Relevant perturbations at the spin quantum Hall transition
aa r X i v : . [ c ond - m a t . d i s - nn ] J a n Relevant perturbations at the spin quantum Hall transition
Shanthanu Bhardwaj
Department of Physics, The University of Chicago, Chicago, Illinois 60637, USA
Ilya A. Gruzberg
Department of Physics, The Ohio State University, Columbus, OH 43210, USA
Victor Kagalovsky
Shamoon College of Engineering, Beer-Sheva 84105, Israel (Dated: January 27, 2014)We study relevant perturbations at the spin quantum Hall critical point using a network modelformulation. The model has been previously mapped to classical percolation on a square lattice,and we use the mapping to extract exact analytical values of the scaling dimensions of the relevantperturbations. We find that several perturbations that are distinct in the network model formulationcorrespond to the same operator in the percolation picture. We confirm our analytical results bycomparing them with numerical simulations of the network model.
PACS numbers: 72.15.Rn, 73.20.Fz, 73.43.-f
INTRODUCTION
Anderson localization of a quantum particle [1] or aclassical wave in a random environment is a vibrant re-search field [2]. One of its central research directions isthe physics of Anderson transitions [3], quantum criticalpoints tuned by disorder. These include metal-insulatortransitions and transitions of quantum Hall type sepa-rating distinct phases of topological insulators. Whilesuch transitions are conventionally observed in electronic(metallic and semiconductor) structures, there is also aconsiderable number of other experimental realizationsactively studied in recent and current works. These in-clude localization of light [4] and microwaves [5], coldatoms [6] (see a recent review [7]), ultrasound [8], andoptically driven atomic systems [9].From the theoretical point of view, symmetries playa central role in determination of universality classes ofcritical phenomena. This idea was applied to Andersonlocalization by Altland and Zirnbaueer (AZ) [10] whoidentified ten distinct symmetry classes. In three of theseclasses, classes A, C, and D in AZ classification, the time-reversal invariance is broken, and there is a possibility fora quantum Hall transition in two dimensions.The transition in class A is the usual integer quantumHall (IQH) transition in a two-dimensional (2D) elec-tronic system in a strong perpendicular magnetic field(see Ref. [11] for a review). Class A also includes themodel of electrons in a random magnetic field, where allstates are believed to be localized [12].Class C is one of the four Bogolyubov-de Gennesclasses which describe transport of quasiparticles in dis-ordered superconductors at a mean field level, and pos-sess the particle-hole symmetry. In this class the spin-rotation invariance is preserved, the quasiparticles haveconserved spin, and one can study spin transport. The corresponding Hall transition is known as the spin quan-tum Hall (SQH) transition [13, 14], at which the systemexhibits a jump in the spin Hall conductance from 0 to2 in appropriate units.In spite of tremendous efforts, most models of Ander-son transitions have resisted analytical treatment. TheIQH transition is one prominent example where only re-cently some analytical progress has been achieved [15].On the other hand, the SQH transition enjoys a spe-cial status, since a network model of this transition wasmapped exactly to classical percolation on a square lat-tice [16]. The original mapping used the supersymme-try (SUSY) method of Efetov [17] adapted to networks[18, 19]. An alternative way to obtain the mapping wasfound later [20, 21]. It was also extended to networkmodels in class C on arbitrary graphs [22]. Many ex-act results are known for classical percolation. Thus, themapping has lead to a host of exact critical propertiesat the SQH transition [16, 21–24]. However, these re-sults are not exhaustive, since not all possible relevantperturbations were considered in Refs. [16, 21, 23, and24]. Several critical exponents have been obtained nu-merically in Refs. [13, 14, 25–27].In this paper we reexamine the relevant perturbationsat the SQH critical point. As our main tool we use theSUSY method applied to the simplest network model inclass C describing the SQH effect. We introduce all pos-sible perturbations that are relevant at the critical pointof the SHQ network model. One of them preserves thesymmetries of the model and drives the SQH transition.Other relevant perturbations break symmetries specificto class C and lead to a crossover to class A. We use thepercolation mapping of Ref. [16] to extract analyticalvalues of the scaling dimensions of all relevant perturba-tions. As a result, we find that one of the results of Ref.[16] does not hold, and find the correct value of the cor-responding critical exponent. In addition, we find thatseveral microscopically distinct perturbations all have thesame scaling dimension related to a single operator in thepercolation picture.The paper is organized as follows. In Sec. we de-scribe the network model appropriate for the study ofthe spin quantum Hall transition in class C and rele-vant perturbations near it, and summarize our results.In Sec. we briefly describe the SUSY method for thenetwork, and derive the second-quantized supersymmet-ric transfer matrices. These matrices are then averagedover quenched disorder. In Sec. we take an anisotropiclimit, thereby mapping the network model to a superspinchain. The superspin chain contains a critical point, andseveral relevant perturbations. All terms in the super-spin chain Hamiltonian are interpreted in terms of theclassical percolation picture of Ref. [16], and this inter-pretation allows us to extract dimensions of all relevantperturbations and the corresponding critical exponents.In Sec. we present our recent numerical results, dis-cuss results of other numerical simulations of the net-work model, and compare all these with our analyticalpredictions. We then conclude. For completeness, we re-view details of the SUSY method for the class C networkmodel in a series of Appendices.
THE MODEL AND A SUMMARY OF RESULTS
A scattering theory description of Anderson localiza-tion and Anderson transitions in terms of random net-work models was introduced in Ref. [28]. For systemsexhibiting quantum Hall effects one can use semiclassi-cal drifting orbits [29, 30] scattered at saddle points of asmooth random potential to provide an intuitive deriva-tion for network models. The resulting networks are chi-ral, reflecting the breaking of time reversal invariance instrong magnetic fields. The simplest such model is thethe Chalker-Coddington (CC) model originally proposedto describe the IQH effect [31].Here we consider a generalization of the CC modelshown in Fig. 1. In this network each link supportstwo co-propagating channels which we label by σ = ↑ , ↓ .The corresponding doublets of complex fluxes propagatealong links, and their components get mixed by scatter-ing matrices S link , which relate the incoming and outgo-ing fluxes. In class A, the symmetry class of the IQHeffect, the scattering matrices on the links are generalunitary U(2) matrices, and can be parameterized as S δ = e iδ S , (1)where the matrix S ∈ SU(2). The link matrices are inde-pendent identically distributed random variables whosedistribution is chosen depending on a specific physicalsituation.
