Self-duality triggered dynamical transition

We start with a description of the SM model. Consider a spin $1/2$ quantum particle moving on a $2$-torus ${\mathbb{T}}^{2}$ (i.e., a $2$-rotor), with a constant angular velocity ${\bm{\omega}}\equiv(\omega_{1},\omega_{2})$. The particle’s angular coordinates are denoted by ${\bm{\theta}}\equiv(\theta_{1},\theta_{2})$, and their conjugate momentum operators are denoted ${\hbar_{e}\bm{N}}\equiv-i\hbar_{e}(\partial_{\theta_{1}},\partial_{\theta_{2}})$. The state vector of the particle at time $t$ is a spinor with two complex components,

$$\boldsymbol{\psi}_{t}({\bm{\theta}})=\left(\begin{array}[]{c}\psi_{1,t}({\bm{\theta}})\\ \psi_{2,t}({\bm{\theta}})\\ \end{array}\right),$$

(1)

which belongs to the Hilbert space ${\cal H}\equiv L^{2}({\mathbb{T}}^{2})\otimes{\mathbb{C}}^{2}$. The particle is kicked periodically in time; with the period set to unity, at all integer times $t\in\mathbb{Z}$ it undergoes instantaneous pulses, or kicks, that prompt sudden jumps of its momentum and simultaneously rotate its spin in a way, that depends on the angular coordinates. Such kicks are produced by switching on an external “potential” $\boldsymbol{V}({\bm{\theta}})=\sum_{k=1}^{3}V_{k}({\bm{\theta}})\boldsymbol{\sigma}_{k}$. Here $\boldsymbol{\sigma}_{k}$ are the Pauli matrices and the $V_{k}$ are some functions $\mathbb{T}^{2}\rightarrow\mathbb{R}$. The quantum dynamics follows the Schrödinger equation:

	$\displaystyle i\hbar_{e}\partial_{t}\boldsymbol{\psi}_{t}({\bm{\theta}})=\bm{H}(t)\boldsymbol{\psi}_{t}({\bm{\theta}})\,,\qquad\qquad\qquad$		(2)
	$\displaystyle\bm{H}(t)=\hbar_{e}{\bm{\omega}}\boldsymbol{\cdot}\bm{N}+V({\bm{\theta}})\sum_{s\in\mathbb{Z}}\delta(t-s).$

The first term of the Hamiltonian $\bm{H}(t)$ corresponds to rotation over the torus $\mathbb{T}^{2}$ with constant angular velocity and the second to the kick. The dynamics from immediately after a kick to immediately after the next is described by:

$$\forall t\in\mathbb{Z}:\,\boldsymbol{\psi}_{t+1}({\bm{\theta}})\;=\;{\bm{U}}_{\alpha,{\bm{\omega}}}\boldsymbol{\psi}_{t}({\bm{\theta}}),\quad{\bm{U}}_{\alpha,{\bm{\omega}}}:=\bm{M}_{\alpha}\bm{T}_{{\bm{\omega}}},$$

(3)

which defines the Floquet operator ${\bm{U}}_{\alpha,{\bm{\omega}}}$. Here $\bm{T}_{{\bm{\omega}}}$ is the translation operator: $\bm{T}_{{\bm{\omega}}}\boldsymbol{\psi}({\bm{\theta}})\equiv\boldsymbol{\psi}(\boldsymbol{\tau}_{\boldsymbol{\omega}}^{-1}{\bm{\theta}})$ where $\boldsymbol{\tau}_{\boldsymbol{\omega}}:{\bm{\theta}}\mapsto\boldsymbol{{\bm{\theta}}}+\boldsymbol{\omega}$ is translation by $\boldsymbol{\omega}$ along the torus $\mathbb{T}^{2}$. Instead $\bm{M}_{\alpha}=e^{-i\frac{\alpha}{\pi}\boldsymbol{V}({\bm{\theta}})}$ is a smooth map ${\mathbb{T}}^{2}\to SU(2)$; so $\boldsymbol{V}({\bm{\theta}})$ is a smooth map from $\mathbb{T}^{2}$ to the $2\times 2$ self-adjoint matrices with null trace. It will be assumed to have a constant Hilbert-Schmidt norm such that ${\rm tr}(\boldsymbol{V}({\bm{\theta}})^{2})=2\pi^{2}$ , so it can be written in the general form:

$$\boldsymbol{V}({\bm{\theta}})=\pi\sum_{k=1}^{3}d_{k}({\bm{\theta}})\boldsymbol{\sigma}_{k}\;,$$

(4)

which with some abuse of notation will be concisely written in the form $\pi\bm{d}({\bm{\theta}})\boldsymbol{\cdot}\bm{\sigma}$. Here ${\boldsymbol{d}}({\bm{\theta}})=(d_{1}({\bm{\theta}}),d_{2}({\bm{\theta}}),d_{3}({\bm{\theta}}))$ is a fixed map from $\mathbb{T}^{2}$ to the unit real $2$-sphere $\mathbb{S}^{2}$. Equation (2) or its discrete-time version Eq. (3) define our general SM model.

The following basic property will be extensively used in the following:

$${\bm{\rho}}\bm{M}_{\alpha}{\bm{\rho}}^{\dagger}\;=\;\bm{M}_{\alpha},\quad{\bm{\rho}}{\bm{U}}_{\alpha,{\bm{\omega}}}{\bm{\rho}}^{\dagger}\;=\;{\bm{U}}_{\alpha,{\bm{\omega}}}\;$$

(5)

where ${\bm{\rho}}$ is the standard time-reversal operator for spin-$1/2$ particles; ${\bm{\rho}}:=-i\boldsymbol{\sigma}_{2}C$ (where $C$ denotes complex conjugation in the coordinate representation). It is the antiunitary map that is defined in $\cal H$ by

$\displaystyle{\bm{\rho}}:\left(\begin{array}[]{c}\psi_{1}\\ \psi_{2}\end{array}\right)\mapsto\left(\begin{array}[]{c}-\overline{\psi_{2}}\\ \overline{\psi_{1}}\end{array}\right)$

(10)

The ordinary Maryland model Fishman82 ; Fishman84 is recovered from the present one, whenever the potential $\bm{V}({\bm{\theta}})$ is invariant under spin rotations, i.e., $\bm{V}({\bm{\theta}})$ does not couple the angular and spin degrees of freedom. As shown below, thanks to this difference the SM model exhibits a much more intriguing phenomenology than the ordinary Maryland model.

On the other hand, the SM model crucially differs from the spinful QKR Tian14 ; Tian16 in that its free Hamiltonian between kicks is linear in $n_{1}$, while for the spinful QKR the free Hamiltonian is quadratic in $n_{1}$. It is well known that kicked systems with a quadratic free Hamiltonian may exhibit a chaotic behavior (in some sense), which cannot be observed with a linear one.

