rule out the possibility that the model has a unique gapped ground state that is topologically trivial. To be precise, we assume that the models on open finite chains with a boundary magnetic field have unique ground states with a uniform gap and prove that the ground state of the infinite chain has a nontrivial topological index. This further implies the presence of a gapless edge excitation in the model on the half-infinite chain and the existence of a topological phase transition in the model that interpolates between the Heisenberg chain and the trivial model.Under the widely accepted but unproven assumption that the one-dimensional S=1 antiferromagnetic Heisenberg model has a unique gapped ground state, we prove that the model belongs to a nontrivial symmetry-protected topological phase. In other words, we rigorously rule out the possibility that the model has a unique gapped ground state that is topologically trivial. To be precise, we assume that the models on open finite chains with a boundary magnetic field have unique ground states with a uniform gap and prove that the ground state of the infinite chain has a nontrivial topological index. This further implies the presence of a gapless edge excitation in the model on the half-infinite chain and the existence of a topological phase transition in the model that interpolates between the Heisenberg chain and the trivial model.

arXiv:2407.17041Physical Review LettersFeb 19, 20254 citations

Ground States of the Antiferromagnetic Heisenberg Chains

Hal Tasaki

Graduate texts in physicsJan 1, 20200 citations

Topological effects in short antiferromagnetic Heisenberg chains

Jizhong Lou, Shaojin Qin, Tai Kai Ng +1 more

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 $5/2.$ For open chains with spin magnitudes $S=5/2$ to $S=5,$ 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 $S>~1.$ 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 $5/2.$ For open chains with spin magnitudes $S=5/2$ to $S=5,$ 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 $S>~1.$

arXiv:cond-mat/0006409Physical review. B, Condensed matterJan 28, 200223 citations

String Order in Half-Integer-Spin Antiferromagnetic Heisenberg Chains

Jizhong Lou, Shaojin Qin, Changfeng Chen

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.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.

Physical Review LettersAug 21, 200316 citations

Spin dynamics of a Heisenberg antiferromagnetic chain

Roger Arthur Cowley, D. M. Tennant, T. G. Perring +2 more

Physica A Statistical Mechanics and its ApplicationsMar 1, 19933 citations

5/2.

For open chains with spin magnitudes

S=5/2

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.