Adiabatic transition from a BCS superconductor to a Fermi liquid and phase dynamics

We investigate the physics of an adiabatic transition from a BCS superconductor to a Fermi liquid for an exponentially slow decreasing pairing interaction. We show that, depending on the order of the thermodynamic limit and large times, a situation can arise in which the Fermi liquid keeps a memory of the parent BCS state. Furthermore, a time inversion of the interaction, supplemented by a manipulation analogous to a spin-/photon-echo experiment, allows us to recover the parent BCS state. Moreover, we study the evolution of the order parameter phase $\varphi$ in transforming the BCS superconductor to a conventional metal. Since the global phase is the conjugate variable of the density, we explicitly show how to use the dynamics of $\varphi$ together with gauge invariance to build up the noninteracting chemical potential away from particle-hole symmetry. We further analyze the role of $\varphi$ in restoring the gauge-invariant current response when the noninteracting Fermi liquid is approached starting from a BCS superconductor in the presence of an external vector field.

Figure 1

Time evolution of the Gorkov function $\left|f\right(t\left)\right|$ (solid lines) for different adiabatic timescales $T=1000$ (black), $T=2000$ (blue), and $T=3000$ (red). Circles correspond to the equilibrium values $|f_0\left(t\right)|$ obtained from the BCS equation for the interaction value $U\left(t\right)$ from Eq. (3), and the dashed lines are the corresponding parametrization from Eq. (19). The inset shows $1-f_0\left(t\right)/f\left(t\right)$ for the different adiabatic timescales in comparison with the time at which the adiabaticity condition $d\mathrm{\Delta }/dt<{\mathrm{\Delta }}^2$ starts to get violated (dashed). Results are for a semielliptic DOS $\rho \left(\omega \right)=\frac{2}{\pi }\sqrt{B^2-{\omega }^2}$ with $B\equiv 1$ at density $n=0.875$ and an initial value of $U(t=0)=0.5$.

Figure 2

Time dependence of the energy per site for different adiabatic timescales. The horizontal gray line is the equilibrium value for $U=0$. System and parameters are the same as in Fig. 1.

Figure 3

Dependence of $J_k^{\parallel }\equiv \sqrt{\left(J_k^x\right)+\left(J_k^y\right)}$ (a), the momentum-dependent phase ${\varphi }_k=arctan(J_k^y/J_k^x)$ [main panel (b) and inset], and $J_k^z$ (c) on ${\varepsilon }_k$ at times $t=300$ and 3000 for an adiabatic decay time $T=2000$. The inset to (c) details the behavior of $J_k^z\approx 0$. Results are for a semielliptic DOS $\rho \left(\omega \right)=\frac{2}{\pi }\sqrt{B^2-{\omega }^2}$ with $B\equiv 1$ at density $n=0.875$ and we have used a time-independent chemical potential parameter $\mu ={\mu }_0$. The initial value of the interaction $U(t=0)=0.5$.

Figure 4

Same as panels (a),(c) of Fig. 3 but for a small adiabatic decay time $T=200$ and $t=3000$.

Figure 5

Time dependence of the Gorkov function where the evolution is reversed after $t_0=3000$ time steps with adiabatic timescales $T=2000$ (black) and $T=3000$ (red). Solid (dashed) lines correspond to the reversed time evolution with (without) inversion of the $J_k^{x,y}$ component. Interaction parameter $U(t=0)=0.5$; density $n=0.875$.

Figure 6

(a) Phase of the Gorkov function for the case $\mu =\text{const}$ (black) and after transforming to the Larmor frame (red). (b) Time-dependent chemical potential parameter from transforming to the Larmor frame (red), from the zeros of $J_z$ (black), and thermodynamic chemical potential from the relation $\mu \left(t\right)=dE(n,t)/dn$ (blue). Density $n=0.875, U_0/t=1$, decay time $T=500$.

Figure 7

Main panel: static response to an external vector potential. Black: transverse response to $A_x\left(q_y\right)$; blue: (gauge invariant) longitudinal response to $A_x\left(q_x\right)$; red: longitudinal response in the BCS approximation. Dashed lines correspond to Eqs. (36) and (39) with ${\xi }_t=1.88, {\xi }_l=1.96$. Inset: spatial variation of the order-parameter phase for the (gauge-invariant) longitudinal response. Parameters: $U/t=2.0, n=0.875$. $72\times 72$ lattice.

Figure 8

Finite-size effects for the adiabatic transition of the Gorkov function with $U\left(t\right)=U_0{e}^{-t/T}$ and $T=500$. The dynamics for the infinite lattice (blue dashed line) has been computed with the 2D DOS. Parameters: $U_0/t=2.0, n=0.875$.

Figure 9

Time dependence of the longitudinal response when the initial state is computed in the BCS approximation. (a) Dia- (red) and paramagnetic (black) currents for $q_x=2\pi /72$. (b) Total response $D_l=(j_{\text{para}}+j_{\text{dia}})/A_0$ for $q_x=2\pi /72$ (black) and $q_x=8\pi /72$ (red). Parameters: $U_0/t=2.0, n=0.875$. $72\times 72$ lattice.

Figure 10

Phase mode along a selected momentum cut in the Brillouin zone. The linear approximation at small momenta is ${\omega }_q=0.25q_x$ and fits to the observed phase excitations in the current shown in Fig. 9. The horizontal line marks the superconducting gap $2{\mathrm{\Delta }}_0/t=0.18$. Parameters: $U/t=2.0, n=0.875$.

Figure 11

Time dependence of the transverse response. (a) Dia- and paramagnetic currents for different momenta $q_y$. (b) Total response $D_t(q_y,t)=(j_{\text{para}}+j_{\text{dia}})/A_0$. The inset compares the time-dependent correlation length (solid) with the linear-response expression Eq. (36). Parameters: $U_0/t=2.0, n=0.875$. $72\times 72$ lattice.

Figure 12

Time dependence of the Gorkov function with the initial density matrix given by Eqs. (26) and (27) and a time-dependent interaction $U\left(t\right)=U_0[exp(t/T)-1]$. Parameters: $\phi =0, U_0=0.5$, and $T=1000$.