Thanks to the linear momentum dependence of the free Hamiltonian, the SM model Eq. (3), though of quantum origin, can be translated into a classical dynamical system, with the usual technical meaning, i.e. a discrete time dynamics which is generated by iterating a measure preserving transformation in some phase space. Though mathematically trivial, this quantum-classical juxtaposition illustrates that in SM models wavepacket propagation and motion along phase-space trajectories are equivalent pictures of the same dynamics. The phase space is $\Omega\equiv{\mathbb{T}}^{2}\times{\mathbb{C}}_{1}^{2}$, where ${\mathbb{C}}_{1}^{2}$ denotes the unit sphere in $\mathbb{C}^{2}$, and the map is defined as follows:

$$\displaystyle{\bm{{\cal S}}}_{\alpha,{\bm{\omega}}}:({\bm{\theta}},\boldsymbol{\phi})\mapsto(\bm{\tau}_{\boldsymbol{\omega}}{\bm{\theta}},\bm{M}_{\alpha}(\bm{\tau}_{\boldsymbol{\omega}}{\bm{\theta}})\boldsymbol{\phi})\,.$$

(11)

This map preserves the natural measure $d\mu$ that is defined in $\Omega$ by the product of the Lebesgue measure $dm$ on $\mathbb{T}^{2}$, normalized to unity, and the uniform measure on ${\mathbb{C}}^{2}_{1}$ [or equivalently the Haar measure on $SU(2)$]. In mathematical terms the map described by Eq. (11) defines a skew product action of $SU(2)$ Eliasson98 ; Eliasson02 ; Aldecoa15 ; Karaliolios18 . The corresponding Perron-Frobenius operator, denoted by ${\bm{{\cal U}}}_{\alpha,{\bm{\omega}}}$, acts on functions $F\in{\mathfrak{H}}\equiv L^{2}(\Omega,d\mu)$ as follows:

	$\displaystyle{\bm{{\cal U}}}_{\alpha,{\bm{\omega}}}F({\bm{\theta}},\boldsymbol{\phi})$	$\displaystyle=$	$\displaystyle F({\bm{{\cal S}}}_{\alpha,{\bm{\omega}}}^{-1}({\bm{\theta}},\boldsymbol{\phi}))$		(12)
		$\displaystyle=$	$\displaystyle F(\bm{\tau}^{-1}_{\boldsymbol{\omega}}{\bm{\theta}}\;,\;\boldsymbol{M}_{\alpha}({\bm{\theta}})^{-1}\boldsymbol{\phi})\;.$

It preserves the scalar product in $\mathfrak{H}$, i.e. $\forall F^{\prime},F^{\prime\prime}\in\mathfrak{H}$,

	$\displaystyle\langle F^{\prime}|F^{\prime\prime}\rangle_{\mathfrak{H}}$	$\displaystyle:=$	$\displaystyle\int_{\Omega}d\mu({\bm{\theta}},\boldsymbol{\phi})\;\overline{F^{\prime}({\bm{\theta}},\boldsymbol{\phi})}F^{\prime\prime}({\bm{\theta}},\boldsymbol{\phi})$		(13)
		$\displaystyle=$	$\displaystyle\langle{{\bm{{\cal U}}}_{\alpha,{\bm{\omega}}}F^{\prime}}|{{\bm{{\cal U}}}_{\alpha,{\bm{\omega}}}F^{\prime\prime}}\rangle_{\mathfrak{H}}\;.$

The actual link between the “quantum” SM model and its “classical” equivalent is as follows. Given a $\boldsymbol{\psi}\in\cal H$, let

$$F_{\boldsymbol{\psi}}({\bm{\theta}},\boldsymbol{\phi}):=\boldsymbol{\psi}({\bm{\theta}})\boldsymbol{\cdot}\boldsymbol{\phi}.$$

(14)

Here the dot denotes the canonical scalar product of vectors in ${\mathbb{C}}^{2}$: $\boldsymbol{\psi}\boldsymbol{\cdot}\boldsymbol{\phi}:=\overline{\psi_{1}}\phi_{1}+\overline{\psi_{2}}\phi_{2}$. Then

$\displaystyle{\bm{{\cal U}}}_{\alpha,{\bm{\omega}}}F_{\boldsymbol{\psi}}({\bm{\theta}},\boldsymbol{\phi})\;=\;{\bm{U}}_{\alpha,{\bm{\omega}}}\boldsymbol{\psi}({\bm{\theta}})\boldsymbol{\cdot}\boldsymbol{\phi}\;=\;F_{\boldsymbol{\psi}^{\prime}}({\bm{\theta}},\boldsymbol{\phi})\;,$

(15)

where $\boldsymbol{\psi}^{\prime}={\bm{U}}_{\alpha,{\bm{\omega}}}\,\boldsymbol{\psi}$. This equation formalizes the quantum-classical juxtaposition within the same model. It is important to emphasize that the quantum system (3) and the classical dynamical system (11) are fully equivalent, and the latter is not related to any notion of classical limit whatsoever: the subscript $\alpha$ – i.e. the inverse Planck constant – in ${\bm{{\cal S}}}_{\alpha,{\bm{\omega}}}$ is a bookkeeping of this fact. The existence of a classical equivalent of the SM model implies that quantum interference does not play any role in its phenomenology. As mentioned in the introduction, the same is true for the original Maryland model, where the observed localization is not related to the Anderson-like dynamical localization that typically occurs in quantum systems exhibiting chaotic diffusion in the classical limit and results from quantum interference.

The spectral and dynamical properties of the SM models crucially depend on the arithmetic type of the triple $(\omega_{1},\omega_{2},\pi)$. In particular, the following well-known fact is of capital importance to the present work: the ${\bm{\omega}}$-shift $\bm{\tau}_{\boldsymbol{\omega}}$ is ergodic in ${\mathbb{T}}^{2}$ whenever ($\omega_{1},\omega_{2},\pi$) are an incommensurate triple of real numbers. More detailed arithmetic properties of the triple play important spectral dynamical roles; a few more comments on this theme are deferred to Sec. IV.3. However, for the purposes of the present paper sheer incommensuration will be sufficient. From now on, we shall always consider incommensurate triples, unless explicitly mentioned otherwise.

In this work we implement the quantum-classical juxtaposition to investigate both mathematically and numerically the dynamical properties of SM. In numerical simulations, we used two different incommensurate frequency vectors $\boldsymbol{\omega}$, which are specified in Sec. III.2.1.

