Integrable many-body quantum Floquet-Thouless pumps

TL;DR

This paper introduces an exactly solvable interacting quantum Floquet model with topologically nontrivial quasiparticles, revealing new integrable structures and operator dynamics in periodically driven quantum systems.

Contribution

It presents a novel integrable Floquet model with topological quasiparticles, solvable via simplified Bethe equations, and demonstrates unique operator spreading properties.

Findings

Exact solution of the Bethe equations for the model

Construction of operators with no butterfly effect

Illustration of a new class of solvable Floquet quantum systems

Abstract

We construct an interacting integrable Floquet model featuring quasiparticle excitations with topologically nontrivial chiral dispersion. This model is a fully quantum generalization of an integrable classical cellular automaton. We write down and solve the Bethe equations for the generalized quantum model, and show that these take on a particularly simple form that allows for an exact solution: essentially, the quasiparticles behave like interacting hard rods. The generalized thermodynamics and hydrodynamics of this model follow directly. Although the model is interacting, its unusually simple structure allows us to construct operators that spread with no butterfly effect; this construction does not seem to be possible in other interacting integrable systems. This model illustrates the existence a new class of exactly solvable, interacting quantum systems specific to the Floquet setting.

Equations

$$\hat{F}\left(\lambda \right)=e^{-i\lambda \hat{H}}\prod _{j\text{even}}{\hat{U}}_{j-1,j,j+1}^{}\prod _{j\text{odd}}{\hat{U}}_{j-1,j,j+1}^{},$$

$$\hat{H}$$

$$\hat{F}\left(\lambda \right)|k,\nu ⟩={\mathrm{e}}^{-i\nu k-2i\lambda \cos k}|k,\nu ⟩,$$

$$e^{i{k}_j^{+}L}$$

$$e^{i{k}_j^{-}L}$$

$$\Theta =\frac{2\pi N_{\theta }+(N_{+}-N_{-}-L)K}{L+N_{-}+N_{+}},$$

$$k_j^{\nu }=\frac{\pi (2m_j^{\nu }+N_{\nu }-1)-\nu \Theta }{L-N_{\nu }+N_{\bar{\nu }}},$$

$$2\pi {\rho }_{\pm }^{\mathrm{tot}}(q)=$$

$${ϵ}_{\pm }={\mu }_{\pm }+\log \left(\frac{1+e^{-{ϵ}_{\pm }}}{1+e^{-{ϵ}_{\mp }}}\right).$$

$$v_{\pm ,k}=v_{\pm ,k}^0+\int dq(v_{\pm ,k}-v_{\pm ,q}){\rho }_{\pm ,q}-\int dq(v_{\pm ,k}-v_{\mp ,q}){\rho }_{\mp ,q}.$$

