Radical chiral Floquet phases in a periodically driven Kitaev model and beyond

1.1 MB·

by  Hoi Chun Po

Radical chiral Floquet phases in a periodically driven Kitaev model and beyond
Hoi Chun Po,1,2 Lukasz Fidkowski,3,4 Ashvin Vishwanath,1,2 and Andrew C. Potter5
1Department of Physics, University of California, Berkeley, CA 94720, USA
2Department of Physics, Harvard University, Cambridge MA 02138, USA
3Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794, USA
4Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA
5Department of Physics, University of Texas at Austin, Austin, TX 78712, USA
TimeperiodicdrivingservesnotonlyasaconvenientwaytoengineereffectiveHamiltonians,but
also as a means to produce intrinsically dynamical phases that do not exist in the static limit. A
recentexampleofthelatterare2DchiralFloquet(CF)phasesexhibitinganomalousedgedynamics
that pump discrete packets of quantum information along one direction. In non-fractionalized
systemswith onlybosonic excitations, thispumpingis quantifiedbya dynamical topological index
7
that is a rational number – highlighting its difference from the integer valued invariant underlying
1
equilibriumchiralphases(e.g. quantumHallsystems). Here,weexploreCFphasesinsystemswith
0
2 emergent anyon excitations that have fractional statistics (Abelian topological order). Despite the
absence of mobile non-Abelian particles in these systems, external driving can supply the energy
n
topumpotherwiseimmobilenon-Abeliandefects(sometimescalledtwistdefectsorgenons)around
a
the boundary, thereby transporting an irrational fractional number of quantum bits along the edge
J
during each drive period. This enables new CF phases with chiral indices that are square roots of
5
rational numbers, inspiring the label: “radical CF phases”. We demonstrate an unexpected bulk-

boundary correspondence, in which the radical CF edge is tied to bulk dynamics that exchange
]
l electricandmagneticanyonexcitationsduringeachperiod. Weconstructsolvable,stroboscopically
e drivenversionsofKitaev’shoneycombspinmodelthatrealizetheseradicalCFphases,anddiscuss
-
r theirstabilityagainstheatinginstronglydisorderedmany-bodylocalizedsettingsorinthelimitof
t rapid driving as an exponentially long-lived pre-thermal phenomena.
s
.
t
a
m Time-periodic driving provides a powerful tool to clockwise around the edge[16]. For this reason, we dub
shape the properties of quantum matter, from changing these phases “rational” CF phases.
-
d its topological phase[1–3], to enabling entirely new non- In equilibrium settings, strong interactions can ef-
n equilibrium dynamical phases. These dynamical phases fectively fracture the original microscopic particles into
o are characterized by properties forbidden in the thermal emergent excitations with fractional (anyonic) statistics,
c
equilibrium, such as spontaneous time-translation sym- leadingtonewtypesoftopologicalbehaviorlikethefrac-
[
metry breaking[4–7] or dynamical topological order[8– tionalquantumHalleffect. Giventheroughparallelsbe-

1 14]. Thelatteropensthedoortoanewrealmofquantum tween rational CF phases and the integer quantum Hall
v topological phenomena, which has only barely begun to effect, it is natural to ask: Can strong interactions also
0
be explored. produce new “fractional” CF phases? In this paper, we
4
AstrikingexamplearechiralFloquet(CF)phases[15– answer this question in the affirmative. We construct
4
1 17] – 2D dynamical topological phases with inert bulk examples (including solvable lattice models) of systems
0 dynamics, but whose edges act as chiral quantum chan- whoseedgeschirallypumpanirrationalnumberofquan-
. nels that pump discrete quanta of quantum informa- tum states per driving period, which we dub radical chi-
1
tion along the edge in a chiral fashion each driving pe- ral Floquet phases. Here we choose the nomenclature
0
7 riod. These are, in a sense, dynamical analogs, of in- “radical” rather than “irrational” as these phases have
1 teger quantum Hall phases familiar from thermal equi- chiral unitary index of the form ν = log√r where r is a
: librium settings, which can also occur in non-interacting positive rational number. The existence of these radical
v
i fermion systems, have ordinary bulk properties, and chi- CF phases relies crucially on the emergence of a particu-
X rallypropagatingedgesthatareprotectedevenintheab- lartypeofFloquet-enrichedtopologicalorder(FET)[18],
r sence of any symmetry. However, CF phases are sharply withanyonexcitationswhosetopologicalchargesaredy-
a
distinct from integer quantum Hall systems. For ex- namically transmuted during each driving period.
ample, their edge states exhibit a discrete pumping of Oursetupthroughoutwillbea2Dlatticewithafinite
quantuminformationratherthancontinuousflowofheat numberofquantumdegreesoffreedomoneachsite,sub-
and charge. Moreover, CF phases are not captured by jected to a time-periodic Hamiltonian H(t) = H(t+T)
the usual equilibrium concepts like Chern numbers and with period T, and associated Floquet operator (time-
Hall conductance[15], but are instead characterized by evolution operator for one period):
a new type of intrinsically dynamical topological invari-
ant, the chiral unitary invariant ν. In interacting but U(T)= e−i(cid:82)0TH(t)dt, (1)
T
non-fractionalizedboson(spin)systems,ν takestheform
logr where r is a positive rational fraction representing where denotes time-ordering. An important concern
T
the ratio of states being pumped clockwise to counter- is whether driving will induce heating and destroy quan-
2
tum coherence. In generic systems, rapid driving can
postpone heating for an exponentially long time, leading
toalong-livedmetastable“pre-thermal”regime[19]with
quantum coherent non-equilibrium dynamics[20]. Even
more strikingly, in isolated systems such as cold-atoms PL PR
P
andtrappedions,heatingcanbeavoidedentirelybysyn-
thesizingsufficientlystrongdisorderdrivethesysteminto
a many-body localized (MBL) regime[21], where it can- by bz
c
not absorb energy from the drive and remains quantum x z bx
y
coherent, in principle, indefinitely[22]. We will focus on
the latter MBL case, though our discussion also enables a)
a construction of pre-thermal versions of these phases.
