Sensing Floquet-Majorana fermions via heat transfer

4.3 MB·

by  Paolo Molignini

Sensing Floquet-Majorana fermions via heat transfer
Paolo Molignini,1 Evert van Nieuwenburg,1 and R. Chitra1
1Institute for Theoretical Physics, ETH Zu¨rich, 8093 Zurich, Switzerland
(Dated: January 24, 2017)
TimeperiodicmodulationsofthetransversefieldintheclosedXYspin-1 chaingenerateaveryrich
2
dynamicalphasediagram,withahierarchyoftopologicalphasescharacterizedbydifferingnumbers
of Floquet-Majorana modes. We show that this rich phase diagram survives when the system is
coupledtodissipativeendreservoirs. Circumventingtheobstacleofpreparingandmeasuringquasi-
energy configurations endemic to Floquet-Majorana detection schemes, we show that stroboscopic
heattransportandspindensityarerobustobservablestodetectboththedynamicalphasetransitions
and Majorana modes. In particular, we find that the derivative of the heat current, with respect
7 to a control parameter, changes sign at the boundaries separating topological phases with different
1 numbers of Floquet Majorana modes. We present a simple scheme to directly count the number
0 of Floquet-Majorana modes in a phase from the Fourier transform of the local spin density profile.
2 Our results are valid provided the anisotropies are not strong and can be easily implemented in
n quantum engineered systems.
a
J
Introduction — Recent developments in quantum en- to lead to novel sum rules for differential conductance
8
1 gineering [1] offer the remarkable possibility to probe evaluated over all the quasi-energies in driven topologi-
physics in the strongly out-of-equilibrium regime. Par- cal insulators [22, 23]. However, practical realizations of

] ticularlyinterestingfromthisperspectiveareperiodically transport based schemes to evaluate these sum rules are
l
e driven quantum systems also known as Floquet systems. hampered by difficulties in preparing the system in the
- Floquet systems, on the one hand open up fundamental appropriate energy interval [8], garnering a clear knowl-
r
t questionsaboutnon-equilibriumsteadystates[2]andon edge of chemical potential bias [24] and extracting all
s
. the other offer a rich toolbox to explore new dynamical quasi-energies simultaneously [10]. Proposals for clear
t
a phasesofmatter. Afewexamplesofthelatterrangefrom and universal signatures of Majorana modes, both in-
m dynamically induced superfluid-Mott insulating transi- and out-of-equilibrium are thus very desirable.
- tions in optical lattices [3], to coherent destruction of InthisLetter, westudyadrivendissipativespinchain
d
tunneling[4, 5] as well as dynamical many-body phases inwhichahierarchyofFloquetMajoranaexcitationsand
n
of parametrically driven systems which do not have any associatedtopologicalphasetransitionscanbeinducedin
o
c static counterpart [6]. acontrolledmanner. Thepossibilitytoeasilytuneinand
[ Periodic driving also offers the intriguing possibility out of different topological phases makes them ideal sys-

of dynamically generating exotic topological excitations temsformeasuringexoticexcitations. Weconstructstro-
1
v in otherwise topologically trivial systems [7–10]. One of boscopic observables allowing us not only to distinguish
6 the most well known topological excitations are zero en- phaseswithdifferentFMF’sbutalsotocounttheirnum-
0 ergy Majorana modes which, for example, occur as lo- ber. Our work offers a direct generalization of dynami-
2 calized edge modes in static Kitaev models [11]. The cally generated topology to more realistic open systems
5 non-abelianbraidingstatisticsofMajoranamodesmakes andobviatestheneedforspecialinitialstatepreparations
0
thempromisingcandidatesfortopologicalquantumcom- and fine tuning. In particular, stroboscopic heat/energy
.
1 putation [12, 13]. However, direct observations of these current in this dissipative setup provides very clear sig-
0 Majorana modes in quantum wires are challenging be- natures of the cascade of topological phase transitions
7 cause of their intrinsic weak charge coupling. Indirect between phases with differing numbers of FMF’s. More-
1
observations based on spectroscopy or interferometric over, we show that the number of FMF’s can be directly
:
v measurements in proximitized semiconductor nanowire obtained from the stroboscopic spin density.
i devices [14–18] or hybrid superconducting-quantum in-
X
terference devices [19] are still debated as the signals
r
a are hard to distinguish from the contributions of other
processes like Andreev bound states and the Kondo ef-
fect [18, 20].