Full text

Periodically driven (or “Floquet”) quantum systems have become an important and fruitful theme in condensed matter physics Eckardt (2017); Bukov et al. (2015); Meinert et al. (2016); Holthaus (2016); Goldman and Dalibard (2014); Kitagawa et al. (2010); Lindner et al. (2011); Rudner et al. (2013); Nathan and Rudner (2015); Oka and Aoki (2009); Görg et al. (2018); Bukov et al. (2016); Harper et al. (2019): driving enables one to engineer and stabilize exotic states of matter, as has been experimentally demonstrated Wang et al. (2013); Aidelsburger et al. (2013); Miyake et al. (2013); Jotzu et al. (2014); moreover, driven systems can support phases such as anomalous insulators Lindner et al. (2011); Gu et al. (2011); Rudner et al. (2013); Nathan and Rudner (2015); Kitagawa et al. (2011) and quantum time crystals Khemani et al. (2016); von Keyserlingk et al. (2016); Else et al. (2016, 2017); Yao et al. (2017); Ho et al. (2017); Zhang et al. (2017); Choi et al. (2017) that are absent in equilibrium. Driven free-particle systems are the best understood case: these have band structures that are compactified in both quasi-momentum and quasi-energy, and the fact that quasi-energy is only defined on a ring allows for new topological indices that are unrealizable for local lattice Hamiltonians Oka and Aoki (2009); Kitagawa et al. (2010); Lindner et al. (2011); Gu et al. (2011); Rudner et al. (2013); Nathan and Rudner (2015); Kitagawa et al. (2011). For instance, Floquet systems can have a single chiral mode, which under Hamiltonian dynamics could only exist on the boundary of a higher-dimensional system Kitagawa et al. (2010); Rudner et al. (2013); Titum et al. (2015, 2016). These topological indices are sharply defined for free-particle systems, and are potentially long-lived in some interacting lattice models Kuwahara et al. (2016); Abanin et al. (2015); Lindner et al. (2017); however, in general interactions heat a system up to infinite temperature, unless it is either integrable or many-body localized (MBL) Basko et al. (2006); Oganesyan and Huse (2007); Pal and Huse (2010); Nandkishore and Huse (2015); Altman and Vosk (2015); Vasseur and Moore (2016); Abanin et al. (2018). Although MBL can protect Huse et al. (2013); Bahri et al. (2015); Bauer and Nayak (2013) Floquet topological phases Khemani et al. (2016); Ponte et al. (2015); von Keyserlingk and Sondhi (2016a, b); Else and Nayak (2016); Potter et al. (2016); Roy and Harper (2016); Roy and Sreejith (2016); Roy and Harper (2017); Po et al. (2016); Kolodrubetz et al. (2018), these phases are localized and do not host chiral modes. However, in addition to MBL systems, interacting integrable systems are another broad class of systems—including the canonical Hubbard, Heisenberg, and Lieb-Liniger models—that do not heat up to infinite temperature Kinoshita et al. (2006); Hild et al. (2014); Rigol et al. (2008); whether distinctively Floquet versions of these models exist has been less discussed Gritsev and Polkovnikov (2017); Cubero (2018); Claeys et al. (2018); Vanicat et al. (2018).

This work presents an interacting integrable Floquet model that has quasiparticles with nontrivial winding. This model is thus a many-body realization of a quantum Thouless pump Thouless (1983); Kitagawa et al. (2010); Lindner et al. (2017). Unlike previously proposed interacting integrable Floquet systems, our model is not smoothly connected to any Hamiltonian, and is thus an inherently Floquet model rather than an “integrable Trotterization” Vanicat et al. (2018). This model is a fully quantum extension of an integrable cellular automaton (known as Rule 54, or the Floquet Fredrickson-Andersen (FFA) model Bobenko et al. (1993); Prosen and Mejía-Monasterio (2016); Gopalakrishnan (2018)) that has received much recent attention for its simplicity, which allows one to address explicitly various puzzles concerning hydrodynamics and operator growth in generic interacting integrable systems Prosen and Mejía-Monasterio (2016); Prosen and Buča (2017); Gopalakrishnan and Zakirov (2018); Gopalakrishnan (2018); Gopalakrishnan et al. (2018); Klobas et al. (2018); Buča et al. (2019); Alba et al. (2019). The FFA model can be written as a Floquet unitary comprising local quantum gates; however, it is classical in the sense that it maps computational-basis product states to one another. Although FFA has chiral quasiparticles, they do not disperse, but instead all have one of two group velocities, $\pm 1$. The dispersing FFA (DFFA) generalization we present here involves alternating the FFA dynamics with that of a particular strictly local Hamiltonian. This makes the model fully quantum by restoring dispersion while preserving integrability. As we will show, our generalization also preserves enough of the simplicity of FFA that the Bethe equations can be solved analytically—a remarkable feature for an interacting model. The reason this model is so simple is that the quantization for either species of quasiparticle depends only on the total number of quasiparticles of each species, and not on their rapidities. This simplicity also manifests itself in the existence of special local operators that remain lightly entangled at all times, as in the FFA model Gopalakrishnan et al. (2018); Alba et al. (2019). This model is the first representative of a new class of interacting integrable models specific to the Floquet setting, featuring stable chiral quasiparticles.

