Non-adiabatic Dynamics in d + id-Wave Fermionic Superfluids

Abstract

We consider a problem of non-adiabatic dynamics of a 2D fermionic system with d + id-wave symmetry of pairing amplitude. Under the mean-field approximation, we determine the asymptotic behavior of the pairing amplitude following a sudden change of coupling strength. We also study an extended d + id pairing system for which the long-time asymptotic states of the pairing amplitude in the collisionless regime can be determined exactly. By using numerical methods, we have identified three non-equilibrium steady states described by different long-time asymptotes of the pairing amplitude for both the non-integrable and the integrable versions of d + id-wave models. We found that despite of its lack of integrability, long-time dynamics resulting from pairing quenches in the non-integrable d + id model are essentially similar to the ones found for its exactly integrable extended d + id model. We also obtain the long-time phase diagram of the extended d + id model through the Lax construction that exploits underlying integrability showing that the dynamic phases obtained by numerics are consistent with the dynamics of the exactly integrable approach. Both models describe a topological fermionic system with a topologically non-trivial BCS phase appearing at weak coupling strength. We show that the presence of oscillating order parameter region in the chiral d + id pairing dynamics differs from the d-wave ($d_{x^{2}-y^{2}}$), which may be used to probe pairing symmetries of chiral superconductors.

Appendix: Integrals of Motion for the Extended d + id Model

In order to derive the integrals of motion, we use the method of Lax construction. The components of the Lax vector ${\vec L}(u)$ (u is a parameter) are given by (4.1) in the main text. These quantities satisfy the algebra

$$ \begin{array}{@{}rcl@{}} \{L^+(u),L^-(v)\}&=&-2\left[\frac{uL^z(u)-vL^z(v)}{u-v}\right], \\ \{L^z(u),L^+(v)\}&=&-\left[\frac{uL^{+}(u)-vL^{+}(v)}{u-v}\right], \\ \{L^z(u),L^-(v)\}&=&-\left[\frac{uL^{-}(u)-vL^{-}(v)}{u-v}\right]. \end{array} $$

(A1)

Note that all three commutation relations retain the same form as in the s- and chiral p-wave cases.

Our main task now is to define the “Casimir” of the Lax vector, L₂(u), which will be conserved by the evolution. The dynamics of the Lax vector components is described by the following equations which can be obtained from the equations of motion for the pseudospins together with (4.1). For the dynamics of $\vec {L}(u)$, we find

$$ \begin{array}{@{}rcl@{}} \frac{d{\vec{L}}}{dt}&=& {\det} \left[\begin{array}{lll} \hat{x} & \hat{y} & \hat{z} \\ -2u{\Delta}_x & -2u{\Delta}_y & u\left( 1-\frac{G}{\nu_F^{-1}}\sum\limits_{\mathbf{p}}\varepsilon_{\mathbf{p}} S_{\mathbf{p}}^z\right) \\ L^x(u) & L^y(u) & L^z(u) \end{array}\right]\\ \end{array} $$

(A2)

where $\vec {L}\equiv \hat {x}L^{x}+\hat {y}L^{y}+\hat {z}L^{z}$ and L^± = L^x ±iL^y. It is easy to see that quantity—the Lax norm—

$$ {L_2(u)= L^+(u)L^{-}(u)+\left[L^z(u)\right]^2 } $$

(A3)

is conserved by the evolution, i.e.,

$$ {\frac{dL_2(u)}{dt}=0.} $$

(A4)

In addition, the Poisson bracket which involves L₂(u) is

$$ \{L_2(u),L_2(v)\}=0. $$

(A5)

We will use this relation to show that the Hamiltonian (2.1) is exactly integrable.

To show that number of the integrals of motion equals exactly to the number of the degrees of freedom, let us introduce the discreet mesh of momenta

$$ \varepsilon_j=k_j^2/2, \quad (j=1,...,N) $$

(A6)

so that summation over the discreet energy levels ε_j in the continuum limit becomes

$$ \sum\limits_{j=1}^Nf(\varepsilon_j)\to\nu_F\int\limits_0^{{k_{\Lambda} ^2}/{2}}f(\varepsilon) d\varepsilon, $$

(A7)

where $\nu _{F}=\frac {\mathcal {A}}{8\pi }$ is the two-dimensional single-particle density of states at the Fermi level. Hamiltonian in (2.5) can now be written as a spin chain

$$ H=\sum\limits_j\varepsilon_j2s_j^z-g\sum\limits_j\varepsilon_js_j^+\sum\limits_l\varepsilon_ls_l^- -g\sum\limits_j\varepsilon_js_j^z\sum\limits_l\varepsilon_ls_l^z $$

(A8)

and g = Gν_F. With these conventions, the pseudospins are normalized:

$$ \left( {\vec s}_j\right)^2=\frac{1}{4}. $$

(A9)

For the Lax norm (A3), we find

$$ \begin{array}{@{}rcl@{}} L_2(u)&=&\sum\limits_{j=1}^{N}\sum\limits_{l=1}^{N}\frac{\varepsilon_j\varepsilon_l\left( s_j^{+}s_l^{-}+s_j^zs_l^z\right)}{(\varepsilon_j-u)(\varepsilon_l-u)}+\frac{2}{ug}\sum\limits_{j=1}^N\frac{\varepsilon_js_j^z}{\varepsilon_j-u}\\ &&+\frac{1}{u^2g^2}\\ &=&\sum\limits_{j=1}^N\frac{H_j}{u-\varepsilon_j}+\frac{1}{4}\sum\limits_{j=1}^N\frac{\varepsilon_j^2}{(u-\varepsilon_j)^2}+\frac{J_z}{gu}+\frac{1}{u^2g^2},\\ \end{array} $$

(A10)

where H_j denotes the Hamiltonian

$$ H_j=-\frac{2s_j^z}{g}+\sum\limits_{l\not=j}\frac{\varepsilon_j\varepsilon_l}{\varepsilon_j-\varepsilon_l}\left( s_j^+s_l^-+s_j^-s_l^++2s_j^zs_l^z\right) $$

(A11)

and J_z gives the total pseudospin projection on z-axis

$$ J_z=2\sum\limits_{j=1}^Ns_j^z=\text{const}. $$

(A12)

Since L₂(u) is conserved by the evolution so that {L₂(u),H_j} = 0, (A5) implies that {H_i,H_j} = 0, i.e., H_i’s are mutually conserved. There are N independent H_j’s in a system of N spins. Therefore, we have identified N integrals of motion for a system of N spins. Furthermore, our initial Hamiltonian (A8) can be expressed in terms of H_j’s as follows

$$ {H=-g\sum\limits_{j=1}^N \varepsilon_j H_j}+\textrm{const.} $$

(A13)

This equation means that H Poisson commutes with L₂(u) and we have identified all integrals of motion. Hence, the extended d + id model is exactly integrable.