Local quantum criticality of a one-dimensional Kondo insulator model

The continuous quantum phase transition and the nature of the quantum critical point (QCP) in a modified Kondo lattice model with Ising anisotropic exchange interactions are studied within the density-matrix renormalization group algorithm. We investigate the effect of quantum fluctuations on critical Kondo destruction QCP by probing static and dynamic properties of the magnetic order and the Kondo effect. In particular, we identify that local Kondo physics itself becomes critical at the magnetic phase transition point, providing unbiased evidences for local quantum criticality between two insulators without involving the Fermi surface.

Figure 1

(Top) One-dimensional Kondo lattice model with an Ising-type interaction between nearest-neighbor localized spins. Red dots and blue squares represent the conduction electrons and localized spins, respectively. (Bottom) The global phase diagram as a function of $J_z$ and $J_K$ by setting $t=0.25$ (bandwidth of conduction electron is $4t=1$). The phase transition is determined to be continuous (see main text).

Figure 2

(a) Energy spectrum evolution as a function of $J_z$, obtained on an $L=8$ periodic chain by ED calculations. (b) Energy gaps (${\mathrm{\Delta }}_S$, ${\mathrm{\Delta }}_N$, ${\mathrm{\Delta }}_C$ defined in main text) as a function of $J_z$, obtained on $L=36$ open chain by DMRG calculations. Here we set $J_K=1.0$.

Figure 3

(a) Log-linear plot of neutral gap ${\mathrm{\Delta }}_N$ near the critical point $J_z^c\approx 1.825$ by setting $J_K=1.0$. Various system sizes are labeled by different symbols. (b) Neutral gap ${\mathrm{\Delta }}_N$ as a function of $J_K$ by setting $J_z=J_z^c=1.825$. Inset: Log-linear plot of ${\mathrm{\Delta }}_N$ and exponential fitting.

Figure 4

Antiferromagnetic order parameter $m_{AF}=\frac{1}{L}{\sum }_i\left|\langle S_i^z\rangle \right|$ (black triangular) and spin-density-wave order parameter $\mathrm{\Delta }{n}_{\mathrm{SDW}}=\frac{1}{L}{\sum }_i\left|\langle s_i^z\rangle \right|$ (black diamond), and inverse static spin susceptibility $1/\chi (Q=\pi ,\omega =0)$ (red cross) as a function of $J_z$. Blue dashed line marks the transition point $J_z^c\approx 1.825$.

Figure 5

Frequency dependence of the local spin susceptibility at various values of $J_z$ around the magnetic transition: (left panel) imaginary part and (right panel) real part. Here we choose $J_K=1.0$.

Figure 6

Semilog plot of the real part of the local spin susceptibility around the magnetic transition: (a) $J_z>J_z^c$ and (b) $J_z<J_z^c$. (c) Inverse of the real part of the local spin susceptibility $\Re {\chi }^{-1}(\omega =0)$ versus $\eta$. Red line shows the polynomial function fitting: $\Re {\chi }^{-1}(\omega =0)=A{\eta }^2+B\eta +C$, with nonzero $A$, $B$, and $C=-0.0013\pm 0.002$.

Figure 7

(a) Local electron spectrum density as a function of $\omega$ for $J_z=1.65$ (blue) and $J_z=1.95$ (red). Solid and dotted line represents spin-down and spin-up component, respectively. (b) In the Ising AFM phase, the $J_K$ dependence of magnetization $m_{AF}$ and charge polarization $\mathrm{\Delta }{n}_{\mathrm{SDW}}$. The parameter $J_z$ is set to be $J_z=J_K+1$.