Universal quantum criticality in static and Floquet-Majorana chains

The topological phase transitions in static and periodically driven Kitaev chains are investigated by means of a renormalization group (RG) approach. These transitions, across which the numbers of static or Floquet Majorana edge modes change, are accompanied by divergences of the relevant Berry connections. These divergences at certain high-symmetry points in momentum space form the basis of the RG approach, through which topological phase boundaries are identified as a function of system parameters. We also introduce several aspects to characterize the quantum criticality of the topological phase transitions in both static and Floquet systems: a correlation function that measures the overlap of Majorana-Wannier functions, the decay length of the Majorana edge mode, and a scaling law relating the critical exponents. These indicate a common universal critical behavior for topological phase transitions, in both static and periodically driven chains. For the latter, the RG flows additionally display intriguing features related to gap closures at non-high-symmetry points due to momentarily frozen dynamics.

Figure 1

The curvature function $F(k,\mathbf{M})$ near the HSP $k_0=0$ for the static Kitaev chain plotted for several values of ${\mu }_0$ at $\mathrm{\Delta }=0.7$. The critical point is located at ${\mu }_0=1$. As the critical point is approached, the curvature function develops a divergence at the HSP (compare the orange line and the red line), and the divergence flips sign as the system crosses the critical point (compare the red line and the blue line). The CRG procedure requires that the $F(k_0+\delta k,\mathbf{M})$ (red dot) be equal to $F(k_0,{\mathbf{M}}^{\prime })$ (orange dot), as indicated by the dashed line, through which one obtains the CRG flow $\mathbf{M}\to {\mathbf{M}}^{\prime }$ along which the divergence is reduced.

Figure 2

RG flow of the static Kitaev chain described by Eqs. (16) and (18), using $k_0=0$ and $k_0=\pi$. The flow direction is indicated by the arrows, and the color scale indicates the flow rate in log scale. The yellow lines are the critical points ${\mu }_0=\pm t$, where the flow rate diverges, with $t=1$ set to be the energy unit. The blue ellipses are the fixed points described by Eq. (17) where the flow rate vanishes, which are stable in some regions and unstable in the other.

Figure 3

Illustration of the number of FMF's of the driven Kitaev chain for ${\mu }_0=0.1$, plotted as a function of the driving parameters $T$ and ${\mu }_1$. The driving was performed starting from a static topological region. Note that the topological phase diagrams are independent of $\mathrm{\Delta }$. (a) The FMF's with Floquet eigenvalues $+1$. (b) The FMF's with Floquet eigenvalues $-1$. (c) The phase diagram of the system according to the total number of FMF's for each phase. (d) The winding number $W$ stemming from the Berry connection. One sees that the winding number $W$ does not fully coincide with the correct number of FMF's.

Figure 4

Depiction of quasienergy spectra—i.e., eigenvalues of the Floquet operator—as a function of the driving intensity ${\mu }_1$ for an open chain of $N=100$ fermions. The other system's parameters are chosen as follows: (a) ${\mu }_0=0.1,\mathrm{\Delta }=0.1,T=1.0$; (b) ${\mu }_0=0.5,\mathrm{\Delta }=0.9,T=1.9$. The TPT's generating or removing 0-FMF's ($\pi$-FMF's) are marked by gap closings at 0 ($\pi$) with the corresponding appearance or disappearance of eigenvalues at 0 ($\pi$). Note that there are instances of gap closings, marked by red circles, not associated with a change in the topological invariants. Those phase transitions appear in observables such as correlators and are detected by the CRG scheme.

Figure 5

Top: Gap-closing manifests in the quasienergy dispersion ${\theta }_k$, and bottom: the corresponding divergence of the curvature function $F(k,\mathbf{M})$ in the periodically driven Kitaev chain: (a) At $\mathrm{\Delta }=0.5,{\mu }_0=0.1,{\mu }_1=0.9,T=2.0$, which is the critical point of gap-closing at $k_0=0$ and creating a 0-FMF. The inset of the quasienergy plot shows that the gap-closing at $k_0=0$ is in fact linear at low energy, although it looks quadratic at larger scale. (b) At $\mathrm{\Delta }=0.1,{\mu }_0=0.1,{\mu }_1=0.78,T=2.0$, where the gap closes at non-HSPs (see also Fig. 9).

