Quantum quench in a $p+ip$ superfluid: Winding numbers and topological states far from equilibrium

We study the nonadiabatic dynamics of a two-dimensional $p+ip$ superfluid following an instantaneous quantum quench of the BCS coupling constant. The model describes a topological superconductor with a nontrivial BCS (trivial BEC) phase appearing at weak- (strong-) coupling strengths. We extract the exact long-time asymptotics of the order parameter $\Delta \left(t\right)$ by exploiting the integrability of the classical $p$-wave Hamiltonian, which we establish via a Lax construction. Three different types of asymptotic behavior can occur depending upon the strength and direction of the interaction quench. We refer to these as the nonequilibrium phases {I, II, III}, characterized as follows. In phase I, the order parameter asymptotes to zero due to dephasing. In phase II, $\Delta \to {\Delta }_{\infty }$, a nonzero constant. Phase III is characterized by persistent oscillations of $\Delta \left(t\right)$. For quenches within phases I and II, we determine the topological character of the asymptotic states. We show that two different formulations of the bulk topological winding number, although equivalent in the BCS or BEC ground states, must be regarded as independent out of equilibrium. The first winding number $Q$ characterizes the Anderson pseudospin texture of the initial state; we show that $Q$ is generically conserved. For $Q\ne 0$, this leads to the prediction of a “gapless topological” state when $\Delta$ asymptotes to zero. The presence or absence of Majorana edge modes in a sample with a boundary is encoded in the second winding number $W$, which is formulated in terms of the retarded Green's function. We establish that $W$ can change following a quench across the quantum critical point. When the order parameter asymptotes to a nonzero constant, the final value of $W$ is well defined and quantized. We discuss the implications for the (dis)appearance of Majorana edge modes. Finally, we show that the parity of zeros in the bulk out-of-equilibrium Cooper-pair distribution function constitutes a ${\mathbb{Z}}_2$-valued quantum number, which is nonzero whenever $W\ne Q$. The pair distribution can in principle be measured using rf spectroscopy in an ultracold-atom realization, allowing direct experimental detection of the ${\mathbb{Z}}_2$ number. This has the following interesting implication: topological information that is experimentally inaccessible in the bulk ground state can be transferred to an observable distribution function when the system is driven far from equilibrium.

Figure 1

Anderson pseudospin description of (a) a normal Fermi liquid and (b) an $s$-wave superconductor, in any number of spatial dimensions at zero temperature. In this figure, $k$ measures the radial coordinate along any direction in momentum space.

Figure 2

Momentum-space pseudospin texture of the $p+ip$ ground state (a) in the topological BCS phase $Q=1$, (b) at the BCS-BEC quantum phase transition, and (c) in the topologically trivial BEC phase $Q=0$. Here, $Q$ denotes the pseudospin winding number (skyrmion charge), defined via Eq. (1.6). For a $p+ip$ state, $Q=1$ ($Q=0$) when the $\mathrm{k}=0$ spin ${\vec{s}}_0$ is up (down). By contrast, $Q$ is ill defined at the quantum phase transition.

Figure 3

Zero-temperature chemical potential $\mu$ as a function of the ground-state order-parameter amplitude ${\Delta }_0$ in the $p+ip$ ground state, for fixed particle density $n$. The point $\{{\Delta }_0,\mu \}=\{{\Delta }_{\mathrm{QCP}},0\}$ is a quantum phase transition between the topologically nontrivial BCS and trivial BEC phases. The quasiparticle spectrum has a gapless Dirac node in the bulk at $\mathrm{k}=0$ when ${\Delta }_0={\Delta }_{\mathrm{QCP}}$.

