Multifractal nature of the surface local density of states in three-dimensional topological insulators with magnetic and nonmagnetic disorder
aa r X i v : . [ c ond - m a t . d i s - nn ] F e b Multifractal nature of the surface local density of states in three-dimensionaltopological insulators with magnetic and nonmagnetic disorder
Matthew S. Foster ∗ Center for Materials Theory, Department of Physics and Astronomy,Rutgers University, Piscataway, NJ 08854, USA (Dated: May 8, 2018)We compute the multifractal spectra associated to local density of states (LDOS) fluctuationsdue to weak quenched disorder, for a single Dirac fermion in two spatial dimensions. Our resultsare relevant to the surfaces of Z topological insulators such as Bi Se and Bi Te , where LDOSmodulations can be directly probed via scanning tunneling microscopy. We find a qualitative differ-ence in spectra obtained for magnetic versus non-magnetic disorder. Randomly polarized magneticimpurities induce quadratic multifractality at first order in the impurity density; by contrast, nooperator exhibits multifractal scaling at this order for a non-magnetic impurity profile. For thetime-reversal invariant case, we compute the first non-trivial multifractal correction, which appearsat two loops (impurity density squared). We discuss spectral enhancement approaching the Diracpoint due to renormalization, and we survey known results for the opposite limit of strong disorder. PACS numbers: 73.20.-r, 73.20.Jc, 64.60.al, 72.15.Rn
I. INTRODUCTION
The defining attribute of a 3D Z topological insulator (TI) is the presence of an odd number of 2D masslessDirac bands at the material surface. Unlike the Diracelectrons that can appear in a purely 2D system (no-tably in graphene), the surface states of a (strong) 3D TIare robustly protected from the opening of gap, so longas time-reversal symmetry is preserved. The protectioncan be viewed as a consequence of the parity anomaly, which “holographically” links surface states separated bya topologically non-trivial bulk, and gives rise to the sig-nature properties of the Z TI state: the half-integerquantum Hall effect, quantized magnetoelectric coupling,“axion” electrodynamics, etc.
As stressed by Schny-der et al. in Ref. 7, the robust character of the surfacestates in the presence of quenched disorder can also betaken as a principal characteristic of a topological insu-lator. In particular, these states are protected from An-derson localization, even in the presence of a “strong”impurity potential, so long as time-reversal invariance ispreserved. With its 2D Dirac band pinned to an exposed surface,a 3D TI is ideally suited to local probes such as scanningtunneling microscopy (STM). In spectroscopic mode, anSTM captures an areal map of the local density of states(LDOS). There are several ways of analyzing such data.One is to look for quasiparticle interference (QPI) inthe LDOS Fourier transform. This method is useful fordetermining short-distance details, and contains similarinformation as an analysis of LDOS Friedel oscillationsin the presence of a single impurity. It has been appliedin TIs to experimental data and analyzed theoretically inRefs. 12,13 and 14,15, respectively. In QPI, the disorderis employed primarily as a facilitator to gleam informa-tion about the clean system. Multifractal analysis provides a complementary method better suited to extracting large-distance,disorder-dominated features in the same LDOS datafield. It is a standard tool for assaying quantum interfer-ence phenomena, and is employed in the analysis of wave-functions near a metal-insulator transition as well as
FIG. 1: Sketch of disorder “flavors” on the surface of a Z topological insulator. In the time-reversal invariant case,the impurities are neutral adatoms or charged dopant ions,depicted as spheres in (a) . The effects of these on the surfaceDirac theory [Eq. (3.1)] are encoded in the scalar potential V ( r ). In the case of magnetic disorder, the impurity spinsare indicated by the arrows in (b) and (c) . In the limit thatthe spins reside in the plane of the surface, (b) , the disorderappears as a vector potential A ( r ). The opposite case of out-of-plane polarization, (c) , gives the random mass M ( r ). Thecase of generic time-reversal breaking disorder has all threepotentials present. mesoscopic fluctuations in diffusive metallic systems. In this paper, we derive new results for LDOS multifrac-tal spectra associated to disordered topological insulatorsurface states. In particular, we extend the pioneering re-sults of Ref. 24 to the generic cases of time-reversal ( T )preserving and breaking impurities. Our calculations areperformed in the near-ballistic limit, wherein weak dis-order enters as a perturbation to the clean Dirac bandstructure. A key characteristic of 2D Dirac fermions isthat this weak disorder regime is continuously connectedto more conventional domains of multifractal analysis,i.e. the diffusive (symplectic) metal and the inte-ger quantum Hall plateau transition. These ap-pear at strong coupling (many impurities) for dirty Diracfermions.
We consider the case of a single flavor Dirac surfaceband, relevant to (e.g.) the TIs Bi Se and Bi Te (Refs. 2,3,30). The different kinds of T -preserving and T -breaking disorder are sketched in Fig. 1. We demon-strate that the LDOS multifractal spectra observed inthe absence of time-reversal symmetry breaking (i.e., fornon-magnetic disorder) is qualitatively weaker than thatinduced by magnetic impurities. In particular, the firstmultifractal correction obtains at first order in the im-purity density for the case of broken T , while the firstnon-trivial amplitude appears at second order in the T -invariant case. We compute the leading terms via one-and two-loop calculations, respectively. We also computeunnormalized spectra for the spin LDOS in the case ofmagnetic impurities. We show that renormalization ef-fects can enhance multifractality near the Dirac point.Finally, we summarize prior results on various strong-coupling regimes. Our goal is to sketch the full portraitof quantum interference physics on the surface of a TI,valid when interparticle interactions can be neglected.Our results indicate that the long-distance, disorder-dominated features captured by the multifractal analysisbehave in many cases opposite to the short-distance char-acteristics that appear in quasiparticle interference. In Ref. 14, the authors observed that QPI is strongestfor the spin LDOS response to magnetic impurities, whilethe unpolarized LDOS pattern vanishes for magnetic dis-order (in the first Born approximation). The QPI re-sponse of the LDOS to non-magnetic disorder is weakbut non-zero. By contrast, in this work we find thatthe LDOS multifractality is strongest for magnetic im-purities, while the spin LDOS spectrum comparativelyexhibits the same or weaker strength fluctuations, de-pending upon the polarization direction.The weak influence of non-magnetic disorder is tied tothe intrinsic spin-orbit coupling that defines the mass-less Dirac kinetic term. Multifractality is suppressed atone loop due to interference mediated by the Dirac pseu-dospin, which is proportional to the physical spin on a Z insulator surface. The spin is also responsible for thesuppression of backscattering from a single non-magneticimpurity. On the TI surface, magnetic disorder Zeemancouples directly to the Dirac spin, enabling backscatter- ing in near-ballistic transport, and inducing multifractalLDOS fluctuations at the lowest order in the impuritydensity.A notable problem in experiments probing topologicalinsulator surface states has been the unintentional dop-ing of carriers into the bulk bands, which then dominatetransport measurements in large samples. Even if thechemical potential is moved into the gap, it may residefar from the Dirac point, making it difficult to observesurface state carrier dynamics at low densities. In thisrespect, STM offers several advantages over transport ex-periments. First, the position of the chemical potential isno barrier to probing states at the Dirac point, since thelatter can always be reached by tuning the bias voltage(although the Dirac point is not guaranteed to reside inthe bulk gap).
