We report on a new mechanism for heat conduction in the neutron star crust.We find that collective modes of superfluid neutron matter, called superfluidphonons (sPhs), can influence heat conduction in magnetized neutron stars. Theycan dominate the heat conduction transverse to magnetic field when the magneticfield $B \gsim 10^{13}$ G. At density $\rho \simeq 10^{12}-10^{14} $ g/cm$^3$the conductivity due to sPhs is significantly larger than that due to latticephonons and is comparable to electron conductivity when temperature $\simeq10^8$ K. This new mode of heat conduction can limit the surface anisotropy inhighly magnetized neutron stars. Cooling curves of magnetized neutron starswith and without superfluid heat conduction could show observationallydiscernible differences.
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