Weak-coupling continuous-time quantum Monte Carlo study of the single impurity and periodic Anderson models with $s$-wave superconducting baths

We apply the unbiased weak-coupling continuous time quantum Monte Carlo method to review the physics of a single magnetic impurity coupled to $s$-wave superconducting leads described by the BCS reduced Hamiltonian. As a function of the superconducting gap $\Delta$, we study the signature of the first-order transition between the singlet and doublet (local-moment) states on various quantities. In particular, we concentrate on the Josephson current with 0 to $\pi$ phase shift, the crossing of the Andreev bound states in the single-particle spectral function, as well as the local dynamical spin structure factor. Within dynamical mean-field theory, this impurity problem provides a link to the periodic Anderson model (PAM) with superconducting conduction electrons (BCS-PAM). The first-order transition observed in the impurity model is reproduced in the BCS-PAM and is signalized by the crossing of the low-energy excitations in the local density of states. The momentum resolved single-particle spectral function in the singlet state reveals the coherent, Bloch-type, superposition of Andreev bound states. In the doublet or local-moment phase the single-particle spectral function is characterized by incoherent quasiparticle excitations.

Figure 1

(Color online) Eigenenergies of the effective Hamiltonian (14) for varying $U$. The fixed parameters are given by $V=0.5$ and $\Delta =1$. The crossing of the two lowest levels is clearly seen at $U\approx 1.7$. The ground state for $U<1.7$ is a singlet state. For larger values of $U$, the twofold-degenerate doublet state becomes energetically more favorable.

Figure 2

(Color online) For $\Delta <1.412$ the ground state is the singlet state from Eq. (15). If $\Delta$ is increased, the weight $\alpha$ of the single occupied states $|\tilde{\uparrow },\tilde{\downarrow }⟩$ and $|\tilde{\downarrow },\tilde{\uparrow }⟩$ decreases in favor of the states with a double occupied quantum dot, corresponding to the weights $\beta$ and $\gamma$. At $\Delta =1.412$ the ground state changes to the twofold-degenerate doublet state given in Eq. (16) and the weight of the states with a single occupied quantum dot $b$ increases with $\Delta$. The parameters in this plot are $V=0.5$ and $U=1.0$.

Figure 3

(Color online) Double occupancy $⟨{\tilde{d}}_{\uparrow }^{†}{\tilde{d}}_{\uparrow }{\tilde{d}}_{\downarrow }^{†}{\tilde{d}}_{\downarrow }⟩$ of the quantum dot in the effective model at $\beta =200$ and $V=0.5$. This plot can be understood as a phase diagram of the effective model, as the phase boundary is accompanied by a sharp decay of the double occupancy.

Figure 4

(Color online) Spectral function $A_{\uparrow \uparrow }\left(\omega \right)$ of the effective model for different values of $\Delta$ at $\beta =200$, $U=1$, and $V=0.5$. The $\delta$ peaks have been broadened by a Gaussian function of width $\sigma =0.04$ for better visibility.

Figure 5

(Color online) Dynamical spin structure factor $S\left(\omega \right)$ of the effective model at $\beta =200$. The phase transition from the singlet phase to the doublet phase for $U=1$ and $V=0.5$ occurs at $\Delta \approx 1.412$. At this point a transition from a gapped excitation to a peak at $\omega =0$ corresponding to a local magnetic moment in the doublet phase is observed. To visualize the $\delta$ functions, a Gaussian broadening of width $\sigma =0.05$ has been applied.

Figure 6

(Color online) Dynamical charge structure factor $N\left(\omega \right)$ of the effective model at $\beta =200$. We have used the same parameters as for Fig. 4.

Figure 7

(Color online) Josephson current in the 0 junction regime.

Figure 8

(Color online) Josephson current in the $0^{\prime }$ and ${\pi }^{\prime }$ junction regimes.

Figure 9

(Color online) Josephson current in the $\pi$ junction regime.

Figure 10

(Color online) Josephson current for different temperatures $1/\beta$. The current-phase relations do not intersect at one single point as suggested by the NRG results of Karrasch et al. (Ref. 23).