Figure 4

Exact interaction strength quench phase diagram, extracted from the isolated roots of the spectral polynomial. The vertical axis measures the initial pairing amplitude ${\Delta }_0^{\left(i\right)}$. ${\Delta }_{\mathrm{QCP}}$ marks the equilibrium BCS-to-BEC topological quantum phase transition. The horizontal axis specifies the post-quench Hamiltonian through ${\Delta }_0^{\left(f\right)}$, which is the order parameter the system would exhibit in its ground state. The diagonal line ${\Delta }_0^{\left(i\right)}={\Delta }_0^{\left(f\right)}$ is the case of no quench. Off-diagonal points to the left (right) describe strong-to-weak (weak-to-strong) pairing quenches, wherein the dynamic variable $\Delta \left(t\right)$ evolves away from its initial value. Similar to the $s$-wave case (Refs. 6, 7, 8, 9, 10, 11, 12, 13, 14), the $p+ip$ system exhibits three different dynamical phases defined by the long-time asymptotics ($t\to \infty$) of $\Delta$. In phase I, $\Delta \left(t\right)\to 0$ due to dephasing (Refs. 13 and 14) in II, $\Delta \left(t\right)\to {\Delta }_{\infty }$, a nonzero constant (Refs. 10 and 12), and in phase III, $\Delta \left(t\right)$ shows persistent oscillations (Refs. 6, 12, and 13). The dashed purple line is the nonequilibrium continuation of the quantum phase transition, in the sense that the asymptotic value of the chemical potential ${\mu }_{\infty }\equiv {\lim }_{t\to \infty }\mu \left(t\right)$ vanishes. This leads to a change in the retarded Green's function winding number $W$ (Fig. 9, below). For this plot and all subsequent figures, we choose the Fermi energy ${\mathcal{E}}_F=2\pi n=5.18$ and the energy cutoff $\Lambda =50{\mathcal{E}}_F$, so that ${\Delta }_{\mathrm{QCP}}=1.54$.

Figure 5

Asymptotic values of the nonequilibrium order parameter induced by various quenches. In (a), we plot ${\lim }_{t\to \infty }\Delta \left(t\right)={\Delta }_{\infty }$ as a function of the post-quench ground-state amplitude ${\Delta }_0^{\left(f\right)}$, for fixed values of the initial ${\Delta }_0^{\left(i\right)}$. Curves (i)–(iv), respectively, correspond to ${\Delta }_0^{\left(i\right)}/{\Delta }_{\mathrm{QCP}}=\{1.2,0.5,0.3,0.00651\}$; each gives ${\Delta }_{\infty }$ along a horizontal cut across the quench phase diagram in Fig. 4. The value of ${\Delta }_{\infty }$ is determined by the isolated roots in Eq. (4.3). For the quench specified by $\{{\Delta }_0^{\left(i\right)},{\Delta }_0^{\left(f\right)}\}$, the roots are computed from the conserved spectral polynomial [Eqs. (3.34)]. The dashed vertical line marks the boundary separating phases III and II at ${\Delta }_0^{\left(i\right)}=0$. The portion of curve (iv) to the left of this line in fact represents quenches in phase III, while the portion to the right resides in phase II. Instead of asymptoting to a constant, $\Delta \left(t\right)$ executes periodic amplitude and phase motion in phase III. For a given ${\Delta }_0^{\left(f\right)}$ to the left of the dashed line, the value of the curve (iv) specifies the average $\left(\right|{\Delta }_{+}|+|{\Delta }_{-}\left|\right)/2$, where $|{\Delta }_{\pm }|$ denote the turning points of the orbit in the complex $\Delta$ plane. Phase III orbits (Ref. 59) associated to the quenches marked A and B appear in (b).

Figure 6

Asymptotic value of the nonequilibrium chemical potential ${\lim }_{t\to \infty }\mu \left(t\right)={\mu }_{\infty }$ for quenches in phase II, as a function of the post-quench ground-state amplitude ${\Delta }_0^{\left(f\right)}$, for fixed values of the initial ${\Delta }_0^{\left(i\right)}$. Curves (i)–(iii) correspond to the associated quenches in Fig. 5a. The dashed line is the ground-state curve [Eq. (1.5)].

Figure 7

Example of persistent order-parameter oscillations following a quench. The coordinates $\{{\Delta }_0^{\left(i\right)},{\Delta }_0^{\left(f\right)}\}=\{0.00503,0.108\}$ place this quench in phase III of Fig. 4. The blue solid curve is the result of numerical simulation for classical pseudospins (5024 spins). The red dashed curve obtains from the effective two-spin analytical solution, the parameters of which are extracted from the isolated roots of the spectral polynomial ${\mathcal{Q}}_{2N}(u;\beta )$ (see text).

