Bulk-edge correspondence and new topological phases in periodically driven spin-orbit coupled materials in the low frequency limit
BBulk-edge correspondence and new topological phases in periodically driven spin-orbitcoupled materials in the low frequency limit
Ruchi Saxena , Sumathi Rao and Arijit Kundu Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad 211 019, India Department of Physics, Indian Institute of Technology - Kanpur, Kanpur 208 016, India
We study the topological phase transitions induced in spin-orbit coupled materials with bucklinglike silicene, germanene, stanene, etc, by circularly polarised light, beyond the high frequency regime,and unearth many new topological phases. These phases are characterised by the spin-resolvedtopological invariants, C ↑ , C ↓ , C ↑ π and C ↓ π , which specify the spin-resolved edge states traversingthe gaps at zero quasi-energy and the Floquet zone boundaries respectively. We show that for eachphase boundary, and independently for each spin sector, the gap closure in the Brillouin zone occursat a high symmetry point. PACS numbers:
I. INTRODUCTION
Dynamical control of topological phases is one of themost intensely researched topics in recent times[1–10].Proposals have involved periodic driving in semiconduc-tor systems[4], cold atom (or optical lattice) systems[5],graphene[1–3, 9] and systems with spin-orbit couplinglike silicene[10], with a variety of analytical and numer-ical methods. Apart from band-structure control of asystem by renormalization of its dynamical parametersvia a periodic drive, in a number of recent papers [11–16], novel non-trivial topological phases, which do nothave any analog in static systems, have been explored.Silicene [17], and other spin-orbit coupled [18] materi-als like germanene, stanene, etc are recently synthesizedmaterials which have shot into prominence because theirbuckled nature allows them to be tuned by an electricfield through a transition between a band insulator anda topological insulator [20–22]. In their pristine forms,they consist of a honeycomb lattice of the appropriateatoms. However, unlike graphene, the lattice is buckleddue to their large spin-orbit coupling. Hence, although,like in graphene [23], the low energy dynamics is governedby Dirac electrons at the K and K (cid:48) points at the oppo-site corners of a hexagonal Brillouin zone, there exists agap - i.e., the Dirac electrons are massive. However, thismass can be tuned by an external electric field, becausethe electric field acts differently on the electrons in the A and B sub-lattices due to the buckling. Thus, the transi-tion between a topological insulator to a band insulatorthrough a metallic phase in the middle can be controlledby an external electric field. This tunability, and in par-ticular, the experimental realisation [24] of silicene-basedtransistors has led to extensive work [25–30] on the in-terplay of topology and transport in these materials.More recently, the question of whether topologicalphases can be controlled in silicene and similar mate-rials when there are time dependences in the problem, inparticular periodic time dependences, has been studied.Although the system is now driven, it is often possible,when the driving frequency is the largest energy scale of the problem, to describe the dynamics of the system interms of an effective Hamiltonian. Ezawa [10] concen-trated on silicene and showed that at high frequenciesand for small amplitudes of driving, new phases suchas quantum Hall insulator, spin-polarised quantum Hallinsulator and spin and spin-valley polarised metals canbe realised. Further, it was shown [31] that many moretopological phases could be realised by performing a sys-tematic Brillouin-Wigner (BW) expansion of the Hamil-tonian to second order in the inverse of the frequency, notonly in silicene, but in other spin-orbit coupled materials.But, as was discussed also in earlier references [31, 32],the BW expansion breaks down when the frequency ω be-comes smaller than the band-width of the effective Hamil-tonian. The real constraint on the applicability of theBW expansion is a combined bound on both ω and theamplitude of driving and in fact, the validity of BW in-creases, even for lower frequencies when the amplitude in-creases. However, the physical reason for the breakdownof the earlier studies at low frequencies is because, at fre-quencies comparable to the band-width, it is no longerpossible to neglect the topology of the quasienergy space,which forms a periodic structure with the single valued-ness of the eigenfunction requiring the