Decomposition of split-step quantum walks for simulating Majorana modes and edge states
Wei-Wei Zhang, Sandeep K. Goyal, Christoph Simon, Barry C. Sanders
DDecomposition of split-step quantum walks for simulatingMajorana modes and edge states
Wei-Wei Zhang,
1, 2, 3
Sandeep K. Goyal,
2, 4
Christoph Simon, and Barry C. Sanders
1, 2, 5, 6 Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics,University of Science and Technology of China, Hefei, Anhui 230026, China Institute for Quantum Science and Technology, and Department ofPhysics and Astronomy, University of Calgary, Canada, T2N 1N4 State Key Laboratory of Networking and Switching Technology,Beijing University of Posts and Telecommunications, Beijing, 100876, China Department of Physical Sciences, Indian Institute of Science Educationand Research Mohali, Sector 81, SAS Nagar, Punjab, 140306, India ∗ Shanghai Branch, CAS Center for Excellence and SynergeticInnovation Center in Quantum Information and Quantum Physics,University of Science and Technology of China, Shanghai 201315, China Program in Quantum Information Science, Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada
We construct a decomposition procedure for converting split-step quantum walks into ordinaryquantum walks with alternating coins, and we show that this decomposition enables a feasible linearoptical realization of split-step quantum walks by eliminating quantum-control requirements. Assalient applications, we show how our scheme will simulate Majorana modes and edge states.
I. INTRODUCTION
Topologically ordered quantum states demonstratemany interesting properties such as fractional statis-tics, spin liquids, and robust ground-state degeneracy,which are the basis of topological and fault-tolerantquantum computation [1–14]. One-dimensional discretetime quantum walks and their one- and two-dimensionalgeneralizations called split-step quantum walk (SSQW)exhibit a rich class of topological phases and exoticphases such as Majorana modes and edge states [15–22]. We present a procedure to decompose one- and two-dimensional SSQWs into ordinary one-dimensional quan-tum walks (OQWs) with alternating coins. Using this de-composition we propose simple implementation schemesto realize one- and two-dimensional SSQWs in the linearoptical setup.The interface of two distinct topological phases canhost topologically protected bounded states such as Ma-jorana modes and edge states [6, 15, 22, 23]. Majoranamodes [23] are quasiparticles which are their own antipar-ticles and the edge states are the low energy conductingstates which exist on the surface (or the edges) of an in-sulating material [10, 24, 25]. Edge states have been usedto understand topological insulators and the Hawking ra-diations in black holes [26, 27]. Discrete time quantumwalks provide controllable platforms to simulate and ma-nipulate these exotic phases [15–21].The OQW and the SSQW exhibit a large class oftopological phases, the Majorana modes and the edgestates [15, 17, 22]. Despite these advantages, the two-dimensional SSQW has never been implemented, whereasthe realization of the one-dimensional SSQW was re- ∗ [email protected] ported only in Ref. [28]. This is partly due to the dif-ficulty in implementing quantum walks in more than onedimension in a controllable way and partly due to aninadequate understanding of the SSQW.In this article, we show that in spite of having verydifferent propagators, the one- and the two-dimensionalSSQWs and the OQW are closely related; each step ofthe one-dimensional SSQW can be decomposed into twosteps of the OQW with alternating coin-flip operations.Similarly, every step of the two-dimensional SSQW canbe decomposed into two steps of one-dimensional SSQWperformed over two independent degrees of freedom insequence with the same coin.The decomposition of the SSQW in terms of theOQW presented here yields a direct relation betweenthe Hamiltonian of the OQW and the Hamiltonian forthe SSQW. This decomposition shows that the SSQWcan be thought of as a special case of alternate quantumwalks [29–31]. It also paves the way to simulate morecomplicated Hamiltonians using only the