The nature of dark matter is still mysterious despite various astronomical evidence. As a possible candidate, self-interacting dark matter (SIDM) can potentially resolve some issues appearing in the cold dark matter paradigm. Here we investigate how SIDM around supermassive black holes (SMBHs) in galaxy centers may form a density spike and imprint in the spectrum shape of stochastic gravitational-wave background from SMBH binaries (SMBHBs). Employing a refined dynamical friction formula and consistently evolving the orbital dynamics, we demonstrate that current pulsar timing array (PTA) data are sensitive to the cross section of SIDM with $\sigma (v)/m_{\chi }\lesssim 0.66{\mathrm{cm}}^2/\mathrm{g}$, comparable to other astrophysical probes. We also highlight the importance of including the eccentricity of SMBHBs in the parameter inference, which would affect the results significantly. Our findings reveal the promising potential of PTA observations in probing the nature of dark matter.

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• Gravitational waves
• Particle dark matter
• Astronomical black holes