Model.—We consider a chain of $2L$ qubits (spins-$\frac{1}{2}$) with dynamics generated by the repeated application of the unitary evolution (Floquet) operator

[TABLE]

with gates $\hat{U}^{\vphantom{\dagger}}_{j-1,j,j+1}\equiv\,\mathrm{CNOT}(1\rightarrow 2)\,\mathrm{CNOT}(3\rightarrow 2)\,\mathrm{Toff.}(1,3\rightarrow 2)$, in terms of controlled NOT (CNOT) and Toffoli gates Nielsen and Chuang (2011); and $H$ is a Hamiltonian perturbation that we will specify below. In simpler terms, $\hat{U}^{\vphantom{\dagger}}_{j-1,j,j+1}$ is the instruction “flip spin $j$ if one or both of its nearest neighbors is up.” When $\lambda=0$ this model reduces to the FFA model, $\hat{F}\left(0\right)=\hat{F}^{\vphantom{\dagger}}_{0}$.

FFA limit.—On its own, $\hat{F}^{\vphantom{\dagger}}_{0}$ hosts two species of chiral quasiparticle excitations above the vacuum state $\mathinner{|{0}\rangle}=\mathinner{|{\downarrow^{\vphantom{\dagger}}\,\downarrow^{\vphantom{\dagger}}\dots\downarrow^{\vphantom{\dagger}}}\rangle}$, indexed $\nu=+1$ for “right movers” and $\nu=-1$ for “left-movers”. We regard the $2L$ physical sites as $L$ unit cells: the $n^{\rm th}$ unit cell contains the $A$ site $2n-1$ and $B$ site $2n$. If both of these sites are $\uparrow^{\vphantom{\dagger}}$, then there is a $\nu=+1$ right-moving doublon in cell $n$; if the $B$ site of cell $n-1$ and $A$ site of $n$ are both $\uparrow^{\vphantom{\dagger}}$, there is a $\nu=-1$ left-moving doublon in cell $n$. Additionally, we refer to isolated $\uparrow^{\vphantom{\dagger}}$’s as molecules, which contain one of each mover: a molecule on the $A$ site of cell $n$ corresponds to both $\nu=\pm 1$ movers in cell $n$; a $B$ molecule in cell $n$ corresponds to a $+$ at $n$ and $-$ at $n+1$. The molecule states $\downarrow^{\vphantom{\dagger}}\uparrow^{\vphantom{\dagger}}\downarrow^{\vphantom{\dagger}}$ arise during collisions between the two species. Apart from these collisions, FFA acts by changing the positions (in unit cells) of the $\pm$ particles by $\pm 1$, and conserves independently the number of each, $N_{\pm}$.

Thus in the FFA model all quasiparticles on top of a given state have the same velocity. The structure of conservation laws in this model differs from that of generic interacting integrable models, in which a generalized Gibbs ensemble (GGE) Vidmar and Rigol (2016) can be fully specified through the distribution of quasiparticle velocities. In the FFA model, there are only two velocities, which do not fully specify a state. The remaining conservation laws correspond to asymptotic “spacings” between adjacent quasiparticles of the same species Gopalakrishnan (2018). In the zero-density limit, the bare spacings between same-species quasiparticles are conserved, since all such quasiparticles have the same velocity. At nonzero densities, one can define an asymptotic spacing by accounting for interaction effects: e.g., suppose we have two $+$ quasiparticles that are $n$ steps apart; the quasiparticle on the right collides first with a $-$ quasiparticle and is time delayed by one step: therefore, while there is a $-$ quasiparticle between them, the two $+$ quasiparticles will be exactly $n-1$ steps apart if their asymptotic spacing is $n$. Given a spin configuration, its asymptotic spacings can be found numerically by simulating its free expansion into vacuum Gopalakrishnan (2018).

