The Asymmetric Valence-Bond-Solid States in Quantum Spin Chains: The Difference Between Odd and Even Spins
| Autor: | Maekawa, Daisuke, Tasaki, Hal |
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| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | J. Math. Phys. 64, 031901 (2023) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1063/5.0123743 |
| Popis: | The qualitative difference in low-energy properties of spin $S$ quantum antiferromagnetic chains with integer $S$ and half-odd-integer $S$ discovered by Haldane can be intuitively understood in terms of the valence-bond picture proposed by Affleck, Kennedy, Lieb, and Tasaki. 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. 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 $\mathbb{Z}_2\times\mathbb{Z}_2$, 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. Comment: 21 pages, 8 figures. A discussion about spin pumping has been added to section 4.4 in version 2. Many improvements (thanks to the referee) in version 3. There is a 24 minutes video in which the essence of the present work is discussed: https://youtu.be/URsf9e_PLlc |
| Databáze: | arXiv |
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