Fermi surface topology of the two-dimensional Kondo lattice model: Dynamical cluster approximation approach

We report the results of extensive dynamical cluster approximation calculations, based on a quantum Monte Carlo solver, for the two-dimensional Kondo lattice model. Our particular cluster implementation renders possible the simulation of spontaneous antiferromagnetic symmetry breaking. By explicitly computing the single-particle spectral function both in the paramagnetic and antiferromagnetic phases, we follow the evolution of the Fermi surface across this magnetic transition. The results, computed for clusters up to 16 orbitals, show clear evidence for the existence of three distinct Fermi surface topologies. The transition from the paramagnetic metallic phase to the antiferromagnetic metal is continuous; Kondo screening does not break down and we observe a backfolding of the paramagnetic heavy-fermion band. Within the antiferromagnetic phase and when the ordered moment becomes large the Fermi surface evolves to one which is adiabatically connected to a Fermi surface where the local moments are frozen in an antiferromagnetic order.

Figure 1

(Color online) (a) Definition of the antiferromagnetic unit cell and (b) DCA patching of the Brillouin zone. The lattice vectors ${\mathbf{a}}_1=\left(1,1\right)$ and ${\mathbf{a}}_2=\left(1,-1\right)$ connect the AF unit cells. The two inequivalent $c$ orbitals are denoted by filled (empty) circles and the localized $f$ orbitals with arrows. The reciprocal-space lattice vectors ${\mathbf{b}}_1$ and ${\mathbf{b}}_2$ span the magnetic Brillouin zone. The MBZ is uniformly discretized in $N_p=4$ patches, corresponding to real-space clusters each containing $N_p$ magnetic unit cells. The color coding refers to constant momentum dependency of the self-energy and cluster Green’s function. The dashed square denotes the extended Brillouin zone.

Figure 2

The staggered magnetization $m_s^f$ of the local moment spins in the ground state as a function of coupling $J/t$ with next-nearest-neighbor hopping $t^{\prime }/t=-0.3$ and $t^{\prime }/t=0$, respectively. The low-temperature limit is reached by performing simulations at various inverse temperatures $\beta$. Convergence to the ground state has been achieved for the following inverse temperatures: $\beta t=20$ $\left(2.5\ge J/t\ge 1.7\right)$, $\beta t=40$ $\left(1.6\ge J/t\ge 1.1\right)$, $\beta t=80$ $\left(J/t=1\right)$, and $\beta t=100$ $\left(J/t=0.8\right)$.

Figure 3

(Color online) Single-particle spectral function $A^{cc}\left(\mathbf{k},\omega \right)$ at half filling $\left(n_c=1\right)$ and with particle-hole symmetry $\left(t^{\prime }/t=0\right)$ close to the magnetic phase transition on the paramagnetic side, $J/t=2.2$.

Figure 4

(Color online) Single-particle spectral function $A^{cc}\left(\mathbf{k},\omega \right)$ at half filling $\left(n_c=1\right)$ and with particle-hole symmetry $\left(t^{\prime }/t=0\right)$ close to the magnetic phase transition on the AF side, (a) $J/t=2.0$ and (b) closeup. The onset of magnetic order and concomitant emergence of the MBZ is signaled by the appearance of shadow bands. The staggered magnetization for this point is measured to be $m_s^f=0.244\pm 0.001$.

Figure 5

(Color online) Comparison of the DCA single-particle spectral function $A^{cc}\left(\mathbf{k},\omega \right)$ with the $T=0$ spectral function. Data (a) taken from Ref. 39 which used a projective auxiliary field Monte Carlo (BSS) method for the KLM on a $12\times 12$ lattice and our DCA results (b) at $\beta t=40$, both for the same parameter set—$t^{\prime }/t=0$, $n_c=1$, and $J/t=1.2$.

Figure 6

(Color online) Single-particle spectral functions $A^{cc}\left(\mathbf{k},\omega \right)$ at half filling $\left(n_c=1\right)$ with $t^{\prime }/t=-0.3$. As the Kondo coupling $J/t$ is decreased, the spectrum changes from a paramagnetic Kondo insulator with indirect gap (a) to an antiferromagnetic ordered state (d).