A B
FIG. 1. Two-channel chiral network model. Dots representscattering matrices on the links (1) and squares represent thenodal scattering matrices (2).
As in the ordinary CC network there are two sublat-tices, A and B , on which the nodes are related by a 90 ◦ rotation. Scattering of the fluxes at the nodes (blacksquares) is described by orthogonal matrices diagonal inspin indices: S S = S S ↑ ⊕ S S ↓ , S Sσ = (cid:18) r Sσ t Sσ − t Sσ r Sσ (cid:19) , r Sσ ≡ (1 − t Sσ ) / , (2)where S = A , B labels the sublattice. Usually the scat-tering amplitudes on the two sublattices t Sσ are assumedto be non-random. The network has a critical point at t A ↑ = t B ↑ = t A ↓ = t B ↓ . (3)Depending on the choice of parameters and their prob-ability distributions, the generalized two-channel networkmodel in class A can be used to describe various physi-cal systems: spin-degenerate Landau levels and localiza-tion in a random magnetic field [12], the IQH effect ina double-layer system [32, 33], and the splitting of delo-calized states due to the valley mixing in graphene [34].Using the SUSY method, in Ref. [35] we have provided acomparative study of the relevant networks and relatedmodels.Let us now consider class C, the symmetry class ofthe SQH effect. Similar to previous works, we study theSQH transition in (the mean field description of) a sin-glet superconductor after a particle-hole transformationon the down-spin particles [36]. The transformation in-terchanges the roles of particle number and z compo-nent of spin, and so particle number is conserved ratherthan spin. This somewhat obscures the spin-rotationsymmetry, but makes it possible to use a single parti-cle description and, in particular, a network model. Thesingle-particle energy ( E ) spectrum has a particle-holesymmetry [10], so, when states are filled up to E = 0,the positive-energy particle and hole excitations becomedoublets of the global SU(2) symmetry. In this picture, auniform Zeeman magnetic field B z for the quasiparticlesmaps onto a simple shift in the Fermi energy to E ∝ B z [14], splitting the degeneracy.In the network model description a particle of eitherspin and with E = 0 is represented by a doublet of com-plex fluxes that can propagate in one direction along eachlink (Fig. 1). The global spin-rotation symmetry of classC requires the scattering matrices to be unitary sym-plectic. Thus in the two-channel model the link matricesbelong to Sp(2) ∼ = SU(2), and in the parametrization (1)we have to set the overall phase δ = 0. The absence ofan additional (random or deterministic) U(1) phase hereis crucial. Taking the link matrices S to be uniformlydistributed over the Haar measure on SU(2) we obtainthe model that maps to a classical bond percolation onthe square lattice [16]. Both the absence of the overallphase and the uniform distribution over SU(2) are essen-tial technical ingredients of the mapping (as explained inthe Appendix).Let us describe how a non-zero energy E enters thenetwork model description. A state of the network (thecollection of the fluxes on all channels joining scatteringmatrices) evolves in discrete time steps under the actionof a unitary evolution operator U which has nonzero ma-trix elements only between pairs of incoming and outgo-ing channels scattered on a link or at a node, the matrixelements simply being the scattering amplitudes relatingthe corresponding fluxes. The main object of study isthe Green’s function, or the resolvent, of the evolutionoperator: G ( e ′ , e ; z ) = h e ′ | (1 − z U ) − | e i , (4)where e and e ′ are two channels (edges) of the network.In a closed network U is unitary, and the resolvent hassingularities on the unit circle in the complex plain ofthe spectral parameter z . Roughly speaking, if we write U = e i H and z = e i ( E + iη ) , H can be thought of as theHamiltonian for the network, and E + iη as the energywith a finite level broadening η . The level broadeningmay be induced by attaching ideal leads that make thenetwork open and break unitarity. Scaling of various ob-servables with energy E close to the critical point is thesame as with its imaginary part η [21, 24]. This factallows us to use the real z = e − η .If we expand the Green’s function (4) into a powerseries in z , it is clear that a factor of z is associated witheach scattering event. In fact, it is sufficient to assignthe factors of z only to scattering at the nodes or on thelinks. We choose the latter option. This leads to themodified link scattering matrices S φ = e iφ S , φ ≡ δ + iη. (5)The notation we use stresses the fact that the phase δ andthe level broadening η combine to form a single complexparameter φ = δ + iη where δ plays the role of the energy E . As expected, the modified scattering matrices are notunitary, since a finite level broadening leads to decay ofthe states of the network and breaks current conserva-tion. We also note here that while δ can be random, thelevel broadening η will be taken the same for every link.The class C network model can be driven away fromits (multi)critical point given by Eq. (3) (and δ = 0) in Exponent ν ν B µ ∆ µ p µ δ Analytical predictionsRef. [16] 4 / / / ≈ . ≈ .
57 = 1 . / / / ≈ . ≈ . ≈ . .
12 – 1 .
45 – –Ref. [14] 1 . . .
12 – 1 .
45 1 .
17 –Ref. [26] 1 .
12 – 1 .
45 – 0 . . .