Recently,itwasshownthatahierarchyofFloquetMa-
joranafermions(FMF)couldbegeneratedinanisolated
spin-1 (fermionic) chain subject to a periodically vary-
2
ing magnetic field (chemical potential) [13, 21]. These
exotic emergent and out-of-equilibrium modes are dy-
namically generated in the Floquet quasi-energy spec-
trum but retain familiar topological characteristics, like Figure 1: Schematic illustration of the driven spin-1 chain
2
winding numbers. Toy models of FMF’s were shown coupled to dissipative baths at the ends.
2
2(γcosk−µ )
+sin(2µ ) 0 sin(TE ) (4)
1 E k,0
Model — We consider a spin-1 chain described by the k,0
2
XY model in a periodically driven transverse magnetic (cid:113)
with E = 2 (γcos(k)−µ )2+∆2sin2(k). These re-
field. The system’s Hamiltonian reads k,0 0
sults can easily be generalized to the case of multi-step
H(t)=−(cid:88)(cid:2)J σxσx +J σyσy +µ(t)σz(cid:3), (1) driving.
x n n+1 y n n+1 n
The eigenvalue equation for θ can be understood as
n k
a counterpart of the equation for Floquet quasi-energies
where σna with a={x,y,z} is the Pauli matrix at site n. (cid:15)α(k). At topological transitions, quasi-energy gap clos-
The exchange couplings are parametrized as Jx = γ−2∆ ings in (cid:15)α(k) translate into the appearance of non-trivial
and J = γ+∆ and µ is the time dependent transverse stationary points in θ . We find that the number p of
y 2 k
magneticfield. ViaaJordan-Wignertransformation[25], stationary points k∗, defined as dθ /dk| = 0 in the
k k=k∗
thespinchainHamiltonian(1)canbemappedontothat interval 0 < k ≤ π directly yields the number of FMF’s.
of a fermionic model describing a p-wave superconduc- Forgivenvaluesof∆andµ ,mappingthenumberofsta-
0
tor [11] or equivalently, can be rewritten in terms of 2N tionarypointsasafunctionof(T,µ )providesacomplex
1
Majorana fermions w phase diagram shown in Fig. 2a) and e) with each phase
i
beingcharacterizedbyitsownnumberofFMF’s. Conse-
N(cid:88)−1(cid:20)γ−∆ γ+∆ (cid:21) quently,differentFMFsectorsarelinkedbyatopological
H(t)=i w w − w w
2 2n 2n+1 2 2n−1 2n+2 phases transition of the Lifshitz kind [28].
n=1 Thesedynamicalphasediagramsclearlyillustratethat
(cid:88)N (cid:88)2N Floquet systems exhibit a rich and varied topology that
+i µ(t)w w =i w A (t)w , (2)
2n−1 2n m mn n has no counterparts in the undriven system. However,
n=1 m,n any study of FMF’s in an experimental context requires
taking into account dissipation and also identifying ac-
where the Majorana operators w satisfy the anti-
i
cessible physical observables. To this end, we couple the
commutation relations {w ,w } = 2δ . In the absence
i j ij
chain to two Markovian baths at the ends (see figure 1).
of driving, the closed XY-model in a field exhibits three
Foraweakcouplingbetweenthechainandthetworeser-
distinct phases, two of which have nontrivial and oppo-
voirs, the system realizes a non-equilibrium steady state
site topology. Specifically, in the topologically nontrivial
phase for |µ| < 1, zero-energy Majoranas appear at the (NESS).Inthisweakcouplinglimit, weexpectthetime-
γ evolutionofthesystem’sdensitymatrixρtobegoverned
ends of the chain [11, 21]. For the case of the trans-
by the master equation in Lindblad form
verse field Ising model, corresponding to γ =1, periodic
modulations of the transverse field were recently shown ρ˙(t)=−i[H(t),ρ]+Dˆ (ρ)+Dˆ (ρ), (5)
L R
to induce a multitude of FMF’s for a wide range of sys-
tem parameters, even when the undriven phase has triv- where H(t) is the periodically driven Hamiltonian of
ial topology [21]. This is analogous to the generation of Eq. 2. The relevant dissipators Dˆ =(cid:80) (2L ρL† −
L µ=1,2 µ µ
Floquettopologicalinsulatorsfromnon-topologicalband (cid:8)L†L ,ρ(cid:9))andDˆ =(cid:80) (2L ρL† −(cid:8)L†L ,ρ(cid:9))de-
insulators [7, 26]. µ µ R µ=3,4 µ µ µ µ
scribe the effect of the coupling to the baths in terms
Typically, the number of generated FMF’s is obtained (cid:113)
of jump-operators L = ΓL (w ±iw ) (left bath)
by a direct evaluation of the topological winding num- 1,2 1,2 1 2
(cid:113)
ber. Here, we show that the generation of FMF’s can and L = ±(−i)N ΓR (w ±iw ) (right bath).
