Dynamical correlations in the spin-half two-channel Kondo model

Dynamical correlations of various local operators are studied in the spin-half two-channel Kondo (2CK) model in the presence of channel anisotropy or external magnetic field. A conformal field theory–based scaling approach is used to predict the analytic properties of various spectral functions in the vicinity of the two-channel Kondo fixed point. These analytical results compare well with highly accurate density-matrix numerical renormalization-group results. The universal crossover functions interpolating between channel-anisotropy- or magnetic-field-induced Fermi-liquid regimes and the two-channel Kondo, non-Fermi-liquid regimes are determined numerically. The boundaries of the real 2CK scaling regime are found to be rather restricted and to depend both on the type of the perturbation and on the specific operator whose correlation function is studied. In a small magnetic field, a universal resonance is observed in the local fermion’s spectral function. The dominant superconducting instability appears in the composite superconducting channel.

Figure 1

(Color online) Spectral function ${\varrho }_F$ of the composite fermion operator $F_{0,1,\uparrow }$ as a function of $\omega$, and the definitions of the scales $T_K$ and $T^{\ast }$. $T_K$ is defined by the relation ${\varrho }_F\left(\omega =T_K,T=0,K_R=0\right)\equiv \frac{1}{2}{\varrho }_F\left(\omega =0,T=0,K_R=0\right)$. For nonzero $K_R$ the scale $T^{\ast }$ is defined through ${\varrho }_F\left(\omega =T^{\ast },T=0,K_R\right)\equiv \frac{3}{4}{\varrho }_F\left(\omega =0,T=0,\left|K_R\right|\right)$.

Figure 2

(Color online) (Top) Sketch of the dimensionless spectral function ${\hat{\varrho }}_f=D_F{\varrho }_f$ of $f_{0,1,\sigma }^{†}$, and (bottom) the real part of its dimensionless Green’s function, $\text{Re}{\widehat{g}}_f=D_F\text{Re}{\mathcal{G}}_f$, for $T>0$ and $K_R=0$ and $B=0$ as a function of $\text{log}\left(\omega /T_K\right)$. Asymptotics indicated for $\omega <T_K$ were derived through scaling arguments. The large-$\omega$ behavior is a result of perturbation theory. The features of the spectral functions at $\omega \sim D_F$ are nonuniversal and depend on the realization of the model.

Figure 3

(Color online) (Top) Sketch of the dimensionless spectral function of $f_{0,1,\sigma }^{†}$, ${\hat{\varrho }}_f=D_F{\varrho }_f$, and (bottom) the real part of its dimensionless Green’s function, $\text{Re}{\widehat{g}}_f=D_F\text{Re}{\mathcal{G}}_f$, for $T=0$, $K_R>0$, and $B=0$ as a function of $\text{log}\left(\omega /T_K\right)$. Asymptotics indicated for $\omega <T_K$ were derived through scaling arguments. The large-$\omega$ behavior is a result of perturbation theory. The features of the spectral functions for $\omega \sim D_F$ are nonuniversal and depend on the realization of the model.

Figure 4

(Color online) (a) Dimensionless spectral function of $f_{0,1,\sigma }$, ${\hat{\varrho }}_f\left(\omega \right)=D_F{\varrho }_f\left(\omega \right)$, as a function of $\omega /T_K$ for different values of $K_R$. [(b)–(e)] Numerical confirmations of the low-frequency asymptotics derived through scaling arguments in Sec. 5. Dashed straight lines are for demonstrating deviations from the expected [(b) and (c)] $\sqrt{\omega }$-like, (d) $1/\sqrt{\omega }$-like, and (e) ${\omega }^2$-like behaviors. $T^{\ast }/T_K=2.4\times {10}^{-4}$ in plots (c)–(e).

Figure 5

(Color online) Universal collapse of the dimensionless spectral functions ${\hat{\varrho }}_f=D_F{\varrho }_f$ (with $f$ in channel 1) to two scaling curves ${\mathcal{K}}_f^{\pm }$ as a function of $\omega /T^{\ast }$ for (a) negative and (b) positive values of $K_R$.

Figure 6

(Color online) Real part of the dimensionless Green’s function, $\text{Re}{\widehat{g}}_f=D_F\text{Re}{\mathcal{G}}_f$ (with $f^{†}$ in channel 1), as a function of $\omega /T_K$ for different values of $K_R$. From among the three peaks sketched in Fig. 3 only the two peaks around $T^{\ast }$ and $T_K$ are shown.

