The Z_2 network model for the quantum spin Hall effect: two-dimensional Dirac fermions, topological quantum numbers, and corner multifractality
Shinsei Ryu, Christopher Mudry, Hideaki Obuse, Akira Furusaki
aa r X i v : . [ c ond - m a t . m e s - h a ll ] D ec The Z network model for the quantum spin Halleffect: two-dimensional Dirac fermions, topologicalquantum numbers, and corner multifractality Shinsei Ryu , Christopher Mudry , Hideaki Obuse and AkiraFurusaki Department of Physics, University of California, Berkeley, CA 94720, USA Condensed Matter Theory Group, Paul Scherrer Institute, CH-5232 Villigen PSI,Switzerland Department of Physics, Kyoto University, Kyoto 606-8502, Japan Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, JapanE-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract.
The quantum spin Hall effect shares many similarities (and someimportant differences) with the quantum Hall effect for the electric charge. As withthe quantum (electric charge) Hall effect, there exists a correspondence between bulkand boundary physics that allows to characterize the quantum spin Hall effect indiverse and complementary ways. In this paper, we derive from the network modelthat encodes the quantum spin Hall effect, the so-called Z network model, a DiracHamiltonian in two dimensions. In the clean limit of this Dirac Hamiltonian, weshow that the bulk Kane-Mele Z invariant is nothing but the SU(2) Wilson loopconstructed from the SU(2) Berry connection of the occupied Dirac-Bloch single-particle states. In the presence of disorder, the non-linear sigma model (NLSM)that is derived from this Dirac Hamiltonian describes a metal-insulator transition inthe standard two-dimensional symplectic universality class. In particular, we showthat the fermion doubling prevents the presence of a topological term in the NLSMthat would change the universality class of the ordinary two-dimensional symplecticmetal-insulator transition. This analytical result is fully consistent with our previousnumerical studies of the bulk critical exponents at the metal-insulator transitionencoded by the Z network model. Finally, we improve the quality and extend thenumerical study of boundary multifractality in the Z topological insulator. We showthat the hypothesis of two-dimensional conformal invariance at the metal-insulatortransition is verified within the accuracy of our numerical results.PACS numbers: 73.20.Fz, 71.70.Ej, 73.43.-f, 05.45.Df ONTENTS Contents1 Introduction 22 Definition of the Z network model for the quantum spin Hall effect 53 Two-dimensional Dirac Hamiltonian from the Z network model 8 θ = 0 . . . . . . . . . . . . . . . . . . . . . . 103.3 Z topological number . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Z network model 23 Spin-orbit coupling has long been known to be essential to account for the band structureof semiconductors, say, semiconductors with the zink-blende crystalline structure.Monographs have been dedicated to reviewing the effects of the spin-orbit coupling onthe Bloch bands of conductors and semiconductors [1]. Electronic transport properties ofmetals and semiconductors in which impurities are coupled to the conduction electronsby the spin-orbit coupling, i.e., when the impurities preserve the time-reversal symmetrybut break the spin-rotation symmetry, are also well understood since the predictionof weak antilocalization effects [2]. Hence, the prediction of the quantum spin Halleffect in two-dimensional semiconductors with time-reversal symmetry but a sufficientlystrong breaking of spin-rotation symmetry is rather remarkable in view of the maturityof the field dedicated to the physics of semiconductors [3, 4, 5, 6]. The quantumspin Hall effect was observed in HgTe/(Hg,Cd)Te quantum wells two years later [7].Even more remarkably, this rapid progress was followed by the prediction of three-dimensional topological insulators [8, 9, 10] and its experimental confirmation for Bi-based compounds [11, 12, 13, 14, 15].The quantum spin Hall effect, like its relative, the quantum (electric charge) Halleffect, can be understood either as a property of the two-dimensional bulk or as aproperty of the one-dimensional boundary. The bulk can be characterized by certainintegrals over the Brillouin zone of Berry connections calculated from Bloch eigenstates.These integrals are only allowed to take discrete values and are examples of topologicalinvariants from the mathematical literature. As is well known, the topological number ν takes integer values for the quantum (electric charge) Hall effect [16]. By contrast,it takes only two distinct values ( ν = 0 or 1) for time-reversal invariant, Z topologicalband insulators [4, 8, 9, 10, 17]. Because they are quantized, they cannot change under a ONTENTS Z topological band insulator with ν = 1,there exists helical edge states, a Kramers’ pair of counter propagating modes, whichinterpolates between the bulk valence band and the bulk conduction band. If onechanges the Fermi energy from the center of the band gap to lower energies throughthe conduction band, one should observe a transition from a Z topological insulatorto a metal, and then from a metal to a trivial band insulator ( ν = 0) without helicaledge states. Since both helical edge states and a metallic phase are stable against(weak) disorder (due to the quantized topological number and to weak anti-localization,respectively), the same sequence of phases should appear as the Fermi energy is variedeven in the presence of disorder, as confirmed recently by numerical simulations [18, 19].A question one can naturally ask is then whether there is any difference between thecritical phenomena at the metal-to- Z -topological-insulator transition and those at themetal-to-trivial-insulator transition. This is the question which we revisit in this paper,extending our previous studies [19, 20]. It will become clear that one needs to distinguishbetween bulk and boundary properties in the universal critical phenomena.For the quantum (electric charge) Hall effect, the Chalker-Coddington networkmodel serves as a standard model for studying critical properties at Anderson transitionbetween different quantum Hall states [21]. The elementary object in the Chalker-Coddington network model is chiral edge states. These edge states are plane wavespropagating along the links of each plaquette which represents a puddle of a quantumHall droplet formed in the presence of spatially slowly varying potential. They arechiral as they represent the mode propagating along equipotential lines in the directiondetermined by the external magnetic field. The Chalker-Coddington network model isa unitary scattering matrix that scales in size with the number of links defining thenetwork, and with a deterministic parameter that quantifies the relative probability foran incoming mode to scatter into a link rotated by + π/ − π/
2. By tuning thisparameter through the value 1 /
2, one can go through a transition from one insulatingphase to another insulating phase, with the topological number ν changed by one.This remains true even when the phase of an edge state along any link is taken tobe an independent random number to mimic the effects of static local disorder. TheChalker-Coddington model is a powerful tool to characterize the effects of static disorderon the direct transition between two successive integer quantum Hall states. It hasdemonstrated that this transition is continuous and several critical exponents at thistransition have been measured from the Chalker-Coddington model [21, 22].The present authors have constructed in [19] a generalization of the Chalker-Coddington model that describes the physics of the two-dimensional quantum spin Hall ONTENTS Z network model, which will be brieflyreviewed in section 2. As with the Chalker-Coddington model, edge states propagatealong the links of each plaquette of the square lattice. Unlike the Chalker-Coddingtonmodel there are two edge states per link that form a single Kramers’ doublet, whichcorresponds to helical edge states moving along a puddle of a quantum spin Hall droplet.Kramers’ doublets undergo the most general unitary scattering compatible with time-reversal symmetry at the nodes of the square lattice. The Z network model is thusa unitary scattering matrix that scales in size with the number of links defining thenetwork and that preserves time-reversal symmetry. The Z network model supportsone metallic phase and two insulating phases, as we discussed earlier ‡ . The metallicphase prevents any direct transition between the insulating phases and the continuousphase transition between the metallic and any of the insulating phases belongs to thetwo-dimensional symplectic universality class of Anderson localization [2].Numerical simulations have shown that bulk properties at metal-insulator transitionin the Z network model are the same as those at conventional metal-insulator transitionsin the two-dimensional symplectic symmetry class [19, 20]. In fact, one can understandthis result from the following general argument based on universality. The non-linearsigma model (NLSM) description is a very powerful, standard theoretical approach toAnderson metal-insulator transition [23]. A NLSM can have a topological term if thehomotopy group of the target manifold, which is determined by the symmetry of thesystem at hand, is nontrivial. Interestingly, in the case of the symplectic symmetry class,as is called the statistical ensemble of systems (including quantum spin Hall systems)that are invariant under time reversal but are not invariant under SU(2) spin rotation,the NLSM admits a Z topological term [24, 25, 26]. Moreover, the NLSM in thesymplectic symmetry class with a Z topological term cannot support an insulatingphase. This can be seen from the fact that this NLSM describes surface Dirac fermionsof a three-dimensional Z topological insulator which are topologically protected fromAnderson localization [27, 28, 29]. This in turn implies that any two-dimensional metal-insulator transition in time-reversal-invariant but spin-rotation-noninvariant systemsshould be in the same and unique universality class that is encoded by the NLSMwithout a topological term in the (ordinary) symplectic class.Whereas bulk critical properties at the transition between a metal and a Z topological insulator do not depend on the topological nature of the insulating phase,there are boundary properties that can distinguish between a topologically trivial andnon-trivial insulating phases. Boundary multifractality is a very convenient tool toprobe any discrepancy between universal bulk and boundary properties at Andersontransition [30, 31]. To probe this difference, the present authors performed a multifractalanalysis of the edge states that propagate from one end to the other in a network modelat criticality with open boundary condition in the transverse direction [20]. It was foundthat boundary multifractal exponents are sensitive to the presence or absence of a helical ‡ The presence or absence of a single helical edge state in an insulating phase is solely dependent onthe boundary conditions which one imposes on the network model.