OQW. Further-more, it enables simple schemes to implement compli-cated quantum walks on any accessible systems. Usingour decompositions we present implementation schemesto realize one- and two-dimensional SSQWs using a linearoptical setup.In our scheme, the one-dimensional quantum walk isperformed over the orbital angular momentum (OAM)states of a single photon (or a light pulse) whereas weuse the time-bins along with the OAM of light to realizetwo-dimensional SSQWs. The polarization of light servesas the coin in our scheme. The proposed setup requiresa simple combination of wave plates, q -plate, polarizingbeamsplitters, and mirrors, and a ring interferometer isused to implement progressive steps in the walk.Since both classical light pulses as well as single pho-tons can possess the OAM and the time-bin degrees offreedom (DoFs), and the proposed setup for the imple- a r X i v : . [ qu a n t - ph ] M a y mentation consists of only linear optical elements, ourscheme works equally well for single photons and classi-cal light pulses. Simulating quantum protocols with clas-sical light offers advantages over the single photons suchas noninvasive and real-time measurements, which is notpossible with single photons [32–34]. Furthermore, clas-sical light is robust against losses and easy to produce.Our scheme is capable of simulating the Majorana modesand the edge states with classical light, and the setup sizedoes not increase with increasing number of steps in thewalk, as was the case in the earlier implementation [28].This is the first scheme where the realization of such ex-otic modes in two dimensions is addressed.The article is organized as follows: In Sec. II we de-scribe the one- and two-dimensional SSQWs, and thetopological properties of the underlying Hamiltonians.Section III deals with the decomposition of SSQWsinto OQWs. In Sec. IV we present our implementa-tion schemes and the methods to simulate the Majoranamodes and the edge states. We conclude in Sec. V. II. BACKGROUND
In this section, we present the relevant background ofthe OQW and SSQW, and the topological nature of thesequantum walks.
A. Ordinary and split-step quantum walk
We start with the OQW where the coin-flip operator C θ and the conditional propagator S are C θ ≡ C ( θ ) = cos θ − i sin θσ y , (1) S = F ⊗ |↑(cid:105) (cid:104)↑| + F † ⊗ |↓(cid:105) (cid:104)↓| . (2)Here, {|↑(cid:105) , |↓(cid:105)} are the two orthogonal states in the coinspace, {| x (cid:105) , x ∈ Z } are the position states of the walkersuch that F = (cid:80) x | x + 1 (cid:105) (cid:104) x | is the forward propagator,and σ y is the Pauli spin matrix along the y axis. Thecoin parameter θ ∈ ( − π, π ].Repeated action of the propagator Z ( θ ), Z ( θ ) = ( ⊗ C θ ) S, (3)on the states of the walker results in the quantum walkevolution. Note that the operator ¯ Z ( θ ) = S ( ⊗ C θ ) alsoyields the same quantum walk dynamics as the propa-gator Z ( θ ) for different initial states which are relatedto each other by the unitary operator S . Thus, the twooperators, Z ( θ ) and ¯ Z ( θ ), are equivalent. We shall callit the cyclic property of the quantum walk propagator.In a one-dimensional SSQW [15] the conditional propa-gator S (2) is divided into the left propagator T − and theright propagator T + which are separated by a coin-flipoperation C θ . Thus, the new quantum walk propagator S S S FIG. 1. Schematics representation of two-dimensional SSQWon a triangular lattice. Here we show the directions of the con-ditional propagators S and S defined in Eqs. (9) and (10),and S = S S . reads Z ss ( θ , θ ) = ( ⊗ C θ ) T − ( ⊗ C θ ) T + ≡ ( ⊗ C θ ) T θ , (4)where T + = F ⊗ |↑(cid:105) (cid:104)↑| + ⊗ |↓(cid:105) (cid:104)↓| , (5) T − = ⊗ |↑(cid:105) (cid:104)↑| + F † ⊗ |↓(cid:105) (cid:104)↓| , (6) T θ = T − ( ⊗ C θ ) T + . (7)The propagator Z ( θ , θ ) for the two-dimensionalanalog of a SSQW on a triangular lattice consists of threeconditional propagators S i applied in series, separated bythe coin operations C θ [15], Z ( θ , θ ) = S C θ S C θ S C θ , (8)where (see Fig. 1) S = ( F ⊗ ) ⊗ |↑(cid:105) (cid:104)↑| + ( F † ⊗ ) ⊗ |↓(cid:105) (cid:104)↓| , (9) S = ( ⊗ F ) ⊗ |↑(cid:105) (cid:104)↑| + ( ⊗ F † ) ⊗ |↓(cid:105) (cid:104)↓| (10)are the conditional propagators on the two principal axesin the triangular lattice and S = S S . The coin opera-tor C θ i = ⊗ ⊗ C θ i . B. Topological phases in quantum walk
