Long-time rigidity to flux-induced symmetry breaking in quantum quench dynamics

We investigate how the breaking of charge conjugation symmetry $\mathcal{C}$ impacts on the dynamics of a half-filled fermionic lattice system after global quenches. We show that, when the initial state is insulating and the $\mathcal{C}$ symmetry is broken nonlocally by a constant magnetic flux, local observables, and correlations behave as if the symmetry were unbroken for a time interval proportional to the system size $L$. In particular, the local particle density of a quenched dimerized insulator remains pinned to $1/2$ in each lattice site for an extensively long time, while it starts to significantly fluctuate only afterwards. Due to its qualitative resemblance to the sudden arrival of rapidly rising ocean waves, we dub this phenomenon the “tsunami effect.” Notably, it occurs even though the chiral symmetry is dynamically broken right after the quench. Furthermore, we identify a way to quantify the amount of symmetry breaking in the quantum state, showing that in insulators perturbed by a flux, it is exponentially suppressed as a function of the system size, while it is only algebraically suppressed in metals and in insulators with locally broken $\mathcal{C}$ symmetry. The robustness of the tsunami effect to weak disorder and interactions is demonstrated, and possible experimental realizations are proposed.

Figure 1

Quantum dynamics resulting from a quench applied to a dimerized ring-shaped model (14) with $L=80$ cells (160 sites), where the tunneling amplitudes, with initial values $v^i=1\text{and}{w}^i=0.5$, are exchanged $w\leftrightarrow v$. The deviation $\delta n$ from 1/2 of the on-site density is shown as a function of time. (a) Case of a staggered potential $U=1/10$ breaking the charge conjugation symmetry $\mathcal{C}$ locally. (b) Case of a nonlocal $\mathcal{C}$-symmetry breaking induced by a flux phase $\phi =\pi /2$, where $\phi =2\pi \mathrm{\Phi }/{\mathrm{\Phi }}_0$, with ${\mathrm{\Phi }}_0$ denoting the flux quantum.

Figure 2

Time evolution of the on-site density deviation [see Eq. (21)] in a SSH ring with $L=12$ cells (=24 sites), threaded by a magnetic flux $\phi =\pi /2$, undergoing the quench protocol (24), with dimerization $r=0.1$. The analytical expression (27) obtained in the strongly dimerized limit (red dashed curve) perfectly reproduces the numerically exact evolution (blue curve). The deviation $\delta n$ is exponentially small in the system size $L$ [see Eq. (28)] for an extensively long time.

Figure 3

Correlations between the reference point $P^{*}$ (red dot) and any other point $P$ along the ring (black dots) evolve in time, as a result of the quench because pairs of counter-propagating excitations depart from intermediate ring points (grey dots) and reach the envisaged red and black points. The nonlocal presence of the flux can be observed only when two pairs of excitations, departed from intermediate points $Q_1$ and $Q_2$, have covered the entire ring and have reached the two points (red and black) simultaneously. The two panels illustrate two types of gauges describing the presence of the magnetic flux threading the ring. (a) The flux phase $\phi$ is uniformly distributed along the ring, so that the vector potential along each link is $O(\phi /L)$. (b) The flux phase $\phi$ is accumulated only along the link located oppositely to the reference point $P^{*}$.

Figure 4

Dynamics of the SSH ring [Eq. (14) with $U=0$], with $L=80$ cells ($=160$ sites), threaded by a flux ($\phi =\pi /4$), quenched with the protocol (24), with dimerization parameter $r=0.7$. (a) density plot of the space-time behavior of correlations, with $l\in [-L/2,L/2]$ denoting the distance between two sites. (b) The dynamical behavior of the site density deviation $\delta n$ from $1/2$ [see Eq. (21)]. (c) The time evolution of the maximally localized Wannier functions. (d) Time-evolution of the quantifiers of local breaking for chiral symmetry (green curve) and charge-conjugation symmetry (red curve).

Figure 5

(a) The global quantifier $Q_{\mathrm{glob}}^C$ of $\mathcal{C}$-symmetry breaking, Eq. (40), is shown in a log-log scale as a function of the number of cells $L$, for the ground states of four different Hamiltonians: the metallic case ($v=w$) and the insulating SSH case ($v=1$ and $w=0.7$), where the $\mathcal{C}$ symmetry is broken locally by an on-site potential $u=1$ (red and cyan curves, respectively), and broken nonlocally by a flux $\phi =\pi /2$ (black and blue curves, respectively). (b) The behavior of the local quantifier $Q_{\mathrm{loc}}^C$ of $\mathcal{C}$-symmetry breaking, Eq. (43), is shown in a log-log scale as a function of $L$, for the same ground states as in panel (a). For the case of the SSH insulator with $\mathcal{C}$ symmetry broken nonlocally by the flux (blue curve), $Q_{\mathrm{loc}}^C\sim exp[-L/\mathrm{\Lambda }]$ decreases exponentially with the system size $L$ (here $\mathrm{\Lambda }\simeq 5.3$), in striking contrast with the algebraic decay of the other local quantifiers and of all global quantifiers.

Figure 6

Effects of disorder (51). The dynamical behavior of $\delta n$, the local density deviation from $1/2$, is plotted as a function of time for a quenched SSH ring, for various values of the disorder strength $\sigma$ [see Eq. (51)], namely, $\sigma =0$ (clean case, green curve), $\sigma =0.1$ (orange curve), and $\sigma =0.2$ (purple curve). The number of cells is $L=12$ ($=24$ sites), the dimerization is $r=0.1$, the flux phase is $\phi =\pi /2$ and quench protocol is specified in Eq. (24).

Figure 7

Effects of interaction term Eq. (52). The dynamical behavior of the local density deviation $\delta n$ is plotted as a function of time for a SSH ring with $L=7$ cells ($=14$ sites), threaded by a flux ($\phi =\pi /2$) and undergoing a quench (24) with dimerization $r=0.1$, for various values of the interaction strength $V$, namely, $V=0$, (green curve), $V=+0.1$ (red curve), and $V=-0.1$ (black curve).

Figure 8

Sketch of the structure of the single-particle density matrix $\rho$ of the ground state of the SSH model with flux, evaluated in real space for two different $\left\{\theta \right\}$ phase sets. The ivory areas identify the exponentially small entries of $\rho$, while the cyan pattern highlights how the flux phase $\phi$ is distributed over the finite-value entries, here marked as the orange areas. (a) The case of the phase set $\left\{{\theta }_u\right\}$ in Eq. (D1), corresponding to the gauge illustrated in Fig. 3: the flux is uniformly distributed over the various $\rho$ entries. (b) The case of the phase set $\left\{{\theta }_{\ell }\right\}$ in Eq. (20), corresponding to the gauge illustrated in Fig. 3, where the flux phase is accumulated in the farthest link with respect to the considered reference point $P^{*}$. The dashed box highlights the $(j^{*},{\alpha }^{*})$ row, which determines $Q_{\mathrm{loc}}^C$ in Eq. (43) and describes the correlations between the chosen reference point [specifically, the site $P^{*}=(L/2,A)$, depicted as a red spot] and any other lattice site. An insulator is locally insensitive to a flux because in the in the local $\mathcal{C}$-breaking quantifier $Q_{\mathrm{loc}}^C$, the flux phase only appears in the exponentially small tails located at the box ends, since the localization length $\lambda$ is much smaller than the system size $L$.