Upon expanding $\boldsymbol{\psi}_{t}({\bm{\theta}})$ in the $2$D Fourier basis we move from the ${\bm{\theta}}$-coordinate representation to the momentum $\bm{N}$-representation, where the SM dynamics can be pictured as propagation of a quantum wavepacket over the $2$D integer lattice $\mathbb{Z}^{2}$. This defines quantum transport for the SM models in the same way as for the QKR Casati79 and for the ordinary Maryland model Fishman82 ; Fishman84 . A basic probe of the quantum transport associated with the evolution of a state ${\boldsymbol{\psi}}_{t}$ from an initial state ${\boldsymbol{\psi}}_{0}$ is

	$\displaystyle E_{j}({\boldsymbol{\psi}}_{t}):=\langle{\boldsymbol{\psi}}_{t}|n_{j}^{2}|{\boldsymbol{\psi}}_{t}\rangle=\|\partial_{\theta_{j}}\boldsymbol{\psi}_{t}\|^{2}_{\cal H},\,j=1,2$
	$\displaystyle E({\boldsymbol{\psi}}_{t})\;=\;E_{1}({\boldsymbol{\psi}}_{t})\;+\;E_{2}({\boldsymbol{\psi}}_{t}),\;$		(16)

which characterizes how the wavepacket spreads in the course of time. Like QKR systems Casati79 ; Fishman82a ; Tian14 ; Tian16 , this probe provides an important link between SM models and condensed matter systems. Notably, when $E({\boldsymbol{\psi}}_{t})$ remains bounded as $t\to\infty$, the wavepacket does not spread at long time and remains localized in a finite region in momentum space; no quantum transport arises and this simulates a quantum insulator in condensed matter physics. When either or both $E_{1,2}({\boldsymbol{\psi}}_{t})$ display unbounded growth, the wavepacket propagation is delocalized; and if the asymptotic growth is linear in time, then quantum diffusive transport arises, and this simulates a quantum normal metal in condensed matter physics. Such different transport properties are mirrored by the spectral properties of the Floquet operators ${\bm{U}}_{\alpha,{\bm{\omega}}}$; a detailed analysis is given in Sec. IV.

We will show that quantum transport in the SM models finds a “classical” counterpart in the stability of trajectories, i.e. in the way initially close trajectories separate in time. This is a fundamental token in the theory of classical dynamical systems, and a commonly accepted definition of ”chaos” applies whenever the divergence of trajectories is exponentially fast. Specifically, the quantum transport in momentum space, probed by $E_{1,2}({\boldsymbol{\psi}}_{t})$ defined by Eq. (II.3), has an equivalent description in terms of the instability of trajectories $({\bm{\theta}}_{t},\boldsymbol{\phi}_{t}):={\bm{{\cal S}}}_{\alpha,{\bm{\omega}}}^{t}({\bm{\theta}},\boldsymbol{\phi})$ in the phase space $\Omega$, in the following sense. The (linear) stability of a trajectory $({\bm{\theta}}_{t},\boldsymbol{\phi}_{t})$ is determined by the behavior in time of the derivatives $\partial_{\theta_{j}}\boldsymbol{\phi}_{t}({\bm{\theta}},\boldsymbol{\phi})$ with $j=1,2$; see Sec. V, where we prove that if the initial spinor ${\boldsymbol{\psi}}_{0}({\bm{\theta}})$ is localized at the origin of momentum space then the quantum transport which is defined by Eq. (II.3) is related to trajectory stability by:

$\displaystyle E_{j}({\boldsymbol{\psi}}_{-t})$

$\displaystyle=$

$\displaystyle\int_{\Omega}d\mu({\bm{\theta}},\boldsymbol{\phi})\;\boldsymbol{|}\partial_{\theta_{j}}\boldsymbol{\phi}_{t}({\bm{\theta}},\boldsymbol{\phi})\boldsymbol{|}^{2}.$

(17)

Spectral implications of this result are presented in Sec. IV.3.

In this paper we study three different SM models. Their Floquet operators ${\bm{U}}_{\alpha,{\bm{\omega}}}$ share the general form described in Sec. II.1. Differences arise from the explicit form of $\bm{d}({\bm{\theta}})$:

• •

  SM Model I:

  $\displaystyle\bm{d}({\bm{\theta}})=p({\bm{\theta}})(\sin(\theta_{1}),\sin(\theta_{2}),\beta(1-\cos(\theta_{1})-\cos(\theta_{2}))),$

  (18)

  where $p({\bm{\theta}})$ is the normalization factor, chosen such that $\bm{d}({\bm{\theta}})$ is a unit vector, and $\beta>0$ is a real parameter.

• •

  SM Model II:

  $\displaystyle\bm{d}({\bm{\theta}})=p({\bm{\theta}})(\sin(\theta_{1}),\sin(\theta_{2}),\mbox{\rm const.}\neq 0);$

  (19)

• •

  SM Model III:

  $\displaystyle\begin{array}[]{c}\bm{d}({\bm{\theta}})=p({\bm{\theta}})(\sin(\theta_{1}),\sin(\theta_{2}),\cos(\theta_{1})+\cos(\theta_{2}))\\ {\rm or}\,\bm{d}({\bm{\theta}})=p({\bm{\theta}})(\sin(\theta_{1}),\sin(\theta_{2}),0).\end{array}$

  (22)

An important part of our study is how the transport properties of such models depend on the parameter $\alpha$. In this respect, the following properties:

$${\bm{U}}_{\alpha+2\pi,{\bm{\omega}}}={\bm{U}}_{\alpha,{\bm{\omega}}}\,,\,\,\,\,{\bm{U}}_{\alpha+\pi,{\bm{\omega}}}=\,-\,{\bm{U}}_{\alpha,{\bm{\omega}}}$$

(23)

which directly follow from the definition of ${\bm{U}}_{\alpha,{\bm{\omega}}}$, hold true for all SM models; so, analysis of SM Models I, II, III can be restricted to $\alpha\in[0,\pi]$. The three models share certain symmetries, which are described in Sec. IV.2. A fundamental fact is that such symmetries in physical space give rise to a symmetry in parameter space; notably, quantum SM systems whose $\alpha\in[0,\pi]$ are symmetric with respect to $\pi/2$ are in a “duality” relation nature . This duality has strong spectral implications, as shown in Sec. IV.2. Besides, at $\alpha=\pi/2$ it forces the system to be invariant under a unitary symmetry transformation and thus to be self-dual.