Review of rational Chiral Floquet phases – In
the absence of fractionalized anyonic excitations, and in ... ...
the MBL regime, the Floquet evolution operator decom-
b)
poses into commuting bulk and edge pieces. The edge
evolution acts within a finite strip at the system bound-
ary, and describes an effectively one-dimensional evolu- FIG. 1. Spin-1/2 Honeycomb model – (a) Depiction
tion. Suchlocality-preserving1Dunitarytimeevolutions of solvable lattice for a radical chiral Floquet phase and Ma-
are exhaustively characterized by a rational fraction, r, jorana fermion description. (b) The topological and trivial
characterizingtheratioofquantumstatesbeingpumped phases of a chain of complex fermions (blue boxes indicate
right versus left across any spatial cut at the edge[23]. fermionsites)canbeviewedastwotopologicallydistinctways
In a pure 1D system there must be an exact balance of to pair (black lines) adjacent Majorana fermions (open cir-
cles). Underanopenboundaryconditionthesetwophaseare
quantum state flow, r =1, so that states cannot pile up
distinguishedbythepresenceorabsenceofunpairededgeMa-
orbedepletedfromtheendsofthesystem(otherwisethe
jorana fermions, but with periodic boundary condition they
quantumdynamicscannotrespectbothunitarityandlo-
are related by a chiral translation of the Majorana fermions
cality). However, the boundary of a 2D system forms
(arrows).
a closed loop and does not face this difficulty, and can
realize any rational index, r[16]. We remark that it is
convenient to convert the “number of quantum states”
order that each carry an irrational amount, log√2, of
into an additive measure of chiral quantum “entropy”
quantum information. We will first construct an ideal-
flow, by taking the logarithm of r, which defines the
ized fixed-point drive with uniform couplings. Weakly
chiral unitary index: ν = logr. A non-zero value of ν
perturbingthisuniformmodelproducesalong-livedpre-
necessarily arises from having a topologically nontrivial
thermal phase upon weak perturbations. Alternatively,
bulkdynamics. Further,ν cannotbealteredbyanylocal
we may add strong disorder to convert it into a stable
edge perturbation, but only via a bulk dynamical phase
MBL phase (see Appendix B).
transition.
InspiredbyKitaev’sconstruction[26],wetakespin-1/2
SolvablemodelswithsuchCFbehaviorwereoriginally
degreesoffreedom,S(cid:126) ,sittingonsitesr ofahoneycomb.
constructed for free fermion systems[8, 15, 24] which re- r
The three distinct types of bonds of the honeycomb are
alize a subset of chiral unitary indices: ν = nlog2 for
labeled by x, y, and z respectively, and we embed the
integer n. CF phases were subsequently shown to be
fermionic degrees of freedom as shown in Fig. 1a. The
stable to interactions[16, 17], and generalized to generic
idealized drive consists of a three-step stroboscopic time
interacting bosonic systems that can realize ν =logr for
evolution obtained by cycling through the Hamiltonians
any rational fraction r[16]. These models share a com-
H(t)= 3h[j(t)] applied in the sequence: j =x,y,z, each
mon structure in which bulk degrees of freedom traverse T
for time T/3, so that the Floquet evolution operator is:
circles in short closed loops and edge particles or spins
are pumped chirally around the sample boundary[25].
U(T)=e−ih[z]e−ih[y]e−ih[x], h[j] = πJ (cid:88) SjSj
Spin model – We begin by constructing a solvable 4 r r(cid:48)
lattice model, based on a stroboscopically driven ver- (cid:104)rr(cid:48)(cid:105)∈j
(2)
sion of Kitaev’s honeycomb spin model[26]. This model
willenablecontrolledinsightintothegeneralstructureof
radical CF phases. Starting from an ordinary lattice of Inthelimitofweakdriving(J 1),U(T)realizesacon-
spin-1/2 degrees of freedom with two states per site, our ventionalstaticphasewithZ2 t(cid:28)opologicalorderfeautring
strategywillbetodynamicallyinduceZ topologicalor- an emergent gapless Majorana fermion[26]. However, we
2
dertherebyliberatingemergentfermionexcitations. The will instead consider the strong driving limit: J =1.
edge of the model will then act as a chiral edge pump To analyze this model, Eq. (2), following[26], we
for non-Abelian (Majorana) defects of this topological rewrite each spin-1/2 in terms of four Majorana fermion
3
operators, c ,bx,y,z , as: using strings of spin operators Sx,y,z as a basis for oper-
{ r r } i
ators, shows that ν =log2, i.e.:
2T
Si =ic bi, (3)
r r r
1
which introduces a local Z2 gauge redundancy gener- ν = 2ν2T =log√2, (5)
ated by (c ,(cid:126)b ) ( 1)(c ,(cid:126)b ), and must be subjected
r r r r
→ − confirming that this model indeed realizes a radical CF
to the gauge-neutral sector via the on-site constraints
( iSxSySz) = (c bxbybz) = 1 to faithfully describe phase.
− r r r r r r r
the spin-1/2 Hilbert space. We can draw the Majorana Bulk properties – To characterize the bulk dynam-
fermion degrees of freedom such that c resides on the ics, we can pair the bulk Majoranas on the left and right
r
honeycomb sites, and bi reside onthe links oftype i (see sides of each hexagonal plaquette into complex fermion
r
Fatiogr.s1ain)t.oItZ2isgcaoungveenliiennktvtaoripaabilresthσer,(cid:126)brr(cid:48) M=aijbojrrbajrn(cid:48),awohpeerre- iosrbaitgaalus,geψPstr=ing12(ccoPnLn+ecitWin(cid:120)PgctPhRe)s,itwehsePreR/WL(cid:120)Pvia≡aσc(cid:46)oσu←ntσe(cid:45)r-
j x,y,z according to the type of link r,r(cid:48) , and clockwiseloopoverthetopofplaquetteP (dashedcurved
wh∈ere{we ta}ke an arbitrary fixed orientation o(cid:104)f of r(cid:105) r(cid:48) arrow in Fig. 1a). The plaquette fermion orbitals can be
→
on each type of bond. either occupied or empty, corresponding respectively to
Each factor of e−ih[j] drives the hopping of Majorana local fermion parity: P =( 1)ψP†ψP = 1.
fermions: eih[j]cre−ih[j] = cr+eˆjσr,r+eˆj where eˆj is the After one FloquetPperiod,−PP → FP±PP due to the
oriented unit vector along the type-j bonds, and leaves Aharonov-Bohmphasementionedabove. Inotherwords,
the gauge links σr,r(cid:48) unaffected. the plaquette Fermion parity is conserved on gauge-flux-
The gauge link variables σrr(cid:48) are invariant under the free plaquettes (FP = +1), and is flipped on plaquettes
Floquetevolution,whichwecanexpressasaconservation with a flux ( P = 1).