ONTENTS Z network model and a Hamiltoniandescription of the Z topological insulator perturbed by time-reversal symmetriclocal static disorder.(ii) to improve the quality and extend the numerical study of boundary multifractalityin the Z topological insulator.For item (i), in section 3, we are going to relate the Z network model to a problemof Anderson localization in the two-dimensional symplectic universality class that isencoded by a stationary 4 × × Z insulating phases in the 4 × Z topological invariant. In particular, we show that an SU(2) Wilson loop of Berryconnection of Bloch wave functions is equivalent to the Z index introduced by Kaneand Mele [4]. The 4 × Z network model and the NLSM description of two-dimensional Anderson localizationin the symplectic universality class derived 30 years ago by Hikami et al. in [2]. Inour opinion, this should remove any lingering doubts that the metal-insulator transitionbetween a two-dimensional metallic state and a two-dimensional Z insulator that isdriven by static disorder is anything but conventional.For item (ii), besides improving the accuracy of the critical exponents for one-dimensional boundary multifractality in the Z network model, we compute criticalexponents for two zero-dimensional boundaries (corners) in section 4. We shall usethese critical exponents to verify the hypothesis that conformal invariance holds at themetal-insulator transition and imposes relations between lower-dimensional boundarycritical exponents.
2. Definition of the Z network model for the quantum spin Hall effect The Z network model is defined as follows. First, one draws a set of corner sharingsquare plaquettes on the two-dimensional Cartesian plane. Each edge of a plaquette isassigned two opposite directed links. This is the network. There are two types S and S ′ of shared corners, which we shall call the nodes of the network. Second, we assign toeach directed link an amplitude ψ , i.e., a complex number ψ ∈ C . Any amplitude ψ iseither an incoming or outgoing plane wave that undergoes a unitary scattering processat a node. We also assign a 4 × S to each node of the network. The setof all directed links obeying the condition that they are either the incoming or outgoingplane waves of the set of all nodal unitary scattering matrices defines a solution to the Z network model. ONTENTS Figure 1. (a) The Z network model. The solid and dashed lines represent the linksfor up and down spin electrons, respectively. The electrons are unitarily scattered atthe nodes S and S ′ . The choice for the scattering basis at the nodes S and S ′ is shownin (b) and (c), respectively. To construct an explicit representation of the Z network model, the center of eachplaquette is assigned the coordinate ( x, y ) with x and y taking integer values, as isdone in figure 1. We then label the 8 directed links ψ nσ ( x, y ) of any given plaquette bythe coordinate ( x, y ) of the plaquette, the side n = 1 , , , σ = ↑ or σ = ↓ if the link is directedcounterclockwise or clockwise, respectively, relative to the center of the plaquette. The4 × S -matrix is then given by ψ ↑ ( x, y ) ψ ↓ ( x, y ) ψ ↑ ( x + 1 , y − ψ ↓ ( x + 1 , y − =: S ψ ↑ ( x, y ) ψ ↓ ( x, y ) ψ ↑ ( x + 1 , y − ψ ↓ ( x + 1 , y − (1)at any node of type S or as ψ ↑ ( x + 1 , y + 1) ψ ↓ ( x + 1 , y + 1) ψ ↑ ( x, y ) ψ ↓ ( x, y ) =: S ′ ψ ↑ ( x + 1 , y + 1) ψ ↓ ( x + 1 , y + 1) ψ ↑ ( x, y ) ψ ↓ ( x, y ) (2)at any node of type S ′ , with S = U ( x, y ) S V ( x, y ) , S ′ = U ′ ( x, y ) S V ′ ( x, y ) . (3)Here, the 4 × S := rs tQ − tQ † rs ! (4)is presented with the help of the unit 2 × s and of the 2 × Q := s sin θ + s cos θ = cos θ sin θ sin θ − cos θ ! , (5) ONTENTS s , s , and s are the 2 × σ = ↑ , ↓ , together with the real-valued parameters r := tanh X, t := 1cosh
X , (6)with { ( X, θ ) | ≤ X ≤ ∞ , ≤ θ ≤ π/ } . (7)For later use, we shall also introduce the real-valued parameter β ∈ [0 , π ] through r = cos β, t = sin β. (8)The parameter θ controls the probability of spin-flip scattering, sin θ . The unitarymatrices U, V, U ′ , V ′ are defined as U ( x, y ) = diag(e i χ ( x,y ) , e i χ ( x,y ) , e i χ ( x +1 ,y − , e i χ ( x +1 ,y − ) , (9 a ) V ( x, y ) = diag(e i χ ( x,y ) , e i χ ( x,y ) , e i χ ( x +1 ,y − , e i χ ( x +1 ,y − ) , (9 b ) U ′ ( x, y ) = diag(e i χ ( x +1 ,y +1) , e i χ ( x +1 ,y +1) , e i χ ( x,y ) , e i χ ( x,y ) ) , (9 c ) V ′ ( x, y ) = diag(e i χ ( x +1 ,y +1) , e i χ ( x +1 ,y +1) , e i χ ( x,y ) , e i χ ( x,y ) ) , (9 d )where 2 χ n ( x, y ) equals a (random) phase that wave functions acquire when propagatingalong the edge n of the plaquette centered at ( x, y ).The Z network model is uniquely defined from the scattering matrices S and S ′ .By construction, the S -matrix is time-reversal symmetric, i.e., i s
00 i s ! S ∗ − i s − i s ! = S † , (10)and a similar relation holds for S ′ .In [19], we obtained the phase diagram of the Z network model shown schematicallyin figure 2(a). Thereto, ( X, θ ) are spatially uniform deterministic parameters that canbe changed continuously. On the other hand, the phases χ n of all link plane waves inthe Z network model are taken to be independently and uniformly distributed randomvariables over the range [0 , π ). The line θ = 0 is special in that the Z network modelreduces to two decoupled Chalker-Coddington network models [19]. Along the line θ = 0, the point X CC = ln(1 + √ ⇐⇒ β = π θ can also be chosen to be randomly and independently distributed ateach node with the probability sin(2 θ ) over the range (0 , π/ X as the soledeterministic parameter that controls the phase diagram as shown in figure 2(b). Whenperforming numerically a scaling analysis with the size of the Z network model, onemust account for the deviations away from one-parameter scaling induced by irrelevantoperators. The Z network model with a randomly distributed θ minimizes such finite-size effects (see [19]). ONTENTS Figure 2. (a) Schematic phase diagram from the analysis of the Z network modelwith the constant X and θ . The metallic phase is surrounded by the two insulatingphases with the critical points X s and X l ( > X s )) for 0 < θ < π/
2. The fixed pointdenoted by a filled (green) square along the boundary θ = 0 is the unstable quantumcritical point located at X CC = ln(1 + √
2) separating two insulating phases in theChalker-Coddington model. The fixed point denoted by the filled (blue) rhombus atthe upper left corner is the unstable metallic phase. The shape of the metallic phase iscontrolled by the symmetry crossover between the unitary and symplectic symmetryclasses. (b) The phase diagram for Z network model with randomly distributed θ over the range (0 , π/
