In the last decades, multifaceted manifestations of the breakdown of the self-consistent perturbation theory have been identified for the many-electron problem. Yet, the investigations have been so far mostly limited to paramagnetic states, where symmetry breaking is not allowed. Here, we extend the analysis to the spontaneously symmetry-broken antiferromagnetic (AF) phase of the repulsive Hubbard model. To this aim, we calculated two-particle quantities using dynamical mean-field theory for the AF-ordered Hubbard model and studied the possible occurrence of divergences of the irreducible vertex functions in the charge and spin sectors. Our calculations pinpoint the divergences in the AF phase diagram, showing that while the onset of AF order mitigates the breakdown of the perturbation expansion, it does not fully prevent it. Moreover, we have been able to link the changes in the dynamical structure of the corresponding generalized susceptibilities to the physical crossover from a weak-coupling (Slater) to a strong-coupling (Heisenberg) antiferromagnet, which takes place as the interaction strength is gradually increased. Finally, we discuss possible physical consequences of the irreducible vertex divergences in triggering phase-separation instabilities within the AF phase and elaborate on the implications of our findings for two-dimensional systems, where the onset of a long-range AF order is prevented by the Mermin-Wagner theorem.
In the last decades, multifaceted manifestations of the breakdown of the self-consistent perturbation theory have been identified for the many-electron problem. Yet, the investigations have been so far mostly limited to paramagnetic states, where symmetry breaking is not allowed. Here, we extend the analysis to the spontaneously symmetry-broken antiferromagnetic (AF) phase of the repulsive Hubbard model. To this aim, we calculated two-particle quantities using dynamical mean-field theory for the AF-ordered Hubbard model and studied the possible occurrence of divergences of the irreducible vertex functions in the charge and spin sectors. Our calculations pinpoint the divergences in the AF phase diagram, showing that while the onset of AF order mitigates the breakdown of the perturbation expansion, it does not fully prevent it. Moreover, we have been able to link the changes in the dynamical structure of the corresponding generalized susceptibilities to the physical crossover from a weak-coupling (Slater) to a strong-coupling (Heisenberg) antiferromagnet, which takes place as the interaction strength is gradually increased. Finally, we discuss possible physical consequences of the irreducible vertex divergences in triggering phase-separation instabilities within the AF phase and elaborate on the implications of our findings for two-dimensional systems, where the onset of a long-range AF order is prevented by the Mermin-Wagner theorem.
Starting from the (Hubbard) model of an atom, we demonstrate that the uniqueness of the mapping from the interacting to the noninteracting Green function, G→G_{0}, is strongly violated, by providing numerous explicit examples of different G_{0} leading to the same physical G. We argue that there are indeed infinitely many such G_{0}, with numerous crossings with the physical solution. We show that this rich functional structure is directly related to the divergence of certain classes of (irreducible vertex) diagrams, with important consequences for traditional many-body physics based on diagrammatic expansions. Physically, we ascribe the onset of these highly nonperturbative manifestations to the progressive suppression of the charge susceptibility induced by the formation of local magnetic moments and/or resonating valence bond (RVB) states in strongly correlated electron systems.
In this work, we analyze in detail the occurrence of divergences in the irreducible vertex functions for one of the fundamental models of many-body physics: the Anderson impurity model (AIM). These divergences, a surprising hallmark of the breakdown of many-electron perturbation theory, have been recently observed in several contexts, including the dynamical mean-field solution of the Hubbard model. The numerical calculations for the AIM presented in this work, as well as their comparison with the corresponding results for the Hubbard model, allow us to clarify several open questions about the properties of vertex divergences in a particularly interesting context, the correlated metallic regime at low temperatures. Specifically, our analysis (i) rules out explicitly the transition to a Mott-insulating phase, but not the more general suppression of charge fluctuations (proposed in [O. Gunnarsson et al., Phys. Rev. B 93, 245102 (2016)]), as a necessary condition for the occurrence of vertex divergences, (ii) clarifies their relation with the underlying Kondo physics, and, eventually, (iii) individuates which divergences might also appear on the real-frequency axis in the limit of zero temperature, through the discovered scaling properties of the singular eigenvectors.