F −
of gauge flux: To physically interpret this result, first note that the
fact that bulk gauge fluxes have no quantum fluctua-
(cid:89)
P = σr,r(cid:48), (4) tions, which indicates the bulk Floquet operator has a
F dynamical Z topological order, with three topologically
(cid:104)rr(cid:48)(cid:105)∈∂P 2
distincttypesofanyonexcitations: afermionψ( = 1,
P
P −
through each hexagonal plaquette, P. =+1), a bosonic flux m ( = 1, =+1), and their
P P P
F P − F
In the bulk, the c Majorana fermions are driven in bosonic bound state: e = m ψ, each having mutual
×
small counter-clockwise loops, encircling their respective statistics ( 1) with the others. The Floquet evolution
−
plaquettes after two driving periods (Fig. 1a) and ac- conserves the number of ψ particles on each plaquette,
cumulates a Z -valued Aharonov-Bohm phase along n , and the total number gauge fluxes (either e or m
2 P ψ,P
F
the way. While seemingly innocuous on first glance, this particles): n = (n +n ). During each driving
F,p e,P m,P
phase will play a crucial role in enabling the radical CF period, e anyons are transmuted into m anyons and vice
edge physics. versa,aphenomenadubbedFloquetenrichedtopological
order (FET)[18].
Edge chiral index – Incontrasttotheirbulkcounter-
Since the bulk Floquet evolution does not produce ki-
parts, the c Majoranas at the edge are driven in a large
neticmotionofanyofthebulkanyonparticles,itappears
clockwise loop around the entire system boundary, such
naturally amenable to MBL. Indeed, in Appendix B, we
that an odd number of Majorana fermions cross any cut
argue that a disordered version of this Hamiltonian pro-
through the edge during each driving cycle. Since each
ducesastableMBLphase, albeitonethatexhibitstime-
Majorana degree of freedom has “half” the number of
degrees of freedom as a fermion, it corresponds to √2 translation symmetry breaking[6] due to the continual
period-2T flip-flopping of e and m particles[18]. Having
quantum states, and hence we expect a radical chiral
unitary index ν =log√2. solved the bulk dynamics of this model, we turn to its
edge state properties, and examine the correspondence
Wecanconfirmthisexpectationbythefollowingtrick:
between bulk-FET order and edge radical CF order.
instead of computing the index ν for U(T) directly, we
can consider the time evolution for two periods, U(2T), Bulk-boundary correspondence – We saw that the
whose edge turns out to be governed by a rational chiral radical chiral Floquet nature of the edge was accompa-
unitaryindexν =2ν. Theindex,ν canbecomputed niedbyFETorderinthebulk. Next,wearguethatthese
2T 2T
by the algebraic method of Refs. [16, 23]. The quantum two phenomena are inextricably linked.
informationpumpedacrossacutintheedgeduringeach Tobuildsomeintuition, wefirstrecallthatthee m
↔
period is quantified by taking a basis of observables on exchanging FET order can be viewed as arising from a
one side of a cut in the edge, evolving them forward in dynamical pumping of loops of 1D topological chains
time, and evaluating its overlap with observables on the of the ψ-fermions onto the systems boundary during
othersideofthecut. Thechiraltopologicalindexisthen each Floquet period[18], which toggles the 1D chain of
thelogarithmoftheratioofthequantuminformationbe- fermions at the edge between the topological and trivial
ing pumped right to that being pumped left per period. phases. Since the fermion parity of the 1D topological
A relatively straightforward computation (Appendix A), fermion chain is flipped by insertion of a π-flux[27], this
4
pumping adds a fermion to bulk gauge fluxes, and in- viewed as the ends of topological chains of the emergent
terchanges e and m particles. By formally decomposing fermionic quasi-particles, (or equivalently as “twist” de-
each complex fermion degree of freedom at the edge into fectsthatexchangeeandmparticles[30,31]),whichhave
a pair of Majorana fermions, one can see that the chi- irrational quantum dimension d=√2.
ral Majorana translation at the edge of the radical CF We can readily generalize the spin-1/2 construction
phase toggles the 1D topological invariant of the edge to obtain other radical CF phases characterized by chi-
(Fig. 1b). In this picture, the chiral translation modi- ral edge pumping of defects of more general topologi-
fiesthepairingpatternsoftheMajoranafermions,which cal orders. For simplicity, and to retain the possibility
amounts to changing the topological phase of this 1D of MBL[32], we will restrict ourselves to Floquet drives
complex fermion chain[27]. More generally, this picture with Abelian bulk topological order and vanishing Hall
immediately shows that any odd number of Majorana conductance[33]. A simple extension that captures the
translations at the edge will produce the accompanying generalstructureofsuchAbelianradicalCFphasesisto
FETorder,whereasevennumberoftranslationswillnot. consideradrivenZ topologicalorderforgeneralinteger
N
This suggests that the radical chiral edge index always N, whoseexcitationsconsistofbosonicZ gaugefluxes,
N
appears with bulk FET order. m, (satisfying mN = 1), and anyonic gauge charge, ψ,
Tofurthersupportthisconjecture, wecanworkinthe withfractionalstatisticsθ =e2πi/N,andvariousbound
ψ
limit of well-localized gauge fluxes, so that the above states such as e = m ψ−1. Following the footsteps of
problem reduces to that of non-interacting Majorana the Z example above×, we can fractionalize the ψ par-
2
fermions in a static gauge-flux background. In this non- ticles into pairs of parafermionic defects with quantum
interacting limit, we can directly show that the radical dimension√N (whichcanbeviewedasedgestatesof1D
part of the chiral edge invariant is related to the bulk topological chains of ψ particles, generalizing the Majo-
FET order as (see Appendix C): rana case for N =2, or as twist defects that interchange
e and m particles). One can consider a regular lattice
e2πiνedge/log2 = FET (6) of these parafermionic defects, subjected to a Floquet
I
drive that pumps them around closed loops in the bulk
where FET is the Z2-valued bulk FET invariant defined and around large open chiral orbits at the boundary, re-
I
as 1 in the FET phase and +1 in the trivial phase. We sulting in a radical chiral edge index ν = log√N. An
−
note that, this relation can be rigorously established be- explicit solvable lattice model with such properties can
yond the free fermion limit by generalizing the rigorous be readily obtained by a straightforward generalization
machinery for chiral unitary invariants[16, 23] to incor- ofthedrivenhoneycombspinmodelwithspins-N−1 (see
2
porate fermionic super-algebras[28]. Appendix D).