3. Two-dimensional Dirac Hamiltonian from the Z network model The Chalker-Coddington model is related to the two-dimensional Dirac Hamiltonian aswas shown by Ho and Chalker in [32]. We are going to establish the counterpart ofthis connection for the Z network model. A unitary matrix is the exponential of aHermitian matrix. Hence, our strategy to construct a Hamiltonian from the Z networkmodel is going to be to view the unitary scattering matrix of the Z network model as aunitary time evolution whose infinitesimal generator is the seeked Hamiltonian. To thisend, we proceed in two steps in order to present the Z network model into a form inwhich it is readily interpreted as a unitary time evolution. First, we change the choice ofthe basis for the scattering states and select the proper unit of time. We then performa continuum approximation, by which the Z network model is linearized, so to say.This will yield an irreducible 4-dimensional representation of the Dirac Hamiltonian in(2 + 1)-dimensional space and time, a signature of the fermion doubling when deriving acontinuum Dirac Hamiltonian from a time-reversal symmetric and local two-dimensionallattice model. Our goal is to reformulate the Z network model defined in Sec. 2 in such a way that thescattering matrix maps incoming states into outgoing states sharing the same internaland space labels but a different “time” label. This involves a change of basis for thescattering states and an “enlargement” of the Hilbert space spanned by the scatteringstates. The parameter θ is assumed to be spatially uniform. We choose the plaquette ONTENTS x, y ) of the network.At node S of the plaquette ( x, y ), we make the basis transformation and write the S -matrix (1) in the form ψ ↓ ψ ↓ ψ ↑ ψ ↑ =: M S ψ ↑ ψ ↑ ψ ↓ ψ ↓ , M S = U N S U , (12)where we have defined N S = − t t x − t y + sin θ t t x − t y + cos θ rt t x + t y − sin θ r − t t x + t y − cos θt t x + t y − cos θ r t t x + t y − sin θr − t t x − t y + cos θ − t t x − t y + sin θ (13)and U ( x, y ) = diag(e i χ ( x,y ) , e i χ ( x,y ) , e i χ ( x,y ) , e i χ ( x,y ) ) . (14)Here given n = 1 , , , σ = ↑ , ↓ , we have introduced the shift operators acting on ψ nσ ( x, y ), t x ± ψ nσ ( x, y ) := ψ nσ ( x ± , y ) , t y ± ψ nσ ( x, y ) := ψ n ( x, y ± , (15)and similarly on the phases χ n ( x, y ) ∈ [0 , π ). We note that the scattering matrix N S is multiplied by the unitary matrix U from the left and the right in (12), because theKramers’ doublet acquires exactly the same phase χ n when traversing on the edge n ofthe plaquette ( x, y ) before and after experiencing the scattering N S at the node S .At node S ′ of the plaquette ( x, y ), we make the basis transformation and rewritethe scattering matrix S ′ (2) into the form ψ ↑ ψ ↑ ψ ↓ ψ ↓ =: M S ′ ψ ↓ ψ ↓ ψ ↑ ψ ↑ , M S ′ = U N S ′ U , (16)where we have defined N S ′ = − t t x + t y + sin θ r − t t x + t y + cos θt t x − t y − sin θ t t x − t y − cos θ rr t t x + t y + cos θ − t t x + t y + sin θ − t t x − t y − cos θ r t t x − t y − sin θ . (17)As it should be M † S M S = M † S ′ M S ′ = 1 . (18)Next, we introduce the discrete time variable l ∈ Z as follows. We define theelementary discrete unitary time evolution to be ψ + ↓ ψ −↑ ψ + ↑ ψ −↓ l +1 := M S M S ′ ! ψ + ↓ ψ −↑ ψ + ↑ ψ −↓ l . (19) ONTENTS S and S ′ , we have enlarged the scatteringbasis with the introduction of the doublets ψ + := ψ ψ ! , ψ − := ψ ψ ! . (20)Due to the off-diagonal block structure in the elementary time evolution, it is moreconvenient to consider the “one-step” time evolution operator defined by ψ + ↓ ψ −↑ ψ + ↑ ψ −↓ l +2 = M S M S ′ M S ′ M S ! ψ + ↓ ψ −↑ ψ + ↑ ψ −↓ l ≡ M SS ′ M S ′ S ! ψ + ↓ ψ −↑ ψ + ↑ ψ −↓ l . (21)The two Hamiltonians generating this unitary time evolution are then H SS ′ := +i ln M SS ′ , H S ′ S := +i ln M S ′ S . (22)Evidently, the additivity of the logarithm of a product implies that H SS ′ = H S ′ S . (23)From now on, we will consider H SS ′ exclusively since M S ′ S = exp(i H S ′ S ) merelyduplicates the information contains in M SS ′ = exp(i H SS ′ ). θ = 0In this section, we are going to extract from the unitary time-evolution (21)–(23) ofthe Z network model a 4 × θ, β ) CC := (0 , π/ . (24)To this end and following [32], it is convenient to measure the link phases χ n ( n =1 , , ,
4) relative to their values when they carry a flux of π per plaquette. Hence, weredefine χ → χ + π H SS ′ = +i ln M SS ′ = +i (ln M S + ln M S ) = +i ln M S ′ S = H S ′ S (26)defined in (22) to leading order in powers of θ, m ≡ β − π , ∂ x,y ≡ ln t x,y + , χ n (27) ONTENTS n = 1 , , , ∂ x,y is the generator of infinitesimal translation on the network(the two-dimensional momentum operator).When θ = 0, the unitary time-evolution operator at the plaquette ( x, y ) is given by ψ + ↓ ψ −↑ ! l +2 = M (0) SS ′ ψ + ↓ ψ −↑ ! l , (28) M (0) SS ′ = A (0) D (0) B (0) C (0) ! , (29)whereby M (0) SS ′ = M (0) S M (0) S ′ , (30) M (0) S = A (0) B (0) ! , M (0) S ′ = C (0) D (0) ! , (31)with the 2 × A (0) := e i χ t x − t y + e i χ sin β ie i( χ + χ ) cos β e i( χ + χ ) cos β − ie i χ t x + t y − e i χ sin β ! , (32) B (0) := e i χ t x + t y − e i χ sin β e i( χ + χ ) cos β ie i( χ + χ ) cos β − ie i χ t x − t y + e i χ sin β ! , (33) C (0) := e i( χ + χ ) cos β − ie i χ t x + t y + e i χ sin β e i χ t x − t y − e i χ sin β ie i( χ + χ ) cos β ! , (34) D (0) := e i( χ + χ ) cos β e i χ t x + t y + e i χ sin β − ie i χ t x − t y − e i χ sin β ie i( χ + χ ) cos β ! . (35)Observe that in the limit θ = 0, the Z network model reduces to two decoupled U(1)network models where each time evolution is essentially the same as the one for theU(1) network model derived in [32].In the vicinity of the Chalker-Coddington quantum critical point (24), we find the4 × H (0) SS ′ = D + D − ! (36)where the 2 × × σ and of the Pauli matrices σ x , σ y , and σ z according to D + = σ z ( − i ∂ x + A x ) − σ x (cid:0) − i ∂ y + A y (cid:1) − σ y m + σ A , (37)and D − = − σ y ( − i ∂ x − A x ) + σ z (cid:0) − i ∂ y − A y (cid:1) + σ x m + σ A . (38)Thus, each 2 × A := − ( χ + χ + χ + χ ) , ( A x , A y ) := ( − χ + χ , χ − χ ) , (39)enter as a scalar gauge potential and a vector gauge potential would do, respectively. ONTENTS θ from θ = 0 lifts the reducibility of (36). To leading order in θ and close to the Chalker-Coddington quantum critical point (24), M SS ′ = (cid:16) M (0) S + θ M (1) S + · · · (cid:17) (cid:16) M (0) S ′ + θ M (1) S ′ + · · · (cid:17) = M (0) SS ′ + θ (cid:16) M (1) S M (0) S ′ + M (0) S M (1) S ′ (cid:17) + · · · (40)with M (1) S = A (1) B (1) ! , A (1) = 1 √ −
11 0 ! , B (1) = 1 √ − i 0 ! , (41) M (1) S ′ = C (1) D (1) ! , C (1) = 1 √ −
11 0 ! , D (1) = 1 √ − ii 0 ! , (42)where we have set m = χ n = 0 and t x,y ± = 1. We obtain H SS ′ = D + D θ D † θ D − ! , D θ := θ − i 1i 1 ! (43)to this order.Next, we perform a sequence of unitary transformation generated by U = e i πσ y /
