Floquet topological transition by unpolarized light
FFloquet topological transition by unpolarized light
Bhaskar Mukherjee Theoretical Physics Department, Indian Association for the Cultivation of Science, Jadavpur, Kolkata-700032, India. (Dated: August 10, 2018)We study Floquet topological transition in irradiated graphene when the polarization of incidentlight changes randomly with time. We numerically confirm that the noise averaged time evolutionoperator approaches a steady value in the limit of exact Trotter decomposition of the whole periodwhere incident light has different polarization at each interval of the decomposition. This steady limitis found to coincide with time-evolution operator calculated from the noise-averaged Hamiltonian.We observe that at the six corners (Dirac( K ) point) of the hexagonal Brillouin zone of graphenerandom Gaussian noise strongly modifies the phaseband structure induced by circularly polarizedlight whereas in zone-center (Γ point) even a strong noise isn’t able to do the same. This can beunderstood by analyzing the deterministic noise averaged Hamiltonian which has a different Fourierstructure as well as lesser no of symmetries compared to the noise-free one. In 1D systems noise isfound to renormalize the drive amplitude only. PACS numbers:
I. INTRODUCTION
Realizing topological phenomenon in solid state sys-tem has been one of the major topic in condensed mat-ter physics since the discovery of IQHE in 2D semicon-ductor devices . These materials are model system for2D non-interacting electron gas which under the appli-cation of strong magnetic field forms highly gapped Lan-dau levels at low temperature. This results in very pre-cise quantization of Hall conductance and supports ro-bust conducting chiral states at the edges . Later itwas shown that the magnetic field is not necessary andone can also observe such phenomenon in systems de-scribed by tight-binding Hamiltonians . The so called“Haldane Model”describe electrons hopping in a hon-eycomb lattice threaded by periodic magnetic flux withzero net flux. The resulting complex hopping is difficultto implement experimentally and it is only recently thatthe advancement in ultra-cold atomic systems have madesuch experiments possible . To avoid such complicatedimplementation of the Haldane model and thus realizeChern insulating states more easily, a possible alterna-tive way, namely “irradiation of electromagnetic wave ongraphene”, is proposed recently to achieve the essentialgoal of time reversal symmetry breaking.Graphene is a gapless 2D Dirac system which can openup a gap at the Dirac Point under irradiation of circu-larly polarized light . This resulting new state, termedas Floquet topological insulator was found later in manyother systems . It is also detectable by various trans-port signatures . These are steady states of peri-odically driven non-equilibrium systems which re-cently gained tremendous attention because of it’s po-tential to create new phases. These phases can hardlybe found in their equilibrium counterparts. Traditionalbulk-boundary correspondence was extended to Floquettopological systems taking into account the periodicityof the Floquet spectrum . Experimental verificationof such states has already been achieved using both time and angle resolved photoemission spectroscopy(PES) and also in photonic systems .Throughout the last decade a large number of studiesof real time dynamics in closed quantum systems haveextended the notions of universality from equilibriumto non-equilibrium via Kibble-Zurek scaling . Furtherstudies show that the qualitative nature of these scalingscan be completely reversed by introducing noise in thedrive . In these studies the Heisenberg equation of mo-tion picks up a dephasing term due to averaging overdifferent noise realizations which leads to non-unitarydynamics. Recently in equilibrium systems it has beenshown that periodicity in space (i.e the crystal structure)is not necessary to get topological behavior and one canalso see it in amorphous systems . Analogously one canask at this point that