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The problem of a quantum Ising degree of freedom coupled to a gapless bosonicmode appears naturally in many one dimensional systems, yet surprisingly littleis known how such a coupling affects the Ising quantum critical point. Weinvestigate the fate of the critical point in a regime, where the weak couplingrenormalization group (RG) indicates a flow toward strong coupling. Using arenormalization group analysis and numerical density matrix renormalizationgroup (DMRG) calculations we show that, depending on the ratio of velocities ofthe gapless bosonic mode and the Ising critical fluctuations, the transitionmay remain continuous or become fluctuation-driven first order. The two regimesare separated by a tri-critical point of a novel type.

Fate of the one-dimensional Ising quantum critical point coupled to a gapless boson