00 e i πσ z / ! e − i πσ x /
00 e − i πσ x / ! e − i π/
00 e i π/ ! , (44)yielding H := U † H SS ′ U = H + ασ ασ H − ! (45)with α = √ θ and H ± = σ x ( − i ∂ x ± A x ) + σ y (cid:0) − i ∂ y ± A y (cid:1) ± σ z m + σ A . (46)The 2 × H + and H − describe a Dirac fermion with mass ± m in the presence ofrandom vector potential ± ( A x , A y ) and random scalar potential A , each of which is aneffective Hamiltonian for the plateau transition of integer quantum Hall effect [32, 33].The H ± sectors are coupled by the matrix element ασ .The 4 × H can be written in the form H = ( − i ∂ x σ x − i ∂ y σ y ) ⊗ τ + ( A x σ x + A y σ y + mσ z ) ⊗ τ z + A σ ⊗ τ + α σ ⊗ τ x , (47)where τ is a unit 2 × τ x , τ y , and τ z are three Pauli matrices. TheHamiltonian (47) is invariant for each realization of disorder under the operation T H ∗ T − = H , T := i σ y ⊗ τ x , (48)that implements time-reversal for a spin-1/2 particle.The Dirac Hamiltonian (47) is the main result of this subsection. It is an effectivemodel for the Anderson localization of quantum spin Hall systems, which belongs to ONTENTS Z network model. In the presence ofthe “Rashba” coupling α , there should appear a metallic phase near m = 0 which issurrounded by two insulating phases. In the limit α →
0, the metallic phase shouldshrink into a critical point of the integer quantum Hall plateau transition.The 4 × H should be contrasted with a 2 × H = − i ∂ x σ x − i ∂ y σ y + V ( x, y ) σ , (49)which has the minimal dimensionality of the Clifford algebra in (2 + 1)-dimensionalspace time and is invariant under time-reversal operation, σ y H ∗ σ y = H . The 2 × Z topological insulator. After averaging over the disorder potential V , the problem of Anderson localization of the surface Dirac fermions is reduced toa NLSM with a Z topological term [25, 26]. Interestingly, this Z topological termprevents the surface Dirac fermions from localizing [27, 28]. It is this absence of two-dimensional localization that defines a three-dimensional Z topological insulator [29].In contrast, the doubling of the size of the Hamiltonian (47) implies that the NLSMdescribing the Anderson localization in the 4 × Z topological term, because two Z topological terms cancel each other. We can thusconclude that the critical properties of metal-insulator transitions in the Z networkmodel are the same as those in the standard symplectic class, in agreement with resultsof our numerical simulations of the Z network model [19, 20].Before closing this subsection, we briefly discuss the Dirac Hamiltonian (47) in theclean limit where A = A x = A y = 0. Since the system in the absence of disorder istranslationally invariant, momentum is a good quantum number. We thus consider theHamiltonian in momentum space H ( k ) = k x σ x + k y σ y + mσ z ασ ασ k x σ x + k y σ y − mσ z ! , (50)where the wave number k = ( k x , k y ). When α = 0, the Hamiltonian (50) becomes adirect sum of 2 × Z topological number We now discuss the topological property of the time-reversal invariant insulator whichis obtained from the effective Hamiltonian (50) of the Z network model in the absenceof disorder. The topological attribute of the band insulator is intimately tied to theinvariance ˆΘ − H ( − k ) ˆΘ = H ( k ) (51) ONTENTS σ y ⊗ τ x ) K = − ˆΘ − , (52)where K implements complex conjugation. We are going to show that this topologicalattribute takes values in Z , i.e., the Z index introduced by Kane and Mele [4].We begin with general considerations on a translation-invariant single-particlefermionic Hamiltonian which has single-particle eigenstates labeled by the wave vector k taking values in a compact manifold. This compact manifold can be the first Brillouinzone with the topology of a torus if the Hamiltonian is defined on a lattice andperiodic boundary conditions are imposed, or it can be the stereographic projectionbetween the momentum plane R and the surface of a three-dimensional sphere if theHamiltonian is defined in the continuum. We assume that (i) the antiunitary operationˆΘ = − ˆΘ − = − Θ † that implements time-reversal leaves the Hamiltonian invariant,(ii) there exists a spectral gap at the Fermi energy, and (iii) there are two distinctoccupied bands with the single-particle orthonormal eigenstates | u ˆ a ( k ) i and energies E ˆ a ( k ) labeled by the index ˆ a = 1 , × m is nonvanishing.Because of assumptions (i) and (ii) the 2 × w ˆ a ˆ b ( k ) defined by w ˆ a ˆ b ( k ) := h u ˆ a ( − k ) | (cid:16) ˆΘ | u ˆ b ( k ) i (cid:17) ≡ (cid:10) u ˆ a ( − k ) (cid:12)(cid:12) Θ u ˆ b ( k ) (cid:11) , ˆ a, ˆ b = 1 , , (53)i.e., the overlaps between the occupied single-particle energy eigenstates with momentum − k and the time reversed images to the occupied single-particle energy eigenstates withmomentum k , plays an important role [17]. The matrix elements (53) obey w ˆ a ˆ b ( k ) ≡ h u ˆ a ( − k ) | (cid:16) ˆΘ | u ˆ b ( k ) i (cid:17) = h u ˆ b ( k ) | (cid:16) ˆΘ † | u ˆ a ( − k ) i (cid:17) = −h u ˆ b ( k ) | (cid:16) ˆΘ | u ˆ a ( − k ) i (cid:17) ≡ − w ˆ b ˆ a ( − k ) , ˆ a, ˆ b = 1 , . (54)We used the fact that ˆΘ is antilinear to reach the second equality and that it isantiunitary with ˆΘ = − × w ( k ) with the matrix elements (53) can be parametrized as w ( k ) = w ( k ) w ( k ) − w ( − k ) w ( k ) ! = − w T ( − k ) (55)with the three complex-valued functions w ( k ) = − w ( − k ) , w ( k ) = − w ( − k ) , w ( k ) . (56)We observe that w ( k ) reduces to w ( k ) = e i f ( k ) − ! (57) ONTENTS f ( k ) at any time-reversal invariant wave vector k ∼ − k (time-reversal invariant wave vectors are half a reciprocal vector for a lattice model, and 0 or ∞ for a model in the continuum).As we shall shortly see, the sewing matrix (53) imposes constraints on the U(2)Berry connection A ˆ a ˆ b ( k ) := (cid:10) u ˆ a ( k ) (cid:12)(cid:12) d u ˆ b ( k ) (cid:11) ≡ (cid:28) u ˆ a ( k ) (cid:12)(cid:12)(cid:12)(cid:12) ∂∂k µ u ˆ b ( k ) (cid:29) d k µ ≡ A µ ˆ a ˆ b ( k )d k µ , (58)where the summation convention over the repeated index µ is understood (we donot make distinction between superscript and subscript). Here, at every point k inmomentum space, we have introduced the U(2) antihermitian gauge field A µ ( k ) withthe space index µ = 1 , A µ ˆ a ˆ b ( k ) = − (cid:16) A µ ˆ b ˆ a ( k ) (cid:17) ∗ (59)labeled with the U(2) internal indices ˆ a, ˆ b = 1 ,
2, by performing an infinitesimalparametric change in the Hamiltonian. We decompose the U(2) gauge field (58) intothe U(1) and the SU(2) contributions A µ ( k ) ≡ a µ ( k ) ρ
