Thermodynamics and criticality of supersymmetric spin chains with long-range interactions

We study the thermodynamics and critical behavior of su$(m|n)$ supersymmetric spin chains of Haldane-Shastry type with a chemical potential term. We obtain a closed-form expression for the partition function and deduce a description of the spectrum in terms of the supersymmetric version of Haldane's motifs, which we apply to obtain an analytic expression for the free energy per site in the thermodynamic limit. By studying the low-temperature behavior of the free energy, we characterize the critical behavior of the chains with $1\le m,n\le 2$, determining the critical regions and the corresponding central charge. We also show that in the su$(2|1)$, su$(1|2)$ and su$(2|2)$ chains the bosonic or fermionic densities can undergo first-order (discontinuous) phase transitions at $T=0$, in contrast with the previously studied su$(2)$ case.