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arXiv:2401.04042 · Physical review. B./Physical review. B

Second-order phase transitions and divergent linear response in dynamical mean-field theory

Erik G. C. P. van Loon

Published Jun 10, 2024• 5 citations
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Abstract

Second-order phase transitions appear as a divergence in one of the linear response functions. For a system of correlated electrons, the relevant divergent response can and does involve many-particle observables, most famously the double occupancy. Generally, evaluating the linear response function of many-particle observables requires a many-particle generalization of the Bethe-Salpeter equation. However, here I show that the divergence of linear response functions in dynamical mean-field theory is governed by a -particle Bethe-Salpeter equation, even for many-particle observables. The reason for this is that the divergence at the second-order phase transition is produced by the self-consistent feedback of the dynamical mean field. Published by the American Physical Society 2024

DOI
10.1103/physrevb.109.l241110

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Second-order phase transitions appear as a divergence in one of the linear response functions. For a system of correlated electrons, the relevant divergent response can and does involve many-particle observables, most famously the double occupancy. Generally, evaluating the linear response function of many-particle observables requires a many-particle generalization of the Bethe-Salpeter equation. However, here I show that the divergence of linear response functions in dynamical mean-field theory is governed by a -particle Bethe-Salpeter equation, even for many-particle observables. The reason for this is that the divergence at the second-order phase transition is produced by the self-consistent feedback of the dynamical mean field. Published by the American Physical Society 2024
A Bethe-Salpeter equation for the three-particle correlation function is derived, representing a second-order response generalization of the usual Bethe-Salpeter equation. The equation can be solved formally, giving the three-particle correlation function in terms of the two-particle correlation function and many-body interaction kernels. The similarity to second-order response time-dependent density-functional theory can be used to improve the understanding of the higher-order exchange and correlation kernels of this theory. An exact expression for such a kernel is derived.
Dynamical mean-field theory describes the impact of strong local correlation effects in many-electron systems. While the single-particle spectral function is directly obtained within the formalism, two-particle susceptibilities can also be obtained by solving the Bethe-Salpeter equation. The solution requires handling infinite matrices in Matsubara frequency space. This is commonly treated using a finite frequency cutoff, resulting in slow linear convergence. A decomposition of the two-particle response in local and nonlocal contributions enables a reformulation of the Bethe-Salpeter equation inspired by the dual boson formalism. The reformulation has a drastically improved cubic convergence with respect to the frequency cutoff, considerably facilitating the calculation of susceptibilities in multi-orbital systems. This improved convergence arises from the fact that local contributions can be measured in the impurity solver. The dual Bethe-Salpeter equation uses the fully reducible vertex which is free from vertex divergences. We benchmark the approach on several systems including the spin susceptibility of strontium ruthenate Sr2RuO4 , a strongly correlated Hund's metal with three active orbitals. Published by the American Physical Society 2024
Introduction Our discussion of critical phenomena surrounding a second-order phase transition has thus far focused only on qualitative features. We have now examined three systems, fluids, magnets and random percolation, each of which displays an abruptly sharp transition from a less ordered to a more ordered phase as the respective transition point is encountered. Each shows a similar pattern of developing structure just in advance of the critical point. Fluctuations in the respective order parameter display a self-similar structure that is limited only by a single relevant length scale, the correlation length ξ, which diverges on approach to the transition point. For fluids and magnets, this structure arises from inherent fluctuations that are amplified by either a diverging compressibility or susceptibility, respectively. In this chapter, we explore more quantitative, theoretical approaches taken to understand the features of second-order phase transitions. The simplest of these are the mean field theories , in which the pairwise interaction (needed to produce a phase transition) is introduced in the form of an average field. In this approach, the effects of the growing fluctuations of the order parameter near the critical point are ignored. Although the mean field approach does meet many of the requirements and does predict divergences of certain quantities near the transition, the critical exponents predicted by the theory do not match those seen experimentally. Obtaining correct exponents requires a more advanced approach involving renormalization techniques that exploit the self-similar structure of the fluctuations near the critical point and allow all of the various critical exponents to be inter-related, such that knowledge of any two yields all the others.
Strong repulsive interactions between electrons can lead to a Mott metal-insulator transition. The dynamical mean-field theory (DMFT) explains the critical end point and the hysteresis region usually in terms of single-particle concepts, such as the spectral function and the quasiparticle weight. In this Letter, we reconsider the critical end point of the metal-insulator transition on the DMFT's two-particle level. We show that the relevant eigenvalue and eigenvector of the nonlocal Bethe-Salpeter kernel in the charge channel provide a unified picture of the hysteresis region and of the critical end point of the Mott transition. In particular, they simultaneously explain the thermodynamics of the hysteresis region and the iterative stability of the DMFT equations. This analysis paves the way for a deeper understanding of phase transitions in correlated materials.
Identifying the fingerprints of the Mott-Hubbard metal-insulator transition may be quite elusive in correlated metallic systems if the analysis is limited to the single particle level. However, our dynamical mean-field calculations demonstrate that the situation changes completely if the frequency dependence of the two-particle vertex functions is considered: The first nonperturbative precursors of the Mott physics are unambiguously identified well inside the metallic regime by the divergence of the local Bethe-Salpeter equation in the charge channel. In the low-temperature limit this occurs for interaction values where incoherent high-energy features emerge in the spectral function, while at high temperatures it is traceable up to the atomic limit.