Here we analyze the divergences of the irreducible vertex function in dynamical mean field theory, which may indicate either a nonphysical breakdown of the perturbation theory or a response to some physical phenomenon. To investigate this question, we construct a quasiparticle vertex from the diverging irreducible vertex functions. This vertex describes the scattering between quasiparticles and quasiholes in a Fermi liquid. We show that the quasparticle vertex does not diverge in the charge channel, wherein the irreducible vertex does diverge; and we show that the quasiparticle vertex does diverge in the spin channel, wherein the irreducible vertex does not diverge. This divergence occurs at the Mott transition, wherein the Fermi-liquid theory breaks down. Both the half filled Hubbard and Anderson lattices are investigated. In general, our results support that the divergences of the irreducible vertex function do not indicate a nonphysical failure of the perturbation theory. Instead, the divergences are the mathematical consequence of inverting a matrix (the local charge susceptibility) which accumulates increasingly negative diagonal elements as the Hubbard interaction suppresses charge fluctuations. Indeed, we find that the first divergences of the irreducible vertex in both Hubbard and Anderson lattices occurs near the maximum magnitude of the (negative) vertex-connected part of the charge susceptibility.
We discuss the general formalism and validity of weak-coupling perturbation theory as an impurity solver for nonequilibrium dynamical mean-field theory. The method is implemented and tested in the Hubbard model, using expansions up to fourth order for the paramagnetic phase at half filling and third order for the antiferromagnetic and paramagnetic phase away from half filling. We explore various types of weak-coupling expansions and examine the accuracy and applicability of the methods for equilibrium and nonequilibrium problems. We find that in most cases an expansion of local self-energy diagrams including all the tadpole diagrams with respect to the Weiss Green's function (bare-diagram expansion) gives more accurate results than other schemes such as self-consistent perturbation theory using the fully interacting Green's function (bold-diagram expansion). In the paramagnetic phase at half filling, the fourth-order bare expansion improves the result of the second-order expansion in the weak-coupling regime, while both expansions suddenly fail at some intermediate interaction strength. The higher-order bare perturbation is especially advantageous in the antiferromagnetic phase near half filling. We use the third-order bare perturbation expansion within the nonequilibrium dynamical mean-field theory to study dynamical symmetry breaking from the paramagnetic to the antiferromagnetic phase induced by an interaction ramp in the Hubbard model. The results show that the order parameter, after an initial exponential growth, exhibits an amplitude oscillation around a nonthermal value followed by a slow drift toward the thermal value. The transient dynamics seems to be governed by a nonthermal critical point, associated with a nonthermal universality class, which is distinct from the conventional Ginzburg-Landau theory.
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 analyze the highly nonperturbative regime surrounding the Mott-Hubbard metal-to-insulator transition (MIT) by means of dynamical mean field theory (DMFT) calculations at the two-particle level. By extending the results of Sch\"afer et al. [Phys. Rev. Lett. 110, 246405 (2013)] we show the existence of infinitely many lines in the phase diagram of the Hubbard model where the local Bethe-Salpeter equations, and the related irreducible vertex functions, become singular in the charge as well as the particle-particle channel. By comparing our numerical data for the Hubbard model with analytical calculations for exactly solvable systems of increasing complexity [disordered binary mixture (BM), Falicov-Kimball (FK), and atomic limit (AL)], we have (i) identified two different kinds of divergence lines; (ii) classified them in terms of the frequency structure of the associated singular eigenvectors; and (iii) investigated their relation to the emergence of multiple branches in the Luttinger-Ward functional. In this way, we could distinguish the situations where the multiple divergences simply reflect the emergence of an underlying, single energy scale ${\ensuremath{\nu}}^{*}$ below which perturbation theory is no longer applicable, from those where the breakdown of perturbation theory affects, not trivially, different energy regimes. Finally, we discuss the implications of our results on the theoretical understanding of the nonperturbative physics around the MIT and for future developments of many-body algorithms applicable in this regime.