easily be understood via an analysis of the stationary 3,4 1,2 2N−1 2N
points of the underlying Floquet energy dispersion. We The rates ΓL1,,2R completely characterize the effect of the
consider the case of delta-function driving, modeled as bath on the system[29]. This choice of bath imposes a
µ(t)=µ +µ (cid:80) δ(t−nT) where T is the period of net magnetization of the end spins along the z direction
0 1 n∈Z
the drive. The ensuing Floquet time-evolution operator in the absence of any spin-spin interactions. We men-
overoneperiodintheMajoranabasisisgivenbyatime- tion that the directions of this imposed magnetization
(cid:104) (cid:16) (cid:82)T (cid:17)(cid:105) can generally lead to very different scenarios for certain
ordered exponential U(T,0) = T exp −i H(t)dt .
0 observables in the steady state.
By assuming periodic boundary conditions or an infi-
Rewriting the dissipators described above in the Ma-
nitechain,theFloquetoperatorthendecouplesintotwo-
jorana representation[29, 30], we find that equation (5)
dimensional matrices described by the quasi-momentum
for the time evolution of the density matrix can be re-
k [21, 27]:
cast as an equation for the covariance matrix C (t) ≡
ij
U (T,0)=eiµ1σze−i2T[(γcosk−µ0)σz+∆sinkσy]eiµ1σz (3) Tr[wiwjρ(t)]−δij. The covariance matrix C satisfies,
k
(cid:104) (cid:105)
The eigenvalues eiθk of the Floquet operator Uk can be C˙ij(t)=−iTr[wiwj[H(t),ρ(t)]]+Tr wiwjDˆ(ρ) , (6)
compactly written as
where the r.h.s can be evaluated using Wick’s theorem
cosθ =cos(2µ )cos(TE )+ sincebothH andDˆ arequadraticinMajoranafermions.
k 1 k,0
3
Figure2: StationarypointphasediagramoftheFloquetoperator(left-mostcolumn), residualcorrelatorC (secondcolumn
res
fromtheleft)andheatcurrentJ (thirdandfourthcolumnsfromtheleft)fortwodifferentstartingpointsinthe(µ ,∆)plane.
L 0
Figures 2a)-2d) correspond to the quasi-isotropic case in a low static field (∆=0.1, µ =0.1), while 2e)-2g) show the results
0
formoderateanisotropy(∆=0.5,µ =0.1). Thecutsdisplayedin2d)and2h)areindicatedbyverticallinesofcorresponding
0
color in Figs.2d) and 2h) .
Although the full time dependent equation is not easy easily accessible experimental observable. A natural ob-
to solve, the stroboscopic behavior of the covariance ma- servablewouldbestroboscopicspincorrelationfunctions,
trix in the steady state can be obtained using Floquet which can be easily obtained from C . However, these
F
theory. Firstly, in the steady state, the stroboscopic co- are not good trackers of the phase transitions as the as-
variance matrix C = C(0) = C(T) will no longer de- sociated signatures are weak. Charge transport, on the
F
pend on the initial conditions and will fully exhibit the otherhandhascaveatsashighlightedintheintroduction.
periodicity of the underlying drive. Following the treat- Wenowshowthatagoodcandidatetoprobethehier-
ment of Ref [29], the steady state behavior of the covari- archyoftopologicalphasetransitionsandobtainobserv-
ancematrixcanbeshowntobegovernedbythediscrete able signatures related to the number of FMF’s is the
Lyapunov equation [31] heat transport across the chain in the non-equilibrium
steady state (NESS). The heat currents from and to the
Q(T)C −C Q−T(T)=iP(T), (7)
F F reservoirs can be obtained from the first law of thermo-
wherethematricesQandP dependonthenatureofthe dynamicsdU =δQ+δW,whereδQisthechangeinheat
driving[30]. Solving this Lyapunov equation then helps andδW isthechangein work. Therate ofchangeofthe
us obtain various stroboscopic observables as a function internal energy is given in terms of ρ is given by
of C .
