Periodically, quasiperiodically, and randomly driven conformal field theories

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Periodically, quasiperiodically, and randomly driven conformal field theories

Xueda Wen, Ruihua Fan, Ashvin Vishwanath, and Yingfei Gu

Phys. Rev. Research 3, 023044 – Published 14 April 2021

Abstract

In this paper and its upcoming sequel, we study nonequilibrium dynamics in driven ( $1 + 1$ )-dimensional conformal field theories (CFTs) with periodic, quasiperiodic, and random driving. We study a soluble family of drives in which the Hamiltonian only involves the energy-momentum density spatially modulated at a single wavelength. The resulting time evolution is then captured by a Möbius coordinate transformation. In this paper, we establish the general framework and focus on the first two classes. In periodically driven CFTs, we generalize earlier work and study the generic features of entanglement and energy evolution in different phases, i.e., the heating and nonheating phases and the phase transition between them. In quasiperiodically driven CFTs, we mainly focus on the case of driving with a Fibonacci sequence. We find that (i) the nonheating phases form a Cantor set of measure zero; (ii) in the heating phase, the Lyapunov exponents (which characterize the growth rate of the entanglement entropy and energy) exhibit self-similarity, and can be arbitrarily small; (iii) the heating phase exhibits periodicity in the location of spatial structures at the Fibonacci times; (iv) one can find exactly the nonheating fixed point, where the entanglement entropy and energy oscillate at the Fibonacci numbers, but grow logarithmically and polynomially at the non-Fibonacci numbers; (v) for certain choices of driving Hamiltonians, the nonheating phases of the Fibonacci driving CFT can be mapped to the energy spectrum of electrons propagating in a Fibonacci quasicrystal. In addition, another quasiperiodically driven CFT with an Aubry-André–type sequence is also studied. We compare the CFT results to lattice calculations and find remarkable agreement.

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Received 28 June 2020
Revised 18 January 2021
Accepted 22 January 2021

DOI:https://doi.org/10.1103/PhysRevResearch.3.023044

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Nonequilibrium statistical mechanics

Condensed Matter, Materials & Applied PhysicsStatistical Physics & ThermodynamicsQuantum Information, Science & Technology

Authors & Affiliations

Xueda Wen ¹, Ruihua Fan², Ashvin Vishwanath², and Yingfei Gu ²

¹Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
²Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

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Vol. 3, Iss. 2 — April - June 2021