15 –TABLE I. A summary of previous and new results for criticalexponents at the SQH transition. different ways. Taking t Aσ = t Bσ (but keeping t S ↑ = t S ↓ )is the only perturbation that preserves the class C sym-metries. It drives the system through a SQH transitionbetween an insulator and a SQH state. Introducing auniform Zeeman field (or a non-zero chemical potential)breaks the global spin-rotation symmetry, and splits thetransition into two ordinary IQH transitions, each in classA. The same effect is achieved by making t S ↑ = t S ↓ .To describe these relevant perturbations in a quanti-tative way, let us parametrize the node scattering ampli-tudes in the vicinity of the critical point (3) as follows: t A ↑ = t (1 + ǫ + ∆) , t A ↓ = t (1 + ǫ − ∆) ,t B ↑ = t (1 − ǫ + ∆) , t B ↓ = t (1 − ǫ − ∆) . (6)Then nonzero ǫ , Zeeman field B z (or η ), and ∆ are all rel-evant perturbations that induce finite localization lengthsscaling as ξ ∼ | ǫ | − ν , ξ B ∼ | η | − ν B , ξ ∆ ∼ | ∆ | − µ ∆ . (7)Thus defined critical exponents have been analyticallydetermined in Ref. [16], where the authors suggestedthat the parameter ∆ may describe a random Zeemanfield. The exponents were numerically studied in Refs.[13, 14, 25–27]. The results are summarized in the firstthree columns of Table I.A microscopically distinct perturbation of the class Cnetwork that induces a crossover to class A is the intro-duction of a nonzero phase δ in the link matrices (1). Thecase of a random extra phase δ with zero mean and vari-ance p was numerically studied in Ref. [25], and thatof a constant phase δ — in Ref. [26]. Both perturba-tions appeared to be relevant, as expected on symmetrygrounds, and resulted in finite localization lengths thatscaled as ξ p ∼ p − µ p , ξ δ ∼ | δ | − µ δ . (8) - t A τ t A τ e i δ - η e i δ - η t A τ e i δ - η t A τ e i δ - η ~ ~ - t B τ t B τ e i δ - η e i δ - η t B τ e i δ - η t B τ e i δ - η ~ ~ FIG. 2. The scattering amplitudes at the nodes on the twosublattices. The node scattering matrices are diagonal in thespin indices, so we only show one channel per link.
From our comments above it should be clear that a con-stant phase δ is exactly equivalent to a uniform nonzeroZeeman field, which immediately implies µ δ = ν B . (9)By the same token, a random phase δ is equivalent to arandom Zeeman field. In the rest of the paper we assumethat the phases δ on the links are independent identicallydistributed random variables with the mean δ and vari-ance p , where both quantities are small: δ , p ≪
1. Inthe following sections we will show that the small param-eters η , δ , and p always appear in the combination λ ≡ η − iδ + p . (10)This immediately implies that µ p = 2 µ δ = 2 ν B . (11)In subsequent sections we will use the SUSY methodand the mapping to percolation to obtain the exact val-ues of the exponents µ δ and, therefore, µ p . In addition,our analysis uncovers a subtle mistake made in Ref. [16]that led to a wrong prediction for the exponent µ ∆ . Af-ter correcting the mistake, we obtain the values of theexponents shown in Table I. We also show in the Table anumerical value for the exponent µ p obtained by a directcomputer simulation of the class C network model withan additional random phase δ on the links. SUPERSYMMETRIC TRANSFER MATRICES
In this section we apply the SUSY method [18, 19] toour network model in the vicinity of its critical point tomap it to classical percolation. Before we describe thetechnical steps, let us present our strategy. As we men-tioned in the previous section, the mapping to percola-tion is only possible when the scattering on the links isdescribed by SU(2) matrices. Thus, a direct applicationof this method to our system, where the link matricesare given by Eq. (5), is impossible. We circumvent thisdifficulty as follows. We can always remove the factors e iφ = e iδ − η from the link scattering matrices and reas-sign them to the nodal matrices in such a way that theGreen’s function (4) is not affected. The redefined nodalmatrices become S Sσ = r Sσ e iφ t Sσ e iφ ′ − t Sσ e iφ r Sσ e iφ ′ ! . (12)Here δ and δ ′ in φ and φ ′ are independent, since theycome from two different links incoming at a node, seeFig. 2. Having shifted the factors e iφ onto the nodes,we are now free to perform the SU(2) average on thelinks. Subsequently, we can perform the average over thephases δ .In the SUSY method the vertical direction in Fig. 1 isregarded as the (imaginary) time τ . The vertical zig-zagsof links that go up (along the time direction) correspondto sites of a quantum one-dimensional chain with an oddlabel i . The down-going links correspond to even sites.At each odd site i there is a Fock space F i = F i ↑ ⊗ F i ↓ of fermions and bosons, and at each even site the Fockspace is ¯ F i = ¯ F i ↑ ⊗ ¯ F i ↓ . The spaces on the odd and evensites differ by the commutation relations for creation andannihilation operators of the up and down particles, seeAppendix .Scattering of fluxes on links of the network is repre-sented by the second-quantized transfer matrices T i − and T i which describe the evolution of states in F i − or ¯ F i between two discrete imaginary time slices throughthe lower and upper half-link. Scattering at a node onsublattice A is represented by the transfer matrix T i − , i which evolves states in the tensor product F i − ⊗ ¯ F i between two discrete imaginary time slices (below andabove the node), and similarly for the B sublattice. Allsecond-quantized transfer matrices are exponentials ofquadratic forms in creation and annihilation operators,see details in Appendix A.As is known from Ref. [16] (and reviewed in Appendix), in the spin-rotation invariant case ( t S ↑ = t S ↓ and δ = 0), the transfer matrices commute with the sumover sites of the eight generators (superspin components)of the superalgebra osp (2 | ∼ = sl (2 | osp (2 |