2i + a µ ( k ) · ρ , (60)where ρ is a 2 × ρ is a 3 vector made of the Pauli matrices ρ x , ρ y ,and ρ z . Accordingly, A U(2) ( k ) = A U(1) ( k ) + A SU(2) ( k ) . (61)Combining the identity ˆΘ = − P ˆ a =1 , | u ˆ a ( k ) ih u ˆ a ( k ) | for the occupied energy eigenstates with momentum k yields X ˆ a =1 , ˆΘ | u ˆ a ( k ) ih u ˆ a ( k ) | ˆΘ = − , (62)where the proper restriction to the occupied energy eigenstates is understood for the unitoperator on the right-hand side. Using this identity, we deduce the gauge transformation A µ ( − k ) = − (cid:18)(cid:28) u ˆ a ( − k ) (cid:12)(cid:12)(cid:12)(cid:12) ∂∂k µ u ˆ b ( − k ) (cid:29)(cid:19) ˆ a, ˆ b =1 , = − w ( k ) A ∗ µ ( k ) w † ( k ) − w ( k ) ∂ µ w † ( k )= + w ( k ) A T µ ( k ) w † ( k ) − w ( k ) ∂ µ w † ( k ) (63)that relates the U(2) connections at ± k . For the U(1) and SU(2) parts of the connection, a µ ( − k ) = a µ ( k ) − ∂ µ ζ ( k ) , (64) a µ ( − k ) · ρ = a µ ( k ) · ˜ w ( k ) ρ T ˜ w † ( k ) −
2i ˜ w ( k ) ∂ µ ˜ w † ( k ) , (65)where we have decomposed w ( k ) into the U(1) (e i ζ ) and SU(2) ( ˜ w ) parts according to w ( k ) = e i ζ ( k ) ˜ w ( k ) , (66)(note that this decomposition has a global sign ambiguity, which, however, will notaffect the following discussions). ONTENTS W U(2) [ C ] := 12 tr P exp I C A U(2) ( k ) = W U(1) [ C ] × W SU(2) [ C ] , (67)where the U(1) Wilson loop is given by W U(1) [ C ] := exp I C A U(1) ( k ) , (68)while the SU(2) Wilson loop is given by W SU(2) [ C ] := 12 tr P exp I C A SU(2) ( k ) . (69)The symbol P in the definition of the U(2) Wilson loop represents path ordering, while C is any closed loop in the compact momentum space.By construction, the U(2) Wilson loop (67) is invariant under the transformation A µ ( k ) → U † ( k ) A µ ( k ) U ( k ) + U † ( k ) ∂ µ U ( k ) (70)induced by the local (in momentum space) U(2) transformation | u ˆ a ( k ) i → | u ˆ b ( k ) i U ˆ b ˆ a ( k ) (71)on the single-particle energy eigenstates. Similarly, the SU(2) and U(1) Wilson loopsare invariant under any local SU(2) and U(1) gauge transformation of the Bloch wavefunctions, respectively.When C is invariant as a set under k → − k , (72)the SU(2) Wilson loop W SU(2) [ C ] is quantized to the two values W SU(2) [ C ] = ± W SU(2) [ C ] = K ∼− K Y K ∈C Pf (cid:16) ˜ w ( K ) (cid:17) , (74)which we will prove below, follows. Here, the symbol Pf denotes the Pfaffian of anantisymmetric matrix, and only the subset of momenta K ∈ C that are unchangedunder K → − K contribute to the SU(2) Wilson loop. According to (54), the sewingmatrix at a time-reversal symmetric wave vector is an antisymmetric 2 × × ρ y up to asign). Hence, its Pfaffian is a well-defined and nonvanishing real-valued number.Before undertaking the proof of (74), more insights on this identity can be obtainedif we specialize to the case when the Hamiltonian is invariant under any U(1) subgroup ONTENTS z -component of spin σ z . In this case we can choose the basis stateswhich diagonalize σ z ; σ z | u ( k ) i = + | u ( k ) i , σ z | u ( k ) i = −| u ( k ) i . Since the time-reversal operation changes the sign of σ z , the sewing matrix takes the form w ( k ) = − i χ ( k ) − e − i χ ( − k ) ! , (75)which, in combination with (64) and (65) implies the transformation laws a µ ( − k ) = + a µ ( k ) + ∂ µ [ χ ( k ) + χ ( − k )] , (76) a zµ ( − k ) = − a zµ ( k ) + ∂ µ [ χ ( k ) − χ ( − k )] . (77)We conclude that when both the z component of the electron spin and the electronnumber are conserved, we can set a xµ ( k ) = a yµ ( k ) = 0 , A U(2) µ ( k ) = a µ ( k ) σ
2i + a zµ ( k ) σ z , (78)and use the transformation law A U(2) ν, ( − k ) = 12i (cid:2) a ν ( − k ) + a zν ( − k ) (cid:3) = A U(2) ν, ( k ) − i ∂ ν χ ( k ) . (79)With conservation of the z component of the electron spin in addition to that ofthe electron charge, the SU(2) Wilson loop becomes W SU(2) [ C ] = 12 tr P exp I C A SU(2) ( k ) (80)= 12 tr exp I C a zµ ( k ) σ z
2i d k µ = cos I C a zµ ( k )d k µ . (81)We have used the fact that σ z is traceless to reach the last line. This line integral canbe written as the surface integral I C a zµ ( k )d k µ = Z D d k ε µν ∂ µ a zν ( k ) (82)by Stokes’ theorem. Here, D is the region defined by ∂ D = C , and covers a half of thetotal Brillouin zone (BZ) because of the condition (72). In turn, this surface integral isequal to the Chern number for up-spin fermions,Ch ↑ := Z BZ d k π i ε µν ∂ µ A U(2) ν, ( k ) (83) ≡ Z BZ d k π i F U(2)11 ( k )= Z D d k π i h F U(2)11 ( k ) + F U(2)11 ( − k ) i ONTENTS Z D d k π i ε µν ∂ µ h A U(2) ν, ( k ) − A U(2) ν, ( k ) i = − i Z D d k π i ε µν ∂ µ a zν ( k ) , (84)where we have used the transformation law (79) to deduce that F U(2)11 ( − k ) = − F U(2)22 ( k ) (85)to reach the fourth equality.To summarize, when the z component of the spin is conserved, the quantized SU(2)Wilson loop can then be written as the parity of the spin Chern number (the Chernnumber for up-spin fermions, which is equal to minus the Chern number for down-spinfermions) [3, 4, 5], W SU(2) [ C ] = ( − Ch ↑ . (86)Next, we apply the master formula (74) to the 4 × m by the k -dependent mass, m k = m − C k , C > , (87)and parametrize the wave number k as k x + i k y = k e i ϕ , −∞ < k < ∞ , ≤ ϕ < π. (88)Without loss of generality, we may assume α >
0. The mass m k is introduced so thatthe SU(2) part of the sewing matrix is single-valued in the limit | k | → ∞ .We then perform another series of unitary transformation with e U = σ
00 i σ z ! σ √ σ √ − σ √ σ √ ! e i πσ z /
00 e − i πσ z / ! , (89)to rewrite the Hamiltonian (50) in the form e H ( k ) := e U † H ( k ) e U = k x σ x + k y σ y + ( α − i m k ) σ k x σ x + k y σ y + ( α + i m k ) σ ! . (90)The four eigenvalues of the Hamiltonian (90) are given by E ( k ) = ± λ + k , ± λ − k , where λ ± k = q ( k ± α ) + m k . (91)The occupied eigenstate with the energy E ( k ) = − λ − k reads | u ( ϕ, k ) i = 12 λ − k − λ − k λ − k e − i ϕ − k + α + i m k ( k − α − i m k )e − i ϕ , (92) ONTENTS Table 1. θ ± k at the time-reversal invariant momenta k = 0 and k = ±∞ , when m − Cα < m − Cα > ≤ arctan( | m | /α ) ≤ π/ (a) m − Cα < k −∞ ∞ θ + k π/ − m/α ) π/ θ − k − π/ − m/α ) − π − π/ i( θ + k − θ − k ) / i i i(b) m − Cα > k −∞ ∞ θ + k − π/ − arctan( m/α ) π/ θ − k π/ π − arctan( m/α ) − π/ i( θ + k − θ − k ) / i − i iand the occupied eigenstate with the energy E ( k ) = − λ + k is | u ( ϕ, k ) i = 12 λ + k − λ + k − λ + k e − i ϕ k + α + i m k ( k + α + i m k )e − i ϕ . (93)Notice that | u ( ϕ, k ) i = | u ( ϕ + π, − k ) i .The 2 × w ( k ) is obtained from the eigenstates (92)–(93) as w ( ϕ, k ) := (cid:16) h u ˆ a ( ϕ, − k ) | ˆΘ | u ˆ b ( ϕ, k ) i (cid:17) ˆ a, ˆ b =1 , = − e i ϕ λ + k ( k + α − i m k )1 λ − k ( k − α + i m k ) 0 , (94)which is decomposed into the U(1) part,i exp (cid:16) i ϕ + i( θ + k + θ − k ) / (cid:17) , (95)and the SU(2) part,˜ w ( k ) = i( θ + k − θ − k ) / ie − i( θ + k − θ − k ) / ! , (96)of the sewing matrix. Here, we have defined θ ± k through the relatione i θ ± k = 1 λ ± k [ k ± ( α − i m k )] . (97)For the SU(2) sewing matrix (96), there are two momenta which are invariant underinversion k → − k , namely the south K = 0 and north K = ∞ poles of the stereographicsphere. The values of θ ± k at these time-reversal momenta are listed in table 1. The ONTENTS (a) (b) QSH QSH