The OQW (3) exhibits two distinct topological phaseswhich can be characterized by the sign of the parameter θ [15, 17]. The interface of these two topological phasessupports two bound states corresponding to (quasi-) en-ergy E = 0 , π [15, 17, 22]. The underlying Hamilto-nian of this dynamics possesses the particle-hole symme-try which implies that creating a particle with energy E is equivalent to annihilating a hole with energy − E .Since the two bound states in the quantum walk satisfy E = − E , the creation and the annihilation operatorsfor these states are the same. Thus, the correspondingmodes are the Majorana modes [22].Both one- and two-dimensional SSQWs exhibit topo-logical phases in the parameter space of θ , θ . Moreover,choosing site-dependent θ while keeping θ uniform overthe lattice can result in a boundary such that we ob-serve different topological phases on both sides of theboundary. Such boundaries support topologically pro-tected bound states in the one-dimensional SSQW andedge states in the two-dimensional case [15].Although SSQWs are known to simulate a large class oftopological phases, schemes to implement these quantumwalks in a controllable manner are not known. In thenext section, we present a procedure for converting theSSQW into the OQW. III. DECOMPOSING SSQW
In this section, we present the decomposition proce-dure for a single step of the one- and the two-dimensionalSSQWs in terms of OQWs. First, we decompose eachstep of the one-dimensional SSQW into two steps of anOQW with different coin operators (Sec. III A), and thenwe decompose the two-dimensional SSQW in terms oftwo one-dimensional SSQWs being performed on twodifferent degrees of freedom or two different lattices(Sec. III B).
A. Decomposing one-dimensional SSQW
Here, we show that a single step of the one-dimensionalSSQW is isomorphic to two steps of the OQW. This iso-morphism can be established easily by replacing the leftand right propagators T − and T + in Eqs. (5) and (6)by T − and T . Thus, the new one-dimensional SSQWpropagator reads˜ Z ss ( θ , θ ) = ( ⊗ C θ ) T − ( ⊗ C θ ) T . (11)Qualitatively, there is no difference between the propa-gator Z ss ( θ , θ ) (4) and the propagator ˜ Z ss ( θ , θ ) (11).The only difference is in the former one, in which thewalker jumps on the neighboring sites and in the latterone the walker skips one site in every jump.Since the one-dimensional SSQW is translation invari-ant, we can write˜ Z ss ( θ , θ ) = −→ T ˜ Z ss ( θ , θ ) ←− T = ( ⊗ C θ ) −→ T T − ( ⊗ C θ ) ←− T T = Z ( θ ) Z ( θ ) . (12)Here the coin-independent translation operators −→ T = ←− T † = F ⊗ commute with ˜ T ± operators. The operator S = −→ T T − = ←− T T is the conditional shift operator (2).Thus, the one-dimensional SSQW operator can be de-composed into two steps of the OQW propagators withalternating coin operators C θ and C θ . In other words,we can perform the one-dimensional SSQW on a latticewhich consists of only the even- (or the odd-) numberedlattice sites by performing two steps of the OQW withalternatively changing coin operators. B. Decomposing two-dimensional SSQW
A decomposition similar to (12) can also be ob-tained for the two-dimensional SSQW propagator Z ( θ , θ ) (8) in terms of two one-dimensional SSQWsperformed on independent one-dimensional lattices. Us-ing the definitions (9), (10), S = S S , and the cyclicproperty of the quantum walk propagators we can sim-plify the propagator Z ( θ , θ ) as Z ( θ , θ ) = S C θ S C θ S C θ S (13)= Z (2)ss (0 , θ ) Z (1)ss ( θ , θ ) , (14)where we have used the definition of the OQW propa-gator Z ( θ ) (3) and the decomposition (12). Here thesuperscript ( i ) ∈ { (1) , (2) } denotes the DoF the oper-ator is acting on. Equation (14) clearly shows that thetwo-dimensional SSQW on a triangular lattice can be de-composed into two one-dimensional SSQWs performed inseries on two different DoFs.One of the advantages of the