Tuesday 15 September 2020

condensed matter - Detailed derivation and explanation of the AKLT Hamiltonian


I am trying to read the original paper for the AKLT model,



Rigorous results on valence-bond ground states in antiferromagnets. I Affleck, T Kennedy, RH Lieb and H Tasaki. Phys. Rev. Lett. 59, 799 (1987).



However I am stuck at Eq. $(1)$:



we choose our Hamiltonian to be a sum of projection operators onto spin 2 for each neighboring pair: $$ \begin{align} H &= \sum_i P_2 (\mathbf{S_i} + \mathbf{S_{i+1}}) \\ &= \sum_i \left[\frac{1}{2}\mathbf{S_i}\cdot\mathbf{S_{i+1}} + \frac{1}{6}(\mathbf{S_i}\cdot\mathbf{S_{i+1}})^2 + \frac{1}{3}\right] \end{align}\tag{1} $$




In the equation, $H$ is the proposed Hamiltonian for which the authors intend to show that the ground state is the VBS ground state: the (unique) state with a single valence bond connecting each nearest-neighbor pair of spins, i.e. a type of spin-$1$ valence-bond-solid. Moreover, $\mathbf{S_i}$ and $\mathbf{S_{i+1}}$ are spin-$1$ operators, and $P_2$ is the projection operator onto spin-2 for the pair $(i,i+1)$.


I have several questions here.




  • First, how did the authors come up with the first line by observing the spin-1 valence-bond-solid state as below (Fig. 2 of the above paper)?








  • Why do they use a Hamiltonian which is "a sum of projection operators onto spin 2 for each neighboring pair"?



  • What does it mean exactly to project spin-$1$ pairs to spin $2$, and why do they want to project to spin $2$?


I feel there are quite a few steps skipped between here and standard QM textbooks. It would be great if somebody could recommend me some materials bridging them.



  • Secondly, I want to know how to go from the first line to the second line of equation $(1)$. However, I couldn't find the explicit form of $P_2$ either in the paper or by googling. Could somebody give me a hint?


Edit: I found a chapter of the unfinished book "Modern Statistical Mechanics" by Paul Fendley almost directly answers all my questions.




No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...