TheconcurrentappearanceofFETorderalsoexplains As for the Z case, the action of translating edge
2
how this model can exhibit irrational values of the chi- parafermions by one (or more generally an odd num-
ral edge index. So long as we may factorize the Flo- ber of) sites interchanges the trivial and topological
quet evolution into commuting bulk and edge pieces, the parafermion phases on the 1D boundary (see Fig. 1b),
chiral unitary index will be rational [16, 23].[29] How- and hence produces bulk FET order characterized by a
ever, this decomposition fails in a radical CF phase pre- dynamical interchanging of e m particles. The bulk
ciselyduetothepresenceoftheFETorder. Specifically, FET nature again causes a fa↔ilure of the Floquet evo-
in the sector with an odd number Z2 gauge fluxes in lution to factorize into commuting bulk and edge parts,
the bulk, the Floquet evolution transfers an odd number since in the sector with j gauge fluxes in the bulk, j
of fermions from bulk to boundary, such that the bulk anyonic ψ particles are transferred from bulk to edge.
and boundary factors in U(T) would be anti-commuting The anyonic character of the bulk action of U(T) al-
fermionic operators. This failure to factorize into com- lows the system to elude the restriction to rational edge
mutingbulkandedgepiecesexposesaloopholeinthera- indices[16,23],sincetheevolutiondoesnotfactorizeinto
tionalclassification[16],andallowsforradicalchiraledge commuting bulk and edge pieces, but rather into a sum
invariants. over bulk-gauge flux sectors, with non-commuting bulk
Parafermionic generalizations – In the driven Z - and edge pieces:
2
topologically ordered example above, we found that, de-
(cid:16) (cid:17)
spitethatthesystemhasonlyinteger-dimensionAbelian U(T)=V N Ψ(j) Ψ(j) V† (7)
⊕j=0 bulk edge
anyon excitations, it exhibited a chiral edge pumping of
effectively non-Abelian objects with irrational quantum
dimension√2. Theresolutiontothisapparentcontradic- where Ψ(ejd)ge/bulk are non-commuting anyonic operators
tion is this: since we are explicitly driving the system in with topological spin e±2πj/N respectively, j denotes the
a time-dependent fashion, energy is not conserved, and bulk gauge flux, and V = e−iH is a local unitary trans-
we may pump certain confined defects of the Abelian formation generated by local Hamiltonian (i.e. a local
H
topological order around the edge, without these defects basis transformation). Again, we notice that time evo-
appearing as deconfined bulk quasi-particles. Namely, lution by two periods returns all bulk anyons to their
the Majorana fermions in the above example can be originalstate,sothatU(2T)canbewrittenasaproduct
5
ofbosonicbulkandedgeterms,whicharehencegoverned σ P , UσU† P ,i.e.underconjugationbyU ‘Pauli
n n
∀ ∈ ∈
by a rational chiral index[16]. products’ remain as ‘Pauli products’. Up to an irrele-
vant overall U(1) phase, a Clifford operation is uniquely
Discussion – We have so far considered a system with
determined by its action on the Pauli group.
bosonic (spin) degrees of freedom, where the emergence
To connect with the problem at hand, first we note
of a radical chiral edge requires Majorana fermion de-
thatallthetwo-qubitgatesinvolvedinU(T)areClifford
fects arising from emergent fermion degrees of freedom.
operations. For example:
In fermionic systems, such Majorana defects are already
present, and a radical CF phase with ν = log√2 can i
be obtained without any accompanying bulk topolog- e−iπX1X2/4Z2eiπX1X2/4 = X1[Z2,X2]= X1Y2,
2 −
ical order[16, 28]. An important caveat is that, this (A1)
phase requires the breaking of fermion-number conser-
vation through pair condensation of the fermions. This andsimilarlyforallothercombinationsofgatesandPauli
paircondensationpreventsMBL[32],butmayberealized operators. This implies U(2T) is a Clifford circuit, and
as a prethermal phenomena[20]. by the previous discussion it is specified by its action on
While we have focused on the case of Abelian bulk the Pauli operators. The action of U(2T) on single-site
topological order, with an eye towards leveraging MBL operators σr Xr,Yr,Zr are represented graphically
∈ { }
to stabilize the system against bulk heating, one could in Fig. 2. Together with the symmetries of the problem,
also consider metastable chiral Floquet phases arising in this completely specifies U(2T) as a Clifford circuit.
systemswithnon-Abelianbulktopologicalorderinapre-
thermal regime[19, 20]. A direct anyonic generalization
of the bosonic SWAP model of Ref. [16] is obtained by
taking a square lattice of non-Abelian particles, and re-
placingtheSWAPtermsbypair-wisebraidings,resulting
in a chiral translation of non-Abelian anyons at the sys-
temboundary. Intuitively,suchconstructiongivesriseto
aphasewithchiralunitaryindexν =logd,wheredisthe
quantum dimension of the anyon in question. However,
we leave a more systematic study of such non-Abelian
generalizations of CF phases for future work.
Acknowledgements – LF is supported by NSF DMR-
1519579 and by Sloan FG-2015- 65244. AV acknowl- ( 1) ( 1)
− −
edges support from a Simons Investigator Award and
AFOSR MURI grant FA9550-14-1-0035. This research
was supported in part by the Kavli Institute of Theoret-
ical Physics and the National Science Foundation under
Grant No. NSF PHY11-25915.
Appendix A: Computation of the chiral index for
the spin-1/2 honeycomb model
In this appendix, we solve U(2T) exactly by recogniz-
ing a connection of the model to the stabilizer formalism
in quantum computing. We will then establish that the FIG. 2. Evolution of Pauli operators. The three Pauli
operators associated to each site (filled or open circle) are
chiralunitaryindexν ofU(2T),whichiswell-definedde-
representedgraphicallyasacolored, thickenedlinealongthe
spite the MBL intrinsic topological order in the bulk, is
corresponding bond. The same color scheme as in Fig. 1 is
ν[U(2T)]=log2.
used,wherered,greenandbluerespectivelyrepresentX ,Y
r r
and Z . The triangles indicate the positions of the original
r
sites, and the arrows indicate time evolution by U(T).