Figure 7

(Color online) (Top) Dimensionless spectral function of $f_{0,1,\uparrow }$, ${\hat{\varrho }}_{f,\uparrow }=D_F{\varrho }_{f,\uparrow }$, for different values of $B$ as a function of $\omega /T_K$ on linear scale. (Bottom) Dimensionless spectral functions (a) of $f_{0,1,\uparrow }$ and (b) of $f_{0,1,\downarrow }$ for different values of $B$ as a function of $\omega /T_K$ on logarithmic scale.

Figure 8

(Color online) Universal collapse of the dimensionless spectral functions ${\hat{\varrho }}_{f,\uparrow }=D_F{\varrho }_{f,\uparrow }$ and ${\hat{\varrho }}_{f,\downarrow }=D_F{\varrho }_{f,\downarrow }$ to two scaling curves ${\mathcal{B}}_{f,\uparrow }$ and ${\mathcal{B}}_{f,\downarrow }$ for sufficiently small, nonzero values of $B$ as a function of $\omega /T_h$.

Figure 9

(Color online) Universal collapse of the dimensionless spectral functions of the composite fermion operator, ${\hat{\varrho }}_{F,\uparrow }=D_F{\varrho }_{F,\uparrow }$ and ${\hat{\varrho }}_{F,\downarrow }=D_F{\varrho }_{F,\downarrow }$, to two scaling curves ${\mathcal{B}}_{F,\uparrow }$ and ${\mathcal{B}}_{F,\downarrow }$ for sufficiently small, nonzero values of $B$ as a function of $\omega /T_h$.

Figure 10

(Color online) Sum of the dimensionless spectral functions ${\hat{\varrho }}_{f,\uparrow }=D_F{\varrho }_{f,\uparrow }$ and ${\hat{\varrho }}_{f,\downarrow }=D_F{\varrho }_{f,\uparrow }$ for different values of $B$ as a function of $\omega /T_K$.

Figure 11

(Color online) (Top) Sketch of the dimensionless spectral function of $\vec{S}$, ${\hat{\varrho }}_S=T_K{\varrho }_S=T_K\text{Im}{\chi }_S\left(\omega \right)/\pi$, for $T>0$ and $K_R=0$ and $B=0$ as a function of $\text{log}\left(\omega /T_K\right)$. (Bottom) Sketch of ${\hat{\varrho }}_S=T_K{\varrho }_S=T_K\text{Im}{\chi }_S\left(\omega \right)/\pi$ for $T=0$ and $K_R\ne 0$ and $B=0$ as a function of $\text{log}\left(\omega /T_K\right)$. Asymptotics indicated for $\omega <T_K$ were derived through scaling arguments. The large-$\omega$ behavior is a result of perturbation theory (Ref. 56).

Figure 12

(Color online) Sketch of the real part of the dimensionless Green’s function of $\vec{S}$, $\text{Re}{\widehat{g}}_S=-T_K\text{Re}{\chi }_S\left(\omega \right)\equiv T_K\text{Re}{\mathcal{G}}_S\left(\omega \right)$, for $T,T^{\ast }>0$ as a function of $\text{log}\left(\omega /T_K\right)$.

Figure 13

(Color online) (a) Dimensionless spectral function of $\vec{S}$, ${\hat{\varrho }}_S=T_K{\varrho }_S=T_K\text{Im}{\chi }_S\left(\omega \right)/\pi$, as a function of $\omega /T_K$ for different values of $K_R$. (b) Minute $\text{log}\left(\omega \right)$ dependence at the lowest frequencies diminishing as a function of the number of kept multiplets. [(c)–(e)] Numerical confirmations of the low-frequency asymptotics derived from scaling arguments in Sec. 6. Straight dashed lines are for demonstrating deviations from the expected (c) $\sqrt{\omega }$-like, (d) $\omega$-like, and (e) $1/\omega$-like behaviors. $T^{\ast }/T_K=7\times {10}^{-2}$ in plots (d) and (e).

Figure 14

(Color online) Universal collapse of the dimensionless spectral function of $\vec{S}$, ${\hat{\varrho }}_S=T_K{\varrho }_S$, to the scaling curve ${\mathcal{K}}_S$ as a function of $\omega /T^{\ast }$ for sufficiently small, nonzero values of $K_R$.

Figure 15

(Color online) Real part of the dimensionless Green’s function (susceptibility) of $\vec{S}$, $\text{Re}{\widehat{g}}_S=-T_K\text{Re}{\chi }_S\left(\omega \right)$, as a function of $\omega /T_K$ for different values of $K_R$.

Figure 16

(Color online) Dimensionless spectral functions (a) of $S^{+}$, ${\hat{\varrho }}_{S,+}=T_K{\varrho }_{S,+}$; (b) of $S^z$, ${\hat{\varrho }}_{S,z}=T_K{\varrho }_{S,z}$; and (c) of $S^{-}$, ${\hat{\varrho }}_{S,-}=T_K{\varrho }_{S,-}$, for different values of $B$ as functions of $\omega /T_K$.

