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We study the spin-boson model (SBM) with two spins in staggered biases by anumerically exact method based on variational matrix product states. Severalobservables such as the magnetization, the entanglement entropy between the twospins and the bosonic environment, the ground-state energy, as well as thecorrelation function for two spins are calculated exactly. The characteristicsof these observables suggest that the staggered biases can drive the 2nd-orderquantum phase transition (QPT) to the 1st-order QPT in the sub-Ohmic SBM, whilethe Kosterlitz-Thouless QPT in the Ohmic SBM goes directly to the 1st-orderone. A quantum tricritical point, where the continuous QPT meets the 1st-orderone, can then be detected. It is found that the staggered biases would notchange the universality of { the phase transition in this model} below thequantum tricritical point.

Quantum tricritical point emerging in the spin-boson model with two dissipative spins in staggered biases