quasienergies tobe within a “Floquet zone”. Low frequency driving canlead to crossings between the bottom of one Floquet bandand the top of the next Floquet band. These crossingsare neglected in the BW expansion and hence, the studyof the driving at low frequencies requires a new formal-ism which goes beyond the effective static approximationof a dynamical Hamiltonian.It is well understood that the Chern number by it-self cannot fully describe the topological nature of thesesystems [11, 14]. In a time periodic system the Floquetspectrum can be organized into quasienergy bands, andthe Chern numbers of these bands can be computed. Butactually, the Chern number of a particular band, whichis computed by integrating the Berry curvature over thewhole Brillouin zone, is the difference between the num-ber of chiral edge modes leaving the band from above andthose entering the band from below. In a static system,the spectrum is bounded from below and the edge states a r X i v : . [ c ond - m a t . m e s - h a ll ] D ec entering the band were always zero; hence, the Chernnumber of the band was sufficient to determine the edgespectrum. But for Floquet systems, this is no longertrue and it is possible to have edge states even whenthe Chern number of the band is zero. So for a char-acterization of the topological nature of the system thatwould satisfy the edge-bulk correspondence, one needs tohave access to full time-dependent bulk evolution opera-tor U ( t ), evaluated for all intermediate times within thedriving period [11]. The invariants thus computed pre-dict the complete Floquet edge-state spectrum. Similar Z valued indices for periodically driven time-reversal in-variant two dimensional indices have also been found [14].For a geometry with edges, the number of the edgestates, counted with a sign corresponding to their chiral-ity, is related to the winding number of the bulk timeevolution operator. It was also shown that the differencein the winding numbers at two different energies was pre-cisely equal to the sum of all the Chern numbers thatlie between these energies. More specifically, for a twoband model with the Fermi energy at zero quasi-energy,it was shown that the gaps at zero quasi-energy and atthe zone boundary ω/ C and C π , whose difference gave the Chern number of theband. In other words, a Floquet topological insulator ischaracterized by two integers, in contrast to the singleChern number for static topological insulators. Whilethis formalism is, of course, applicable both in the highfrequency as well as the low frequency regime, at highfrequencies, since the frequency is much larger than theband-width, the zone boundary is not accessible. Hence,in the high frequency limit, computation of the Chernnumbers at zero quasi-energy is sufficient to characterizeall the phases.However, in most cases, the high frequency, strong am-plitude limit needed for obtaining new topological phasesis currently experimentally unattainable and in recenttimes, the focus on low frequencies has increased. Earlywork in this area focussed on graphene [9] and showednot only the existence of several new phases, but alsohow disorder could enhance conductance by several or-ders of magnitude. Broad dips in the conductance atresonances between valence and conduction bands ingraphene nano-ribbons have been predicted [6] and morerecently studied [33] in detail. New states with opti-cally induced changes of sub-lattice mixing have beenidentified [15]. Quantum resonances have been studiedin irradiated graphene n - p - n junctions [16]. The roleof the symmetries of the instantaneous Hamiltonian andthe time-evolution operator in determining the phase di-agram at ultra-low frequencies in irradiated graphene us-ing the adiabatic impulse method has also been recentlyemphasized [34]. More recently, universal fluctuations ofthe topological invariants have also been studied [35].In this paper, we will compute the Chern numbers ofa silicene [44] band, both at zero quasi-energy and atthe zone boundary, and for both spin up and spin downelectrons, since the up-down symmetry is broken in the (+ + ) ( ) ( + , + ) ( + , + ) ( + , + ) ( + , + ) (+ + ) ( + , + ) (− − ) (+ − ) (− − ) (− − ) ( ) αω ' (+ )( − ) (− ) (− − ) FIG. 1: The phase diagram for the model Hamiltonian inEq. 1 as we vary the amplitude α and the drive frequency ω , with the external electric field fixed at lE z = 0 . t . Eachphase is characterized by the spin resolved quantum numbers( C ↑ , C ↓ ). We label the phases by calligraphic letters. Thedotted lines P i indicate the topological phase boundaries, in-ferred from the gap closing in momentum space, shown onlyfor the ↑ spin sector. presence of spin-orbit coupling. We will work in the lowfrequency regime, where the static approximation doesnot hold; however, it is still possible to reliably computeChern numbers using numerical methods. We will showexplicitly that the bulk-boundary correspondence holds,by checking that the C σ and C σπ as obtained by countingthe number of edge states at the right and left edges ofthe sample, agrees with the Chern number of the bulkobtained from C σπ − C σ . II. COMPUTATION OF THE DYNAMICALBAND STRUCTURE
