#  Floquet simulators for topological surface states in isolation

Kun Woo Kim1,2, Dmitry Bagrets1, Tobias Micklitz3, Alexander Altland1
1Institut für Theoretische Physik, Universität zu Köln, Zülpicher Straße 77,
50937 Köln, Germany
2 Department of Physics, Chung-Ang University, 06974 Seoul, Republic of Korea
3Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, 22290-180, Rio
de Janeiro, Brazil

###### Abstract

We propose dynamical protocols allowing for the engineered realization of
topological surface states in isolation. Our approach builds on the concept of
synthetic dimensions generated by driving systems with incommensurate
frequencies. As a concrete example, we consider $3d$ topological surface
states of a $4d$ quantum Hall insulator via a
$(1+2_{\mathrm{syn}})$-dimensional protocol. We present first principle
analytical calculations demonstrating that no supporting $4d$ bulk phase is
required for a $3d$ topological surface phase. We back the analytical approach
by numerical simulations and present a detailed blueprint for the realization
of the synthetic surface phase with existing quantum linear optical network
device technology. We then discuss generalizations, including a proposal for a
quantum simulator of the $(1+1_{\mathrm{syn}})$-dimensional surface of the
common $3d$ topological insulator.

##  I Introduction

Surface states of topological insulators (TI) define one of the most
fascinating forms of quantum matter. Depending on their symmetries and
dimensionality, they conduct charge, spin, or heat with topological protection
against the detrimental effects of impurity scattering or interactions. These
features make the TI surface distinct from any other form of quantum matter,
and are believed to harbor far-reaching potential future device applications.
At the same time, our understanding of the TI surface physics remains
incomplete, both experimentally and theoretically. For example, even in the
absence of interactions their conduction properties are not known
quantitatively, and according to recent numerical work [1] even enigmatic. The
experimental analysis of surface transport is hindered by the inevitable
presence of an “insulating” bulk, with quotation marks because heat or
electric currents easily leak away from the surface hindering a clear
separation of surface and bulk currents.

According to the bulk–boundary principle, no _lattice_ quantum system in
isolation can be in the universality class of the TI surface. The necessity of
a supporting bulk follows from topological band theory or, more fundamentally,
as a consequence of anomaly inflow. The main message of this paper is that
this no-go theorem can be sidestepped within the wider framework of Floquet
quantum matter. Specifically, we will propose realizations of (dynamical)
synthetic matter in universality classes indistinguishable from those of
_isolated_ (static) TI surfaces in the presence of effective disorder. Our
work includes three novel conceptual elements: (i) the encoding of two and
three dimensional surface state topologies in multi-frequency dynamical
protocols, (ii) the first principle demonstration of the equivalence between
the quantum states engineered in this way and insulator surface states, and
(iii) the formulation of a detailed experimental blueprint suggesting that
this program can be implemented in realistic devices within the framework of
current date technology.

Previous work [2, 3] indeed pointed out the realizability of topological
metallic phases in dynamically driven lattice systems. However, the presence
of a lattice structure made these systems subject to the notorious fermion
doubling principle, which requires an even number of Dirac cones in the
Floquet Brillouin zone. In the presence of impurities these mutually gap out,
spoiling the surface state analogy. In order to realize a genuine surface
state in isolation, a more radical departure from the solid state crystal
paradigm is required. In this paper, we demonstrate that the toolbox of
quantum optics contains platforms that are up to this task, optical lattices
[4, 5], or linear optical networks [6, 7, 8] driven by multiple incommensurate
frequencies. The driving of $d$-dimensional realizations of such systems by
$d_{\mathrm{syn}}$ incommensurate frequencies is microscopically identical to
a time periodic (Floquet) dynamics acting in an effective system of
dimensionality $d+d_{\mathrm{syn}}$ [9], where the structure of the Floquet
operator in the $d$ physical and $d_{\mathrm{syn}}$ synthetic dimensions
depends on the driving protocol. Importantly, the correlations in the
synthetic directions are not confined by the fermion doubling theorem, and
this will be key to the engineering of topological surface states in
isolation. We will label the $d+d_{\mathrm{syn}}$-dimensional Floquet metallic
(FM) systems realized in this way as FM${}_{d+d_{\mathrm{syn}}}$ throughout.

The simulation of higher dimensional systems via driven low dimensional
physical platforms is experimental reality. In breakthrough experiments it was
applied to extend one-dimensional Anderson localization in the quantum kicked
rotor [10, 11, 12, 13] to higher dimensions. This defined an effectively
disordered FM1+2 and led to the first high precision observation of a three-
dimensional Anderson transition under parametrically controlled conditions [4,
14].

However, the realization of the TI surface states addressed in this paper
requires the additional structure of an internal bi-valued degree of freedom
or ’spin’. (For earlier proposals to realize topological quantum matter with
synthetic dimensions via the driving of systems with internal degrees of
freedom, see Refs. [15] and [16]. Specifically, we need full control over the
lattice nearest neighbor hopping for a system with two internal degrees of
freedom (“spin”). The required technology is not yet realized for the optical
lattice [17] but is available in the alternative platform of linear optical
networks [18]. We will therefore focus on this hardware and discuss the
implementation of a FM${}_{1+1_{\rm syn}}$ and a FM${}_{1+2_{\rm syn}}$ TI
surface state. We will demonstrate by numerical control simulations that
unique signatures of surface state delocalization in the synthetic disordered
system are observable for experimentally accessible time and length scales.

A further hallmark of our approach is that it realizes surfaces in effectively
“disordered” phases lacking translational invariance. The reason is that the
generation of synthetic dimension requires non-commuting operators both in
synthetic and physical space. The simultaneous presence of these operators in
the dynamics leads to non-integrability and chaotic fluctuations, physically
equivalent to tunable disorder at mesoscopic length scales. Our approach,
thus, simulates the surfaces of disordered topological quantum matter, which
one may take as an added element of realism.

The plan of the paper is as follows. In Section II, we present dynamical
protocols of 1$d$ quantum walks that utilize synthetic dimensions to simulate
higher dimensional systems. In Section III we introduce a quantum simulator of
topological insulator surface states in isolation at the example of the $4d$
quantum Hall insulator. We introduce a $1d$ quantum walk protocol, discuss its
topological property, and report on numerical simulations of the protocol, all
supporting the idea that the surface states of the $4d$ quantum Hall insulator
can be simulated by the $1d$ quantum walk. We then discuss a concrete
blueprint, realizing the quantum walk within existing optical linear network
set-up. Section IV provides further details and discusses generalizations.
Specifically, we introduce a simulator of surface states of a $3d$ quantum
spin Hall insulator. We conclude in Section VI. Aiming to keep the
presentation as non-technical as possible, the details of various derivations
are relegated to the Appendices.

Figure 1:  A $1d$ quantum walk with time-dependent spin quantization axis
$\hat{r}_{t}$ can simulate dynamics of a $2d$ systems when the period of
$r_{t}$ is incommensurate with the discrete time-steps of the evolution
operator. This engineering of synthetic dimensions allows to sidestep the
fermion doubling principle, and to simulate e.g. the $2d$ surface states in
isolation of a $3d$ quantum spin Hall insulator.

##  II Dynamical protocols

Consider the quantum walk of a spin-$1/2$ particle on a $1d$ lattice,
generated by successive applications of translations and spin rotations. The
single time-step evolution operator is of the general form

$\displaystyle\hat{U}_{t}=\sum_{m}\hat{R}_{m}(t)\otimes\hat{T}_{m},$ (1)

where $\hat{T}_{m}$ shifts the walker by $m$ lattice sites, and
$\hat{R}_{m}(t)=\vec{r}_{m}(t)\cdot\vec{\sigma}$ rotates its spin. Here and in
the following $\vec{\sigma}=(\sigma_{0},i\boldsymbol{\sigma})$ and
$\vec{r}_{m}=(r_{m0},\boldsymbol{r}_{m})$ are four component vectors such that
$\sigma_{0}=\mathds{1}_{2}$,
$\boldsymbol{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})$ and
$\vec{r}_{m}\in\mathds{C}^{4}$. Central for our proposal is the time-dependent
spin rotation axes $r_{m}(t)$ which are dynamically changed in the course of
the walk. As we show below, using dynamical protocols with periods that are
incommensurate with the discrete time step of the evolution operator Eq. (1)
enables the simulation of dynamics in higher dimensional systems. In the
following we will focus on quantum walks with short-range hops to the nearest
neighbors, $m\in\\{0,-1,+1\\}$. The unitary operator Eq. (1) then simplifies
to

$\displaystyle\hat{U}_{t}=\vec{r}_{0}\cdot\vec{\sigma}+(\vec{r}_{+}\cdot\vec{\sigma})\otimes\hat{T}_{+}+(\vec{r}_{-}\cdot\vec{\sigma})\otimes\hat{T}_{-},$
(2)

and unitarity sets the following constraint on $\vec{r}_{0}$, $\vec{r}_{\pm}$:
Expressing $\vec{r}_{\pm}=\vec{r}_{r}\pm i\vec{r}_{i}$, with
$\vec{r}_{r,i}\in\mathds{R}^{4}$ real four component vectors, the latter are
orthogonal, $\vec{r}_{r}\cdot\vec{r}_{i}=0$,
$\vec{r}_{0}\cdot\vec{r}_{r,i}=0$, and equal in magnitude,
$|\vec{r}_{r}|=|\vec{r}_{i}|=\frac{1}{2}\sqrt{1-|\vec{r}_{0}|^{2}}$ at each
time step $t$ (see Appendix A for details).

Finally, we add spin-dependent spatial disorder to the dynamics. To this end
we introduce the unitary matrix
$(\hat{U}_{\text{dis}})_{nn^{\prime}}=\hat{U}_{\mathrm{dis}}(n)\delta_{nn^{\prime}}$
where $\hat{U}_{\mathrm{dis}}(n)$ are independent random spin rotation
matrices, acting locally on each site $n$. The single time-step evolution
$|\psi_{t}\rangle=\mathcal{U}_{t,t-1}|\psi_{t-1}\rangle$ generating the $1d$
quantum walk is then composed of the combined operator,

$\displaystyle{\mathcal{U}}_{t,t-1}=\hat{U}_{t}\hat{U}_{\text{dis}},$ (3)

and we next discuss its potential to simulate higher dimensional dynamics.

###  II.1 Synthetic dimensions from multi-frequency dynamical protocols

Let us specify the protocol Eq. (1) to rotations $\hat{R}_{m}$ which depend on
$d_{\rm syn}$ time-dependent functions,

$\displaystyle\hat{R}_{m}(t)\equiv\hat{R}_{m}(\varphi_{2,t},\ldots\varphi_{{d_{\rm
syn}}+1,t}),$ (4)

and where the time-dependence for each of the functions is of the form
$\varphi_{i,t}=k_{i}+\omega_{i}t$ with frequencies
$\omega_{2},...,\omega_{d_{\rm syn}+1}$ incommensurate to $2\pi$ and among
themselves. Further, $k_{\mathrm{syn}}\equiv(k_{2},..,k_{d_{\rm syn}+1})$ are
arbitrary initial phases which we consider averaged over in our dynamical
protocols below. The mapping to an effectively $1+d_{\rm syn}$ dimensional
Floquet system is achieved by extending the Hilbert space of the system and
interpreting these phases as momenta conjugate to integer valued coordinates
$n_{\mathrm{syn}}=(n_{2},...,n_{d_{\rm syn}+1})$, with canonical commutation
relations $[\hat{n}_{i},\hat{k}_{j}]=-i\delta_{ij}$ between the corresponding
operators. We note that $\hat{n}_{i}=-i\partial_{k_{i}}$ in the phase-momentum
representation of the theory. These coordinates extend the lattice in
$1+{d_{\rm syn}}$ dimensions with sites $\textbf{n}=(n_{1},n_{\mathrm{syn}})$,
where $n_{1}=n$ is the physical lattice coordinate, conjugate to a phase
$k_{1}$. In the same notation, $\textbf{k}=(k_{1},k_{\rm syn})$.

Using the general relation
$e^{ia\hat{n}}f(\hat{k})e^{-ia\hat{n}}=f(\hat{k}+a)$, the time dependence in
the arguments of the rotation operator can be removed by the gauge
transformation

$\displaystyle\hat{R}_{m}(t)=e^{i\omega_{j}t\hat{n}_{j}}R_{m}(0)e^{-i\omega_{j}t\hat{n}_{j}},$
(5)

where a summation over $j=2,\ldots,1+d_{\rm syn}$ is implicit. This enables us
to express the time evolution operator
$\mathcal{U}_{t,0}\equiv\mathcal{U}_{t,0}\equiv\prod_{\tau=0}^{t-1}\mathcal{U}_{\tau+1,\tau}$
as

$\displaystyle|\psi_{t}\rangle$
$\displaystyle=\mathcal{U}_{t,t-1}\mathcal{U}_{t-1,t-2}\cdots\mathcal{U}_{1,0}|\psi_{0}\rangle$
$\displaystyle=e^{i\omega_{j}t\hat{n}_{j}}\left[\mathcal{U}_{0,-1}e^{-i\omega_{j}\hat{n}_{j}}\right]^{t}|\psi_{0}\rangle.$
(6)

We notice that the time evolution is governed by powers of the single Floquet
operator
$\mathcal{U}_{F}\equiv\mathcal{U}_{0,-1}e^{-i\omega_{j}\hat{n}_{j}}=\hat{U}_{t=0}\hat{U}_{\mathrm{dis}}e^{-i\omega_{j}\hat{n}_{j}}\equiv
U_{\textbf{k}}W_{\textbf{n}}$. Here,
$\hat{W}_{\textbf{n}}\equiv\hat{U}_{\mathrm{dis}}(n_{1})e^{-i\omega_{j}\hat{n}_{j}}$
is diagonal in the coordinate representation, while
$\hat{U}_{\textbf{k}}=\hat{U}_{0}(\textbf{k})$ is momentum-diagonal. To
understand this last statement, we note that in Eq. (1) the coordinate
translation operator $\hat{T}_{m}f(n_{1})=f(n_{1}-m)$ affords the
representation $\hat{T}_{m}=e^{im\hat{k}_{1}}$ while $\hat{R}_{m}(0)$ depends
on the phases $k_{\mathrm{syn}}$.

To summarize, our dynamics is governed by the effective multi-dimensional
Floquet operator $\mathcal{U}_{F}=\hat{U}_{\mathbb{k}}\hat{W}_{\textbf{n}}$
factoring into two pieces which are individually diagonal in coordinates and
momenta, respectively. Our numerical simulations below demonstrate that the
combined action of these operators induces integrability breaking, physically
equivalent to static disorder, in all $1+d_{\rm syn}$ dimensions. However,
before introducing quantum simulators for the combined effects of disorder and
topology in this setting, we briefly introduce observables probing topological
surface states in an experimentally accessible way.

###  II.2 Observable

The spreading after $t$ time steps of a wave packet, describing a quantum
walker initially prepared at site $n_{1}=0$ with spin $\sigma$, can be
expressed as

$\displaystyle\langle\Delta X^{2}\rangle$
$\displaystyle\equiv\sum_{n_{1}}\sum_{\sigma^{\prime},\sigma}n_{1}^{2}\overline{|\langle
n_{1},\sigma^{\prime}|\mathcal{U}_{t,0}|0,\sigma\rangle|^{2}}.$ (7)

Here the sum is over spin orientations $\sigma=\uparrow,\downarrow$ and
$\overline{(...)}$ refers to the average over both, an ensemble of
realizations of the random rotations $\hat{U}_{\rm dis}$ and the initial
momenta $k_{\mathrm{syn}}$. In a mixed coordinate-momentum representation,
basis states of the extended Hilbert space are defined by the kets
$|n_{1},\sigma\rangle\to|n_{1},k_{\mathrm{syn}},\sigma\rangle$. Specifically,
the initial state of the quantum walker is confined to $n_{1}=0$, and
independent of synthetic momenta. Upon Fourier transformation to a full
coordinate representation,
$|n_{1},k_{\mathrm{syn}},\sigma\rangle\to|n_{1},n_{\mathrm{syn}},\sigma\rangle\equiv|\mathbf{n},\sigma\rangle$
this translates to localization at $|\boldsymbol{0},\sigma\rangle$ in both
physical and synthetic space. The spreading of the quantum walker is thus
given by (see further details in Appendix B)

$\displaystyle\langle\Delta X^{2}\rangle$
$\displaystyle\equiv\sum_{\boldsymbol{n}}\sum_{\sigma^{\prime},\sigma}n_{1}^{2}\overline{|\langle\boldsymbol{n},\sigma^{\prime}|\mathcal{U}_{F}^{t}|\boldsymbol{0},\sigma\rangle|^{2}}.$
(8)

The correlation function Eq. (8) describes the width in the physical
$n_{1}$-direction of the wave packet initially prepared at $n_{1}=0$, and its
finite time scaling encodes information on the quantum walk dynamics.