Assuming that the Dirac point or thelow density regime can be accessed by tuning the tunnel-ing bias, the advent of a finite, even large doping of thesurface and/or bulk states may actually play a beneficialrole in facilitating the observation of disorder-inducedquantum interference effects. This is because a finitecarrier density screens the long-range Coulomb potentialintroduced by charged defects. The potential landscapeformed by screened impurities is short-range correlatedon scales larger than the screening length. Good screen-ing eliminates the problem of electron and hole puddleformation, which has until recently occluded trans-port and other properties of Dirac carriers in graphenenear the Dirac point. On the other hand, a low density ofpoorly-screened bulk dopants induces a long-range corre-lated potential and puddle formation, as in graphene. LDOS fluctuations in the puddle regime are an importanttopic for future work.Three-dimensional topological insulators provide uswith an interesting paradigm flip for quantum interfer-ence phenomena. Isolating the surface state contributionin transport measurements is problematic. By compari-son, direct LDOS imaging is easier than in conventionalsemiconductor systems, wherein the 2D electron gas istypically buried in a layered material stack. Moreover,the amount of surface disorder can to some extent becontrolled; for example, magnetic impurities can be de-posited across the surface of an otherwise high-qualitybulk 3D sample. These can be charge-neutral adatomsor charged dopants; an example of the former (latter) isprovided by iron (manganese) in Bi Se .This paper is organized as follows. We begin in Sec. IIwith a lightning review of multifractal composite and spinLDOS measures. In Sec. III, we present new results formultifractal LDOS fluctuations in TI surface states, inthe presence of weak disorder. We also show how renor-malization can enhance multifractality close to the Diracpoint. Finally, in Sec. IV, we review previous results onvarious strong disorder regimes relevant to the Z TI sur-face states and LDOS statistics. In particular, we discussthe symplectic metal, the integer quantum Hall plateautransition, and the Anderson insulator. Various technicaldetails are relegated to appendices. In Appendix A, wereview the symmetry classes of Anderson (de)localizationthat appear in the disordered Dirac surface theory. InAppendix B, we supply some details of our perturbativecalculations.
II. MULTIFRACTAL LDOS MEASURESA. Definitions
We suppose that the tunneling local density of states(LDOS) ν ( ε, r ) is imaged at a fixed energy ε over an L × L field of view. The field is finely partitioned into a grid ofboxes. The box edge length a ≪ L must be chosen largerthan any “microscopic” scale l m , such as the correlationlength of the random potential. One introduces the boxprobability µ n ( ε ) ≡ R A n d r ν ( ε, r ) P l " R A l d r ν ( ε, r ) , (2.1)where A n denotes the n th box. LDOS multifractalityis defined through the inverse of the participation ratio(IPR), P q ( ε ) ≡ X n µ qn ( ε ) ∼ (cid:16) aL (cid:17) τ ( q,ε ) . (2.2)The right-hand side (scaling limit) obtains when l m ≪ a ≪ L ; corrections are down by higher powers of a/L . The exponent τ ( q, ε ) is the multifractal momentspectrum for LDOS fluctuations at energy ε .The construction in Eqs. (2.1) and (2.2) is useful forcharacterizing a system with extended states, or for anAnderson localized system in which L ≪ ξ loc ( ε ); ξ loc de-notes the localization length. In what follows, we as-sume experiments are performed at sufficiently low tem-peratures so that inelastic cutoffs to quantum interfer-ence can be ignored. A clean system with plane wavestates at energy ε has τ ( q, ε ) = 2( q − Multifractality refers to the incorporation of corrections non-linear in q .Physically, these arise due to quantum interference viamultiple scattering of electron waves in a dirty environ-ment, processes that serve as the precursor to Andersonlocalization. For weak disorder, the spectrum is typically dominatedby the quadratic correction τ ( q, ε ) = 2( q − − θ ( ε ) q ( q − , (2.3)where θ ≥ L ≪ ξ loc for the orthogonal or unitary classes), one finds θ = β − π N ( ε ) D , (2.4) where N ( ε ) denotes the average density of states, D isthe classical (Drude) diffusion constant, and β ∈ { , , } ,depending upon the presence or absence of time-reversalsymmetry and spin-orbit scattering. At stronger dis-order, higher order corrections in θq must be retained; forthe diffusive metals, results are known to four loops. An alternative characterization of LDOS multifractal-ity is provided by the singularity spectrum f ( α ):Over a subset of the sample grid area that scales as( L/a ) f ( α ) , the box probability µ ∼ ( a/L ) α . The sin-gularity spectrum is the Legendre transform of τ ( q ), f ( α ) = qα − τ ( q ) , dτ ( q ) dq = α. For the quadratic spectrum in Eq. (2.3), one obtains f ( α ) = 2 − θ ( α − − θ ) . (2.5)In this “parabolic approximation,” the strength ofthe multifractality is encoded in the peak position α [ f ( α ) = 2], and the width α W of the spectrum such that f ( α ± α W /
2) = 0, α = 2 + θ, α W = 4 √ θ. (2.6)Part of the power of multifractal analysis for disorderedquantum systems derives from the fact that the spectra[ τ ( q ) or f ( α )] typically depend only upon a few grossmeasures of the impurity potential. In the case of dirtymetals, the entire spectrum can be computed as an ex-pansion in one parameter, the inverse conductance (con-sistent with scaling theory). At a non-interactingAnderson localization transition, τ ( q ) and f ( α ) becomeuniversal functions, so that the critical point is charac-terized by an infinite set of critical exponents [e.g., theexpansion coefficients for τ ( q )].The spectra above have been defined for data collectedin a single fixed realization of the disorder. Strictlyspeaking, Eq. (2.3) then applies only for | q | ≤ q c , where q c = p /θ . Outside of this range, the τ ( q ) associatedto a fixed disorder realization is linear, a phenomenonknown as spectral termination. [This assumes thathigher order corrections can be ignored for q ≥ q c . Re-gardless, the τ ( q ) spectrum is always linear for suffi-ciently large q ]. Termination can be viewed as a con-sequence of the restriction to positive sample measures f ( α ) ≥ In the localized regime, the states contributing to theLDOS at a given position in the sample have a discreteenergy spectrum, quantized by the typical localizationvolume ξ loc . As a result, all non-unity LDOS moments diverge in the absence of level smearing. In a tunnelingexperiment, smearing can appear due to inelastic scatter-ing (temperature), open sample boundary conditions, ordue to the finite energy resolution of the instrument. Tocharacterize an Anderson insulating state over an L × L field of view with L ≫ ξ loc , the full LDOS distributionshould be examined; sensitive dependence of the dis-tribution shape to smearing can serve as a telltale sign ofthe localized regime. LDOS fluctuations in the Andersoninsulator are reviewed in more detail in Sec. IV B. B. Spin LDOS spectra
By restricting the character of the tunneling species, itmay be possible to measure individual LDOS componentsseparately. For example, in the case of a spin-polarized(ferromagnetic) STM tip, the spin-projected compo-nents ν ↑ , ↓ can be separately resolved. The use of an un-polarized tip recovers the composite LDOS ν = ν ↑ + ν ↓ .We define the spin LDOS along the spin space directionˆ ι , ν ˆ ι ( ε, r ) ≡ ν ˆ ι ↑ ( ε, r ) − ν ˆ ι ↓ ( ε, r ) . (2.7)For a time-reversal invariant system (with or withoutspin-orbit scattering and/or disorder), one has ν ˆ ι ( ε, r ) =0. In a system with broken time-reversal (e.g., magneticimpurities), but zero average spin polarization, the in-tegral of ν ˆ ι ( ε, r ) over a sufficiently large region becomesarbitrarily small; we cannot use the normalized construc-tion in Eqs. (2.1) and (2.2) to characterize spin LDOSmultifractals. Instead, we employ the un-normalized in-verse spin participation ratio (ISPR) P ˆ ιq ( ε ) ≡ X n (cid:0) µ ˆ ιn (cid:1) q , µ ˆ ιn ≡ Z A n d r ν ˆ ι ( ε, r ) . (2.8)In the scaling limit, P ˆ ιq ( ε ) ∼ c q (cid:16) aL (cid:17) x ˆ ιq − , (2.9)where the exponent x ˆ ιq is the scaling dimension for thecorresponding moment operator in the disorder-averagedfield theory description, and c q = 0 for even q . III. WEAK DISORDER MULTIFRACTALITYA. Model. Short- and long-range correlatedpotential landscapes
