Phase diagram and dynamics of the $\mathrm{SU}\left(N\right)$ symmetric Kondo lattice model

In heavy-fermion systems, the competition between the local Kondo physics and intersite magnetic fluctuations results in unconventional quantum critical phenomena which are frequently addressed within the Kondo lattice model (KLM). Here we study this interplay in the $\mathrm{SU}\left(N\right)$ symmetric generalization of the two-dimensional half-filled KLM by quantum Monte Carlo simulations with $N$ up to 8. While the long-range antiferromagnetic (AF) order in $\mathrm{SU}\left(N\right)$ quantum spin systems typically gives way to spin-singlet ground states with spontaneously broken lattice symmetry, we find that the $\mathrm{SU}\left(N\right)$ KLM is unique in that for each finite $N$ its ground-state phase diagram hosts only two phases—AF order and the Kondo-screened phase. The absence of any intermediate phase between the $N=2$ and large-$N$ cases establishes adiabatic correspondence between both limits and confirms that the large-$N$ theory is a correct saddle point of the KLM fermionic path integral and a good starting point to include quantum fluctuations. In addition, we determine the evolution of the single-particle gap, quasiparticle residue of the doped hole at momentum $(\pi ,\pi )$, and spin gap across the magnetic order-disorder transition. Our results indicate that increasing $N$ modifies the behavior of the coherence temperature: while it evolves smoothly across the magnetic transition at $N=2$ it develops an abrupt jump—of up to an order of magnitude—at larger but finite $N$. We discuss the magnetic order-disorder transition from a quantum-field-theoretic perspective and comment on implications of our findings for the interpretation of experiments on quantum critical heavy-fermion compounds.

Figure 1

Ground-state phase diagram of the $\mathrm{SU}\left(N\right)$ KLM as a function of the inverse of the number of fermionic flavors $N$ and Kondo coupling $J$ with antiferromagnetic (AF) and Kondo insulating (KI) phases at half filling; empty squares indicate magnetic order-disorder transition points extracted from the behavior of the staggered moment Eq. (21) in the thermodynamic limit; solid line is a fit to a finite-$N$ functional form of the critical coupling $J_c\propto 1/\text{ln}N$ obtained by comparing the Kondo temperature $T_K$ to the magnetic energy scale $T_{\mathrm{RKKY}}$; see Appendix pp1. Color-coded circles correspond to the quasiparticle (QP) residue $Z_{\mathit{k}}$ of the doped hole at momentum $\mathit{k}=(\pi ,\pi )$, see Sec. 4b for raw data with error bars.

Figure 2

Staggered moment $m_{}^{\alpha }$ of the (a) $f$ electrons and (b) $c$ electrons as well as (c) the spin gap ${\mathrm{\Delta }}_s$ of the $\mathrm{SU}\left(N\right)$ KLM after extrapolation the QMC data to the thermodynamic limit; see Appendixes pp3-s2 and pp3-s3. For $N=6$ with $J/t<0.5$ and for $N=8$ for all values of $J$, we were not able to distinguish $m^c$ from zero.

Figure 3

(a) Expectation value of the Kondo interaction term ${\widehat{\mathcal{H}}}_J$ corresponding to the derivative of the free energy with respect to $J$ obtained on the $14\times 14$ lattice. The arrows indicate magnetic order-disorder transition points. Bottom panels show a close-up around the transition point: (b) $N=4$, (c) $N=6$, and (d) $N=8$.

Figure 4

(a) Single-particle gap ${\mathrm{\Delta }}_{qp}$ at momentum $\mathit{k}=(\pi ,\pi )$ and (b) the corresponding QP residue $Z_{(\pi ,\pi )}$ after extrapolation the QMC data to the thermodynamic limit; see Appendix pp3-s4. Insets show second-order polynomial fits to the QMC data in order to extract ${\mathrm{\Delta }}_{qp}$ and $Z_{(\pi ,\pi )}$ in the $N\to \infty$ limit; the extrapolated values ${\mathrm{\Delta }}_{qp}^{N\to \infty }=0.109\left(3\right)$ and $Z_{(\pi ,\pi )}^{N\to \infty }=0.0256\left(3\right)$ at $J/t=2$ and ${\mathrm{\Delta }}_{qp}^{N\to \infty }=0.051\left(2\right)$ and $Z_{(\pi ,\pi )}^{N\to \infty }=0.0125\left(2\right)$ at $J/t=1.6$ match well those obtained using the large-$N$ approximation (triangles).