what would happen in Floquetsystems if time periodicity of the Hamiltonian is bro-ken due to the presence of noise in the drive. Severalstudies in this direction in models decomposable in freefermions have already revealed that the nature of theasymptotic steady state depends on the type of aperi-odic protocol . Further some analytical studies showthat disorder-averaging can be avoided for a special classof protocols .Influenced by this kind of works we plan to study thefate of the Floquet topological systems when the smoothtime variation of incident electromagnetic wave is bro-ken by the insertion of a random phase in one of thecomponent of vector potential. This kind of noise is al-ways there in a typical experiment if the setup to pro-duce polarized light isn’t calibrated properly. Moreoversuch noise can also be generated artificially using syn-thetic gauge fields. We term this kind of monochromaticwave as unpolarized light in the sense that the associ-ated Lissajous figures keeps on changing with time. Thecentral results of this work can be summarized as fol-lows. We show that depending on the spatial dimensionof the problem Floquet topological transitions can be in-fluenced by the random change in polarization of inci- a r X i v : . [ c ond - m a t . o t h e r] A ug dent light. For graphene we find that the transitions atDirac(K) point are significantly modified compared to Γpoint. The origin of this effect can be understood to bedue to a fundamental change in Fourier structure of thenoise-averaged time-dependent Hamiltonian at K point.At low frequencies of the incident radiation, it is wellknown that symmetries of the underlying Hamiltonianis crucial for topological transition . In the presence ofnoise, we find such symmetries to be broken. Interest-ingly, in contrast to standard expectation, we find thatfew of these symmetries are restored in the noise-averagedHamiltonian. This symmetry restoration has impact onthe self-averaging limit in this parameter regime. Finallyfor a 1D model( p -wave superconducting wire), using anon-trivial drive protocol, we show that even a strongnoise (large standard deviation) can’t prohibit the tran-sition.The rest of the paper is planned as follows. In Sec.II weintroduce our protocol for irradiated graphene and plotthe results(phasebands) for numerical disorder averaging.In Sec.II A we establish the existence of self-averaginglimit which suggests numerical averaging is meaningfuland can be mimicked by the ensemble averaged Hamil-tonian. This is followed by possible explanation of thedeviation from noise free (circularly polarized case) be-havior separately in high and low frequency regime inSec.II B and Sec.II C respectively. Next, in Sec.III, weshows results for 1D systems. Finally we conclude anddiscuss possible experimental scenarios in Sec.IV. II. IRRADIATED GRAPHENE
We consider graphene irradiated by electromagneticwave defined by the vector potential A = A (cos( ωt + φ ( t )) , sin( ωt )). One have to further assume it to be spaceindependent in graphene plane to keep the integrabilityof the problem intact. The φ = 0(circularly polarized)case is well studied in the literature . We allow φ tobe a normally distributed random variable with mean µ and standard deviation σ at each instant of time whichgives rise to its unpolarized nature. If one wish to pro-duce this vector potential in lab then this kind of noisewill be inherently present as random experimental error.The normalized probability distribution of φ at each timeinstant t is given by P ( φ ) = 1 √ πσ e − ( φ − µ )22 πσ (1) µ can be any real number within the interval ( − π ≤ µ ≤ π ). Here we will concentrate on the special value µ = 0(i.e this is the value of µ in all plot). This will allow usto directly compare the result with circularly polarizedcase.The time-dependent graphene Hamiltonian(for each k-mode) after Peierls’s substitution with this protocol be- < φ ( Τ ) > σ=0σ=π/10σ=π/4σ=π/2 < φ ( Τ ) > σ=0σ=π/10σ=π/4σ=π/2 FIG. 1: Noise averaged phasebands vs T for Γ point(left) (at α = 1 .
5) and for K point(right) (at α = 2 .
