This is an excellent and subtle question which requires a nuanced answer regardless of whether one if a physicalist, non-reductive physicalist (which, at least in the philosophical literature is often the same or pretty close to what scientists mean by "physicalist"), duelist, etc.
However, it seems to me that concepts do not exist apart from the physical processes that create them.
First nuance: let's grant that (and not just because I think it to be true). The physical processes that create them are in constant flux and there is no one-to-one correspondence between any concept and any physical process (that is, imagine that all concepts are the result of neural activity, as I believe. In order to refer to the activity that represents the concept "car", one has to involve so many networks and the intra- & inter-network activity that in doing so one is also referring to large parts of thousands of concepts, memories of different types, and more). In fact, essential to conceptual processing is the ways in which related concepts overlap in how neural activity is able to represent them.
Thus there is never any actual physical representation of a concept, even though all concepts are represented physically. It is a small (albeit dangerous, epistemically radical, and ontologically questionable) step to say that even though concepts do not exist apart from the physical processes that create them, all concepts exist apart from physical processes. One way to think about this (it's not the best analogy but it's 2 in the morning and I've been sick) is to think about the number two. Is it identical to 2? How about II? 8/4? Here, different representations correspond to the same entity. In the brain, we have the reverse in some sense. The same concept is represented in no small part in how it relates to related concepts, memories we have, sensorimotor experiences, and so forth. We can't identify any physical processes that represent any concepts, but we can understand concepts abstractly as well as the result of physical processes.
If I have in my mind an image of a barn, it is because at the very moment I have that image, certain physical processes are creating it.
Second nuance: some of those physical processes involve light, barns you've seen (in images, in real life, both, in movies, etc.), the conceptual network you rely on to create the image which itself relies on passed experiences an related concepts, and the representation via neural activity of this network. Most importantly, the representation cannot exist apart from the perceptual experiences you've had, and these require physical processes that are not in the brain (like those that have allowed you to see, hear, smell, read, and/or touch some external example of a barn).
If the physical processes cease at any moment to function, my image of a barn comes to an abrupt end.
Most of the physical processes that allow your mental image do not cease when your mental image does.
To posit that concepts exist to some extent separately or distinct from the processes that create them is to risk lending concepts a metaphysical or trans-physical status, although it might subjectively appear to us that they do exist separately.
Third nuance: although many concepts have no basis in the external world the way barns and cars do, they are often still based on sensorimotor experience and made abstract via metaphor (in the scientific, not literary, sense). Additionally, none exist in isolation and I doubt any exist that are not in some way connected to bodily experience. However, how we categorize perceptual stimuli (such as calling something a barn instead of a house, a building, wooden blanks and a roof, etc.) is itself a conceptual process almost as much as it is a perceptual process. But there is no universal way that humans categorize perceptual experiences such that given any two people exposed to the same stimuli, they would conceptualize it the same way. Put simply, someone who has never heard of computers, TV, or similar devices would, having seen my laptop, perceive it in terms of parts in a way that I do not, and subsequently form concepts I don't have. Basically, just as we develop concepts based upon perceptual experiences, so to do concepts exist in many ways as perceptual experience. As perceptual experiences require physical processes outside the brain, the concepts we have exist as they do because of other concepts we have formed from prior perceptual experiences in a sort of loop that always involves processes outside the brain.