###  II.3 Topological invariants

All our topological FMs to be discussed below are characterized by integer
valued invariants. These numbers afford two different interpretations:

Topological invariants and FM classification: The first relates to a
classification of FM phases in terms of the periodic table of Hamiltonian
insulators[3]. Its idea is to map the translation-invariant part of the
Floquet operator $U_{\textbf{k}}$ onto a block off-diagonal ‘auxiliary’
Hamiltonian

$H_{\textbf{k}}=\left(\begin{array}[]{cc}&U_{\textbf{k}}\cr
U_{\textbf{k}}^{\dagger}&\end{array}\right).$ (9)

This Hamiltonian inherits the symmetries of $U_{\textbf{k}}$, but in addition
possesses a ‘chiral’ symmetry due to its off-diagonality; it belongs to a
symmetry class different from the class of the Floquet theory. For example, if
the latter is in class A (just unitary), $H$ will be in class AIII (chiral, no
further symmetries.) Bott periodicity then implies that a class A Floquet
theory realizes a FM state in odd effective dimension $D=d+d_{\rm syn}$ if the
associated $D$-dimensional Hamiltonian in class AIII is also topologically non
trivial. Further, the presence of topologically non trivial phases of the
Hamiltonian theory is signaled by invariants mathematically identical to those
constructible for the Floquet theory. For example, in the above case, these
invariants are ‘winding numbers’ defined by a unitary map from odd-dimensional
Brillouin zones $\textbf{k}\mapsto U_{\textbf{k}}$ into the unitary group.
These winding numbers classify class AIII insulating phases in odd dimensions
and class A Floquet metallic phases in even dimensions.

Topological invariants and localization theory: To understand this statement
in more concrete terms, we note that our Floquet theories are categorically
disordered or chaotic. Their physical description requires real space methods,
as defined by the nonlinear $\sigma$-models of disordered conductors. In these
theories protection against the effects of Anderson localization, i.e.
topological metallicity, is introduced via topological terms (see Eq.(25) for
a concrete example). These terms take physical effect provided their coupling
constants are not vanishing. Below, we will demonstrate in two concrete case
studies that the momentum space invariants responsible for the ‘abstract’
classification of topological FMs indeed feature as coupling constants in the
topological field theories. In this way, they serve a double function in the
classification and the localization theory of FMs. In the latter context, they
protect topological FMs from developing a ‘mobility gap’ and force them to
remain metallic, including FMs in low dimensions which would otherwise show
strong localization.

##  III Three dimensional topological Floquet metal FM${}_{1+2_{\rm syn}}$

The concept of synthetic dimensions is general and can be realized for a wide
class of driven or kicked Floquet systems [19, 20, 21, 16, 22]. In the
following, we introduce a specific realization in $1+2_{\mathrm{syn}}$
dimensions, physically equivalent to the surface of a four dimensional
topological insulator in symmetry class ${\rm A}$ (aka ’four-dimensional
quantum Hall insulator’).

###  III.1 Model

We consider a one dimensional quantum walker, whose forward and backward
hopping amplitudes are time-dependent matrices coupling to the internal
degrees of freedom (see Fig. 1). In the notation of the previous section, its
time evolution from one discrete time step, $t$, to the next is defined
through

$\displaystyle\hat{U}_{t}$
$\displaystyle=\frac{1}{2}(\sigma_{0}+\mathbb{r}_{t}\cdot\boldsymbol{\sigma})\otimes\hat{T}_{+}+\frac{1}{2}(\sigma_{0}-\mathbb{r}_{t}\cdot\boldsymbol{\sigma})\otimes\hat{T}_{-},$
(10)

where the specific choice

$\mathbb{r}_{t}=(\cos\varphi_{2,t}\sin\varphi_{3,t},\sin\varphi_{2,t}|\sin\varphi_{3,t}|,\cos\varphi_{3,t}),$
(11)

will be motivated momentarily, and the time dependence of the phase arguments
is defined below Eq.(4). Turning to the gauge-equivalent representation in
terms of a Floquet operator acting in a space with one physical and two
synthetic dimensions, we describe the dynamics through the Floquet operator
$\hat{\mathcal{U}}_{F}=\hat{U}_{\mathbb{k}}\hat{W}_{\mathbb{n}}$ with
$U_{\textbf{k}}=U_{t=0}$ and $\hat{T}_{\pm}=e^{\pm ik_{1}}$. For later
reference, we note that the unitary $U_{\textbf{k}}$ affords different
representations, each useful in its own right. First, it is straightforward to
verify that

$\displaystyle\hat{U}_{\mathbb{k}}$
$\displaystyle=\exp\left[ik_{1}({\mathbb{r}}(\mathbb{k})\cdot\boldsymbol{\sigma})\right],$
(12) $\displaystyle{\mathbb{r}}(k)$ $\displaystyle=(\cos k_{2}\sin k_{3},\sin
k_{2}|\sin k_{3}|,\cos k_{3}).$

Alternatively, we may represent the spin matrices as rotations acting upon a
translation operator with $z$-axis polarization:
$\hat{U}_{\mathbb{k}}=R_{3}(-k^{\prime}_{2})R_{2}(-k_{3})\hat{T}_{C}R_{2}(k_{3})R_{3}(k^{\prime}_{2})$
with $\hat{T}_{C}=e^{ik_{1}\sigma_{3}}$, the momentum
$k^{\prime}_{2}=k_{2}\,{\rm sgn}\,k_{3}$, and spin rotation operators
$R_{j}(\varphi_{l})=\exp\left(i\varphi_{l}\sigma_{j}/2\right)$.

A crucial feature of this realization is its non-analytic dependence on the
momentum variables through $|\sin k_{3}|$. In the Fourier conjugate
representation it translates to long ranged hopping
$(\mathcal{U}_{F})_{n_{3}n^{\prime}_{3}}\sim|n_{3}-n^{\prime}_{3}|^{-2}$
111For $n\neq\pm 1$, $\int_{-\pi}^{\pi}e^{ik_{2}n}|\sin
k_{2}|dk_{2}=\frac{2}{n^{2}-1}((-1)^{n+1}-1)$. At this point, the synthetic
dimensions begin to play an essential role: power law hopping in physical
dimensions is difficult to engineer. More importantly, the $|\sin k_{3}|$ non-
analyticity is the essential resource allowing us to sidestep the fermion
doubling theorem and to realize a synthetic topological metal.

Figure 2:  The spread of the quantum walker as a function of time steps for
the two protocols and different values of bandwidth $w$ introduced in Eq.
(14). To facilitate observation of the metal-to-insulator transition, the
width of wave packet is rescaled by the critical scaling at the transition,
$\langle\Delta X^{2}\rangle\sim t^{2/3}$. Left panel: The topologically
trivial protocol, $W=0$, shows a metal-to-insulator transition for $w\simeq
0.1$. Right panel: The topological non trivial variant, $W=4$, shows no
signatures of localization, i.e. $\langle\Delta X^{2}\rangle\sim t$ for all
values $w$ and up to $t\sim 5\times 10^{5}$ time steps.

Winding number:—To elucidate this last point, we interpret Eq. (12) as a
mapping from the $3d$ Brillouin zone torus to the two-dimensional special
unitary group $\mathbb{T}^{3}\to{\rm SU}(2)$ (with unit determinant,
$\det[U_{\mathbb{k}}]=1$), and assign the topological invariant

$\displaystyle W$ $\displaystyle=\frac{1}{24\pi^{2}}\int
d^{3}\mathbb{k}\,\epsilon^{\mu\nu\rho}\mathrm{tr}\left[(U_{\mathbb{k}}^{\dagger}\partial_{\mu}U_{\mathbb{k}})(U_{\mathbb{k}}^{\dagger}\partial_{\nu}U_{\mathbb{k}})(U_{\mathbb{k}}^{\dagger}\partial_{\rho}U_{\mathbb{k}})\right]$
$\displaystyle=\frac{1}{2\pi^{2}}\int d^{3}\mathbb{k}\sin^{2}k_{1}|\sin
k_{3}|=4,$ (13)

where $\mu,\nu,\rho\in\\{1,2,3\\}$. As anticipated above, it is the non-
analyticity $|\sin k_{3}|$ that leads to a non-vanishing winding number.
Conversely, a model with analytic $k$-dependence would necessarily lead to a
vanishing winding number, in accordance with the fermion doubling theorem.

Driving protocols with non-analytic functions, simulating power law hoppings
in synthetic space, allow to construct Floquet operators with even winding
numbers $W\in 2{\mathbb{Z}}$: The starting point of our construction are
realizations of unitary operators which display finite winding numbers over
certain subsets of a Brillouin zone, say $W=n$ in region ${\rm I}$ and $W=-n$
in region ${\rm II}$. (The numbers must add to zero by virtue of the fermion
doubling theorem.) The trick to generate a finite winding number then is to
modify the momentum dependence in region ${\rm II}$ via a sign change in the
momentum dependence which inverts the winding number to $W|{\rm II}=+n$ and
$W=2n$ in total. We also constructed an alternative model with the minimal
winding possible in this scheme, $W=2$. Since it was rather involved we opted,
however, for the discussion of the simpler $W=4$ variant, Eqs. (10) and (11).
We also emphasize that winding numbers for topological Floquet systems cannot
be simply understood from their low energy effective Hamiltonians [24]. $W=4$
of Eq. (12) is e.g. not related to four Weyl cones in a low energy
description, but rather stores information on the entire Brillouin zone (see
also discussion on dispersion of Eq. (12) in the next subsection).

The invariant Eq.(III.1) indicates topological metallicity of our Floquet
system. Within the alternative interpretation discussed above, the non-
vanishing winding number, $W$, signals topological non-triviality of the
‘auxiliary’ class ${\rm AIII}$ Hamiltonian. Bott periodicity implies non
trivial phases of $4d$ class ${\rm A}$ systems (the four-dimensional quantum
Hall effect), and the original Floquet system describes the physics of its
three dimensional metallic surface state. In more concrete terms, we will see
in section V that the winding number $W$ features as a building block in our
construction of a gapless effective field theory equivalent to that of a
three-dimensional topological metal.

###  III.2  Numerical simulations

To independently verify the topological metallic nature of the $3d$ dynamics
simulated by the protocol Eq. (10) we have run numerical simulations. More
specifically, we have simulated the quantum walk with trivial and non trivial
winding numbers for varying effective disorder strengths by introducing
additional bandwidths in the model, as we discuss next. This allows to test
our main prediction, that is, the absence of Anderson localization for all
disorder strength for finite windings $W$, contrasting Anderson localization
at large disorder for vanishing winding number $W=0$.

Simulation details:—We numerically study the time evolution of an initially
localized wave packets, under influence of the $1d$ quantum walk operator Eq.
(10). To allow for a comparison of topologically trivial and non trivial
quantum walks with same energy-momentum dispersion relation of the clean
system, we implement the walk with $|\sin\varphi_{3,t}|$, as indicated in Eq.
(12), and a second protocol with $|\sin\varphi_{3,t}|$ replaced by
$\sin\varphi_{3,t}$. A metal-to-insulator transition with increasing disorder
strength is expected for the second protocol. The static spatial disorder
$\hat{U}_{\rm dis}$ is implemented by randomly drawing spin rotation matrices
from the uniform Haar measure. That is, the disorder strength is fixed and we
need to introduce some tunable parameter allowing to drive a (possible) metal-
to-insulator transition. We then notice that Eq. (12) and its topologically
trivial cousin have no energy dispersion in $k_{2,3}$ direction, which makes
the latter always prone to localization. At the same time, we can perturb the
original models to generate a dispersion with tunable bandwidth $w$ in
$k_{2,3}$-direction. This then allows to study a delocalization transition as
a function of $w$. To realize this idea, we multiply the original single time-
step evolution operators by the unitary operator

$\displaystyle\hat{U}_{w}$ $\displaystyle=\exp\left[iw(\sin
k_{2,t}\sigma_{1}+\sin k_{3,t}\sigma_{2})\right],$ (14)

which does the job. We then study the time scaling of the average spread
$\langle\Delta X^{2}\rangle$ of the initial wave packet in the physical
dimension.

Results:—Fig. 2 shows the time evolution of $\langle\Delta X^{2}\rangle$ for
the two protocols and different values of the bandwidth $w$, as indicated in
the legend. The vertical axis is rescaled by the time-dependence
$\langle\Delta X^{2}\rangle\sim t^{2/3}$ expected at the metal-to-insulator
transition [5]. For the trivial protocol $W=0$ (left panel) one clearly sees
the metallic ($w>0.1$) and insulating ($w<0.1$) regimes separated by critical
scaling at $w\simeq 0.1$. In contrast, the topological non trivial protocol
with $W=4$ (right panel) shows metallic behavior for all values of $w$, with
$\langle\Delta X^{2}\rangle\sim t$ up to the largest time steps $5\times
10^{5}$. The two incommensurate frequencies here were chosen as
$\omega_{2}=2.4\sqrt{5}$, $\omega_{3}=2.4\sqrt{15}$, and disorder averaging is
over $~{}50$ realizations for each data point. We also notice that the long-
time numerical results are independent of the specific value for the
incommensurate frequencies.

In case of the trivial protocol $W=0$, deviations from classical scaling
$\langle\Delta X^{2}\rangle\sim t$ signals Anderson localization, and
eventually the dynamics will entirely freeze, $\langle\Delta X^{2}\rangle\sim
t^{0}$, on longer time and length scales. The important observation for us is
that a clear distinction between quantum simulators of trivial and topological
metallic phases are noticeable already for a small number of ${\cal O}(20)$
time steps, see Fig. 3(c) and (d). This is crucial, once it comes to an
experimental implementation of the quantum walks, as we discuss next.

Figure 3:  (a,b) Blueprint of an optical linear network simulating the
topological surface states of a $4d$ quantum Hall insulator in the quantum
walk setting. The feedback loop is build of the step operator $\hat{T}_{\cal
C}$ (upper and middle arms) followed by the coin operator
$\hat{R}(t)=R_{t+1}\hat{U}_{\text{dis}}R^{\dagger}_{t}$ (lower arm), for
details see main text. To the left and right of the loop, source and detection
units are connected. The source consist of a laser and a polarizing beam
splitter (PBS) that allows for the preparation of the initial state. The
passage to the detection unit can be activated by the dynamically tunable EOMs
following the fiber lines of the step operation. In the detection unit photons
are registered by avalanche photo-diode (APD). (c,d) Numerical simulations of
wave packet spreading for trivial and topological quantum walks, here for
varying incommensurate frequencies. The latter are chosen as
$\omega_{2}=C\sqrt{5}$ and $\omega_{3}=C\sqrt{15}$ for several values $2.0\leq
C\leq 2.6$ (see legends). Differences between the two systems become visible
already after $t\gtrsim 10$ time steps: the topological metal (d) shows robust
diffusion for all values of $C$, while a crossover from diffusive to
subdiffusive dynamics is observed as $C>2.4$ for the trivial metal (c).

###  III.3  Experimental realization: A FM${}_{1+2_{\rm syn}}$ simulator

So far we have shown how the freedom of choosing operators of arbitrary
$k$-dependence in the synthetic momentum space allows for the engineering of
topological Floquet operators that cannot exist in autonomous lattice
environments. Specifically, we have (i) proposed a concrete dynamical protocol
based on a $1d$ quantum walk, (ii) shown that this simulates a topological
metal e.g. realized on the isolated surface of a $4d$ quantum Hall insulator,
and (iii) demonstrated that its most characteristic feature—absence of
Anderson localization at strong disorder—can be observed already after ${\cal
O}(20)$ time steps. The final piece of our proposal is to indicate an
experimental platform that offers the required flexibility to implement
dynamical protocols for spin-1/2 walkers. We here argue that linear optical
networks are ideally suited to realize the proposed quantum simulators. After
a brief review of their principal elements, we suggest a blueprint for the
quantum Hall simulator.

Principle elements:—In the typical optical realization of a quantum walk,
photons propagate through a network of linear elements, viz. beam splitters,
phase shifters and polarization plates, realizing the “step” and “coin”
operations. The step operation typically implements chiral hopping (here as a
matrix in spin space)

$\displaystyle\hat{T}_{\cal C}\equiv\begin{pmatrix}\hat{T}_{+}&\\\
&\hat{T}_{-}\end{pmatrix},$ (15)

translating the walker to the right ($T_{+}$) or ($T_{-}$) left according to
its spin being in the up (first component), respectively, down state (second
component). The dynamical coin operations realize spin rotations,
$\hat{R}(t)$. Using an Euler angle decomposition they can be generated from
repeated application of elementary rotations around any two of the three
internal axes

$\displaystyle\hat{R}_{j}(\varphi_{l,t})$
$\displaystyle=\exp\left(i\varphi_{l,t}\sigma_{j}/2\right),\quad j=y,z,\quad
l=2,3,$ (16)

with Pauli matrices operating in spin space. The great flexibility offered by
optical linear networks is that Euler angles $\varphi_{l,t}=k_{l}+\omega_{l}t$
can be changed dynamically during the realization of the quantum walk.