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Figure 1
Typical features of the time evolution of entanglement entropy in different phases of a periodically driven CFT. The entanglement entropy grows linearly in time in the heating phase, grows logarithmically at the phase transition, and simply oscillates in the nonheating phase.
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Figure 2
A cartoon of the entanglement pattern and the energy-momentum density distribution in real space in the heating phase of a periodically driven CFT, where we drive the system with $H_{0} (x) = \int_{0}^{L} h (x) d x$ and $H_{1} (x) = \int_{0}^{L} 2 {sin}^{2} \frac{q π x}{L} h (x) d x$ with $q = 4$ here. Red and blue colors stand for two different chiralities. Each peak is entangled with its nearest neighbor with the same chirality and color. Periodic boundary conditions are assumed here.
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Figure 3
A local view of the operator evolution on a Riemann surface. By choosing suitable coordinates, each step of the driving can be characterized by a Möbius transformation that is determined by the ${SL}_{2}$ deformed Hamiltonian.
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Figure 4
Conformal map $z = e^{\frac{2 π q w}{L}}$ from the $w$ cylinder (where $w = τ + i x$ , and $x = L$ and 0 are identified) to the $q$ -sheet Riemann surface $z$ .
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Figure 5
A general protocol for a periodically driven CFT. There are $p$ steps of driving within each driving period. In the $i th$ step of driving, we consider the driving with $(H_{i}, T_{i})$ , where $H_{i}$ is a ${SL}_{2}$ deformed Hamiltonian in Eq. (7) and $T_{i}$ is the corresponding time interval of driving.
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Figure 6
Illustration for the locations of fixed points of Möbius transformation in the three phases. In the nonheating phase, the two fixed points are inside and outside the unit circle, respectively. They will merge at the same point on the unit circle at the phase transition. Then, the two fixed points will split but still sit on the unit circle in the heating phase.
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Figure 7
Lyapunov exponent with (from left to right) $θ = 0.2$ , 0.5, 1, and $\infty$ . The regime with $λ_{L} > 0$ corresponds to the heating phase, while the dark blue regime with $λ_{L} = 0$ corresponds to nonheating phase, and boundary between them is the phase transition.
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Figure 8
Trajectories of $(ρ_{n p} ζ_{n p})$ on the unit disk $D$ in a periodically driven CFT driven with $H_{0}$ and $H_{1} = H_{θ}$ in Eq. (71). Here we choose $θ = 0.2$ , $T_{0} = L / 2$ , and $T_{1} / [L cosh (2 θ)] = 0.3$ in the nonheating phase (left), $T_{1} / [L cosh (2 θ)] = 0.3758$ in the nonheating phase near the phase transition (middle, the phase transition happens at $\frac{1}{π} arcsin [1 / cosh (2 θ)] ≃ 0.3759$ ), and $T_{1} / [L cosh (2 θ)] = 0.376, 0.4$ , and 0.45 (in the counterclockwise order) in the heating phase (right).
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Figure 9
Comparison of the CFT and lattice calculations on the entanglement entropy (left) and the total energy (right) evolution in the heating phase of a periodically driven CFT. The CFT is periodical driving with $H_{0}$ and $H_{θ}$ with time intervals $T_{0} = L / 2$ and $T_{1} = L_{eff} (θ) / 2$ , respectively. From bottom to top, we choose $θ = 0.03$ , 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, and 0.1. The CFT results are plotted according to Eq. (83).
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Figure 10
A Fibonacci driving is generated by two unitaries $U_{A} = e^{- i H_{A} T_{A}}$ and $U_{B} = e^{- i H_{B} T_{B}}$ following the pattern of the Fibonacci bitstring $10110101 ...$ .
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Figure 11
Two-dimensional manifolds $M$ determined by Eq. (93), with $I = 0.5$ , 0, and $- 0.1$ , respectively. The manifold with $I = 0$ is called the Cayley cubic.
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Figure 12
Return orbits and initial conditions: here we plot $M$ with $I = \frac{1}{4}$ fixed. The right plot is the same as the left one with a different angle of view. The six blue solid dots correspond to the period-6 returning orbit or equivalently speaking the fixed points of $T^{6}$ action. This orbit can be viewed as the limit of a family of initial conditions: for each fixed $θ$ , the allowed initial condition forms a line [two parameters $T_{0}, T_{1}$ with one constraint (97)]. In the figure, we set $θ = 1$ , 0.4, 0.27, and 0.245 for four loops from big to small. Then, if we further decrease $θ$ to a critical value $θ^{*}$ such that $I = {cosh}^{2} (2 θ^{*}) - 1$ , then we have essentially only one possible initial condition $T_{0} = L / 2$ and $T_{1} = L_{eff} / 2$ which will generate the fixed points.
