Geometric approach to inhomogeneous Floquet systems

We present a new geometric approach to Floquet many-body systems described by inhomogeneous conformal field theory in $1+1$ dimensions. It is based on an exact correspondence with dynamical systems on the circle that we establish and use to prove existence of (non)heating phases characterized by the (absence) presence of fixed or higher-periodic points of coordinate transformations encoding the time evolution: Heating corresponds to energy and excitations concentrating exponentially fast at unstable such points while nonheating to pseudoperiodic motion. We show that the heating rate (serving as the order parameter for transitions between these two) can have cusps, even within the overall heating phase, and that there is a rich structure of phase diagrams with different heating phases distinguishable through kinks in the entanglement entropy, reminiscent of Lifshitz phase transitions. Our geometric approach generalizes previous results for a subfamily of similar systems that used only the $\mathfrak{sl}\left(2\right)$ algebra to general smooth deformations that require the full infinite-dimensional Virasoro algebra, and we argue that it has wider applicability, even beyond conformal field theory.

Figure 1

Illustrations for our inhomogeneous Floquet drive.

Figure 2

Phase diagrams in $({\tau }_1,{\tau }_2)$-space for $H_1$ with $v_1\left(x\right)=v_1$ and $H_2$ with $v_2\left(x\right)=v_2{w}_2(x/L)$ in Table 1: (a, b) Ex. 1, (c) Ex. 2, and (d) Ex. 3. Blue lines signify transitions between nonheating (white regions) and heating (pink regions) [41], obtained by solving (4) and (5) analytically. Black dashed lines are given by ${\tau }_2=m/k_1{w}_{2,0}-(k_2{w}_{1,0}/k_1{w}_{2,0}){\tau }_1$, $m\in \mathbb{Z}$.

Figure 3

Phase diagram and heating rate for $H_1$ given by homogeneous CFT and $H_2$ by Ex. 4 in Table 1 with $a=6$, $b=3$, and $k=2$. (a) Phase diagram in $({\tau }_1,{\tau }_2)\in [0,1]\times [0,1/w_{2,0}]$ (cf. Fig. 2) due to fixed points (center leaf) computed analytically and periodic points with period 2 (leaves on each side of the center) computed numerically. Regions with two and four unstable periodic points are light and dark pink, respectively. (There are regions due to periodic points with period 3 not shown, and possibly also due to ones with period $p>3$.) (b, c) Rescaled heating rate $\nu t_{\mathrm{cyc}}$ as function of $({\tau }_1,{\tau }_2)$ and along rays with fixed ${\tau }_2$, respectively.

Figure 4

Illustration of fixed points as intersections between the (black solid) curve $y=f_{+}\left(x\right)$ and the straight (black dotted) line $y=x$ (up to integer multiples of $L$). Stable and unstable fixed points are indicated by empty blue and filled red circles, respectively. [The plotted curve is for the case in Fig. 3 at $({\tau }_1,{\tau }_2)=(0.10,0.45)$, which corresponds to a point in one of the darker leaves within the center leaf in Fig. 3, see also column (i) in Figs. 5 and 7.]

Figure 5

Plots for the case in Fig. 3 at (i) $({\tau }_1,{\tau }_2)=(0.10,0.45)$ in the left column and (ii) $({\tau }_1,{\tau }_2)=(0.35,0.08)$ in the right column. In (i) and (ii) there are eight fixed points and eight periodic points with period 2, respectively. (a, b) $x_t^{\mp }\left(x\right)$-trajectories. (c) Energy-density expectation $\langle \mathcal{E}(x;t)\rangle$ for $\mathcal{E}(x;t)={\mathcal{E}}_1(x;t)+(c{v}_1/24\pi )[\{x_t^{-}\left(x\right),x\}+\{x_t^{+}\left(x\right),x\}]$ given by (15) with respect to an arbitrary translation-invariant state. (d) Correlation functions $\langle 0\left|{\mathrm{\Phi }}_r^{†}(x,t){\mathrm{\Phi }}_r(x_0,0)\right|0\rangle$ with respect to the ground state $|0\rangle$ of $H_1$ for primary fields ${\mathrm{\Phi }}_r(x,t)$ with ${\mathrm{\Delta }}_r^{\pm }={\delta }_{r,\pm }$ for $r=\pm$, averaged over $x_0\in [-L/2,L/2]$ uniformly distributed, setting $L=50$. Stable and unstable periodic points are indicated by empty blue and filled red circles, respectively.

Figure 6

Illustrations of the two cases (a) and (b) described in the text for the entanglement entropy $S_A\left(t\right)$ in the heating phase. Stable and unstable periodic points are indicated by empty blue and filled red circles, respectively. For simplicity, we illustrate only the contribution from the right-moving component. (Flows to periodic points with period $p$ should be interpreted as those under $f_{+}^p$.)

Figure 7

Plots of the entanglement entropy $S_{[-L/2,x]}\left(t\right)$ given by (17), setting $L=50$, for the case in Fig. 3 at (i) $({\tau }_1,{\tau }_2)=(0.10,0.45)$ and (ii) $({\tau }_1,{\tau }_2)=(0.35,0.08)$. See the caption to Fig. 5 for further details.

Figure 8

Illustrations of the two cases (a) and (b) described in the text for the mutual information $I_{A;B}\left(t\right)$ in the heating phase. See the caption to Fig. 6 for further details.

Figure 9

Phase diagrams showing heating phases due to only fixed points for the special case when $H_1$ is homogeneous and $H_2$ is given by Ex. 4 with $a=6$, $b=3$, and (a) $k=2$, (b) $k=3$, or (c) $k=4$. In all diagrams, the center leaf-shaped region in light pink contains two unstable fixed points (one for each of the $\pm$-components), while leaves in dark pink contain four or more (an intersection of two leaves with $m$ unstable fixed points each yields a new leaf with $m+2$ unstable fixed points). See the caption to Fig. 2 for further details.

Figure 10

Plots of $f_{+}^{\prime }\left(x_{*,1}^{-}\right)$ (blue curve) and $f_{-}^{\prime }\left(x_{*,2}^{+}\right)$ (red curve) obtained using (C12) as a function of ${\tau }_2$ for ${\tau }_1=1/2$ fixed. The black dashed curve is the maximum, corresponding to the heating rate through (6), and has a cusp at ${\tau }_2=1/4$.

Figure 11

Illustration of the additional case (c) described in the text [cf. (D3)] for the entanglement entropy $S_A\left(t\right)$ in the heating phase. See the caption to Fig. 6 for further details.