The successive application of chiral step and coin operations composing the
quantum walk protocol is implemented in a “feedback loop”, see Fig. 3(b).
Typically, a coherent laser pulse attenuated to an average single-photon per
pulse injects photons into the linear network. Horizontal and vertical
polarizations of the photon constitute the internal “spin”-states. The step
operation $\hat{T}_{\cal C}$ is realized in time, employing a polarizing beam
splitter in combination with fiber delay lines. That is, horizontally and
vertically polarized photons are separated by the beam splitter and send
through fiber lines of different lengths. The length mismatch of the fibers
introduces a well-defined delay between the two polarization components. When
coherently recombined, the temporal separation of the two components is
equivalent to the spatial separation by two lattice sites induced by the
chiral translation to left and right neighbors of the $1d$ lattice. Dynamical
coin operations $\hat{R}_{j}(\varphi_{l,t})$ are realized via tunable
polarization rotations. In practice, the dynamical control over only one
rotation-axis, e.g. the $z$-axis is required, and rotations around remaining
axes are realized by the combination with suitable polarization controllers,
i.e. half- and quarter-wave plates 222 For example, rotation around y-axis,
$\hat{R}_{2}(\varphi)=e^{i\pi\sigma_{1}/4}\hat{R}_{3}(\varphi)e^{-i\pi\sigma_{1}/4}$,
is implemented using the quarter-wave plates aligned at an angle $\pi/4$,
$\hat{C}_{\text{QWP}}(\pi/4)=e^{-i\pi\sigma_{1}/4}$ [8]. While the rotation
around x-axis,
$\hat{R}_{1}(\varphi)=e^{-i\pi\sigma_{2}/4}\hat{R}_{3}(\varphi)e^{i\pi\sigma_{2}/4}$,
is implemented by using the two half wave plates:
$\hat{C}_{\text{HWP}}(\pi/8)\hat{C}_{\text{HWP}}(0)=e^{-i\pi\sigma_{2}/4}$
[18]..

The dynamical control is achieved via control voltages applied to fast-
switching electro-optic modulators (EOMs) that change rotation-angles on time
scales shorter than a step operation. Recent progress allows to operate the
latter without high additional losses and walks up to $t=30-40$ time-steps
have been reported within this set up, see e.g. Refs. [18, 8]. From the
numerical simulations of the previous section, we expect this to be sufficient
to distinguish the dynamics of a topological from a trivial Floquet metal.
After the light pulses have been fed back into the loop of step and coin
operations, realizing a single time step operation, for the desired number of
time steps they are released to the detection unit, see Fig. 3(b). Repeating
the procedure for varying numbers of time-steps and different realizations of
coin operations, one obtains the walker’s probability distribution which
allows for the full characterization of its dynamics.

Blueprint for the FM${}_{1+2_{\rm syn}}$:—A detailed blueprint for an optical
linear network simulating the topological FM${}_{1+2_{\rm syn}}$ is shown in
Fig. 3. A crucial observation here is that the chiral translational $U_{t}$
(10) with tunable spin-quantization axis $\mathbb{r}_{t}$ can be implemented
via a step operation $T_{\cal C}$ dressed by coin operations,
$\hat{U}_{t}=\hat{R}^{\dagger}_{t}\hat{T}_{\cal C}\hat{R}_{t}$, with the coin
matrix $\hat{R}_{t}$ being a product of two elementary rotations,
$\hat{R}_{t}=\hat{R}_{y}(\varphi_{3,t})\hat{R}_{z}(\varphi_{2,t})$, where

$\displaystyle\varphi_{3,t}$ $\displaystyle=k_{3}+\omega_{3}t,$ (17)
$\displaystyle\varphi_{2,t}$
$\displaystyle=\begin{cases}k_{2}+\omega_{2}t,\quad\,\sin(\varphi_{3,t})\geq
0,\\\ -k_{2}-\omega_{2}t,\quad\,\sin(\varphi_{3,t})<0,\end{cases}$ (18)

(note the conditional value of $\varphi_{2,t}$ depending on the sign of
$\sin(\varphi_{3,t})$, which follows from the definition (11) of the vector
$\mathbb{r}_{t}$). The role of $\hat{R}_{t}$ is to rotate $\mathbb{z}$-axis
into the instantaneous spin quantization axis $\mathbb{r}_{t}$.

Figure 4:  Dispersion relations of the quasi-energy spectrum of the two-band
model along the high symmetry lines $k_{2}=0$ (left) and $k_{2}=\pi$ (middle),
respectively, see also discussion in the main text. The former contributes to
the topological invariant by the factor
$\text{Pf}\left[w_{\mathbb{\Lambda}_{1}}\right]\text{Pf}\left[w_{\mathbb{\Lambda}_{2}}\right]=-1$,
while the latter gives the factor
$\text{Pf}\left[w_{\mathbb{\Lambda}_{3}}\right]\text{Pf}\left[w_{\mathbb{\Lambda}_{4}}\right]=1$.
Here $w_{\mathbb{k}}$ is the antisymmetric sewing matrix. Right panel:
Visualization of dispersions along the high-symmetry lines in the $2d$
Brillouin zone.

In the linear network set-up it is convenient to start the feedback loop with
a step operation. Reorganizing thus spin rotations and the local disorder
potential in the originally defined one step evolution operator
$\mathcal{U}_{t,t-1}=\hat{U}_{t}\hat{U}_{\rm dis}$ (with
$\hat{U}_{\text{dis}}$ specified below), we construct the equivalent one as
the following succession of step and coin operations,

$\displaystyle\mathcal{U}_{t+1,t}$ $\displaystyle=\hat{R}(t)\hat{T}_{\cal
C},\qquad\hat{R}(t)=\hat{R}_{t+1}\hat{U}_{\text{dis}}\hat{R}^{\dagger}_{t}.$
(19)

This sequence is then iterated for the desired number of time steps.

Fig. 3(a) schematically shows the elements of quantum walk operations to be
applied to an initial localized wave packet before the detection after $t_{f}$
time steps. The actual implementation of the linear optical network can be
prepared as in Fig. 3(b), realizing chiral quantum walk and coin operators.
The EOMV,H are equipped for the initiation and the readout of the quantum walk
simulation. For the static disorder in real space, we suggest to follow the
protocol used in the numerical simulations with fixed bandwidth. That is,
choosing $\hat{U}_{\text{dis}}=R_{z}(\phi_{n_{1}})$ with static local angles
$\phi_{n_{1}}$, randomly drawn from the unit circle
$-\pi\leq\phi_{n_{1}}<\pi$, and frequencies $\omega_{2}$, $\omega_{3}$
indicated in the previous section.

Fig. 3(c) and (d) shows numerical results for the quantum simulators of the
trivial ($W=0$) and topological ($W=4$) metal over a range of experimentally
accessible time steps. Here the bandwidth is set to $w=0$, and incommensurate
frequencies are varied as $\omega_{2}=C\sqrt{5},\omega_{3}=C\sqrt{15}$ with
values of $C$ indicated in the legend. There are notable differences between
the two systems already after $t\gtrsim 10$ time steps: Dynamics for the
topological metal is diffusive for all values of $C$, while the the trivial
system shows a $C$-dependent behavior reminiscent of a metal-insulator
transition as $C$ is increased. Notice, however, that the metallic behavior
for small values $C=2,2.1$ only holds for short time, and localization sets in
at longer times (i.e. for the trivial protocol there is no true metallic phase
at $w=0$, as discussed above).

This finalizes our discussion of a quantum simulator for the surface states of
a $4d$ quantum Hall insulator. We next discuss generalizations to other
dimensions and symmetry classes.

##  IV Two dimensional Floquet topological metal $FM_{1+1_{\rm syn}}$

Figure 5:  The scaled width of a wave packet $\langle\Delta X^{2}/t\rangle$
for different quantum walks in (1+1syn)D and monitored over $5\times 10^{5}$
time steps. (Left) Topological quantum walk showing anomalous super diffusion,
(middle) critical quantum walk of a class ${\rm A}$ model fine tuned to a
topological phase transition, (right) topologically trivial quantum walk
subject to Anderson localization. Colors correspond to incommensurate
frequencies $\omega_{2}=\sqrt{5}C$ with values $C$ indicated in the legend.

To illustrate the generality of our approach, we next discuss the example of a
quantum simulator for surface states in a symmetry class different from the
quantum Hall insulators. Specifically, we propose a simulator for the $2d$
surface states of a class ${\rm AII}$ quantum spin Hall insulator in $d=3$.

To this end, we start out from a $1d$ quantum walk, Eq. (2), with dynamical
protocol,

$\displaystyle\vec{r}_{0}$
$\displaystyle=\frac{1}{2}(1-\cos\varphi_{2,t},0,0,\sin\varphi_{2,t}),$
$\displaystyle\vec{r}_{r}$
$\displaystyle=\frac{1}{4}(-1-\cos\varphi_{2,t},0,0,\sin\varphi_{2,t}),$
$\displaystyle\vec{r}_{i}$
$\displaystyle=\frac{1}{4}(0,|\sin\varphi_{2,t}|,-1-\cos\varphi_{2,t},0),$
(20)

where $\varphi_{2,t}=k_{2}+\omega_{2}t$ and the frequency $\omega_{2}$
incommensurate to $2\pi$. The three vectors Eq. (IV) are orthogonal to each
other, and unitarity of the single time-step evolution operator follows from
$|\vec{r}_{0}|=|\sin\varphi_{2,t}/2|$ and
$|\vec{r}_{r,i}|=\frac{1}{2}|\cos\varphi_{2,t}/2|$, see also Appendix A. Upon
Fourier transform in the physical coordinate and gauge transformation to
eliminate time dependence of the driving protocol, we arrive at the Floquet
operator

$\displaystyle\hat{U}_{\mathbb{k}}=(\hat{r}_{0}\cdot\vec{\sigma})+(\vec{r}_{+}\cdot\vec{\sigma})e^{ik_{1}}+(\vec{r}_{-}\cdot\vec{\sigma})e^{-ik_{1}},$
(21)

where $k_{2}=\varphi_{2,t=0}$ and $\vec{r}_{\pm}=(\vec{r}_{r}\pm
i\vec{r}_{i})$. We notice that the specific choice of the driving protocol has
lead to the non-analytical $n_{2}$-component $\propto|\sin k_{2}|$. It is
again this unusual dependence, impossible to realize on a lattice with finite
range hopping, which allows us to sidestep the fermion doubling theorem. It is
readily verified that Eq. (21) satisfies the time reversal relation
$\sigma_{2}\hat{U}_{\mathbb{k}}^{T}\sigma_{2}=\hat{U}_{-\mathbb{k}}$ of class
${\rm AII}$ systems. The latter host topological insulating $\mathbb{Z}_{2}$
phases in $3d$, and thus topological metallic Floquet phases in $2d$.

###  IV.1 Topological invariant

To demonstrate the topological nature of the protocol Eq. (IV), we focus on
the translational invariant part $U_{\mathbb{k}}$, and consider the latter as
a map from the $2d$ Brillouin zone torus to the special unitary group
$\mathbb{T}^{2}\to{\rm SU}(2)$ ($\det[\hat{U}_{\mathbb{k}}]=1$). Time reversal
symmetry imposes that $n(\mathbb{k})\parallel e_{0}$ at the four time-reversal
invariant momenta, $\mathbb{\Lambda}_{1}=(0,0)$,
$\mathbb{\Lambda}_{2}=(0,\pi)$, $\mathbb{\Lambda}_{3}=(\pi,0)$, and
$\mathbb{\Lambda}_{4}=(\pi,\pi)$. At these points the so-called sewing matrix
$w_{\mathbb{k}}=-i\sigma_{2}\hat{U}^{T}_{\mathbb{k}}$ is anti-symmetric and
the map $\hat{U}_{\mathbb{k}}$ is thus characterized by the $\mathbb{Z}_{2}$
topological index,

$\displaystyle W_{\mathbb{Z}_{2}}$ $\displaystyle=\prod_{i=1}^{4}{\rm
Pf}[-i\sigma_{2}\hat{U}^{T}_{\mathbb{\Lambda}_{i}}]=-1,$ (22)

where in the last identity we used that for Eq. (21)
$\hat{U}_{\mathbb{\Lambda}_{1}}=\sigma_{0}$, while
$\hat{U}_{\mathbb{\Lambda}_{i}}=-\sigma_{0}$ for $i=2,3,4$. Notice that the
non-triviality of the index follows from the specific choice of the
$n_{2}$-component.

Building on the alternative interpretation of the topological invariant
discussed earlier, $W_{\mathbb{Z}_{2}}$ signals topological non-triviality of
its auxiliary class ${\rm DIII}$ Hamiltonian [3]. Indeed, time-reversal
symmetry of the latter is inherited from the Floquet operator, while block
off-diagonal structure induces the additional chiral structure. In this
interpretation Eq. (22) then encodes topological properties of the class ${\rm
DIII}$ system in $2d$ [26].

For a more intuitive interpretation of Eq. (22), we prove in Appendix D that
the Pfaffians can be expressed as ${\rm
Pf}(w_{\mathbb{\Lambda}_{j}})=-\exp{(i\epsilon_{\mathbb{\Lambda}_{j}})}$,
where $\epsilon_{\mathbb{\Lambda}_{j}}$ are the quasi-energies of $U_{\bf k}$
at the time-reversal invariant momenta $\mathbb{\Lambda}_{j}$. The
$\mathds{Z}_{2}$ topological invariant thus affords the alternative
representation

$W_{\mathbb{Z}_{2}}=\exp\bigl{(}i\sum_{j=1}^{4}\epsilon_{\Lambda_{j}}\bigr{)},$
(23)

which has a simple intuitive visualization. To this end, consider the $1d$
dispersion relations $E^{0}_{\pm}(k_{1})=\epsilon_{\pm}(k_{1},k_{2}=0)$ and
$E_{\pm}^{\pi}(k_{1})=\epsilon_{\pm}(k_{1},k_{2}=\pi)$ of the quasi-energy
spectrum $\epsilon_{\pm}(\mathbb{k})$ of the two-band model along the two high
symmetry lines $k_{2}=0,\pi$, respectively. As shown in Fig. 4, bands
$E^{0}_{\pm}(k_{1})$ touch at $\mathbb{\Lambda}_{1}$ and are split by energy
$2\pi$ at $\mathbb{\Lambda}_{2}$. Bands $E^{\pi}_{\pm}(k_{1})$, on the other
hand, touch in both momenta $\mathbb{\Lambda}_{3}$ and $\mathbb{\Lambda}_{4}$.
This different pattern of the dispersion along the two high-symmetry lines
results in the negative topological index $W_{\mathbb{Z}_{2}}=-1$, as
formalized by Eq. (23).

###  IV.2 Numerical simulations

We simulate the time evolution of an initially localized wave packet in
(1+1syn) dimensions for three different Floquet operators (all involving
maximal disorder in the real coordinate, viz. random Haar unitaries): The
first simulates the topological Floquet metal, described in Eq. (IV). Sharing
the low energy physics of $2d$ class ${\rm AII}$ topological metallic surface
states, we expect anomalous super-diffusion, $\langle\Delta X^{2}\rangle\sim
t\ln t$ [27], which is confirmed in Fig. 5 left panel. Numerical calculations
are performed for the incommensurate frequency $\omega_{2}=\sqrt{5}C$ (where
the value $C$ is indicated in the legend), and each data point is obtained
from averaging over 50 disorder realizations. The second Floquet operator
simulates a critical state in class ${\rm A}$. That is, replacing the non
analytic function in Eq. (IV) by an analytic function,
$|\sin\varphi_{2,t}|\rightarrow\sin\varphi_{2,t}$, we obtain a (1+1syn)
dimensional class ${\rm A}$ model fine-tuned to a quantum critical point
separating two topologically distinct Anderson insulating phases [28]. The
presence of a topological $\theta$-term fine-tuned to the angle $\theta=\pi$
in this case protects against Anderson localization. Fig. 5 middle panel
indeed indicates sub-diffusion on all accessible time scales in our numerics.
Notice that the energy dispersions for the first and second Floquet operator
are identical, and differences in the dynamics therefore root in the different
topological terms. For the third Floquet operator, we tune the second Floquet
operator away from the quantum critical point. The low energy physics in this
case has a $\theta$-term, however, with topological angle $\theta$ detuned
from the critical value (see also next section). At long distances/times we
then expect conventional Anderson insulating behavior, which is confirmed in
the right panel of Fig. 5.

###  IV.3 Incommensurability and synthetic dimensions

So far we have discussed idealized quantum walk protocols with irrational
driving frequencies and frequency ratios. However, in view of our proposed
experimental implementations a comment on rational approximations is due. A
driving frequency $\omega=2\pi\frac{p}{q}$ generates a synthetic dimension of
finite extension $\sim q$. If the dynamics on length scales $\lesssim q$ is
diffusive, a dimensional crossover takes place on time scales comparable to
the diffusion time associated to distance scales $\sim q$. On larger scales,
the system behaves as if it lived in one dimension lower. The precision by
which frequencies have to be chosen thus depends on the experimentally probed
time scales: the above crossover should remain invisible in that it occurs on
scales larger than the above crossover scales. (The precise value of these
scales depends on system specific parameters, notably the effective diffusion
constant.)

To make these general considerations more quantitative we numerically studied
the protocol for the FM${}_{1+1_{\rm syn}}$ of the previous section,
substituting $\omega_{2}=2.6\sqrt{5}$ by $\omega_{2}=2\pi\alpha$ with rational
approximations of increasing periodicity $\alpha=0.9,0.92,0.925,0.9253$. Left
Fig. 6 shows the width of a wave packet normalized by the width expected for
diffusive dynamics, $\Delta X^{2}/t$, as a function of $t$ on a
$\log$-$\log$-scale. At the crossover scale to one-dimensional dynamics the
(approximately) constant profile for diffusive dynamics turns into a linear
slope, characteristic for localized wave packets. As anticipated, the
characteristic time scale increases with the periodicity of the rational
approximation, i.e. the number of decimals kept in $\alpha$. Right Fig. 6
shows the corresponding numerical results for the $2d$ critical class-A metal,
also discussed in the previous section.