2) on each site appear as all bilinears inthe fermions and bosons and their adjoints, which aresinglets under the random SU(2). These are denoted by[37] B , Q , Q ± , V ± , W ± for the up sites (and with barsfor the down sites) and have similar expressions for thetwo types of sites. We combine the generators on eachsite into a single eight-component object, a superspin,and call it J i − and ¯ J i for up and down sites. Breakingthe spin-rotation invariance by either of the symmetry-breaking perturbations, breaks the SUSY of the transfermatrices down to gl (1 |
1) generated on each up site by K = { B, Q , V − , W + } (similarly for the down sites).Averaging over the random SU(2) matrices on thelinks projects each Fock space F i − ( ¯ F i ) onto a three-dimensional subspace which is the fundamental (dual tothe fundamental) representation of osp (2 | π ( ± , ). For a single transfer matrix the projection(see Appendix for details) results inˆ P T ˆ P = 1 + (cid:0) r ↑ r ↓ e iφ − (cid:1) ( B + Q ) − (cid:0) r ↑ r ↓ e i ¯ φ − (cid:1) ( ¯ B + ¯ Q ) − (cid:0) r ↑ r ↓ e iφ − (cid:1)(cid:0) r ↑ r ↓ e i ¯ φ − (cid:1) ( B + Q )( ¯ B + ¯ Q )+ e i ( ¯ φ + φ ) h t ↑ t ↓ ( B + Q )( ¯ B + ¯ Q ) − t ↑ + t ↓ K · ¯ K − B − Q + ¯ B + ¯ Q ) i − t ↑ t ↓ e i ¯ φ ( Q + ¯ Q − + V + ¯ W − ) − t ↑ t ↓ e iφ ( Q − ¯ Q + − W − ¯ V + ) . (13)Here we have suppressed the sublattice index for brevity,and also used the gl (1 | K and ¯ K : K · ¯ K = 2 Q ¯ Q − B ¯ B − V − ¯ W + + W + ¯ V − . (14)We can now carry out averages (that we denote byangular brackets) over the independent random phases δ . Using the notation h e iφ i = h e i ¯ φ i ≡ Λ , (15)and the osp (2 | J · ¯ J = 2 Q ¯ Q − B ¯ B − V − ¯ W + + W + ¯ V − + V + ¯ W − − W − ¯ V + + Q + ¯ Q − + Q − ¯ Q + , (16)we can write the average of the projected transfer matrixon the sublattice S as h ˆ P T S ˆ P i = 1 + c S J · ¯ J + c S K · ¯ K + c S ( B + Q )( ¯ B + ¯ Q ) + c S ( B + Q − ¯ B − ¯ Q ) , (17)where the coefficients are given by c S = − Λ t S ↑ t S ↓ ,c S = −
12 Λ ( t S ↑ + t S ↓ ) + Λ t S ↑ t S ↓ ,c S = Λ t S ↑ t S ↓ − (Λ r S ↑ r S ↓ − ,c S = 12 Λ ( t S ↑ + t S ↓ ) + Λ r S ↑ r S ↓ − . (18)When the extra phases φ vanish (Λ = 1), the coeffi-cients in this expression simplify, and it reduces to theone studied before in Ref. [16]: h ˆ P T S ˆ P i = 1 − t S ↑ t S ↓ J · ¯ J − ( t S ↑ − t S ↓ ) K · ¯ K − ( r S ↑ − r S ↓ ) (cid:2) B + Q )( ¯ B + ¯ Q ) + B + Q − ¯ B − ¯ Q (cid:3) . (19)In particular, if no class C symmetries are broken ( t S ↑ = t S ↓ ), the average transfer matrix reduces to h ˆ P T S ˆ P i = 1 − t S J · ¯ J, (20)an expression that was interpreted in terms of classicalbond percolation on a square lattice in Ref. [16]. SUPERSPIN CHAIN AND CRITICALEXPONENTS
So far everything was exact. Now we will perform anadditional step that is useful in the study of networkmodels. This is to consider an anisotropic limit, whenall amplitudes t Sσ are small [which can be achieved bytaking t ≪ τ : U = exp (cid:16) − Z dτ H (cid:17) , (21)where the effective Hamiltonian H describes a 1D su-perspin chain, with alternating π ( ± , ) representations(superspins) on each site along the chain. The spin chainhas a critical point, and various deviations from it appearat this step as perturbations of the critical Hamiltonian.When passing to the anisotropic limit, we will ex-pand the coefficients in expressions for average trans-fer matrices to leading order in all small parameters( t, ǫ, ∆ , η, δ , p ).First, consider the maximally symmetric case (20): h ˆ P T i − , i ˆ P i ≈ − t (1 + 2 ǫ ) J i − · ¯ J i , h ˆ P T i, i +1 ˆ P i ≈ − t (1 − ǫ ) ¯ J i · J i +1 . (22)Combining all transfer matrices, we obtain the effective1D Hamiltonian H = H + H , (23)where H = t X i (cid:0) J i − · ¯ J i + ¯ J i · J i +1 (cid:1) (24)describes the critical superspin chain, and the staggeredterm H = 2 t ǫ X i D i , D i = J i − · ¯ J i − ¯ J i · J i +1 , (25)represents a relevant perturbation. As was argued in Ref.[16], the dimer operator D i represents the two-hull oper-ator in the critical percolation picture, with dimension x = 5 /
4. The corresponding critical exponent is ν = (2 − x ) − = 43 . (26)Having identified the role of the sublattice asymmetry ǫ , let us turn to the general case, Eqs. (17) and (18). Theterms with the coefficients c S and c S contain bilinearsin the superspins, and can be thought of as introduc-ing two kinds of anisotropy in the superspin space. Thelast term (with the coefficient c S is linear in superspins,ans corresponds to the one-hull operator in the criticalpercolation picture, with dimension x = 1 /
4. This isthe lowest among the dimensions of operators in criticalpercolation. Thus, even without the knowledge of thedimensions of the anisotropic terms, we can claim thatthe last term in Eq. (17) is the most relevant perturba-tion. We will see that all symmetry-breaking perturba-tions couple to this term, and it, therefore, determinesthe corresponding critical exponents.In the anisotropic limit and close to the critical pointwe have, first of all Λ ≈ − λ, (27)where λ is given in Eq. (10). Expanding the coefficients c S in Eq. (17) we have c A ≈ − t (1 + 2 ǫ − ∆ − λ ) ,c B ≈ − t (1 − ǫ − ∆ − λ ) . (28)We see that the symmetry-breaking perturbations sim-ply renormalize the coupling constant t of the criticalHamiltonian H .In the other terms it is sufficient to set ǫ = 0. Thenthe coefficients on the two sublattices coincide, and theirexpansions look like c S ≈ − t (∆ − λ ) ,c S ≈ − t ∆ − t λ,c S ≈ − t ∆ − t ) λ. (29)The last expression confirms the conclusion of Ref. [16]that the non-zero energy η couples to the most relevantperturbation, the one-hull operator B + Q − ¯ B − ¯ Q (that happens to represent the local density of states),and leads to a localization length that has a power-lawbehavior with the exponent ν B = (2 − x ) − = 47 . (30)Moreover, since η enters all expressions in the combina-tion λ = η − iδ + p , we immediately obtain the equalitiesbetween critical exponents given in Eq. (11).Next we see that the square of the spin-rotationsymmetry-breaking parameter ∆ also couples to the one-hull operator. This immediately implies that µ ∆ = µ p = 87 . (31) This value is different from the result µ ∆ = 3 / enters the coefficient of the one-hull term B + Q − ¯ B − ¯ Q in the combination t ∆ . In taking the anisotropic limit t → µ ∆ isdetermined by the dimension of the superspin anisotropyoperator K · ¯ K , which was conjecturally found (and waslarger than x = 1 / ξ ∆ for any finite t . NUMERICAL RESULTS
