Emergent gapless spiral phases and conformal Lifshitz criticality in the cluster Ising model with off-diagonal interactions

Abstract We perform a comprehensive analytical study of the exotic quantum phases and phase transitions emerging from the cluster-Ising model with off-diagonal Gamma interactions. Specifically, we map out the ground-state phase diagram by analyzing both local and nonlocal order parameters, together with the energy spectra. The results reveal one pair of gapped phases—namely antiferromagnetic (AFM) long-range ordered phase and symmetry-protected topological (SPT) phase—as well as two distinct gapless spiral phases induced by the off-diagonal interactions. Remarkably, the phase diagram contains four distinct transition lines. Two of them, separating different gapped or gapless phases, are governed by the Ising and three-copy Ising conformal field theories, respectively. In contrast, the remaining two transition lines—between the gapless spiral and gapped phases—belong to a non-conformal Lifshitz criticality with dynamical critical exponent $z = 2$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>z</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:math> . More importantly, the intersection of these four transition lines gives rise to a new Lifshitz multicritical point exhibiting emergent conformal symmetry, marking a fundamental departure from all previously known non-conformal Lifshitz points. This work provides a valuable reference for future investigations of exotic gapless phases and their transitions in exactly solvable many-body systems.