Dynamical quantum phase transitions through the lens of mode dynamics

Abstract

We study the mode dynamics of a generic quadratic fermionic Hamiltonian under a sudden quench protocol in momentum space. Modes with zero energy at any given time, $t$, are referred to as dynamical critical modes. Among all zero-energy modes, spin-flip symmetry is restored in the eigenvector corresponding to selected zero-energy modes. This symmetry restoration is used to define the dynamical quantum phase transition (DQPT). This shows that the occurrence of these dynamical critical modes is necessary but not sufficient for a DQPT. We show that the conditions on the quench protocol and time for such dynamical symmetry restoration are the same as the divergence of the rate function and integer jump in the dynamical topological order parameter, which have been the traditional identifiers of a DQPT. This perspective also naturally explains when one or both of DQPT and ground-state quantum phase transitions will occur.

Figures (5)

• Figure 1: Plot of the post-quench parameters $(\mu, \Delta)$ for which DQPT occurs with either a single or double dynamical critical mode, for fixed pre-quench parameters $(\mu_0,\Delta_0)=(0.5,-1)$.
• Figure 2: In the left column (top panel), we plot $\mathcal{R}(t)$ (left axis) and DTOP ($\nu_D(t)$) (right axis) as a function of time $t$ for the quench protocol $(\mu_0,\Delta_0)=(0.5,-1)\rightarrow (\mu,\Delta)=(1.5,1)$, where a single mode satisfies the condition $\cos\!\left(2\delta \theta_{k_c}\right)=0$. In the right column, we show the same plot for $(\mu_0,\Delta_0)=(0.5,-1) \rightarrow (\mu,\Delta)=(0.8,1)$, where two modes satisfy the DQPT condition. In the left column (bottom panel), for the same quench protocol as in the left column (top panel), we plot the dynamical mode energy $\Tilde{\lambda}_{k_n}(t)$ as a function of momentum $k$ at $t=t_c$ from Eq. \ref{['eq:dqpt_cond']}, with $m_2=0$ (red circles) and $m_2=2$ (magenta circles). In the right column, for the same quench protocol as in the right column (top panel), we plot the same at two different $t=t_c$ corresponding to two different $k_c$ values.
• Figure 3: DME zeros in the $k-t$ plane for the same quench protocol as in the left column of fig. \ref{['fig:rfvsttddispvsk']}. The three horizontal black lines indicate the momentum modes satisfying $\cos\left(2\delta \theta_{k}\right) = 0$ and $\cos\left(2\delta \theta_{k}\right) = \pm \frac{1}{\sqrt{2}}$. The Black vertical lines denote the times at which $\Phi_{k_c}(t_c)=0$.
• Figure S.1: Plot of $\mathcal{R}(t)$ (red solid line) and $2r(t)$ (blue circles) as functions of time for the quench protocol $(\mu_0,\Delta_0)=(0.5,-1) \rightarrow (\mu,\Delta)=(1.5,1)$, where only a single mode $k=k_c$ exists such that $\cos\left(2\delta \theta_{k_c}\right)=0$. The vertical dotted lines correspond to the critical times $t=t_c$ given in Eq. (11) of main text, with $m_2 \in [0,6]$.
• Figure S.2: (Top left), plot of $\mathcal{R}(t)$ (left axis) and the DTOP (right axis) as a function of time for the single-quench protocol $\mu_0=0.5,\mu=1.5$ and $\Delta=\Delta_0=1$. In the right column, we plot the same for the quench protocol where $\mu=\mu_0=0.5$ and $\Delta_0=-1, \Delta=1$. In this case, two momentum modes satisfy the dynamical symmetry restoration condition, and the associated critical times are indicated by solid and dashed vertical lines. For the same quench protocol as in the top-left panel, in the bottom-left panel, we plot the DME as a function of momentum modes at the critical times with $m_2=0$ (red circles) and $m_2=2$ (magenta circles), corresponding to the single critical momentum mode that satisfies $\cos(2 \delta \theta_{k_c})=0$. This mode is indicated by the vertical dotted line. In the right column, we plot the same for a fixed chemical potential $\mu_0=\mu=0.5$ and performing the quench in $\Delta$ from $\Delta_0=-1$ to $\Delta=1$ at two critical times with $m_2=0$ associated with the two critical momentum modes.