In this section we compare the exact values of criti-cal exponents obtained above with results of numericalsimulations.First we report our numerical results for the exponent µ p . We have simulated the SQH network model withonly one relevant perturbation: extra random phases onthe links with the mean δ = 0 and variance p . Theother perturbations ( ǫ , η , ∆) were set to zero. We usedthe standard transfer matrix method in the quasi-one-dimensional geometry with periodic boundary conditionsin the transverse direction (cylinder) [39, 40]. Our systemlengths reached 10 , and the circumferences M rangedfrom 32 to 192 with various random phase variances p .Without symmetry-breaking perturbations, all Lya-punov exponents of the transfer matrix product are dou-bly degenerate due to the presence of time-reversal in-variance (Kramers degeneracy). It was suggested in Ref.[13] that when the time-reversal symmetry is broken by asmall perturbation, the renormalized localization length(the inverse of the smallest positive Lyapunov exponent)and the deviation from Kramers degeneracy ¯ ξ (the differ-ence between the two smallest positive Lyapunov expo-nents multiplied by the circumference M ) exhibit scalingbehavior characterized by the same exponent. This ideawas further supported in Refs. [25] and [26]. It turnsout that the deviation from Kramers degeneracy ¯ ξ is asuperior way to extract critical exponents in this case,since we know its exact value ¯ ξ = 0 at the critical point.Thus, in Fig. 3 we present a one-parameter scalingresults for ¯ ξ ≡ ( λ M/ − − λ M/ ) M as a function of thescaling variable x ≡ pM /µ p : ¯ ξ = f ( x ). In order toimprove the accuracy we do not use data for systems withsmall circumferences, and obtain the value of the criticalexponent using an optimization program that ensures thebest scaling collapse. The routine determines the least-squares approximation to the scaling function f ( x ) interms of the Chebyshev polynomials by minimizing thesum of squares of the deviations of the data points fromthe corresponding values of the polynomial, while alsovarying the exponent µ p . The result µ p = 1 .
15 is in pM p M=32 M=48 M=64 M=96 M=128 M=144 M=192 p =1.15 ( M/2-1 - M/2 )M FIG. 3. (Color online) Deviation from Kramer’s degeneracyas function of pM /µ p with µ p = 1 .
15 for ǫ = ∆ = 0. excellent agrement with our analytical prediction µ p =8 / ≈ . µ p . As we have shownabove, the values of µ p and µ ∆ must be the same. Onthe other hand, a numerical result found in Ref. [13]was µ ∆ ≈ .
45. We believe that the reason for this dis-crepancy is that only large values of ∆ were used in Ref.[13]. Indeed, in that paper it was impossible to resolvetwo separate critical states for ∆ . ν reported in different papers. In theoriginal paper [13] a broad range of ǫ ∈ [0 ,
1] was used(including the values of ǫ far from the critical point), andthe result was ν ≈ .
12. In a more recent study [27]the authors used only data for ǫ < .
05 (very close tothe critical point), and obtained ν ≈ . ν = 4 /
3. Thesame arguments explain the discrepancy between the ex-act value µ B = µ δ = 4 / ≈ .
57 and the numericalresult µ δ ≈ . CONCLUSIONS
In conclusion, we have studied relevant perturbationsat the spin quantum Hall (SQH) transition critical point.Many critical exponents at the transition have beenfound before. We have derived several new exponents,and corrected a subtle error in an earlier prediction.All (present and older) results are summarized in Ta-
FIG. 4. (Color online) Schematic plot of the phase boundary ǫ = ± ( a ∆ + bp ) ϕ/ . ble I. Our analysis demonstrates that several symmetry-breaking perturbations, which are distinct in the micro-scopic network model of the SQH transition, correspondto the same relevant perturbation at the critical point.In particular, the variance p of the extra random phaseof the scattering matrices on the links plays exactly thesame role as the spin-rotation symmetry-breaking param-eter ∆. Both happen to represent the effect of a randomZeeman magnetic field and drive the system to a localizedphase.Our results allow us to represent the phase diagram forour system in the three-dimensional space of parameters ǫ , ∆, and p . Indeed, the last two parameters appear inthe combination a ∆ + bp with some non-universal co-efficients. By the standard scaling argument, the criticalsurface is described by the equation ǫ = ± ( a ∆ + bp ) ϕ/ , (32)where the crossover exponent ϕ = µ p /ν = 6 / . (33)The critical surface is schematically shown in Fig. 4,where, for illustration purposes, we chose a = 1, b = 2.Finally, it is interesting to note that when we set ǫ = 0in our model, then for any nonzero p it becomes formallyequivalent to the network model proposed in Refs. [12]as a tool to study localization of electrons in a randommagnetic field. While the physics of the random mag-netic field problem and the spin quantum Hall effect isvery different, the equivalence of the models is seen, inparticular, in the absence of extended states in the ∆- p plane except for the critical point at the origin. ACKNOWLEDGEMENTS
We thank Mikhail Raikh for many fruitful discussions.This work was supported in part by US - Israel BinationalScience Foundation (BSF) Grant No. 2010030 and by theShamoon College of Engineering (SCE) under internalGrant No. 5368911113.
SUSY method for the SU(2) network in class C
In this Appendix we provide details of the SUSYmethod for the class C network.Usually in the SUSY approach one needs two types ofbosons and fermions, retarded and advanced, to be ableto obtain two-particle properties. However, the particle-hole symmetry relates retarded and advanced Greensfunctions [14]. Hence, for the study of mean values ofsimple observables, we need only one fermion and oneboson per spin direction per site. Let us now considertransfer matrices on the up- and down-going links sepa-rately.