The Dirac surface states of a Z topological insula-tor (TI) are guaranteed to appear in an odd number offlavors. In this paper, we consider the simplest case ofa single flavor, relevant to (e.g.) Bi Se and Bi Te . TheHamiltonian is (in units such that ~ = 1) H = Z d r ψ † (cid:26) v F ˆ σ µ [ − i∂ µ + A µ ( r )]+ M ( r ) ˆ σ + V ( r ) (cid:27) ψ, (3.1)where µ ∈ { , } , and repeated indices are summed. Thecoordinates r = { x, y } chart the TI surface, while the topological bulk resides in the perpendicular z direction.In Eq. (3.1), v F denotes the Fermi velocity, and the Diracpseudospin Pauli matrices ˆ σ are related to the physicalspin 1 / S via { ˆ σ µ , ˆ σ } = 2 { ǫ µν S ν , S } . Thevector, scalar, and mass potentials { A , V, M } describethe effects of external electromagnetic fields and/or sur-face impurities. In the absence of time-reversal ( T ) sym-metry breaking, A = M = 0. (See Appendix A for anenumeration of discrete symmetry operations.) Thus, amass gap is explicitly forbidden so long as T remains agood symmetry, a consequence of the protection affordedby the topologically nontrivial bulk. When T is brokenby an external magnetic field B , the vector and masspotentials are A µ = − eA ( orb ) µ − γ k v F ǫ µν B k ,ν ,M = − γ ⊥ B z , (3.2)where γ k ( γ ⊥ ) denotes the Zeeman coupling to the in-plane field B k (out-of-plane field B z ), and the orbitaleffect is embedded in A ( orb ) α via ǫ αβ ∂ α A ( orb ) β = B z .Non-magnetic adatoms or charge traps are encoded inthe scalar potential V ( r ). In-plane (out-of-plane) polar-ized magnetic impurities additionally induce point ex-change coupling to the vector A ( r ) [mass M ( r )] fields. The different types of disorder leading to V , A , and M are sketched in Fig. 1. Assuming a random surface distri-bution of impurities and spatial rotational invariance onaverage, the disorder potentials can be taken as Gaussianwhite noise distributed variables, V ( r ) V ( r ′ ) = ∆ V v F δ ( r − r ′ ) ,A α ( r ) A β ( r ′ ) = ∆ A v F δ αβ δ ( r − r ′ ) ,M ( r ) M ( r ′ ) = ∆ M v F δ ( r − r ′ ) . (3.3)The dimensionless variances ∆ V,A,M quantify the disor-der strength. In the first Born approximation, these areof the form ∆ v F = n imp | ˜ u (0) | , (3.4)where n imp is the impurity density, and ˜ u ( q ) denotes theFourier transform of the single impurity potential. Wenote that a net in-plane magnetization of the surface im-purities A µ = 0 can be removed by a gauge transforma-tion, while the average scalar potential V is absorbed intothe chemical potential. We will assume that there is nonet magnetization perpendicular to the surface, M = 0,or that we only probe LDOS fluctuations on energy scalesmuch larger than the induced gap 2 v F M .In 2D, the single impurity potential u ( r ) [Eq. (3.4)]must decay faster than 1 /r (or oscillate rapidly enough)so that the limit ˜ u ( q →
0) exists; otherwise, the whitenoise assumption in Eq. (3.3) is invalidated by long rangeimpurity potential correlations. This causes a problemfor charged impurities, which can become poorly screenedfor a small surface doping relative to the Dirac point.In graphene, the long-range correlated potential undu-lations induced by poorly-screened substrate impuritiesleads to a smearing of the Dirac point over an energyscale k B T rms ∝ v F √ n imp , and to the breakup of the sam-ple into electron and hole puddles. The advent ofelectron-hole puddles has until recently prevented the ob-servation of various “intrinsic” phenomena associated tothe Dirac carriers in graphene experiments such as veloc-ity renormalization and hydrodynamic transport nearthe Dirac point. In this respect, a large surface or bulkdoping actually improves the situation for STM mea-surement of disorder-induced quantum interference, sincethese carriers screen the potential of surface charges. Thedisorder potential can be considered short-range corre-lated for scales larger than the screening length.If we consider only surface doping, with an insulatingbulk, then the Thomas-Fermi wavelength due to a finitesurface carrier density n is given by λ TF = 1 α r πn , (3.5)where α ≡ e /ǫv F is the effective “fine structure con-stant.” The permittivity ǫ = (1 + ǫ TI ) /
2, the averageof the bulk TI below and vacuum above the surface. ForBi Se with a surface density of n = 7 × cm − (corre-sponding to a doping level of 0 . v F = 5 × m/s (Ref. 30), and permittivity ǫ TI = 113, one obtains λ TF ∼
90 nm. This is very large,and indicates that the surface state carrier density is in-adequate to screen charged impurities. A smaller screen-ing length is possible for bulk doping, or by performingexperiments on thin film samples exfoliated over a metal-lic gate. Alternatively, one can restrict the deposition ofsurface impurities to non-doping adatoms, e.g. iron inBi Se . The disorder variance associated to Thomas-Fermi screened charged impurities is∆ V = π n imp n . (3.6)Finally, we note that the appearance in isolationof any of the three disorder potentials in Eq. (3.1)realizes three different symmetry classes of Anderson(de)localization, see Appendix A for a review. The T -invariant case with ∆ A,M = 0 belongs to the spin-orbitclass AII, which is also the class of the Z topologicalbulk [Fig. 1 (a) ]. In the case of broken T , ∆ V,M = 0realizes the random vector potential model in class AIII[Fig. 1 (b) ], while ∆
V,A = 0 gives the random mass modelin class D [Fig. 1 (c) ]. All three classes exhibit delocal-ized states in 2D, although this occurs only at the Diracpoint for class AIII. In the T -invariant symplectic case,the unpaired single Dirac flavor avoids the usual spin-orbit metal-insulator transition, remaining delocalizedeven for strong disorder due to a topological term. Thegeneric case of broken- T with all three disorder poten-tials non-zero realizes the unitary class A, and is believed to flow under renormalization to the plateau transitionin the integer quantum Hall effect. (See Sec. IV A 2for a review).Because in-plane (out-of-plane) Zeeman coupling ap-pears in the vector (mass) potential [Eq. (3.2)], one istempted to identify class AIII (class D) with the limitof an otherwise clean surface, dusted with charge neutralmagnetic impurities randomly polarized in-plane (per-pendicular to the TI surface). However, a magneticadatom is expected to also induce a local scalar potentialdeformation V ( r ). For example, it can dope the surfaceor bulk, as occurs for a manganese impurity in Bi Se (Ref. 37)]. As discussed in Appendix A, the advent ofany two flavors of disorder destroys the additional dis-crete symmetries enjoyed by the special class D and AIIIHamiltonians. The asymptotic long-distance LDOS scal-ing is then governed by the unitary class A, discussedabove. Nevertheless, depending upon the relative mi-croscopic strength of the magnetic versus potential per-turbations induced by polarized magnetic impurities, theclass AIII or D model may provide an adequate approxi-mation for broken- T LDOS fluctuations on intermediatescales.
B. Results
To compute the scaling of LDOS moments in a quan-tum theory with quenched disorder, one employs a pathintegral Z to express products of fermion Green’s func-tions as functionals of the disorder configuration. Usinga trick (replicas, supersymmetry, or Keldysh )to normalize Z = 1, the Green’s functions are for-mally averaged over disorder configurations (typicallywith a Gaussian weight). The result is a translationally-invariant, but “interacting” field theory, where the disor-der strength ∆ appears as a coupling constant. Per-turbative calculations are controlled via loop expansionfor small ∆.To determine the scaling, one decomposes the q th LDOS moment into projections upon the renormaliza-tion group (RG) eigenoperators of the disorder-averagedtheory.