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Figure 13
Phase diagrams in a periodically driven CFT with the sequence generated by finitely truncated Fibonacci bit string, i.e., ${X_{j}}$ with $ω_{n} = F_{n - 1} / F_{n}$ . Here we choose $n = 2$ , 4, 5, 6, 10, 20, 100, and 1000 for eight plots, respectively. The two Hamiltonians we use are $H_{0} (θ = 0)$ and $H_{1} (θ = \infty)$ in (71). The blue (yellow) regions correspond to the heating (nonheating) phases.
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Figure 14
Phase diagrams in a periodically driven CFT with the sequence generated by finitely truncated Fibonacci bit string, i.e., ${X_{j}}$ with $ω_{n} = F_{n - 1} / F_{n}$ . Here we choose $n = 2$ , 4, 5, 6, 10, 20, 100, and 1000 for eight plots, respectively. The two Hamiltonians we use are $H_{0} (θ = 0)$ and $H_{1} (θ = 0.5)$ in (71). The phase diagram is periodic in the $T_{0}$ direction with period $L$ and in the $T_{1}$ direction with period $L cosh (2 θ) ≃ 1.54 L$ . The blue (yellow) regions correspond to the heating (nonheating) phases.
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Figure 15
Phase diagrams in a periodically driven CFT with the sequence chosen in (101) where $ω_{n} = F_{n - 1} / F_{n}$ . The parameters are the same as those in Fig. 13 except that now we change the variables to $V_{CFT}$ and $E_{CFT}$ [see Eq. (112)]. The blue (yellow) regions correspond to the heating (nonheating) phases.
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Figure 16
Initial conditions in Eqs. (108) and (112) on the two-dimensional manifolds $M$ determined by Eq. (93), with $I = V^{2} = 0 . 1^{2}$ , $0 . 5^{2}$ , and $1 . 5^{2}$ , respectively. The green solid lines correspond to the initial condition line $l_{(E, I)}$ in Eq. (108) for a quasicrystal, and the red solid lines correspond to $l_{(E_{CFT}; I)}$ of a fixed length 2 in Eq. (112) for a quasiperiodically driven CFT. For smaller $I$ , $l_{(E_{CFT}; I)}$ overlaps with $l_{(E, I)}$ mainly in the region with $| x |$ , $| y | ⩽ 1, z = 1$ . As $I$ increases, $l_{(E_{CFT}; I)}$ overlaps with $l_{(E, I)}$ mainly in the region with $| x |$ , $| y | > 1, z = 1$ . That is, as $I$ increases, $l_{(E_{CFT}; I)}$ moves from the middle “island” into the noncompact funnel. This behavior agrees with the feature of the phase diagram in Fig. 15 where the nonheating phases vanish for larger $V_{CFT}$ .
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Figure 17
Left: measure of the nonheating phases $σ_{n} (I, θ)$ in the phase diagram in Figs. 15 ( $θ = \infty$ ), 18 ( $θ = 0.5$ ), and 40 ( $θ = 0.2$ ) as a function of $n$ for different $I$ . From top to bottom, we consider $I = 0 . 04^{2}$ , $0 . 06^{2}$ , $0 . 1^{2}$ , $0 . 15^{2}$ , $0 . 2^{2}$ , $0 . 25^{2}$ , and $0 . 3^{2}$ , respectively. One can find that the measure is $σ_{n} (I) \propto e^{- λ (I, θ) n}$ , where $n$ corresponds to the subscript in $F_{n}$ . Right: the escape rate $λ (I, θ)$ as a function of $I$ . It is found that $λ (I, θ)$ with different $θ$ collapse to the same line described by $y = a x^{b}$ , where $a ≃ 0.285$ and $b ≃ 0.552$ .
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Figure 18
Phase diagrams in a periodically driven CFT with the sequence generated by finitely truncated Fibonacci bit string, i.e., ${X_{j}}$ with $ω_{n} = F_{n - 1} / F_{n}$ . The parameters are the same as those in Fig. 14 except that now we change the variables to $V_{CFT}$ and $E_{CFT}$ [see Eq. (112)]. The blue (yellow) regions correspond to the heating (nonheating) phases.
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Figure 19
Self-similarity in the distribution of Lyapunov exponents along $E_{CFT}$ in Figs. 15 and 18. Left: the CFT is driven by $H_{0}$ and $H_{θ = \infty}$ . We fix $I = V_{CFT}^{2} = 0 . 3^{2}$ , and scan the Lyapunov exponents along $E_{CFT} - 3 V_{CFT}$ . Right: the CFT is driven by $H_{0}$ and $H_{θ = 0.5}$ . We fix $I = [{cosh}^{2} (2 θ) - 1] V_{CFT}^{2} = 0 . 3^{2}$ and scan the Lyapunov exponents along $E_{CFT}$ . Each curve in the lower panel is the zoom-in plot of the region in blue dashes in the upper panel. We choose $n = 1000$ in $ω_{n} = F_{n - 1} / F_{n}$ here.
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Figure 20
Self-similarity in the distribution of Lyapunov exponents along $T_{1} / L$ in Figs. 13 and 14. The CFT is driven by $H_{0}$ and $H_{θ = \infty}$ (left), and $H_{0}$ and $H_{θ = 0.5}$ (right). Fixing $T_{0} / L = \frac{1}{2}$ , we scan the Lyapunov exponents along $T_{1} / L$ . Each curve in the lower panel is the zoom-in plot of the region in blue dashes in the upper panel. We choose $n = 1000$ in $ω_{n} = F_{n - 1} / F_{n}$ here.
