how to make my trigger function to perform asyncronously to update tsvector field?

Question

I am new to postgres. Suppose i have two models AnimalType and Animal. And i have search_vector field of type tsvector in Animal model. i created trigger that whenever i update a AnimalType row, the searchvector should update itself for every Animal related to that AnimalType.

            CREATE OR REPLACE FUNCTION update_animal_type()
            RETURNS TRIGGER
            LANGUAGE plpgsql AS $$
            BEGIN
                UPDATE animal SET search_vector = NULL
                WHERE animal_type_id = NEW.id;
                RETURN NEW;
            END;
            $$; 
            
            CREATE TRIGGER animal_type_update_trigger
            AFTER UPDATE OF animal_type_name ON animal_type
            FOR EACH ROW EXECUTE PROCEDURE update_animal_type();

It takes a lot of time to update a single animal type row when we have large number of animals related to that particular animal type. Is this behaviour is synchronous or asynchronous? If it is synchronous how can i make it to behave asynchronous?