Figure 3. (a) Quantum spin Hall droplet immersed in the reference vacuum [in realspace ( x, y ) ∈ R ]. (b) The Z network model or its tight-binding equivalent when x < x > x = 0[in real space ( x, y ) ∈ R ]. Pfaffian of the sewing matrix at the south and north poles of the stereographic sphereare Pf ˜ w (0) = − sgn( m − Cα )Pf ( − i ρ y ) , (98)Pf ˜ w ( ∞ ) = Pf ( − i ρ y ) , (99)respectively. Hence, W SU(2) [ C ] = − sgn( m ) (100)for any time-reversal invariant path C passing through the south and north poles, wherewe have suppressed Cα by taking the limit Cα / | m | → m . This ambiguity is a mere reflection of the fact that,as noted in [20], the topological nature of the Z network model is itself defined relativeto that of some reference vacuum. Indeed, for any given choice of the parameters ( X, θ )from figure 2 that defines uniquely the bulk properties of the insulating phase in the Z network model, the choice of boundary conditions determines if a single helical Kramers’doublet edge state is or is not present at the boundary of the Z network model. In viewof this, it is useful to reinterpret the Z network model with a boundary as realizing aquantum spin Hall droplet immersed in a reference vacuum as is depicted in figure 3(a).If so, choosing the boundary condition is equivalent to fixing the topological attribute ofthe reference vacuum relative to that of the Z network model, for the reference vacuumin which the quantum spin Hall droplet is immersed also has either a trivial or non-trivial Z quantum topology. A single helical Kramers’ doublet propagating unhinderedalong the boundary between the quantum spin Hall droplet and the reference vacuumappears if and only if the Z topological quantum numbers in the droplet and in thereference vacuum differ.In the low-energy continuum limit (50), a boundary in real space can be introducedby breaking translation invariance along the vertical line x = 0 in the real space ONTENTS kk y x
20 468141210
Figure 4.
Momentum space ( k x , k y ) ∈ R is discretized with the help of a rectangulargrid on which two paths are depicted. The red path that is restricted to the upper leftquadrant is not invariant as a set under the inversion ( k x , k y ) → − ( k x , k y ). The bluepath with its center of mass at the origin is. This path is assembled out of 16 links:( i , i ) = − ( i , i ), ( i , i ) = − ( i , i ), ( i , i ) = − ( i , i ), ( i , i ) = − ( i , i ),( i , i ) = − ( i , i ), ( i , i ) = − ( i , i ), ( i , i ) = − ( i , i ), ( i , i ) = − ( i , i ). Sites i = i = i along the path are the only ones invariant under ( k x , k y ) → − ( k x , k y ). ( x, y ) ∈ R through the profile [see figure 3(b)] m ( x, y ) = m ( x ) = − m, if x → −∞ ,+ m, if x → + ∞ , (101)for the mass.We close Sec. 3 with a justification of the master formula (74). To this end, weregularize the continuum gauge theory by discretizing momentum space (figure 4). Weuse the momentum coordinate i ∈ Z on a rectangular grid with the two lattice spacings∆ k µ >
0. To each link from the site i to the nearest-neighbor site i + µ of the grid, weassign the SU(2) unitary matrix U i,i + µ ≡ e A i,i + µ ∆ k µ , (102)which is obtained by discarding U(1) part of the U(2) Berry connection. Consistencydemands that U i + µ,i = U † i,i + µ ⇐⇒ A i + µ,i = A † i,i + µ . (103)We define the SU(2) Wilson loop to be W SU(2) ( i , · · · , i N − ) := 12 tr (cid:16) U i ,i U i ,i · · · U i N − ,i (cid:17) (104) ONTENTS i n and i n +1 are nearest neighbors, i.e., their difference i n +1 − i n = η n is a unitvector η n . The Wilson loop is invariant under any local gauge transformation by which U i,i + µ → V † i U i,i + µ V i + µ (105)where the V i ’s are U(2) matrices. Observe that the cyclicity of the trace allows us towrite W SU(2) ( i , · · · , i N − ) = 12 tr (cid:18) U i N ,i N · · · U i N − ,i U i ,i U i ,i · · · U i N − ,i N (cid:19) . (106)To make contact with the master formula (74), we assume that the closed path withvertices i ℓ parametrized by the index ℓ = 0 , , · · · , N − i N − n is the wave vector − P nm =1 η m ∆ k µ m , ... i N − is the wave vector − η ∆ k µ ,i is the wave vector 0 , (107) i is the wave vector + η ∆ k µ , ... i n is the wave vector + P nm =1 η m ∆ k µ m , ...with η m = ± m = 1 , · · · , N/ w i ) ˆ a ˆ b := h u ˆ a ( − i ) | ˆΘ | u ˆ b ( i ) i , (108)which obeys the condition w − i = − w T i , (109)i.e., the counterpart to the relation (54). This implies that w i and w i N/ areantisymmetric unitary 2 × w i must alsoobey the counterpart to (63), namely U − j, − i = w j U T i,j w † i . (110)It now follows from (107) and (110) that U i N − ,i = w i U T i ,i w † i , ... U i N − − n ,i N − n = w i n +1 U T i n ,i n +1 w † i n , (111)... U i N ,i N = w i N U T i N − ,i N w † i N − . ONTENTS U i N − ,i U i ,i = w i U T i ,i w † i U i ,i = w i w † i U † i ,i U i ,i = w i w † i , (112)since w i is a 2 × w i is the second Pauli matrixup to a phase factor, while( ρ · n ) T ρ = − ρ ( ρ · n ) (113)holds for any three-vector n contracted with the three-vector ρ made of the three Paulimatrices. By repeating the same exercise a second time, U i N − ,i N − (cid:16) U i N − ,i U i ,i (cid:17) U i ,i = w i U T i ,i w † i (cid:16) w i w † i (cid:17) U i ,i = w i w † i , (114)one convinces oneself that the dependences on the gauge fields A i ,i and A i N − ,i , A i ,i and A i N − ,i N − , and so on until A i n − ,i n and A i N − n ,i N − n +1 at the level n of this iterationcancel pairwise due to the conditions (107)–(110) implementing time-reversal invariance.This iteration stops when n = N/
2, in which case the SU(2) Wilson loop is indeed solelycontrolled by the sewing matrix at the time-reversal invariant momenta correspondingto ℓ = 0 and ℓ = N/ W SU(2) ( i , · · · , i N − ) = 12 tr (cid:16) w i N/ w † i (cid:17) . (115)Since i and i N/ are invariant under momentum inversion or, equivalently, time-reversal invariant, w i N/ = e i α N/ i ρ , w i = e i α i ρ (116)with α N/ , α = 0 , π . Here, the Z phases e i α N/ and e i α are none other than the Pfaffianse i α N/ = Pf (cid:16) w i N/ (cid:17) , e i α = Pf (cid:16) w i (cid:17) , (117)respectively. Hence, W SU(2) ( i , · · · , i N − ) = 12 tr h Pf (cid:16) w i N/ (cid:17) i ρ × Pf (cid:16) w † i (cid:17) ( − i ρ ) i = Pf (cid:16) w i N/ (cid:17) Pf (cid:16) w i (cid:17) (118)is a special case of (74). (Recall that w i and w i N/ are real-valued.)
