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arXiv:2605.02625 · Physical Review Research

Flow of local quantum fluids: Conservation laws and vertex corrections from many-body linear-response theory with local self-energy

Davide Valentinis

Published Oct 9, 2026• 0 citations
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Abstract

In non-diffusive conduction regimes of strongly correlated quantum electron systems, electromagnetic perturbations simultaneously probe the electronic dynamics in time and space: the exchanged energy $\hbar ω$ excites retarded, i.e., frequency-dependent, many-body interactions, while the probing spatial modulation renders the response spatially nonlocal, i.e., dependent on the external wave vector $\vec{q}$. This work derives the nonlocal electrodynamic response of such dynamical quantum fluids assuming local but frequency-dependent self-energies and particle-hole irreducible two-particle vertex functions, and preserving charge/mass conservation. The latter is ensured by Bethe-Salpeter equations for renormalized interaction vertices, entering the Kubo formalism for two-particle correlation functions (e.g., for density, currents, momentum, stress). Within such a framework, exact symmetry criteria for the absence of vertex corrections are inferred. In particular, vertex corrections vanish at $q=0$ for single-particle dispersions that are even and bare interaction vertices that are odd with respect to specific momentum-space point group transformations, including inversion for vector vertices, and mirror reflections or two- or higher-fold rotations for tensor vertices. If the dispersion is isotropic, vertex corrections vanish from the transverse part ($\perp\vec{q}$) of vectorial vertices (such as for electric/particle current) at any finite $\hbar ω$ and $\vec{q}$. These cancellations extend to the full current-current correlation function for quadratic isotropic dispersion. Further symmetry breaking, multiband effects, and the additional imposition of momentum conservation, are discussed, with application to the Hall viscosity of Landau levels. Explicit expressions for generic nonlocal correlation functions are derived for Fermi liquids and non-Fermi liquids.

DOI
10.1103/drq4-95tc

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In non-diffusive conduction regimes of strongly correlated quantum electron systems, electromagnetic perturbations simultaneously probe the electronic dynamics in time and space: the exchanged energy ℏω\hbar ωℏω excites retarded, i.e., frequency-dependent, many-body interactions, while the probing spatial modulation renders the response spatially nonlocal, i.e., dependent on the external wave vector q⃗\vec{q}q​. This work derives the nonlocal electrodynamic response of such dynamical quantum fluids assuming local but frequency-dependent self-energies and particle-hole irreducible two-particle vertex functions, and preserving charge/mass conservation. The latter is ensured by Bethe-Salpeter equations for renormalized interaction vertices, entering the Kubo formalism for two-particle correlation functions (e.g., for density, currents, momentum, stress). Within such a framework, exact symmetry criteria for the absence of vertex corrections are inferred. In particular, vertex corrections vanish at q=0q=0q=0 for single-particle dispersions that are even and bare interaction vertices that are odd with respect to specific momentum-space point group transformations, including inversion for vector vertices, and mirror reflections or two- or higher-fold rotations for tensor vertices. If the dispersion is isotropic, vertex corrections vanish from the transverse part (⊥q⃗\perp\vec{q}⊥q​) of vectorial vertices (such as for electric/particle current) at any finite ℏω\hbar ωℏω and q⃗\vec{q}q​. These cancellations extend to the full current-current correlation function for quadratic isotropic dispersion. Further symmetry breaking, multiband effects, and the additional imposition of momentum conservation, are discussed, with application to the Hall viscosity of Landau levels. Explicit expressions for generic nonlocal correlation functions are derived for Fermi liquids and non-Fermi liquids.
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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.
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