Adding dispersion.— We now construct $\hat{H}$, the Hamiltonian part of (1), to generate dispersion while maintaining integrability. Conservation of particle number automatically precludes many simple terms, i.e. most single spin processes. Even a more complicated pair-hopping term like $\hat{\sigma}^{+}_{i}\hat{\sigma}^{+}_{i+1}\hat{\sigma}^{-}_{i+2}\hat{\sigma}^{-}_{i+3}$ will not always conserve $N_{\pm}$: it can adjoin two doublons of the same species, producing another of the opposite type. The simplest $N_{\pm}$-conserving operator that disperses quasi-particles is $\hat{h}^{\vphantom{\dagger}}_{j}\equiv\hat{d}^{\vphantom{\prime}}_{j-1}\hat{\sigma}^{+}_{j}\hat{\sigma}^{+}_{j+1}\hat{\sigma}^{-}_{j+2}\hat{\sigma}^{-}_{j+3}\hat{d}^{\vphantom{\prime}}_{j+4}$, where $\hat{d}^{\vphantom{\prime}}_{j}\equiv\frac{1}{2}(1-\hat{\sigma}^{z}_{j})$ [analogously, $\hat{u}^{\vphantom{\prime}}_{j}\equiv\frac{1}{2}(1+\hat{\sigma}^{z}_{j})$]. This term “checks” that it would not create any new quasiparticles before moving one.

Setting $\hat{H}=\sum_{j}(\hat{h}^{\vphantom{\dagger}}_{j}+\hat{h}^{\dagger}_{j})$ would give a simple dispersive extension of FFA; however, we cannot confirm that this preserves integrability. Hence, we add other terms:

[TABLE]

where “refl.” indicates that one should reverse the sequence of indices in the previous term. In the quasiparticle language, $\hat{H}$ (2) maps a configuration $\sigma$ to a uniform superposition of all configurations $\sigma^{\prime}$ with a single quasiparticle moved by one unit cell, provided $N_{\pm}$ are preserved. Although this is precisely what one expects of a generic dispersing term, in this system it requires multiple microscopic processes.

We remark that (2) commutes with the FFA unitary, $\hat{F}^{\vphantom{\dagger}}_{0}$. It acts nontrivially, regardless, because $\hat{F}^{\vphantom{\dagger}}_{0}$ has exponentially degenerate eigenstates: for a given $N_{\pm}$ in a system of size $L$, there are only $O(L^{2})$ eigenvalues, but exponentially many basis states, corresponding to different quasiparticle positions. The perturbation (2) lifts the degeneracy in this subspace, and thus makes the dynamics fully quantum. This perturbation cures many pathological features of the FFA model that are due to these degeneracies, such as its failure to equilibrate to the diagonal ensemble Gopalakrishnan (2018).

Single-quasiparticle sectors.—We start by finding the eigenstates of (1) for a single $\pm$ quasiparticle, $\left|j,\pm\right>=\sigma^{x}_{2j}\sigma^{x}_{2j\pm 1-1}\left|0\right>$. The Fourier transform is an eigenstate of $\hat{F}\left(\lambda\right)$,

[TABLE]

where $\nu=\pm 1$. Here, $\lambda$ controls the strength of the dispersing term, and $k=2\pi m/L$ for integer $m$, with $L$ the system size in unit cells. This model thus has two chiral bands (Fig. 1). For $\lambda<1/2$, all $+$ ($-$) quasiparticles have right- (left-) moving group velocities, but for $\lambda>1/2$, both species have left- and right-moving quasiparticles. The group velocities of $\pm$ quasiparticles are given by $v^{0}_{\pm,k}=\pm 1-2\lambda\sin k$. These chiral bands are characterized by a quantized winding number $\nu=\int_{-\pi}^{\pi}\frac{dk}{2\pi}v_{\pm,k}^{0}=\pm 1$, which is the topological invariant characterizing Thouless’ quantized charge pump Thouless (1983); Kitagawa et al. (2010); Lindner et al. (2017).