0) for various valuesof standard deviation( σ ). N=1000, no of sample=1000 and α = eA /c comes H ( k , t ) = (cid:18) Z ( k , t ) Z ∗ ( k , t ) 0 (cid:19) where Z ( k , t ) = − γ (2 e i ˜ kx cos( √ k y ) + e − i ˜ k x ) and ˜k = k + e A Next we calculate the time-evolution operator over onetime period( T ) for each k-mode by dividing the periodin N parts U k ( T,
0) = T t e − i (cid:82) T H k ( t (cid:48) ) dt (cid:48) = e − iH k ( T − δt ) δt e − iH k ( T − δt ) δt .....e − iH k (2 δt ) δt e − iH k ( δt ) δt (2)where T t denotes time-ordered product and δt = T /N is a very small but fixed time interval. Such decompo-sition introduces Trotter error which gets reduced withincreasing N and reproduces the exact U for the cho-sen continuous drive in the N → ∞ limit. We calcu-late the time-dependent Hamiltonian at each partitionby drawing φ from a normal distribution and using Eq.2get U ( T,
0) for one particular noise realization. We thenaverage over several such realizations numerically and getthe noise averaged time evolution operator (cid:104) U k ( T, (cid:105) = (cid:104) T t e − i (cid:82) T H k ( t (cid:48) ) dt (cid:48) (cid:105) . (3)Eq.3 has a self-averaging limit , in the sense that allfour elements of (cid:104) U ( T, (cid:105) goes to some steady value withincreasing no of partitions (N). We shall discuss this inmore details in the next sub-section.In Fig.1 we plot the phasebands (Φ( T )) obtained usingcos(Φ( T )) = Re [ (cid:104) U ( T ) (cid:105) ]. One can see with increasingmagnitude of random noise the phasebands gets modifiedbut we recover the results for pure circularly polarizedlight in σ → K point, they arestrongly modified by the noise. We calculate Chern num-ber of the lower Floquet band using the eigenfunctionsof (cid:104) U ( T ) (cid:105) in a discretized Brillouin zone. The plot isshown in Fig.2. We find that the transitions (position of C σ=0σ=π/10 C σ=0σ=π/3 FIG. 2: Chern number of the noise averaged lower Floquetband for Γ (left) and K (right) point. Others parameters aresame as in Fig.1 integer jump in Chern number) can sustain an apprecia-ble amount of temporal noise and merely gets shifted inparameter space but very strong noise (large σ ) abolishthem. A. Ensemble averaged Hamiltonian
In this subsection we explore the possibility of con-structing a deterministic Hamiltonian such that time- evolution operator constructed using it resembles thenoise averaged time-evolution operator. In a recentwork Lobejko et al have showed rigorously that thedifference of ensemble averaged time-evolution operatorand the time-evolution operator constructed by the en-semble averaged Hamiltonian scales as O ( N ) for a cer-tain class of protocols. For these protocols the ensem-ble averaged Hamiltonian at two different time commuteswhich they have termed as “commutation in statisticalsense”. They further extends the applicability of abovetheorem to some simple non-commuting Hamiltonian bynumerical simulations. But unlike those cases irradiatedgraphene contains the noise term within the argumentof complicated trigonometric functions. Hence the en-semble averaged Hamiltonian can not be obtained heresimply by substituting φ by it’s mean value. Thereforewe explicitly calculate the ensemble-averaged Hamilto-nian for irradiated graphene at time t (cid:104) H k ( t ) (cid:105) = (cid:90) ∞−∞ P ( φ ) H k ( φ, t ) dφ (4)with P ( φ ) in Eq.1 we get using Jacobi-Anger relations . (cid:104) Z ( k , t ) (cid:105) = − γ (2 e i kx cos( √ k y + α sin( ωt ))2 )[ J ( α ∞ (cid:88) n =1 i n J n ( α e − n σ cos( n ( ωt + µ ))] + e − ik x [ J ( α ) +2 ∞ (cid:88) n =1 ( − i ) n J n ( α ) e − n σ cos( n ( ωt + µ ))]) (5)Using this we numerically calculate the Frobenius normof the distance between (cid:104) U ( H ( t )) (cid:105) and U ( (cid:104) H ( t ) (cid:105) ) D N = (cid:107) (cid:104) T e − i (cid:82) T H ( t (cid:48) ) dt (cid:48) (cid:105) − T e − i (cid:82) T (cid:104) H ( t (cid:48) ) (cid:105) dt (cid:48) (cid:107) (6)and the same norm for the corresponding variance matrix S N = (cid:107) (cid:104) ( T e − i (cid:82) T H ( t (cid:48) ) dt (cid:48) − T e − i (cid:82) T (cid:104) H ( t (cid:48) ) (cid:105) dt (cid:48) ) (cid:105) (cid:107) (7)where N is the no of partitions used to calculate (us-ing Eq.2 and 3) each quantities inside the norm. Theseare two appropriate quantities to measure the deviationof the time-evolution operator in different noise realiza-tions. We see power law fall of both D N and S N in noof partitions(N)(see Fig.3) which suggest self-averaginglimit exists here. It is only in this limit that the disorderaveraging is meaningful in dynamical systems. This is inclose analogy to equilibrium disordered systems where foreach disorder realization some amount of deviation (fromthe mean) is introduced in all physical observable due tothe finite size of the system but these deviations get can-celed when averaged out over several disorder realizationsand thus helps to achieve the thermodynamic result fast. Here in dynamical system finite no of partition(N) playthe role of finite system size and the thermodynamic limitcorresponds to the continuous drive ( N → ∞ ). Vanish-ing of S N in large N also implies the equivalencecos( (cid:104) Φ( T ) (cid:105) ) ≡ (cid:104) cos(Φ( T )) (cid:105) (8)which we have used throughout the paper. In Fig.3 notethat D N and S N have larger values at K point comparedto Γ point for small N. This is related to the fact thattime dependent Hamiltonian of irradiated graphene at K point is more complicated than at Γ point due to thepresence of lesser no of symmetries . Larger the com-plexity larger N one need to use to reduce these errors.This power law fall suggests that the time consumingnumerical disorder averaging can be avoided by the useof ensemble averaged Hamiltonian to calculate U ( T, K point) where as in some other cases(as in Γ point) even a l n ( D N ) -1.78-1.84 D N ( × − ) l n ( S N ) -1.98-1.97 S N ( × − ) l n ( D N ) -1.9-2.03 D N ( × − ) l n ( S N ) -1.92-1.88 S N ( × − ) FIG. 3: Fall of D N (upper left panel) and S N (upper rightpanel) with N for Γ point at α = 1 . T = 4 .
