Continuous ferromagnetic quantum phase transition on an anisotropic Kondo lattice

Motivated by the recent discovery of ferromagnetic quantum criticality in the heavy fermion compound ${\mathrm{CeRh}}_6{\mathrm{Ge}}_4$, we develop a numerical algorithm of infinite projected entangled pair states for the anisotropic ferromagnetic Kondo-Heisenberg model in two dimensions and study the ferromagnetic quantum phase transitions with varying magnetic and hopping anisotropy. Our calculations reveal a continuous ferromagnetic quantum phase transition in the large anisotropic region and first-order quantum phase transitions for smaller anisotropy. Our results highlight the importance of magnetic anisotropy on ferromagnetic quantum criticality in Kondo lattice systems and provide a possible explanation for the experimental observation in ${\mathrm{CeRh}}_6{\mathrm{Ge}}_4$ with a quasi-one-dimensional magnetic structure. Our work opens the avenue for future studies of the rich Kondo lattice physics using state-of-the-art tensor network approaches.

Figure 1

(a) Illustration of the anisotropic FM Kondo-Heisenberg model studied in this paper. The orange arrows represent local spins and the blue droplets with white arrows denote conduction electrons. The reduced width of the solid lines along the $x$ direction reflects the reduced strength of the Heisenberg exchange interaction and the hopping parameter due to lattice anisotropy. (b) Illustration of the $2\times 2$ unit cell of the iPEPS ansatz with four bulk tensors: A, B, C, D. Each has one physical bond ($p$) and four virtual bonds ($l$, $u$, $r$, $d$) as noted for B. The yellow diamonds represent the four-leg swap gates $S$ used for the fermionization of iPEPS, which is introduced at the crossing point of two legs [33, 34].

Figure 2

Comparison of the FM QPTs in the isotropic ($\alpha =1$) and quasi-1D ($\alpha =0.01$) limits based on the $J_{\mathrm{K}}$ dependence of (a), (b) $m_{\mathrm{FM}}^2$, the FM order parameter defined in Eq. (2), (c), (d) $E_{\mathrm{g}}$, the ground state energy per site, (e), (f) the derivative $\partial E_{\mathrm{g}}/\partial J_{\mathrm{K}}=\langle {\mathbf{s}}_i·{\mathbf{S}}_i\rangle$, and (g), (h) the second derivative ${\partial }^2{E}_{\mathrm{g}}/\partial J_{\mathrm{K}}^2$. The red solid and blue dashed lines represent two different routes of calculations in the hysteresis strategy with gradually increasing (IK) and decreasing (DK) $J_{\mathrm{K}}$, respectively. The yellow solid or dotted lines are DMRG results on the 1D Kondo chain for comparison. The purple and cyan dashed-dotted vertical lines mark the continuous or first-order (energy crossing) transition point, respectively. The virtual bond dimension of iPEPS is set to $D=12$.

Figure 3

Evolution of $m_{\mathrm{FM}}^2\text{−}{J}_{\mathrm{K}}$ curves for (a) $\alpha =0.01$ and (b) $\alpha =1.0$ with varying iPEPS virtual bond dimensions $D=8$, 10, 12. The DMRG results (black dashed-dotted line) on the Kondo chain are also shown in (a) for comparison. The inset of (b) plots the extrapolated lower (upper) hysteresis boundary $J_{\mathrm{K}}^{\mathrm{L}}$ ($J_{\mathrm{K}}^{\mathrm{U}}$) as $D^{-6}$ for $\alpha =1.0$.

Figure 4

Comparison of $m_{\mathrm{FM}}^2\text{−}{J}_{\mathrm{K}}$ curves for $\alpha =1.0$, 0.4, 0.15, 0.05, 0.03, 0.01. The red solid and blue dashed lines represent the results for increasing (IK) and decreasing (DK) $J_{\mathrm{K}}$, respectively. The virtual bond dimension is set to $D=10$ to reduce the computational cost. The dashed-dotted vertical lines mark the continuous or first-order (energy crossing) transition point for each $\alpha$, respectively.

Figure 5

The overall phase diagram on the $\alpha \text{−}{J}_{\mathrm{K}}$ plane for the anisotropic FM Kondo lattice model. The purple solid line represents the continuous (second-order) QPT for large anisotropy. The shaded area marks the hysteresis region for the first-order QPT. The blue and red dashed lines denote its boundaries and the cyan solid line marks the energy crossing point of two coexisting phases. The inset shows an enlarged view for $\alpha \le 0.2$.