Exactly solvable model for a deconfined quantum critical point in 1D

Abstract

We construct an exactly solvable lattice model for a deconfined quantum critical point (DQCP) in (1+1) dimensions. This DQCP occurs in an unusual setting, namely at the edge of a (2+1) dimensional bosonic symmetry protected topological phase (SPT) with $\mathbb{Z}_2\times\mathbb{Z}_2$ symmetry. The DQCP describes a transition between two gapped edges that break different $\mathbb{Z}_2$ subgroups of the full $\mathbb{Z}_2\times\mathbb{Z}_2$ symmetry. Our construction is based on an exact mapping between the SPT edge theory and a $\mathbb{Z}_4$ spin chain. This mapping reveals that DQCPs in this system are directly related to ordinary $\mathbb{Z}_4$ symmetry breaking critical points.

Figures (3)

• Figure 1: (a)-(b) The two degenerate ground states of the Hamiltonian (\ref{['Hssb']}) that spontaneously breaks $\mathbb{Z}_{2a}$. The blue arrows represent the $\sigma_j$ spins and the black arrows represent the $\tau_{j+1/2}$ spins. Both states are eigenstates of $U_b$ with eigenvalue $+1$. (c) Domain walls occur at the boundaries between these states. A state with two $\mathbb{Z}_{2a}$ domain walls (indicated by the dashed lines) has eigenvalue $-1$ under $U_b$, meaning two $\mathbb{Z}_{2a}$ domain walls fuse to a $\mathbb{Z}_{2b}$ charge.
• Figure 2: A mapping between the four kinds of domain walls in the SPT edge theory and the four kinds of domain walls in the $\mathbb{Z}_4$ spin chain, which are labeled by their eigenvalues $\{1,i,-1,-i\}$ under $C_j^\dagger C_{j+1}$. As discussed in the main text, two $\mathbb{Z}_{2a}$ domain walls (second configuration) fuse to a $U_b$ charge, which is equivalent to a $\tau_{j+1/2}$ spin flip (third configuration).
• Figure 3: The action of $S_j$ on domain wall states in the SPT edge theory: $S_j$ shifts the domain wall measured by $C_{j-1}^\dagger C_j$ by $i$ and the domain wall measured by $C_j^\dagger C_{j+1}$ by $-i$. Here, $j$ labels the spin in the middle of each 5-spin configuration.