0. Same forthe Dirac point in lower left and right panel at α = 2 and T = 4 .
0. No of sample=1000 and σ for blue and red curve is π/
10 and π/ φ ( Τ ) φ ( Τ ) FIG. 4: Comparison of phasebands obtained by numericallydisorder averaging of U ( T,
0) operator (black-solid line) andby using the ensemble averaged H ( t ) to calculate U ( T,
0) op-erator (red-dashed line) for Γ point (left panel) and for K point (right panel). Relevant parameters are same as in Fig.3. strong noise just causes a shift of the crossing positionsand nothing more than that. We will do it by analyzingthe ensemble averaged Hamiltonian (Eq.5) in two differ-ent frequency regime. B. High frequency, Floquet formalism
The Floquet formalism allows one to treat a periodictime-dependent problem as a time-independent eigen-value problem. The cost of this is to deal with an in-finite dimensional Hilbert space which is a vector spaceof T periodic functions and also known as Sambe space.The representation of Floquet Hamiltonian (related to U ( T,
0) by U ( T,
0) = e − iH F T ) in this basis is defined bythe following matrix elements H m,ni,j = mωδ mn δ ij + 1 T (cid:90) T e − i ( m − n ) ωt (cid:48) H ij ( t (cid:48) ) dt (cid:48) (9) where ( m, n ) is row and column index of different squareblocks each of size ( H × H ) where H is the Hilbertspace dimension of the equilibrium problem (2 for eachk-mode in our case) and ( i, j ) denotes position of eachmatrix element within one such block. For numerical pur-poses one can truncate this matrix after some order whichdepends on details of the problem especially the absolutevalue of maximum order of the Fourier components (oftime-dependent Hamiltonian) with non-vanishing coeffi-cient. One also needs to increase the truncation dimen-sion with decreasing frequency. Following this prescrip-tion one can safely truncate the Floquet Hamiltonian inzero-th order at Γ point(where one has a 2 × H F ) andin 1st order at K point( where one has a 6 × H F )for high frequencies and low Amplitude of radiation .Thus one gets expressions of Floquet conduction band(Φ( T )) in 1st quasi-energy BZ for the noise free (circu-larly polarized) case with hopping-amplitude( γ ) set tounity Φ(Γ , T ) = 3 J ( α ) T (10)Φ( K, T ) = (cid:112) π + 36 J ( α ) T − π φ appears as argument of trigonometric func-tions due to it’s random nature at each instant of time φ [ µ, σ ] and φ [ µ + 2 nπ, σ + 2 pπ ] will not give same timeevolution operator. Using e − n σ ≈ σ in Eq.5we get (cid:104) Z ( k , t ) (cid:105) | σ (cid:29) ≈ − γ (2 J ( α e ikx cos( √
32 ( ky + α sin( ωt ))) + J ( α ) e − ikx ) (12)for Γ point this gives a Hamiltonian proportional to σ x only and hence one simply gets the phasebandΦ(Γ , T ) = (cid:90) T (cid:104) Z (Γ , t (cid:48) ) (cid:105) dt (cid:48) (13)the integrand is difficult but again using Jacobi-Angerrelations we get(taking γ = 1)Φ(Γ , T ) = (2 J ( α J ( √ α J ( α )) T +4 J ( α ∞ (cid:88) n =1 J n ( √ α (cid:90) T cos(2 nωt (cid:48) ) dt (cid:48) = (2 J ( α J ( √ α J ( α )) T (14)similarly for K point we getΦ( K, T ) = ( J ( α ) − J ( α J ( √ α T (15) C o s ( φ ( Τ )) C o s ( φ ( Τ )) FIG. 5: Comparison of cosines of Eq.10(black-solid) and14(red-dashed) for Γ point (left panel) and of Eq.11(black-solid) and 15(red-dashed) for K point (right panel). All pa-rameters are same as before. we compare cosines of Floquet bands for circularlypolarized( σ = 0) and unpolarized( σ (cid:29)