1. Clifford circuits
We begin by reviewing the pertinent preliminaries of
the stabilizer formalism. Consider a quantum system of
n qubits (spin-1/2’s) labeled by r = 1,...,n. For each 2. Factorization of U(2T) into bulk and edge pieces
qubit we have the Pauli operators X , Y and Z , and
r r r
we consider the Pauli group of n qubits P ασ As discussed in the main text, U(2T) admits a com-
n 1
≡ { ⊗
σ σ : α 1,i, 1, i ,σ 1,X ,Y ,Z . plete set of local integrals of motion , , where
2 n r r r r P P
⊗···⊗ ∈{ − − } ∈{ }} {F P }
We say a unitary operator U is a Clifford operation if P runs over all plaquettes in the system. This suggests
6
U(2T) factorizes into a mutually commuting product of From the discussion in Ref. [16], when the system is
local unitary operations. Here, we demonstrate this by put on a geometry with OBC one can write U(2T) =
writing down a closed-form expression of U(2T), which Y U with exponential accuracy, where Y is a
edge bulk edge
then allows us to compute its chiral edge invariant via quasi-1Dunitaryactingnontriviallyonlyneartheedges.
the GNVW formalism[16, 23]. Note that this procedure is unaffected by the fact that
Since U(2T) conserves , , we can construct it U(2T) features intrinsic topological order in the bulk,
P P
{F P }
from the projection operator onto the plaquette flux: and so the chiral unitary index of Y is well-defined
edge
(1 )/2. Recall that U(2T) applies a phase and remains as a diagnostic of the chiral nature of the
P P
V ≡ −F
to the Majorana operators c . Together with the model. Inaddition, thecomputedindexisstableagainst
P r
F
translation invariance of Eq. (2), this property suggests: small perturbation that maintains the MBL nature of
the bulk – and in the present case such robustness can
U(2T)=? G (cid:89)(cid:0)(1 )+eiφ (cid:1), beachievedbyappendingtothedrivingprotocolafourth
φ P P P (A2)
≡ −V V P
P disordering step. (See also Appendix B for a elaborated
discussion on disordering.)
where eiφ is a physical U(1) phase that is to-be-
determined,andtheequalityholdsuptoanunimportant
overall phase.
ToverifyEq.(A2),wecomputetheevolutionofaspin 3. Chiral unitary index of U(2T)
operator σ under conjugation by G . As and
r φ P P
F P
are also product of Pauli operators, we have σ =
P r The chiral unitary index ν of U(2T) can be computed
F
ζ( ,σ )σ and σ = ζ( ,σ )σ , where ζ =
P r r P P r P r r P following the discussion in Ref. [16], where the edge uni-
F F P P P
1. Therefore, given σ and a particular plaquette P
r 0 tary Y is exposed by ‘undoing’ the bulk dynamics
± edge
thereareonlyfourpossibleresults,astabulatedinTable
using the truncation of the periodic-boundary-condition
I.
evolution to the disk geometry. (For notational clarity
we drop the subscript ‘edge’ on Y henceforth.) As a mi-
nor technical remark, we note that the truncation of G
+
TABLEI.Transformationofaspinoperatorσ underasingle-
plaquette factor, labeled by P , in G =(cid:81) Gr . and G− do not agree, as they differ by the global flux
0 φ P φ,P operator on the disk. However, this mismatch, being a
ζ(FP0,σr) ζ(PP0,σr) Gφ,P0σrG†φ,P0 Pauliproductalongtheboundary,ismanifestlyalocally-
+ + σr generated unitary, and therefore ν remains well-defined.
+ − F σ
P0 r In the following, Y is defined using G+.
− + exp(−iφF )P σ
P0 P0 r Toefficientlyevaluate ν(Y), we first tailorthe original
− − −exp(−iφF )F P σ
P0 P0 P0 r index formula in Refs. [16 and 23] into a form optimized
for a Clifford circuit. Recall the overlap η of two local
operator algebras and is defined as
A B
Computing the evolution of X , one sees that G could
r φ
(cid:118)
bnveoertiiedfyernetsthpifiaetecdtbivwoetilhtyhGbUy±(2GrTe+)praoonnddlyucGief−φt.h=eFu0crotorhrreeπrcm,twoehrveioc,lhuotnwieoencdaoen-f η(A,B)≡ √ppΛapb(cid:117)(cid:117)(cid:117)(cid:116)(cid:88)p2a (cid:88)p2b (cid:12)(cid:12)(cid:12)TrΛ(cid:16)eaµ†ebν(cid:17)(cid:12)(cid:12)(cid:12)2, (A4)
µ=1ν=1
all Pauli operators in Fig. 2, and from the discussion in
the previous subsection the guess in Eq. (A2) is verified
–exceptthatwehavefoundtwo solutionsevenwhenthe where µ (ν) indexes a complete set of basis for ( ).
A B
Clifford circuit is uniquely determined up to the global To take advantage of the Clifford structure, we choose
phase. In addition, one can check that the π-rotation a standard basis for an interval with l sites labeled by
aboutthecenterofahexagon,asymmetryofthemodel, the multi-index µ ≡ (µ1,...,µl), defined through σµL ≡
ceaxnchbaengaessoGlu+tio↔n.GT−h,esaendpasroadaopxpeasreanretlryesnoelivtehderboyfntohteimc- Pσµa1ul⊗i mσµa2tr⊗ice·s··in⊗tσhµels,twanhdearerdµcio∈nv{en0,ti1o,n2.,3} labels the
ing the following: The chiral unitary index is then defined as
(cid:89) (cid:89)
G+G†− = ((1−VP)−VP)= FP =1, (A3) ν(Y) logη(Y(AL),AR), (A5)
P P ≡ η( ,Y( ))
L R
A A
wherethelastequalityholdsforperiodicboundarycondi-
tion. This establishes G+ =G−, and hence we conclude where AL and AR respectively denote the operator alge-
U(2T) = G . Note that G is a product of mutually bras (with a sufficiently large size) on the left and right
± ±
commuting local unitaries, and therefore U(2T) mani- of a specified spatial cut, and Y( ) YeY† : e
A ≡{ ∈A}
festly factorizes into bulk commuting and edge pieces, is the transformed algebra.
and that its edge is characterized by a rational chiral As σ is traceless for µ = 1,2,3, only terms with
µi i
unitary invariant. σa = σb can contribute in the trace in Eq. (A4). In
7
(a) Edge addition, as Y is a Clifford circuit, generally one finds
(cid:32) (cid:33) (cid:32) (cid:33)  
(cid:79) (cid:79) (cid:79)
Y σµi Y† =± σµ(cid:48)i ⊗ σνj, (A6)
i∈L i∈L j∈R
(cid:78)
and we say σ is ‘transported across the cut’ if
Bulk µ(cid:48) = 0 i, i.e.i∈(cid:0)LYσµLiY†(cid:1) = 1 (this includes, in par-
i ∀ µ |L
(b) ticular, the identity). These are the only operators that
can contribute in η(Y Y†, ), and each such term
L R
A A
contribute with the same weight as the identity. There-
fore the index formula for a Clifford circuit is simply a
counting formula:
(cid:115) (cid:0) (cid:1)
σL : YσLY† =1
Yedge ν(Y)Cli=ffordlog |{ µ (cid:0) µ (cid:1)|L }|. (A7)
σR : YσRY† =1
|{ ν µ |R }|
The computation is detailed in Fig. 3, which shows
that ν[U(2T)] = log2 – the minimal possible value in
a spin-1/2 system. As discussed in the main text, this
implies ν[U(T)] = 1ν[U(2T)] = log√2, which falls out-
2
side of the original GNVW classification. Such a radical
index is allowed because, unlike U(2T), U(T) cannot be
factorized into commuting bulk and edge pieces due to
(c)
the bulk FET order.
