Are there deterministic private-key encryption schemes that are CPA-secure?
Summarizing fkraiem's comment, a CPA-secure encryption scheme can not be deterministic.

The reason is simple: the attacker is challenged to distinguish between the encryption of $m_0$ and $m_1$, but he also has access to an encryption oracle (and he can query the oracle at whatever input he wants!). If the encryption scheme yields the same ciphertexts each time it's queried on the same plaintexts, the attacker can simply ask the oracle for an encryption of $m_0$ and an encryption of $m_1$, and compare which one is equal to the given challenge ciphertext.