Figure 8

Quench phase diagram: Pseudospin winding number $Q$. We find that $Q$ is unchanged from its initial value, such that $Q=1$ ($Q=0$) for an initial BCS ${\Delta }_0^{\left(i\right)}<{\Delta }_{\mathrm{QCP}}$ (BEC ${\Delta }_0^{\left(i\right)}>{\Delta }_{\mathrm{QCP}}$) state. The highlighted region B is a “topological gapless” phase. Throughout phase I of Fig. 4, $\Delta \left(t\right)$ vanishes as $t\to \infty$. In subregion B, the pseudospin texture nevertheless retains a nonzero winding $Q=1$ [Fig. 12 and Eq. (2.8)]. The state can be visualized as a skyrmion texture as in Fig. 2a, but now the tilted pseudospins precess at different frequencies about $\widehat{z}$ in spin space [see Eq. (2.8)]. The notion of smooth topology for the evolving texture remains well defined for times up to the inverse level spacing.

Figure 9

Quench phase diagram: Retarded Green's function winding number $W$ and Majorana edge modes. In phase II of Fig. 4, $\{\Delta \left(t\right),\mu \left(t\right)\}\to \{{\Delta }_{\infty },{\mu }_{\infty }\}$ as $t\to \infty$, with ${\Delta }_{\infty }$ a nonzero constant. Along the dashed purple line, ${\mu }_{\infty }=0$; this is a nonequilibrium extension of the ground-state quantum phase transition; see also Fig. 6. To the left (right) of this line, ${\mu }_{\infty }>0$ (${\mu }_{\infty }<0$), leading to $W=1$ ($W=0$). $W$ is ill defined in the gapless phase I. Phase III, wherein $\Delta \left(t\right)$ exhibits persistent amplitude and phase oscillations [Figs. 5b and 7] is topologically nontrivial (Ref. 46). Both phase III and the $W=1$ region of phase II support gapless Majorana edge modes. The former is confirmed by the Floquet analysis in Ref. 46, while the latter is established here via Eq. (2.11) and the surrounding discussion.

Figure 10

Sectioned phase diagram. Regions C and H include strong-to-weak and weak-to-strong quenches across the quantum critical point at ${\Delta }_0={\Delta }_{\mathrm{QCP}}$.

Figure 11

Asymptotic (infinite-time) pseudospin distribution function for a representative quench in the gapless region A of Fig. 8. The solid blue curve is the result of numerical simulation for 5024 classical pseudospins; the red dashed curve is the analytical solution obtained from the Lax construction. The dotted gray line gives the initial ground-state distribution. Although the initial and asymptotic distributions share the same winding $Q=0$, in the latter case $\Delta \left(t\right)$ has decayed to zero, due to the dephasing of precessing pseudospins [Eqs. (2.8) and (2.9)]. By contrast, in the ground state each pseudospin is aligned along its magnetic field. The quench coordinates are $\{{\Delta }_0^{\left(i\right)},{\Delta }_0^{\left(f\right)}\}=\{1.56,0.00211\}$, and $k_D\equiv 4\sqrt{4\pi n}=12.9$.

Figure 12

The same as Fig. 11, but for a point in the gapless region B of Fig 8. In this case $Q=1$, and the decay of $\Delta \left(t\right)$ to zero implies that the fluctuating region B is a “topological gapless” phase. The quench coordinates are $\{{\Delta }_0^{\left(i\right)},{\Delta }_0^{\left(f\right)}\}=\{0.750,0.00224\}$.

Figure 13

Nonequilibrium winding number: parity of zeros in the Cooper-pair distribution function of phase II, for representative quenches in regions C and D (Fig. 10). The Cooper-pair distribution function is the pseudospin projection $\gamma \left(k\right)$ at each momentum $k$ onto the effective field $-\vec{B}(\varepsilon =k^2)$. The latter encodes the asymptotic global parameters $\{{\Delta }_{\infty },{\mu }_{\infty }\}$ [see Eqs. (2.14) and (2.15)]. Distributions C (blue solid line) and D (red dashed line) belong to quenches in the corresponding regions highlighted in Fig. 10. Whenever $Q\ne W$ ($Q=W$), $\gamma \left(k\right)$ exhibits an odd (even) number of zeros, illustrated here by curve C (D) (cf. Figs. 8 and 9). The parity of these zeros is a ${\mathbb{Z}}_2$-valued quantum number that can in principle be extracted from a rf spectroscopic measurement in a cold-atomic realization of the quench. The quench coordinates for curves C and D are $\{{\Delta }_0^{\left(i\right)},{\Delta }_0^{\left(f\right)}\}=\{1.65,0.359\}$ and $\{1.47,0.365\}$, respectively. Both curves were obtained from the analytical solution.

