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
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
In many statistical models (but not all) one can introduce a local order parameter associated with a finite, discrete or continuous group of symmetries. The higher the temperature, the more important the fluctuations. One expects therefore in general to find pure phases at low temperature, with a reduced symmetry group, as a consequence of a nonvanishing expectation value of some order parameter. This situation is referred to as spontaneous symmetry breaking. A typical example is the classical Heisenberg model describing short-range interactions of an n -vector field ϕ, with an orthogonal O(n) symmetry group. For n = 1, 2, 3, this can account for a Curie transition from a ferromagnetic to a paramagnetic phase. In particle physics the σ-model of Gell-Mann and Levy involves a spontaneous symmetry breaking of chiral invariance, typical of massless spinor fields, accompanied by soft excitation modes, the so-called Goldstone modes, associated to a π-meson triplet and leading to a nonvanishing dynamical fermion mass. In a first and rather crude approximation, one can analyse the action itself or an effective one incorporating some fluctuation effects, and look for extrema as a function of field configurations, generally translationally invariant. The remaining fluctuations are then treated perturbatively. This mean field method is common to a great variety of domains, ranging from the Clausius–Mossotti formula for a polarizable medium, the Weiss molecular field in the theory of magnetism, Landau's effective action in various statistical contexts, the effective medium approximation in disordered systems, the Hartree–Fock method in atomic or many-body physics, to the semiclassical approximation in the study of quantum systems.
Spontaneous symmetry breaking of interacting fermion systems constitutes a major challenge for many-body theory due to the proliferation of new independent scattering channels once absent or degenerate in the symmetric phase. One example is given by the ferro/antiferromagnetic broken symmetry phase (BSP) of the Hubbard model, where vertices in the spin-transverse and spin-longitudinal channels become independent with a consequent increase in the computational power for their calculation. We generalise the non-perturbative Two-Particle-Self-Consistent method (TPSC) to address broken SU(2) magnetic phases in the Hubbard model, offering an efficient approach that incorporates strong correlations. We show that in the BSP, the sum-rule enforcement of susceptibilities must be accompanied by a modified gap equation resulting in a renormalisation of the order parameter, vertex corrections and the preservation of the gap-less feature of the Goldstone modes. We then apply the theory to the antiferromagnetic phase of the Hubbard model in the cubic lattice at half-filling. We compare our results of double occupancies and staggered magnetisation to the ones obtained using Diagrammatic Monte Carlo showing excellent quantitative agreement. We demonstrate how vertex corrections play a central role in lowering the Higgs resonance with respect to the quasi-particle excitation gap in the spin-longitudinal susceptibility, yielding a well visible Higgs-mode.
We investigate spontaneous U(1) symmetry breaking and the associated phase transitions in rotating interacting Bose gases. Using a theoretical framework that combines mean-field analysis with rotational dynamics, we analyze how rigid rotation modifies the condensate structure and critical behavior. The study identifies the emergence of Goldstone modes and clarifies their role in the low-energy excitation spectrum. The results provide insight into the interplay between symmetry, rotation, and many-body interactions, contributing to a deeper theoretical understanding of phase structures in Bose systems.We investigate spontaneous U(1) symmetry breaking and the associated phase transitions in rotating interacting Bose gases. Using a theoretical framework that combines mean-field analysis with rotational dynamics, we analyze how rigid rotation modifies the condensate structure, critical behavior, and low-energy excitation spectrum. We identify the emergence of Goldstone modes (massless rotons and massive phonons) in the symmetry-broken phase and clarify their role in mediating low-energy excitations—findings that remain robust at low momentum regardless of rotation. A key result is the angular velocity (Ω) dependence of the critical temperature (Tc) for U(1) phase transition, where Tc scales as Ω^(1/3), distinct from the Ω^(2/5) (nonrelativistic) and Ω^(1/4) (ultrarelativistic) scaling observed in noninteracting rotating Bose gases. Rotation also alters the temperature dependence of the thermodynamic potential minima, changing the characteristic factor from (1 - t) (t = T/Tc for nonrotating systems) to (1 - t³) for rotating gases. We further demonstrate that rotation preserves the second-order nature of the phase transition, while modifying the critical exponents and reducing the discontinuity in heat capacity with increasing Ω. Additionally, we define a σ meson dissociation temperature (Tdiss) characterized by mσ(Tdiss) = 2mπ(Tdiss), showing that Tdiss is always lower than Tc. Thermal mass corrections are shown to ensure the validity of Goldstone’s theorem in the rotating frame, even in the chiral limit. These results deepen our understanding of the interplay between symmetry, rotation, and many-body interactions, with implications for interpreting extreme conditions in heavy-ion collisions and compact astrophysical objects, while advancing the theoretical framework for phase structures in rotating Bose systems.
A brief description of excited and ground states in two-dimensional quantum dots and rotating nuclei is presented within a mean field approach and a random-phase approximation (RPA). We discuss the procedure to restore the rotational symmetry broken at the mean field, which can be extended for other symmetry breaking cases. We propose to consider a disappearance of collective excitations in the rotating frame as a manifestation of symmetry breaking phenomena of the rotating mean field. In particular, we demonstrate that the disappearance of a collective octupole mode in the rotating frame in 162 Yb gives rise to the nonaxial octupole deformation.
We investigate the magnetic orders and excitations in a half-filled Hubbard model that continuously interpolates between the Lieb and kagome lattices. Using the self-consistent Hartree-Fock approximation combined with the Bethe-Salpeter equation in the random phase approximation, we map the U − t ′ phase diagram of the Lieb-kagome lattices and compute real-frequency transverse spin susceptibilities, i.e., the magnon spectra. Dynamical mean-field theory and exact diagonalization are used as complementary checks of static magnetic phases. In particular, besides gapless Goldstone magnons, the symmetry-broken ferrimagnetic and antiferromagnetic phases also exhibit gapped Higgs magnon bands, which originate from amplitude fluctuations in the order parameter characterizing spontaneous symmetry breaking.