Bethe Ansatz solution.—We now move on to multi-particle sectors. We note, first, that in the absence of left-movers, the FFA evolution is just a trivial global translation. In this purely right-moving sector, the dynamics of $+$ quasiparticles consists of hopping and hardcore nearest-neighbor repulsion. The scattering phase shift between particles of the same species is thus $S_{\rm++}(k_{2},k_{1})=S_{\rm--}(k_{2},k_{1})=\mathcal{S}(k_{2},k_{1})=-{\rm e}^{i(k_{2}-k_{1})}$. Meanwhile, the scattering between left and right movers is engineered to retain the FFA form such that the phase shift after a collision is $S_{-+}(k_{+},k_{-})=\tilde{\mathcal{S}}(k_{+},k_{-})=+e^{i\left(k_{+}-k_{-}\right)}$, and no meaning is ascribed to the order of the arguments. Higher-body collisions factorize onto the two-body scattering processes, ensuring integrability. For a many-body state with fixed $(N_{+},N_{-})$, where $\{k^{\pm}_{j}\}$ refer to the momenta of the $\pm$-quasiparticles, we find the following quantization condition sup

[TABLE]

These quantization conditions have the same form as Bethe equations familiar in Hamiltonian systems. Translational invariance and the recurrence properties of the FFA model (with which the Hamiltonian (2) commutes) impose two further constraints. First, we require that $\sum_{j}k_{j}^{+}+\sum_{j}k_{j}^{-}=K$, where $K=2\pi m/L$ with $m$ an integer is one of the allowed global momenta. Second, we require that the relative momentum $\sum_{j}k_{j}^{+}-\sum_{j}k_{j}^{-}=\Theta$, where

[TABLE]

with $1\leq N_{\theta}\leq(L+N_{-}+N_{+})$ an integer, unless $L+N_{-}+N_{+}$ is even, in which case $N_{\theta}$ must be as well sup . Finally, no two quasiparticles of the same species may occupy the same momentum state. With these constraints the solutions (4) fully characterize the eigenstates in a finite system, and the corresponding quasi-energy ${\rm e}^{-i\varepsilon}$ of the Floquet unitary (1) reads $\varepsilon=\sum_{\nu=\pm}\sum_{n=1}^{N_{\nu}}(\nu k^{\nu}_{n}+2\lambda\cos k_{n}^{\nu})$.

Remarkably, these equations are simple enough that they can be solved exactly for any finite system. The set of allowed momenta for either species, $\nu$ is

[TABLE]

with $\bar{\nu}=-\nu$ and $1\leq m^{\nu}_{j}\leq L-N_{\nu}+N_{\bar{\nu}}$. The number of available $m^{\pm}_{j}$ decreases with the total number of $\pm$ movers because neighboring unit cells cannot both host $\pm$’s without a $\mp$ between them. We also note that the quantization condition depends on the total number and momentum of the $\pm$ quasiparticles, not on the details of their distribution. Relatedly, (4) and (6) do not depend on $\lambda$, and thus also apply to $\hat{F}^{\vphantom{\dagger}}_{0}$, though in that model, the phase shift between quasiparticles of the same species is ill-defined as they move in unison and never collide.

The DFFA model corrects several pathological features of FFA. This can be seen numerically by analyzing the (quasi-)energy level statistics, which does not show level repulsion (Fig. 2), consistent with integrability. We also checked that the value of the “$r$-ratio” Oganesyan and Huse (2007) is consistent with a Poisson distribution for all $\lambda>0$.