Up-links
Let us denote the single boson and fermion per spindirection σ for an up site i as f iσ , b iσ . In our scheme oflabeling the up sites have an odd index i . Propagation ofa doublet of complex fluxes on a link is governed by anSU(2) scattering matrix S , which relates the doublets ofincoming I and outgoing O fluxes: o ↑ o ↓ ! = S i ↑ i ↓ ! = α β − β ∗ α ∗ ! i ↑ i ↓ ! . (34)This propagation looks identical to the propagation offluxes on two adjacent links of a directed network whichwas considered in detail in Ref. [19]. Thus, we can eas-ily borrow the second-quantized supersymmetric form ofthe transfer matrix from that reference by omitting theadvanced particles and replacing the site indices by thespin indices: T = : exp (cid:18) βα ∗ ( f †↑ f ↓ + b †↑ b ↓ ) − β ∗ α ( f †↓ f ↑ + b †↓ b ↑ ) (cid:19) : × α n f ↑ + n b ↑ ( α ∗ ) n f ↓ + n b ↓ , (35)where n bσ = b † σ b σ , etc., and the colons stand for normalordering.It is easy to obtain the commutation relations between T and fermions and bosons: T c †↑ = ( αc †↑ − β ∗ c †↓ ) T, T c †↓ = ( α ∗ c †↓ + βc †↑ ) T,T c ↑ = ( α ∗ c ↑ − βc ↓ ) T, T c ↓ = ( αc ↓ + β ∗ c ↑ ) T. (36) −1/2 3 V + V − W − W + BQ + Q −1/2 1/2−1/2 Q FIG. 5. The weights of the adjoint representation of osp (2 | | Here and later by c, c † we denote either b or f and theirconjugates. These relation are conveniently interpretedas giving the evolution of states created by c † σ in theSchr¨odinger representation, or the operators c † σ them-selves in the Heisenberg representation, where the op-erators are time ordered from right to left. Equations(36) may be written in a short form as T c † σ = c † σ ′ S σ ′ σ T, T c σ = S † σσ ′ c σ ′ T. (37)Relations (37) imply that under the commutation with T the bosons and fermions transform as spinors (in thefundamental representation) of the SU(2) group of thescattering matrices S . Then the SU(2) singlet bilin-ear combinations of our fermionic and bosonic operatorscommute with the T . There are 8 such combinations,which we denote following Ref. [37] as B = 12 ( b †↑ b ↑ + b †↓ b ↓ + 1) , Q = 12 ( f †↑ f ↑ + f †↓ f ↓ − ,Q + = f †↑ f †↓ , Q − = f ↓ f ↑ ,V + = 1 √ b †↑ f †↓ − b †↓ f †↑ ) , W − = ( V + ) † ,V − = − √ b †↑ f ↑ + b †↓ f ↓ ) , W + = − ( V − ) † . (38)We combine these generators into a single eight-component object J , or superspin. These operators sat-isfy the (anti)commutation relations of the osp (2 |
2) Liesuperalgebra: [
B, Q ] = [ B, Q ± ] = 0 , [ B, V ± ] = 12 V ± , [ B, W ± ] = − W ± , [ Q , Q ± ] = ± Q ± , [ Q + , Q − ] = 2 Q , [ Q , V ± ] = ± V ± , [ Q , W ± ] = ± W ± , [ Q + , V − ] = V + , [ Q + , W − ] = W + , [ Q − , V + ] = V − , [ Q − , W + ] = W − , [ Q + , V + ] = [ Q + , W + ] = [ Q − , V − ] = [ Q − , W − ] = 0 , { V + , V − } = { W + , W − } = 0 , { V + , W + } = Q + , { V + , W − } = B − Q , { V − , W − } = − Q − , { V − , W + } = − B − Q . (39)The components B and Q , Q ± of the superspin gen-erate the even subalgebra u (1) ⊕ su (2). An importantsub-superalgebra is the gl (1 |
1) formed by Q , B, V − , W + ,which we will call collectively the components of the su-perspin K .The algebra osp (2 |
2) has rank two (it has two Cartangenerators: B and Q ), and its representations are la-beled by two quantities. We use the u(1) “charge” (thevalue of B in a representation) b , and the value q of the“spin” of su(2) generated by the Q i . Representation withthe highest weight ( b, q ) is denoted by π ( b, q ) [38]. Forexample, the adjoint representation of osp (2 |
2) is π (0 , osp (2 |
2) is C ( J ) = Q − B + 12 (cid:0) Q − Q + + Q + Q − + V + W − − W − V + − V − W + + W + V − (cid:1) , (40)and in the representation π ( b, q ) it takes the value q − b .The quadratic Casimir of the gl (1 |
1) subalgebra is C ( K ) = Q − B + 12 (cid:0) W + V − − V − W + (cid:1) (41)As follows from Eqs. (37) the action of T decomposesthe Fock space of the bosons and fermions into irreduciblerepresentations of SU(2). When we average over theSU(2), any non-trivial representation is projected out.Thus, the averaged link transfer matrix acts as the pro-jection operator to the subspace of the SU(2)-singlets: h T i SU(2) = P. (42)There are only three singlets in this subspace which wedenote as | m i , m = 0 , ,
2, and define as | i = | vacuum i , (43) | i = V + | i = 1 √ b †↑ f †↓ − b †↓ f †↑ ) | i , (44) | i = Q + | i = f †↑ f †↓ | i . (45) ½ ½-½ -½ BQ | 〉 | 〉 | 〉 | 〉 -2 | 〉 -1 | 〉 - W - + V + Q - Q + W - - V + - W - - - V - + V - - - W - - Q- - + Q- - FIG. 6. The weights of the fundamental representation π ( , ) of osp (2 |
2) and its dual π ( − , ). These singlets form the fundamental representation π ( , ) of the osp (2 |
