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arXiv:2206.01222 · Physical Review Letters

Exactly Solvable Model for a Deconfined Quantum Critical Point in 1D

Carolyn Zhang, Michael Levin

Published Jan 11, 2023• 41 citations
arXivHTMLPDF

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 (SPT) phase with ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry. The DQCP describes a transition between two gapped edges that break different ${\mathbb{Z}}_{2}$ subgroups of the full ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ 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.

DOI
10.1103/physrevlett.130.026801

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We study the boundary states of the archetypal three dimensional topological order, i.e. the three dimensional \mathbb{Z}_2 ℤ 2 toric code. There are three distinct elementary types of boundary states that we will consider in this work. In the phase diagram that includes the three elementary boundaries there may exist a multi-critical point, which is captured by the so-called deconfined quantum critical point (DQCP) with an “easy-axis” anisotropy. Moreover, there is an emergent \mathbb{Z}_{2,d} ℤ 2 , d symmetry that swaps two of the boundary types, and it becomes part of the global symmetry of the DQCP. The emergent \mathbb{Z}_{2,d} ℤ 2 , d (where d represents “defect”) symmetry on the boundary is originated from a type of surface defect in the bulk. We further find a gapped boundary with a surface topological order that is invariant under the emergent symmetry.
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Deconfined quantum critical points (DQCPs) are putative phase transitions beyond the Landau paradigm with emergent fractionalized degrees of freedom. The original example of a DQCP is the spin-12\frac{1}{2}21​ quantum antiferromagnet on the square lattice which features a second-order transition between valence bond solid (VBS) and N\'eel order. The VBS order breaks a lattice symmetry, and the corresponding VBS order parameter may couple to lattice distortion modes (phonons) at appropriate momenta. We investigate a field-theoretic description of the DQCP in the presence of such a spin-lattice coupling. We show that treating phonons as classical lattice distortions leads to a relevant monopole-phonon interaction inducing an instability towards a distorted lattice by an analogous mechanism to the spin-Peierls instability in one dimension. Consequently, there is a breakdown of the DQCP which generally becomes a strong first-order transition. Taking into account the full quantum nature of the phonons, we argue that the continuous DQCP persists above a critical phonon frequency. Lastly, we comment on the connection to general gapless, deconfined gauge theories.