We start with two dimensional Dirac systems whichare buckled due to the large ionic radius of the siliconatoms and consequently have a non-coplanar structureunlike graphene. These materials can be described bya four-band tight binding model in a hexagonal latticegiven by H = − t (cid:88) (cid:104) i,j (cid:105) ,σ c † iσ c jσ + iλ √ (cid:88) (cid:104)(cid:104) i,j (cid:105)(cid:105) ,σ σν i,j c † iσ c jσ + lE z (cid:88) iσ ξ i c † iσ c iσ . (1)Here, the first term is the kinetic term where t isthe hopping parameter. The second term representsthe spin-orbit coupling term where the value of λ de-pends on the material and ν i,j = ± A =( A cos( ωτ ) , A sin( ωτ ) ,
0) is introduced into the Hamil-tonian using Peierls substitution. ω is the frequency oflight and A is its amplitude. In the Fourier transformedspace, this is written as H ( τ ) = lE z − δ λ δ t δ ∗ t − lE z + δ λ lE z + δ λ δ t δ ∗ t − lE z − δ λ (2)where δ λ ( τ ) = 2 λ √ (cid:34) √ a sin ˜ k x − sin (cid:32) √ a k x + 3 a k y (cid:33) − sin (cid:32) √ a k x − a k y (cid:33)(cid:35) (3)with ˜ k x = k x + A cos ωτ and ˜ k y = k y + A sin ωτ and δ t ( τ ) = t [exp( − iα sin ωτ )+ T + exp iα ( √ ωτ + sin ωτ )2+ T − exp iα ( −√ ωτ + sin ωτ )2 (cid:35) (4)with T ± = exp( ia ( ±√ k x + 3 k y / α = Aa , where a is the lattice constant.For the bulk system, the vector potential and hencethe Hamiltonian is periodic in both the x and y direc-tions. This implies that we can rewrite the Hamiltonianin terms of a Floquet eigenvalue problem with the Hamil-tonian given by H F = − i ∂∂τ + H ( τ ) , (5)the eigen functions given by ψ k ,b ( x, y, τ ) = u b ( k x , k y , τ ) e i r · k − i(cid:15) b τ (6)with u b ( k x , k y , τ ) = u b ( k x , k y , τ +2 π/ω ), and where (cid:15) b arethe quasienergies or the eigenvalues of H F . The Hamil-tonian can now be solved numerically as a function ofthe amplitude A , frequency ω and the sub-lattice po-tential E z , both for the quasienergy eigenvalues and forthe wave-functions.At high frequencies, ω constitutes a large gap be-tween unperturbed subspaces, and the extended FloquetHilbert space splits into decoupled subspaces with differ-ent photon numbers. Since the perturbation scale of theHamiltonian, which is the band-width t , is much smaller FIG. 2: The phase diagram as a function of the amplitude α and the external electric field lE z . A low drive frequency ischosen ( ω = 3 . t ) since we wish to study the system in the lowfrequency limit. All the phases are the same as those found inFig.1 except for three new phases - H , I and J . The labellingof the phases follows the same convention as in Fig. 1. than ω , one can use systematic perturbation theory toinclude virtual processes of emitting and absorbing pho-tons, and upto a given order in perturbation theory, onecan obtain an effectively static Hamiltonian as shown inRef. 31. The Chern numbers for the model can then becomputed by integrating the Berry curvature over thewhole Brillouin zone [36] using the eigenvectors of theeffective Hamiltonian. However it is expected that suchan expansion in 1 /ω would fail to predict the correctChern numbers once the frequency of the drive, ω , be-comes comparable to the bandwidth. This is the part ofthe phase diagram that we shall complete in this paper. A. The phase diagram of the Floquet Hamiltonian
As the frequency of the drive becomes comparable tothe effective bandwidth of the system, it is essential tonow consider the complete nature of the quasi-energybands in the computation of the topological invariantsof the system. As was mentioned in the introduction,the quasi-energy bands (of the two band system) are nowidentified with two topological invariants, C and C π andthe net Chern number of a band is given by C = C − C π (independently for each of the spins).The Fourier-transformed time-dependent Hamiltonian(Eq. 2) is block-diagonal in the spin space. For eitherthe ↑ or the ↓ spin, it is a 2 × FIG. 3: Gap closing points in the Brillouin zone along the P i ( i = 1 . . .