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Figure 21
Lyapunov exponents as a function of $T_{0} / L$ and $T_{1} / L$ for different choices of drivings (from left to right): $H_{0}$ and $H_{θ = \infty}$ , $H_{0}$ and $H_{θ = 0.5}$ , $H_{0}$ and $H_{θ = 0.2}$ . The Lyapunov exponents are obtained by choosing $ω_{n} = F_{n - 1} / F_{n}$ in Eq. (118), with $n = 1000$ .
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Figure 22
Time evolution of the entanglement entropy of $A = [0, L / 2]$ at the nonheating fixed point in Eq. (96) for $θ = 0.5$ (left) and $θ = 0.2$ (right). The red solid lines correspond to the entanglement entropy at the Fibonacci numbers $n = F_{j}$ , with the expression given in Eq. (123), where we choose $c = 1$ . The points in rectangles, downward triangles, upward triangles, and diamonds correspond to the entanglement entropy evolution at the non-Fibonacci numbers, with the expressions given in Eqs. (129), (133), (137), and (140), respectively.
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Figure 23
Time evolution of the total energy at the nonheating fixed point in Eq. (96) for $θ = 0.5$ (left) and $θ = 0.2$ (right). The red solid lines are the total energy at the Fibonacci numbers $n = F_{j}$ , with the expression given in Eq. (123), where we choose $c = 1$ and $L = 1$ . The points in rectangles, downward triangles, upward triangles, and diamonds correspond to the entanglement entropy evolution at the non-Fibonacci numbers, with the expressions given in Eqs. (129), (133), (137), and (140), respectively.
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Figure 24
Comparison of the lattice and CFT calculations at the nonheating fixed point for the entanglement entropy evolution (left) and the total energy evolution (right). Here we choose $c = 1$ , $A = [0, L / 2]$ , $θ = 0.02$ , and $n_{max} = F_{11} = 144$ . In the lattice calculation we consider $L = 2000$ . A plot of the entanglement entropy for a larger $n_{max}$ can be found in Fig. 25.
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Figure 25
Comparison of the CFT (top left) and lattice calculations on the entanglement entropy evolution $S_{A} (n)$ at the nonheating fixed point. Here we choose $c = 1$ , $A = [0, L / 2]$ , $θ = 0.02$ , and $n_{max} = F_{15} = 987$ . In the lattice calculation, we consider $L = 500$ (top right), 1000 (bottom left), and 2000 (bottom right), respectively.
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Figure 26
Group walking of $ρ$ (top) and $(ρ ζ)$ (bottom) in the heating phase of a Fibonacci driven CFT. We consider a Fibonacci quasiperiodical driving with $H_{0}$ and $H_{θ = 0.5}$ . The parameters are $T_{0} / L = \frac{1}{2}$ , and (from left to right) $T_{1} / L_{eff} = 0.041$ , 0.04, and 0.0401, respectively. The total number of driving steps is taken as $F_{20} = 10 946$ .
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Figure 27
Group walking of $ρ$ (top) and $(ρ ζ)$ (bottom) in the heating phase of a Fibonacci driven CFT. We consider a Fibonacci quasiperiodical driving with $H_{0}$ and $H_{θ = 0.5}$ . The parameters are $T_{0} / L = 0.6$ , and (from left to right) $T_{1} / L_{eff} = 0.06$ , 0.05, and 0.055, respectively. The total number of driving steps is taken as $F_{20} = 10 946$ .
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Figure 28
Time evolution of the total energy (top) and the entanglement entropy (bottom) of $A = [0, L / 2]$ . The parameters are the same as those in Fig. 26. Note that the total energy and entanglement entropy (in the left plot) are plotted in a small window to see the detailed oscillating structure.
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Figure 29
Time evolution of the total energy (top) and the entanglement entropy (bottom) of $A = [0, L / 2]$ of the system. The parameters are the same as those in Fig. 27. Note that the total energy and entanglement entropy (in the left plot) are plotted in a small window to see the detailed oscillating structure.
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Figure 30
(Top) Group walking of $(ρ ζ)$ at all numbers (green dots) and at Fibonacci numbers (purple). The two driving Hamiltonians are $H_{0}$ and $H_{θ = 0.5}$ . The parameters (from left to right) are $T_{0} / L = \frac{1}{2}$ and $T_{1} / L_{eff} = 0.041$ , $T_{0} / L = \frac{1}{2}$ and $T_{1} / L_{eff} = 0.04$ , $T_{0} / L = 0.6$ and $T_{1} / L_{eff} = 0.06$ , $T_{0} / L = 0.6$ and $T_{1} / L_{eff} = 0.05$ , respectively. The first (last) two plots have the same parameters as those in Fig. 26 (Fig. 27). The total number of driving steps are $F_{25} = 121 393$ . This means there are in total 25 steps of moving (purple lines) at the Fibonacci numbers. (Bottom) $Arg (ρ ζ) \in (- π, π]$ as a function of $n$ , where $n$ denotes the $n th$ Fibonacci numbers $F_{n}$ . The parameters are the same as the top panel, but with a larger driving number $F_{32}$ .