Figure 6:  Wave packet spread for quantum simulators of finite size class
${\rm AII}$ topological metals FM${}_{1+1_{\rm syn}}$ (left), and $2d$ class
${\rm A}$ critical metals (right). The size of the compact dimension increases
with the number of decimals kept in the rational approximation $\alpha$, and
the dimensional crossover in the dynamics is observed at increasingly later
times, see also discussion in main text.

##  V Low energy physics

###  V.1 Class A

To support our claim that the protocol Eq. (10) simulates the isolated surface
of a topological $4d$ quantum Hall insulator, we next apply field theory
methods for disordered systems. Our aim in this section is to show that the
low energy physics of both systems is described by the same effective field
theory. Readers interested in more background material on field theories of
disordered systems are invited to look into the supplemental material before
or while plunging into this section.

To begin with, let us recall that the dispersion of the low energy excitations
simulated by the clean contribution to the Floquet operator (10) is linear in
$k_{1}$ and flat in all other directions. The emergence of Weyl fermions at
low energies then results from the interplay of a non trivial topology and
non-integrable chaotic fluctuations induced by the protocol. Indeed, we will
see that the presence and stability of the Weyl fermions is topologically
protected by the winding number, precisely in the same manner as the four Weyl
cones of surface states in a $4d$ class ${\rm A}$ insulator at the $\nu=4$
integer quantum Hall plateau.

Applying methods for disordered systems [29, 30, 31, 32, 33, 34], we can
evaluate correlation functions for the dynamical quantum walk defined by (10)
in the effective quantum field theory (QFT) framework. Its matrix degree of
freedom $T$ acts in a replica space supplemented by additional causal
(“retarded” and “advanced”) structure. Within such QFT the physics at long
time and length scales is described by an effective action
$S[T]=S_{\sigma}[T]+S_{\mathrm{top}}[T]$, consisting of two contributions

$\displaystyle S_{\sigma}$
$\displaystyle=\frac{1}{8}\sum_{i,j=1}^{3}\sigma_{ij}^{(0)}\,{\rm
Tr}\left(\partial_{i}Q\partial_{j}Q\right),$ (24) $\displaystyle
S_{\mathrm{top}}$ $\displaystyle=iW\times S_{\mathrm{CS}}[T].$ (25)

Here the winding number $W$ is defined in Eq. (III.1), with $U$ the
translational invariant part of the Floquet operator. The matrix field
$Q=TQ_{0}T^{-1}$, is expressed as rotations around
$Q_{0}\equiv\sigma_{3}\otimes\openone_{R}$ with Pauli matrix $\sigma_{3}$
operating in the causal sector of the $2R$-dimensional vector space, and
‘${\rm Tr}=\int d^{3}x\,{\rm tr}$’ involves the trace over the latter and $3d$
space of physical and synthetical dimensions. Readers interested in further
details are invited to look into the supplemental material where we explain
the mathematical structures and outline a derivation of the above action. We
here restrict ourselves to a discussion of the physical implications of Eqs.
(24), (25).

The first observation is that $S_{\sigma}$ is the standard model for Anderson
localization in disordered single particle systems, here in $3d$. For
sufficiently strong disorder, viz. sufficiently small “bare” values
$\sigma_{ij}^{(0)}$, the model flows to an Anderson insulating fixed point at
long length scales with vanishing coupling constant. We here consider Haar
random disorder, for which

$\sigma_{ij}^{(0)}=\frac{1}{2}\int
d^{3}\mathbb{k}\,\mathrm{tr}(\partial_{k_{i}}U_{\mathbb{k}}\partial_{k_{j}}U_{\mathbb{k}}^{-1})$
(26)

is purely determined by the translational invariant part of the Floquet
operator. As we detailed above, we can then drive a metal-to-insulator
transition by tuning the bandwidth of the system, see discussion around Eq.
(14).

A game changer to the Anderson insulating scenario at strong disorder is
provided by the second, topological term with Chern-Simons action $S_{\rm CS}$

$\displaystyle S_{\mathrm{CS}}$ $\displaystyle=\frac{1}{8\pi}\sum_{s=\pm}s{\rm
Tr}\left(A_{s}\wedge dA_{s}+\frac{2}{3}A_{s}\wedge A_{s}\wedge A_{s}\right).$
(27)

Here $A_{s}=T^{-1}dTP^{s}$, where
$P^{s}=\frac{1}{2}(1+s\sigma_{3})\otimes\openone_{R}$ are projectors onto the
retarded ($s=+$) and advanced ($s=-$) sector of the $2R$-dimensional vector
space. The key observation then is that the combined action defined by Eqs.
(24) and (25) realizes a $3d$ topological metal, which unlike systems with
$W=0$ has a conductance growing in system size even at strong disorder. In the
field theory language this means that the “bare” coupling $\sigma_{ij}^{(0)}$
grows under renormalization and correlation functions show similar behavior.
The same action, $S[T]=S_{\sigma}[T]+S_{\mathrm{top}}[T]$, has previously been
identified as describing (at length scales exceeding the mean free path) $W$
disordered Weyl cones realized on the surface of a $4d$ class ${\rm A}$
quantum Hall insulator surface [35, 36]. We have thus established the
equivalence between the protocol Eqs. (10) for driven synthetic matter and
quantum Hall insulator surface states, demonstrating that both belong to the
same universality class.

###  V.2 class AII

The non trivial $\mathbb{Z}_{2}$ index Eq. (22) indicates that the low energy
physics of the dynamical protocol Eq. (IV) is dominated by a single Weyl
fermion, similar to the isolated surface of a $3d$ quantum Spin Hall
insulator. The single Weyl cone is not immediate from the low energy
dispersion of the clean Floquet operator Eq. (IV), but rather emerges as a
consequence of the non trivial topology in combination with the chaotic
fluctuations induced by the protocol.

Applying field theory methods of disordered systems, we can again derive a low
energy effective theory $S[T]=S_{\rm\sigma}[T]+S_{\rm top}[T]$, that allows
for the calculation of correlation functions at long time and length scales.
Here $S_{\rm\sigma}$ is the $\sigma$ model action, already introduced in Eq.
(24), now for the $2d$ system in the symplectic class ${\rm AII}$. The latter
alone predicts a metal-to-insulator transition for strong disorder, a scenario
changed by the second, topological contribution

$\displaystyle S_{\mathrm{top}}$
$\displaystyle=i\frac{\theta}{\pi}\times\left.\Gamma[g]\right|_{g(0,\mathbb{x})=Q(\mathbb{x})}.$
(28)

Here $\theta=\pi W_{\mathbb{Z}_{2}}[U]$ is the topological angle from (22),
and

$\displaystyle\Gamma[Q]$ $\displaystyle=\frac{1}{24\pi}\int_{\cal
M}\mathrm{tr}\left(\Phi_{g}\wedge\Phi_{g}\wedge\Phi_{g}\right)$ (29)

is “half” of a Wess-Zumino-Witten action, involving the usual deformation of
the field degree of freedom $T$. Specifically, $\Phi_{g}\equiv g^{-1}dg$ with
$g(x_{0}=0,\mathbb{x})=Q(\mathbb{x})$, and integration is over half the
$3$-torus ${\cal M}=[0,1]\times[-1,1]^{2}$. This topological action was
previously identified [37, 38] for the description of the $3d$ disordered
quantum Hall insulator. We refer the interested reader to the accompanying
supplemental material for further explanations, and here only focus on a
discussion of the physical implications. These are similar to those of the
Chern Simons action encountered in the unitary class. For $\theta=\pi$ the
system flows to a conformally invariant quantum critical point, where the
coupling constant of $S_{\sigma}$ assumes a disorder independent value (26),
implying the absence of Anderson localization also for strong disorder. The
same principle of delocalization is at work on $2d$ surfaces of $3d$
topological spin quantum Hall insulators. The latter are indeed described by
the same effective action [39, 37], and we have thus shown that the protocol
Eq. (IV) belongs to the same universality class as the isolated surface of a
quantum spin Hall insulator.

###  V.3 More on topological terms

The possibility for disordered systems to escape the fate of Anderson
localization is signaled by topological terms in their low energy field theory
description. Whether the latter are allowed depends on the dimension of the
system and the target space of the field degree of freedom. The effective
action of the aforementioned $3d$ class ${\rm A}$ system contains a Chern-
Simons action, with coupling constant that is the winding number of the
Floquet operator. The winding number and Chern Simons action signal the
presence of topologically inequivalent classes of mappings

$\displaystyle U_{\mathbb{k}}:\,\,$ $\displaystyle\mathbb{T}^{3}\mapsto{\rm
SU}(2),$ (30) $\displaystyle Q(x_{0},\mathbb{x}):\,\,$
$\displaystyle\mathbb{T}^{(3+1)}\mapsto\mathrm{U}(2R)/[\mathrm{U}(R)\times\mathrm{U}(R)].$
(31)

Here the boundary configuration $Q(x_{0}=0,\mathbb{x})$ is parametrized by the
$3d$ field $T(\mathbb{x})$, used in the Chern-Simons action in Eq.(27).
Introducing the deformation parameter $0\leq x_{0}\leq 1$, continuously
transforming the boundary value $Q(x_{0}=0,\mathbb{x})$ to the constant matrix
$Q(x_{0}=1,\mathbb{x})=\sigma_{3}\otimes\openone_{R}$, the Chern-Simons action
can be expressed as a Wess-Zumino-Witten (WZW) term. The latter is precisely
what is necessary for a $3d$ class ${\rm A}$ system to avoid Anderson
localization (see supplemental material). The winding number of
$U_{\mathbb{k}}$ given in (III.1), on the other hand, defines the coupling
constant of the WZW term. That is, the combination of non-trivial homotopy
groups $\pi_{3}({\rm SU}(2))=\mathbb{Z}$ and $\pi_{4}({\rm
U}(2R)/[\mathrm{U}(R)\times\mathrm{U}(R)])=\mathbb{Z}$ allows for non-
vanishing coupling constants weighting topologically non-trivial field
configurations $T(\mathbb{x})$.

For the $2d$ quantum spin Hall surface states it is the emergence of a WZW
action, weighted by a $\mathbb{Z}_{2}$ topological angle in the effective low
energy description that allows for topological metallic phases. The involved
maps in momentum- and real-space read

$\displaystyle U_{\mathbb{k}}:\,\,$ $\displaystyle\mathbb{T}^{2}\mapsto{\rm
U}(2)/{\rm Sp}(2),$ (32) $\displaystyle Q(\mathbb{x}):\,\,$
$\displaystyle\mathbb{T}^{2}\mapsto{\rm O}(4R)/[{\rm O}(2R)\times{\rm
O}(2R)],$ (33)

with non trivial homotopy groups, $\pi_{2}({\rm U}(2)/{\rm
Sp}(2))=\mathbb{Z}_{2}$ and $\pi_{2}({\rm O}(4R)/[{\rm O}(2R)\times{\rm
O}(2R)])=\mathbb{Z}_{2}$, respectively. Both maps are thus characterized by
non trivial $\mathbb{Z}_{2}$ indices, and we already introduced a topological
$\mathbb{Z}_{2}$ invariant for the Floquet operator Eq. (32) in Eq. (22).
Alternatively, one can express the $\mathbb{Z}_{2}$ index as “half” of a Wess-
Zumino-Witten term, which is readily extended to Eq. (33), see also
supplemental material for more details.

We conclude remarking that the above discussion only relies on general
structures, such as the softmode manifold identified in the course of the
construction of the effective field theory, and applies as a matter of
principle. Whether there exist physical systems characterized by non-trivial
couplings is an independent issue. The models we propose in the earlier
sections are one option how to realize non trivial mappings utilizing the idea
of engineered synthetic dimension. That is, in the present work we provide the
field theories, physical models, and numerical confirmation of topological
Floquet metals for both complex and real symmetry classes, which are
characterized by $\mathbb{Z}$ and $\mathbb{Z}_{2}$ topological indices,
respectively. The presented structures encompass topological Floquet metal in
other dimension and symmetry classes.

##  VI Discussion

In this paper, we have introduced quantum simulators for topological surface
states in isolation. Our proposal sidesteps the bulk boundary principle and
overcomes the fermion doubling theorem, impeding the realization of isolated
surface states in generic solid state (lattice) systems. The key element of
our proposal is the dynamical generation of physical dimensions via external
driving, using incommensurate frequencies. The simulation of extra dimensions
via driving physical platforms has already been used in cold atom systems to
measure the Anderson localization-delocalization transition in three
dimensions to a degree of resolution not reachable in solid state materials.
We here apply the idea to one-dimensional quantum walks of a spin-1/2 particle
with time dependent spin rotation matrices, viz. “coin operations”, following
multi-frequency dynamical protocols. The latter provide a flexibility absent
in lattice systems, which allows for the simulation of (gauge) equivalent
real-space dynamics involving long range hopping.

We have illustrated the general idea on two specific examples, the three-
dimensional topological surface states of a four-dimensional quantum Hall
insulator, and the two-dimensional surface states of a three-dimensional spin
quantum Hall insulator. An inherent feature of both protocols is that the
artificial generation of “synthetic” dimensions induces diffusive dynamics in
all (gauge) equivalent space directions after already a few iterations of the
protocol, as verified in numerical simulations. Our approach, thus, simulates
the surfaces of “disordered” phases lacking translational invariance, which
adds an element of realism. For both examples, we identified topological
invariants showing the non trivial topological nature of the dynamical
protocols. Comparing simulations of the latter to that of topologically
trivial parents with variable disorder strengths (respectively bandwidth)
clearly shows the impact of a non trivial topology. While strong disorder
turns the simulators of trivial metals into Anderson insulators, no signature
of localization is found for the topological non trivial protocols for all
disorder strengths, respectively, bandwidths. Importantly, the numerical
simulations show differences in the dynamics simulated by the different
protocols already after an experimentally accessible number of $\sim{\cal
O}(20)$ time steps. This also sets the precision to which frequencies have to
be chosen in experiment. Approximating irrational numbers by rational
generates finite rather than infinitely extended synthetic dimensions. As long
as the corresponding diffusion time (i.e. the time required to explore the
finite dimension) exceeds the time scales probed in experiment, protocols with
rational numbers can be used for all practical purposes.

Employing field theory methods, we have shown that the quantum simulators
generate dynamics within the same universality class as the corresponding
topological insulator surface states. Specifically, we demonstrated that the
universal long-time dynamics of the dynamical protocol is described by
precisely the same topological field theory also proposed for the simulated
surface states. The field theory construction builds on the color-flavor
transformation, and can be readily generalized to other symmetry classes and
dimensions. Generalizing e.g. the simulator of topological quantum Hall
surface states to other (odd) dimensions, different from three, one can derive
the corresponding Chern Simons actions. Similarly, a topological field theory
with Chern Simons action can be derived for the simulator probing the surface
states of a quantum spin Hall insulator (‘class ${\rm AII}$’) in four
dimensions, and different from the $\mathbb{Z}_{2}$ field theory in three
dimensions discussed here in detail. An exception is provided by class ${\rm
AIII}$ systems. These cannot be simulated within the proposed scheme, since
the gauge transformation, establishing the equivalence between the
periodically driven and higher dimensional system, breaks chiral symmetry.

Our proposal requires full dynamical control over a two-state internal degree
of freedom (“spin”), which at the current state may be difficult to achieve in
optical lattices. We, therefore, focused on the alternative platform of linear
optical networks, similar to that used in Ref. [8]. Specifically, time-
multiplexing networks with fast switching electro-optic modulators seem
promising candidates for the implementation of the quantum simulators. We
provided detailed blueprints for the experimental implementation of the two
protocols within existing set ups, realizing the quantum simulators of the
surface states of a four dimensional quantum Hall insulator and a three
dimensional quantum spin Hall insulator. We have shown in our numerical
simulations that the experimental signature, viz. absence of Anderson
localization, is observable within the experimentally realizable number of
time steps. A tunable quantum simulator of topological surface states in
isolation, would open fascinating experimental possibilities. Specifically, it
would provide a new, direct window into the intriguing physics resulting from
the interplay of disorder and non trivial topology.

Acknowledgments:—T. M. acknowledges financial support by Brazilian agencies
CNPq and FAPERJ. K.W.K. acknowledges financial support by Basic Science
Research Program through the National Research Foundation of Korea (NRF)
funded by the Ministry of Education (20211060) and Korea government(MSIT)
(No.2020R1A5A1016518). A.A. and D.B. were funded by the Deutsche
Forschungsgemeinschaft (DFG) Projektnummer 277101999 TRR 183 (project
A01/A03).