F dU (cid:104) (cid:105)
Intheabsenceofanorderparametertotrackthetopo- =Tr H˙(t)ρ(t) +Tr[Hρ˙(t)], (8)
dt
logical phase transitions in our spin chains, a weighted
sum of the covariance matrix called the residual corre- The first term is related to the power of the system,
lator C ∝ (cid:80) |C | has been shown to play while the second corresponds to the change in heat. The
res |j−k|≥N/2 j,k
the role of an effective order parameter which tracks the Lindblad equation for the density matrix (5) leads to
stationary point phase diagram[29]. The structure of the following definition of the heat current: JL,R ≡
(cid:104) (cid:105)
Cresshowsaone-to-onecorrespondencetothestationary Tr DˆL,R(ρ)H [32]. Note that the direction of the cur-
points phase diagram [29]. Figs. 2b) and f) show that
rent is implicitly contained in J and J by defining the
the underlying stationary point phase diagram seen in L R
quantitiesastheflowofheatfromthereservoirsinto the
the closed system survives even in the presence of dissi-
system. Using the Majorana basis and (6), the strobo-
pation, though the phase boundaries are mildly shifted.
scopic heat current of the left reservoir can be simply
Unfortunately, although C indicates the boundaries
res expressed in terms of the covariance matrix
delineating regions with differing FMF’s, it does not in-
dicate the number of FMF’s in each zone nor is it an J =4ΓL[iJ C −iJ C +2iµ C ]+8µ ΓL (9)
L + x 3,2 y 4,1 0 2,1 0 −
4
(cid:20)
J =4ΓR iJ C −iJ C + sidered here, the spin profile away from the edges is dis-
R + x 2N−1,2N−2 y 2N,2N−3 tinctlynon-uniforminallthenontrivialFloquettopolog-
(cid:21) ical phases. This is because the FMF’s are not localized
+2iµ0C2N,2N−1 +8µ0ΓR−. (10) in the spin language. The Fourier transform of the spin
profileisplottedinFig.3forseveralvaluesofthedriving
intensity across the cut at period T = 1.0. We see that
Our results for the stroboscopic heat current J are
L
thepresenceofFMF’sismanifestedbytheappearanceof
showninFig. 2fordifferentdrivingparametersandweak
pronounced peaks which correspond to superpositions of
static field µ . In Figs. 2a) and e), the corresponding
0
the modulations of the spin profile at different k vectors.
Lifshitz points phase diagrams indicating the number of
Note that in Fig. 3, the central peak corresponding to
FMF’sareplotted. Figs. 2b),2f)resp. 2c),2g)showthe
the uniform background has been removed to facilitate
behaviours of C resp. J across the phase diagram.
res L
the visualization of the other peaks.
Clearly, C delineates the different topological phases
res
As the control parameter is varied, these symmetric
eveninthepresenceofdissipation. Reversalsoftheheat
pairs peaks move smoothly either move towards the ori-
current flow in topological phases with nonzero FMF’s
gin or away from it. The green arrows in figure 3 in-
are clearly seen. This is essentially due to driving and
depends on the choice of bath parameters ΓL,R within dicate the direction of the movement as the intensity of
1,2 the driving is increased. Peaks are destroyed or created
a FMF phase. To see specific features of the heat cur-
at zero momentum, indicating that coherent spin mod-
rent at these topological phase boundaries we consider
ulations appear/disappear above a uniform background
the vertical cuts plotted in Figs. 2d) and h). Typically,
signaling the creation or annihilation of FMF’s. The to-
amplitudesoftheheatcurrentdecreasesasthenumberof
tal number of peaks with k (cid:54)= 0, obeys N = 2p where p
FMF’s increases. We find that, at the phase boundaries
is the number of FMF’s in the given phase determined
between two phases with differing non-trivial topology,
by the control parameters. As for the heat current, this
the slope of the heat current, with respect to the tuning
countingofFMFfromthespindensityisrobustforsmall
parameter, changes sign. On the other hand, transitions
anisotropies. At higher anisotropy (∆>0.5) and higher
between a zero and a nonzero FMF phase are tracked by
periods (T > 2.0) the correspondence between localized
either changes in sign or discontinuities in the slope of
peaksandFMF’slosesitsprecisionasnewpeaksappear
the heat current.
at fixed momenta, possibly corresponding to other forms
Away from the transitions, the quasi-energy spectra is
of excited many-body states.