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Figure 31
Aubry-André quasiperiodic driving. In the $n th$ cycle, $H_{0}$ is applied for time $T_{0} = n ω L$ and $H_{1}$ is applied for time $T_{1}$ .
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Figure 32
Phase structure and Lyapunov exponents for the sequence of periodically driven CFT determined by $ω_{n} = F_{n - 1} / F_{n}$ , with $n = 3$ (top left) and $n = 5$ (top right). In the top left and top right, the blue (white) regions are the region where the system is nonheating (heating). In the bottom left where $θ = \frac{1}{5}$ , for a given $n$ , the blue lines (blank region) correspond to the nonheating (heating) phase. In the bottom right (where $θ = \frac{1}{5}$ and $n = 8$ ) is the distribution of Lyapunov exponents.
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Figure 33
(Left) The measure of the nonheating phase $σ_{n} (θ)$ . (Right) The decay rate $λ (θ)$ as a function of $θ$ . For the curve with $θ = 0.1$ , only the data for $n \geq 8$ are used for the fitting.
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Figure 34
The absolute value of the trace of the matrix for one period with $θ = 0$ (left) and $θ = 0.1$ (right). We choose $p = 2$ , $q = 5$ for both plots. In (b), the blue dots are the approximated result in Eq. (164). It matches the exact value quite well.
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Figure 35
The phase diagram for the Aubry-André quasiperiodic driving CFT. The plot uses $θ = 0.2$ and includes all the rational number whose denominators are equal or less than 20. The usage of different colors is only for the purpose of presentation and has no physical meaning. We shift the origin of $T_{1} / L_{eff}$ by $\frac{1}{2}$ when presenting the data.
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Figure 36
The nested structure of the phase diagram for the Aubry-André quasiperiodic driving CFT, with the principal series (left) and the first descendant (right).
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Figure 37
The group walking of $ρ$ (left) and $ρ ζ$ (right). We choose $ω$ being $(\sqrt{5} - 1) / 2$ and $θ = 0.1$ . The choice for $T_{1} / L_{eff}$ is not special, and one can choose any other generic values. In the numerics, we choose $ω = F_{29} / F_{30}$ as an approximation to plots (a) and (b). In (b), the scattered dots in the circle represent ${(ρ ζ)}_{n}$ for $n \leq 500$ . The inset of (b) is $Arg (ρ ζ) / π$ as a function $n$ , with $n$ denoting the $n th$ Fibonacci number $F_{n}$ .
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Figure 38
Comparison of the CFT and lattice calculations on the entanglement entropy (left) and the total energy (right) evolution in the heating phase of a periodically driven CFT. The numerical data in $\circ$ ( $\times$ ) correspond to protocol I (II) in Eq. (A22). The CFT is periodically driven with $H_{0}$ and $H_{θ}$ with time intervals $T_{0} = L / 2$ and $T_{1} = L_{eff} / 2$ , respectively. From bottom to top (in the right plot), we choose $θ = 0.03$ , 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, and 0.1. The CFT results are plotted according to Eqs. (A25) and (A26).
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Figure 39
Phase diagrams in a periodically driven CFT with the sequence generated by finitely truncated Fibonacci bit string, i.e., ${X_{j}}$ with $ω_{n} = F_{n - 1} / F_{n}$ . Here we choose $n = 2$ , 4, 5, 6, 10, 20, 100, and 1000, respectively. The two Hamiltonians we use are $H_{0} (θ = 0)$ and $H_{1} (θ = 0.2)$ in (71). The phase diagram is periodic in $T_{0}$ direction with period $L$ and in $T_{1}$ direction with period $L cosh (2 θ) ≃ 1.08 L$ . The blue (yellow) regions correspond to the heating (nonheating) phases.
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Figure 40
Phase diagrams in a periodically driven CFT with the sequence generated by finitely truncated Fibonacci bit string, i.e., ${X_{j}}$ with $ω_{n} = F_{n - 1} / F_{n}$ . Here we choose $n = 2$ , 4, 5, 6, 10, 20, 100, and 1000, respectively. The two Hamiltonians we use are $H_{0} (θ = 0)$ and $H_{1} (θ = 0.2)$ in (71). The phase diagram is periodic in $T_{0}$ direction with period $L$ and in $T_{1}$ direction with period $L cosh (2 θ) ≃ 1.08 L$ . The blue (yellow) regions correspond to the heating (nonheating) phases.
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Figure 41
An illustration for the characteristic function $χ (t)$ for the Fibonacci sequence. The blue regime stands for value 1 and the yellow for 0.
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