Heartcarving 1.0 can be done on any ski really, but ideally a slalom ski - as it has the most responsive sidecut. It's not crucial though. You could try it on your present ski, and if you like that, you could rent some SLs. I was doing it on different skis in 2002, 2003 and 2007, but
found SL skis work best.
This is a video of HC 1.0 at Mt Buller on the SL skis (2007):
and here's another about 6 months later Whistler on the same skis:
But really it's just a feeling-based style of skiing. So put the poles away and move to a wide-flat-beginner or wide-flat-intermediate run, and aim to do S turns all the way down, without spraying out the tails of the ski. Let the ski determine your line down the run. So you really have to work with your ski. If you remember, take a deep breath into your chest and think about your heart whenever you can. Also don't be afraid to let your head angulate with your body (in fact you are meant to) and be comfortable about looking to the sides of the run as you snake down it. You are naturally looking to the sides of the run anyhow as that's the way your chest is pointing. The head angulations can be the most disconcerting aspect of the style/technique, but once you get used to it, it is liberating, and you can feel the energy efficiency at work - energy which you can channel elsewhere...
But if you want to practice this right now, try riding down a hill on a bike, and doing long s-turns. This a similar feeling to heartcarving, especially with the head.
This video from 2007 shows the head angulations (obviously the later videos also demonstrate it too.)
Heartcarving 1.5 was done in 2012 on 125 skiboards, and HC 2.0 was done a year later on custom skis - both have videos on the homepage.
Best of luck, and feel free to ask questions whenever you need to.