The dynamical behaviors of the three SM models are very different:

• •

  SM Model I:
  We find that for $\alpha\in[0,\pi]$, on the left or on the right of the critical point $\alpha=\pi/2$, both $E_{1,2}({\boldsymbol{\psi}}_{t})$ remain bounded in time, so the system is dynamically localized, and localization is exponential in momentum space. At the critical point $E_{1,2}({\boldsymbol{\psi}}_{t})\propto t$ approximately, so the system is dynamically delocalized. In the classical system, the deviation $\delta_{t}$ of initially close trajectories grows as $\sqrt{t}$ (approximately) at half-integer $\alpha/\pi$, whereas $\delta_{t}$ saturates at long time away from the critical point. The transition in quantum transport is thus mirrored by a change of stability of classical trajectories. Interestingly, sublinear divergence of trajectories is what one should expect if the classical SM model were known to be ergodic; see Sec. V. We explain delocalization at the critical value, proving that self-duality is incompatible with dynamical localization. As we will show in Sec.V, this dynamical transition mirrors the transition in the spectral structure of Floquet operator ${\bm{U}}_{\alpha,{\bm{\omega}}}$.

  Moreover, it turns out that the dynamically localized phases which are separated by the critical points can be obtained from topologically distinct phases which are found in the spinful QKR Tian14 ; Tian16 by deforming that QKR continuously into the considered SM.

• •

  SM Model II:
  As $\alpha$ varies, dynamical localization is always observed.

• •

  SM Model III:
  As $\alpha$ varies, dynamical delocalization is always observed except at integer $\alpha/\pi$, where dynamical localization trivially occurs because $\bm{M}_{\alpha}=\pm\mathbb{I}$ there.

We see that, due to the presence of spin, SM models exhibit rich dynamical phenomena. This is in sharp contrast to the ordinary Maryland model for which only dynamical localization can occur. In addition, the Maryland model inherits dynamical localization from the integrability of the classical dynamics in the limit $\hbar_{e}\to 0$ Berry84 ; this mechanism for dynamical localization does not apply here, because the SM model is ill-defined in that limit.

Direct numerical simulation of the $2$D quantum evolution requires a $2$D Fourier basis. Thus it cannot be pushed too far in time. This difficulty is greatly softened by a trick which was first used in Ref. Casati89 . It consists in replacing the $2$D model described by Eq. (3) by two $1$D models, which allow to separately compute $E_{1,2}({\boldsymbol{\psi}}_{t})$ [abbreviated as $E_{1,2}(t)$ throughout this section]. This possibility crucially rests on the ${\bm{\omega}}$-shift $\bm{\tau}_{\bm{\omega}}$ in $\mathbb{T}^{2}$, namely, the first term of $\bm{H}(t)$ given by Eq. (2), as we will show below.

By performing the transformation:

	$\displaystyle{\bm{U}}_{\alpha,{\bm{\omega}}}$	$\displaystyle\rightarrow$	$\displaystyle e^{t\omega_{2}\partial_{\theta_{2}}}{\bm{U}}_{\alpha,{\bm{\omega}}}e^{-(t-1)\omega_{2}\partial_{\theta_{2}}}$
		$\displaystyle=$	$\displaystyle\bm{M}_{\alpha}(\theta_{1},\theta_{2}+\omega_{2}t)e^{-\omega_{1}\partial_{\theta_{1}}}=:\tilde{{\bm{U}}}_{t},$
	$\displaystyle\boldsymbol{\psi}_{t}$	$\displaystyle\rightarrow$	$\displaystyle e^{t\omega_{2}\partial_{\theta_{2}}}\boldsymbol{\psi}_{t}=:\tilde{\boldsymbol{\psi}}_{t},$		(24)

we rewrite Eq. (3) as

$$\tilde{\boldsymbol{\psi}}_{t}\;=\;_{\rm T}\!\prod_{s=1}^{t}\tilde{{\bm{U}}}_{s}\tilde{\boldsymbol{\psi}}_{0},$$

(25)

where ${}_{\rm T}\!\prod$ denotes a time-ordered product, with the index $s$ increasing from right to left. In Eqs. (III.1) and (25), $\theta_{2}$ is no longer a dynamical variable, rather it is an angular parameter. Thus the quantum evolution (25) is $1$D. When the $2$D model is traded for such $1$D model, we have

$$E_{1}(t)=-\langle\langle\tilde{\boldsymbol{\psi}}_{t}|\partial_{\theta_{1}}^{2}\tilde{\boldsymbol{\psi}}_{t}\rangle_{\tilde{\cal H}}\rangle_{\theta_{2}},$$

(26)

where the Hilbert space $\tilde{{\cal H}}\equiv L^{2}({\mathbb{T}}^{1})\otimes{\mathbb{C}}^{2}$ and $\langle\cdot\rangle_{\theta_{2}}$ denotes the average over $\theta_{2}$. The whole derivation is fully symmetric with respect to the indices $1,2$, so similar equations are obtained for $E_{2}(t)$ by just interchanging indices in the above equations. Such equations will share the same reference numbers.

Throughout this paper in both $1$D and $2$D simulations we choose initial states which have only the spin-up ($s=1$) component and are zero momentum state.

The same numerical procedures were repeated with the constant in Eq. (19) set to unity. The observed dynamical behaviors of SM Model II are totally different from SM Model I. In particular, regardless of the value of $\alpha$, $E_{1}(t)$ always saturates at long times and the long-time diffusion coefficient $\lim_{t\rightarrow\infty}\frac{E_{1}(t)}{t}$ always vanishes (not shown). Thus, dynamical localization is always observed.

The same numerical procedures were repeated for SM Model III with $d_{3}=p({\bm{\theta}})(\cos(\theta_{1})+\cos(\theta_{2}))$ and $d_{3}=0$, respectively. The observed dynamics are totally opposite to those of SM Model II. In particular, dynamical delocalization is observed regardless of the value of $\alpha$, with the one exception of the points $\alpha/\pi\in\mathbb{Z}$, for which ${\bm{U}}_{\alpha,{\bm{\omega}}}=\pm\bm{T}_{{\bm{\omega}}}$ leading to trivial localization.

We first discuss the spectral properties of general SM models, without imposing any special features on the $V_{k}({\bm{\theta}})$. Such properties crucially depend on arithmetic properties of the frequency vector ${\bm{\omega}}$. We always require $(\omega_{1},\omega_{2},\pi)$ to be an incommensurate triple. A general result about the spectrum of the Floquet operator ${\bm{U}}_{\alpha,{\bm{\omega}}}$ is stated below, and is proven in Sec.VI. Exact operator-theoretic terminology will be confined to those proofs, so here we fix a few terms of a more informal language. The spectrum of a unitary operator ${\bm{U}}$ in a Hilbert space $\cal H$ consists of complex numbers of the form $e^{i\chi}$ with $\chi\in{\mathbb{R}}$, and it cannot be empty. The point spectrum of ${\bm{U}}$ is that part of the spectrum which consists of proper eigenvalues, i.e., of eigenvalues which are associated with normalizable eigenfunctions – if there are any. Such eigenvalues are a countable set at most, and the corresponding eigenfunctions span a subspace ${\cal H}_{p}$ of $\cal H$, which is invariant under ${\bm{U}}$; so its orthogonal complement subspace ${\cal H}_{c}$ is also invariant under ${\bm{U}}$. If ${\cal H}_{p}$ is not the whole of $\cal H$, then ${\bm{U}}$, restricted to ${\cal H}_{c}$, is still unitary, so it has a spectrum, which by construction cannot contain any proper eigenvalue. This is, in fact, the continuous spectrum of ${\bm{U}}$, and it can be informally pictured as the set of generalized eigenvalues, which are associated with eigenfunctions that are not normalizable, because they are, in some sense, extended. The spectral type of ${\bm{U}}$ is said to be pure, if either ${\cal H}_{p}$ or ${\cal H}_{c}$ is the whole $\cal H$; in the former case it is called pure point, and in the latter case it is called purely continuous. The set of all proper eigenvalues of ${\bm{U}}$ will be denoted ${\rm Eig}({\bm{U}})$.

