Zooming in on heavy fermions in Kondo lattice models

Resolving the heavy fermion band in the conduction electron momentum resolved spectral function of the Kondo lattice model is challenging since, in the weak coupling limit, its spectral weight is exponentially small. In this article we consider a composite fermion operator, consisting of a conduction electron dressed by spin fluctuations that shares the same quantum numbers as the electron operator. Using approximation free auxiliary field quantum Monte Carlo simulations we show that, for the SU(2) spin-symmetric model on the square lattice at half filling, the composite fermion acts as a magnifying glass for the heavy fermion band. In comparison to the conduction electron residue that scales as $e^{-W/J_k}$ with $W$ the bandwidth and $J_k$ the Kondo coupling, the residue of the composite fermion tracks $J_k$. This result holds down to $J_k/W=0.05$ and confirms the point of view that magnetic ordering, present below $J_k/W=0.18$, does not destroy the heavy quasiparticle. We furthermore investigate the spectral function of the composite fermion in the ground state and at finite temperatures, for $\mathrm{SU}\left(N\right)$ generalizations of the Kondo lattice model, as well as for ferromagnetic Kondo couplings, and compare our results to analytical calculations in the limit of high temperatures, large-$N$, large-$S$, and large $J_k$. Based on these calculations, we conjecture that the composite fermion operator provides a unique tool to study the destruction of the heavy fermion quasiparticle in Kondo breakdown transitions. The relation of our results to scanning tunneling spectroscopy and photoemission experiments is discussed.

Figure 1

(a) Sketch of the square KLM with the nearest neighbor hopping $t$ and Kondo exchange coupling $J_k$: the conduction and localized orbitals are indicated by red and blue dots. (b) First Brillouin zone and its high symmetry points: $\mathrm{\Gamma }=(0,0)$, $X=(\pi ,0)$, and $M=(\pi ,\pi )$.

Figure 2

Conjectured ground state phase diagram for a square KLM at half filling. For $J_k/t>0$, a quantum critical point (red dot) separates the Kondo-screened and antiferromagnetically ordered phases. In the latter, the composite fermion spectral function is consistent with the coexistence of Kondo screening and the RKKY interaction; for $J_k/t<0$, the RKKY interaction is the only relevant energy scale.

Figure 3

Spectral weight $A_f(\mathit{k},\omega )$ as a function of momentum $\mathit{k}$ and energy $\omega /t$ for $J_k/t=2$ as obtained in the large-$N$ limit. The line thickness reflects the $f$ character of the band.

Figure 4

Composite fermion $A_{\psi }(\mathit{k},\omega )$ (left) and conduction electron $A_c(\mathit{k},\omega )$ (right) spectral functions in the $12\times 12$ KLM at $\beta t=36$ with (a),(b) $J_k/t=2$, (c),(d) $J_k/t=1.6$, (e),(f) $J_k/t=1.2$, and (g),(h) $J_k/t=1$.

Figure 5

Static spin structure factor $S\left(\mathit{Q}\right)$ at $\mathit{Q}=(\pi ,\pi )$ for the localized spins as a function of temperature $T/t$.

Figure 6

Emergence of the heavy fermion band structure as seen in $A_{\psi }(\mathit{k},\omega )$ (left) and $A_c(\mathit{k},\omega )$ (right) upon decreasing temperature in the $12\times 12$ KLM at $J_k/t=1$: (a),(b) $\beta t=2$, (c),(d) $\beta t=6$, (e),(f) $\beta t=12$, and (g),(h) $\beta t=24$.

Figure 7

(a) Local spin-spin correlation function $S^{cf}=\frac{1}{N_u}{\sum }_{\mathit{i}}\langle {\widehat{\mathit{S}}}_{\mathit{i}}^c·{\widehat{\mathit{S}}}_{\mathit{i}}^{}\rangle$ and (b) estimate of the local density of states at the Fermi level $A_{\psi }(\omega =0)\approx \frac{1}{\pi }\frac{\beta }{N_u}{\sum }_{\mathit{k}}{G}_{\psi }(\mathit{k},\tau =\beta /2)$ as a function of temperature $T/t$ and for various Kondo couplings $J_k/t$. The formation of local singlets seen as a decrease in $S^{cf}$ induces a depletion of the spectral weight and opens a gap in $A_{\psi }(\omega =0)$ in the low-$T$ limit.

Figure 8

Temperature evolution of the momentum integrated composite fermion spectral function $A_{\psi }\left(\omega \right)=\frac{1}{N_u}{\sum }_{\mathit{k}}{A}_{\psi }(\mathit{k},\omega )$ at $J_k/t=1$.

Figure 9

Composite fermion $A_{\psi }(\mathit{k},\omega )$ (left) and conduction electron $A_c(\mathit{k},\omega )$ (right) spectral functions at $T=0$ of the $12\times 12\mathrm{SU}(N$) KLM in the Kondo-screened phase at $J_k/t=2$: (a),(b) $N=2$, (c),(d) $N=4$, (e),(f) $N=6$, and (g),(h) $N=8$.

Figure 10

Same as in Fig. 9 but in the magnetically ordered phase at $J_k/t=0.4$.

Figure 11

Zero temperature composite fermion $G_{\psi }(\mathit{k},\tau )$ and conduction electron $G_c(\mathit{k},\tau )$ Green's functions at $\mathit{k}=(\pi ,\pi )$ in the $\mathrm{SU}(N$) KLM with $J_k/t=0.4$. Both $G_{\psi }(\mathit{k},\tau )$ and $G_c(\mathit{k},\tau )$ follow at large values of $\tau t$ an exponential law $Z{e}^{-{\mathrm{\Delta }}_{qp}\tau }$ (solid lines), thus indicating the existence of the heavy fermion band in their respective spectral functions. The latter have identical supports set by ${\mathrm{\Delta }}_{qp}$ but differ, by nearly three orders of magnitude, in the quasiparticle weights $Z_{\mathit{k}}^{\psi }$ and $Z_{\mathit{k}}^c$.

Figure 12

(a) Quasiparticle residue $Z_{(\pi ,\pi )}^{\psi }$ and (b) single particle gap ${\mathrm{\Delta }}_{qp}$ of the composite fermion spectral function obtained from $T=0$ QMC simulations of the $\mathrm{SU}(N$) KLM. For comparison we also show $Z_{(\pi ,\pi )}^{\psi }$ predicted by the large-$N$ approximation (dashed line). Inset shows a second-order polynomial fit to the QMC data in order to extract $Z_{(\pi ,\pi )}^{\psi }$ in the $N\to \infty$ limit; the extrapolated values $Z_{(\pi ,\pi )}^{\psi }(N\to \infty )=0.463\left(4\right)$ at $J_k/t=2$ and $Z_{(\pi ,\pi )}^{\psi }(N\to \infty )=0.345\left(1\right)$ at $J_k/t=1.6$ match well those obtained using the large-$N$ approximation (triangles).

Figure 13

Temperature dependence of $A_{\psi }(\mathit{k},\omega )$ (left) and $A_c(\mathit{k},\omega )$ (right) in the $12\times 12$ KLM with ferromagnetic coupling $J_k/t=-3$: (a),(b) $\beta t=6$, (c),(d) $\beta t=12$, (e),(f) $\beta t=24$, and (g),(h) $\beta t=36$.

Figure 14

Static spin structure factor $S\left(\mathit{Q}\right)$ at $\mathit{Q}=(\pi ,\pi )$ for the localized spins as a function of temperature $T/t$ in the ferromagnetic KLM.