2) algebra. It is shown in Fig. 6together with its dual π ( − , ).It is easy to find the action of the generators of osp (2 | Q | i = − | i , Q | i = 0 , Q | i = 12 | i ,B | i = 12 | i , B | i = | i , B | i = 12 | i ,Q + | i = | i , Q − | i = | i ,V + | i = | i , V − | i = −| i ,W + | i = | i , W − | i = | i . (46) Down-link
Consider next a down-going link. Fermions and bosonson such links will be denoted by bars: ¯ f σ , ¯ b σ . On adown link the incoming and outgoing channels are in-terchanged. Then if we want to think of the evolutionof the states on the link as going up in the vertical timedirection, we need to relate the doublet I to the doublet O . Inverting relations (34), we get i ↑ i ↓ ! = S † link o ↑ o ↓ ! = α ∗ − ββ ∗ α ! o ↑ o ↓ ! . (47)Then the bosonic part of the transfer matrix on a downlink is¯ T ¯ b = : exp (cid:18) − βα ¯ b †↑ ¯ b ↓ + β ∗ α ∗ ¯ b †↓ ¯ b ↑ (cid:19) : ( α ∗ ) n ¯ b ↑ α n ¯ b ↓ . (48)This transfer matrix gives the following commutationrelations for the bosons:¯ T ¯ b ¯ b † σ = ¯ b † σ ′ S † σ ′ σ ¯ T ¯ b , ¯ T ¯ b ¯ b σ = S σσ ′ ¯ b σ ′ ¯ T ¯ b . (49)0These relations are again easily interpreted from thepoint of view of evolution of states. Comparing themwith Eq. (37), we see that on the down links the statesat later times are related to the states at earlier times inthe opposite way to what happens on the up links, whichis natural.Now we want to add fermions. This is somewhat tricky,since the cancellation of closed loops in the SUSY for-malism requires the presence of negative norm states inthe same way as in the case of the Chalker-Coddingtonmodel. So far we used the canonical bosons ¯ b . Then thefermions on the down links should satisfy { ¯ f , ¯ f † } = − . (50)Then the states with odd number of ¯ f will have negative(squared) norms. For such fermions the operator count-ing the number of them in a state has to be defined as n ¯ f = − ¯ f † ¯ f . (51)We also want the fermions to satisfy the same com-mutation relations (49) with ¯ T as the bosons. This isachieved by the following transfer matrix:¯ T = : exp (cid:18) − βα ( ¯ f ↓ ¯ f †↑ + ¯ b ↓ ¯ b †↑ ) + β ∗ α ∗ ( ¯ f ↑ ¯ f †↓ + ¯ b ↑ ¯ b †↓ ) (cid:19) : × ( α ∗ ) n ¯ f ↑ + n ¯ b ↑ α n ¯ f ↓ + n ¯ b ↓ . (52)Notice that the bosonic part of this operator is the sameas Eq. (48). It is now easy to check that the commutatorswith bosons and fermions have the same form:¯ T ¯ c † σ = ¯ c † σ ′ S † σ ′ σ ¯ T , ¯ T ¯ c σ = S σσ ′ ¯ c σ ′ ¯ T . (53)As on the up links, these relations imply that thefermions and bosons on the down links transform asSU(2) spinors under commutation with ¯ T . Their sin-glet bilinear combinations again form the generators ofthe osp (2 |
2) superalgebra, and we define them as¯ B = −
12 (¯ b †↑ ¯ b ↑ + ¯ b †↓ ¯ b ↓ + 1) , ¯ Q = 12 ( ¯ f †↑ ¯ f ↑ + ¯ f †↓ ¯ f ↓ + 1) , ¯ Q + = ¯ f ↓ ¯ f ↑ , ¯ Q − = ¯ f †↑ ¯ f †↓ , ¯ V + = − √ b ↑ ¯ f ↓ − ¯ b ↓ ¯ f ↑ ) , ¯ W − = ( ¯ V + ) † , ¯ V − = 1 √ f †↑ ¯ b ↑ + ¯ f †↓ ¯ b ↓ ) , ¯ W + = − ( ¯ V − ) † . (54)These operators satisfy the same commutation relations(39) as the ones on the up links.The quadratic Casimirs for the dual superspins ¯ J and¯ K are defined in the same way as for J and K , see Eqs.(40) and (41). Using the quadratic Casimirs we can in- troduce the invariant products of superspins: J · ¯ J ≡ ¯ J · J = C ( J + ¯ J ) − C ( J ) − C ( ¯ J )= 2 Q ¯ Q − B ¯ B − V − ¯ W + + W + ¯ V − + V + ¯ W − − W − ¯ V + + Q + ¯ Q − + Q − ¯ Q + , (55) K · ¯ K ≡ ¯ K · K = C ( K + ¯ K ) − C ( K ) − C ( ¯ K )= 2 Q ¯ Q − B ¯ B − V − ¯ W + + W + ¯ V − . (56)The transfer matrix ¯ T averaged over random SU(2)scattering matrices again gives the projector h ¯ T i SU(2) = ¯
P . (57)onto the space of SU(2) singlets | ¯ m i , m = 0 , , | ¯0 i = | vacuum i , (58) | ¯1 i = − ¯ W − | ¯0 i = 1 √ b †↑ ¯ f †↓ − ¯ b †↓ ¯ f †↑ ) | ¯0 i , (59) | ¯2 i = − ¯ Q − | i = − ¯ f †↑ ¯ f †↓ | ¯0 i . (60)These singlets form the representation π ( − , ) of the osp (2 |
2) algebra dual to the fundamental π ( , ). Notethat the state | ¯1 i contains odd number of fermions, and,therefore, has negative square norm: h ¯1 | ¯1 i = − . (61)The action of the generators on the states in the rep-resentation π ( − , ) is easily found to be¯ Q | ¯0 i = 12 | ¯0 i , ¯ Q | ¯1 i = 0 , ¯ Q | ¯2 i = − | ¯2 i , ¯ B | ¯0 i = − | ¯0 i , ¯ B | ¯1 i = −| ¯1 i , ¯ B | ¯2 i = − | ¯2 i , ¯ Q + | ¯2 i = −| ¯0 i , ¯ Q − | ¯0 i = −| ¯2 i , ¯ V + | ¯1 i = | ¯0 i , ¯ V − | ¯1 i = −| ¯2 i , ¯ W + | ¯2 i = −| ¯1 i , ¯ W − | ¯0 i = −| ¯1 i . (62) Nodal transfer matrices
