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.
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.
The two-particle self-consistent approach (TPSC) is a method for the one-band Hubbard model that can be both numerically efficient and reliable. However, TPSC fails to yield physical results deep in the renormalized classical regime of the bidimensional Hubbard model where the spin correlation length becomes exponentially large. We address the limitations of TPSC with improved approaches that we call TPSC+ and TPSC+SFM (spin fluctuation mediated). In this work, we show that these improved methods satisfy the Mermin-Wagner theorem and the Pauli principle. We also show that they are valid in the renormalized classical regime of the two-dimensional Hubbard model, where they recover a generalized Stoner criterion at zero temperature in the antiferromagnetic phase. We discuss some limitations of the TPSC+ approach with regards to the violation of the f-sum rule and conservation laws, which are solved within the TPSC+SFM framework. Finally, we benchmark the TPSC+ and TPSC+SFM approaches for the one-band Hubbard model in two dimensions and show how they have an overall better agreement with available diagrammatic Monte Carlo results than the original TPSC approach.
A new approach to the single-band Hubbard model is described in the general context of many-body theories. It is based on enforcing conservation laws, the Pauli principle and a number of crucial sum-rules. More specifically, spin and charge susceptibilities are expressed, in a conserving approximation, as a function of two constant irreducible vertices whose values are found self-consistently. The Mermin-Wagner theorem in two dimensions is automatically satisfied. The effect of collective modes on single-particle properties is then obtained by a paramagnon-like formula that is consistent with the two-particle properties in the sense that the potential energy obtained from TrΣG is identical to that obtained using the fluctuation-dissipation theorem for susceptibilities. The vertex corrections are included through constant irreducible vertices. The theory is in quantitative agreement with Monte Carlo simulations for both single-particle and two-particle properties. In the two-dimensional renormalized classical regime, spin fluctuations lead to precursors of antiferromagnetic bands (shadow bands) and to the destruction of the Fermi-liquid quasiparticles in a wide temperature range above the zero-temperature phase transition. The analogous phenomenon of pairing pseudogap can occur in the attractive model in two dimensions when the pairing fluctuations become critical. Other many-body approaches are critically compared. It is argued that treating the spin fluctuations as if there was a Migdal's theorem can lead to wrong predictions, in particular with regard to the the single-particle pseudogap.
Based on the self-energy-functional approach proposed recently [M. Potthoff, Eur. Phys. J. B 32, 429 (2003)], we present an extension of the cluster-perturbation theory to systems with spontaneously broken symmetry. Our method applies to models with local interactions and accounts for both short-range correlations and long-range order. Short-range correlations are accurately taken into account via exact diagonalization of finite clusters. Long-range order is described by variational optimization of a ficticious symmetry-breaking field. In comparison with related cluster methods, our approach is more flexible and, for a given cluster size, less demanding numerically, especially at zero temperature. An application of the method to the antiferromagnetic phase of the Hubbard model at half-filling shows good agreement with results from quantum Monte Carlo calculations. We demonstrate that the variational extension of the cluster-perturbation theory is crucial to reproduce salient features of the single-particle spectrum.
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.
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.
In systems of indistinguishable particles, correlations between the quantum degrees of freedom can give rise to collective modes where particles behave coherently together over some correlation length and time, thereby forming quasiparticles. One example of such quasiparticle would be the π-ton [66], describing the binding of two particle-hole pairs via the exchange of momentum corresponding to the antiferromagnetic (AFM) wave vector. These quasiparticles would emerge in the vicinity of an AFM ordering instability [66, 119, 120] and nonlocal correlations are crucial to describe them. Especially, in systems out of equilibrium, nonlocality can give rise to prethermal states wherein the distribution function can be momentum-dependent [92, 119], and nonthermal exotic physics can be observed and manipulated [40, 68, 79, 84, 93, 132, 147, 157–159]. To be able to model and simulate systems out of equilibrium is a challenge. A few numerical methods such as nonequilibrium Dynamical Mean Field Theory (DMFT) [44] have been used extensively [5]. Nonequilibrium DMFT is designed to treat nonperturbatively the local time-dependent single-particle correlations. Since phenomena such as quantum criticality, high-Tc superconductivity or the creation of π-tons via light-matter coupling stem from nonlocal two-particle correlations, one needs to resort to diagrammatic extensions to DMFT [103] and other methods to effectively describe them, such as the post-processing DMFT treatment of vertex corrections, which will be introduced. The Two-Particle Self-Consistent approach (TPSC) and its variants figure among those methods that include nonlocal two-particle and single-particle correlations in a self-consistent way [121, 122]. TPSC and its variants have the ability to capture the effect of the growing spin fluctuations (paramagnons) in the self-energy when close to amagnetic phase transition leading to the breakdown of the Fermi-liquid quasiparticles at the Fermi level: this would mark the onset of a pseudo-gap [145, 152]. It fulfils the Mermin-Wagner theorem in two dimensions, the Pauli principle, and various two-particle local sum-rules [152]. In the works embodied in this thesis, numerical methods have been designed to access nonequilibrium regimes, by changing model parameters across time, allowing for nonlocal correlations to be properly included. With these new algorithms at hand, the π-tons were studied in the optical conductivity and magnetic response [119, 120] as well as spin and charge fluctuations in the 2D and 3D single-band Hubbard model [121, 122]. It is demonstrated by means of those methods that nonlocal two-particle vertex corrections can underpin prethermal states and could generate transient nonthermal states such as low-energy charge excitations associated with nonthermal negative temperatures. It is also shown that states lying close to the Fermi level control the thermalization timescale and that lattice hopping quenches changing the unit-cell dimensionality could allow one to play around with the charge and spin scattering channels.