##  Appendix A  Unitarity of quantum walk operator

The general single time-step operator Eq. (1) simplifies to Eq. (2) when
focusing on quantum walks with short range hopping $m=\\{-1,0,+1\\}$. In
momentum-representation,

$\displaystyle\hat{U}_{\mathbb{k}}$ $\displaystyle=$
$\displaystyle\sum_{m=0,\pm}(\vec{r}_{m}\cdot\vec{\sigma})e^{imk_{1}},$
$\displaystyle=$ $\displaystyle\left[\vec{r}_{0}+(\vec{r}_{+}+\vec{r}_{-})\cos
k_{1}+i(\vec{r}_{+}-\vec{r}_{-})\sin k_{1}\right]\cdot\vec{\sigma},$

where $\vec{r}_{m}$ is a four-component vector and
$\vec{\sigma}=(\sigma_{0},i\boldsymbol{\sigma})$. To satisfy unitarity, the
vector multiplying $\vec{\sigma}$ must be real valued, that is,
$\vec{r}_{+}=(\vec{r}_{-})^{*}$. Expressing
$\vec{r}_{+}=(\vec{r}_{r}+i\vec{r}_{i})$ in terms of two real vectors
$\vec{r}_{r}$, $\vec{r}_{i}$,

$\displaystyle\\!\\!\\!\hat{U}_{\mathbb{k}}=\left[\vec{r}_{0}+(\vec{r}_{r}+i\vec{r}_{i})e^{ik_{1}}+(\vec{r}_{r}-i\vec{r}_{i})e^{-ik_{1}}\right]\cdot\vec{\sigma},$
(35)

and requiring further that
$\hat{U}_{\mathbb{k}}\hat{U}_{\mathbb{k}}^{\dagger}=\mathds{1}$, the following
relations can be verified

$\displaystyle|\vec{r}_{0}|^{2}+|\vec{r}_{-}|^{2}+|\vec{r}_{+}|^{2}=1,$
$\displaystyle|\vec{r}_{0}|^{2}+2|\vec{r}_{r}|^{2}+2|\vec{r}_{i}|^{2}=1,$
$\displaystyle\vec{r}_{r}\cdot\vec{r}_{i}=0,\vec{r}_{r}\cdot\vec{r}_{0}=0,\vec{r}_{i}\cdot\vec{r}_{0}=0,$
$\displaystyle|\vec{r}_{r}|=|\vec{r}_{i}|=\frac{1}{2}\sqrt{1-|\vec{r}_{0}|^{2}}.$
(36)

These are stated below Eq. (2) in the main text.

##  Appendix B Spreading of a wave packet

In this Appendix we demonstrate the equivalence of Eqs. (7 and (8). To this
end we introduce the initial density matrix

$\rho_{0}=\frac{1}{N_{\rm syn}}\sum_{k_{\rm syn},\sigma}|0,k_{\rm
syn},\sigma\rangle\langle 0,k_{\rm syn},\sigma|,\quad\rho_{0}^{2}=\rho_{0},$
(37)

(here $n_{1}=0$ refers to the origin in the physical space and $N_{\rm syn}\gg
1$ is the number of initial phases) and note that Eqs. (7) for $\Delta X^{2}$
can can be cast in the basis independent form

$\langle\Delta X^{2}\rangle={\rm
tr}\overline{\left(\hat{\rho}_{0}\mathcal{U}_{t,0}^{\dagger}\hat{n}_{1}^{2}\mathcal{U}_{t,0}\right)},$
(38)

where $\overline{(...)}$ refers to a disorder average. The rationale behind
this expression is the following. The average over initial phases (momenta
$k_{\rm syn}$) implies the trace operation in the extended Hilbert space, and
we discretize the corresponding momentum integral so that it becomes a sum
over $N_{\rm syn}$ terms.

Applying further the time-dependent gauge transformation introduced in section
II.1 one writes

$\mathcal{U}_{t,0}=e^{it\sum_{j\geq
2}\omega_{j}\hat{n}_{j}}\,\mathcal{U}_{F}^{t},$ (39)

where the Floquet operator $\mathcal{U}_{F}$ was defined in Eq. (II.1). This
ansatz gives us the equivalent expression for the width of a wave packet,

$\langle\Delta X^{2}\rangle={\rm
tr}\left(\hat{\rho}_{0}(\mathcal{U}_{F}^{\dagger})^{t}\,\hat{n}_{1}^{2}\,\mathcal{U}_{F}^{t}\right).$
(40)

Lastly, to evaluate the trace above one can use a full coordinate
representation, which gives us

$\displaystyle\langle\Delta X^{2}\rangle$ $\displaystyle=\frac{1}{N_{\rm
syn}}\sum_{\mathbb{n}\,\mathbb{n}^{\prime},\sigma\sigma^{\prime}}n_{1}^{2}\overline{|\langle\mathbb{n}^{\prime},\sigma^{\prime}|\mathcal{U}_{F}^{t}|\mathbb{n},\sigma\rangle|^{2}},$
(41)

with $|\mathbb{n},\sigma\rangle\equiv|n_{1},n_{\rm syn},\sigma\rangle$ and
$|\mathbb{n}^{\prime},\sigma\rangle\equiv|n_{1},n^{\prime}_{\rm
syn},\sigma^{\prime}\rangle$. We then notice that upon a disorder average the
transition probability depends only on the difference in position,
$\mathbb{n}^{\prime}-\mathbb{n}$, and thereby the expression (8) in the main
text is recovered.

Figure 7:  Blueprint of an optical linear network simulating the topological
surface states of a $3d$ quantum spin Hall insulator in the quantum walk
setting. (a) The feedback loop is build of the step operator $\hat{T}_{\cal
C}^{\pm\frac{1}{2}}$ (upper and middle arms, respectively) and the coin
operators $\hat{R}_{a}$ and $\hat{R}_{b}$ (right and left arm, respectively),
for details see main text. To the left and right of the loop, source and
detection units are connected. (b) Numerical simulation of a topological metal
($\mathbb{Z}_{2}=-1$) in a $(1+1_{\rm syn})D$ quantum walk for short time
steps accessible by experiments. The width of the wave packet scaled by time
is plotted on a $\log$-scale, showing that diffusion is anomalously fast. (c)
Critical quantum walk in $(1+1_{\rm syn})D$ class ${\rm A}$ at a topological
quantum phase transition, showing the same scaling with the classical
diffusion $\langle\Delta X^{2}\rangle\sim t$. (d) Quantum walk in $(1+1_{\rm
syn})D$ class ${\rm A}$ without topological term, showing Anderson
localization. Colors correspond to incommensurate frequencies
$\omega_{2}=\sqrt{5}C$ with values for $C$ as indicated in the legends.

##  Appendix C  Blueprint for the FM${}_{1+1_{\rm syn}}$ simulator

A detailed blueprint for the optical linear network simulating the topological
FM${}_{1+1_{\rm syn}}$ is shown in Fig. 7. The dynamical protocol, Eq. (IV),
involving all three components $\hat{R}_{\pm,0}$ requires a more complex set-
up in comparison to FM${}_{1+2_{\rm syn}}$ in class A, which now has to be
build from two chiral half-step and two coin operations. Therefore we start by
summarizing the optical scheme in Fig. 7 and then provide its justification.
To this end we decompose the translational invariant part of the single time
step evolution operator, Eq. (IV), into the product

$\displaystyle\mathcal{U}_{t+1,t}$ $\displaystyle=\hat{U}_{\rm
dis}\hat{R}_{a}(t)\hat{T}_{\cal C}^{\frac{1}{2}}R_{b}(t)\hat{T}_{\cal
C}^{\frac{1}{2}},$ (42)

of a chiral half-step, $\hat{T}^{\frac{1}{2}}_{\cal C}$, and coin operators,
$\hat{R}_{a/b}$. (The former are positioned in the four horizontal arms,
notice the half fiber lengths $\tau_{H}/2$ and $\tau_{V}/2$, and the latter
are placed in the vertical arms.) Dynamical EOMs after fiber lines of the
first step operation allow to terminate the walk by sending the photons to the
detection unit. The coin operators are chosen as

$\hat{R}_{b}(t)=YR_{z}(-\varphi_{2,t})Y^{\dagger}\equiv R_{y}(\varphi_{2,t}),$
(43)

where $Y=e^{i\pi\sigma_{1}/4}$ is the matrix of $y$-basis change
††footnotemark: , and
$\hat{R}_{a}(t)=\hat{R}_{a,t+1}^{\text{(II)}}\hat{R}_{a,t}^{\text{(I)}}$, with

$\displaystyle\hat{R}_{a,t}^{\text{(I)}}$
$\displaystyle=\begin{cases}R_{z}(\varphi_{2,t})\hat{Y}^{\dagger},\quad\sin\varphi_{2,t}\geq
0,\\\ \hat{Y}^{\dagger},\quad\sin\varphi_{2,t}<0,\end{cases}$ (44)

and

$\displaystyle\hat{R}_{a,t+1}^{\text{(II)}}$
$\displaystyle=\begin{cases}\hat{Y},\quad\sin\varphi_{2,t+1}\geq 0,\\\
\hat{Y}R_{z}(\varphi_{2,t+1}),\quad\sin\varphi_{2,t+1}<0,\end{cases}$ (45)

with $\varphi_{2,t}=k_{2}+\omega_{2}t$. Finally, disorder is introduced by
placing in between $R_{a,t}^{\text{(I)}}$ and $R_{a,t+1}^{\text{(II)}}$ the
local, time reversal invariant random potential
$U(n_{1})=e^{i\phi_{n_{1}}\sigma_{0}}$, with position dependent angles
$\phi_{n_{1}}$, randomly drawn from the unit circle
$-\pi\leq\phi_{n_{1}}<\pi$.

Coming back to the justification of (42) we note that expressing the time
evolution operator in Eq. (IV) as the product of elementary chiral
translations and coin operators is not immediately straightforward. To start
with, we first notice that upon replacing $|\sin k_{2}|\rightarrow\sin k_{2}$
in Eq.(21), the model is reduced to the familiar $2d$ class ${\rm A}$ Floquet
insulator [40, 28], which is built by the multiplication of simpler operators.
For an implementation of the protocol with absolute value $|\sin k_{2}|$, we
separate cases $\sin k_{2}\leq 0$ and $\sin k_{2}>0$. Specifically, a class
${\rm A}$ $2d$ Floquet model with homotopy parameter $s$ is written as [28]
the product of four unitary operators
$\hat{U}_{\text{A}}=\hat{U}_{4}\hat{U}_{3}\hat{U}_{2}\hat{U}_{1}$, with

$\displaystyle\hat{U}_{i}(k)$
$\displaystyle=\cos(s)+i\sin(s)\begin{pmatrix}&e^{-i\mathbb{k}\cdot\mathbb{v}_{i}}\\\
e^{i\mathbb{k}\cdot\mathbb{v}_{i}}&\end{pmatrix}.$

Here
$(\mathbb{v}_{1},\mathbb{v}_{2},\mathbb{v}_{3},\mathbb{v}_{4})=(0,-\mathbb{e}_{1},-\mathbb{e}_{1}+\mathbb{e}_{2},\mathbb{e}_{2})$,
with $\mathbb{e}_{1,2}$ lattice unit vectors in the horizontal/vertical
direction. At $s=\pi/4$, the product of unitaries can be expanded in the
quantum walk form

$\displaystyle\hat{U}_{\text{A}}=(\hat{r}_{0}\cdot\vec{\sigma})+(\vec{r}_{+}\cdot\vec{\sigma})e^{ik_{1}}+(\vec{r}_{-}\cdot\vec{\sigma})e^{-ik_{1}},$

where

$\displaystyle\vec{r}_{0}$ $\displaystyle=\frac{1}{2}(1-\cos k_{2},0,0,\sin
k_{2}),$ $\displaystyle\vec{r}_{r}$ $\displaystyle=\frac{1}{4}(-1-\cos
k_{2},0,0,\sin k_{2}),$ $\displaystyle\vec{r}_{i}$
$\displaystyle=\frac{1}{4}(0,\sin k_{2},-1-\cos k_{2},0).$

Following the general recipe outlined in Sec. II.1, we simulate the $2d$
dynamics as a $1d$ quantum walk with time dependent protocol, replacing
momentum $k_{2}\rightarrow\varphi_{2,t}$ by a time dependent angle. We then
notice that for $\sin\varphi_{2,t}>0$ vectors $\vec{r}_{0,r,i}$ of
$\hat{U}_{\text{A}}$ are identical to that of $\hat{U}_{\text{AII}}$ in
Eq.(IV). Hence, when $\sin\varphi_{2,t}>0$, the quantum walk operator in
Eq.(IV) can be written as the product of the four unitary operators,
$\hat{U}_{\text{AII}}(\sin\varphi_{2,t}>0)=\hat{U}_{\text{A}}(k_{1},\varphi_{2,t})$.
When $\sin\varphi_{2,t}<0$, on the other hand, one can verify that
$\hat{U}_{\text{AII},\sin\varphi_{2,t}<0}(k_{1},\varphi_{2,t})=\hat{U}^{T}_{\text{A}}(-k_{1},\varphi_{2,t},s=\frac{\pi}{4})$.

Next, we express $\hat{U}_{j=1,2,3,4}$ as a combination of shift and rotation
operators,

$\displaystyle\hat{U}_{1}$ $\displaystyle=e^{is\sigma_{1}},$
$\displaystyle\hat{U}_{2}$
$\displaystyle=\hat{T}_{C}^{-\frac{1}{2}}e^{is\sigma_{1}}\hat{T}_{C}^{\frac{1}{2}},$
$\displaystyle\hat{U}_{3}$
$\displaystyle=e^{i\varphi_{2,t}\sigma_{3}/2}\hat{T}_{C}^{-\frac{1}{2}}e^{is\sigma_{1}}\hat{T}_{C}^{\frac{1}{2}}e^{-i\varphi_{2,t}\sigma_{3}/2},$
$\displaystyle\hat{U}_{4}$
$\displaystyle=e^{i\varphi_{2,t}\sigma_{3}/2}e^{is\sigma_{1}}e^{-i\varphi_{2,t}\sigma_{3}/2},$

with ‘half’ translation operator
$\hat{T}_{C}^{\frac{1}{2}}=e^{ik_{1}\sigma_{3}/2}$. We stress that the full
Floquet operator $\hat{U}_{\text{A}}$ is $2\pi$ periodic in $k_{1}$ and the
appearance of a ‘half’ translation operator does not imply a doubling of the
unit cell. The same is true for the topological Floquet metal model.

The $2d$ class ${\rm AII}$ model is expressed as

$\displaystyle\hat{U}_{\text{AII},\sin\varphi_{2,t}\geq
0}(k_{1},\varphi_{2,t})$ (46)
$\displaystyle=\hat{U}_{\text{A}}(k_{1},\varphi_{2,t},s=\frac{\pi}{4})$
$\displaystyle=e^{i\varphi_{2,t}\sigma_{3}/2}Y\hat{T}_{C}^{-\frac{1}{2}}Ye^{-i\varphi_{2,t}\sigma_{3}/2}Y\hat{T}_{C}^{\frac{1}{2}}Y,$
$\displaystyle=-e^{i\varphi_{2,t}\sigma_{3}/2}Y^{\dagger}\hat{T}_{C}^{\frac{1}{2}}Y^{\dagger}e^{-i\varphi_{2,t}\sigma_{3}/2}Y\hat{T}_{C}^{\frac{1}{2}}Y,$

and to arrive at this result we used that
$Y\hat{T}_{C}^{-\frac{1}{2}}Y=-Y^{\dagger}\hat{T}_{C}^{\frac{1}{2}}Y^{\dagger}$
and commutativity of operators $\hat{T}_{C}^{\frac{1}{2}}$ and
$e^{-i\varphi_{2,t}\sigma_{3}/2}$. On the other hand,

$\displaystyle\hat{U}_{\text{AII},\sin\varphi_{2,t}<0}(k_{1},\varphi_{2,t})$
(47)
$\displaystyle=\hat{U}^{T}_{\text{A}}(-k_{1},\varphi_{2,t},s=\frac{\pi}{4})$
$\displaystyle=Y\hat{T}_{C}^{-\frac{1}{2}}Ye^{-i\varphi_{2,t}\sigma_{3}/2}Y\hat{T}_{C}^{\frac{1}{2}}Ye^{i\varphi_{2,t}\sigma_{3}/2},$
$\displaystyle=-Y^{\dagger}\hat{T}_{C}^{\frac{1}{2}}Y^{\dagger}e^{-i\varphi_{2,t}\sigma_{3}/2}Y\hat{T}_{C}^{\frac{1}{2}}Ye^{i\varphi_{2,t}\sigma_{3}/2},$

and from above relations (46) and (47) we notice that $\hat{U}_{\rm
AII}=R_{a}^{\rm(I)}\hat{T}_{C}^{\frac{1}{2}}R_{b}\hat{T}_{C}^{\frac{1}{2}}R_{a}^{\rm(II)}$.
Up to a cyclic permutation of the operator $R_{a}^{\rm(II)}$ this is
equivalent to the clean part of ${\cal U}_{t+1,t}$, see Eq. (42). Notice that
in both cases $\hat{U}_{\text{AII}}$ involves the same unitary sandwiched
between the two $\hat{T}_{C}^{\frac{1}{2}}$, which simplifies the
implementation of
$\hat{R}_{b}(t)=Y^{\dagger}e^{-i\varphi_{2,t}\sigma_{3}/2}Y$.
$\hat{R}_{a}(t)$, on the other hand, depends on the sign of
$\sin\varphi_{2,t}$.