Figure 7

(Color online) Closeup for low energies around the Fermi energy (given by $\omega /t=0$). Development of the single-particle spectral function $A^{cc}\left(\mathbf{k},\omega \right)$ at half filling with $t^{\prime }/t=-0.3$ which shows a shift of the maximum in the lower band from $\mathbf{k}=\left(\pi ,\pi \right)$ to $\mathbf{k}=\left(\pi /2,\pi /2\right)$ with decreasing $J/t$.

Figure 8

(Color online) Schematic representation of the position of the upper band minima (top row) and lower band maxima (bottom row) of the conduction electron single-particle spectral function with next-nearest-neighbor hopping $t^{\prime }/t=-0.3$ and for half filling. The squares represent the first Brillouin zone (for the PM phase) or the extended Brillouin zone (for the AF phase), with the bottom left corner and top right corner of each square given by $\mathbf{k}=\left(-\pi ,-\pi \right)$ and $\mathbf{k}=\left(\pi ,\pi \right)$, respectively.

Figure 9

The quasiparticle gap ${\Delta }_{qp}/t$ as a function of Kondo coupling $J/t$. At $t^{\prime }=-0.3t$, the magnetic order-disorder transition takes place at $J_c^{\prime }/t\simeq 1.85$. In the coupling range $0.8<J/t<J_c^{\prime }/t$ Kondo screening coexists with magnetism and is at the origin of the quasiparticle gap.

Figure 10

The staggered magnetization $m_s^f$ of the $f$ electrons as a function of $n_c$ at different constant couplings $J/t$.

Figure 11

(Color online) Ground-state magnetic phase diagram of the hole-doped KLM showing simulation results for the staggered magnetization $m_s^f$ (color coded) as a function of coupling $J/t$ and conduction electron occupancy $n_c$. Triangles: PM region, large FS. Squares: AF, large FS. Circles: AF, small FS. Here $t^{\prime }/t=-0.3$ and the calculations are carried out with the $N_p=1$ cluster. Below, the FS topologies corresponding to the numbered regions are shown schematically.

Figure 12

(Color online) Single-particle spectra $A^{cc}\left(\mathbf{k},\omega \right)$ of the conduction electrons for $J/t=1$ and $t^{\prime }/t=-0.3$ at $\beta t=80$. The conduction electron density is reduced progressively from half filling (a) $n_c=1$ to (h) $n_c=0.880$. The respective FS topologies are shown in Fig. 11.

Figure 13

(Color online) Single-particle spectra $A^{cc}\left(\mathbf{k},\omega \right)$ of the conduction electrons for $J/t=1.4$ and $t^{\prime }/t=-0.3$ at $\beta t=40$. The shadow features present in (a) vanish as the occupation number $n_c$ is reduced.

Figure 14

(Color online) Single-particle spectrum $A^{cc}\left(\mathbf{k},\omega \right)$ of the conduction electrons obtained using cluster size $N_p=4$ confirming that the small Fermi surface topology in the lightly doped AF phase with coupling $J/t=1$ is not an artifact of the smaller cluster results. The staggered magnetization of the $f$ electrons is $m_s^f=0.657\pm 0.005$.

Figure 15

Mean-field band structures $E_n\left(\mathbf{k}\right)$ with $J/t=1$ $\left(t^{\prime }/t=-0.3\right)$. (a) $V=0.5,m_s^c=0.066,m_s^f=0.257$, $n_c=0.898$. The line intensity is proportional to the quasiparticle spectral weight, $A_n\left(\mathbf{k},\omega \right)$. The Fermi surface topology [see closeup (b)] results from backfolding of the large-$N$ mean-field bands. This corresponds to the Fermi surface (2) in Fig. 11. (d) $V=0.260,m_s^c=0.190,m_s^f=0.693$, $n_c=0.975$. The Fermi surface topology [see closeup (c)] corresponds to that of hole pockets centered around $\mathbf{k}=\left(\pm \pi /2,\pm \pi /2\right)$ as depicted by the Fermi surface (3) in Fig. 11.

Figure 16

(Color online) Topology change (a) in the Fermi surface as obtained from the mean-field modeling $E_n\left(\mathbf{k}\right)$ of the DCA results ($J/t=1$, $t^{\prime }/t=-0.3$ $m_s^c=0.066$, $m_s^f=0.257$, and $n_c=0.898$). (b) As apparent on the scale of the closeup, hole pockets around $\mathbf{k}=\left(\pm \pi /2,\pm \pi /2\right)$ as well as around $\mathbf{k}=\left(\pi ,\pi \right)$ are present at $V=0.314$.