Discrepancy between microscopic quantum dynamics and macroscopic order in strongly correlated electron systems

We disclosed in this paper a long-neglected problem of matching microscopic quantum many-body dynamics with the macroscopic long-range order. The latter is presently derived from the former within the canonical Baym-Kadanoff construction of thermodynamically consistent conserving approximations. Two two-particle vertices, responsible for the system’s response to the external perturbations, must be introduced: microscopic dynamic and macroscopic conserving vertices. The former is determined from the diagrammatic perturbation theory, while the latter is obtained from the respective Ward identity, guaranteeing macroscopic conservation laws. The divergence of each vertex indicates an instability. 

We revealed an ambiguity of the Baym-Kadanoff construction in determining the critical behavior with a transition to a thermodynamic long-range order. We demonstrated that each vertex leads to incomplete and distinct critical behavior with distinct instability points. The diagrammatically controlled dynamic vertex from the Schwinger-Dyson equation cannot directly be linked with a macroscopic long-range order, since it does not obey the Ward identity. Consequently, it cannot be continued beyond its instability. On the other hand, the divergence in the macroscopic vertex, obeying the conservation laws, has no direct microscopic consequences and does not invoke critical behavior of the spectral function and the specific heat as derived from the dynamic vertex. Consequently, the description of the critical behavior of correlated electrons becomes consistent and reliable only if the fluctuations of the order parameter in the conserving vertex lead to a divergence coinciding with that of the dynamical one. None of the existing approximate schemes fulfills this condition.  

The ambiguity in the definition of phase instabilities of the Baym-Kadanoff scheme disclosed by us has a firm conclusion that an entirely consistent theory beyond the weak-coupling static Hartree-Fock approximation of the critical behavior of strongly correlated electrons is to be devised.

Description

Demonstration of the ambiguity of the definition of phase instability. The dependence on the interaction strength U of the dimensionless Kondo scale a, measuring the distance to the magnetic instability from the dynamic vertex of the microscopic dynamic theory, red curves, and the Kondo scale from the macroscopic susceptibility a_m=χ₀/χ, blue curves, plotted for different values of the ultraviolet cutoff in the polar approximation, accurate in determining the critical point, Ω; χ₀,χ are zero-temperature bare and full susceptibilities. The black solid line is the Hartree static solution (1-U/π). Notice that the exact solution suppresses the magnetic instability, as obtained for the dynamic solution, red lines. The thermodynamic macroscopic criterion, blue lines, leads to a spurious instability, disqualifying the credibility of the dynamic approximations in determining consistently macroscopic instabilities of strongly correlated electrons.

Contact person: Václav Janiš