One-dimensional symmetry protected topological phases and their transitions

We present a unified perspective on symmetry protected topological (SPT) phases in one dimension and address the open question of what characterizes their phase transitions. In the first part of this work, we use symmetry as a guide to map various well-known fermionic and spin SPTs to a Kitaev chain with coupling of range $\alpha \in \mathbb{Z}$. This unified picture uncovers new properties of old models, such as how the cluster state is the fixed point limit of the Affleck-Kennedy-Lieb-Tasaki state in disguise, and elucidates the connection between fermionic and bosonic phases, with the Hubbard chain interpolating between four Kitaev chains and a spin chain in the Haldane phase. In the second part, we study the topological phase transitions between these models in the presence of interactions. This leads us to conjecture that the critical point between any SPT with $d$-dimensional edge modes and the trivial phase has a central charge $c\ge {log}_{2}d$. We analytically verify this for many known transitions. This agrees with the intuitive notion that the phase transition is described by a delocalized edge mode, and that the central charge of a conformal field theory is a measure of the gapless degrees of freedom.

Figure 1

SPT models related to the $\alpha$-chain with $\alpha =2$. In the case of the AKLT model, it is at the level of the ground state, whereas for the other three models it is at the level of the full Hamiltonian.

Figure 2

Schematically representing the $\alpha$-chain (4) for $\alpha =0,1,2$. The left black dot denotes the Majorana mode $\gamma$, the right one $\tilde{\gamma }$.

Figure 3

SSH model (9) with fermionic edge modes for $\lambda =1$.

Figure 4

The bipartite Hubbard chain, Eq. (13), with single occupancy. For $U=0$ this is a double copy of the SSH state ($\lambda =1$) in Fig. 3. For $U\to \infty$, this reduces to the AKLT fixed point state in Fig. 5.

Figure 5

The AKLT state with dashed lines denoting spin singlets. The AKLT state has spin-1 projectors on the gray ovals [35], disappearing in the fixed point limit [44].

Figure 6

Phase diagram for the noninteracting Hamiltonian (25).

Figure 7

Phase diagram for the interacting Hamiltonian (26).

Figure 8

Phase diagram for the Hamiltonian (25) in the presence of the interactions described in the main text.

Figure 9

Phase diagram for the spin model (27), which does not correspond to any fermionic model if $h_y\ne 0$. The central charges can be predicted from our conjecture and are confirmed by iDMRG.