The multifractal spectrum τ ( q ) is deter-mined by the most relevant (negative) scaling dimen-sion x q exhibited by an eigenoperator in this decomposi-tion, and is given by τ ( q ) = 2( q −
1) + x q − qx . (3.7)
1. Broken T : random vector potential disorder (Class AIII) The properties of the model in Eq. (3.1) with short-range correlated disorder [Eq. (3.3)] were originally stud-ied in Ref. 24. In this work, the exact multifractal spec-trum τ ( q ) was calculated for the broken- T , random vec-tor potential ( ∼ in-plane polarized magnetic impurity) class AIII model, to all orders in ∆ A . Technically, thisresult obtains because the disorder-averaged AIII modelis conformally invariant at the Dirac point, and the ex-act LDOS moment spectra can be extracted using anAbelian bosonization treatment. The exact spectrum is quadratic in q , and takes the form of Eq. (2.3), with θ A = ∆ A π . (3.8)Subsequent work on the random vector potentialmodel elucidated the mechanisms of termination andfreezing, transitions that occur in the spectral statis-tics for large moments q > q c (∆ A ) or strong disorder∆ A ≥ π .For this broken- T class, we can also examine the spinLDOS fluctuations, utilizing the same nonperturbativebosonization treatment employed in Ref. 24. The spinLDOS ν ˆ ι ( ε, r ) taken along an axis ˆ ι in spin space wasdefined by Eq. (2.7). Moment fluctuations are charac-terized by the inverse spin participation ratio (ISPR) inEq. (2.8), the scaling limit of which is controlled by thedimension x ˆ ιq that appears in Eq. (2.9). The out-of-planeISPR P ˆ3 q ( ε ) is associated to the “mass” fermion bilinear ν ˆ3 = ψ † ˆ σ ψ . For the random vector potential model,the most relevant contribution to P ˆ3 q ( ε ) carries the samescaling dimension that gives the composite LDOS scalingin Eqs. (2.3) and (3.8), x ˆ3 q = q − ∆ A π q . (3.9)The chiral components of the in-plane spin LDOS are theenergy-resolved U (1) Dirac current operators ν ± ≡ ν ˆ1 ± iν ˆ2 = ψ † ˆ σ ± ψ. (3.10)Moments of these are RG eigenoperators that receive nocorrections. The scaling of the associated ISPR is gov-erned by the disorder-independent (tree level) exponent x ± q = q. (3.11)Eqs. (3.8), (3.9), and (3.11) are exact results that holdto all orders in ∆ A .
2. Broken T : random mass disorder (Class D) In the rest of this section, we provide new resultsfor the broken- T , random mass ( ∼ out-of-plane polar-ized magnetic impurity) class D model, the T -invariantclass AII model, and the generic broken- T unitary classA model. For weak disorder, none of these are confor-mally invariant, and we resort to perturbation theory. Inthis section we summarize results; some technical aspectsare sketched in Appendix B. The results obtained belowhold only for small ∆ V,M ≪
1, wherein the disorder ap-pears as a weak marginal perturbation (at tree level) tothe clean Dirac surface band structure. For the broken- T case of random mass disorder (with∆ V = ∆ A = 0), one obtains quadratic multifractality atone loop, again governed by Eq. (2.3), with θ M = ∆ M π + O (cid:0) ∆ M (cid:1) . (3.12)Moments of the out-of-plane spin LDOS operator ν ˆ3 = ψ † ˆ σ ψ , as well as of the chiral in-plane [ U (1) current]operators ν ± = ψ † ˆ σ ± ψ constitute RG eigenoperators atone loop, with scaling dimensions x ˆ3 q = q + ∆ M π q + O (cid:0) ∆ M (cid:1) , (3.13) x ± q = q + O (cid:0) ∆ M (cid:1) . (3.14)Note that the first correction in Eq. (3.13) is positive (andlinear in q ); this should be contrasted with the AIII case,Eq. (3.9) above. On general grounds, the anomalous scal-ing dimension associated to the q th ≥ through Eqs. (2.1) and (2.2).For a quadratic τ ( q ) spectrum, this leads in particular to θ ≥ difference be-tween two orthogonal projections [Eq. (2.7)]; for this rea-son, the first disorder correction to the scaling dimensionin Eq. (3.13) is not required to appear with a particularsign.
3. Non-magnetic disorder (Class AII)
In the T -invariant case of scalar potential disorder, itturns out that no local operator (without derivatives) ex-hibits multifractal scaling to first order in ∆ V . For Diracfermions, this applies to both LDOS and energy-resolvedcurrent moments. Physically, the weak influence of non-magnetic disorder is due to interference mediated by theDirac pseudospin (equivalent to physical spin 1 / Technically, this result is derivedby mapping the one-loop renormalization process of localoperators to the action of a certain spin-1 / H ( eff ) V , and identifying renormalization group eigenopera-tors with states that diagonalize H ( eff ) V (see Appendix B).As a result, to lowest order one observes plane wave scal-ing in the LDOS IPR [Eq. (2.2)]. The spin LDOS van-ishes exactly, due to T .The first non-trivial correction to the LDOS τ ( q ) ap-pears at two loops. To this order, the spectrum is againquadratic as in Eq. (2.3). A straight-forward but labori-ous calculation gives the coefficient in this equation, θ V = 3∆ V π + O (cid:0) ∆ V (cid:1) . (3.15) FIG. 2: Quadratic multifractality for isolated disorder fla-vors. The Renyi dimension d q = 2 − θ q [Eqs. (2.3) and(3.16)] is plotted for the exact vector potential (AIII), one-loop mass (D), and two-loop scalar potential (symplectic AII)results [Eqs. (3.8), (3.12), and (3.15)]. The disorder strengthis ∆ = 0 .
05 for each case. The broken time-reversal class Dand AIII corrections appear at order ∆
M,A , while the (muchweaker) time-reversal invariant class AII correction begins atorder ∆ V . Since ∆ V ∝ n imp [Eqs. (3.4) or (3.6)], we find that thenon-trivial multifractal scaling begins at second orderin the impurity density. This is qualitatively weakerthan any of the broken- T regimes, where the quadraticmultifractality appears already at first order, Eqs. (3.8)and (3.12). This distinction between T -invariant and T -broken surfaces is our primary result, and can betested directly in STM experiments by varying the con-centration of deposited surface disorder. Although the T -invariant case is not conformally invariant (for a dis-cussion of renormalization effects, see Sec. III C, below),the multifractal τ ( q ) and f ( α ) spectra depend only upona single parameter, the variance ∆ V . Eq. (3.15) canbe extended to higher loops, allowing ever more pre-cise tests against numerics or experimental data withinthe perturbatively accessible regime. The multifractalspectrum therefore provides a unique fingerprint for thetime-reversal invariant Dirac surface state of the Z topo-logical insulator, in the presence of weak but otherwisegeneric non-magnetic disorder. The opposite limit ofstrong disorder for the T -invariant case is discussed be-low in Sec. IV A 1.
4. Broken T : generic disorder (Class A) When T is broken and any two disorder flavors appear,the system resides in the unitary class A. The third disor-der flavor is always generated under renormalization—seeSec. III C, below. The results of Eqs. (3.8) and (3.12) forthe LDOS τ ( q ) spectrum in the random vector and masspotential models suggest that the unitary case also ex-hibits multifractality to first order in the impurity density n imp , since ∆ A,M,V ∝ n imp .With multiple flavors of the disorder, solving the op-erator mixing problem for the q th LDOS moment re- quires the diagonalization of an effective spin Hamilto-nian H ( eff ) , transcribed in Eq. (B10) of Appendix B. InFigs. 3 and 4, we present the results obtained by numer-ically diagonalizing this matrix for various combinationsof { ∆ V , ∆ M , ∆ A } . In these figures we plot the Renyidimension d q , defined for q = 1 via d q ≡ τ ( q ) q − . (3.16)Figs. 3 and 4 show that the generic broken- T case ismultifractal at one loop, and easily distinguished fromthe two-loop T -invariant result, in the limit of weak dis-order. [Note that Fig. 4 indicates that the τ ( q ) spectrumis not purely quadratic in this general case.] It shouldtherefore be possible to precisely distinguish the broken- T and T -invariant spectra experimentally, by observingthe dependence of the deviation 2 − d q on n imp . Thesingle-disorder flavor results for comparable strengths areplotted in Fig. 2 for reference.For the multidisorder unitary model, the same RGeigenoperators dominate the scaling of composite ν andout-of-plane spin ν ˆ3 LDOS moments. The dimension x ˆ3 q that determines the spin LDOS scaling via Eq. (2.9) alsoenters into the LDOS spectrum in Eq. (3.7), leading toFigs. 3 and 4. By contrast, moments of the chiral spinLDOS components ν ± [Eq. (3.10)] remain eigenoperatorsthat acquire no corrections at one loop, x ± q = q + O (∆ α ∆ β ) , (3.17) α, β ∈ { A, M, V } .As reviewed in the subsequent Sec. IV A 2, for M =0, the generic broken- T model is believed to flow tothe critical state at the integer quantum Hall plateautransition. This state exhibits strong multifractalitythat has been extensively studied in numerics.
