[Paper Review] Long-term memory magnetic correlations in the Hubbard model: A dynamical mean-field theory analysis

Clemens Watzenböck, Martina Fellinger+2 more|arXiv (Cornell University)|Dec 6, 2021

Physics of Superconductivity and Magnetism60 references27 citations

TL;DR

This paper introduces a robust method to quantify long-term memory effects in magnetic correlations of the Hubbard model using dynamical mean-field theory (DMFT), by computing the difference C between the zero-frequency Kubo susceptibility and the static isothermal susceptibility. It finds that C ≠ 0 in the Mott insulating phase due to ground-state degeneracy, signaling persistent spin fluctuations, with abrupt onset at the first-order metal-insulator transition and gradual emergence in the crossover regime near the Widom line.

ABSTRACT

We investigate the onset of a not-decaying asymptotic behavior of temporal magnetic correlations in the Hubbard model in infinite dimensions. This long-term memory feature of dynamical spin correlations can be precisely quantified by computing the difference between the zero-frequency limit of the Kubo susceptibility and the corresponding static isothermal one. Here, we present a procedure for reliably evaluating this difference starting from imaginary time-axis data, and apply it to the testbed case of the Mott-Hubbard metal-insulator transition (MIT). At low temperatures, we find long-term memory effects in the entire Mott regime, abruptly ending at the first order MIT. This directly reflects the underlying local moment physics and the associated degeneracy in the many-electron spectrum. At higher temperatures, a more gradual onset of an infinitely-long time-decay of magnetic correlations occurs in the crossover regime, not too far from the Widom line emerging from the critical point of the MIT. Our work has relevant algorithmic implications for the analytical continuation of dynamical susceptibilities in strongly correlated regimes and offers a new perspective for unveiling fundamental properties of the many-particle spectrum of the problem under scrutiny.

Motivation & Objective

• To develop a reliable numerical procedure for computing the long-term memory parameter C from imaginary-time correlation functions.
• To investigate the emergence of non-decaying magnetic correlations in the Hubbard model using DMFT.
• To clarify the physical origin of C ≠ 0 in the context of the Mott-Hubbard metal-insulator transition (MIT).
• To analyze the connection between C and the many-body energy spectrum, particularly local moment formation and degeneracy.
• To assess algorithmic implications for analytic continuation of dynamical susceptibilities in strongly correlated systems.

Proposed method

• Formalize the difference C = χR(ω=0) − χT as a measure of long-term memory in spin correlations.
• Use the Lehmann representation to express the imaginary-time spin correlation function C(τ) in terms of spectral weights and energy levels.
• Apply numerical analytic continuation to Matsubara-frequency data χSzSz(iωn) to reconstruct real-frequency spectral functions.
• Employ a minimal peak width constraint and error estimation to stabilize the analytic continuation process.
• Compute C from the spectral weight at zero frequency and the static susceptibility, using high-accuracy DMFT calculations.
• Map C(β, U) across the phase diagram to identify regions of persistent correlations.

Experimental results

Research questions

• RQ1 What is the physical origin of a non-vanishing long-term memory parameter C in the Hubbard model?
• RQ2 How does C behave across the Mott-Hubbard metal-insulator transition, particularly in the coexistence region?
• RQ3 Is the onset of long-term memory effects abrupt or continuous, and how does it relate to the critical point of the MIT?
• RQ4 To what extent does C reflect the presence of degenerate many-electron eigenstates in the Mott insulating phase?
• RQ5 How does the behavior of C at high temperatures compare to that at low temperatures, and what does it reveal about the crossover regime?

Key findings

• C ≠ 0 is found in the entire Mott insulating phase at low temperatures, indicating persistent magnetic correlations due to ground-state degeneracy.
• The long-term memory effect ends abruptly at the first-order Mott metal-insulator transition, signaling a sharp change in the many-body spectrum.
• In the crossover regime near the Widom line, C emerges gradually with increasing U, indicating a continuous onset of long-time correlations.
• At high temperatures, the onset of C ≠ 0 is smooth and not tied to a phase transition, suggesting a crossover-driven enhancement of slow fluctuations.
• The value of C in the insulating phase is nearly independent of temperature, confirming its origin in the degeneracy of the Mott-insulating ground state.
• The method successfully extracts C from imaginary-time data, demonstrating its viability for analyzing dynamical susceptibilities in strongly correlated systems.