0) case in Fig.5.The functional behavior of these two bands do not changemuch for Γ point whereas for K point they show dras-tically different behavior. This huge change for K pointis due to the fact that strong noise (highly unpolarizedlight) changes the lowest non-vanishing Fourier compo-nent of (cid:104) H K ( t ) (cid:105) from 1 to 0 and thus reduces the effectiveSambe space dimension from 6 to 2. These changes makethe Floquet band at K point to depend on J s only abol-ishing J s. Note that J and J has completely differentbehavior when the argument is small, the former is a de-creasing function but the later is an increasing functionof the argument. C. Low frequency
At low frequencies (and also at high radiation ampli-tudes) one need to take into account the higher Fouriercomponents of the time-dependent Hamiltonian and con-sequently the truncation dimension of the Floquet Hamil-tonian increases. This is why at low frequencies onecan’t have simple analytical expression of Floquet bandsin terms of Bessel functions and one needs to considerother methods like the adiabatic-impulse which givesgood matching with numerics in low to moderate fre-quencies and high amplitudes . Symmetries of H ( t ) alsoplay a crucial role in predicting the existence of phase-band crossings at different high symmetry points. Butbefore going into the details of that we investigate thebehavior of D N and S N as a function of N at low frequen-cies. Generally low ω and hence a high period ( T ) ne-cessitates a proportional increase of no of partitions butnumerics suggests that the convergence of these quanti-ties to zero is much slower than that in this parameterregime. In Fig.6(a)-(c) we demonstrate this. We see fora typical high σ one needs to increase N nearly quadrat-ically (instead of linearly) with T to make the value of D N go below some particular threshold. We, thereforeto reduce the numerical cost, keep our all calculationsconfined within small σ values at low frequencies.It was shown in ref[33] that there exists 6 fold sym- D N l n ( D N ) -1.77-1.79-1.54 0 2000 4000 6000 8000 10000N012345 S N l n ( S N ) -2.03-1.07 T log(T) l og ( N * ) N * ( × ) T -1-0.500.51 C o s ( φ ( Τ )) FIG. 6: Fall of D N (upper left panel) and S N (upper rightpanel) with N for Γ point at α = 2 . T = 60. σ for theblue,red and green curve is π/ π/
10 and π/ N ∗ (for which D N ∗ fall below 10 − ) vs T in lower left panelfor σ = π/
3. Slope of linear fit in log-log plot is 1.83(inset).Lower right panel shows matching of phaseband from numer-ically averaged U(T) and U(T) calculated from averaged Hfor Γ point at α = 2 . σ = π/ H ( T − t ) = H ( t ) (cid:52) (cid:52) H ( T ± t ) = τ x H ( t ) τ x (cid:52) (cid:52) H ( T ± t ) = τ x H ( t ) τ x (cid:52) (cid:56) H ( T ± t ) = H ( t ) (cid:52) (cid:56) H ( T ± t ) = H ( t ) (cid:52) (cid:56) H ( T ± t ) = τ x H ( t ) τ x (cid:52) (cid:56) TABLE I: Symmetries of Γ point for circularly polarized andunpolarized light. metries at Γ point of graphene irradiated by circularlypolarized light. This was shown to be responsible forphaseband crossing simultaneously at
T /
3, 2
T / T .But here for unpolarized light typically all these sym-metries are absent for any disorder-realization. Conse-quently, disorder averaging also leads to avoided cross-ing. Here also the ensemble averaged Hamiltonian cancapture the essential physics but interestingly two of thesymmetries get restored in it. We chart out the symme-tries of Γ point under the irradiation of CP and unpolar-ized(ensemble averaged H ( t )) light in detail in Table.1.This kind of symmetry mismatch between the two quan-tities inside the norm of Eq.6 has significant impact onfall of D N at low frequencies. We find that D N falls veryslowly with N (see Fig.6) here.In Fig.7 we show this symmetry mismatch between CP -1-0.500.51 E ( t ) E ( t ) (a) 148 148.5 149 149.5 150T0.960.981 C o s ( φ ( Τ / )) σ = 0σ = π/100 (b)148 148.5 149 149.5 150T0.9970.9980.9991 C o s ( φ ( Τ / )) σ = 0σ = π/100σ = π/80σ = π/60 (C) 111 111.5 112T-1-0.998-0.996 C o s ( φ ( Τ / )) σ = 0σ = π/100σ = π/80σ = π/60 (d) FIG. 7: (a)Instantaneous energies vs t/T for CP(upper panel)and for unpolarized( σ = π/