FIG. 3: One-loop Renyi dimensions [Eq. (3.16)] in the bro-ken T , multidisorder unitary class A case, for various disorderstrength combinations. These results were obtained by nu-merically extracting the largest positive eigenvalue from theeffective spin Hamiltonian H eff in Eq. (B10) (restricting thesearch to operators invariant under spatial rotations and re-flections); see Appendix B for details. The two-loop result forthe T -invariant case is shown for reference. FIG. 4: The same as Fig. 3, but for different unitary classdisorder strength combinations. In the data presented here,∆ V > ∆ M,A . Regardless, the one-loop spectrum obtained foreither ∆
M,A > T -invariantcase with ∆ M = ∆ A = 0. The latter is also shown for refer-ence. C. Renormalization effects
As discussed at the beginning of the previous section,the disorder-averaged Dirac surface state theory used tocompute LDOS multifractal spectra is an “interacting”field theory, wherein the disorder strengths ∆
V,A,M ap-pear as coupling constants (c.f. Appendix B). Becausethese parameters are dimensionless, at weak coupling thedisorder constitutes a marginal perturbation of the cleanDirac band structure. The one-loop RG equations forthese parameters are given by d ∆ A dl = 1 π ∆ M ∆ V , (3.18a) d ∆ M dl = 1 π (2∆ A − ∆ M ) (∆ M + ∆ V ) , (3.18b) d ∆ V dl = 1 π (2∆ A + ∆ V ) (∆ M + ∆ V ) , (3.18c)where l = log L denotes the log of the RG length scale(e.g., the system size). Energy ε scales as d ln εdl = z ( l ) , (3.19)where the (scale-dependent) dynamic critical exponent is z = 1 + 12 π (2∆ A + ∆ M + ∆ V ) + O (∆ α ∆ β ) , (3.20) α, β ∈ { A, M, V } .In this section, we use Eqs. (3.18)–(3.20) to derive thedynamical scaling of the disorder parameters ∆ V,A,M ( ε );here energy ε is measured relative to the Dirac point, not the Fermi energy. (From the point-of-view of the disor-dered Dirac theory, a finite energy above the Dirac pointconstitutes a relevant perturbation. ) Using the resultsobtained in the previous section, we thereby determinethe enhancement or suppression of LDOS multifractalityapproaching the Dirac point, due to renormalization.
1. Broken T : random vector potential disorder (Class AIII) For the random vector potential model with ∆ V =∆ M = 0, Eq. (3.18) implies d ∆ A dl = 0 , so that ∆ A = ∆ ( ◦ ) A (constant), where ∆ ( ◦ ) A is the “mi-croscopic” value derived from the randomly polarizedin-plane magnetic impurity distribution. This result infact holds to all orders in ∆ A ; in this case, the theorydescribing LDOS fluctuations at the Dirac point is con-formally invariant. Multifractality is neither enhancednor suppressed as one moves away from the Dirac point,defined as ε = 0. However, for non-zero energies ε = 0,in an infinite size sample all states are in fact localized. The localization length diverges upon approaching theband center as ξ loc ( ε ) ∼ ε − /z , with z = 1 + ∆ A /π [Eq. (3.20)]. Eqs. (2.3) and (3.8) for τ ( q ) hold on scalessmaller than ξ loc ( ε ).
2. Broken T : random mass disorder (Class D) For the random mass model with ∆ V = ∆ A = 0, d ∆ M dl = − ∆ M π + O (cid:0) ∆ M (cid:1) , so that the disorder is marginally irrelevant atweak coupling. Integrating this equation and usingEqs. (3.19) and (3.20), we can compute the scaling of ∆ M with energy. At energy scale Υ, we define ∆ ( ◦ ) M ≡ ∆ M (Υ);then for the smaller energy scale ε (relative to the Diracpoint), we obtain the logarithmic suppression∆ M ( ε . Υ) ∼ ∆ ( ◦ ) M − (cid:0) ∆ ( ◦ ) M (cid:1) π log (cid:18) Υ ε (cid:19) + O (cid:26)h ∆ ( ◦ ) M (cid:16) − ε Υ (cid:17)i , (cid:0) ∆ ( ◦ ) M (cid:1) (cid:27) . (3.21)This equation holds for | − ε/ Υ | ≪
1. In the limit as ε →
0, the disorder strength vanishes as∆ M ( ε → ∼ π (cid:20) ln (cid:18)r π ∆ ( ◦ ) M Υ ε (cid:19)(cid:21) − + O (cid:20) ( ◦ ) M ln − (cid:18)r π ∆ ( ◦ ) M Υ ε (cid:19)(cid:21) . (3.22)For small ∆ ( ◦ ) M , Eq. (3.22) applies only at very small en-ergies ε . Υ exp( − / ∆ ( ◦ ) M ).
3. Non-magnetic disorder (Class AII)
Now we consider the T -invariant model. The flowequation for ∆ V is d ∆ V dl = ∆ V π + O (cid:0) ∆ V (cid:1) . (3.23)In contrast to the random mass, the random scalar po-tential is a marginally relevant perturbation to the cleanband structure. Examining lower and lower energyscales approaching the Dirac point, one observes strongereffects of the disorder. In the asymptotic scaling limitwherein the impurity potential strength becomes “large”(∆ V & results and analytical argumentsimply that the disordered T -invariant Dirac theory renor-malizes into the “conventional” symplectic metal. Themetal is distinguished from the Dirac theory by its non-zero (and non-critical) density of states at zero energy, and by its τ ( q ) spectrum. We discuss the strong cou-pling LDOS multifractality below in Sec. IV A 1. If atenergy Υ, ∆ V (Υ) ≡ ∆ ( ◦ ) V ≪
1, then for a somewhatsmaller energy ε we obtain the logarithmic enhancement∆ V ( ε . Υ) ∼ ∆ ( ◦ ) V + (cid:0) ∆ ( ◦ ) V (cid:1) π log (cid:18) Υ ε (cid:19) + O (cid:26)h ∆ ( ◦ ) V (cid:16) − ε Υ (cid:17)i , (cid:0) ∆ ( ◦ ) V (cid:1) (cid:27) . (3.24)Eq. (3.24) implies that renormalization strengthensmultifractality approaching the Dirac point ε = 0, forthe T -invariant case. We emphasize that this has nothing to do with weak (anti-)localization. The latter occurs inthe diffusive metallic regime with k F l mfp ≫
1, where l mfp denotes the elastic mean free path. The diffusive regimeobtains at strong coupling near the Dirac point k F →
4. Broken T : generic disorder (Class A) In the generic case of broken T , with multiple disor-der coupling strengths non-zero, the system flows towardstrong coupling ∆ V,M,A → ∞ . As a result, multifractal-ity is enhanced approaching the Dirac point. The RGflow ultimately terminates at a strong coupling criticalpoint, or in the Anderson insulator, discussed in the nextsection.
IV. STRONG COUPLING REGIMES
In this section, we review prior results on strong cou-pling regimes relevant to the disordered Dirac Z topo-logical insulator surface states, LDOS fluctuations andassociated multifractal spectra. These are not new, but provide complimentary information to the new resultsderived in the previous section.In both generic cases of T -invariant, and T -breakingimpurities, the disordered Dirac description used inSec. III fails on the largest length and lowest energyscales (approaching charge neutrality). For a sufficientlydilute concentration impurities, the results obtained inthe previous section characterize the start of the scalingregime, over energy and length scales such that the dis-order strengths remain weak ∆ V,A,M ( L, ε ) ≪
1. Whenthese parameters become order one (due to renormal-ization down to lower energies and longer lengths), thesystem crosses over to one of the strong coupling regimesdiscussed below.