Figure 6

(a) The topological phase boundaries in the $\mathbf{M}=(T,{\mu }_1)$ parameter space for the periodically driven Kitaev chain at ${\mu }_0=0.1$. White lines signal the creation of a 0-FMF and black lines the creation of a $\pi$-FMF. (b) The magnitude of the residual correlator $C_{\text{res}}$ in the same parameter space, taken from Ref. [30]. (c) The RG flow obtained from $k_0=0$. The color codes are log of the numerator $log\left[{\partial }_k^2{F(k,\mathbf{M})|}_{k=0}\right]$ in Eq. (7), with orange the high value and blue the low value. The bright lines (critical points of 0-FMF) coincide with the white lines in (a), and blue lines (stable or unstable fixed points) are close to the white lines in (b) (minimum of $C_{\text{res}}$). (d) The RG flow obtained from $k_0=\pi$, whose bright lines (critical points of $\pi$-FMF) correspond to the black lines in (a).

Figure 7

(a) The CRG flow along the $T$ direction at fixed ${\mu }_1=0.1$ close to the critical point ${\mathbf{M}}_c=(T_c,{\mu }_{1c})\approx (1.9635,0.1)$. The flow has been obtained from $k_0=0$ by numerically evaluating the derivatives with a very find grid $\mathrm{\Delta }k=0.0001$ and $\mathrm{\Delta }T=0.00001$. One sees that the critical point ${\mu }_{1c}\approx 1.9635$ at which $dT/dl$ diverges and the fixed point ${\mu }_{1f}\approx 1.9637$ are extremely close. (b) The inverse of the Wannier state correlation length ${\xi }_{k_0}^{-1}$ and that of the curvature function at HSP, $F{(k_0,T,{\mu }_1)}^{-1}$, both vanish linearly as $T\to T_c$, indicating their critical exponents $\gamma =\nu =1$.

Figure 8

Plot of the Berry connection $F(k,\mathbf{M})$ across the FL for $\mathrm{\Delta }=0.1,{\mu }_0=0.1$, and at (a) ${\mu }_1=1.5$ and (b) ${\mu }_1=1.0$. Panel (a) shows the Berry connection for a region with one FMF (see also Fig. 3). The same number of positive and negative peaks flips sign. Hence, the topological index $W=-1$ is unchanged during this transition. Panel (b) shows instead a region with two FMF's, while the topological index $W$ changes from 0 to $+2$. Note that the smaller peaks centered around $k\approx 1.4$ have the same weight (area under the peak) as the diverging peaks that flip sign.

Figure 9

Backfolding of the upper band of (a) the static energy dispersion into (b) the first Floquet-Brillouin zone corresponding to the LCFLs where the dynamics is frozen. This backfolding procedure induces apparent gap closings at non-HSPs. The parameters are $\mathrm{\Delta }=0.1,{\mu }_0=0.1,{\mu }_1=1.26$, and $T=2.5$.

Figure 10

Comparison between the topological invariant $\mathcal{M}=N_0+N_{\pi }$ and the residual correlator $C_{\text{res}}$ for the $(T,{\mu }_1)$-phase diagram of the driven Kitaev chain with $\mathrm{\Delta }=0.1,{\mu }_0=0.5$. The topological invariant seems to indicate additional phase boundaries at higher periods, where the number of FMF's jumps by 2. These additional phase boundaries coincide with the fixed lines of the CRG flow and the lines of low correlation in $C_{\text{res}}$.

Figure 11

Normalized quasienergy spectrum ${\epsilon }_i/\pi$ across the LCFL at ${\mu }_0=0.5,T=2.4,{\mu }_1=0.6$–0.7 for open boundary conditions [panels (a) and (b)] and periodic boundary conditions [panels (c) and (d)]. The spectrum is shown in proximity of quasienergy 0 [panels (b) and (d)] and $\pi$ [panels (a) and (c)]. A small but sizable gap closes at ${\mu }_1\approx 0.65$ in correspondence to the fixed point of the CRG flow, also for periodic boundary conditions.

Figure 12

Illustration of the eigenfunctions corresponding to the quasienergies $\pi$ (${\psi }_1$ and ${\psi }_2$) and 0 (${\psi }_3$) below the FL at $T=2.4,{\mu }_1=0.6$ for a chain of $N=2000$ fermions. The other parameters are $\mathrm{\Delta }=0.1$ and ${\mu }_0=0.5$. Note that the additional $\pi$-modes tend to localize at the edges, but their localization length is much larger than that of the 0-mode (bottom panel). These modes are expected to localize asymptotically as $N\to \infty$.