When a SM model is viewed in the classical picture, as described by Eq. (12), its spectral analysis is based on the Perron-Frobenius operator ${\bm{{\cal U}}}_{\alpha,{\bm{\omega}}}$. The functions on the phase space $\Omega$ which depend only on ${\bm{\theta}}$ are a closed subspace ${\mathfrak{H}}_{0}$ of $\mathfrak{H}$, that can be identified with $L^{2}(\mathbb{T}^{2},dm)$. This subspace is invariant under ${\bm{{\cal U}}}_{\alpha,{\bm{\omega}}}$. In Sec. VI we prove the following result:

Pure continuity of the spectrum of ${\bm{U}}_{\alpha,{\bm{\omega}}}$ does not by necessity entail pure continuity of the spectrum of ${\bm{{\cal U}}}_{\alpha,{\bm{\omega}}}$ in the orthogonal complement ${\mathfrak{H}}_{0}^{\bot}$ of ${\mathfrak{H}}_{0}$; the latter may still have a pure point component. This is somehow related to ergodicity of the classical description of SM models. Ergodicity is a capital notion in the theory of classical dynamical systems, and can be stated in purely spectral terms: ${\bm{{\cal S}}}_{\alpha,{\bm{\omega}}}$ is ergodic, if and only if $1$ is a simple proper eigenvalue of ${\bm{{\cal U}}}_{\alpha,{\bm{\omega}}}$ Sinai . Since SM models bear a quantum–classical juxtaposition, a natural question arises, how would ergodicity in the classical description of SM models be mirrored in the quantum description of SM models. The following result presents a partial answer. Loosely speaking, an operator in $\cal H$ is fibered, if it does not couple different position eigenstates; more properly, if it acts as in $\boldsymbol{\psi}(\boldsymbol{\theta})\mapsto\bm{O}(\boldsymbol{\theta})\boldsymbol{\psi}(\boldsymbol{\theta})$, where $\bm{O}$ is a map from $\mathbb{T}^{2}$ to the $2\times 2$ matrices.

Though not necessary for the elaborations that follow, it can be shown that the spectral properties of the Floquet operator ${\bm{U}}_{\alpha,{\bm{\omega}}}$ are related to the cohomology of the classical dynamical system $(\Omega,{\bm{{\cal S}}}_{\alpha,{\bm{\omega}}})$; see Appendix A.

We now focus on a special class of SM models, which are exemplified by those considered in numerical experiments. A SM model belongs to this class, if it is defined by maps $\bm{d}$ that enjoy the following special symmetries:

1. (i)

   $d_{1}({\bm{\theta}})$ is odd wrt $\theta_{1}$ and even wrt $\theta_{2}$;

2. (ii)

   $d_{2}({\bm{\theta}})$ is even wrt $\theta_{1}$ and odd wrt $\theta_{2}$;

3. (iii)

   $d_{3}({\bm{\theta}})$ is even wrt both $\theta_{1}$ and $\theta_{2}$.

It has been previously found Tian14 ; Tian16 that such symmetries are crucial to the IQHE-like phenomenon in the spinful QKR. SM models endowed with such symmetries will be termed class-$*$ models. The SM Models I, II and III, as given in Eqs. (18), (19) and (22), are class-$*$ models.

A key role will be played by the following Proposition 4, that crucially rests on the above conditions (i), (ii) and (iii), and its corollaries.

Proof : The first identity in Eq. (30) will be proven first; the proof of the second is essentially identical. First of all, from the general definition of $\bm{M}_{\alpha}$ it follows that

$\displaystyle{\boldsymbol{M}}_{2\pi-\alpha}={\boldsymbol{M}}_{\alpha}^{\dagger},\,\,\boldsymbol{M}_{\pi+\alpha}=-{\boldsymbol{M}}_{\alpha},\,\,{\boldsymbol{M}}_{\alpha}^{\dagger}=-\boldsymbol{M}_{\pi-\alpha},$

(31)

which hold true independently of the symmetries of $\bm{d}$. Let $\boldsymbol{R}_{{\boldsymbol{\eta}}}$ denote the reflection operator in a point $\boldsymbol{\eta}\in\mathbb{T}^{2}$: that is, $(\boldsymbol{R}_{{\boldsymbol{\eta}}}\boldsymbol{\psi})({\bm{\theta}})=\boldsymbol{\psi}(2\boldsymbol{\eta}-{\bm{\theta}})$. From $\boldsymbol{U}^{\dagger}_{\alpha,{\bm{\omega}}}=\boldsymbol{T}_{-\boldsymbol{\omega}}\boldsymbol{M}_{\alpha}^{\dagger}$ and the third identity in Eq. (31), we obtain that

$$\boldsymbol{U}^{\dagger}_{\alpha,\boldsymbol{\omega}}\,=\,-\;\boldsymbol{T}_{-\boldsymbol{\omega}}\,\boldsymbol{U}_{\pi-\alpha,-\boldsymbol{\omega}}\,\boldsymbol{T}_{\boldsymbol{\omega}}.$$

(32)

Thanks to the assumed symmetries (i), (ii) and (iii), the unitary operator $\boldsymbol{B}=\boldsymbol{\sigma}_{3}\boldsymbol{R}_{\boldsymbol{0}}$ commutes with $\boldsymbol{M}_{\alpha}$, and $\boldsymbol{B}\boldsymbol{T}_{\boldsymbol{\omega}}\boldsymbol{B}^{\dagger}=\boldsymbol{T}_{-\boldsymbol{\omega}}$. As a result,

$$\boldsymbol{U}_{\alpha,-\boldsymbol{\omega}}\,=\,\boldsymbol{B}\,\boldsymbol{U}_{\alpha,\boldsymbol{\omega}}\,\boldsymbol{B}^{\dagger}\,.$$

(33)

