Proving a negative is impossible, but right now, the best answers to your questions are: 
>**No, federal statutes do not give the Feds authority to *police cities.*** Federal law gives the Feds power to **protect federal property** and *enforce federal and Oregon law on Federal property.*   

>**No, the Feds have not cited any statutes authorizing them** to do more than protect federal property.  

Since you should not believe me, let me quote Professor Steven Vladeck of the University of Texas. He has [impeccable][1] credentials, he wrote about this topic on a well-known and respected[web site][2], and his article is [referred to][3] by other [lawyers][4] writing about Portland. 

Here is his answer to the question, **which federal laws are being enforced?" 

>This is a significant question, and **the answer is hardly obvious.** Unlike D.C., which is entirely federal territory (and where even “local” law is technically “federal”), **Portland is principally subject to the sovereignty of the state of Oregon.** There is **some federal property in Portland,** but not much. And **federal law enforcement officers do not generally have the authority (under federal law, anyway) to enforce state laws on nonfederal property.** 

>Some of Acting Secretary Wolf’s statements have referred to vandalism of “courthouses,” including the three federal courthouses in downtown Portland. And there are a handful of other federal buildings in the city. 

>But this is where the Portland situation starts to look very different from D.C. **Protecting a federal courthouse from vandalism is an easy case for the use of federal law enforcement authorities.** *If, as media reports have suggested, federal officers are patrolling streets a significant distance from federal buildings* (and arresting protestors who pose no imminent threat thereto), **that would be far murkier.** 

The fact that he does not cite any federal law *answers your question.* He didn’t just “end up” not providing cites to federal law. In the article, he explains why he can't cite any relevant federal laws: 
>1) The Feds haven’t cited any statutory authority beyond protecting Federal buildings. 

>2) As he (implicitly) says in the cited section, he can’t think of any statutes they could be enforcing. 

>3) In order to enforce Oregon law off of federal property, Oregon law requires the Feds to do a bunch of stuff they haven’t done.

There may be other experts who have found federal statutes that give the Fed authority to "restore order" in Portland, but I have not found any. 

In the rest of the article, Vladeck discusses what authority Feds have to detain or arrest people off of Federal property. (Answer: Not the sort it sounds like they are exercising). 

In related [article][5] he looks at Federal plans to surveil people who might be threatening monuments, including monuments that are not on Federal property. 


  [1]: https://law.utexas.edu/faculty/stephen-i-vladeck
  [2]: https://www.lawfareblog.com/what-heck-are-federal-law-enforcement-officers-doing-portland
  [3]: https://reason.com/2020/07/20/whats-happening-in-portland/
  [4]: https://thebulwark.com/what-we-know-and-dont-know-about-portland-and-the-dhs/
  [5]: https://www.lawfareblog.com/dhs-authorizes-domestic-surveillance-protect-statues-and-monuments