SO(n) AKLT Chains as Symmetry Protected Topological Quantum Ground States

Abstract

This thesis studies a pair of symmetry protected topological (SPT) phases which arise when considering one-dimensional quantum spin systems possessing a natural orthogonal group symmetry. Particular attention is given to a family of exactly solvable models whose ground states admit a matrix product state description and generalize the AKLT chain. We call these models ``$SO(n)$ AKLT chains'' and the phase they occupy the ``$SO(n)$ Haldane phase''. We present new results describing their ground state structure and, when $n$ is even, their peculiar $O(n)$-to-$SO(n)$ symmetry breaking. We also prove that these states have arbitrarily large correlation and injectivity length by increasing $n$, but all have a 2-local parent Hamiltonian, in contrast to the natural expectation that the interaction range of a parent Hamiltonian should diverge as these quantities diverge. We extend Ogata's definition of an SPT index for a split state for a finite symmetry group $G$ to an SPT index for a compact Lie group $G$. We then compute this index, which takes values in the second Borel group cohomology $H^2(SO(n),U(1))$, at a single point in each of the SPT phases. The two points have different indices, confirming the two SPT phases are indeed distinct. Chapter 1 contains an introduction with a detailed overview of the contents of this thesis, which includes several chapters of background information before presenting new results in Chapter 7 and Chapter 8.

Figures (16)

• Figure 1.1: The phase diagram for $O(n)$-invariant nearest-neighbor interactions (adapted from bjornberg2021dimerization).
• Figure 1.2: Illustration of dimerization at the south pole point. Depending on whether $\ell$ is even or odd, the site $x=0$ is more entangled with either its left or its right neighbor. This gives rise to a pair of 2-periodic ground states in the thermodynamic limit (Figure from bjornberg2021dimerization).
• Figure 3.1: The "ladder operators" of the spin-$s$ irrep.
• Figure 3.2: The defining representation $V=\mathbb{C}^4$ of $\mathfrak{so}(4)$. The Cartan subalgebra $\mathfrak{h}$ from (\ref{['eq:cartan subalgebra so(n)']}) acts as diagonal matrices when we pick the basis $\{f_1,f_2,\widetilde{f}_1,\widetilde{f}_2\}$ of $V$. Given a root $\alpha$, an element of the root space $X_\alpha$ maps between the linear subspaces spanned by the written basis elements. For instance, $X_{L_1-L_2} f_2 \in \mathbb{C} f_1$. In this sense roots "generalize" the raising and lowering operators of $\mathfrak{su}(2)$.
• Figure 5.1: The spin-1 bilinear biquadratic phase diagram (Figure 2 in bjornberg2021dimerization). The red phase corresponds to the Haldane phase, inhabited by the AKLT chain.

Theorems & Definitions (199)

• Example 2.2.1
• Example 2.2.2
• Example 2.2.3
• Example 2.2.4
• Example 2.2.5
• Theorem 2.3.1
• Theorem 2.3.2
• Corollary 2.3.1
• Example 2.3.3
• Definition 3.1.1