Appendix B: Radical CF phase at strong disorder
Y
edge
In the main text we have examined a zero-correlation
length “fixed point” honeycomb spin-1/2 model for a
radical CF phase with no disorder. We can attempt
to extend this into a stable MBL phase by adding a
fourth driving step with strongly random disorder cou-
pling to the local conserved quantities of the clean driv-
ing steps in Eq. (2), so that U(T) becomes U˜(T) =
FIG. 3. Chiral unitary index ν of U(2T). (a) The e−ihdise−ih[z]e−ih[y]e−ih[x], with:
index is computed for the indicated edge, where the boxed
(cid:88) (cid:88)
region (of depth being one lattice constant) corresponds to h = µ n (B1)
dis a,P a,P
where the edge unitary Y acts nontrivially. Note that we −
edge P a=e,m,ψ
have rotated the lattice by 90◦ relative to Fig. 1. (b) As
the chosen edge retains lattice translation invariance along Here, µ is a random potential for an anyon excitation
a,P
the parallel direction, Y (also a Clifford circuit) is fully
edge of type a on plaquette P.
specifiedbycomputingtheevolutionofthesixPaulioperators
Thelocalplaquettefermionnumber,n ,andtheto-
associated with the two inequivalent sites in a unit cell. The ψ,P
tal gauge flux n +n are conserved by the clean
PaulioperatorsarerepresentedinthesamewayasFig.2,and e,P m,P
partofthedrive, U(T). However, thedifferencebetween
we do not keep track of the global phase as it does not enter
the index computation. The vertical dashed line indicates a the number of e and m particles is flipped by U(T) due
fixedspatialcut. (c)Usingtheevolutionin(b),oneseesthat to the FET order. Despite that hdis does not fully com-
exactlyfourPaulioperatorsare‘transported’fromtheleftto mute with U(T), we can still readily write down exact
therightofthecut,andonlyon√e(theidentity)fromrightto eigenstates of the disordered drive, U˜(T).
left. This gives ν[U(2T)]=log 4=log2.
Denoteafixedconfigurationofanyonexcitationsas: ,
C
anditsenergy,andtherelatedconfigurationobtainedby
interchangingalleandmparticlesin as (cid:48). Theenergy
differencebetween and (cid:48) withrespCecttCoh is: δE =
dis
(cid:80) C C
(µ µ )n ( ), Denoting the the number of
P e,P − m,P e,P C
anyons of type a on plaquette P in configuration as
C
n ( ), wecanwritedowntheenergyofconfigurations
a,P
C
8
and (cid:48) with respect to the disorder Hamiltonian h : beoccupiedorempty. TheperturbationV inducesquan-
dis
C C
tumfluctuationsthatmixthesedegeneratestates,which
(cid:88)
E = µ n ( ) E ( )+∆E( ) can be viewed as virtual anyon particle-hole pairs being
C a,P a,P 0
− C ≡ C C
a,P created out of the “vacuum” and either annihilating.
(cid:88)
EC(cid:48) = µa,Pna,P( ) E0( ) ∆E( ) Toobtainacontrolleddescription,wewillassumethat
− C ≡ C − C
there is a low density of e/m particles in , typically
a,P
C
(cid:88) µe,P +µm,P separated by distance r, which is much greater than the
E0(C)=− µψ,Pnψ,P(C)+ 2 ne,P(C) strength of quantum fluctuations, Γ0 compared to the
P RMSvariationsofµ ,whichdefinesalocalizationlength
ψ
∆E = (cid:88)µe,P −µm,Pne,P( ) (B2) scale: ξ ≈1/log(∆µpsi/Γ0).
− 2 C
P There are two distinct types of these virtual processes
that we will be interested in: First, virtually excited
From this, we can readily identify a pair of Floquet
fermions land on an e or m particle toggling it to an
eigenstates of U˜(T):
m or e particle (such processes must occur in pairs to
1 (cid:16) (cid:17) stay within the degenerate manifold of states associated
ψ± = ei∆E/2 e−i∆E/2 (cid:48) (B3) with ). With this process alone, we can model the sys-
| (cid:105) √2 |C(cid:105)± |C (cid:105) C
tem as a free fermion system, with a lattice of fermion
which have quasi-energies (cid:15) = E ( ), (cid:15) = E + π, sites corresponding to either e or m particles, which we
+ 0 − 0
which differ exactly by π. This formCof eigenstates are will label by sites i, governed by the free fermion Hamil-
exactlythoseofaFloquettime-crystalwithspontaneous tonian Hψ ≈ (cid:80)ijΓijψiψj +h.c., where Γij ≈ Γ0e−rij/ξ
perioddoubling[6],asasysteminitiallypreparedinstate aregenericallyexponentiallydecayinginthedistancebe-
will oscillate between and (cid:48) from one period to tween i and j, and ψi destroys a fermion on site i.
|C(cid:105) |C(cid:105) |C (cid:105)
the next. The second type of virtual processes of interest are
However,thisisnottheendofthestory,becausethere those in which a pair of virtually excited e parti-
are actually an extensive number of other pairs of eigen- cles (or equivalent pair of virtual m particles) encir-
states that also have quasi-energies (E0,E0+π), which cles a pair of fermion “sites”, which gives a different
can be obtained simply by flipping any number of e par- topological phase depending on whether or not both
ticlesin withm-particles,resultinginanoveralldegen- of the sites have the same occupation number. This
C
eracy: corresponds to an interaction term between fermions
(cid:16) (cid:17)(cid:16) (cid:17)
D( )=2(cid:80)P(ne,P(C)+nm,P(C)−1) (B4) Hint ≈(cid:80)ijVij ψi†ψi− 12 ψi†ψi− 12 +..., withVij ≈
C
Γ0e−2rij/ξ, and where the +... indicates the contribu-
ofstateswithquasienergyE0 andE0+π respectively,as tions from virtual fluctuations that encircle higher num-
if the e and m particles have been effectively promoted bersoffermionsites,whicharesuppressedbyexponential
to quantum dimension 2 objects[18]. distance factors compared to the leading term.