Figure 5

Dynamical spin structure factor $S(\mathit{q},\omega )$ along a high-symmetry path in the Brillouin zone in the (a)–(c) AF phase at $J/t=0.8$ and (d)–(f) KI phase at $J/t=1.6$ obtained on the $14\times 14$ lattice with increasing $N$ (from left to right). In panels (a)–(c), the particle-hole continuum threshold (dashed) lies above the Goldstone modes ${\omega }_{\mathrm{ph}}/t\simeq 0.68$, 0.32, and 0.22, respectively, while in panels (d)–(f) it drops from ${\omega }_{\mathrm{ph}}/t\simeq 1$ through 0.4 to 0.28.

Figure 6

Single-particle spectral function $A(\mathit{k},\omega )$ of the conduction electrons in the (a)–(c) AF phase at $J/t=0.8$ and (d)–(f) KI phase at $J/t=1.6$ obtained on the $14\times 14$ lattice with increasing $N$ (from left to right).

Figure 7

Single-particle spectral function $A(\mathit{k},\omega )$ in the KI phase at $J/t=1.6$ from (a) QMC simulations with $N=8$ and (b) the large-$N$ approach.

Figure 8

VBS correlation function $S_{\mathrm{VBS}}^f\left(\mathit{q}\right)={\sum }_{\delta }{\left[S_{\mathrm{VBS}}^f\left(\mathit{q}\right)\right]}_{\mathit{\delta },\mathit{\delta }}$ for the $f$ electrons for various lattice sizes measured across the magnetic order-disorder transition point in the (a),(b) AF and (c),(d) KI phases for $N=2$ (left) and $N=4$ (right).

Figure 9

Same as in Fig. 8 but for $N=6$ (left) and $N=8$ (right).

Figure 10

(a),(d),(g),(j) Convergence of the spin structure factor $S^f[\mathit{Q}=(\pi ,\pi )]$ for the $f$ electrons at representative values of $J/t$ as a function of the projection parameter $\mathrm{\Theta }t$ for the $L=12$ system. Panels (b), (e), (h), (k) and (c), (f), (i), (l) show finite-size extrapolation of the spin structure factor for the $f$ and $c$ electrons, respectively, on approaching the magnetic order-disorder transition point; solid lines are linear and second-order polynomial fits to the QMC data. From top to bottom: $N=2,4,6,\text{and}8$.

Figure 11

Finite-size extrapolation of the (a),(d),(g),(j) spin gap ${\mathrm{\Delta }}_s$, (b),(e),(h),(k) single-particle gap ${\mathrm{\Delta }}_{qp}$ at momentum $\mathit{k}=(\pi ,\pi )$, and (c),(f),(i),(l) QP residue $Z_{(\pi ,\pi )}$ at representative values of $J/t$. Solid lines are linear fits to the QMC data. From top to bottom: $N=2,4,6,\text{and}8$.

Figure 12

Imaginary-time Green's function for conduction electrons at momentum $\mathit{k}=(\pi ,\pi )$ obtained from QMC simulations for a given linear system size $L$ at $J/t=0.6$ for (a) $N=4$, (b) $N=6$, and (c) $N=8$. Finite-size estimates of the single-particle gap ${\mathrm{\Delta }}_{qp}\left(\mathit{k}\right)$ and QP residue $Z_{\mathit{k}}$ are extracted by fitting the tail of the Green's function to the form $Z_{\mathit{k}}{e}^{-{\mathrm{\Delta }}_{qp}\left(\mathit{k}\right)\tau }$ (solid lines).

Figure 13

VBS correlation function $S_{\mathrm{VBS}}^f\left(\mathit{q}\right)={\sum }_{\delta }{\left[S_{\mathrm{VBS}}^f\left(\mathit{q}\right)\right]}_{\mathit{\delta },\mathit{\delta }}$ for the $f$ electrons for various lattice sizes along a high-symmetry path in the Brillouin zone across the magnetic order-disorder transition point in the (a)–(c) AF and (d)–(f) KI phases for increasing $N$ (from left to right). Here, a twice smaller imaginary-time step $\mathrm{\Delta }\tau t=0.05$ in the Trotter-Suzuki decomposition in Eq. (9) is used as compared to Figs. 8 and 9.