decompositions (12)and (14) is that now we can write the Hamiltonian H ss = i ln[ Z ss ( θ , θ )] and H = i ln[ Z ( θ , θ )] whichgovern the dynamics in one- and two-dimensional SSQWsas a function of the Hamiltonian H θ = i ln[ Z ( θ )] ofOQWs (see Appendix). Hence, these decompositions of-fer an alternative way to express complicated Hamiltoni-ans in terms of simple well-understood one-dimensionalquantum walk Hamiltonians.So far, we have shown a procedure to express one- andtwo-dimensional SSQWs using only OQWs on differentDoFs or on different one-dimensional lattices. In thefollowing, we present optical implementation schemes tosimulate an SSQW in the OAM and the time-bin spaceof light. We also propose methods to simulate Majo-rana modes, edge states, and the topologically protectedbound states in these systems. IV. OPTICAL IMPLEMENTATION SCHEMESFOR SSQW
Here we present optical implementation schemes tosimulate the one- and the two-dimensional SSQWs. Wealso present a scheme to simulate the exotic phases suchas Majorana modes and edge states. The schemes pre-sented here are based on the decompositions constructedin the previous section, which make use of the one-dimensional OQW. Hence, our implementation schemefor a SSQW uses earlier schemes for OQWs [34–39].This section is organized as follows: In Sec. IV Awe detail the implementation of one-dimensional SSQWin OAM and in time-bins space. The implementationscheme for the two-dimensional SSQW is presented inSec. IV B. Section IV C contains the method to simulateMajorana modes and edge states in optical systems.
A. One-dimensional SSQW
The one-dimensional SSQW propagator Z ss (4) in theOAM space of light can be implemented by two propa-gators Z ( θ ) and Z ( θ ) in series (12). To implement theOQW propagator Z ( θ ) in the OAM space we can use thescheme presented in [34]. In this scheme, the OAM states {| (cid:96) (cid:105) , (cid:96) ∈ Z } represent the lattice sites and the right- ( | R (cid:105) )and the left-handed ( | L (cid:105) ) circular polarization states oflight represent the two orthogonal states of the coin [34].The coin-flip operator C θ is realized using the Simon-Mukunda polarization (SMP) gadget which consists ofa combination of two half-wave plates and two quarter-wave plates mounted in series [40]. By rotating the waveplates one can realize an arbitrary SU(2) rotation in thepolarization states of light.The conditional propagator S (2) in this scheme is re-alized by a combination of a half-wave plate and an op-tical device called q -plate. The q -plate is a linear opticaldevice which couples the OAM of light with its polar-ization. It is a birefringent plate made of a thin liquidcrystal film sandwiched between glass substrates with aphase retardation δ . Its inhomogeneous birefringence op-tical axis is distributed in space according to a singularpattern characterized by the topological charge q whichis the nematic-order defect exhibited in the center of theplate. Here q can be an integer or half-integer number[41, 42]. The action of the q -plate on the state of a lightbeam can be represented by the operator Q ( q ) δ as Q ( q ) δ = cos δ − i sin δ (cid:0) F q ⊗ | L (cid:105) (cid:104) R | + F † q ⊗ | R (cid:105) (cid:104) L | (cid:1) , (15)where F q = (cid:80) (cid:96) | (cid:96) + 2 q (cid:105) (cid:104) (cid:96) | represents the forward shiftoperator in the OAM of light. The phase retardation δ of the q -plate can be controlled by applying an exter-nal electric potential [41, 43, 44] and can take any valuebetween 0 and π .The conditional propagator S (2) is realized by setting q = 1 / δ = π/ q -plate, and a half-wave plate we canrealize the OQW propagator Z ( θ ) (3) which can performa single step of quantum walk on the OAM of light. Plac-ing two such setups in series in a ring interferometer, onecan simulate the one-dimensional SSQW.The one-dimensional SSQW in the time-bins spaceof light can be performed using the scheme presentedin [35, 36]. In this scheme, the time-bins form the one-dimensional lattice and the polarization serves as thecoin. The conditional propagator S (2) is realized bysplitting the incoming light pulse (or