8) in lower panel. α = 2 . T )) vs T. The red curve is achieved by exact-numericalaveraging with no of sample=1000, N=10000. α = 2 . T )) vs T for α = 2 .
35 (d) cos(Φ( T )) vs T for α = 2 .
28. Phasebands at (c)-(d) are calculated using theensemble averaged Hamiltonian and unpolarized light pictorially (a large σ is used forthis purpose in Fig.7(a)) and its consequences. Fig.7(b)shows for exact numerical disorder averaging a small σ is sufficient to abolish the crossing at T /
3. Fig.7(c)-(d)shows for ensemble averaged Hamiltonian the crossings at
T /
T / σ . III. 1D SYSTEMS
One-dimensional interacting spin chains whose Hamil-tonian can be expressed in terms of free fermions viaJordan-Wigner transformation have attracted a lot oftheoretical attention in last decades due to their inte-grable structure, existence of topological transition aswell as possibility of experimental realization using ion-traps and ultracold atom systems. Non-equilibrium dy-namics in these models is equally interesting because non-trivial topology can be induced by periodic drive of dif-ferent terms in the Hamiltonian . This can be inde-pendently done using multiple lasers with different am-plitudes and frequency. In these experiments phase dif-ferences between different drive terms can be randomlychanged in a time scale t (cid:28) /ω where ω is the fre-quency of drive. This constitutes a 1D platform to studysimilar physics as studied in previous section for 2D sys-tems using unpolarized light. The survival of the topo-logical transition under such noisy drive is the key is-sue we would like to address. To this end, we con-sider a p-wave superconductor described by the following Hamiltonian H = L − (cid:88) i =1 [( γc † i c i + H.c )+∆( c i c i +1 + H.c )] − µ L (cid:88) i =1 (2 c † i c i − XY chainin perpendicular magnetic field via Jordan-Wignertransformation . After a Fourier transformation definedby c k = L (cid:80) Lj =1 c i e ikj we can write this as H = 2 (cid:88) ≤ k ≤ π ψ † k H k ψ k (17)where ψ k = ( c k , c †− k ) T is a two component vector. Thuseach k-mode of such systems can be described by thefollowing Hamiltonian(we scale everything by γ ) H ( k, t ) = ( µ − cos( k )) σ z + ∆ sin( k ) σ x (18)and we use the following drive protocol µ = A cos( ωt + φ ( t )) and ∆ = cos( rωt ) where r is an integer and φ is as usual a random variable at each time t. The dy-namics of this model is non-trivial for r > . This model(with φ ( t ) = 0) has aphaseband crossing for k = π/ t = T / φ is a random Gaussianvariable with zero mean. Below we mention the schemefor partitioning a full period to calculate the noise aver-aged U ( t,
0) now at any time t ≤ Tδt = tN = const (19)i.e we increase no of partitions proportionally as the time t gets closer to T keeping the duration of constant timeevolution( δt ) fixed. Thus we calculate noise averagedphaseband at all time t within a period for different noisestrength ( σ ) and compare it with noise free case in Fig-ure.8(a). Interestingly noise modifies the phaseband atall times except at t = T / t = T / (cid:104) H ( k = π , t ) (cid:105) = A cos( ωt ) e − σ / σ z + cos( rωt ) σ x (20)In Fig.8(left panel) we see time evolution governed bythis averaged H mimics the numerically disorder aver-aged U operator as like before. We note that this nu-merical agreement leads to the following statement “Theeffect of random noise is just to renormalize the laseramplitude ” ˜ A = Ae − σ / (21) C o s ( φ ( t )) σ = 0σ = π/10σ = π/5σ = π/3 E ( t ) σ = 0σ = π/3 FIG. 8: Phasebands from numerically averaged U operator(continuous line) and from the averaged Hamiltonians(dots)for 1D model(in Eq.15) in left panel. A=1.5, ω = 1 .