6) phase boundaries (drawn in Fig.1) as describedin the caption of Fig.1. as U ( k , π/ω ) = T e − i (cid:82) π/ω H ( k ,τ ) dτ . (7)and the Floquet states u b ( k x , k y ,
0) are the eigenstates ofthis operator. The Chern number of each Floquet bandis then defined by integrating the Berry curvature of theFloquet states over the whole Brillouin zone - C = 12 π (cid:90) BZ dk x dk y ( ∇ × A lower ( k )) , (8)where A lower is the Berry connection in terms of Floquetstates of the quasi-energy band with quasienergy lyingbetween ( − ω/
2, 0). We numerically compute the Chernnumbers of the lower band (of both ↑ and ↓ spins) fol-lowing the work by Fukui et al [36].When the parameter ranges are such that a high fre-quency approximation would be valid, the Chern numbercomputed using the effective static Hamiltonian wouldexactly match the one obtained by considering the Flo-quet states. In this sense, the following phase diagramthat we present complements what has been obtainedearlier in Ref. 31, and completely specifies the topologi-cal phases of the system for all parameter regimes.The phase diagrams for both the up spin and the downspin bands are presented in Figs. 1 and 2. In Fig. 1, weshow the Chern number of the lower quasienergy bandas a function of the amplitude of the drive versus thefrequency, whereas in Fig. 2 we show it as a function ofthe amplitude of the drive versus the sub-lattice poten-tial. For lower frequencies, many different phases appearand appear to follow a fractal structure, as was seen forgraphene in Ref.[32]. But as such phases are not ex- pected to be protected by a large enough band-gap, wehave only shown phases which are ‘large enough’ (oc-cupy enough area in the phase diagram) and we haveignored tinier phases. As α → ω →
6, these phasessmoothly go over to the high frequency phases in Ref. 31.We have also chosen to name only those phases that arelarge enough to be possible stable phases in calligraphicletters as A , B . . . J , with A , B , C , E , F being present inboth Figs. 1 and 2, and B (cid:48) , D , G in Fig.1 and H , I , J inFig. 2. Note that there are two phases B and B (cid:48) whichhave identical values of the Chern numbers for both the ↑ spin band and the ↓ spin band. Nevertheless, they aretwo distinct phases since they occur for different valuesof ω and α and are not continuously connected to eachother and they could have different edge state structures.Note also the existence of a phase A which has zero Chernnumbers for both spin ↑ and spin ↓ electrons. We will seelater in the next section, that this is a topological phaseand has edge states despite having zero Chern numbers.The lines that separate the phases are when the gapcloses and the gap closing typically occurs at the highsymmetry points of the Brillouin zone as shown in Fig. 3.For the lines P , P and P , the gap closes at the Γ pointwhereas for the P and P lines, it closes at the K pointand for the P line, the closure happens at the half-waypoint between the Γ point and the K point. Note that wehave concentrated on the spin ↑ bands and hence havelines separating region C from E , which have differentChern numbers for ↑ spin, but no line separating regions C from B , which have the same Chern number for ↑ spin.A similar analysis can be done for the ↓ spin case.We note that the Chern number changes by ± P crossing, which essentially implies a quadratic touch-ing of the bands. This is similar to the transition ex-plained in Ref. 9 where the Hamiltonian for the first Γpoint transition at the Floquet zone boundary was ob-tained perturbatively, and was shown to lead to a Chernnumber change of ±
2. This can only happen at the spher-ically symmetric Γ point. Along P , P , P and P , thechange in the Chern number is ± K points. Along P , however,the change in the Chern number is ±
