In terms of representation, does the United States work by the delegate model or the trustee model?
Legislators in the United States are not required to adhere to either of these models. They are free to pick whichever they prefer. They are also free to act as delegates in some cases and as trustees in others.
There are no laws in the United States requiring legislators to act as trustees or delegates. In practice, what happens is that legislators act differently in different cases. They tend to act as delegates on issues where their constituents have strong beliefs and as trustees on issues where beliefs are not as strong or on which they constituents are divided. It is rare for legislators to go against their constituents on an issue where the constituents have a strong and clear opinion that is important to them. Legislators who do so too often tend to be voted out of office.