Incommensurate gapless ferromagnetism connecting competing symmetry-enriched deconfined quantum phase transitions

We present a scenario in which a gapless extended phase serves as a “hub” connecting multiple symmetry-enriched deconfined quantum critical points. As a concrete example, we construct a lattice model with <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:msubsup><a:mi mathvariant="double-struck">Z</a:mi><a:mn>2</a:mn><a:mspace width="0.16em"/></a:msubsup><a:mo>×</a:mo><a:msubsup><a:mi mathvariant="double-struck">Z</a:mi><a:mn>2</a:mn><a:mspace width="0.16em"/></a:msubsup><a:mo>×</a:mo><a:msubsup><a:mi mathvariant="double-struck">Z</a:mi><a:mn>2</a:mn><a:mspace width="0.16em"/></a:msubsup></a:mrow></a:math> symmetry for quantum spin-1/2 degrees of freedom that realizes four distinct gapful phases supporting antiferromagnetic long-range order and one extended incommensurate gapless ferromagnetic phase. The quantum phase transition between any two of the four gapped and antiferromagnetic phases goes through either a (deconfined) quantum critical point, a quantum tricritical point, or the incommensurate gapless ferromagnetic phase. In this phase diagram, it is possible to interpolate between four deconfined quantum critical points by passing through the extended gapless ferromagnetic phase. We identify the phases in the model and the nature of the transitions between them through a combination of analytical arguments and density matrix renormalization group studies.