Applying this to the right-hand side of Eq. (32) gives

$$\boldsymbol{U}^{\dagger}_{\alpha,\boldsymbol{\omega}}\,=\,-(\boldsymbol{T}_{-\boldsymbol{\omega}}\boldsymbol{B})\,\boldsymbol{U}_{\pi-\alpha,\boldsymbol{\omega}}\,(\boldsymbol{T}_{-\boldsymbol{\omega}}\boldsymbol{B})^{\dagger}\;.$$

(34)

Therefore, the first identity in Eq. (30) is true with $\bm{A}$ given by:

$$\boldsymbol{A}\;=\;\boldsymbol{T}_{-\boldsymbol{\omega}}\boldsymbol{B}\;=\;\boldsymbol{\sigma}_{3}\boldsymbol{T}_{-\boldsymbol{\omega}}\boldsymbol{R}_{\boldsymbol{0}}\;=\;\boldsymbol{\sigma}_{3}\boldsymbol{R}_{{-\boldsymbol{\omega}/2}}\;\,\,\,.$$

(35)

To prove the second identity in Eq. (30), we use the first identity in Eq. (31) instead of the third. $\Box$

The symmetry $\bm{A}$ establishes a ”duality”  nature between models whose $\alpha$ values in $[0,\pi]$ are symmetric with respect to $\pi/2$. In other words, the model at a value $\alpha_{0}$ of the parameter $\alpha$ is dual to the model at $\alpha=\pi-\alpha_{0}$. The most immediate consequence of this duality is:

$${\rm Eig}(\boldsymbol{U}_{{\pi-\alpha,\,\boldsymbol{\omega}}})\;=\;-\;{\rm Eig}(\boldsymbol{U}_{{\alpha,\,\boldsymbol{\omega}}})\,,$$

(36)

because ${\rm Eig}(\boldsymbol{U}_{{\alpha,\,\boldsymbol{\omega}}})$ is closed under complex conjugation, thanks to the symmetry relation (5).

Models with $\alpha=\pi/2$ are self-dual. This self-dual symmetry, i.e. the aforementioned time-reversal invariance has nontrivial consequences:

This corollary imposes quite demanding conditions on self-dual eigenfunctions (if any). In the case of SM Model III, they force the absence of proper eigenfunctions, so pure spectral continuity follows:

For SM Model I we cannot assess the spectral type of ${\bm{U}}_{\frac{\pi}{2},\,\boldsymbol{\omega}}$. However the following weaker result has crucial dynamical consequences:

A proper mathematical statement should be: $\boldsymbol{u}$ does not belong to the domain of the momentum operators.

The exact spectral results presented above provide a basis for the study of the dynamical delocalization which is observed in numerical experiments. It is important to point out that while such exact results hold for any incommensurate triple $(\omega_{1},\omega_{2},\pi)$, the same is not true of other more detailed results to be discussed in this subsection, which strongly rely on the value of $\boldsymbol{\omega}$ chosen as in Sec. III.2.1, and can reasonably be taken as representatives of a wide class of strongly incommensurate situations.

Based on the definition (II.3) of $E_{j}$ ($j=1,2$) and $E$, we give a precise definition of dynamical delocalization/localization in momentum space. For given $\alpha,{\bm{\omega}}$, the quantum dynamics ${\boldsymbol{\psi}}_{t}={\bm{U}}^{t}_{\alpha,{\bm{\omega}}}\,{\boldsymbol{\psi}}$ ($t\in\mathbb{Z}$) will be said to exhibit dynamical delocalization if for any ${\boldsymbol{\psi}}$ such that $E({\boldsymbol{\psi}})<+\infty$ the following relation holds:

$$\limsup_{t\to\infty}\,E({\boldsymbol{\psi}}_{t})\,=\,+\infty$$

(39)

On the opposite, it will be said to exhibit dynamical localization whenever the right-hand side is finite. It is important to note that if $E_{j}({\boldsymbol{\psi}})<+\infty$ , then $E_{j}({\boldsymbol{\psi}}_{t})<+\infty$ at all times whenever the map $\boldsymbol{M}_{\alpha}(\boldsymbol{\theta})$ that defines a SM model is smooth. This is not the case, e.g. with SM Model III, where dynamical delocalization is somewhat trivial, as it will be shown below in Sec.  VI.5. Next we recall that dynamical localization implies pure point spectra, and, equivalently, continuous spectra imply dynamical delocalization scho . So, for SM Model I (away from critical points) and Model II, we have strong numerical evidence of pure point spectra, because dynamical localization is clearly observed numerically. However, we have no mathematical proof (except, of course, in the trivial cases when $\alpha=n\pi,n\in\mathbb{Z}$). Delocalization is produced by continuous spectra, but it can also occur with pure point spectra, for various reasons, e.g. non-uniform localization of proper eigenfunctions in momentum space, or else ”delocalization” of eigenfunctions: meaning that proper eigenfunctions, although normalizable (i.e. not extended) do not yield a finite expectation value for the squared momentum operator. This follows from a general fact, which in our case can be stated as follows:

A compact proof of this standard result is given in Sec. VI. In view of Corollary 3, this may be the case with SM Model I at the critical point $\alpha=\pi/2$, which is self-dual. So, even though we cannot exactly assess the spectral type of that model, we can nevertheless state the following exact result, that provides a rigorous confirmation for numerical observations:

Proof : Should the spectrum of $\boldsymbol{U}_{\frac{\pi}{2},\,\boldsymbol{\omega}}$ be pure point, the claim would immediately follow from Corollary 3 and Proposition 5. The one possible alternative is pure continuity, by spectral dichotomy. If that were the case, then the claim would be trivial. $\Box$

For SM Model III the situation is simpler:

Proof : Delocalization follows from the fact that $\bm{M}_{\alpha}(\boldsymbol{\theta})$ exhibits points of essential discontinuity, i.e. points where it has infinitely many distinct limit values. Such are the points where the normalization factor $p(\boldsymbol{\theta})$ vanishes. Whatever the initial state ${\boldsymbol{\psi}}$, due to the very definition of ${\bm{U}}_{\alpha,\boldsymbol{\omega}}$ such discontinuities are sooner or later transferred to ${\boldsymbol{\psi}}_{t}$ whenever $\alpha\neq n\pi$, and they cannot be removed by re-defining ${\boldsymbol{\psi}}_{t}$ in a set of measure zero. In contrast, a state for which $E_{j}$ has a finite value must be defined by functions which are, almost everywhere at least, equal to everywhere continuous functions. Continuity of the spectrum is proven in Sec. VI.5. $\Box$

This result receives intuitive support in the tight-binding-like formulation of the SM model in Appendix B, because in the presence of discontinuities the hopping terms in the lattice Hamiltonian (B) quite slowly decay at long distances in momentum space.

