Feeling bad about taking time off
Firstly, you are certainly not alone in feeling like you are missing out on work. In the competitive world of academia, everyone feels this way at some point or another, though it is rarely because they are actually not doing enough work. Also, I can only imagine how difficult it must have been to start a PhD while working remotely, as you have no colleagues or peers around you to gauge the &quot;normal&quot; level of work. Once we are all allowed back to our offices, you will see that plenty of PhD time is spent not working at all... my office had regular badminton matches during the working day, or we'd all stop for a cup of tea together, or go out for lunch or to the pub as soon as the clock hit 5pm (or before) :).
You will honestly be more productive if you stop trying to force yourself to work solidly for eight hours a day. One of the great benefits of academic work is the flexibility: if you're having a terrible day, there's nothing to stop you working for a couple of hours and taking the rest of the day off; conversely, if you're having a great day, there's nothing to stop you working late, chasing the solution to your current problem.
Furthermore, you're a first year PhD student, meaning if you started in October you're about five months into your PhD, which is likely going to take you between three and a half and four years to complete. During an undergraduate degree, it's common to work intensively for short bursts, while completing coursework or preparing for exams. A PhD requires a completely different strategy.
It's often said that a PhD is a marathon, not a sprint. In my personal experience, this is very true. You have to learn to pace yourself, since you don't have external deadlines any more (possibly excluding things like submitting a paper to a journal, but that is likely a self-imposed and flexible deadline). The only one you have is your thesis submission, still more than three years away. You can't keep up the high intensity work for that long. As you correctly recognise, you will end up burned out.
My strongest piece of advice is to stop working at the weekend. Your brain needs a break to be able to think productively and creatively when you are at work. Try to prioritise other, non-mathematical activities during the evenings and weekends -- go for a walk or run, read a book, draw, practice a musical instrument -- what ever you can do to relax under the current restrictions.
If you are worried about losing momentum, try writing a short paragraph or list of bullet points every Friday afternoon before you stop work, detailing where you are and what the next step in your work should be. You can even do this at the end of every day, to help you get started the following morning.
Finally, do you have anyone close to you who you can talk to about this? I suggest mentioning it to your PhD supervisor, as they can probably help you with time management or prioritisation strategies (and if they're a good supervisor, I expect they'll reassure you that you are working hard enough, and can afford to relax). If you are not already in touch with the other PhD students in your department, then perhaps send them an email and organise an online meeting, perhaps a virtual pub session on a Friday afternoon, where you can all chat and relax together. You will soon learn that everyone has gripes and complaints about their work, and how busy they are etc. It can be good to let off steam in this way.
In summary: take breaks (as many as you think you need times two), stop working weekends (unless you really, really, really need to, and I mean &quot;PhD thesis is due in a week and I just found a huge mistake in my analysis&quot; need to), and try to talk to your PhD peers at least once a week if you can (and don't just talk about work!).