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arXiv:2106.12601 · Physical review. B./Physical review. B

Dynamical exponent of a quantum critical itinerant ferromagnet: A Monte Carlo study

Yuzhi Liu, Weilun Jiang, Avraham Klein, Yuxuan Wang, Kai Sun, Andrey V. Chubukov, Zi Yang Meng

Published Jan 24, 2022• 17 citations
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Abstract

We consider the effect of the coupling between two-dimensional (2D) quantum rotors near an XY ferromagnetic quantum critical point and spins of itinerant fermions. We analyze how this coupling affects the dynamics of rotors and the self-energy of fermions. A common belief is that near a $q=0$ ferromagnetic transition, fermions induce an $\mathrm{\ensuremath{\Omega}}/q$ Landau damping of rotors (i.e., the dynamical critical exponent is $z=3$) and Landau overdamped rotors give rise to non-Fermi liquid fermionic self-energy $\mathrm{\ensuremath{\Sigma}}\ensuremath{\propto}{\ensuremath{\omega}}^{2/3}$. This behavior has been confirmed in previous quantum Monte Carlo (QMC) studies. Here we show that for the XY case the behavior is different. We report the results of large-scale quantum Monte Carlo simulations, which show that at small frequencies $z=2$ and $\mathrm{\ensuremath{\Sigma}}\ensuremath{\propto}{\ensuremath{\omega}}^{1/2}$. We argue that the new behavior is associated with the fact that a fermionic spin is by itself not a conserved quantity due to spin-spin coupling to rotors, and a combination of self-energy and vertex corrections replaces $1/q$ in the Landau damping by a constant. We discuss the implication of these results to experiments.

DOI
10.1103/physrevb.105.l041111

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We consider the effect of the coupling between two-dimensional (2D) quantum rotors near an XY ferromagnetic quantum critical point and spins of itinerant fermions. We analyze how this coupling affects the dynamics of rotors and the self-energy of fermions. A common belief is that near a q=0q=0q=0 ferromagnetic transition, fermions induce an \ensuremathΩ/q\mathrm{\ensuremath{\Omega}}/q\ensuremathΩ/q Landau damping of rotors (i.e., the dynamical critical exponent is z=3z=3z=3) and Landau overdamped rotors give rise to non-Fermi liquid fermionic self-energy \ensuremathΣ\ensuremath∝\ensuremathω2/3\mathrm{\ensuremath{\Sigma}}\ensuremath{\propto}{\ensuremath{\omega}}^{2/3}\ensuremathΣ\ensuremath∝\ensuremathω2/3. This behavior has been confirmed in previous quantum Monte Carlo (QMC) studies. Here we show that for the XY case the behavior is different. We report the results of large-scale quantum Monte Carlo simulations, which show that at small frequencies z=2z=2z=2 and \ensuremathΣ\ensuremath∝\ensuremathω1/2\mathrm{\ensuremath{\Sigma}}\ensuremath{\propto}{\ensuremath{\omega}}^{1/2}\ensuremathΣ\ensuremath∝\ensuremathω1/2. We argue that the new behavior is associated with the fact that a fermionic spin is by itself not a conserved quantity due to spin-spin coupling to rotors, and a combination of self-energy and vertex corrections replaces 1/q1/q1/q in the Landau damping by a constant. We discuss the implication of these results to experiments.
We analyze the behavior of interacting fermions near a ferromagnetic Stoner instability. We show that the Landau damping of the spin susceptibility is a relevant perturbation near a ferromagnetic quantum-critical point (FQCP). We argue that, as the system approaches FQCP, the fermionic self-energy crosses over from predominantly momentum dependent away from the transition to predominantly frequency dependent in the immediate vicinity of the transition. We argue that due to this self-generated locality, the quasiparticle effective mass does not diverge before FQCP.
We study the role of finite-size effects on critical behavior near a metallic q=0q=0q=0 critical point and compare the results with a recent extensive quantum Monte Carlo (QMC) study [Y. Schattner et al., Phys. Rev. X 6, 031028 (2016)]. This study found several features in both bosonic and fermionic responses, in disagreement with the expected critical behavior with a dynamical exponent z=3z=3z=3. We show that finite-size effects are particularly strong for z=3z=3z=3 criticality and give rise to a behavior different from that of an infinite system, over a wide range of momenta and frequencies. We argue that by taking finite-size effects into account, the QMC results can be explained within z=3z=3z=3 theory.
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We consider spin and electronic properties of itinerant electron systems, described by the spin-fermion model, near the antiferromagnetic critical point. We expand in the inverse number of hot spots in the Brillouin zone, NNN, and present the results beyond the previously studied $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{\infty}$ limit. We found two new effects: (i) Fermi surface becomes nested at hot spots, and (ii) vertex corrections give rise to anomalous spin dynamics and change the dynamical critical exponent from z=2z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2z=2 to z>2z>2z>2. To first order in 1/N1/N1/N we found $z\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2N/(N\ensuremath{-}2)$ which for a physical N=8N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}8N=8 yields $z\ensuremath{\approx}2.67$.
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We analyze the form of the fermionic propagator for two-dimensional fermions interacting with massless overdamped bosons. Examples include a nematic and Ising ferromagnetic quantum-critical points and fermions at a half-filled Landau level. Fermi-liquid behavior in these systems is broken at criticality by a singular self-energy, but the Fermi surface remains well defined. These are strong-coupling problems with no expansion parameter other than the number of fermionic species, NNN. The two known limits, N⪢1N⪢1N⪢1 and N=0N=0N=0, show qualitatively different behavior of the fermionic propagator $G({ϵ}_{k},\ensuremath{\omega})$. In the first limit, $G({ϵ}_{k},\ensuremath{\omega})$ has a pole at some ϵk{ϵ}_{k}ϵk​; in the other it is analytic. We analyze the crossover between the two limits. We show that the pole survives for all NNN, but at small NNN it only exists in a range O(N2)O({N}^{2})O(N2) near the mass shell. At larger distances from the mass shell, the system evolves and $G({ϵ}_{k},\ensuremath{\omega})$ becomes regular. At N=0N=0N=0, the range where the pole exists collapses and $G({ϵ}_{k},\ensuremath{\omega})$ becomes regular everywhere.