3. This can hap-pen at 3 points in the Brillouin zone, symmetric aroundthe Γ point as shown in Fig. 3. We have also checkedthat a change of the chirality of the circularly polarizedlight, besides changing signs of all the Chern numbersalso breaks inversion symmetry with respect to the gapclosing diagram in Fig. 3. The blue points are at K (cid:48) in-stead of K points and the green points are placed so asto complete the smaller hexagon.However, the computation of the Chern number doesnot specify the C and C π invariants individually. As thebulk-boundary correspondence in our system comes fromthese invariants, to discover these two indices, we need toconsider the edge-state structure in a system with edges- e.g ., a ribbon geometry. This is what we shall discussin the following section. k x a x T ( i n un it s o f t ) R t C = 1C C = C - C C = 1C C = C - C RRR (0,0) π/ √ ( ) π/ √ ( )−π π FIG. 4: The quasi-energy band structure of a zigzag nanorib-bon of the periodically driven spin-orbit coupled system forphase A . Both spin sectors, ↑ and ↓ (shown in red and blackrespectively) possesses one pair of chiral edge states both atzero quasi energy and Floquet zone boundary. We also labelthe chirality of the left edge state at the two inequivalent gapsby R or L depending on whether the state is right-moving orleft-moving. The system is finite in the y -direction while the x -direction is periodic.TABLE I: Spin-resolved topological quantum numbers andthe edge states for phases in Figs.1, 2 .Phases ( C ↑ , C ↓ ) C ↑ C ↑ π C ↓ C ↓ π A (0,0) 1 1 1 1 B , B (cid:48) (+2,+2) 0 − − C (+2,+3) 0 − − D (+1,+2) − − − E (+3,+3) 1 − − F (+1,+1) 1 0 1 0 G (+1,+1) − − − − H (+1,+1) 0 − − I ( − , −
2) 0 2 0 2 J ( − , −
1) 0 1 0 1
B. Edge states in a ribbon geometry
In this section, we study the quasi-energy band-structure of the model in an infinite zigzag nanoribbongeometry, with a finite width. We identify the four inte-gers C ↑ , C ↓ , C ↑ π , C ↓ π (defined later) that characterize Flo-quet topological insulators in our model, in each of thephases in Fig. 1 and 2, by choosing appropriate valuesof ω , α and lE z . A representative diagram for the phase A has been shown in Fig. 4 and the remaining diagramshave been relegated to the appendix. The spectrum hasbeen shown slightly beyond the ‘first Floquet-Brillouin zone’, − ω/ < (cid:15) b < ω/
2, so that the edge states at thezone boundary are clearly visible.The first point that we note is the gaps and the edgestates at the zone boundaries (at (cid:15) b = ω/ ≡ − ω/ (cid:15) = ± ω/ L ). The determination of thechirality of the edge state as shown on the graph is madeby actually checking whether the right-moving state (pos-itive slope) is at the left edge or at the right edge andsimilarly whether the left-moving slope (negative slope) isat the left or right edge. This can be done explicitly sincewe have numerically obtained all the wave-functions. Wecan now easily count the number of chiral edge states atthe band-gap at zero, and at the band gap at ω/
2, inthe various plots in the panels in Fig. 3 and in the ap-pendix. We choose a convention where a right-moving(positive slope in the energy versus momentum plot) atthe left L edge state is assigned a winding number orchirality − C σ by takingit to be − / + 1 depending on whether the L state ( orstates) in the band-gap at zero frequency is right-movingor left-moving and adding up the values. Similarly, inthe band-gap at frequency ω/
2, we compute C σπ by tak-ing − / + 1 for each right-moving/left-moving state andadding up the values. For instance, in Fig. 3, for the spin-up band, at zero frequency, there is a single edge state atthe left edge which has negative slope; thus C ↑ = +1. Atthe frequency ω/ C ↑ π = +1 as well. The FIG. 5: The Chern numbers for the ↑ spin sector is shown asa function of the disorder strength w for the phases A . . . G depicted in Fig.1. Static uniform disorder is included in thesystem as an on-site potential. Numerical calculations are car-ried out for a 24 ×
24 lattice with open boundary conditions.The Chern number is averaged over 100 disorder configura-tions and the following parameter values ( α , ω ) - (1.4,2.55),(0.3,4.5), (0.5,2.3), (0.2,2.5), (0.9,3.5), (1.6,5.0) were used forthe phases A , B , B (cid:48) , D , E and F respectively. ( B and C as wellas G and D are the same for ↑ spin as explained in the text). Chern number of the ↑ band in phase ( A ) was computedearlier to be C ↑ = 1 which precisely agrees with C ↑ − C ↑ π ,as expected from Ref. [11].Using the same method, C σ and C σπ can be computedfor each of the phases in Fig. 1 and 2 and the results aretabulated in Table 1. Note that, as expected, the Chernnumber of the band, C σ = C σ − C σπ in each case. Notealso that the phases A , B , C , E , F in the table are presentin both Figs. 1 and 2, whereas B (cid:48) , D and G occur only inFig.1 and H , I and J only in Fig. 2. III. DISCUSSIONS AND CONCLUSIONS
