Broken symmetries in the two Goldstone models

Konstantinos Vasiliou, Yuchi He, Nick Bultinck

We consider electronic systems with a spontaneously broken continuous symmetry. The scattering vertex between electrons and Goldstone modes is calculated over the entire Brillouin zone using the random phase approximation. This calculation reveals two things: (1) electrons always couple to both ϕ and ∂tϕ , where ϕ is the Goldstone field, and (2) quasiparticles in a state with continuous symmetry breaking have to be defined in a rotating frame, which locally follows the fluctuations of the order parameter. The implications of these findings for electron spectral functions in both symmetry-broken and thermally disordered systems are discussed, and the examples of antiferromagnetism in the Hubbard model and spin spiral order in the three-band model are worked out in detail. Published by the American Physical Society 2024

G. Fano

H. Stern

An investigation is made of the relationship between long-wavelength, low-frequency normal modes and broken symmetry in nonrelativistic many-body systems. In particular, the relationship between broken symmetry as manifested through the so-called Goldstone pole and the normal-mode structure is examined. Through the study of various models, we show that the structure of the normal modes is correlated to the Goldstone pole either completely, partially, or not at all, according to the class of symmetry of the Hamiltonian of the system. For example, in the neutral superconductor, the Hamiltonian has such low symmetry that although the Anderson modes restore the symmetry of the ground state, they have no relationship at all to the Goldstone pole. It is observed that as the symmetry decreases, a dynamical sum rule takes the place of such a correlation and in all systems a sum rule gives the normal-mode frequency $\ensuremath{\omega}$. These sum rules also give the $\ensuremath{\omega}$ distribution of states through the Huang-Klein dispersion relation.

Heide Narnhofer, Walter E Thirring

Bo-Ting Chen, Zihan Zhou, Biao Lian

Modulated symmetries are global symmetries with a spatially dependent unit of charge, such as the dipole symmetry and the exponential symmetry. We give the generic condition for a modulated symmetry to be compatible with translationally symmetric Hamiltonians, which we define as a translationally covariant modulated symmetry (TCMS). For Abelian TCMSs, we prove that their units of charge can only contain multipole, exponential and harmonic components. Particularly, we classify all the one-dimensional TCMSs by real Jordan normal form blocks. We further derive the generic Goldstone action for SSB phases of continuous TCMSs, by which we show that a broken multipole symmetry gives higher-order gapless Goldstone modes, a broken harmonic symmetry gives gapless Goldstone modes at finite momenta, and a broken exponential symmetry gives no gapless Goldstone modes, modifying the conventional Goldstone theorem.

Daniel R. Bes