[Paper Review] Exactly solvable model for a deconfined quantum critical point in 1D

Carolyn Zhang, Michael Levin|arXiv (Cornell University)|Jun 2, 2022

Quantum many-body systems4 citations

TL;DR

This paper constructs an exactly solvable 1D lattice model for a deconfined quantum critical point (DQCP) at the edge of a 2+1D $ζ_2\times\mathbb{Z}_2$ symmetry-protected topological (SPT) phase. By mapping the SPT edge theory to a $ζ_4$ spin chain, the authors show that the DQCP arises from a direct transition between two gapped phases breaking different $ζ_2$ subgroups, with criticality governed by a $ζ_4$ symmetry-breaking transition and exhibiting self-duality and a critical line with continuously varying exponents.

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 $\mathbb{Z}_2 imes\mathbb{Z}_2$ symmetry. The DQCP describes a transition between two gapped edges that break different $\mathbb{Z}_2$ subgroups of the full $\mathbb{Z}_2 imes\mathbb{Z}_2$ 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.

Motivation & Objective

• To construct an exactly solvable lattice model for a deconfined quantum critical point (DQCP) in (1+1) dimensions.
• To establish a connection between DQCPs in SPT edge theories and ordinary symmetry-breaking transitions in spin chains.
• To demonstrate that the DQCP arises from a mixed anomaly between $ζ_2\times\u03b6_2$ symmetries, where domain walls of one symmetry carry fractional quantum numbers of the other.
• To show that the critical point is self-dual under a unitary, locality-preserving duality transformation mapping $ζ_{2a}$ and $ζ_{2b}$ order parameters.
• To identify a critical line with continuously varying critical exponents, stabilized by symmetries that suppress chiral perturbations.

Proposed method

• Mapping the $ζ_2\times\u03b6_2$ SPT edge theory to a $ζ_4$ spin chain via an exact correspondence between edge degrees of freedom and spin chain operators.
• Defining the Hamiltonian $H = -\sum_j \sigma^z_j\sigma^z_{j+1} - \sum_j \tau^x_{j+1/2}$, which realizes two degenerate ground states breaking $ζ_{2a}$, with domain walls carrying fractional $ζ_{2b}$ charge.
• Using the fusion rule that two $ζ_{2a}$ domain walls fuse to a $ζ_{2b}$ charge, confirmed via the action of $U_b$ on multi-domain-wall states.
• Establishing a duality transformation $U_c$ that maps $C_j^\dagger C_{j+1}$ to $S_{j+1}$ and $S_j$ to $C_j^\dagger C_{j+1}$, showing self-duality of the critical point.
• Analyzing the critical line by introducing a parameter $\lambda$ in the Hamiltonian, showing that chiral perturbations are irrelevant in a range of $\lambda$, stabilizing the direct transition.
• Demonstrating that the critical theory is equivalent to a $ζ_4$ clock model, with the DQCP corresponding to a continuous transition in the spin chain.

Experimental results

Research questions

• RQ1 Can an exactly solvable 1D lattice model realize a deconfined quantum critical point (DQCP) in the absence of field-theory or numerical approximations?
• RQ2 How does the mixed anomaly between $ζ_2\times\u03b6_2$ symmetries in an SPT edge lead to a DQCP via domain wall fusion?
• RQ3 What is the precise mapping between the SPT edge theory and a $ζ_4$ spin chain that makes the DQCP analytically tractable?
• RQ4 Does the critical point exhibit self-duality, and if so, what are the properties of the duality transformation?
• RQ5 Is the DQCP stabilized by symmetries, and does it exist as part of a critical line with continuously varying exponents?

Key findings

• The DQCP is realized at the edge of a 2+1D $ζ_2\times\u03b6_2$ SPT phase with a mixed anomaly, where two gapped phases breaking different $ζ_2$ subgroups are connected via a direct, symmetry-protected critical point.
• The critical point is exactly solvable and equivalent to a $ζ_4$ clock model, with the DQCP corresponding to a continuous transition in the spin chain that breaks $ζ_4$ symmetry.
• Two $ζ_{2a}$ domain walls carry a $ζ_{2b}$ charge, as confirmed by the action of $U_b$ on multi-domain-wall states, demonstrating the anomaly-induced fusion rule.
• The critical point is self-dual under a unitary, locality-preserving duality $U_c$, which exchanges $ζ_{2a}$ and $ζ_{2b}$ order parameters and is analogous to Kramers-Wannier duality.
• The DQCP exists as part of a critical line parameterized by $\lambda$, where chiral perturbations are irrelevant over a range of $\lambda$, stabilizing the direct transition.
• The critical theory exhibits continuously varying critical exponents along the line, consistent with the existence of a deconfined quantum critical line with non-trivial scaling behavior.