completely gapped and the heat transport is essentially
mediated by FMF’s. The high-frequency oscillations in
theheatcurrentareduetofinitesizeeffectsanddecrease
with increasing N. This change in sign of the slope of
m
teahffneyechtcieuvatetliyncuttrrhareecnkptshtwahsietehpdarireaisgtpyraecomtf.tthoSeitnphcheeactsoheneatnraodcltuipsaavlraaslimigdnetfoeorrf malized power spectru
theheatcurrentisdeterminedbythebathparameters,it nor
is not possible to assign a fixed parity to a phase, rather
hqeevtueeeaerntnst,coluytrhr,oreoednnhdetenaicstuasnmcenunborsetrirteainovstfeceFcoranMtnalyFinin’tsodw.echFehedoatrnhcgechreeasrnatiagngienipvsaebirngainttpyhhw.apCitsahoerniahnsmaeas-- malized power spectrum
giventopologicalphasewithoutaconcomitantchangein nor
the sign of the slope of the current.
Analogousconclusionscanbedrawnfromtheanalysis Figure3: Powerspectrumofthespindensityprofilefordelta-
of the heat current from the right reservoir or the net kick driving with period T =2.0 at increasingly larger inten-
heat flow J +J . It is important to note that the net sityµ . Thesystemwasdrivenoutofaquasi-isotropicregime
L R 1
heat flow is not necessarily zero, since the driving has with ∆ = 0.1 and µ0 = 0.0. The green arrows indicate in
the effect of injecting energy into the system, which can whichdirectionthepeaksshiftuponincreasingµ1. Thetotal
number of peaks is twice the number of FMF’s.
thenpreferentiallyextractordumpexcessheatinoneor
the other reservoir, depending on the physical state of
the chain and the details of the baths. To summarize, We now discuss a plausible physical connection be-
the heat current is a sensitive detector of the topologi- tween heat current and the peaks that appear in the
cal phase transitions for a wide range of static magnetic Fouriertransformofthespindensityprofile. Typicallya
fields, provided the anisotropy ∆≤0.5, whereas the sig- peak in the Fourier transform signals coherent modula-
nals lose their precision for high-anisotropy states. tions of the spin density with a driving-dependent wave-
We now show that the number of FMF’s in any phase length which can carry energy from one end of the chain
can directly be read off from the Fourier transform of to the other, with higher-wavelength modulations car-
the spatial spin profile (cid:104)σz(cid:105). In the driven setup con- rying more energy than lower-frequency ones. However,
i
5
having more peaks does not automatically translate to modes. Such XY spin chains in a transverse field can
increased energy transport as multiple FMF’s can lead be simulated in quantum engineered systems either us-
to destructive interference or standing waves in the spin ing trapped ions[33] or flux qubits[34], where many such
profile modulations, reducing the ability of the system units can be combined to realize potentially long spin
to carry heat. This feature is highlighted for instance at chains. Both systems offer a controllable ways to ap-
T =2.0 and µ in figure 3, where the two Fourier peaks ply periodic magnetic fields. The FMF counting scheme
1
resonate at the same wavelength and simultaneously the can be easily implemented in trapped ions using single
heat current is zero. site fluorescence. The switching of the stroboscopic heat
Conclusions — In conclusion, we have shown that pe- current seen in this system opens up the intriguing pos-
riodic driving can be used to realize interesting topol- sibility of using Floquet Majorana phases to devise both
ogy and Floquet Majorana modes in a dissipative spin quantum heat engines or heat pumps. To establish such
chain. The heat/energy current that flows through the topology driven functionality, more in-depth studies of
chain in its NESS tracks the series of phase transitions the work done during a time period are required, which
generated when Floquet Majorana modes are created or is beyond the scope of the present paper. Future di-
destroyed. Furthermore,thespindensityprofileprovides rections, involve the study of interactions and the role
a simple way of counting the number of Floquet modes played by dephasing of the spins on the robustness of
present in a phase. Direct detection of exotic Majo- theFMF’sdiscussedhereaswellassignaturesspecificto
rana modes is consequently easier, as the ability to tune such dynamically induced topological phase transitions.
through the cascade of phase transitions automatically Acknowledgments This project is supported by SNF,
eliminates the question of distinguishing other modes — Mr. Giulio Anderheggen and the ETH Zu¨rich Founda-
likeAndreevboundstates—whichmimictheMajorana tion.
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