Dynamical properties of the impurity spin-
21 in two-dimensional (2D) and quasi-2D Heisenberg antiferromagnets (AFs) at $T\ensuremath{\geqslant}0$ are discussed. The specific case of an impurity coupled symmetrically to two neighboring host spins is considered. The specific feature of this problem is that the defect is degenerate (frustrated) being located in zero molecular field. It is shown that this problem can be described by the spin-boson model without a tunneling term and with a more complex interaction. We demonstrate that the effect of the host system on the defect is completely described by the spectral function. It is found within the spin-wave approximation that for not too small $\ensuremath{\omega}$ the spectral function is proportional to ${\ensuremath{\omega}}^{2}∕{J}^{3}$, where
J is the exchange constant between the host spins. The defect dynamical susceptibility is derived using Abrikosov's pseudofermion technique and diagrammatic expansion. The calculations are performed within the fourth order of the dimensionless coupling parameter
f. It is found that transverse impurity susceptibility ${\ensuremath{\chi}}_{\ensuremath{\perp}}(\ensuremath{\omega})$ has a Lorenz peak, with the width proportional to
f4J(T∕J)3 which disappears at
T=0, and a nonresonant term. The latter term diverges logarithmically as $\ensuremath{\omega},\phantom{\rule{0.2em}{0ex}}T\ensuremath{\rightarrow}0$. The static susceptibility $\ensuremath{\chi}(0)$ has the free-spin-like contribution
1∕(4T), and a logarithmic correction proportional to
f2ln(J∕T). The influence of finite concentration of the defects
n on the low-temperature properties of AFs is also investigated. A logarithmic correction to spin-wave velocity of the form $n{f}^{4}\mathrm{ln}\ensuremath{\mid}J∕\ensuremath{\omega}\ensuremath{\mid}$ and an anomalous damping of spin waves proportional to $n{f}^{4}\ensuremath{\mid}\ensuremath{\omega}\ensuremath{\mid}$ are obtained. The results of the present paper can be applied to other systems with a frustrated impurity in which the spectral function is proportional to ${\ensuremath{\omega}}^{2}$.
We present a multiloop pseudofermion functional renormalization group (pffRG) approach to quantum spin systems. As a test case, we study the spin-
21 Heisenberg model on the kagome lattice, a prime example of a geometrically frustrated magnet believed to host a quantum spin liquid. Our main physical result is that, at pure nearest-neighbor coupling, the system shows indications for an algebraic spin liquid through slower-than-exponential decay with distance for the static spin susceptibility, while the pseudofermion self-energy develops intriguing low-energy features. Methodologically, the pseudofermion representation of spin models inherently yields a strongly interacting system, and the quantitative reliability of a truncated fRG flow is \textit{a priori} unclear. Our main technical result is the demonstration of convergence in loop number within multiloop pffRG. Through correspondence with the self-consistent parquet equations, this provides further evidence for the internal consistency of the approach. The loop order required for convergence of the pseudofermion vertices is rather large, but the spin susceptibility is more benign, appearing almost fully converged for loop orders
ℓ≥5. The multiloop flow remains stable as the infrared cutoff
Λ is reduced relative to the microscopic exchange interaction
J, allowing us to reach values of
Λ/J on the subpercent level in the spin-liquid phase. By contrast, solving the parquet equations via direct fixed-point iteration becomes increasingly difficult for low
Λ/J. We also scrutinize the pseudofermion constraint of single occupation per site, which is only fulfilled on average in pffRG, by explicitly computing fermion-number fluctuations. Although the latter are not entirely suppressed, we find that they do not affect the qualitative conclusions drawn from the spin susceptibility.