A. Delocalized states at strong disorder T -invariant case: diffusive metal via strong disorder In a random scalar potential field, the Dirac point vac-illates in energy with spatial location; as a result, the den-sity of states near charge neutrality is enhanced by thedisorder. Due to the suppression of pure backscatteringfor Dirac fermions, the state density enhancement morethan compensates for the increased scattering introducedby the additional impurities. As a result, scalar potentialdisorder actually increases the (zero temperature, Lan-dauer) conductance at charge neutrality beyond the cleanballistic result, e /πh (Refs. 9,25). As in Sec. III, herewe assume short-range correlated disorder, due either tocharge neutral impurities or efficient screening by bulkand/or surface carriers. We do not discuss the puddleregime in the present paper.The effective disorder strength ∆ V is enhanced byrenormalization, as indicated by the runaway flow im-plied by Eq. (3.23). The concomitant density of statesand conductance growth suggests that the disorderedDirac theory ultimately crosses over to the ordinary dif-fusive symplectic metal, a result born out by numerics. In the absence of time-reversal symmetry breaking, An-derson localization is prohibited on the surface of a topo-logical insulator.
The symplectic metal possesses a finite (non-critical)average density of states at charge neutrality, and a dis-tinct τ ( q ) spectrum. For a large effective diffusion con-stant D (induced for a Dirac fermion subject to suffi-ciently strong disorder, or for a weakly disordered sys-tem examined on large length scales), the lowest orderresult for the multifractal spectrum appears in Eqs. (2.3)and Eq. (2.4), above. In the latter equation, β = 4 forthe symplectic class. For the T -invariant case, the strongest multifractal-ity is expected at intermediate coupling. Weak disorder∆ V ≪ τ ( q ) in Eqs. (2.3) and (3.15)], while strong dis-order ultimately pushes the system into the symplecticmetal, where a large diffusion constant D suppresses the0first correction in Eq. (2.4).
2. Broken T : IQHP transition For generic T -breaking disorder, i.e. all three ∆ V,M,A non-zero, the disordered Dirac theory is also unsta-ble under renormalization. When the average mass iszero M = 0 (see below), the flow in Eq. (3.18) is be-lieved to terminate at the critical point of the integerquantum Hall plateau transition. This is the delo-calized state separating adjacent Hall plateaux; it ex-hibits strong multifractality that has been extensivelystudied in numerics.
The spectrum is believed tobe universal, and is approximately parabolic as inEqs. (2.3) and (2.5), with θ ∼ .
26 (Refs. 27,28).
B. Anderson insulator
At zero chemical potential relative to the Dirac point,an average out-of-plane spin magnetization at the sur-face of a Z TI corresponds to the presence of a non-zero Dirac mass M for the surface carriers. This in-sulating state resides in a quantum Hall plateau [with σ xy = sgn( M ) e / h ]. In the presence of surfacedisorder, the plateau state will assume the character ofa localized Anderson insulator. In this section we reviewLDOS fluctuations in the Anderson insulator. The dis-cussion is relevant not only to the magnetized surface ofa 3D TI, but also to localized states populating the bulkgap of a disordered TI. Proposals exploiting localizationto realize so-called “topological Anderson insulators” byadding impurities to clean hosts include those in Ref. 55.To understand local density of states fluctuations in anAnderson insulator, it is useful to first study a toy prob-lem. Consider a tight-binding model on a d − dimensionallattice, subject to nearest-neighbor hopping t , and a ran-dom on-site potential V i , distributed uniformly over theregion − W/ ≤ V i ≤ W/
2. We assume the absence ofspatial correlations in the disorder potential. The inverserelative strength of the disorder is measured by the ratio t/W . We consider first the extreme limit of zero hopping, t/W →
0. In that case, the LDOS is the on-site operator ν i ( ε, V i ) = η/π ( ε − V i ) + η , where η denotes an energy-smearing parameter. Physi-cally, smearing is determined by inelastic scattering, opensample boundary conditions, or due to the finite energyresolution of the probing instrument.At the “band center” ε = 0, the distribution function for disorder-averaged LDOS moments evaluates to p ( ν ) ≡ Z W/ − W/ dVW δ [ ν − ν ( ε, V )]= 1 πν W r νν max − ν . (4.1)In this equation, the LDOS is constrained to the interval ν min ≤ ν ≤ ν max , where ν min = 4 ηπ ( W + 4 η ) , ν max = 1 πη . (4.2)Using Eq. (4.1), one can compute the disorder-averagedmoments of the LDOS, ν q = Γ (cid:0) q − (cid:1) W π q − / Γ( q ) η − q . (4.3)The average LDOS is ν = 1 /W ; all higher moments areproportional to η − q , and thus diverge in the limit of zeroenergy smearing η →
0. This not surprising, because theenergy spectrum in our trivial toy model is discrete, sothat the LDOS operator becomes a delta function withill-defined moments as η →
0. The moments are domi-nated by the power-law (Pareto) tail of the distribution,accumulating at the upper limit ν = ν max . By contrast,the typical LDOS, defined as ν typ = exp(log ν ) is domi-nated by the infrared ν typ = 4 ηe πW . This vanishes in the limit η → p ( ν ) ∼ ν − / power-law distribution in Eq. (4.1). The moments are renderedfinite only by the non-zero energy smearing parameter η . This should be compared to the LDOS statistics ina system with extended states and weak multifractality,e.g. that characterized by the quadratic τ ( q ) spectrumin Eq. (2.3), with 0 < θ ≪
1. It is known that thecorresponding LDOS distribution has a Gaussian bulk,with a small amplitude log-normal tail responsible forthe weak multifractality. For the metallic system, theresult is independent of energy smearing, provided thatthe thermodynamic limit is taken before the smearing isset to zero. Returning to the toy insulator model, weobserve that the global density ν G of states (GDOS) isself-averaging in the same limit. The GDOS is definedvia ν G ≡ N N X i =1 ν i ( V i ) , where N denotes the number of sites. In the large N -limit, the cumulant expansion can be evaluated via thesaddle-point. The cumulants of the GDOS then take theform [ ν G ] qc = N − q ( ν q + . . . ) , · · · ] qc denotes the q th cumulant, and the omittedterms are smaller by positive powers of η . Taking theinfinite system size limit N → ∞ before sending the en-ergy smearing to zero η → q > t/W , as per-formed by Anderson in his original 1958 paper. Thisexpansion can be formally summed to all orders in 1Dand on the Bethe lattice, but an explicit solution forthe LDOS statistics is difficult to obtain this way; seeRef. 50 for an alternative approach.Altshuler and Prigodin succeeded in deriving the dis-tribution generating disorder-averaged LDOS momentsin a 1D system, which is exponentially localized for ar-bitrarily weak disorder. In the thermodynamic limit fora closed sample, they obtain the “inverse Gaussian” dis-tribution p (˜ ν ) = r ηπǫ ν / exp (cid:20) − ηǫ (˜ ν − ˜ ν (cid:21) , (4.4)where ˜ ν ≡ ν/ν , and ǫ is the typical energy level spacingin a localization volume; ǫ − is also the elastic scatteringlifetime. In the limit of small smearing η ≪ ǫ , thisdistribution has moments˜ ν q = 4 − q Γ (cid:0) q − (cid:1) √ π (cid:16) ηǫ (cid:17) − q . (4.5)The exact result in Eq. (4.5) for the 1D Anderson in-sulator exhibits the same singular dependence on theenergy smearing η as the single site model moment inEq. (4.3). In fact, the distributions in Eqs. (4.1) and(4.4) are very similar: both feature a ν − / power law atintermediate ν , while the exponential factor in Eq. (4.4)plays the role of the hard cutoffs ν min , max in Eqs. (4.1)and (4.2). The close resemblance of the exact 1D andsingle site model results can be attributed to the discretespectrum of energy levels contributing to the LDOS inan Anderson insulator, with an energy level spacing de-termined by the localization volume ξ d loc in d spatial di-mensions.The take away is that the LDOS distribution in anAnderson insulator becomes very broad, with a power-law tail yielding divergent moments, in the limit of van-ishingly small energy smearing. In an STM experimentperformed at ultra-low temperature, on a large, isolatedAnderson localized surface, the collected LDOS statisticsshould be very sensitive to the smearing induced by theenergy resolution of the measurement itself.The locally discrete energy spectrum of the LDOS inthe Anderson insulator invalidates the use of Eq. (2.2)as a tool to compute the multifractal τ ( q ) spectrum. Asadvocated above, the shape of the LDOS distributionfunction and its sensitivity to smearing can best revealthe insulating phase. If one insists upon computing mo- ments, one must employ τ ( loc ) ( q ) ≡ − dd ln L ln " ν Z L d d d r X i | ψ i ( r ) | q δ ( ε − ε i ) . (4.6)Since the levels are discrete,lim η → ( πη ) q − ν q ( ε, r ) = X i | ψ i ( r ) | q δ ( ε − ε i ) . (4.7)Thus, τ ( loc ) ( q ) = − dd ln L ln (cid:26)Z L d d d r (cid:20) lim η → ( πη ) q − ν q ( ε, r ) (cid:21)(cid:27) + dd ln L ln (cid:26)Z L d d d r ν ( ε, r ) (cid:27) . (4.8)In this equation, we replace averages-of-the-logs withlogs-of-the-average, a procedure that is legitimate herebecause spatial and disorder-averaging are expected toyield the same results on the insulating side. Noting thatthe LDOS moments are L -independent in the insulatorfor L ≫ ξ loc , we obtain the expected result for localizedstates τ ( loc ) ( q ) = 0 , (4.9)computed in a well-defined η → V. ACKNOWLEDGMENTS
The author thanks Kostya Kechedzhi for a collabora-tion that lead to this work, and thanks Andreas Ludwig,Igor Aleiner, Pedram Roushan, Haim Beidenkopf, EmilYuzbashyan, and Deepak Iyer for useful discussions. Theauthor acknowledges support by the National ScienceFoundation under Grant No. DMR-0547769, and by theDavid and Lucile Packard Foundation.