Thermodynamics.— In the thermodynamic limit, one defines densities of quasiparticles at a given species and rapidity, $\rho_{\pm}(k)$, as well as total densities of states $\rho_{\pm}^{\rm tot}(k)$, related via the Bethe equations

[TABLE]

where $n_{\pm}\equiv\int_{-\pi}^{\pi}dq\rho_{\pm}(q)=N_{\pm}/L$. These equations follow from the continuum limit of (4), with the scattering kernels ${\cal K}_{\nu\nu^{\prime}}=\frac{1}{2\pi i}\frac{d}{dk}\,\ln S_{\nu\nu^{\prime}}$ with $\nu,\nu^{\prime}\in\left\{+,-\right\}$ given by $\mathcal{K}_{++}=\mathcal{K}_{--}=1/(2\pi)$, $\mathcal{K}_{+-}=\mathcal{K}_{-+}=-1/(2\pi)$.

Starting with these equations, one can straightforwardly construct generalized equilibrium states of this Floquet system. We emphasize that since the DFFA model is integrable, its dynamics lead to non-trivial steady states that are distinct from featureless infinite temperature states that would be expected for generic interacting Floquet systems. For concreteness we focus on generalized equilibrium states characterized by a given density of $\pm$ quasiparticles via the partition function $\mathcal{Z}=\sum_{\{\sigma\}}e^{-\mu_{-}N_{-}-\mu_{+}N_{+}}$, but our discussion extends naturally to arbitrary GGEs for this model. In terms of quasiparticle densities, the partition function reads $\mathcal{Z}\sim\int{\cal D}\rho^{\vphantom{\prime}}e^{L\int dkS_{\rm YY}}e^{-L\mu_{+}\int dk\rho^{\vphantom{\prime}}_{k,+}-L\mu_{-}\int dk\rho^{\vphantom{\prime}}_{k,-}}$ where $S_{\rm YY}$ is the so-called Yang-Yang entropy Yang and Yang (1969); Takahashi (1999) associated with the occupation of quasiparticle states. In the thermodynamic limit $L\to\infty$, these integrals are dominated by their saddle point, giving rise to thermodynamic Bethe Ansatz (TBA) equations in a manner entirely analogous to Hamiltonian integrable systems where energy is conserved Takahashi (1999). This leads to the following equations for the occupation numbers (Fermi factors) $\theta_{\nu}(k)\equiv\rho_{\nu}(k)/\rho^{\rm tot}_{\nu}(k)\equiv(1+e^{\epsilon_{\nu}(q)})^{-1}$ which turn out to be independent of $k$:

[TABLE]

Together with (7) this forms a complete characterization of the generalized Gibbs ensemble. For $\lambda=0$ (FFA model), the properties of this ensemble can also be derived by a transfer-matrix calculation Gopalakrishnan et al. (2018); these approaches give equivalent results sup .

Hydrodynamics and soliton gas.— The evolution from local to global equilibrium in the DFFA model can be described using the recently developed theory of generalized hydrodynamics (GHD) Castro-Alvaredo et al. (2016); Bertini et al. (2016) – see also Doyon and Yoshimura (2017); Doyon and Spohn (2017); Ilievski and De Nardis (2017a); Bulchandani et al. (2018); Doyon et al. (2018); Piroli et al. (2017); Ilievski and De Nardis (2017b); Collura et al. (2018); Alba and Calabrese (2017); De Luca et al. (2017); Bertini and Piroli (2018); Bulchandani et al. (2017); Bulchandani (2017); Doyon et al. (2017); Caux et al. (2017); Cao et al. (2018); Doyon (2018); Schemmer et al. (2019). This is equivalent to treating this quantum system semiclassically as a gas of solitons Doyon et al. (2018). There are two species of solitons $\pm$ whose bare velocities are given by the dispersion relation (3), so that $v^{0}_{\nu,k}=\nu-2\lambda\sin k$ with $\nu=\pm$. When solitons collide, they interact via a $k$-dependent phase-shift which leads to a semi-classical displacement $\Delta x=2\pi{\cal K}$ (Wigner time delay); i.e. $\Delta x=1$ if the two quasiparticles are of the same species, and $\Delta x=-1$ otherwise (Fig. 1). To leading order (Euler hydrodynamics), this leads to a dressing of the velocities due to collisions Bonnes et al. (2014); Castro-Alvaredo et al. (2016); Bertini et al. (2016), with the effective velocities in a state with quasiparticle densities $\rho_{\pm,k}$ given by