single photon) intotwo spatial modes using a polarizing beam splitter andintroducing different optical paths in the two spatialmodes. The SMP gadget is used as the coin operator C θ . Thus, a single step of the OQW is performed byintroducing different delays corresponding to the differ-ent orthogonal polarization states of light followed by anSMP gadget. Repeating this process twice with different SMP gadgets corresponding to the parameter θ and θ results in the one-dimensional SSQW.Although, the relation (12) between the SSQWsand the OQW makes the implementation of the one-dimensional SSQW feasible in almost any quantum sys-tem, a much-simplified scheme can be achieved while re-alizing the quantum walk in the OAM space. To see this,we recall the SSQW propagator Z ss ( θ , θ ) (4) which con-tains an effective shift operator T θ (7) and a coin-flip op-erator C θ . The operator T θ can be realized using only a q -plate and wave plates by choosing the phase retarda-tion δ = π/ − θ and q = 1 / q -plate; the operator Q ( q ) δ (see Appendix) reads Q (1 / δ = − i( ⊗ σ x ) (cid:2) cos θS + i sin θ ( ⊗ σ x ) (cid:3) . (16)Clearly, by redefining the coin-flip operator C θ in theSSQW propagator Z ss ( θ , θ ) in the following manner˜ C θ = e − i πσ z / C θ σ x e i πσ z / , (17)which can be realized by an SMP device, we can realizethe SSQW using a single q -plate and a single SMP gadgetinstead of using two of each as was done in the previousscheme.A similar scheme was implemented by Cardano et al. in [39] with a different coin operation. Although they alsoshowed the topological order in their experiment, it wasnot clear if their experiment yielded the one-dimensionalSSQW presented in [15]. From our discussion, this ques-tion is now settled.Next, we present an implementation scheme to performtwo-dimensional SSQWs on the triangular lattice. B. Two-dimensional SSQW
In this scheme, we use both the OAM and the time-binsto perform two-dimensional SSQWs. We choose theseDoFs because these are one of the most favored DoFs oflight to perform quantum walks [34–39]. A sketch of ourimplementation scheme is presented in Fig. 2.Here OAM is used as the first principal axis and time-bins are used as the second principal axis in the triangularlattice. Therefore, S represents the conditional propa-gator on OAM and S on time-bins. In order to realizethe Z (1)ss ( θ , θ ) in Eq. (14) we use the q -plate and anSMP device as was done in the one-dimensional SSQWin OAM space. The Z (2)ss (0 , θ ) = S C θ S operation inthe quantum walk is performed over the time-bins spaceof light, as discussed earlier. The details of the schemecan be found in the caption of Fig. 2. C. Simulating Majorana modes and edge states
To simulate the topologically protected bound states inquantum walks we need to create a boundary with two
PBS PBS PBS PBS S S C θ Z (1)ss ( θ , θ )B FIG. 2. Optical implementation scheme for the two-dimensional SSQW based on Eq. (14). Here the operator Z (1)ss ( θ , θ ) is performed over the OAM of light (lower part ofthe ring interferometer) and the operator Z (2)ss (0 , θ ) is per-formed over the time-bins. In this scheme, the incoming lightenters the setup through the beamsplitter B which has a veryhigh reflectance ( r ≈
1) (bottom left corner). The light en-tering the setup is first transformed by a combination of anSMP gadget and a phase-retarded q -plate and then by twotime-shift loops which are separated by an SMP gadget. Thetime-shift loops are realized by sorting the two orthogonal po-larization components of light [using polarizing beamsplitters,(PBS)] in different spatial modes with unequal path lengthswhich can be achieved by multimode optical fibers. One stepof the two-dimensional SSQW is completed upon completionof a full circle in the setup. Upon returning to the beam split-ter B a small fraction of the light will pass through the beamsplitter and the rest will be reflected back in the setup. Thetransmitted part of the light can be used to perform real-timemeasurements on the OAM and the time-bins. distinct topological phases on either side by assigningposition-dependent values to the coin parameter θ . It