##  Appendix D $\mathbb{Z}_{2}$ topological invariant

In this Appendix we further discuss the $\mathbb{Z}_{2}$ topological invariant
for the 2-band model and prove the relation (23). The unitary operator
$\hat{U}_{\mathbb{k}}$ gives rise to the auxiliary Hamiltonian [41, 3],

$\displaystyle\tilde{H}_{U}(\mathbb{k})=\begin{pmatrix}0&U_{\mathbb{k}}\\\
U_{\mathbb{k}}^{\dagger}&0\end{pmatrix},$ (48)

which shares time-reversal and particle-hole symmetries,
$\hat{\Theta}_{1}\tilde{H}_{U}(\mathbb{k})\hat{\Theta}_{1}^{-1}=\tilde{H}_{U}(-\mathbb{k})$,
and
$\hat{\Theta}_{2}\tilde{H}_{U}(\mathbb{k})\hat{\Theta}_{2}^{-1}=-\tilde{H}_{U}(-\mathbb{k})$,
respectively, with $\hat{\Theta}_{1}=\tau_{1}\otimes i\sigma_{2}\mathcal{K}$,
and $\hat{\Theta}_{2}=i\tau_{2}\otimes i\sigma_{2}\mathcal{K}$. That is,
$H_{U}$ belongs to class ${\rm DIII}$. Notice that in Eq.(48) off diagonal
elements are the Floquet unitary operator without band flattening. This allows
us to make a connection between the $\mathbb{Z}_{2}$ invariant and the eigen-
energies at time-reversal invariant momenta.

A way to compute the $\mathbb{Z}_{2}$ topological invariant then is as follows
[26]: The Hamiltonian has two valence bands at energy $E=-1$, and their
eigenvectors are

$\displaystyle u^{-}_{1}(\mathbb{k})=\frac{1}{\sqrt{2}}\begin{pmatrix}-1\\\
0\\\ U^{*}_{\mathbb{k},11}\\\
U^{*}_{\mathbb{k},12}\end{pmatrix},\,\,\,\,\,u^{-}_{2}(\mathbb{k})=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\
-1\\\ U^{*}_{\mathbb{k},21}\\\ U^{*}_{\mathbb{k},22}\end{pmatrix},$ (49)

where $U^{*}_{\mathbb{k},ij}=(U^{\dagger}_{\mathbb{k}})_{ji}$. The sewing
matrix, needed to compute the topological invariant [26], can be obtained from
these two vectors as $(w_{\mathbb{k}})_{ab}=\langle
u^{-}_{a}(-\mathbb{k})|\hat{\Theta}_{1}u^{-}_{b}(\mathbb{k})\rangle$. That is,

$\displaystyle w_{\mathbb{k}}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}-U_{\mathbb{k},12}+U_{-\mathbb{k},12}&-U_{\mathbb{k},22}-U_{-\mathbb{k},11}\\\
U_{\mathbb{k},11}+U_{-\mathbb{k},22}&U_{\mathbb{k},21}-U_{-\mathbb{k},21}\end{pmatrix},$
(50) $\displaystyle=\begin{pmatrix}-U_{\mathbb{k},12}&-U_{\mathbb{k},22}\\\
U_{\mathbb{k},11}&U_{\mathbb{k},21}\end{pmatrix},$ (51)
$\displaystyle=-i\sigma_{2}U^{T}_{\mathbb{k}},$ (52)

where in the second line time-reversal symmetry of the unitary operator was
used, i.e. $U_{\mathbb{k},11}=U_{-\mathbb{k},22}$,
$U_{\mathbb{k},12}=-U_{-\mathbb{k},12}$ and
$U_{\mathbb{k},21}=-U_{-\mathbb{k},21}$ (as follows from
$\sigma_{2}U_{\mathbb{k}}\sigma_{2}=U_{-\mathbb{k}}^{T}$). One can then
readily verify that the sewing matrix is anti-symmetric at time-reversal
invariant momenta $\mathbb{\Lambda}_{1}=(0,0)$,
$\mathbb{\Lambda}_{2}=(\pi,0)$, $\mathbb{\Lambda}_{3}=(0,\pi)$, and
$\mathbb{\Lambda}_{4}=(\pi,\pi)$, i.e.

$\displaystyle
w_{\mathbb{\Lambda}_{j}}=\frac{1}{2}\begin{pmatrix}0&-U_{\mathbb{\Lambda}_{j},11}-U_{\mathbb{\Lambda}_{j},22}\\\
U_{\mathbb{\Lambda}_{j},11}+U_{\mathbb{\Lambda}_{j},22}&0\end{pmatrix},$ (53)

where $U_{\mathbb{\Lambda}_{j},11}=U_{\mathbb{\Lambda}_{j},22}$, and the
Pfaffian is
$\text{Pf}\left[w_{\mathbb{\Lambda}_{j}}\right]=-\frac{1}{2}(U_{\mathbb{\Lambda}_{j},11}+U_{\mathbb{\Lambda}_{j},22})=-\frac{1}{2}\mathrm{tr}[U_{\Lambda_{j}}]=-\exp\left(i\epsilon_{\mathbb{\Lambda}_{j}}\right)$.
Finally, the $\mathds{Z}_{2}$ topological invariant becomes

$\displaystyle W_{\mathbb{Z}_{2}}$
$\displaystyle=\prod_{j=1,2,3,4}\text{Pf}\left[w_{\mathbb{\Lambda}_{j}}\right]=\exp\left(i\sum_{j=1,2,3,4}\epsilon_{\Lambda_{j}}\right),$
(54)

which implies that the condition for a non trivial Floquet topological metal,
$W_{\mathbb{Z}_{2}}=-1$, translates into
$\sum_{j=1,2,3,4}\epsilon_{\Lambda_{j}}=\pi$ (mod $2\pi$).

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  * Note [3] Note that for such construction $U(\pi/2,k)\to-i\sigma_{3}$, which is not yet time-reversal symmetric operator. However, a subsequent rotation $e^{i\psi\sigma_{3}}(-i\sigma_{3})$ with $\psi\in[0,\pi/2]$ brings it to the identity matrix $\sigma_{0}$. At this interval a spectral density in (S3.4) is zero thereby giving no contribution to $\theta$–angle.

Supplemental Materials

In this supplemental material we provide details on the effective field theory
description of the quantum simulator protocols. We start out with a brief
review of the standard effective field theory for disordered single particle
systems, then outline how to map the protocols discussed in the main text to
the latter, and finally deepen our discussion of topological terms.

##  Appendix S1 Field theory review

###  S1.1 Diffusive non-linear $\sigma$ model

The quantum dynamics of disordered single particle systems at long length- and
time-scales is described by a diffusive non-linear $\sigma$ model. The latter
bears similarities to Ginzburg-Landau theories and encodes the physics of
symmetry breaking and (critical) soft-mode fluctuations related to Anderson
localization. Different from the former, the field degree of freedom of the
$\sigma$ model however does not afford the interpretation of an order
parameter. More specifically, the model is formulated in terms of a matrix
degree of freedom $Q$ which satisfies the nonlinear constraint
$Q^{2}=\openone$. In its simplest replica variant the matrix is operating in a
$2R$-dimensional vector space formed by $R$ replicas which exist in two
“causal” variants, a “retarded” and an “advanced”. The replica structure helps
to overcome the notorious problem of disorder averaging the logarithm of the
partition function, which serves as a “generating function” for observables.
The causal structure is introduced to generate the typical observables of
interest (see Section II B in the main text), viz. disordered averaged
probabilities (the product of retarded and advanced propagators), from a
single generating function.

Anderson localization can be viewed as the restoration of rotational symmetry
in causal space. Indeed, the derivation of the field theory builds around a
saddle point that describes the disorder induced level broadening. This is
isotropic in replica space, while causality breaks rotational symmetry in
retarded and advanced components. Formally,
$Q_{0}\equiv\sigma_{3}\otimes\openone_{R}$ where $\sigma_{3}$ operates in
causal space, and the soft mode action of the associated Goldstone modes
$Q=TQ_{0}T^{-1}$ is precisely the diffusive non-linear sigma model, here
restated for convenience of the reader

$S_{\sigma}=\frac{1}{8}\sum_{i,j=1}^{D}\sigma_{ij}^{(0)}\int d^{3}x\,{\rm
Tr}\left(\partial_{i}Q\partial_{j}Q\right).$ (S1)

In the metallic regime, fluctuations around the saddle point are small. At the
onset of localization, on the other hand, fluctuations grow uncontrolled and
rotations start exploring the entire field manifold, restoring thus the
original symmetry in causal space.

While the structure of $S_{\sigma}$ is fixed by general principles, details of
the matrix degree of freedom depend on the system’s symmetries. So far we have
assumed the absence of fundamental symmetries. In the field theory
construction fundamental symmetry are included by “symmetry doubling” of the
matrix dimension. The effective description of a system with time reversal
symmetry is e.g. in terms of a $4R$ dimensional matrix. Its entries are then
constrained by a symmetry relation reminiscent of the symmetry doubling (see
also Section S3.1). The same holds for other symmetries, that is, elements of
the “symmetry doubled” matrices are not independent, and symmetry relations
fix the field manifold to belong to one of ten symmetric spaces [42, 43].

###  S1.2 Sketch of the derivation

A detailed derivation of the effective action $S_{\sigma}$ for the dynamical
protocol realizing the surface states of the $3d$ quantum spin Hall insulator
can be found in Sections S3.1, S3.3, and S3.5, and we here only outline the
basic steps (for the corresponding derivation for the quantum Hall simulator
see e.g. the recent Ref. [28]). Starting out from the Floquet operator after
gauge transformation,

$\displaystyle\hat{U}_{\Phi}{}$
$\displaystyle=\hat{U}_{\mathbb{n}}\hat{U}_{\mathbb{k}},$ (S2)

we focus on the dynamics at long time and length scales. We assume that the
action of the local disorder in physical space, $\hat{U}_{n}$, in combination
with the potential $\hat{\Phi}$, generated by the gauge transformation,
induces non-integrability in all $1+d_{\rm syn}$ dimensions. This assumption
is supported by the numerical simulations, as already discussed in the
previous sections. To capture the universal long time dynamics, we may then
erase system specific details introducing an ensemble of local spin rotations
$U_{\mathbb{n}}$ (sharing fundamental symmetries of the system), and derive a
generating functional for the _ensemble averaged_ correlation function Eq. (8)
in the main text.

For Floquet systems in class AII the derivation can be organized in terms of a
color-flavor transformation (cft), whose details are exposed in Section S3.1.
Its few line summary is as follows. Building on the replica trick and causal
doubling, as discussed above, we lift the Floquet operator from a matrix
operating in $2d$ spin space to a matrix
$\hat{U}_{\Phi}\mapsto\hat{U}_{\Phi}\otimes\openone_{2R}$, operating in the
$2\times 2\times R$ dimensional product space of spin, causal and replica
degrees of freedom. The cft then exchanges integrals over these local spin
rotations, singlet in replica and causal space, for integrals over local
rotations in replica and causal space, and structureless in spin-space. The
latter conveniently accommodate the soft modes of the disordered system, viz.
soft rotations in causal and replica space that are singlet in spin-space.
Formally, the cft is an exact transformation which leads to an alternative
representation of the generating functional in matrices that are the local
coordinates of the matrix degree of freedom $Q(\mathbb{n})$, discussed above.
In a final step, see Section S3.3, the functional is expanded in slow
fluctuations leading to a soft mode action $S=S_{\sigma}+S_{\rm top}$,
consisting of the non-linear diffusive $\sigma$ model action and a topological
term.

##  Appendix S2 Field theory of class A topological Floquet metal

Nonlinear sigma models emerge as effective field theories, capturing the low
energy sector of disordered single particle systems. For static quantum
systems, the low energy sector usually describes a narrow window of energies
around the Fermi level, relevant for the physics at long time and large
distance scales. Effective field theories are then derived from averaging over
disorder ensembles, usually in a Gaussian distribution. For time periodic
quantum systems, on the other hand, a quasi-energy is defined only modulo
$2\pi/T$. Disorder in these systems is usually modeled by random unitaries
drawn from the Haar measure (respectively restrictions of the latter if
symmetries are present), and there is no distinction between different quasi-
energies. Zirnbauer [44] derived the nonlinear sigma model for time periodic
quantum systems with random onsite unitary disorder using the color flavor
transformation, and our discussion below is extensively based on this
approach.

Once the effective field theory of a disordered system is known, one may
naturally ask whether the system is subject to Anderson localization. By now
it is well established that Anderson localization can be avoided if one of two
topological terms, a $\mathbb{Z}_{2}$ topological term or a WZW term, is
present. The possibility of the latter is determined by the system dimension
and symmetries of the nonlinear sigma model target space [26]. Similarly, the
existence of topological insulators and superconductors in a given dimension
and symmetry class can be inferred from the presence of robust metallic
boundary modes. These modes are not subject to Anderson localization and thus
define a topological metal. In the present work we propose quantum simulators
of single copies of topological metals, realized e.g. at the isolated surfaces
of topologically non trivial insulators.

More specifically, $d$-dimensional systems in class A are eligible to
topological metallic phases if $\pi_{d+1}\left(U(N+M)/U(N)\times
U(M)\right)=\mathbb{Z}$. Notice here that target manifolds of nonlinear sigma
models for Floquet and static systems within the same symmetry class are
identical, and table 2 of Ryu et al. [26] applies for both. In our previous
work [28] we derived the nonlinear sigma model action for a $2d$ class A
Floquet system, composed of the conventional diffusive contribution and a
Pruisken $\theta$ term. This system is subject to Anderson localization and
flows (in the thermodynamic limit) to one of the $\mathbb{Z}$ topological
insulating phases of the Quantum Hall class. Its $1d$ boundary mode is chiral
and topologically protected by the winding number of the $2d$ insulating bulk.
The effective theory describing the $1d$ boundary mode can be expressed as a
product of the $1d$ winding number characterizing the chiral edge mode and a
$(1+1)d$ WZW term, in which the matrix field of the $1d$ real space coordinate
is deformed (by introduction of an additional homotopy parameter) into a
trivial field configuration.

In the present work we derive the effective action for a $3d$ topological
metal in class A. Its topological term is composed of a $3d$ winding number
multiplied by $(3+1)d$ WZW term, in which the matrix field parametrized by the
$3d$ real space coordinate is extended (by introduction of an additional
homotopy parameter) to a trivial configuration. Since
$\pi_{4}(U(N+M)/U(N)\times U(M))=\mathbb{Z}$, the possibility of topologically
nontrivial field configurations is guaranteed, however, as already discussed
constructing a physical model with $3d$ winding number is not trivial. We here
achieve such model employing the idea of synthetic dimensions engineered via
time dependent protocols. The proposed model can be realized in a quantum walk
setting, for example, using the optical network shown in Fig. 3 and discussed
in the main text.

An alternative realization of a topological metal is via the emergence of a
$\mathbb{Z}_{2}$ topological term in the effective field theory description.
The latter also protects against Anderson localization, and can be realized in
one of the real symmetry classes, for example, in a $2d$ class AII system for
which $\pi_{2}(O(N+M)/O(N)\times O(M))=\mathbb{Z}_{2}$. In Section D we derive
the nonlinear sigma model for a time reversal symmetric Floquet system using
the color flavor transformation. We verify that the $\mathbb{Z}_{2}$
topological term is written as the product of two $(2+1)d$ WZW terms of matrix
fields, one involving the translational invariant part of the time evolution
operator and the other the sigma model field degree of freedom. In combination
with the $\mathbb{Z}$ example, this completes our derivation of low energy
effective field theories for topological Floquet metals, their model
realizations, and numerical confirmation of the absence of Anderson
localization.

In the remaining part of this Section we discuss the field theories for class
A and class AII systems. We introduce the color flavor transformation for
class A, and the derivation of a $(3+1)d$ WZW term is detailed in Section C.
In Section D we present the color flavor transformation for class AII systems
and the derivation of a $\mathbb{Z}_{2}$ topological term, and its alternative
representation in terms of a WZW term.