Appendix A: Discrete symmetries, random matrixclassification, and disorder
The 10 symmetry classes of disordered Hamiltoni-ans (Hermitian random matrices) can be efficiently dis-tinguished by the presence or absence of time-reversal T , particle-hole P , and chiral/“sublattice” symmetry C . For the two-component Dirac Hamiltonian inEq. (3.1), the definitions of these symmetries are es-sentially unique. In terms of the two-component Diracspinor ψ , these appear as T : ψ → − i ˆ σ ψ, i → − i (A1a) P : ψ → ˆ σ (cid:2) ψ † (cid:3) T , (A1b) C : ψ → ˆ σ (cid:2) ψ † (cid:3) T , i → − i. (A1c)2In the second quantized language, T and C are antiuni-tary transformations; the unitary P can be taken as theproduct of these.The imposition of any one of the discrete symmetriesupon the Hamiltonian in Eq. (3.1) in every disorder real-ization restricts its form, and selects a particular ran-dom matrix symmetry class. (1) T -invariance: A = M = 0, only potential disorder ∆ V ≥ T = −
1, this is the symplectic (spin-orbit) class AII, which is also the symmetry class ofthe (presumed T -invariant) topological Z bulk. (2) P -invariance: V = A = 0, only random mass disorder∆ M ≥ P = +1, this is the bro-ken time-reversal class D. (3) C -invariance: V = M = 0,only random vector potential disorder ∆ A ≥ in the languageof Refs. 7,49.)Class AII is generically realized whenever time-reversalis unbroken. Magnetic impurities randomly polarizedparallel (perpendicular) to the TI surface manifest aspoint exchange sources in the vector (mass) potentialsof Eq. (3.1); we are thus tempted to identify symme-try classes D and AIII with these two limits. How-ever, a magnetic impurity will typically induce a localpotential fluctuation V ( r ) as well. As a consequence,the generic case of broken-time reversal symmetry corre-sponds to the absence of T , P , and C , which gives the uni-tary class A. In fact, for a vanishing average mass M = 0, the surface of a topological insulator with generictime-reversal breaking disorder is expected to flow underrenormalization to the critical point of the integer quan-tum Hall plateau transition. On a different note, theclass AIII and class D versions of H in Eq. (3.1) can berealized on the surface of a bulk T -invariant 3D topolog-ical superconductor, where time-reversal is respectivelypreserved or broken at the surface. Appendix B: Perturbation theory1. Chiral Decomposition and one-looprenormalization
We write a 2+0-D fermion path integral to representcorrelation functions in the disordered Dirac Hamilto-nian transcribed in Eq. (3.1). The fermion operators arereplaced with the Grassmann fields { ψ, ψ † } → { ψ i , ¯ ψ i } ;here i ∈ { , . . . , n } denotes a replica index, and we are tosend n → We employa “chiral decomposition” of the two-component spinors, ψ i = (cid:20) L i R i (cid:21) , ¯ ψ i = (cid:2) ¯ R i ¯ L i (cid:3) (B1) Then the action of the replicated theory is S = Z d r " ε (cid:0) ¯ R i L i + ¯ L i R i (cid:1) + ¯ R i L i φ + ¯ L i R i ¯ φ + ¯ R i ( − i∂ + A ) R i + ¯ L i (cid:0) − i ¯ ∂ + ¯ A (cid:1) L i , (B2)where we have introduced complex coordinates { z, ¯ z } = x ± iy , { ∂, ¯ ∂ } = ( ∂ x ∓ i∂ y ), and disorder potentials { A, ¯ A } = A x ∓ iA y , { φ, ¯ φ } = V ± M . The energy ε is afixed parameter. In Eq. (B2), repeated replica indices aresummed. Assuming the Gaussian white-noise variancesfor the disorder potentials enumerated in Eq. (3.3), thereplicated theory can be averaged over disorder configu-rations. The post-ensemble averaged action is S = S + S A + S + S , (B3)where S is the clean Dirac action, and S A = − A Z d r ¯ R i R i ¯ L j L j , (B4a) S = − ∆ V + ∆ M Z d r (cid:0) ¯ R i L i ¯ R j L j + ¯ L i R i ¯ L j R j (cid:1) , (B4b) S = − (∆ V − ∆ M ) Z d r ¯ R i L i ¯ L j R j . (B4c)Different replicas become coupled through thedisorder. The disorder-averaged composite LDOS ν ( ε, r ) corre-sponds to the fermion bilinear expectation ν = h ¯ ψψ i = h ¯ RL + ¯ LR i . (B5)The spin LDOS ν ˆ ι ( ε, r ) was defined by Eq. (2.7). For theout-of-plane and in-plane (chiral) components, one has ν ˆ3 = h ¯ ψ ˆ σ ψ i = h ¯ RL − ¯ LR i , (B6a) ν ± ≡h ¯ ψ ˆ σ ± ψ i = 2 {h ¯ RR i , h ¯ LL i} , (B6b)The overlines appearing in the left-hand sides ofEqs. (B5) and (B6) denote disorder-averaging, whereasthe angle brackets on the right-hand sides represent in-tegration in the fermion path integral, using the action¯ S in Eq. (B3).A generic local operator corresponding to the q th mo-ment of some fermion bilinear can be viewed as sum of“strings,” where each string consists of 2 q total right (R)and left (L) mover labels, arranged in some order. Forexample, the disorder-averaged q th moment of the LDOSis represented by the the composite operator expectationvalue ν q ( r ) = * q Y i =1 (cid:2) ¯ R i L i ( r ) + ¯ L i R i ( r ) (cid:3)+ . (B7)In this equation, a product is taken over operators carry-ing indices in the first q ≤ n replicas. The q th LDOS mo-ment is computed by placing one copy of the LDOS op-erator into each of q different replicas; before averaging,3this gives ν q in a fixed realization of the disorder. (Plac-ing instead the q copies into the same replica would givethe disorder-averaged first moment of a 2 q -point Green’sfunction.) The operator in Eq. (B7) is an even weightsum of 2 q “strings”, all of length 2 q . I.e., ν q ( r ) = { ¯ RL ; ¯ RL ; . . . ; ¯ RL } + { ¯ LR ; ¯ RL ; ¯ RL ; . . . ; ¯ RL } + { ¯ RL ; ¯ LR ; ¯ RL ; . . . ; ¯ RL } + . . . + { ¯ LR ; ¯ LR ; . . . ; ¯ LR } . (B8)The semicolons separate fermion bilinears in differentreplicas. Each bilinear has two entries, corresponding tothe chiral identity of the barred and unbarred operators.The set of all length 2 q strings forms a complete ba-sis for q th moment local operators (without derivatives).These basis strings mix under renormalization due tothe disorder. In general, the composite operator ( ≡ weighted string sum) corresponding to the q th momentof a bilinear as in Eq. (B7) does not constitute an eigen-operator of the renormalization group. The main task isto (1) identify RG eigenoperators for each disorder typeand compute