0, r=3.Right panel shows the change of instantaneous energies withthe insertion of noise.
The robustness of the transition at t = T / k = π/ r (namely H ( T / − t ) = − H ( t ) )is not de-stroyed by the insertion of noise here (see Fig.8(rightpanel)). This can be used together with the Trotterlike decomposition of U operator (as in Eq.2) to show U − ( T /
2) = U † ( T /
2) = U ( T /
2) signifying that a cross-ing through Floquet zone-center will always be there at t = T / A , ω , φ etc). Furtherright panel of Fig.8 demands that the same adiabatic-impulse method (as done for the noise free case in ref.[33])can be used to show the existence of the crossing at t = T /
IV. DISCUSSION
In this work we have studied the existence of self-averaging limit in graphene irradiated by unpolarizedlight. We see the limit holds in high-frequency regimeand can be captured by the noise-averaged Hamiltonian.In low frequencies the limit is achieved very slowly as apossible consequence of retaining two of the symmetriesin noise-averaged Hamiltonian. This opens up an oppor-tunity to search for some other deterministic Hamilto-nian for speeding up the convergence to asymptotic limit.We hardly found any steady limit at extremely low fre-quencies to the best of our numerical ability. Floquettopological transitions are found to be modified by theinsertion of noise to various degrees depending on the k-point in BZ. These range from a small shift in crossingpositions to complete abolition of the transition depend- ing on the amount of disorder. We find that certain k-points are more affected as a consequence of a change inFourier structure of their time-dependent Hamiltonianinduced by the noise. The presence of a 6-fold sym-metry at Γ point plays a crucial role for the existenceof a special type of crossings which simultaneously hap-pens at
T /
T / T . This kind of crossings are ubiq-uitous in low frequencies but ceases to exist in high fre-quency(scanning the whole parameter regime as much aspossible we found they are absent below T ≈ ◦ with theplane of vibration of the incident plane polarized lightto extract pure circularly polarized light. Now if this an-gle changes randomly (which is always present in smallamount if the experiment is not performed carefully suchas a small vibration of the table on which the set uplies may cause it) then the polarization of the outgoinglight will also fluctuate. One can also use synthetic gaugefields to produce such noisy vector potential. This kindof perturbation is very common in an interference exper-iment if incoherent sources are used. The quantitativelydifferent noise-response from various k-points can be ex-perimentally verified by measuring the photoinduced gapin a momentum resolved manner using pump-probe spec-troscopy as done in ref[22]. The abolition of transitionand hence a change in topological structure of the Flo-quet bands can be detected by analyzing the intensityand angular dependence of ARPES spectra .In conclusion we have shown random noise in the vec-tor potential of incident light has significant impact onFloquet topological transition in graphene. One can an-alyze the symmetries and Fourier structure of the noise-averaged Hamiltonian to understand the modificationsdone by the noise. In 1D systems such noisy drives hasno effect on the transitions. Acknowledgements