###  S2.1 Color-flavor transformation class A Floquet systems

Consider a Floquet system described by the single time step evolution operator
$\hat{U}_{0}$, to which onsite disorder is introduced by adding to the time
evolution locally uncorrelated random phases. The microscopic action is then
of the following general form, $Z=\int D(\psi,\bar{\psi})\exp(-S)$, with

$\displaystyle S$ $\displaystyle=\int
d^{d}\mathbb{x}\,\bar{\psi}_{\mathbb{x}}^{+a}(\hat{G}_{+}^{-1})_{\mathbb{xx^{\prime}}}\psi_{\mathbb{x}^{\prime}}^{+a}+\bar{\psi}_{\mathbb{x}}^{-a}(\hat{G}_{-}^{-1})_{\mathbb{xx^{\prime}}}\psi_{\mathbb{x}^{\prime}}^{-a},$
(S3)
$\displaystyle(\hat{G}_{+}^{-1})_{\mathbb{xx^{\prime}}}=\delta_{\mathbb{xx^{\prime}}}-e^{i\phi_{\mathbb{x}}}(\hat{U}_{0})_{\mathbb{xx^{\prime}}},$
(S4)
$\displaystyle(\hat{G}_{-}^{-1})_{\mathbb{xx^{\prime}}}=\delta_{\mathbb{xx^{\prime}}}-(\hat{U}_{0}^{\dagger})_{\mathbb{xx^{\prime}}}e^{-i\phi_{\mathbb{x}^{\prime}}},$
(S5)

where $\exp(i\phi_{\mathbb{x}})$ are the uniformly distributed phases, and
‘$a$’ carries spin, replica, and particle-hole indices. Upon introduction of
adequate source terms (see e.g. Ref. [28] for details), Eq. (S3) allows for
the convenient generation of products $\langle G^{+}G^{-}\rangle$, e.g.
required for the calculation of Eq. (8) in the main text. The disorder
averaging of the generating function $\langle
Z\rangle=\prod_{\mathbb{x}}\frac{1}{2\pi}\int
d\phi_{\mathbb{x}}Z(\\{\phi_{\mathbb{x}}\\})$ is performed by the color flavor
transformation:

$\displaystyle\frac{1}{2\pi}\int
d\phi_{\mathbb{x}}e^{\bar{\psi}_{\mathbb{x}}^{+a}e^{i\phi_{\mathbb{x}}}\varphi_{\mathbb{x}}^{+a}+\bar{\varphi}_{\mathbb{x}}^{-a}e^{-i\phi_{\mathbb{x}}}\psi_{\mathbb{x}}^{-a}}$
(S6) $\displaystyle=\int
D(Z_{\mathbb{x}},\tilde{Z}_{\mathbb{x}})e^{-\mathrm{tr}\ln(1-\tilde{Z}_{\mathbb{x}}Z_{\mathbb{x}})}e^{\bar{\psi}_{\mathbb{x}}^{+a}Z_{\mathbb{x}}^{ab}\psi_{\mathbb{x}}^{-b}+\bar{\varphi}_{\mathbb{x}}^{-a}\tilde{Z}_{\mathbb{x}}^{ab}\varphi_{\mathbb{x}}^{+b}},$
(S7)

applying for every position $\mathbb{x}$. Here the Floquet operator
$\hat{U}_{0}$ has been absorbed into the newly defined Grassmann fields
$\varphi_{\mathbb{x}}^{+a}=(\hat{U}_{0}\psi)_{\mathbb{x}}^{+a}$ and
$\bar{\varphi}_{\mathbb{x}}^{-a}=(\bar{\psi}\hat{U}_{0}^{\dagger})_{\mathbb{x}}^{-a}$,
and the domain of integration is defined by the condition
$\tilde{Z}=-Z^{\dagger}$. The matrix field $Z_{\mathbb{x}}^{ab}$ connects
retarded (+) and advanced (-) Grassmann field, indicating that its spatial
fluctuation encodes diffusion in the long distance limit. Its role becomes
more clear by introducing $Q=T\tau_{3}T^{-1}$ with

$\displaystyle T\equiv\begin{pmatrix}\openone&Z\\\
\tilde{Z}&\openone\end{pmatrix}_{\rm RA},$

and anticipating that the effective field theory is expressed in terms of the
matrix field $Q$ with nonlinear constraint $Q^{2}=\openone$. More
specifically, $Z$ defines linear coordinates on the symmetric space
$U(2R)/U(R)\times U(R)$ with $Z=0$ representing the ‘north pole’,
$Q=\tau_{3}$, and $Z\rightarrow\infty$ the ‘south pole’, $Q=-\tau_{3}$. After
integration over Grassman fields, we obtain the class A action entirely
expressed in terms of the linear coordinates,

$\displaystyle
S[Z,\tilde{Z}]=-\mathrm{tr}\ln(1-\tilde{Z}Z)+\mathrm{tr}\ln(1-\tilde{Z}UZU^{\dagger}).$

In Ref.[28] we derived the low energy effective action of a $2d$ Floquet
topological insulator, composed of the conventional diffusive term accompanied
by a Pruisken $\theta$ term. This allowed us to confirm that the maximally
disordered Floquet system belongs to the integer quantum Hall universality
class. In the following sections, we derive the effective theory for a $3d$
Floquet system. This turns out to contain a Chern-Simons action, and thus is
within the same universality class as Weyl semimetals without intervalley
scattering.

###  S2.2 Soft-mode actions for class A

A straightforward manipulation of block matrices brings the above
representation of the action into the form [28]

$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{s=\pm}\mathrm{Tr}\,\mathrm{ln}(1+X_{s}P^{s}),$
(S8) $\displaystyle X_{-}$ $\displaystyle\equiv$
$\displaystyle\hat{T}^{-1}[\hat{U}_{0},\hat{T}]\hat{U}_{0}^{\dagger},\,\,\,\,\,X_{+}\equiv\hat{T}^{-1}[\hat{U}_{0}^{\dagger},\hat{T}]\hat{U}_{0},$
(S9)

where $P^{\pm}=(\openone\pm\tau_{3})/2$ are projectors onto retarded and
advanced sectors of the theory. To simplify notation, we will in the following
drop the index ‘$0$’ of the Floquet operator $\hat{U}_{0}$. An expansion of
the log-function to the third order is necessary,

$\displaystyle S$ $\displaystyle=$ $\displaystyle S^{(1)}+S^{(2)}+S^{(3)},$
(S10)

where

$\displaystyle S^{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{s=\pm}\mathrm{Tr}(X_{s}P^{s}),$ $\displaystyle
S^{(2)}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\sum_{s=\pm}\mathrm{Tr}((X_{s}P^{s})^{2}),$
$\displaystyle S^{(3)}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\sum_{s=\pm}\mathrm{Tr}((X_{s}P^{s})^{3}).$

In the following sections, using the Wigner transformation the continuum
representation of the action is obtained.

####  S2.2.1 The first order terms $S^{(1)}$

Consider the ‘-’ contribution to the action $S^{(1)}$,

$\displaystyle S^{(1)-}$
$\displaystyle=\frac{1}{2}\mathrm{Tr}\left((\hat{T}^{-1}\hat{U}\hat{T}\hat{U}^{\dagger}-\mathbb{I})P^{-}\right),$
$\displaystyle=\frac{1}{2}\mathrm{Tr}\left(\int_{0}^{t}ds(\hat{T}^{-1}\partial_{s}\hat{U}\hat{T}\hat{U}^{\dagger}+\hat{T}^{-1}\hat{U}\hat{T}\partial_{s}\hat{U}^{\dagger})P^{-}\right),$
$\displaystyle=\frac{1}{2}\int_{0}^{t}ds\,\mathrm{Tr}\left(\left([\hat{T}^{-1},\partial_{s}\hat{U}\hat{U}^{\dagger}]\hat{U}\hat{T}\hat{U}^{\dagger}\right)P^{-}\right),$
$\displaystyle=-\frac{1}{2}\int_{0}^{t}ds\,\mathrm{Tr}\left(\left([\hat{T}^{-1},\hat{\psi}_{s}^{-}][\hat{U},\hat{T}]\hat{U}^{\dagger}+[\hat{T}^{-1},\hat{\psi}_{s}^{-}]\hat{T}\right)P^{-}\right),$

where in the third equality
$\hat{U}\partial_{s}\hat{U}^{\dagger}=-\partial_{s}\hat{U}\hat{U}^{\dagger}\equiv\psi_{s}^{-}$
is used. Note that $\hat{U}$ has no structure in replica space, thus
$[\partial_{s}\hat{U}^{\dagger},P^{-}]=0$ is used. The commutator in the third
line can be Moyal expanded up to the third derivatives,

$\displaystyle[\hat{T}^{-1},\hat{\psi}_{s}^{-}][\hat{U},\hat{T}]\hat{U}^{\dagger}$
$\displaystyle\simeq$
$\displaystyle(\partial_{i}T^{-1}\partial_{i}\psi_{s}^{-})(\partial_{j}U\partial_{j}TU^{\dagger}),$

and

$\displaystyle\left[\hat{T}^{-1},\hat{\psi}_{s}^{-}\right]\hat{T}$
$\displaystyle\simeq$ $\displaystyle
i(\partial_{i}T^{-1}\partial_{i}\psi_{s}^{-})T-\frac{i}{24}(\partial^{3}_{ijk}\psi_{s}^{-}\partial^{3}_{ijk}T^{-1})T,$

where distinction between an operator and a function is made by hat on
symbols. $\partial_{i}U\equiv\partial_{k_{i}}U$ and
$\partial_{i}T\equiv\partial_{x_{i}}T$ and sum over index $i,j,k$ is
implicitly assumed. By plugging in the above into the action,

$S^{(1)-}=-\frac{i}{2}\int_{0}^{t}ds\,\mathrm{Tr}\left((\partial_{i}T^{-1}T)(\partial_{i}\psi_{s}^{-})P^{-}\right)\\\
+\frac{1}{2}\int_{0}^{t}ds\,\mathrm{Tr}\left((\partial_{i}T^{-1}\partial_{j}T)(\partial_{i}\psi_{s}^{-}\psi_{j}^{-})P^{-}\right)\\\
+\frac{i}{48}\int_{0}^{t}ds\,\mathrm{Tr}\left((\partial^{3}_{ijk}T^{-1}T)(\partial^{3}_{ijk}\psi_{s}^{-})P^{-}\right),$

where the first and third term vanish after the momentum integration because
$\psi_{s}^{-}$ a is a periodic function in momentum. The second term, which is
obtained using
$\partial_{j}UU^{\dagger}=-U\partial_{j}U^{\dagger}=-\psi_{j}^{-}$, is,

$\displaystyle S^{(1)-}_{\mathrm{2nd}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{t}ds\,\mathrm{Tr}(\partial_{i}T^{-1}\partial_{j}TP^{-})\mathrm{tr}(\partial_{i}\psi_{s}^{-}\psi_{j}^{-}).$

Eq. (LABEL:S12) is identical to the one in [28], from which the Pruisken
action and a part of diffusive action is derived.

####  S2.2.2 The second order terms $S^{(2)}$

Using
$X_{-}=\hat{T}^{-1}[\hat{U},\hat{T}]\hat{U}^{\dagger}=\hat{T}^{-1}\hat{U}[\hat{T},\hat{U}^{\dagger}]\rightarrow
i(T^{-1}*U)*\partial_{i}T\partial_{i}U^{\dagger}$, and Moyal expand to the
third derivatives,

$(T^{-1}*U)*\partial_{i}T\partial_{i}U^{\dagger}\\\ \simeq
T^{-1}U\partial_{i}T\partial_{i}U^{\dagger}+\frac{i}{2}(\partial_{k}T^{-1}\partial_{k}U)\partial_{i}T\partial_{i}U^{\dagger}\\\
+\frac{i}{2}(\partial_{k}T^{-1}U)\partial_{i}T\partial^{2}_{ik}U^{\dagger}-\frac{i}{2}(T^{-1}\partial_{k}U)\partial^{2}_{ik}T\partial_{i}U^{\dagger},$

which is then plug in to $S^{(2)-}$,

$\displaystyle
S^{(2)-}=\frac{1}{4}\mathrm{Tr}\left((T^{-1}\partial_{i}T)P^{-}(T^{-1}\partial_{j}T)P^{-}(U\partial_{i}U^{\dagger}U\partial_{j}U^{\dagger})\right)$
$\displaystyle+\frac{i\epsilon^{ijk}}{4}\mathrm{Tr}\left((\partial_{k}T^{-1}\partial_{i}T)P^{-}(T^{-1}\partial_{j}T)P^{-}(\partial_{k}U\partial_{i}U^{\dagger}U\partial_{j}U^{\dagger})\right)$
$\displaystyle+\frac{i\epsilon^{ijk}}{4}\mathrm{Tr}\left((\partial_{k}T^{-1}\partial_{i}T)P^{-}(T^{-1}\partial_{j}T)P^{-}(U\partial^{2}_{ik}U^{\dagger}U\partial_{j}U^{\dagger})\right)$
$\displaystyle-\frac{i\epsilon^{ijk}}{4}\mathrm{Tdr}\left((T^{-1}\partial^{2}_{ik}T)P^{-}(T^{-1}\partial_{j}T)P^{-}(\partial_{k}U\partial_{i}U^{\dagger}U\partial_{j}U^{\dagger})\right).$
(S12)

The Levi-Civita symbol is introduced because only antisymmetric combinations
are nonzero for the 3-dim TFM unitary operator. Thus, the third and fourth
term in (S12) is zero. Introducing $A_{i}=T^{-1}\partial_{i}T$,

$\displaystyle S^{(2)-}$
$\displaystyle=\frac{1}{4}\mathrm{Tr}\left((A_{i}P^{-}A_{j}P^{-})(\psi^{-}_{i}\psi^{-}_{j})\right)$
$\displaystyle+\frac{i}{4}\epsilon^{ijk}\mathrm{Tr}\left((A_{k}A_{i}P^{-}A_{j}P^{-})(\psi_{k}^{-}\psi^{-}_{i}\psi^{-}_{j})\right),$

Note that $\epsilon^{ijk}A_{j}A_{k}=-\epsilon^{ijk}\partial_{j}A_{k}$. On the
other hand,

$\displaystyle S^{(2)+}$
$\displaystyle=\frac{1}{4}\mathrm{Tr}\left((A_{i}P^{+}A_{j}P^{+})(\psi^{+}_{i}\psi^{+}_{j})\right)$
$\displaystyle+\frac{i}{4}\epsilon^{ijk}\mathrm{Tr}\left((A_{k}A_{i}P^{+}A_{j}P^{+})(\psi_{k}^{+}\psi^{+}_{i}\psi^{+}_{j})\right).$

Employing that
$\mathrm{tr}(\psi_{i}^{+}\psi_{j}^{+})=\mathrm{tr}(\psi_{i}^{-}\psi_{j}^{-})$
and
$\mathrm{tr}(\psi_{i}^{+}\psi_{j}^{+}\psi_{k}^{+})=-\mathrm{tr}(\psi_{i}^{-}\psi_{j}^{-}\psi_{k}^{-})$,
the latter two terms of $S^{(2)\pm}$ can be combined as

$\displaystyle S^{(2)}$
$\displaystyle=\frac{1}{4}\sum_{s=\pm}\mathrm{Tr}(A_{i}P^{s}A_{j}P^{j})$ (S13)
$\displaystyle-\frac{i\epsilon^{ijk}}{4}\sum_{s=\pm}s\,\mathrm{Tr}\left(\partial_{k}A_{i}P^{s}A_{j}P^{s}\right)\mathrm{tr}(\psi_{i}^{+}\psi_{j}^{+}\psi_{k}^{+}),$

where the first term becomes part of the diffusive action (see Eq.(30) of Ref.
[28]), and the second term constitutes the first part of the Chern-Simons
action.

####  S2.2.3 The third order terms $S^{(3)}$

The third order expansion of the log yields,

$\displaystyle S^{(3)-}$
$\displaystyle=\frac{i^{3}}{6}\epsilon^{ijk}\mathrm{Tr}\left((T^{-1}U\partial_{i}T\partial_{i}U^{\dagger})P^{-}\right.$
$\displaystyle\times\left.(T^{-1}U\partial_{j}T\partial_{j}U^{\dagger})P^{-}(T^{-1}U\partial_{k}T\partial_{k}U^{\dagger})P^{-}\right),$
$\displaystyle=-\frac{i}{6}\epsilon^{ijk}\mathrm{Tr}(A_{i}P^{-}A_{j}P^{-}A_{k}P^{-})\,\mathrm{tr}(\psi^{-}_{i}\psi^{-}_{j}\psi^{-}_{k}),$

and similarly,

$\displaystyle S^{(3)+}$
$\displaystyle=-\frac{i}{6}\epsilon^{ijk}\mathrm{Tr}(A_{i}P^{+}A_{j}P^{+}A_{k}P^{+})\,\mathrm{tr}(\psi^{+}_{i}\psi^{+}_{j}\psi^{+}_{k}).$

Combining the latter with the 2nd piece in Eq. (S13), we arrive at

$S^{(3)}=-\frac{i\epsilon^{ijk}}{6}\sum_{s=\pm}s\,\mathrm{Tr}(A_{i}P^{s}A_{j}P^{s}A_{k}P^{s})\,\mathrm{tr}(\psi^{+}_{i}\psi^{+}_{j}\psi^{+}_{k}),$

which constitutes the second part of the Chern-Simons action. Introducing the
winding number $\nu_{3}=\frac{1}{24\pi^{2}}\int
d^{3}k\,\epsilon^{\mu\nu\rho}\mathrm{tr}(\psi^{+}_{\mu}\psi^{+}_{\nu}\psi^{+}_{\rho})$,
the Chern-Simons action can be expressed in the following form:

$\displaystyle S_{\rm top}$ $\displaystyle=\nu_{3}\int_{\partial
B_{4}}\left(w_{\rm CS}[AP^{+}]-w_{\rm CS}[AP^{-}]\right),$

where $w_{\rm CS}[A]=\frac{i}{8\pi}\left(A\wedge dA+\frac{2}{3}A\wedge A\wedge
A\right)$. As a result we obtain the topological action summarized in Eqs.
(23) and (25) in the main text. Notice that the presence of the Chern-Simons
action can be anticipated for a $3d$ class A system, noting that the homotopy
group $\pi_{3+1}(U(2R)/U(R)\times U(R))=\mathbb{Z}$ is nontrivial. That is,
the field $T=T(\mathbb{x})$ can be extended from $3d$ space to the matrix
field $Q=Q(x_{0},\mathbb{x})$ in one dimension higher, and the topological
action can then be expressed as a WZW term (see Ref. [36]):

$\displaystyle S_{\rm top}$
$\displaystyle=\frac{i\nu_{3}}{128\pi}\int_{B_{4}}{\rm Tr}\,\left[Q(\wedge
dQ)^{4}\right].$

##  Appendix S3 Field theory of class AII topological Floquet metal

In this Section we provide details on the effective field theory for class AII
Floquet systems. We first present the color flavor transformation for the
latter, from which the disorder averaged effective action is derived in D.1.
The softmode manifold of class AII systems is specified in D.2, and in
sections D.3 and D.4 we present the nonlinear sigma model composed of a
diffusive and a topological term. The latter is expressed as the product of
two WZW terms, measuring the topological content of mappings from momentum and
real space, respectively. We provide the connection of the topological term
and the color flavor action in D.5, completing thus the derivation of
effective field theories for Floquet topological metals. Conventions used
throughout the sections are as follows. We use symbols $\sigma_{j}$ to
indicate sublattice space, $\tau_{j}$ for retarded-advanced space, and $s_{j}$
for particle-hole space. Vectors
$(x_{0},\mathbb{x})=(x_{0},x_{1},\cdots,x_{d})$ summarize the spatial
coordinate and homotopy parameter, and correspondingly,
$(k_{0},\mathbb{k})=(k_{0},k_{1},\cdots,k_{d})$, for momentum coordinates
including a homotopy parameter.

