Vortex loop dynamics and dynamical quantum phase transitions in three-dimensional fermion matter

Over the past decade, dynamical quantum phase transitions (DQPTs) have emerged as a paradigm shift in understanding nonequilibrium quantum many-body systems. However, the challenge lies in identifying order parameters that effectively characterize the associated dynamic phases. In this study we investigate the behavior of vortex singularities in the phase of the Green's function for a broad class of fermion lattice models in three dimensions after an instantaneous quench in both interacting and noninteracting systems. We find that the full set of vortices form one-dimensional dynamical objects, which we call vortex loops. We propose that the number of such vortex loops can be interpreted as a quantized order parameter that distinguishes between different nonequilibrium phases. Our results establish an explicit link between variations in the order parameter and DQPTs in the noninteracting scenario. Moreover, we show that the vortex loops are robust in the weakly interacting case, even though there is no direct relation between the Loschmidt amplitude and the Green's function. Finally, we observe that vortex loops can form complex dynamical patterns in momentum space. Our findings provide valuable insights for developing definitions of dynamical order parameters in nonequilibrium systems.

Figure 1

Exemplary dynamics of vortex loops on a static manifold $\mathcal{M}$ (orange surface) in a 3D BZ. The red and blue colors represent the chirality of the loops. (a) Incontractible loop with opposing chiralities are created in pairs and move in opposite directions; light colors correspond to $t=1.05$ and dark colors to $t=1.1$. (b) Loops of the same chirality can merge if their tangent vectors (black arrows) are antiparallel at the loops' touching points; $t=1.405$ (c) Through loop merging, the loops can change their topological character from incontractible to contractible loops; $t=2$ (light colors), $t=3.5$ (dark colors).

Figure 2

(a) Chirality ($\pm$) of a vortex loop is determined by $\mathrm{\Delta }\varphi =\pm 2\pi$ phase jump while encircling a loop around a tangent vector $\widehat{u}$. (b) The number of contractible $n_c\left(t\right)$ and incontractible $n_{ic}\left(t\right)$ loops can be considered as quantized dynamical order parameters (black dashed and red solid lines, respectively). (c), (d) The first and second derivative of the rate function $\lambda \left(t\right)$ with nonanalytic DQPT points (marked by vertical lines) exactly match the changes of $n_c\left(t\right)$ and $n_{ic}\left(t\right)$.

Figure 3

(a) The phase of the Green's function $g_{\mathbf{k}}\left(t\right)$ in the interacting regime is shown for interaction strength $\eta =0.1$ and time $t=1.05$ with a fixed value of $k_x=\pi /2$. The black circle marks the position of the vortex in the noninteracting case. (b) The first derivatives of the rate function $\lambda \left(t\right)$ (red dashed line) and ${\lambda }_G\left(t\right)$ (black solid), the latter being defined in Eq. (10). Although the two curves almost overlap, their nonanalytic points do not coincide and their mismatch grows in time. Nonanalytic points of $\lambda \left(t\right)$ and ${\lambda }_G\left(t\right)$ are contained within the two shaded areas, as depicted in panels (c) and (d), respectively. (c) Zoom-in of $\dot{\lambda }\left(t\right)$ and ${\dot{\lambda }}_G\left(t\right)$ in a vicinity of the first temporal nonanalytic points at $t_1^{*}, t_2^{*}$, and $t_G^{*}$. (d) Zoom-in of $\ddot{\lambda }\left(t\right)$ and ${\ddot{\lambda }}_G\left(t\right)$ showing the second type of nonanalytic points at $t^{\left(2\right)*}$ and $t_G^{\left(2\right)*}$.

Figure 4

The complex pattern formed by loop vortices at times $t=15$ [panel (a)] and $t=30$ [panel (b)] after the quench to the Weyl semimetal phase, where the number of loops around the Weyl nodes increases linearly in time. Due to the dispersion relation ${\epsilon }_{\mathbf{k}}$ vanishing for some points belonging to the manifold $\mathcal{M}$, the loops never annihilate and instead, their dynamics slows down, leading to their accumulation around the band touching points ${\mathbf{k}}_W$. The resulting dynamical pattern cannot be destroyed due to the system's dynamics.