Keldysh pseudo-fermion functional renormalization group for quantum magnetism

TL;DR

This work presents a Keldysh extension of the pseudo-fermion FRG (PFFRG) to compute the dynamical spin structure factor $S^ ext{Ret}_{ij}( ext{ω})$ directly for frustrated quantum magnets. By formulating flow equations for the one- and two-particle vertices in real time and implementing symmetry constraints, the method yields dynamical information complementary to conventional Matsubara FRG and avoids analytic continuation. Benchmark studies on the AFM dimer, 1D spin chain, frustrated square lattice, triangular lattice, and Kitaev–Honeycomb model demonstrate qualitative improvements over RPA and provide insights into spinon continua and spin-liquid dynamics, while highlighting limitations due to the average constraint and finite-temperature effects. The approach offers a promising route to connect microscopic spin models with neutron-scattering spectra in higher dimensions and near quantum-paramagnetic regimes, with future enhancements such as pseudo-Majorana decompositions and magnetic-field extensions to access symmetry-broken phases.

Abstract

The functional renormalization group (FRG) approach for spin models relying on a pseudo-fermionic description has proven to be a powerful technique in simulating ground state properties of strongly frustrated magnetic lattices. A drawback of the FRG framework is that it is formulated in the imaginary-time Matsubara formalism and thus only able to access static correlations, a limitation shared with most other many-body approaches. A description of the dynamical properties of magnetic systems is the key to bridging the gap between theory and neutron scattering spectra. We take the decisive step of expanding the scope of pseudo-fermion FRG to the Keldysh formalism, which, while originally developed to address non-equilibrium phenomena, enables a direct calculation of the equilibrium dynamical spin structure factors on generic lattices in arbitrary dimension. We identify the principal features characterizing the low-energy spectra of exemplary zero-, one- and two-dimensional spin-$1/2$ Heisenberg models as well as the Kitaev honeycomb model while identifying current limitations of the method that have to be improved upon.

Figures (16)

• Figure 1: Keldysh contour with an operator $\mathcal{O}$ inserted at $t=t_0$. The respective time propagation operators are also given.
• Figure 2: Dynamic susceptibility and resulting spin structure factor [from \ref{['eq:structreFactor']}] for the antiferromagnetic dimer in the $S\to\infty$ limit at $T=0.2J$. The cutoff is chosen such that $\Lambda=0.002J\ll T$. The results of the FRG coincide with the single step RPA calculation. The spin structure factor shows the expected peak around $\omega=0$, falling off relatively quickly.
• Figure 3: (a) Exemplary FRG flow at $T=0.2 J$ for the antiferromagnetic Heisenberg dimer. The flow exhibits a saturation around $\Lambda=0.001J$ but is continued down to $\Lambda_\text{min}=10^{-6}J$ to exclude any influence of the cutoff scale. The dashed lines show the cutoffs, which are used for the comparison in \ref{['fig:DimerSus']}. (b) Temperature scaling of the susceptibilities at lowest cutoff $\Lambda_\text{min}$. The Keldysh PFFRG results match the exact diagonalization results (ED) of the pseudo-fermion dimer down to $T\approx0.2J$. For smaller $T$ the FRG overestimates the susceptibilities.
• Figure 4: Frequency resolved susceptibilities for the AFM dimer at $T=0.2J$ for different points in the flow. The chosen cutoffs are also marked in Fig. \ref{['fig:DimerMisc']}(a). In panel (b) we also show double exponential fits according to Eq. \ref{['eq:doubleExponential']}, which describe the susceptibilities up to some minor differences. The peak positions for the smallest cutoffs are marked by dashed lines for better visibility.
• Figure 5: Spin structure factor for the spin chain with $N=10$ in the $S\to\infty$ limit. The excitation is localized around $q=\pi$ and falls off quickly for $\omega>0$. We do not get any branches, as we would get in spin wave theory since the initial state in the FRG is fully disordered and thus paramagnetic .