利用者:Glayhours/sandbox/AKLT模型

AKLT 模型（英: Affleck–Kennedy–Lieb–Tasaki model; AKLT model）は一次元の量子ハイゼンベルク模型を拡張したスピン系の模型である。AKLT 模型はイアン・アフレック、トム・ケネディ、エリオット・リーブ、田崎晴明らによって提案され、またスピン 1 ハイゼンベルク鎖に関する深い物理学的な洞察から、その厳密解が与えられた。模型の名前は彼らの頭文字をとったものである。AKLT 模型はまた、価電子結合固体秩序 (valence bond solid order, VBS order) や対称性によって保護されたトポロジカル秩序、行列積状態の波動関数などの諸概念をテストする上で重要な役割を果たす。

背景

AKLT 模型に対する主な動機はマジュムダール＝ゴーシュ模型にある。マジュムダール＝ゴーシュ模型において、すべての基底状態の 3 つの隣接スピンの組のうち 2 つは、一重項状態か価電子結合状態になっているために、3 つ組のスピンは決してスピン 3/2 状態にならない。実際、マジュムダール＝ゴーシュ模型のハミルトニアンは

AKLT 論文の主眼は、マジュムダール＝ゴーシュ模型に見られるような構造が、スピンの大きさが 1/2 以外であるような厳密に解ける模型についても一般化できるということである。

Definition

Affleck et al. were interested in constructing a one-dimensional state with a valence bond between every pair of sites. Because this leads to two spin 1/2s for every site, the result must be the wavefunction of a spin 1 system.

For every adjacent pair of the spin 1s, two of the four constituent spin 1/2s are stuck in a total spin zero state. Therefore each pair of spin 1s is forbidden from being in a combined spin 2 state. By writing this condition as a sum of projectors, AKLT arrived at the following Hamiltonian

${\hat {H}}=\sum _{j}{\vec {S}}_{j}\cdot {\vec {S}}_{j+1}+{\frac {1}{3}}({\vec {S}}_{j}\cdot {\vec {S}}_{j+1})^{2}$

This Hamiltonian is similar to the spin 1, one-dimensional quantum Heisenberg spin model but has an additional spin interaction term.

Ground State

By construction, the ground state of the AKLT Hamiltonian is the valence bond solid with a single valence bond connecting every neighboring pair of sites. Pictorially, this may be represented as

Here the solid points represent spin 1/2s which are put into singlet states. The lines connecting the spin 1/2s are the valence bonds indicating the pattern of singlets. The ovals are projection operators which "tie" together two spin 1/2s into a single spin 1, projecting out the spin 0 or singlet subspace and keeping only the spin 1 or triplet subspace. The symbols +, 0 and - label the standard spin 1 basis states (eigenstates of the $S^{z}$ operator).

Spin 1/2 Edge States

For the case of spins arranged in a ring (periodic boundary conditions) the AKLT construction yields a unique ground state. But for the case of an open chain, the first and last spin 1 have only a single neighbor, leaving one of their constituent spin 1/2s unpaired. As a result, the ends of the chain behave like free spin 1/2 moments even though the system consists of spin 1s only.

The spin 1/2 edge states of the AKLT chain can be observed in a few different ways. For short chains, the edge states mix into a singlet or a triplet giving either a unique ground state or a three-fold multiplet of ground states. For longer chains, the edge states decouple exponentially quickly as a function of chain length leading to a ground state manifold that is four-fold degenerate. By using a numerical method such as DMRG to measure the local magnetization along the chain, it is also possible to see the edge states directly and to show that they can be removed by placing actual spin 1/2s at the ends. It has even proved possible to detect the spin 1/2 edge states in measurements of a quasi-1D magnetic compound containing a small amount of impurities whose role is to break the chains into finite segments.

Matrix Product State Representation

The simplicity of the AKLT ground state allows it to be represented in compact form as a matrix product state. This is a wavefunction of the form

$|\Psi \rangle =\sum _{\{s\}}{\text{Tr}}[A^{s_{1}}A^{s_{2}}\ldots A^{s_{N}}]|s_{1}s_{2}\ldots s_{N}\rangle$.

Here the As are a set of 3 matrices labeled by $s_{j}$ and the trace comes from assuming periodic boundary conditions.

The AKLT ground state wavefunction corresponds to the choice:

$A^{+}={\sqrt {\frac {2}{3}}}\ \sigma ^{+}$

$A^{0}={\frac {-1}{\sqrt {3}}}\ \sigma ^{z}$

$A^{-}=-{\sqrt {\frac {2}{3}}}\ \sigma ^{-}$

where the $\sigma {\text{'s}}$ are Pauli matrices.

Generalizations and Extensions

The AKLT model has been solved on lattices of higher dimension, even in quasicrystals . The model has also been constructed for higher Lie algebras including SU(n), SO(n), Sp(n) and extended to the quantum groups SUq(n).