###  S3.1 Color-flavor transformation for class AII Floquet systems

We recall that the $2\times 2$ dimensional time evolution operator in class
${\rm AII}$ satisfies the time-reversal symmetry constraint $\sigma_{2}\,{\cal
U_{F}}^{T}\sigma_{2}={\cal U_{F}}$. It can be written as a product

${\cal
U_{F}}=VU_{0}\bar{V},\quad\bar{V}=\sigma_{2}V^{T}\sigma_{2},\quad\bar{U}_{0}=\sigma_{2}U_{0}^{T}\sigma_{2}\equiv
U_{0},$ (S14)

where $U_{0}$ is a non-random part while $V$ and $\bar{V}$ encode the unitary
disorder. For the simplicity of notation the index zero in the clean part of
the Floquet operator $U_{0}$ will be dropped in the following discussion.
Concentrating e.g. on the contribution of the retarded (+) sector to the
action (S3), the above decomposition of ${\cal U_{F}}$ allows to rewrite

$S_{+}=\bar{\psi}_{+1}\,\psi_{+1}+\bar{\psi}_{+2}\,\psi_{+2}-\bar{\psi}_{+1}e^{i\phi_{+}}V\psi_{+2}-\bar{\psi}_{+2}U\bar{V}\psi_{+1},$
(S15)

where we introduced the two-component spinor
$\bar{\psi}_{+}=(\bar{\psi}_{+1},\bar{\psi}_{+2})$ and similar for $\psi_{+}$.
A constant source term $\phi_{+}$ is introduced. We then arrange contributions
containing $\bar{V}$ using time reversal symmetry,

$\displaystyle\bar{\psi}_{+2}U\bar{V}\psi_{+1}$
$\displaystyle=(\bar{\psi}_{+2}U\sigma_{2}V^{T}\sigma_{2}\psi_{+1})^{T}$
$\displaystyle=-\psi_{+1}^{T}\sigma_{2}V\sigma_{2}U^{T}\bar{V}\bar{\psi}_{+2}^{T}$
$\displaystyle=-(\psi_{+1}^{T}\sigma_{2})\,V\,(U\sigma_{2}\bar{\psi}_{+2}^{T}),$
(S16)

and upon introducing spinors

$\varphi_{+}^{T}=\left(\bar{\psi}_{+1},i\,\psi_{+1}^{T}\sigma_{2}\right),\quad\chi_{+}=\left(\begin{array}[]{c}e^{i\phi_{+}}\psi_{+2}\\\
Ui\sigma_{2}\bar{\psi}_{+2}^{T}\end{array}\right),$ (S17)

we can cast this action into the simpler form

$S_{+}=\bar{\psi}_{+1}\,\psi_{+1}+\bar{\psi}_{+2}\,\psi_{+2}-\varphi_{+}^{T}V\chi_{+}.$
(S18)

Proceeding along the same lines for the contribution of the advanced sector to
the action (S3),

$S_{-}=\bar{\psi}_{-1}\,\psi_{-1}+\bar{\psi}_{-2}\,\psi_{-2}-\bar{\psi}_{-1}\bar{V}^{\dagger}U^{\dagger}\psi_{-2}-\bar{\psi}_{-2}V^{\dagger}e^{-i\phi_{-}}\psi_{-1},$
(S19)

the disorder term $\propto V^{\dagger}$ can be reorganized employing time
reversal symmetry,

$\displaystyle\bar{\psi}_{-1}\bar{V}^{\dagger}U^{\dagger}\psi_{-2}$
$\displaystyle=(\bar{\psi}_{-1}\sigma_{2}V^{*}\sigma_{2}U^{\dagger}\psi_{-2})^{T}$
$\displaystyle=-(\psi_{-2}^{T}U^{*}\sigma_{2})V^{\dagger}(\sigma_{2}\bar{\psi}_{-1})$
$\displaystyle=-(\psi_{-2}^{T}\sigma_{2}U^{\dagger})V^{\dagger}(\sigma_{2}\bar{\psi}_{-1})$
(S20)

and upon introducing the spinors

$\chi_{-}^{T}=(\bar{\psi}_{-2}e^{-i\phi_{-}},i\psi_{-2}^{T}\sigma_{2}U^{\dagger}),\quad\varphi_{-}=\left(\begin{array}[]{c}\psi_{-1}\\\
i\sigma_{2}\bar{\psi}_{-1}^{T},\end{array}\right),$ (S21)

into the form

$S_{-}=\bar{\psi}_{-1}\,\psi_{-1}+\bar{\psi}_{-2}\,\psi_{-2}-\chi_{-}^{T}V^{\dagger}\varphi_{-}.$
(S22)

With these preparations we can now apply the color-flavor transformation to
the action $S=S_{+}+S_{-}$ by averaging over $V$ using (S7). This amounts to
exchanging matrices $V$, $V^{\dagger}$, with structure in spin space, for
matrices $Z$, $\tilde{Z}$, with structure in replica and causal space. That
is,

$\displaystyle S_{1}$
$\displaystyle=\bar{\psi}_{+1}\,\psi_{+1}+\bar{\psi}_{-1}\,\psi_{-1}-\varphi_{+}^{T}Z\varphi_{-},$
$\displaystyle S_{2}$
$\displaystyle=\bar{\psi}_{+2}\,\psi_{+2}+\bar{\psi}_{-2}\,\psi_{-2}-\chi_{-}^{T}\tilde{Z}\chi_{+},$
(S23)

and we notice that $\psi_{\pm 1}$ can be expressed entirely in terms of
$\varphi_{\pm}$, and correspondingly, $\psi_{\pm 2}$ entirely in terms of
$\chi_{\pm}$. The subsequent path integral can therefore be split into two
independent integrals over $\varphi$ and $\chi$ with actions $S_{1}$ and
$S_{2}$, respectively.

To proceed, we organize $\pm$ indices of vectors $\varphi$ and $\chi$ into a
two dimensional space, in the following referred to as ‘particle-hole (ph)’
space, and introduce the matrix $s_{1}\equiv\sigma_{1}^{\rm ph}$, operating in
this space, with $s_{1}^{T}s_{1}=1$. We then notice that

$\displaystyle\varphi_{+}^{T}\,(i\sigma_{2}\otimes s_{1})\,\varphi_{+}$
$\displaystyle=2\bar{\psi}_{+1}\psi_{+1},$
$\displaystyle\varphi_{-}^{T}\,(i\sigma_{2}\otimes s_{1})\,\varphi_{-}$
$\displaystyle=2\bar{\psi}_{-1}\psi_{-1},$
$\displaystyle\chi_{+}^{T}\,e^{-i\phi_{+}}(i\sigma_{2}\otimes
s_{1})U^{\dagger}\,\chi_{+}$ $\displaystyle=2\bar{\psi}_{+2}\psi_{+2},$
$\displaystyle\chi_{-}^{T}\,e^{i\phi_{-}}(i\sigma_{2}\otimes
s_{1})U^{T}\,\chi_{-}$ $\displaystyle=2\bar{\psi}_{-2}\psi_{-2},$ (S24)

which allows us to express actions $S_{1,2}$ in the form

$S_{1}=\frac{1}{2}(\varphi_{+}^{T},\varphi_{-}^{T})\left(\begin{array}[]{cc}(i\sigma_{2}\otimes
s_{1})&-Z\\\ Z^{T}&(i\sigma_{2}\otimes
s_{1})\end{array}\right)\left(\begin{array}[]{c}\varphi_{+}\\\
\varphi_{-}\end{array}\right),$ (S25)

and

$\displaystyle S_{2}$ $\displaystyle=\frac{1}{2}(\chi_{+}^{T},\chi_{-}^{T})$
$\displaystyle\times\left(\begin{array}[]{cc}e^{-i\phi_{+}}(i\sigma_{2}\otimes
s_{1})U^{\dagger}&\tilde{Z}^{T}\\\ -\tilde{Z}&e^{i\phi_{-}}(i\sigma_{2}\otimes
s_{1})U^{T}\end{array}\right)\left(\begin{array}[]{c}\chi_{+}\\\
\chi_{-}\end{array}\right),$

were we used that
$\varphi_{+}^{T}Z\varphi_{-}=-\varphi_{-}^{T}Z^{T}\varphi_{+}$ and similar for
$\chi$. In a final step we then complete Gaussian integrals over $\varphi$ and
$\chi$. Accounting for a nontrivial Jacobian
$J(\phi_{+},\phi_{-})=(e^{-i(\phi_{+}-\phi_{-})})^{-1}$ resulting from the
change of integration variables, we arrive at

$\displaystyle{\cal Z}$ $\displaystyle\propto J(\phi_{+},\phi_{-})\,{\rm
det}^{1/2}\left(\begin{array}[]{cc}i\sigma_{2}\otimes s_{1}&-Z\\\
Z^{T}&i\sigma_{2}\otimes s_{1}\end{array}\right)\times$ (S28)
$\displaystyle\times{\rm
det}^{1/2}\left(\begin{array}[]{cc}e^{-i\phi_{+}}(i\sigma_{2}\otimes
s_{1})U^{\dagger}&\tilde{Z}^{T}\\\ -\tilde{Z}&e^{i\phi_{-}}(i\sigma_{2}\otimes
s_{1})U^{T}\end{array}\right).$ (S31)

Finally, restoring the measure $\mu(Z,\tilde{Z})={\rm det}(1-\tilde{Z}Z)^{-1}$
of the color-flavor transformation, we arrive at ${\cal Z}=e^{S}$, with

$\displaystyle S$ $\displaystyle=-{\rm tr}\ln(1-\tilde{Z}Z)+\frac{1}{2}{\rm
tr}\ln(1-Z^{T}s_{1}Zs_{1}^{T})$ $\displaystyle\quad+\frac{1}{2}{\rm
tr}\ln(1-\tilde{Z}Us_{1}^{T}e^{i\phi_{+}}\tilde{Z}^{T}e^{-i\phi_{-}}s_{1}U^{\dagger}).$
(S32)

Upon color-flavor transformation (enabling the disorder averaging), the
original action for ‘microscopic’ Grassmann fields is expressed in terms of
the matrix fields $Z$, $\tilde{Z}$. These are more convenient to extract the
low energy content of the theory, and to access the low energies we subject
the action to a saddle point analysis. Allowing for (soft) spatial
fluctuations around the saddle points we then derive the low energy effective
action. We notice that the target space of the matrix field is topologically
non trivial and the generating function must include the sum over
topologically distinct configurations, weightening topologically different
sectors by a topological contribution to the action. We present the effective
action in Sections S3.3 and S3.4, and the derivation of the topological term
(starting out from Eq.(S3.1)) in Sec.S3.5.

###  S3.2 Soft-mode manifold for class AII

We then notice that in the zero frequency limit, $\phi_{\pm}\to 0$, action Eq.
(S3.1) is minimized by homogeneous configurations $Z,\tilde{Z}$ related via

$\tilde{Z}=s_{1}^{T}Z^{T}s_{1}.$ (S33)

The soft mode action reads

$S[Z,\tilde{Z}]=-\frac{1}{2}{\rm tr}\ln(1-\tilde{Z}Z)+\frac{1}{2}{\rm
tr}\ln(1-\tilde{Z}UZU^{\dagger}),$ (S34)

which apart from the factor $1/2$ and symmetry constraint Eq. (S33), is the
same as that encountered for class ${\rm A}$ systems. (See e.g. Ref. [28] for
a discussion in the context of the above mentioned AFAI.)

To further elaborate on the soft mode manifold, we introduce matrices,
$\tau_{3}\equiv\sigma_{3}^{\rm RA}$,

$Q=T\tau_{3}T^{-1},\qquad T=\left(\begin{array}[]{cc}1&Z\\\
\tilde{Z}&1\end{array}\right)_{\rm RA},$ (S35)

satisfying the symmetry constraint [34]

$\displaystyle\bar{Q}=Q,\quad\bar{Q}\equiv C^{T}Q^{T}C,\quad C=\tau_{3}\otimes
s_{1}.$ (S36)

The explicit expression for the $Q$-matrix read,

$Q=\left(\begin{array}[]{cc}\displaystyle\frac{1+Z\tilde{Z}}{1-Z\tilde{Z}}&\displaystyle-2Z\frac{1}{1-\tilde{Z}Z}\\\
\displaystyle
2\tilde{Z}\frac{1}{1-Z\tilde{Z}}&\displaystyle-\frac{1+\tilde{Z}Z}{1-\tilde{Z}Z}\end{array}\right)_{\rm
RA},$ (S37)

where we recall the relation $A({1-BA})^{-1}=({1-AB})^{-1}A$. One can now
verify that owing to (S33) the relation $C^{T}Q^{T}C=Q$ holds. This defines
the soft mode manifold of class ${\rm AII}$ systems.

###  S3.3 Soft-mode actions for class AII

Following the steps discussed in Ref. [28], we can apply a gradient expansion
to derive a diffusive $\sigma$ model action for $Q$ fields. The non trivial
properties of the dynamical protocols for the simulators of topological
surface states are encoded in an additional topological contribution to the
soft mode action. Their derivation is discussed in more detail below. The soft
mode action for $Q$-matrix fields satisfying AII symmetry constraint
$Q=\bar{Q}$ therefore reads $S=S_{\sigma}+S_{\rm top}$, where

$\displaystyle S_{\sigma}[Q]$
$\displaystyle=-\frac{1}{8}\sum_{i,j=1}^{2}\sigma_{ij}^{(0)}\int d^{2}x\,{\rm
tr}(\partial_{i}Q\partial_{j}Q),$ (S38) $\displaystyle S_{\rm top}[Q]$
$\displaystyle=\frac{i\theta}{\pi}\,\Gamma[g]\Bigl{|}_{g(x_{0}=0,{\bf
x})=Q({\bf x})},$ (S39)

with conductivity tensor $\sigma_{ij}^{(0)}=\frac{1}{2}\int
d^{2}\mathbb{k}\,\mathrm{tr}(\partial_{k_{i}}U_{\mathbb{k}}\partial_{k_{j}}U_{\mathbb{k}}^{-1})$
and topological angle, $\theta=0,\pi$ defined in Eq. (S44). $\Gamma[g]$ in Eq.
(S39) is a WZW functional

$\displaystyle\Gamma[g]=\frac{1}{24\pi}\int_{0}^{1}dx_{0}\int
d^{2}\mathbb{x}\epsilon^{\mu\nu\rho}{\rm
tr}\left(g^{-1}\partial_{\mu}gg^{-1}\partial_{\nu}gg^{-1}\partial_{\rho}g\right),$
(S40)

here defined for matrices $g$ from the orthogonal group ${\rm O}(4R)$,
satisfying the symmetry relation $\bar{g}=C^{T}g^{T}C=g^{-1}$, and with
constraint $g(x_{0}=1,{\bf x})=\tau_{3}$.

Several comments are here of order. (i) We remind that the WZW action depends
only on the boundary value of the group field, $g(x_{0}=0,{\bf x})$. Given
that the homotopy group $\pi_{2}({\rm O}(4R))=0$ is trivial, an extension to
the third dimension $x_{0}$ with constant boundary value at $x_{0}=1$ is
always possible. (ii) As we show below, $\Gamma[g]$ is only defined modulo
$\pi$. However, recalling that $\theta=0,\pi$, the exponentiated action
$e^{-S_{\rm top}[Q]}=\pm 1$ is well defined. (iii) Finally, as already
discussed in the main text, the possibility to add the topological term
$S_{\rm top}[Q]$ to the diffusive action is related to the nontrivial homotopy
group of the coset space, $\pi_{2}({\rm O}(4R)/{\rm O}(2R)\times{\rm
O}(2R))=\mathbb{Z}_{2}$.

We conclude our discussion, showing that $S_{\rm top}[Q]$ is indeed a