The nouns of our language refer to either concrete entities (like a table) or abstract concepts (like justice or love), and cognitive psychology has established that concreteness influences how words are processed. Accordingly, understanding how concreteness is represented in our mind and brain is a central question in psychology, neuroscience, and computational linguistics. While the advent of powerful language models has allowed for quantitative inquiries into the nature of semantic representations, it remains largely underexplored how they represent concreteness. Here, we used behavioral judgments to estimate semantic distances implicitly used by humans, for a set of carefully selected abstract and concrete nouns. Using Representational Similarity Analysis, we find that the implicit representational space of participants and the semantic representations of language models are significantly aligned. We also find that both representational spaces are implicitly aligned to an explicit representation of concreteness, which was obtained from our participants using an additional concreteness rating task. Importantly, using ablation experiments, we demonstrate that the human-to-model alignment is substantially driven by concreteness, but not by other important word characteristics established in psycholinguistics. These results indicate that humans and language models converge on the concreteness dimension, but not on other dimensions.