Severe regrets because I lied in my PhD application I am an international Physics PhD student who is currently finishing his second year successfully, with one publication and others in preparation. However, I have a problem that has not let me sleep properly since day 1 and I need to vent it out with you.
When I applied, for some stupid reason I mentioned on my CV that I was a teaching assistant for two terms (in an area I am pretty good at) when in reality I was not. I won't go too much into details but I attribute this behavior to extreme low self-esteem and mental health issues that I have had since I was a teenager due to a very traumatic period in my life and more than lying with ill intentions to the admission committee in order to get an advantage I was lying to myself. I did not even realize what a horrendous mistake I had made at the time, and it also did not help that nobody from my former lab checked my CV before I sent it. As a result I have been in therapy since 1 year ago to understand the causes and threat my compulsive lying and me and my therapist believe that I have made important progress.

I don't think the lie was important for my acceptance (the PhD is 100% research based without coursework and teaching duties) and scholarship and every other part of my application was 100% truthful (1 publication at the time, top 5% of my cohort, outstanding letters of recommendation, internships, academic awards, etc) but I still can't get over what I did as I feel extremely bad as a person and also fear every day that I will somehow be caught and get expelled or get my degree revoked in the future. My supervisors are the most competent, intelligent but also the most kind and loving persons I have ever met, so while I have wanted to come clean for a long time I don't want to disappoint them. Besides, I have improved immensely as a person while living abroad in this beautiful country and have never been as happy in my life; I honestly think that getting expelled would have horrible consequences for my mental health, future dreams and finances and God knows how long it would take me to recover completely if ever. 
What do you recommend me to do in this case?
I appreciate the question.  

I think we can all agree that you should not have lied on your graduate application.  It is to your credit that you recognize this.  That said, as a frequent thesis advisor and the recent head of Graduate Studies in my department, I feel that the main victim of this lie is...you.  If a student of mine told me they had done that, I would thank them for telling me, confirm that they should not have done it, make sure they understand that they must never falsify any academic documents again...and then get on with business.  

Cutting the funding of a successful student for this kind of indiscretion that occurred before their arrival is, for me, just not a reasonable thing to do.  It is an interesting thought experiment to wonder what percentage of the graduate students in my department had an undergraduate indiscretion as bad or worse than this.  Rough guess: somewhere between one third and one half?  It is all but impossible to know and also very subjective.   

You have to deal with your bad behavior in the past.  An unfortunate fact of the matter is that it is often not possible or not optimal to directly fix past bad behavior.  That can be an unsettling thought -- for me too, by the way.  Based on the sleepless nights you describe, I am really glad that you are talking to a therapist about this and figuring out what you can do indirectly to move on with your life.

Here's an idea: you say you are doing very well in the program.  That's great.  Is there some optional service task you could take on in the program (or the department or the school or the profession)?  I think that could be beneficial, both in reality and in your own mind.  That way, when you think about the bad thing you did two years ago, you can also think "Yeah, I feel bad about that, but I feel good about X that I am doing now."  By the way, if you do it, you certainly should put it in your CV, but if (and only if) it seems helpful for you to be a bit modest/understated about it, you could pitch it that way.

Good luck.