isoften hard to realize such coin operator. Here we proposea simple linear optical device, a generalized SMP gadget,which can be used to realize two different values of θ fordifferent sections of the OAM lattice.A light beam having the OAM proportional to thevalue (cid:96) (cid:126) has a ring-shaped intensity distribution with ra-dius r (cid:96) max of maximum intensity given by r (cid:96) max = w ( z ) (cid:114) | (cid:96) | , (18)i.e., the radius of the ring is proportional to the squareroot of the absolute value of (cid:96) of the OAM mode [45].Here w ( z ) is the width of the laser beam at the position z . Therefore, an SMP gadget with wave plates of radius r (cid:96) = ( r (cid:96) max + r (cid:96) +1max ) / − (cid:96) and (cid:96) . The rest of the modes will remain unchanged.A generalized SMP gadget consists of two coaxialSMP gadgets, one with radius r (cid:96) and the other, annu-lar shaped, with inner radius r (cid:96) and large outer radius(see Fig. 3). Since the two SMP gadgets used here areindependent of each other, they can be set to realize C θ and C θ (cid:48) coin operations which result in different coin op-erations on different sections of the OAM lattice.Using the generalized SMP device, different values ofthe parameters θ and θ (cid:48) can be chosen which correspond θ θ (cid:48) → → (cid:96)θ FIG. 3. Generalized SMP gadget. A generalized SMP gadgetis realized by coaxial placement of two SMP gadgets contain-ing wave plates of different radii. The radius r (cid:96) of the smallerSMP gadget is chosen such that it affects only the polariza-tion states of light corresponding to the OAM modes between − (cid:96) and (cid:96) . The rest of the polarization is transformed by thelarger SMP gadget. to distinct topological phases, thus making a bound-ary which supports bound states. In one-dimensionalSSQWs, if initially the walker is localized at the bound-ary, it remains localized with large probability. Thesebound states correspond to Majorana modes. Since thetwo-dimensional SSQW can be decomposed into two one-dimensional SSQWs and we need the boundary only inone direction, we can use the same generalized SMPgadget to simulate the edge states in two-dimensionalSSQWs. These bounded states can be observed as par-tially localized states, i.e., localized in the OAM spacebut spreading in the time-bin space. Both Majoranamodes and topologically protected edge states are robustagainst environmental interactions.Although the generalized SMP gadget provides a sharptransition in the value of the parameter θ , it may notbe sharp on the OAM lattice. This is because the ra-dius of the intensity ring for a given OAM mode (cid:96) is notsharp. This may cause an aberration in the bound statesif the parameter θ varies slightly across the boundary.However, if the two distinct topological phases requirewell-separated values of θ across the boundary, we willobserve topologically protected bound states. V. CONCLUSION
To conclude, we have presented schemes to realizeSSQWs on one- and two-dimensional lattices in opticalsystems. We have used the OAM and time-bin DoFs forour schemes. The key finding in this article which madethese implementations feasible is the decomposition ofSSQW in terms of OQWs. We have shown that a sin-gle step in a one-dimensional SSQW as defined in [15] isnothing but two steps of the OQW with alternating coins.Similarly, a two-dimensional SSQW (on a triangular lat-tice) can be decomposed as two one-dimensional SSQWsperformed on two independent DoFs in sequence. Wehave exploited this nature of SSQWs to simulate exotictopologically bound states.We can also interpret the decomposition of SSQWs interms of the OQWs as follows: the Hamiltonian whichgoverns the dynamics of a two-dimensional SSQW canbe simulated by the Hamiltonians of a one-dimensionalSSQW which in turn can be simulated by OQW Hamil-tonians. This decomposition can be extended to realizemore complicated quantum walks by incorporating mul-tiple steps of the one-dimensional OQW with differentcoin parameters. Thus, our decomposition brings us astep closer to realizing a universal quantum simulatorbased purely on quantum walks.