Figure 17

(Color online) Universal collapse of ${\hat{\varrho }}_{S,+}=T_K{\varrho }_{S,+}$, ${\hat{\varrho }}_{S,z}=T_K{\varrho }_{S,z}$, and ${\hat{\varrho }}_{S,-}=T_K{\varrho }_{S,-}$ to the three scaling curves ${\mathcal{B}}_{S,+}$, ${\mathcal{B}}_{S,z}$, and ${\mathcal{B}}_{S,-}$ for sufficiently small, nonzero values of $B$ as a function of $\omega /T_h$.

Figure 18

(Color online) $\varrho \left(\omega \right)/\omega$ (a) of $S^{+}$, $T_K{\hat{\varrho }}_{S,+}/2\omega =T_K^2{\varrho }_{S,+}/2\omega$; (b) of $S^z$, $T_K{\hat{\varrho }}_{S,z}/\omega =T_K^2{\varrho }_{S,z}/\omega$; and (c) of $S^{-}$, $T_K{\hat{\varrho }}_{S,-}/2\omega =T_K^2{\varrho }_{S,-}/2\omega$, for different values of $B$ as functions of $\omega /T_K$.

Figure 19

(Color online) (a) Dimensionless spectral function ${\hat{\varrho }}_{\text{SCC}}=D_F^2/T_K\text{Im}{\chi }_{\text{SCC}}$ of the operator ${\mathcal{O}}_{\text{SCC}}$ as a function of $\omega /T_K$ for different values of $K_R$. (b) The very weak $\text{log}\left(\omega \right)$ dependence at the lowest frequencies. This dependence is suppressed as we increased the number of kept multiplets. [(c)–(e)] Numerical confirmations of the low-frequency asymptotics derived from scaling arguments in Sec. 7. Dashed straight lines are for demonstrating deviations from the expected (c) $\sqrt{\omega }$-like, (d) $\omega$-like, and (e) $1/\omega$-like behaviors. $T^{\ast }/T_K=7\times {10}^{-2}$ in plots (d) and (e).

Figure 20

(Color online) Universal collapse of ${\hat{\varrho }}_{\text{SCC}}$ to the scaling curve ${\mathcal{K}}_{\text{SCC}}$ as a function of $\omega /T^{\ast }$ for sufficiently small, nonzero values of $K_R$.

Figure 21

(Color online) Real part of the dimensionless Green’s function of ${\mathcal{O}}_{\text{SCC}}$, $\text{Re}{\widehat{g}}_{\text{SCC}}=-D_F^2/T_K\text{Re}{\chi }_{\text{SCC}}$, as a function of $\omega /T_K$ for different values of $K_R$.

Figure 22

(Color online) (Top) Dimensionless spectral function ${\hat{\varrho }}_{\text{SCC}}=\left(D_F^2/\pi T_K\right)\text{Im}{\chi }_{\text{SCC}}=-\left(D_F^2/\pi T_K\right)\text{Im}{\mathcal{G}}_{\text{SCC}}$ of the composite superconductor operator ${\mathcal{O}}_{\text{SCC}}$ for different values of $B$ as a function of $\omega /T_K$. (Bottom) Universal collapse of ${\hat{\varrho }}_{\text{SCC}}$ to the scaling curve ${\mathcal{B}}_{\text{SCC}}$ for sufficiently small, nonzero values of $B$ as a function of $\omega /T_h$.

Figure 23

(Color online) (a) Dimensionless spectral function of ${\mathcal{O}}_{\text{SC}}$, ${\hat{\varrho }}_{\text{SC}}=D_F{\varrho }_{\text{SC}}$, as a function of $\omega /T_K$ for different values of $K_R$, and (b) the real part of its dimensionless Green’s function, $\text{Re}{\widehat{g}}_{\text{SC}}=D_F\text{Re}{\mathcal{G}}_{\text{SC}}$.

Figure 24

(Color online) Universal scalings of the dimensionless (top) composite fermion’s and (bottom) local fermion’s spectral functions as a function of $\omega /T_K$ for different values of the exchange coupling $\tilde{J}$ and potential-scattering strength $V$. The insets show $\sqrt{\omega }$-like behavior for low frequencies for both operators.

Figure 25

(Color online) (Top) Sketch of the various 2CK scaling regimes in the presence of channel anisotropy for the local fermions bounded by $T_f^{\ast \ast }$ from below and for the spin bounded by $T_s^{\ast \ast }$. The crossover scale $T^{\ast \ast }$ is also indicated. (Bottom) Sketch of the 2CK scaling regime for the susceptibilities of the highest-weight fields bounded by $T_h^{\ast \ast }$, and the crossover scale $T_h$.