This degeneracy will be resolved by quantum fluctua-
The problem of solving for the excited eigenstates of
tionsuponmovingawayfromthezero-correlationlength
H +H is complicated, but has been studied exten-
limit, by even an infinitesimal amount. However, due to ψ int
sively in analogous 1D models [34], and we may draw
the FET structure, the degenerate states cannot be split
lessons from this previous work. Namely, in 1D it was
in such a way that preserves both their localized proper-
shownviaareal-spacerenormalizationgroup(RG)treat-
ties and time-translation symmetry[18, 32].
ment,that,atstrongdisorder,theinteractiontermswere
At strong disorder, a natural outcome is for quan-
preservedundertheRGflow,whereasthehoppingterms
tum fluctuations to spontaneously select time-crystalline
flowed to zero. In the interaction dominated regime, the
MBL states of the form shown in Eq. (B3). To proceed,
system naturally breaks the particle-hole symmetry, and
letusconsidertheevolutionfortwoperiods,U(2T)which
forms a particle-hole asymmetric, fully localized state.
can be written in the absence of an edge, as evolution
An essentially identical strong disorder RG-based argu-
under some effective topologically ordered Hamiltonian
mentin2Dstronglysuggeststhatthesystemwillflowto
H , with a symmetry between e and m particles, which
eff
the interaction dominated regime, even though the pair-
derives from the dynamical permutation of e and m par-
tunneling amplitudes Γ are typically much larger than
ticlesinU(T)[18]. Letusconsidermovingawayfromthe ij
the interaction strengths V to begin with.
zero-correlation length limit with a generic but weak T- ij
periodic perturbation, which corresponds to an e m In the present context, the spontaneous particle-hole
↔
symmetry preserving perturbation to H H +V. symmetry broken state corresponds to an MBL phase in
eff eff
→
Starting from a localized anyon configuration , we can which the dynamically e m symmetry of the orig-
C ↔
restrict our attention to the degenerate space of anyon inal model is broken, i.e. at strong disorder we ex-
configurations with the same quasi-energy (modulo π) pect an MBL time-crystal with eigenstates close (up to
as , which we can model as a fermionic Hilbert space finite-depth local unitary transformation) to the form in
C
whereeache/mparticlesisafermionsitethatcaneither Eq. (B3).
9
Appendix C: Connection between edge chirality and 1. Chiral edge invariant
bulk FET order
To gain some intuition, we first study the special case
In this section, we show that the minimal radical CF with translation invariance. In this setup we may com-
phase is tied to the bulk FET order in the limit of static pute the number of chiral Majorana edge modes via the
Z gauge fluxes and non-interacting emergent Majorana momentum space winding number (following a straight-
2
fermions. Inthislimit,eachgauge-fluxsectorcanviewed forwardgeneralizationoftheresultsof[15], toMajorana
asafreeproblemwhereaπ-fluxbindsorunbindsacom- fermions instead of complex ones):
plex fermion in every Floquet period. This allows us to
view the FET order, characterized by the dynamical ex- C tran=s.inv.(cid:90) dk tr(cid:16)O˜−1(k)i∂ O˜(k)(cid:17) (C4)
k
changeofmande=m ψ particles,asagaugedversion 2π
×
of a Floquet SPT order of the emergent ψ-fermions.
Wewillfirstderiveexpressionsforthenumberofchiral where O˜α,β(k)=(cid:82) dxeikxOx,α;0,β, and the tr is over the
Majorana edge modes, C Z, related to the CF edge flavor indices α,β.
invariant by: ∈ Sinceweareinterestedindisorderedsystems,wewould
like to reformulate this invariant in a way that does not
rely on momentum conservation. A useful formal tool is
ν =Clog√2. (C1)
CF to replace the integral over momenta in Eq. (C4) by an
adiabatic flow under the insertion of “flux”. Though the
Next we will derive an index for the FET order, FET, fermion charge is not conserved in the present problem
I
which takes value 1 when the e and m anyons are dy- with Majorana fermions, we can still formally define a
−
namically exchanged, and is +1 otherwise. Finally, we version of O with flux θ ( π,π] threaded through the
show that the parity of C is equal to FET: bond between x=0 and∈x=−1 along the edge:
I

(−1)C =IFET. (C2) Oxx(cid:48)eiθ for − L2x <x(cid:48) ≤0<x< L2x
To derive this relation, consider the time evolution on (Oθ)xx(cid:48) =OOxxxx(cid:48)(cid:48)e−iθ footrhe−rwLi2sxe<x≤0<x(cid:48) < L2x ,
a cylinder with periodic boundary condition in the x di- (C5)
rection but open boundary condition in y, and with all
dimensionsmuchlargerthantheLieb-Robinsonlengthof where we have suppressed the flavor index, and L de-
x
theFloquetevolution. Intheabsenceofinteractionsand notes the circumference of the cylinder edge.
gauge-flux dynamics, the Majorana operator at position A minor, but formally necessary technical detail is
r evolves as: thatsometruncationschemeisrequiredtomaketheflux
insertion compatible with periodic boundary-conditions.
(cid:88)
U(T)crU†(T)= Mr,r(cid:48)cr(cid:48) (C3) While various equivalent methods are possible, here, we
have simply turned off the eiθ phase twist at a dis-
r(cid:48)
tance L /2 from the origin. However, the effects of
x
this finite-size truncation can be safely ignored in large
where M is an orthogonal matrix. Note that det(M) =
+1,asrequiredbythetopologicalconservationoffermion systems. Namely, since Oxx(cid:48) results from finite time
| |
evolution with a local (2D) Hamiltonian, the spatial ex-
parity.
tentmatrixelementsareconstrainedbyaLieb-Robinson
Suppose the Floquet evolution can be made trivial in
thebulkbynon-chiralfinite-depthlocalunitarytransfor- bound,i.e. |Ox,x(cid:48)|fallsoffexponentiallyas∼e−|x−x(cid:48)|/(cid:96)LR
for distances much longer than a Lieb-Robinson length
mation(ashappensforboththeMajoranaCFphaseand
x x(cid:48) >(cid:96) . For similar reasons, O will be exponen-
thefermionFSPTphase). Then,after‘undoing’thebulk | − | LR (cid:12)(cid:12) θ (cid:12)(cid:12)
evolutionthenon-trivialactionofU(T)onlyoccursona tiallyclosetoaunitarymatrix: (cid:12)(cid:12)(cid:12)(cid:12)OθOθ†−I(cid:12)(cid:12)(cid:12)(cid:12)≈e−Lx/(cid:96)LR.