Figure 14

Isolated roots and ground-state spectral transitions. (a) shows the positions of the isolated root pair $u_0^{(\pm )}$ [Eq. (3.22)] in the ground-state spectral polynomial, for increasing values of ${\Delta }_0$. Particular root configurations are labeled (1)–(4). For ${\Delta }_0<{\Delta }_{\mathrm{Coh}}$ (1), the isolated roots are a conjugate pair with positive real part. At ${\Delta }_0={\Delta }_{\mathrm{Coh}}$ (2), the roots become purely imaginary. At ${\Delta }_0={\Delta }_{\mathrm{MR}}$ (“Moore-Read,” see text) (3), the roots become negative real and degenerate. For ${\Delta }_{\mathrm{MR}}<{\Delta }_0<{\Delta }_{\mathrm{QCP}}$, the roots split and travel along the negative real axis. At ${\Delta }_0={\Delta }_{\mathrm{QCP}}$ (4), the retreating root hits zero. The thresholds $\{{\Delta }_{\mathrm{Coh}},{\Delta }_{\mathrm{MR}}\}$ lie within the BCS phase ($\mu >0$), while ${\Delta }_{\mathrm{QCP}}$ marks the topological transition. All are defined explicitly in Appendix 7b, Eqs. (A3) and (A4). For ${\Delta }_0>{\Delta }_{\mathrm{QCP}}$ (BEC), both roots are again nondegenerate and negative real. The zero-temperature tunneling density of states $\nu \left(\omega \right)$ is shown for the corresponding root positions in (b). The weak pairing coherence peak visible in the trace $(1\to 2)$ disappears for ${\Delta }_0\ge {\Delta }_{\mathrm{Coh}}$. The difference between the “soft” and “hard” gaps in the BCS ($2\to 3\to 4$) and BEC regimes is a coherence factor effect. The spectrum is gapless at ${\Delta }_0={\Delta }_{\mathrm{QCP}}$ (4).

Figure 15

Detailed quench phase diagram. The dynamical phase boundaries ${\beta }_{\mathsf{c}}^{(\pm )}$ were obtained from a numerical solution to Eqs. (3.40), using Eq. (3.39). The pairing amplitude values ${\Delta }_{\mathrm{Coh}}$, ${\Delta }_{\mathrm{MR}}$, and ${\Delta }_{\mathrm{QCP}}$ mark spectral transitions in the equilibrium ground state, as described in Sec. 3b and illustrated in Fig. 14. Explicit values appear in Eqs. (A3) and (A4). The lines marked ${\beta }_{\text{Coh}}$, ${\beta }_{\text{MR}}$, and ${\beta }_{\text{QCP}}$ are the nonequilibrium extensions of these ground-state spectral transitions, as discussed in Sec. 3f.

Figure 16

Persistent order-parameter oscillations following a quench. The same as Fig. 7, but for quench coordinates $\{{\Delta }_0^{\left(i\right)},{\Delta }_0^{\left(f\right)}\}=\{0.00503,0.961\}$.

Figure 17

The Cooper-pair distribution function $\gamma \left(k\right)$ as in Fig. 13, but plotted for representative quenches in regions E (red dashed line) and H (blue solid line) of phase II. The quench coordinates for E are $\{{\Delta }_0^{\left(i\right)},{\Delta }_0^{\left(f\right)}\}=\{0.972,1.25\}$, while the quench coordinates for H are $\{{\Delta }_0^{\left(i\right)},{\Delta }_0^{\left(f\right)}\}=\{0.972,1.55\}$. Regions E and H are specified in Fig. 10.