Author thanks K. Sengupta for support. K. v. Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett.45, 494(1980). R. B. Laughlin, Phys. Rev. B 23, 5632 (1981). D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M.den Nijs, Phys. Rev. Lett. 49, 405 (1982). R. E. Prange, Phys. Rev. B 23, 4802(R) (1981). B. I. Halperin, Phys. Rev. B 25, 2185 (1982). F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988). Gregor Jotzu et al, Nature volume 515, pages 237240(2014). T Oka, H Aoki, Physical Review B 79 (8), 081406 (2009). Jun-ichi Inoue, Akihiro Tanaka, Phys. Rev. Lett. 105,017401 (2010). NH Lindner, G Refael, V Galitski, Nature Physics 7 (6),490 (2011). J Cayssol, B Dra, F Simon, R Moessner, physica status so-lidi (RRL)-Rapid Research Letters 7 (12), 101-108 (2013). T Kitagawa, T Oka, A Brataas, L Fu, E Demler, PhysicalReview B 84 (23), 235108 (2011). A Kundu, B Seradjeh, Physical Review Letters 111 (13),136042 (2013). I Esin, MS Rudner, G Refael, NH Lindner, Physical Re-view B 97 (24), 245401 (2018). L DAlessio, M Rigol, Physical Review X 4 (4), 041048(2014). L DAlessio, M Rigol, Nature communications 6, 8336(2015). A Lazarides, A Das, R Moessner, Physical Review E 90(1), 012110 (2014). A Lazarides, A Das, R Moessner, Physical review letters112 (15), 150401 (2014). Achilleas Lazarides, Arnab Das, Roderich Moessner, Phys-ical review letters 115 (3), 030402 (2015). T Kitagawa, E Berg, M Rudner, E Demler, Physical Re-view B 82 (23), 235114 (2010). Mark S. Rudner, Netanel H. Lindner, Erez Berg, MichaelLevin, Phys. Rev. X 3, 031005 (2013). Y. H. Wang, H. Steinberg, P. Jarillo-Herrero, and N.Gedik, Science 342, 453 (2013). L. P. Gavensky, G. Usaj, and C. A. Balseiro, Sci. Rep. 6,36577 (2016). Mikael C.Rechtsman et al, Nature volume 496, pages196200 (2013). Sebabrata Mukherjee et al, Nature Communications vol-ume 8, Article number: 13918 (2017). A Polkovnikov, K Sengupta, A Silva, M Vengalattore, Re-views of Modern Physics 83 (3), 863 (2011). Anirban Dutta, Armin Rahmani, Adolfo del Campo, Phys.Rev. Lett. 117, 080402 (2016). Adhip Agarwala and Vijay B. Shenoy, Phys. Rev. Lett.118, 236402 (2017). Sourav Nandy, Arnab Sen, and Diptiman Sen, Phys. Rev.X 7, 031034 (2017). Utso Bhattacharya, Somnath Maity, Uddipan Banik, AmitDutta, Phys. Rev. B 97, 184308(2018). A Kundu, HA Fertig, B Seradjeh, Physical review letters113 (23), 236803 (2014). Marcin Lobejko, Jerzy Dajka and Jerzy Luczka,arXiv:1805.02871 (2018). Bhaskar Mukherjee, Priyanka Mohan, Diptiman Sen andK. Sengupta, Physical Review B 97, 205415 (2018) J. D. Sau and K. Sengupta, Phys. Rev. B 90, 104306(2014). S. Kar, B. Mukherjee, and K. Sengupta, Phys. Rev. B 94,075130 (2016). A. Dutta, G. Aeppli, B. K. Chakrabarti, U. Divakaran, T.F. Rosen- baum, and D. Sen, Quantum phase transitionsin transverse field spin models: from statistical physics toquantum information (Cambridge University Press, Cam-bridge, 2015). V. Mukherjee, U. Divakaran, A. Dutta, and D. Sen,Phys. Rev. B 76, 174303 (2007); C. De Grandi andA. Polkovnikov, in Quantum Quenching, Annealing, andComputation, edited by A. K. Chan- dra, A. Das, andB. K. Chakrabarti, Lecture Notes in Physics, Vol. 802(Springer, Heidelberg, 2010), p. 75. M. Thakurathi, A. A. Patel, D. Sen, and A. Dutta, Phys.Rev. B 88, 155133 (2013); M. Thakurathi, K. Sengupta,and D. Sen, Phys. Rev. B 89, 235434 (2014). E. Lieb, T. Schultz, and D. Mattis, Ann. Phys. (NY) 16,407 (1961).40