Though not proven, continuity of the spectrum of SM Model I at the critical point is numerically supported, as follows. For integer $t$ and $\boldsymbol{\psi}\in\cal H$, let us define:

$$P(\boldsymbol{\psi},t)\;\equiv\;\frac{1}{t}\sum\limits_{s=0}^{t-1}|\langle\boldsymbol{\psi}|\boldsymbol{\psi}_{s}\rangle_{\cal H}|^{2}\;,$$

where $\alpha,\boldsymbol{\omega}$ are left understood, and as usual $\boldsymbol{\psi}_{s}={\bm{U}}_{\alpha,{\bm{\omega}}}^{s}\boldsymbol{\psi}$. This is the time-averaged probability of survival in the initial state $\boldsymbol{\psi}$. Wiener’s theorem scho states that

$$\lim\limits_{t\to\infty}P(\boldsymbol{\psi},t)\;=\;\sum\limits_{k}|\langle\boldsymbol{u}_{k}|\boldsymbol{\psi}\rangle_{\cal H}|^{4}\;,$$

(40)

where the sum on the right-hand side is over all the proper eigenvectors $\boldsymbol{u}_{k}$ of the Floquet operator ${\bm{U}}_{\alpha,{\bm{\omega}}}$ (with an arbitrary numbering). Vanishing of the limit in the left-hand side of Eq. (40) is thus a necessary and sufficient condition for pure continuity of the spectrum. Numerical results of $2$D simulations for $P(\boldsymbol{\psi},t)$ are shown in Fig. 6 for SM Models I and III, with $\boldsymbol{\psi}$ chosen to be the $n_{1}=n_{2}=0$ state, which is constant over $\mathbb{T}^{2}$. As shown in (a), for SM Model I and noncritical $\alpha$, $P(\boldsymbol{\psi},t)$ clearly stabilizes at large $t$, consistently with the pure point spectrum that is implied by the observed dynamical localization. At critical $\alpha$, $P(\boldsymbol{\psi},t)$ appears to decay $\propto t^{-\nu}$ with $\nu$ equal or close to $1$ (a final slight deviation is probably due to finite-basis effects). Such decay is almost identical to the one which is observed for SM Model III at same $\alpha$ [shown in (b)], where continuous spectrum is exactly proven. On such grounds spectral continuity of the critical model I is convincingly supported.

Next we address the subtler question, whether the spectrum of the critical model is absolutely continuous (a.c.) or singular continuous (s.c.). The latter type of spectrum is characterized by the fact that, despite continuity, the spectral expansions of states $\boldsymbol{\psi}\in\cal H$ concede positive (nonzero) weight to regions of the spectrum which have zero measure as subsets of the unit circle $\{e^{i\chi},\chi\in{\mathbb{R}}\}$. It is frequently associated with Cantor-like structures and multifractality. No exact analysis will be attempted here. However, the absence of s.c. spectrum – hence, pure absolute continuity of the spectrum – in the critical model I is numerically supported as follows. It is an exact result dtwo that the decay exponent $\nu$ of $P(\boldsymbol{\psi},t)$ is equal to the multifractal dimension $D_{2}$ of the local density of states (LDOS) associated with the state $\boldsymbol{\psi}$; and $D_{2}=1$ is a strong indication footnote_ph that the continuous spectrum is actually a.c.. This appears to be the case with SM Model I at criticality, because $D_{2}=\nu\approx 1$ according to Fig. 6.

We finally address quantum transport and show that numerical results are consistent with a.c. spectrum. Whenever $\boldsymbol{M}_{\alpha}$ is a smooth function, the propagation over the $2$D $\boldsymbol{N}$-lattice is subject to a ballistic bound; that is, the asymptotic growth of $E(\boldsymbol{\psi}_{t})$ cannot be faster than quadratic in time. This follows from Eq. (II.3), and is equivalent to the fact that classical trajectories can be linearly unstable at most [see the comments after Eq. (56)]. The transport on the $1$D momentum lattice which is associated with the standard Maryland model and the QKR model are also subject to a ballistic bound. The following lower bound is valid for transport over discrete lattices of arbitrary dimension $D$ induced by a unitary evolution group with a.c. spectrum scho ; jmp :

$$\frac{1}{t}\sum\limits_{s=0}^{t-1}E(\boldsymbol{\psi}_{s})\;\geq\;\mbox{\rm const.}\,t^{\frac{2}{D}}\;,\,\,\,(t\to\infty)\,.$$

(41)

where $E(\boldsymbol{\psi})$ is the $D$-dimensional generalization of the definition in Eq. (II.3). With $D=1$, and in the presence of a ballistic bound, this inequality shows that a.c. spectra enforce ballistic transport footnote_bound . Not so in higher dimension; for example, with $D=3$ it has been proved JSP that a.c. spectra can coexist with different diffusion exponents. In SM models the ballistic bound is attained, whenever $(\omega_{1},\omega_{2},\pi)$ is a commensurate triple. Then the quantum evolution enjoys translation invariance in momentum space, leading to band a.c. spectra and quadratic growth of $E(\boldsymbol{\psi}_{t})$. This scenario resembles the well-known ”quantum resonances” of the QKR qkres1 ; qkres2 , but the band structure is likely to be much more complex. This issue will not be investigated in the present paper, which is restricted to the incommensurate case. In the strongly incommensurate situations which were numerically simulated, SM Model I at the critical value of $\alpha$ exhibits diffusive (or close to diffusive) transport, i.e. the long-time growth of $E(\boldsymbol{\psi}_{t})$ and $E_{1,2}(\boldsymbol{\psi}_{t})$ is linear in time. It is not isotropic in momentum space, and the degree of anisotropy depends on the choice of $\boldsymbol{\omega}$; however, it should be noted that diffusion is not restricted to special directions. Diffusive growth is consistent with the bound (41) with $D=2$. The following heuristic argument indicates that indeed it is compatible with a.c. spectrum (in this case). We start with the equation:

$$E_{j}({\boldsymbol{\psi}}_{t})=\bigl{\|}\,\sum_{s=0}^{t-1}{\bm{{\cal U}}}_{\alpha,{\bm{\omega}}}^{s}H_{j}\,\bigr{\|}_{\mathfrak{H}}^{2}\,+\,\bigl{\|}\,\sum_{s=0}^{t-1}{\bm{{\cal U}}}_{\alpha,{\bm{\omega}}}^{s}K_{j}\,\bigr{\|}^{2}_{\mathfrak{H}},$$