ACKNOWLEDGMENTS
SKG and CS acknowledge the support from NSERC.BCS thanks China’s 1000 Talent Plan and the Na-tional Natural Science Foundation of China GrantNo. 11675164, NSERC, and Alberta Innovates for fi-nancial support. WZ appreciates the financial supportfrom China’s 1000 Talent Plan and the National NaturalScience Foundation of China Grant No. 11675164, theChina Scholarship Council (Grant No. 201406470022),and NSERC.
Appendix A: Writing H ss and H in terms of H θ . The Hamiltonian H θ which governs the dynamics inthe ordinary one-dimensional quantum walk can be cal- culated by H θ = i log (cid:2) Z ( θ ) (cid:3) . (A1)Since the propagator Z ( θ ) is translation invariant, theHamiltonian H θ can be block diagonalized in momentumeigenbasis {| k (cid:105)} which are | k (cid:105) = (cid:88) x exp( − i kx ) | x (cid:105) , (A2)where {| x (cid:105)} represents the position basis. In the momen-tum basis the Hamiltonian H θ reads H θ = (cid:77) k ∈ [ − π,π ) H ( k ) , (A3)where H ( k ) = E ( k ) n ( k ) · σ . (A4)The vector n ( k ) = [ n ( k ) , n ( k ) , n ( k )] is a three-dimensional real vector. The explicit form of the vector n ( k ) and the energy E ( k ) is given by n ( k ) = sin θ sin k sin[ E ( k )] , (A5) n ( k ) = sin θ cos k sin[ E ( k )] , (A6) n ( k ) = − cos θ sin k sin[ E ( k )] , (A7) E ( k ) = cos − (cos θ cos k ) . (A8)The relation between the Hamiltonians H ss and H θ canbe derived using the decomposition (12) and the Baker-Campbell-Hausdorff formula [46] H ss ( k ) = H θ ( k ) + H θ ( k ) − i2 [ H θ ( k ) , H θ ( k )] − { [ H θ ( k ) , [ H θ ( k ) , H θ ( k )]] + [ H θ ( k ) , [ H θ ( k ) , H θ ( k )]] } · · · , (A9)whereas H ss = i ln[ Z ss ( θ , θ )] yields H ss ( k ) = E ss ( k )sin[ E ss ( k )] [cos E ( k ) sin E ( k ) n ( k ) + cos E ( k ) sin E ( k ) n ( k ) + sin E ( k ) sin E ( k ) n ( k ) × n ( k )] . σ (A10) ≡ E ss ( k ) N ( k ) · σ , (A11) E ss ( k ) = cos − [cos E ( k ) cos E ( k ) − sin E ( k ) sin E ( k ) n ( k ) · n ( k )] , (A12) E i ( k ) = cos − (cos θ i cos k ) . (A13)Here E i ( k ) n i · σ = H ( θ i ).Similarly, the Hamiltonian H can be written in terms of H ss as H ( k x , k y ) = H ss ( k x ) + H ss ( k y ) − i2 [ H ss ( k x ) , H ss ( k y )] − { [ H ss ( k x ) , [ H ss ( k x ) , H ss ( k y )]] + [ H ss ( k y ) , [ H ss ( k y ) , H ss ( k x )]] } · · · (A14)= E ( k x , k y )sin( E ( k x , k y )) (cos E ss ( k x ) sin E ss ( k y ) N ( k y ) + cos E ss ( k y ) sin E ss ( k x ) N ( k x )+ sin E ss ( k x ) sin E ss ( k y ) N ( k x ) × N ( k y )) . σ , (A15) E ( k x , k y ) = cos − [cos E ss ( k x ) cos E ss ( k y ) − sin E ss ( k x ) sin E ss ( k y ) N ( k x ) · N ( k y )] . (A16) [1] K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev.Lett. , 494 (1980).[2] D. J. Thouless, M. Kohmoto, M. P. Nightingale, andM. den Nijs, Phys. Rev. Lett. , 405 (1982).[3] R. B. Laughlin, Phys. Rev. Lett. , 1395 (1983).[4] X.-G. Wen, Adv. Phys. , 405 (1995).[5] H. L. Stormer, D. C. Tsui, and A. C. Gossard, Rev.Mod. Phys. , S298 (1999).[6] S. Ryu and Y. 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