4. Numerical study of boundary multifractality in the Z network model In [20], we have shown that (i) multifractal scaling holds near the boundary of the Z network model at the transition between the metallic phase and the Z topologicalinsulating phase shown in figure 2, (ii) it is different from that in the ordinary symplecticclass, while (iii) bulk properties, such as the critical exponents for the divergence ofthe localization length and multifractal scaling in the bulk, are the same as those inthe conventional two-dimensional symplectic universality class of Anderson localization.This implies that the boundary critical properties are affected by the presence of the ONTENTS Z -topological-insulatortransition. We thereby support the claim that conformal invariance is present atthe metal-to- Z -topological-insulator transition by verifying that conformal relationsbetween critical exponents at these boundaries hold. To characterize multifractal scaling at the metal-insulator transition in the Z networkmodel, we start from the time-evolution of the plane waves along the links of the networkwith the scattering matrices defined in (1)-(7) at the nodes S and S ′ . To minimize finitesize effects, the parameter θ in (1)-(7) is chosen to be a random variable as explainedin Sec. 2. We focus on the metal-insulator transition at X = X l = 0 .
971 as shown infigure 2(b).When we impose reflecting boundary conditions, a node on the boundary reducesto a unit 2 × S ′ , as shown in figure 5(a), there exists a single helical edge states for X > X l .The insulating phase X > X l is thus topologically nontrivial.For each realization of the disorder, we numerically diagonalize the one-step time-evolution operator of the Z network model and retain the normalized wave function ψ σ ( x, y ), after coarse graining over the 4 edges of the plaquette located at ( x, y ), whoseeigenvalue is the closest to 1. The wave function at criticality is observed to display thepower-law dependence on the linear dimension L of the system, X σ = ↑ , ↓ | ψ σ ( x, y ) | q ∝ L − ∆ ( ζ,ν ) q − dq . (119)The anomalous dimension ∆ ( ζ,ν ) q , if it displays a nonlinear dependence on q , is thesignature of multifractal scaling. The index ζ indicates whether the multifractal scalingapplies to the bulk ( ζ = 2), the one-dimensional boundary ( ζ = 1), or to the zero-dimensional boundary (corner) ( ζ = 0), provided the plaquette ( x, y ) is restrictedto the corresponding regions of the Z network model. For ζ = 1 and 0, theindex ν distinguishes the case ν = O when the ζ -dimensional boundary has no edgestates in the insulating phase adjacent to the critical point, from the case ν = Z when the ζ -dimensional boundary has helical edge states in the adjacent insulatingphase. We ignore this distinction for multifractal scaling of the bulk wave functions,∆ (2 , O) q = ∆ (2 , Z ) q = ∆ (2) q , since bulk properties are insensitive to boundary effects. Wewill also consider the case of mixed boundary condition for which we reserve the notation ν = Z | O.It was shown in [31] that boundary multifractality is related to corner multifractalityif it is assumed that conformal invariance holds at the metal-insulator transition in thetwo-dimensional symplectic universality class. Conversely, the numerical verification of
ONTENTS Figure 5. (a) Boundary multifractality is calculated from the wave functionamplitudes near a one-dimensional boundary. Periodic (reflecting) boundaryconditions are imposed for the horizontal (vertical) boundaries. (b) Cornermultifractality is calculated from the wave function amplitudes near a corner withthe wedge angle ϑ = π/
2. Reflecting boundary conditions are imposed along bothvertical and horizontal directions. The relationship between the scattering matrix ata node of type S ′ and the scattering matrix at a node of type S implies that it is avertical boundary located at nodes of type S that induces an helical edge state when X > X l . this relationship between boundary and corner multifractality supports the claim thatthe critical scaling behavior at this metal-insulator transition is conformal. So we wantto verify numerically if the consequence of the conformal map w = z ϑ/π , namely∆ (0 ,ν ) q = πϑ ∆ (1 ,ν ) q (120)where ϑ is the wedge angle at the corner, holds. Equivalently, f ( ζ,ν ) ( α ), which is definedto be the Legendre transformation of ∆ ( ζ,ν ) q + dq , i.e., α ( ζ,ν ) q = d ∆ ( ζ,ν ) q dq + d, (121) f ( ζ,ν ) ( α q ) = qα ( ζ,ν ) − ∆ ( ζ,ν ) q − dq + ζ , (122)must obey α (0 ,ν ) q − d = πϑ ( α (1 ,ν ) q − d ) , (123) f (0 ,ν ) ( α ) = πϑ (cid:2) f (1 ,ν ) ( α ) − (cid:3) , (124)if conformal invariance is a property of the metal-insulator transition.To verify numerically the formulas (120), (123), and (124), we consider the Z network model with the geometries shown in figure 5. We have calculated wave functionsfor systems with the linear sizes L = 50 , , , , and 180 for the two geometriesdisplayed in figure 5. Here, L counts the number of nodes of the same type along aboundary. The number of realizations of the static disorder is 10 for each system size. ONTENTS Figure 6. (a) The boundary (filled circles, red) and corner with θ = π/ Z -topological-insulator transition. Thesolid curve is computed from (120) by using the boundary anomalous dimension as aninput. The rescaled ∆ − q confirming the reciprocal relation for boundary and cornermultifractality are shown by upper (magenta) and lower (green) triangles, respectively.(b) The multifractal spectra for the boundary (filled circles, red) and the corner (opencircles, blue). The solid curve is computed from (123) and (124). Figure 6(a) shows the boundary anomalous dimensions ∆ (1 , Z ) q (filled circles) andthe corner anomalous dimensions ∆ (0 , Z ) q (open circles). In addition, the anomalousdimensions ∆ ( ζ,ν )1 − q are shown by upper and lower triangles for boundary and corneranomalous dimensions, respectively. They fulfill the reciprocal relation∆ ( ζ,ν ) q = ∆ ( ζ,ν )1 − q (125)derived analytically in [34]. Since the triangles and circles are consistent within errorbars, our numerical results are reliable, especially between 0 < q <
1. If we usethe numerical values of ∆ (1 , Z ) q as inputs in (120) with ϑ = π/
2, there follows thecorner multifractal scaling exponents that are plotted by the solid curve. Since thecurve overlaps with the direct numerical computation of ∆ (0 , Z ) q within the error bars,we conclude that the relation (120) is valid at the metal-to- Z -topological-insulatortransition.Figure 6(b) shows the boundary (filled circles) and corner (open circles) multifractalspectra. These multifractal spectra are calculated by using (119), (121), and (122). Thenumerical values of α ( ζ, Z )0 are α (1 , Z )0 = 2 . ± . , (126) α (0 , Z )0 = 2 . ± . . (127)The value of α (1 , Z )0 is consistent with that reported in [20], while its accuracy isimproved. The solid curve obtained from the relations (123) and (124) by using f (1 , Z ) ( α )as an input, coincides with f (0 , Z ) ( α ). We conclude that the hypothesis of conformalinvariance at the quantum critical point of metal-to- Z -topological-insulator transitionis consistent with our numerical study of multifractal scaling. ONTENTS Figure 7. (a) The z dependence of h ln | Ψ | i z,L at the metal-to- Z -topological-insulator transition in the cylindrical geometry for L = 50 , , , ,
180 from thetop to the bottom. (b) The z dependence of ˜ α ( z ) ( • )and c ( z ) ( (cid:4) ) extrapolatedfrom the system size dependence of h ln | Ψ | i z,L averaged over a small interval of z ’s.˜ α ( z ) and c ( z ) at z = 0 , z ’s are shownby open circles and squares, respectively. The solid line represents the bulk value of α (2)0 = 2 .