In comparison with earlier studies of irradiatedgraphene, the main difference for spin-orbit coupled ma- terials is the fact that the phase boundaries for the spin ↑ electrons and the spin ↓ electrons occur at differentpoints in the parameter space. Besides, due to the buck-ling, an external electric field can be applied which cantune the masses at the K and K (cid:48) points . This externaltuning parameter helps in finding new phases as seen inFig. 2, which do not exist in graphene.We have also studied the robustness of each of thephases in the presence of (uniform) disorder. The dis-order in the system is modeled as an on-site chemi-cal potential which is taken from a uniform distribu-tion [ w/ , w/ w is the strength of the disor-der in terms of the hopping parameter t . In Fig. 5, wehave plotted the Chern numbers of the various phasesin Fig. 1, computed using the coupling matrix approachfollowing Ref. [37] to obtain the real space Chern num-bers. We note that a number of the topological phasesare immune to uniform disorder for a reasonable rangeof the disorder strength, and starts degrading only forlarger values, whereas a few topological phases immedi-ately change their character even for a relatively smalldisorder. For a few of the phases, the robustness againstdisorder can be understood in terms of the respective val-ues of the quasi-energy gap in the system, but in certaincases (such as contrasting phase B and D , see Appendix),the robustness against disorder may not be simply relatedto the quasi-energy gap of the system. This is a surpris-ing outcome and is expected to be related to structureof the time dependent Hamiltonian and is a direction forfuture study. We also note, in passing, that the phase A , characterized by zero value of the topological invari-ant appears to attain the Floquet topological Andersoninsulator phase [38, 39] and exhibits two-lead quantizedcurrent at the infinite bias limit [40]. Further, the robust-ness of a certain phase also implies that any transportphenomena, such as a sum-ruled quantum Hall conduc-tance [41–43], should also be protected and might act assignatures to identify the individual phases. This is ofparticular importance, because the lack of knowledge ofthe occupation of the bands can be circumvented usingsignatures of the edge states. Acknowledgments
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In this appendix, we compute the Floquet bandstructure in a zigzag nano ribbon in all the differentphases which have been shown in Figs.1 and 2 in themain text. In the main text, the band diagram forphase A was already shown; here we show the edge-statespectrum for all the remaining phases. The name of thephase, as well as the values of C σ and C σπ are given inthe figure itself. As described in the main text, C σ and C σπ are computed by taking it to be − / + 1 dependingon whether the L state (or states) in the appropriate band-gap is right-moving or left-moving at the left edgeof the sample and adding up the values. Note that it isnot always to see visually determine whether or not thegap exists and in ambiguous cases, we have explicitlymentioned that it is gapped. Note also that in thediagrams of the phases G , H and I , the edge states areisolated from the bulk states at zero energy even thoughthe spectrum is not gapped ( or has an extremely smallgap). Thus the computation of the Chern numbersby counting edge states is more reliable than the bulkcomputation, which can numerically fail in the absenceof a well-defined gap. k x a x T ( i n un it s o f t ) L t C = 0C C = C - C C = 0C C = C - C LLL L (+2,+2) π/ √ ( ) π/ √ ( )−π π k x a x T ( i n un it s o f t ) L t C = 0 (gapped)C C = C - C C = 0 (gapped)C C = C - C LLL ' (+2,+2) π/ √ ( ) π/ √ ( ) −π π k x a x T ( i n un it s o f t ) L t C = 0C C = C - C C = 1C C = C - C LL L R (+2,+3) π/ √ ( ) π/ √ ( ) π/ √ ( ) π/ √ ( ) −π π k x a x T ( i n un it s o f t ) L t C = -1C C = C - C C = 0C C = C - C LLL L (+1,+2) π/ √ ( ) π/ √ ( ) −π π k x a x T ( i n un it s o f t ) L t C = 1C C = C - C C = 1C C = C - C LLL RR (+3,+3) π/ √ ( ) π/ √ ( ) −π π k x a x T ( i n un it s o f t ) t C = 1C C = C - C C = 1C C = C - C RR (+1,+1) π/ √ ( ) π/ √ ( ) −π π k x a x T ( i n un it s o f t ) L t C = -1C C = C - C C = -1C C = C - C LL LLLL (+1,+1) π/ √ ( )π/ √ ( ) −π π k x a x T ( i n un it s o f t ) L lE z = 0.4 t C = 0C C = C - C C = 0C C = C - C L L (+1,+1) π/ √ ( ) π/ √ ( ) −π π k x a x T ( i n un it s o f t ) L lE z = 0.9 t C = 0C C = C - C C = 0C C = C - C LLL (-2,-2) π/ √ ( ) π/ √ ( ) −π π k x a x T ( i n un it s o f t ) lE z = 0.8 t C = 0C C = C - C C = 0C C = C - C RR (-1,-1) π/ √ ( )π/ √ ( ) −π ππ