Figure 11

(Color online) Double occupancy $⟨{\widehat{n}}_{\uparrow }{\widehat{n}}_{\downarrow }⟩$ of the quantum dot at $\Delta =1.0$. The data show a jump in the double occupancy becoming sharper with decreasing temperature.

Figure 12

(Color online) Double occupancy of the quantum dot as a function of the phase difference $\varphi ={\varphi }_L-{\varphi }_R$ for different values of $\Delta$.

Figure 13

(Color online) Local pair correlation ${\Delta }_d=⟨{\tilde{d}}_{\uparrow }^{†}{\tilde{d}}_{\downarrow }^{†}⟩$ as a function of $\Delta$. We observe the same behavior as Choi et al. (Ref. 16) which is also in very good agreement with the pair correlation expected for the effective model discussed in Sec. 3C.

Figure 14

(Color online) Spectral function $A\left(\omega \right)$ as a function of $\Delta$ for the parameters $\beta =100$, $U=1.0$, and $V=0.5$ at half filling and zero phase difference between the two superconductors.

Figure 15

(Color online) Dynamical spin structure factor $S\left(\omega \right)$ as a function of $\Delta$ for the parameters $\beta =100$, $U=1.0$, and $V=0.5$ at half filling and zero phase difference between the two superconductors. For $\Delta =0$ we can roughly estimate the Kondo temperature $T_K\approx 0.06$ from the peak position of $S\left(\omega \right)$.

Figure 16

(Color online) Charge gap ${\Delta }_c$ as a function of $\Delta$. We calculated the dynamical charge structure factor from the charge-charge correlation function $C_c\left(\tau \right)$ using stochastic analytic continuation and extracted the charge gap using two different methods. First, we read off the charge gap directly from the stochastic analytic continuation data and second, we calculated the charge gap from the charge-charge correlation function. The straight line is a linear fit through the numerical data.

Figure 17

(Color online) Double occupancy of the $f$ sites in the BCS-PAM. In the proximity of the critical value of $U$, we observe two different solutions of the DMFT self-consistency cycle. The upper (thin/red) branch is generated, if we start the DMFT algorithm with a self-energy $\Sigma \equiv 0$ while we obtain the solution shown by the lower (thick/blue) branch if we take the self-energy of the data point at $U=0.44$ as the starting point of the DMFT iterations.

Figure 18

(Color online) Dynamical spin structure factor for the upper and the lower branches of the hysteresis in Fig. 17. Clearly, the upper branch of the hysteresis corresponds to a singlet solution while the lower branch shows a local moment.

Figure 19

(Color online) Density of states for the $f$ electrons as a function of $U$ for the parameters $V=0.5$, $\Delta =2$, $\mu ={ϵ}_f=0$, and $\beta =100$.

Figure 20

(Color online) Trace of the spectral function $A\left(\mathbf{k},\omega \right)$ at $\beta =100$ in the singlet regime. The parameters of the simulation were given by $U=0.125$, $V=0.5$, $\Delta =2$, and $\mu ={ϵ}_f=0$.

Figure 21

(Color online) Trace of the spectral function $A\left(\mathbf{k},\omega \right)$ at $\beta =100$ in the singlet regime for increasing interaction $U$. The width of the $f$ bands clearly decreases and the dispersion becomes weaker. The parameters of the simulations were given by $V=0.5$, $\Delta =2$, and $\mu ={ϵ}_f=0$.

Figure 22

(Color online) Trace of the spectral function $A\left(\mathbf{k},\omega \right)$ at $\beta =100$ in the doublet regime for different values of $U$. Here, we only show the $f$ bands. The parameters of the simulation were given by $V=0.5$, $\Delta =2$, and $\mu ={ϵ}_f=0$.

Figure 23

(Color online) Trace of the spectral function $A\left(\mathbf{k},\omega \right)$ as obtained from using Eq. (49) for the impurity action. The $z$ component of the local moment is sampled from the box distribution $m_z∊\left[-M_z,M_z\right]$. The parameters used for this plot were given by $V=0.5$, $U=0.5$, $\Delta =2$, and $M_z=0.0375$. Here, the calculations are carried out on the real time axis such that no analytical continuation is required.