[TABLE]

Each species of solitons thus behaves as a simple one-dimensional classical hard rod gas with unit length, while collisions between $+$ and $-$ solitons correspond formally to hard rods of negative length. Diffusive corrections to this ballistic picture follow from recent GHD results De Nardis et al. (2018); Gopalakrishnan et al. (2018); Nardis et al. (2019); Gopalakrishnan and Vasseur (2019); De Nardis et al. (2019), and are especially simple for DFFA sup .

Operator dynamics.—The rapidity-independent scattering kernels in the DFFA model have important consequences for operator spreading, which is simpler here than in generic integrable models Gopalakrishnan et al. (2018). In the generic case, any operator creates a “butterfly cone” that fills in at late times: a spatially local operator has a spread of momenta and thus of group velocities, and the velocity-dependence of the scattering kernel implies that perturbing the velocity of one quasiparticle will affect the trajectories of all the others. This does not happen either in the hard rod gas or in the DFFA model, since the scattering kernel in these models is velocity-independent and consequently, any perturbation that preserves $N_{\pm}$ will only affect the state of one quasiparticle. Thus the butterfly cone, measured via the out-of-time-order correlator Larkin and Ovchinnikov (1969); Maldacena et al. (2016); Lin and Motrunich (2018); Khemani et al. (2018) (OTOC) $C\left(x,t\right)\equiv\frac{1}{2}\left|{\rm Tr}\left\{[\,\hat{h}^{\vphantom{\dagger}}_{j=2}\,,\,\hat{\sigma}^{z}_{x}\left(t\right)\,]^{2}\right\}\right|$ does not “fill in” except through the dispersion of the perturbed quasiparticle (Fig. 2). The existence of such operators with simple matrix elements has to do with the structure of the Bethe ansatz equations: in a generic integrable model, changing the rapidity of one quasiparticle would alter the quantization condition for all the others. Therefore, the matrix element from a reference state to a state with one shifted quasiparticle would be suppressed by overlap factors from all the other quasiparticles that have their momenta slightly shifted. In the DFFA model, by contrast, this quantization condition depends only on a few aggregate properties of the quasiparticle distribution, so the matrix element for changing the state of a single quasiparticle is not parametrically suppressed.

Conclusion.— In summary, we present and solve exactly a Floquet model that is the first of its kind in a number of respects. It is the first example of an interacting integrable Floquet model that is not smoothly deformable to Hamiltonian dynamics (integrable Trotterizations Vanicat et al. (2018)), and also not classically simulable (FFA). In fact, our solution of the dispersing model has provided insight into the physics of FFA, which prior to this work was not confirmed to be integrable in the Yang-Baxter sense; and the dispersing model regularizes several pathological features of FFA while making the model more reminiscent of typical quantum systems. Despite the complicated nature of the Hamiltonian terms, the resulting Bethe (4) and TBA equations (8) are the simplest of any interacting integrable model as far as we are aware. This model shows the existence of interacting Floquet models with stable chiral quasiparticles, and suggests a route to finding others, building on integrable cellular automata Bobenko et al. (1993); Bruschi et al. (1992); Grammaticos et al. (1997); Gopalakrishnan and Zakirov (2018); it would be interesting to find other examples in the future.

Acknowledgements.— We thank D. Huse, V. Khemani, A. Nahum, T. Prosen and B. Ware for illuminating discussions. This research was supported by the National Science Foundation via Grants DGE-1321846 (Graduate Research Fellowship Program, AJF), DMR-1455366 (AJF) and DMR-1653271 (SG); and the US Department of Energy, Office of Science, Basic Energy Sciences, under Award No. DE-SC0019168 (RV).