Finite-Volume Scaling of the Quenched Chiral Condensate∗

Surprisingly, there exists a deep connection between finite-volume gauge theories with spontaneous breaking of chiral symmetries, and Random Matrix Theory (RMT). In the chiral limit mi → 0 such that μi ≡ miΣV are kept fixed, the field-theoretic partition functions Zν({μi}) become identical to certain (chiral) RMT partition functions [1]. Here Σ denotes the (conventional) infinite-volume chiral condensate, and ν is the (fixed) topological charge of the gauge fields. This remarkable identity of partition functions is stable under huge perturbations of the RMT “potential”, i.e. universal [2]. RMT plays a role here because the three major (chiral) classes of RMT ensembles correspond to the cosets of spontaneous chiral symmetry breaking. It is crucial that one considers the limit V → ∞ with V 1/mπ. Taken conversely, from reading off the scaling behavior of the finite-volume chiral condensate one deduces the coset of spontaneous chiral symmetry breaking! From the universal relationship between these different expressions for the finite-volume partition functions follows that there are also universal scaling relations for the chiral condensate and higher chiral susceptibilities. Because the pertinent chiral RMT ensembles are presumed relevant also for systems in condensed matter physics,