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We study two-component Bose gases with Raman induced spin-orbit coupling viaa perturbation approach at finite temperature. For weak coupling, free energyis expanded in terms of Raman coupling strength up to the second order, wherethe coefficient (referred to as Raman susceptibility) is determined accordingto linear response theory. The equation of state for the stripe phase and theplane-wave phase are obtained in Popov approximation, and the first ordertransition between these two phases is investigated. As temperature increases,we find the phase boundary bends toward the stripe phase side in mosttemperature regions, which implies the ferromagnetic order is more robust thanthe crystalline order in presence of thermal fluctuations. Our resultsqualitatively agree with the recent experimental observation in rubidium atomicgases. A method to measure the Raman susceptibility through the two-photonBragg scattering experiment is also discussed.

Equation of state and phase transition in spin-orbit-coupled Bose gases at finite temperature: A perturbation approach