173 computed in [31]. The asymmetry with respect to z = 0 . At last, we would like to comment on the dependence on z of h ln | Ψ | i z,L ≡ L L X y =1 ln X σ = ↑ , ↓ | ψ σ ( x, y ) | ! (128)found in [20]. Here, z ≡ ( x − / L , while x and y denote the positions on thenetwork along its axis and along its circumference, respectively (our choice of periodicboundary conditions imposes a cylindrical geometry). The overline denotes averagingover disorder. Figure 7(a) shows the z dependence of h ln | Ψ | i z,L for different values of L in this cylindrical geometry at the metal-to- Z -topological-insulator transition. Weobserve that h ln | Ψ | i z,L becomes a nonmonotonic function of z .We are going to argue that this nonmonotonic behavior is a finite size effect. Wemake the scaling ansatz h ln | Ψ | i z,L = − ˜ α ( ζ, Z )0 ( z ) ln L + c ( z ) , (129)where ζ = 1 if z = 0 , ζ = 2 otherwise, while c ( z ) depends on z but not on L . Tocheck the L dependence of h ln | Ψ | i z,L in figure 7, we average h ln | Ψ | i z,L over a narrowinterval of z ’s for each L . Figure 7(b) shows the z dependence of ˜ α ( z ) ( • ) and c ( z )( (cid:4) ) obtained in this way. In addition, ˜ α ( z ) and c ( z ) calculated for z = 0 , z ’s are shown by open circles and open squares,respectively.We observe that ˜ α ( z ), if calculated by averaging over a finite range of z ’s, is almostconstant and close to α (2)0 = 2 . α ( z = 0 , ≈ .
09, if calculated withoutaveraging over a finite range of z ’s, is close to α (1 , Z )0 = 2 . | c ( z ) | increases near the boundaries. We conclude that it is the nonmonotonic dependence of ONTENTS | c ( z ) | on z that gives rise to the nonmonotonic dependence of h ln | Ψ | i z,L on z . Thisfinite-size effect is of order 1 / ln L and vanishes in the limit L → ∞ . Next, we impose mixed boundary conditions by either (i) coupling the Z networkmodel to an external reservoir through point contacts or (ii) by introducing a long-range lead between two nodes from the Z network model, as shown in figure 8. Inthis way, when X > X l , a single Kramers’ pair of helical edge states indicated by thewavy lines in figure 8 is present on segments of the boundary, while the complementarysegments of the boundary are devoid of any helical edge state (the straight lines infigure 8). The helical edge states either escape the Z network model at the nodes atwhich leads to a reservoir are attached [the green lines in figure 8(a) and figure 8(b)],or shortcut a segment of the boundary through a nonlocal connection between the twonodes located at the corners [figure 8(c)]. These are the only options that accommodatemixed boundary conditions and are permitted by time-reversal symmetry. As shownby Cardy in [35], mixed boundary conditions are implemented by boundary-condition-changing operators in conformal field theory. Hence, the geometries of figure 8 offeryet another venue to test the hypothesis of two-dimensional conformal invariance at themetal-insulator quantum critical point.When coupling the Z network model to an external reservoir, we shall considertwo cases shown in figure 8(a) and figure 8(b), respectively.First, we consider the case of figure 8(a) in which only two nodes from the Z network model couple to the reservoir. At each of these point contacts, the scatteringmatrix S that relates incoming to outgoing waves from and to the reservoir is a 2 × s S ∗ s = S , and it must be proportionalto the unit 2 × Z network model and the reservoir nowsupports two instead of one Kramers’ doublets. The two point-contact scatteringmatrices connecting the Z network model to the reservoirs are now 4 × X = X l for a single realization ofthe static disorder. The two-point-contact conductance is calculated by solving for thestationary solution of the time-evolution operator with input and output leads [36].We choose the cylindrical geometry imposed by periodic boundary conditions along the ONTENTS Figure 8. (a) The system with two point contacts (green curves) attached (a) at thetwo interfaces between different types of boundaries and (b) at the reflecting boundary.The periodic boundary conditions are imposed for the horizontal direction. The thickwavy and solid lines on the edges represent two different types of boundaries, with andwithout a helical edge mode at
X > X l , respectively. (c) Closed network with mixedboundaries. Each dashed line or curve represents a Kramers’ doublet. horizontal directions in figures 8(a) and 8(b) for a squared network with the linear size L = 200. Figure 9(a) shows with the symbol • the dependence on r , the distance betweenthe two contacts in figure 8(a), of the dimensionless two-point-contact conductance g .It is evidently r independent and unity, as expected. Figure 9(a) also shows withthe symbol ◦ the dependence on r of the dimensionless two-point conductance g forleads supporting two Kramers’ doublet as depicted in figure 8(b). Although it is notpossible to establish a monotonous decay of the two-point-contact conductance for asingle realization of the static disorder, its strong fluctuations as r is varied are consistentwith this claim.We turn our attention to the closed geometry shown in figure 8(c). We recallthat it is expected on general grounds that the moments of the two-point conductancein a network model at criticality, when the point contacts are far apart, decayas power laws with scaling exponents proportional to the scaling exponents ∆ ( ζ,ν ) q [36, 37]. Consequently, after tuning the Z network model to criticality, the anomalousdimensions at the node (corner) where the boundary condition is changed must vanish,∆ (0 , Z | O) q = 0 , (130)since the two-point-contact conductance in figure 8(a) is r independent. (130) is anothersignature of the nontrivial topological nature of the insulating side at the Anderson ONTENTS Figure 9. (a) The distance r dependence of the two-point-contact conductance g forthe mixed boundary ( • ) and the reflecting boundary ( ◦ ). (b) The zero dimensionalanomalous dimension ∆ (0 , Z | O) q obtained from the wave function amplitude on the linkwhich connecting the boundary condition changing points. transition that we want to test numerically. Thus, we consider the geometry figure8(c) and compute numerically the corner anomalous dimensions. This is done using theamplitudes of the stationary wave function restricted to the links connecting the twocorners where the boundary conditions are changed. Figure 9(b) shows the numericalvalue of the corner anomalous dimension ∆ (0 , Z | O) q . The linear sizes of the network are L = 50 , , , for each L .We observe that ∆ (0 , Z | O) q is zero within the error bars, thereby confirming the validityof the prediction (130) at the metal-to- Z -topological-insulator transition.
5. Conclusions
In summary, we have mapped the Z network model to a 4 × Z invariantas an SU(2) Wilson loop and computed it explicitly. In the presence of weak time-reversal symmetric disorder, the NLSM that can be derived out of this Dirac Hamiltoniandescribes the metal-insulator transition in the Z network model and yields bulk scalingexponents that belong to the standard two-dimensional symplectic universality class;an expectation confirmed by the numerics in [19] and [20]. A sensitivity to the Z topological nature of the insulating state can only be found by probing the boundaries,which we did numerically in the Z network model by improving the quality of thenumerical study of the boundary multifractality in the Z network model. References [1] Roland Winkler 2003 “Spin-orbit coupling effects in two-dimensional electron and hole systems,” (Springer-Verlag Berlin Heidelberg)[2] Hikami S, Larkin A I and Nagaoka Y 1980
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