With our choice of the scattering at the nodes to bediagonal in the spin index, the node evolution operators T i,i +1 are simple generalizations of the ones used in theSUSY formulation of the CC model. Essentially, we justhave to drop the advanced particles and take the productover the spin indices. As we mentioned in Sec. , the phaseand the damping factors e iφ = e iδ − η have been moved tothe nodes, so we use the nodal scattering matrices (12).This gives the following expression for T : T = Y σ = ↑ , ↓ h exp (cid:16) t Aσ e i ¯ φ (cid:0) f † σ ¯ f † σ + b † σ ¯ b † σ (cid:1)(cid:17) × (cid:0) r Aσ e iφ (cid:1) n f σ + n b σ (cid:0) r Aσ e i ¯ φ (cid:1) n ¯ f σ + n ¯ b σ × exp (cid:16) − t Aσ e iφ (cid:0) ¯ f σ f σ + ¯ b σ b σ (cid:1)(cid:17)i , (63)1and a similar expression for T (obtained by replacing allsubscripts 1 by 3, and the changing the sublattice index A to B ).We now simplify notation by dropping the site indices,since the fermions and bosons on the two sites (as well asthe phases φ ) are differentiated by the overbar. Likewise,we drop the sublattice index. Then we can rewrite theevolution operators for both sublattices as: T = T + T ¯ T T − , (64) T + = Y σ e t σ e i ¯ φ A † σ , A † σ = b † σ ¯ b † σ + f † σ ¯ f † σ , (65) T = Y σ (cid:0) r σ e iφ (cid:1) n fσ + n bσ , ¯ T = Y σ (cid:0) r σ e i ¯ φ (cid:1) n ¯ fσ + n ¯ bσ , (66) T − = Y σ e − t σ e iφ A σ , A σ = ¯ b σ b σ + ¯ f σ f σ . (67) We need to project T to the the tensor product π ( , ) ⊗ π ( − , ). Let us now denote the projectionoperator by ˆ P ≡ P ⊗ ¯ P . We note, first of all, that due toirreducibility of π ( ± , ), the projected transfer matrixmust be a linear combination of products of superspincomponents J or ¯ J (and identity operators) on the twosites. Next we note that when projecting T ± , we needto expand the exponentials in T ± only to linear order ineach A and A † (since higher orders only contain tripletcombinations of bosons and fermions on each site, andwould take us out of the spaces of interest):ˆ P T ˆ P = ˆ P (cid:0) t ↑ e i ¯ φ A †↑ + t ↓ e i ¯ φ A †↓ + t ↑ t ↓ e i ¯ φ A †↑ A †↓ (cid:1) T ¯ T (cid:0) − t ↑ e iφ A ↑ − t ↓ e iφ A ↓ + t ↑ t ↓ e iφ A ↑ A ↓ (cid:1) ˆ P . (68) T and ¯ T act diagonally in the respective irreps, and can be written as 1 + (cid:0) r ↑ r ↓ e iφ − (cid:1) ( B + Q ) and 1 − (cid:0) r ↑ r ↓ e i ¯ φ − (cid:1) ( ¯ B + ¯ Q ), respectively. Then when we develop the products in the last expression, linear and cubic terms in A and A † do not contribute, and neither do products of the type A † σ T ¯ T A − σ , so we haveˆ P T ˆ P = ˆ P (cid:0) T ¯ T − t ↑ e i ( ¯ φ + φ ) A †↑ T ¯ T A ↑ − t ↓ e i ( ¯ φ + φ ) A †↓ T ¯ T A ↓ + t ↑ t ↓ e i ¯ φ A †↑ A †↓ T ¯ T + t ↑ t ↓ e iφ T ¯ T A ↑ A ↓ + t ↑ t ↓ e i ( ¯ φ + φ ) A †↑ A †↓ T ¯ T A ↑ A ↓ (cid:1) ˆ P . (69)Let us number terms in this expression (1) through (6), and consider them one by one. First we have(1) = 1 + (cid:0) r ↑ r ↓ e iφ − (cid:1) ( B + Q ) − (cid:0) r ↑ r ↓ e i ¯ φ − (cid:1) ( ¯ B + ¯ Q ) − (cid:0) r ↑ r ↓ e iφ − (cid:1)(cid:0) r ↑ r ↓ e i ¯ φ − (cid:1) ( B + Q )( ¯ B + ¯ Q ) . (70)We rearrange the next two terms in Eq. (69) noticing that commutation of T or ¯ T with fermions or bosons changesone of the number operators by one. Then we haveˆ P A † σ T ¯ T A σ ˆ P = e − i ( φ + ¯ φ ) r σ ˆ P (cid:0) b † σ b σ T ¯ T ¯ b † σ ¯ b σ − b † σ f σ T ¯ T ¯ b † σ ¯ f σ + f † σ b σ T ¯ T ¯ f † σ ¯ b σ + f † σ f σ T ¯ T ¯ f † σ ¯ f σ (cid:1) ˆ P . (71)Here we need to represent bilinears in bosons and fermions on each site as linear combinations of a singlet and atriplet bilinear, and then the projection operators ˆ P allow us to drop the triplets. This givesˆ P A † σ T ¯ T A σ ˆ P = e − i ( φ + ¯ φ ) r σ ˆ P h(cid:16) Q + 12 (cid:17) T ¯ T (cid:16) ¯ Q − (cid:17) − (cid:16) B − (cid:17) T ¯ T (cid:16) ¯ B + 12 (cid:17) − V − T ¯ T ¯ W + + 12 W + T ¯ T ¯ V − i ˆ P . (72)This expression contains products of superspin components on each site. Due to irreducibility of π ( ± , ), suchproducts can be replaced by linear combinations of superspin components. The easiest way to find these combinationsis to use the matrix representations of the superspin components. The result isˆ P A † σ T ¯ T A σ ˆ P = e i ( φ + ¯ φ ) r − σ h(cid:16) Q + 12 (cid:17)(cid:16) ¯ Q − (cid:17) − (cid:16) B − (cid:17)(cid:16) ¯ B + 12 (cid:17) − V − ¯ W + + 12 W + ¯ V − i , (73)and (2) + (3) = (2 t ↑ t ↓ − t ↑ − t ↓ ) e i ( φ + ¯ φ ) h(cid:16) Q + 12 (cid:17)(cid:16) ¯ Q − (cid:17) − (cid:16) B − (cid:17)(cid:16) ¯ B + 12 (cid:17) − V − ¯ W + + 12 W + ¯ V − i . (74)2In the same way we treat the other terms:ˆ P A †↑ A †↓ T ¯ T ˆ P = ˆ P (cid:0) − f †↑ f †↓ ¯ f †↑ ¯ f †↓ + f †↑ b †↓ ¯ f †↑ ¯ b †↓ + b †↑ f †↓ ¯ b †↑ ¯ f †↓ + b †↑ b †↓ ¯ b †↑ ¯ b †↓ (cid:1) T ¯ T ˆ P = − Q + ¯ Q − − V + ¯ W − , ˆ P T ¯ T A ↑ A ↓ ˆ P = − Q − ¯ Q + + W − ¯ V + , (75)and (4) + (5) = − t ↑ t ↓ e i ¯ φ ( Q + ¯ Q − + V + ¯ W − ) − t ↑ t ↓ e iφ ( Q − ¯ Q + − W − ¯ V + ) . 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