1D strips near the top and bottom of the cylinders, and By introducing the adiabatic flow parameterized by θ,
M can be written as a direct sum of M =O Im O(cid:48). the chiral edge invariant can now be written as [24]:
⊕ ⊕
Here, O actsonlyonastripwithfiniteextentiny atthe
top of the cylinder, I is the identity in the middle, and (cid:90) π dθ (cid:18) ∂ (cid:19)
O(cid:48) acts in a finite y-tmhickness strip at the bottom of the C = 2π tr Oθ†i∂θOθ , (C6)
−π
cylinder.
In the following we will focus on one end of the cylin- wherethetracerunsoverallspatialand(thesuppressed)
der, label the Majorana operators by their x position, flavorindices. OnecanverifythisreproducesEq.(C4)for
and an auxiliary flavor label α. Then, we can consider translation-invariant edges. If O were exactly unitary,
θ
the time-evolution operator restricted to the top of the then C would be a winding number which is precisely
cylinder: Ox,α;x(cid:48),β, which is an orthogonal matrix (i.e. quantized to integer values. For a large but finite Lx,
can have determinant 1, so long as detO =detO(cid:48)). this integer quantization is accurate up to exponentially
±
10
smallcorrectionsofordere−Lx/(cid:96)LR,duetothetruncation 3. Relation between C and IFET
at x= L /2, and becomes exact as L limit.
x x
± →∞
We remark in passing that this invariant can also be Finally,werelatethechiraledgeandbulkFETinvari-
formulateddirectlyinthelimitofaninfinitelylongedge, ants. First observe that, as O is real,
L = , where the flux-threading can then be imple-
x
mented∞by a unitary operator: O =eiθPOe−iθP, where O =O∗ . (C11)
θ θ −θ
P is a projection into the subspace with x>0
This implies
(cid:40)
1 for x>0 (cid:90) π dθ (cid:16) (cid:17) (cid:90) π dθ (cid:16) (cid:17)
Pxx(cid:48) =δxx(cid:48) . (C7) C = tr O†i∂ O = tr O†i∂ O ,
≡ 0 for x 0 2π θ θ θ π θ θ θ
≤ −π 0
(C12)
Putting this form of O into Eq. (C6), one finds
θ
and therefore
C =tr(cid:0)O−1[P,O](cid:1), (C8) =det(cid:0)O O−1(cid:1)
IFET π 0
=exp(tr(logO logO ))
In this form, C is the trace of a difference between two π− 0
projection operators, whose eigenvalues are 0 or 1, and (cid:18) (cid:90) π dθ (cid:19)
=exp iπ i∂ trlogO
thereforeC ispreciselyquantizedtoanintegerthatcan- − π θ θ (C13)
0
not be altered by smooth deformations (local unitary (cid:18) (cid:90) π dθ (cid:16) (cid:17)(cid:19)
transformations of O). Related quantites were identified =exp iπ tr O†i∂ O
− π θ θ θ
in Refs. [23 and 26] as a ‘flow’ index for causal unitary 0
matrices. =e−iπC.
This establishes the claimed bulk-edge correspondence
between the edge chiral unitary invariant and the bulk
2. FET invariant FET invariant, in the limit of vanishing gauge fluctu-
ations and non-interacting emergent fermions. A more
formal proof that also applies to the general interacting
To diagnose the FET order, we would like to compare
case can be obtained using super-algebra methods[28].
the change in fermion parity with and without a π flux
threadingtheedge. IntheFETphase,Floquetevolution
toggles the edge between topological and trivial states,
Appendix D: Spin-N/2 lattice model of chiral
and hence pumps an opposite amount of fermion parity
parafermion Floquet phase
dependent on the presence or absence of a π flux. In a
non-FETphase,theparitypumpedisindependentofthe
flux. To obtain solvable models of radical CF phases whose
edges implement chiral parafermion translation, we can
More precisely, such parity pumping is captured by
adapt the driving protocol of Eq. (2), to the Z gen-
comparing the determinants of O with and without a π N
eralization of Kitaev’s honeycomb model constructed by
fluxinserted. Toseethis,observethatthefermionparity
operator for the system PF =iNsites(cid:81)rcr evolves as: BbuatrkweisthhliNet-satal.te[3sp5i]n.sWraethaegratinhacnon2s-sidtaetreasphionnseoyncoemacbh,
site. The terms of the driving Hamiltonian are formed
U(T)P U(T)† =(detO)P , (C9)
F F out of the unitary single-spin unitary operators τx,y,z,
r
satisfying (cid:0)τi(cid:1)N = 1, τz = (τxτy)†, and τ’s on different
whichfollowsfromtheantisymmetryofthefermionprod- r r r r
sites commute: [τi,τj] = 0 (r = r(cid:48)), and the operators
uctandtheorthogonalityofO. Here,Nsitesisthenumber r r(cid:48) (cid:54)
satisfy the on-site algebra:
ofMajoranasites,andthephasefactorischosentomake
P2 =1.
F τxτy =e2πi/Nτyτx (D1)
From these considerations, we can readily write the r r r r
FET invariant as a comparison between O and O :
0 π plusidenticalrelationsforcyclicpermutationsof(x,y,z)
indices.
IFET =det(cid:0)OπO0−1(cid:1), (C10) Then, simply replacing Sri →τri in Eq. (2) implements
thedesiredirrationalchiralFloquetphase. Inparticular,
which is 1 in the FET phase, and +1 otherwise. We following[35],wemaydescribetheN-statespinsasquar-
−
noteinpassingthatvariousequivalentformsfor like tets of parafermionic twist defects with quantum dimen-
FET
I
(detO /detO ), or(detO detO ), arepossible. How- sion √N (generalizing the Majorana fermion description
π 0 π 0
·
ever, the above formulation is convenient as it remains for N = 2), which we can embed spatially around the
well defined in the infinite system limit. honeycomb in the c,bx,y,z positions shown in Fig. 1.
{ }