(42)

which is derived in Sec. V exploiting the classical picture of SM models. The functions $H_{j}=H_{j}(\boldsymbol{\theta},\boldsymbol{\phi})$, $K_{j}=K_{j}(\boldsymbol{\theta},\boldsymbol{\phi})$ are defined on the phase space $\Omega$ of the classical model and are specified immediately after Eq. (47). Their explicit forms are not important here, beyond the easily checked fact that they are orthogonal to the subspace of the spin-independent functions, where ${\bm{{\cal U}}}_{\alpha,{\bm{\omega}}}$ has a pure point spectrum according to Proposition 2. If the spectrum of $\boldsymbol{U}_{\alpha,{\bm{\omega}}}$ is a.c., then the spectrum of ${\bm{{\cal U}}}_{\alpha,{\bm{\omega}}}$ has an a.c. component too (see the statement in the beginning of Sec.  VI.2). Then a formal elaboration can be implemented. Let $e^{i\chi}$ denote generalized eigenvalues of ${\bm{{\cal U}}}_{\alpha,{\bm{\omega}}}$, and $|\chi,\xi\rangle$ the corresponding generalized eigenvectors in Dirac notation; $\xi$ collectively denotes any additional quantum numbers. Then

$$\|\,\sum\limits_{s=0}^{t-1}\;{\bm{{\cal U}}}_{\alpha,\,\boldsymbol{\omega}}^{s}H_{j}\,\|_{\mathfrak{H}}^{2}\;=\;\int_{0}^{2\pi}d\chi\,\frac{\sin^{2}\bigl{(}\frac{\chi t}{2}\bigr{)}}{\sin^{2}\bigl{(}\frac{\chi}{2}\bigr{)}}\Lambda_{H_{j}}(\chi)\,.$$

(43)

where $\Lambda_{H_{j}}(\chi)\equiv\sum_{\xi}|\langle\chi,\xi|H_{j}\rangle|^{2}$ is the LDOS of $|H_{j}\rangle$ wrt ${\bm{{\cal U}}}_{\alpha,{\bm{\omega}}}$. Under the assumption of a.c. spectrum, this is an ordinary function (i.e. it is not Dirac $\delta$-like or fractal-like), which will be further assumed to be smooth. A fully similar equation holds for the functions $K_{j}$. The fraction under the integral sign in the right-hand side of Eq. (43), divided by $2\pi t$, in the limit $t\to\infty$ tends to a Dirac $\delta$. Then Eq. (42) shows that the asymptotic growth of $E(\boldsymbol{\psi}_{t})$ is linear in time. This result for SM models is not generic for quantum dynamics on $2$D lattices, as it rests on the classical description of SM models which underlies Eq. (42).

The linear stability of a trajectory $({\bm{\theta}}_{t},\boldsymbol{\phi}_{t})$ is determined by the behavior in time of the derivatives $\partial_{\theta_{j}}\boldsymbol{\phi}_{t}({\bm{\theta}},\boldsymbol{\phi})$ with $j=1,2$. From the classical dynamical equation (11) it follows that

$$\partial_{\theta_{j}}\boldsymbol{\phi}_{t+1}\;=\;\boldsymbol{M}_{\alpha}({\bm{\theta}}_{t+1})\partial_{\theta_{j}}\boldsymbol{\phi}_{t}\;+\;(\partial_{\theta_{j}}\boldsymbol{M}_{\alpha}({\bm{\theta}}_{t+1}))\;\boldsymbol{\phi}_{t}\;.$$

(44)

Define vectors $\boldsymbol{\xi}_{j,t}\in{\mathbb{C}}^{2}$ as follows:

$$\partial_{\theta_{j}}\boldsymbol{\phi}_{t}\;=\;_{\rm T}\prod\limits_{s=0}^{t}\;\boldsymbol{M}_{\alpha}({\bm{\theta}}_{s})\;\boldsymbol{\xi}_{j,t}\;,$$

(45)

where $\boldsymbol{\xi}_{j,0}=0$. Then by performing the summation: $\sum_{s=0}^{t-1}\partial_{\theta_{j}}\boldsymbol{\phi}_{s}$, for which $\partial_{\theta_{j}}\boldsymbol{\phi}_{s}$ is substituted by Eq. (45), we obtain that

$$\boldsymbol{\phi}_{0}\,\boldsymbol{\cdot}\;\boldsymbol{\xi}_{j,t}\;=\;\sum\limits_{s=0}^{t-1}H_{j}({\bm{\theta}}_{s},\boldsymbol{\phi}_{s})\;$$

(46)

and, similarly,

$$\boldsymbol{\rho}\boldsymbol{\phi}_{0}\,\boldsymbol{\cdot}\;\boldsymbol{\xi}_{j,t}\;=\;\sum\limits_{s=0}^{t-1}K_{j}({\bm{\theta}}_{s},\boldsymbol{\phi}_{s})\;,$$

(47)

where $H_{j}({\bm{\theta}},\boldsymbol{\phi})=\boldsymbol{\phi}\boldsymbol{\cdot}\boldsymbol{M}_{\alpha}^{\dagger}({\bm{\theta}}+\boldsymbol{\omega})\partial_{\theta_{j}}\boldsymbol{M}_{\alpha}({\bm{\theta}}+\boldsymbol{\omega})\boldsymbol{\phi}$ and $K_{j}({\bm{\theta}},\boldsymbol{\phi})=\boldsymbol{\rho}\boldsymbol{\phi}\boldsymbol{\cdot}\boldsymbol{M}_{\alpha}^{\dagger}({\bm{\theta}}+\boldsymbol{\omega})\partial_{\theta_{j}}\boldsymbol{M}_{\alpha}({\bm{\theta}}+\boldsymbol{\omega})\boldsymbol{\phi}$. Because of $\boldsymbol{\rho}\boldsymbol{\phi}_{0}\boldsymbol{\cdot}\boldsymbol{\phi}_{0}=0$, taking squared moduli in both equations, and summing them together, we obtain that

$$\boldsymbol{|}\partial_{\theta_{j}}\boldsymbol{\phi}_{t}\boldsymbol{|}^{2}=\boldsymbol{|}\boldsymbol{\xi}_{j,t}\boldsymbol{|}^{2}=\bigl{|}\sum\limits_{s=0}^{t-1}H_{j}({\bm{\theta}}_{s},\boldsymbol{\phi}_{s})\bigr{|}^{2}+\bigl{|}\sum\limits_{s=0}^{t-1}K_{j}({\bm{\theta}}_{s},\boldsymbol{\phi}_{s})\bigr{|}^{2},$$

(48)

where $\boldsymbol{|}\cdot\boldsymbol{|}$ stands for the ${\mathbb{C}}^{2}$ norm. As $H_{j},K_{j}$ are by assumption bounded functions, this result shows that trajectories $({\bm{\theta}}_{t},\boldsymbol{\phi}_{t})$ are, at worst, linearly unstable, in sharp contrast to the exponential instability of trajectories for dynamical chaos Chirikov79 .