Quantum dynamics in symmetry-breaking states of correlated electrons: Antiferromagnetic phase

TL;DR

This work addresses the challenge of consistently extending quantum dynamics from the high-temperature symmetric phase to symmetry-broken states, where conventional dynamical self-energies can lead to mismatched transition points with the disordered phase. By employing Baym-Kadanoff thermodynamics in conjunction with the Nambu formalism, the authors identify and suppress anomalous, non-conserving two-particle propagators that distort conservation laws, ensuring a continuous match at the transition. Applying the framework to the antiferromagnetic phase of the Hubbard model in FLEX-DMFT reveals a reduced critical temperature compared to static mean-field theory and uncovers a two-gap spectral structure at zero temperature, with a dynamical gap larger than the static Hartree-Fock-like gap and in-gap states generated by quantum fluctuations. The proposed conserving scheme provides a general, principled approach for incorporating dynamical fluctuations into symmetry-broken phases and suggests analogous spectral features in superconducting states, with broad implications for strongly correlated electron systems.

Abstract

Symmetry-breaking phases in many-fermion systems are characterized by anomalous functions that represent transient processes during which some properties of free particles, such as spin or charge, are not conserved. Connecting the low-temperature symmetry-breaking phase with the high-temperature one within the Baym-Kadanoff scheme, beyond the static mean-field approximation, remains an unresolved, long-standing challenge. We identify the reason why approximations with critical dynamical fluctuations in the Schwinger-Dyson equation lead to a mismatch in the transition temperatures calculated from the high- and low-temperature phases. We propose a solution to this generic problem by excluding anomalous contributions to response functions that do not obey conservation of excitations in their interactions. We illustrate this behavior using the example of an antiferromagnetic state. We reveal that the spectral function in the antiferromagnetic phase exhibits a double-gap structure at zero temperature when the anomalous self-energy is frequency-dependent.

Figures (6)

• Figure 1: Antiferromagnetic order parameter $\Delta$ of the half-filled Hubbard model at $U=2w$, of the second-order approximation with $\Im\Sigma(\omega)<0$, obtained for $\delta\Sigma(\omega)=0$, lower, blue, curve with dots, and the full dynamical solution $\Delta\Sigma(\omega)$, upper, black, curve with dots, compared with the Hatree-Fock solution, red curve. The temperature is normalized to the half bandwidth, $w=1$.
• Figure 2: Dynamical order parameter $\Delta + \Delta \Re\Sigma(\omega)$ as function of frequency at zero tempearture and $U= 2w$. The energy scale was set $w=1$.
• Figure 3: The imaginary part of the normal self-energy $Y(\omega) = - \Im \Sigma(\omega_{+})$ and of the anomalous self-energy $\Delta Y(\omega) = \Im\Delta\Sigma(\omega_{+})$ at zero tempearture and for $U=2w$ with the energy scale $w=1$.
• Figure 4: Frequency dependence of the zero-temperature imaginary parts of the normal self-energy $-Y(\omega)$ and the spectral function $A(\omega)$. The band of occupied states within the self-energy gap is Hartree-Fock-like, with no quantum dynamics.
• Figure 5: Bispinor of the full spin-dependent electron-hole propagator in the ordered phase. The red diagrams conserve the number of electrons and holes and guarantee that the ordered phase continuously matches the high-temperature solution at the critical transition point.