Counting Broken Symmetries
From superconductivity in condensed-matter physics to electroweak symmetry breaking in particle physics, some of the richest phenomena in modern physics can be understood within the paradigm of spontaneous breaking of symmetries, i.e., the quantum mechanical scenario of the ground state of a quantum system being less symmetric than the corresponding Hamiltonian. A telltale sign of broken symmetries are massless bosonic excitations known as Nambu-Goldstone bosons (NGB). The number of such distinct bosonic excitations are generally taken to be a measure of how many of the original symmetries are dynamically broken. Despite decades of research on the subject, a general formula that predicts the number of different NGBs in a given dynamical system with broken symmetries has eluded theorists.
Now, in a paper in Physical Review Letters, Haruki Watanabe and Hitoshi Murayama (both jointly at the University of California, Berkeley, and the University of Tokyo, Japan) provide a general theorem that relates the number of NGBs to that of broken symmetries. The latter are characterized by the number of generators of the symmetry group of the Hamilonian (as well as their mutual commutation relations) that do not leave invariant the vacuum of the dynamical system under consideration. The power of this result lies in the fact that it applies to all dynamical systems subject to spontaneous symmetry breaking (under some minimal assumptions specified in the paper). While the relativistic analog of this result has been known for a long time, this is the first result of its kind that is applicable to nonrelativistic systems of interest in condensed-matter physics. This result should provide crucial insights, particularly in the study of quantum many-body systems and investigations into quantum criticality. – Abhishek Agarwal