Exactly solvable Kondo lattice model in the anisotropic limit

In this paper we introduce an exactly solvable Kondo lattice model without any fine-tuning local gauge symmetry. This model describes itinerant electrons interplaying with a localized magnetic moment via only longitudinal Kondo exchange. Its solvability results from conservation of the localized moment at each site, and is valid for arbitrary lattice geometry and electron filling. A case study on a square lattice shows that the ground state is a Néel antiferromagnetic insulator at half-filling. At finite temperature, paramagnetic phases including a Mott insulator and correlated metal are found. The former is a melting antiferromagnetic insulator with a strong short-range magnetic fluctuation, while the latter corresponds to a Fermi liquidlike metal. Monte Carlo simulation and theoretical analysis demonstrate that the transition from paramagnetic phases into the antiferromagnetic insulator is a continuous two-dimensional Ising transition. Away from half-filling, patterns of spin stripes (inhomogeneous magnetic order) at weak coupling, and phase separation at strong coupling are predicted. With established Ising antiferromagnetism and spin stripe orders, our model may be relevant to a heavy fermion compound $\mathrm{CeCo}{\left({\mathrm{In}}_{1-x}{\mathrm{Hg}}_x\right)}_5$ and novel quantum liquid-crystal order in a hidden order compound ${\mathrm{URu}}_2{\mathrm{Si}}_2$.

Figure 1

(a) The Kondo lattice model in an anisotropic limit [Eq. (1)] and (b) its finite temperature phase diagram on a square lattice from LMC. There exist an antiferromagnetic insulator (AI), a Mott insulator (MI), and a correlated metal (CM). The low-$T$ regime of the phase diagram is dominated by AI. At high $T$, CM in weak coupling is a gapless metallic state while MI in strong coupling exhibits a single-particle excitation gap. The transition from AI to CM or MI is a continuous Ising transition. There only exists a smooth crossover between CM and MI with the opening of a gap at Fermi energy.

Figure 2

(a) The checkerboard order parameter ${\varphi }_c$, (b) SDW structure factor $C_{\mathrm{SDW}}$, (c) specific heat $C_v$, and (d) susceptibility ${\chi }_q$ versus temperature $T$ for different Kondo coupling $J$. Both ${\varphi }_c$ and $C_{\mathrm{SDW}}$ saturate at low $T$, suggesting the existence of AI. The vanishing of $C_{\mathrm{SDW}}$ at high $T$ signals a transition to paramagnetic phases at critical temperature $T_c$. $C_{\mathrm{v}}$ and ${\chi }_q$ also diverge at $T_c\simeq 0.11t,0.29t,0.25t$ for $J/t=2,8,14$, respectively.

Figure 3

The density of state for conduction electrons $N\left(\omega \right)$ in (a) CM ($J/t=8,T/t=0.4$), (b) MI ($J/t=14,T/t=0.4$), and (c) AI ($J/t=14,T/t=0.1$).

Figure 4

The magnetization of conduction electrons $m_c$, localized $f$-electron moment $m_f$, and the total one $m_z$ under external magnetic field $h$ for (a) CM ($J/t=8,T/t=0.4$), (b) MI ($J/t=14,T/t=0.4$), and (c) AI ($J/t=14,T/t=0.1$). (d) The conduction electrons' distribution function $n_c\left(k\right)$ along (0,0) to $(\pi ,\pi )$ direction ($T/t=0.4$).

Figure 5

The energy density $E$ and double occupation $d_c$ versus $J$ at different temperature $T$.

Figure 6

The finite-size scaling behavior of checkerboard order parameter ${\varphi }_c$ for (a) weak coupling ($J/t=2$), (b) intermediate coupling ($J/t=8$), and (c) strong coupling ($J/t=14$). The crossing of the different system size $L=8,10,14$ at $T_c$ and the collapse of data in (d) suggests the critical behavior of our model belongs to a 2D Ising universality class.

Figure 7

The spin stripe orders of the local $f$-electron moment ($\langle {\widehat{S}}_j^z\rangle$) occur in weak coupling regime $J/t=2$ for a different chemical potential $\mu$ ($T/t=0.01$). Here the value of the color bar denotes the strength of $\langle {\widehat{S}}_j^z\rangle$.