the spectrum of scaling dimensions, and (2)compute the projection of the LDOS and (for broken T )spin LDOS moment operators onto this eigenbasis, anddetermine the most relevant contributions. a. Effective Hamiltonian for 1-loop renormalization It is useful to view each string as a configuration of2 q spin 1/2 moments. We associate { ¯ R, R } → / { ¯ L, L } → − / J z = 0. We picture each string as abasis state for a length q chain, with two spins per site.Sites are labeled by the replica index i ∈ { , . . . , q } . Thetwo spins at each site are distinguished by labels “A”and “B,” corresponding to barred and unbarred opera-tors, respectively.Renormalization occurs via the action of the disordervertices appearing in Eq. (B4), employing the clean Diracpropagator in a standard loop expansion. Operator mix-ing at one-loop is encoded in the effective “Hamiltonian” H ( eff ) = ln Λ2 π A q X i,j =1 ( S zAi − S zBi ) (cid:0) S zAj − S zBj (cid:1) + (∆ M + ∆ V ) q X i,j =1 (cid:16) S + Ai S − Bj + S + Bi S − Aj (cid:17) + (∆ M − ∆ V ) q X i,j =1 i = j (cid:16) S + Ai S − Aj + S + Bi S − Bj (cid:17) . (B9) In this equation, S aA/Bi denotes a spin-1/2 operator act-ing on the barred ( A ) or unbarred ( B ) spin in replica i .The prefactor obtains from evaluating the loop integralsusing a hard momentum cutoff Λ. The first, second, andthird lines in the heavy brackets arise through the actionof the disorder vertices in S A , S , and S , respectively.The ∆ A renormalization is diagonal in the ↑ / ↓ ( R/L ) ba-sis. By contrast, S , and S perform single exchanges ofright and left labels. S ( S ) mediates interflavor A ↔ B (intraflavor A ↔ A , B ↔ B ) exchanges. Summing theangular momenta, H ( eff ) = ln Λ2 π A ( J zA − J zB ) + 2 (∆ M + ∆ V ) ( J xA J xB + J yA J yB )+ (∆ M − ∆ V ) × " J A − ( J zA ) + J B − ( J zB ) − q , (B10)where J A,B ≡ P i S A,Bi .In the general case of broken T discussed in Sec. III B 4,all three disorder parameters are present. The most rele-vant eigenvalue of Eq. (B10) determines the scaling of the q th LDOS moment. The first few multifractal momentsfor various disorder configurations were obtained throughnumerical diagonalization; results appear in Figs. 3 and4. Even moments of the out-of-plane spin LDOS ν ˆ3 [Eq. (B6a)] are invariant under spatial rotations andparity. In the multidisorder unitary case, even mo-ments of the composite ν and out-of-plane spin ν ˆ3 LDOSare dominated by the same RG eigenoperator. The di-mension x ˆ3 q that determines the spin LDOS scaling viaEq. (2.9) is the same that enters into the LDOS spec-trum in Eq. (3.7), Figs. 3 and 4. Moments of the chiralspin LDOS components in Eq. (B6b) correspond to thehighest weight states | j = q ; m = ± q i ; here, j ( j + 1) de-notes the eigenvalue of ( J A + J B ) , with 0 ≤ j ≤ q and − j ≤ m ≤ j . These highest weight states are annihilatedby H ( eff ) in Eq. (B10), leading to Eq. (3.17).Below we discuss the special cases of isolated disorderflavors. b. Broken T : random vector potential disorder (Class AIII) For ∆ M = ∆ V = 0, Eq. (B10) reduces to H ( eff ) A = ln Λ π ∆ A ( J zA − J zB ) = ln Λ π ∆ A ( m A − m B ) . (B11)On the second line, we have evaluated H ( eff ) A for theproduct state | j A , j B ; m A , m B i . Since max( j A,B ) = q/ | m A/B | ≤ j A/B , the maximum eigenvalue attainsfor the states | q/ , q/ q/ , − q/ i → { ¯ RL ; ¯ RL ; . . . ; ¯ RL } | q/ , q/ − q/ , q/ i → { ¯ LR ; ¯ LR ; . . . ; ¯ LR } . Thesehave J z = 0, and thus correspond to operators invari-ant under spatial rotations; the symmetric combinationis also parity-invariant. Via standard renormalizationgroup machinery, one obtains the most relevant scal-ing dimension for a q − fold product operator, x ( A ) q = q − q ∆ A π . (B12)Using Eq. (B12) in Eq. (3.7) gives the result for thequadratic τ ( q ) spectrum in Eqs. (2.3) and (3.8). c. Broken T : random mass disorder (Class D) For the random mass case, Eq. (B10) becomes H ( eff ) M = ∆ M ln Λ2 π (cid:2) J − ( J z ) − q (cid:3) = ∆ M ln Λ2 π (cid:2) j ( j + 1) − m − q (cid:3) , (B13)where J ≡ J A + J B . On the second line, we have eval-uated the “Hamiltonian” on a total angular momentumeigenstate | jm i . For 2 q spins, we have max( j ) = q . Themaximum eigenvalue is associated to the non-degenerate j = q , m = 0 state, which is invariant under spatialrotations. The scaling dimension is x ( M ) q = q − q ∆ M π . (B14)The corresponding eigenoperator | j = q, m = 0 i is anequal weight symmetric sum of all permutations of q “ R ” and q “ L ” labels, and has non-zero overlap withthe naive LDOS moment (a q -fold triplet product) inEq. (B7). Using Eq. (B14) in Eq. (3.7), one obtains theresult for the quadratic τ ( q ) LDOS spectrum in Eqs. (2.3)and (3.12). By contrast, the out-of-plane spin LDOS(mass operator) in Eq. (B6a) is a singlet; the disorder-averaged q th moment thus corresponds to the eigenoper-ator | j = 0 , m = 0 i [leading to Eq. (3.13)]. d. Non-magnetic disorder (Class AII) We rotate the “B” composite spin by π around theˆ z -axis, J xB → ˜ J xB ≡ − J xB , J yB → ˜ J yB ≡ − J yB , J zB → ˜ J zB ≡ J zB . The Hamiltonian in Eq. (B10) with ∆ M = ∆ A = 0 be-comes H ( eff ) V = − ∆ V ln Λ2 π h ˜ J − ( ˜ J z ) − q i , (B15)where ˜ J ≡ J A + ˜ J B . Eq. (B15) has the same form asEq. (B13), with ∆ V → − ∆ M . This is consistent with amapping between the random mass and vector potentialmodels identified in Ref. 24. In the case of the scalarpotential, the maximum eigenvalue is associated to thehighly degenerate singlet sector (cid:12)(cid:12) ˜ j = 0 , ˜ m = 0 (cid:11) , leadingto the scaling dimension x ( V ) q = q − q ∆ V π . (B16)Using this result in Eq. (3.7) gives τ ( q ) = 2( q − T -invariant model.
2. Two loop renormalization, T -invariant class AII In the T -invariant class AII model, the first correc-tion to the LDOS τ ( q ) spectrum appears at second orderin ∆ V . We have carried out a two-loop calculation andfound that the naive LDOS moment in Eq. (B7) remainsan eigenoperator. To this order, we find the scaling di-mension x ( V ) q = q − q ∆ V π − ∆ V π [3 q ( q −
1) + q ] + O (cid:0) ∆ V (cid:1) . (B17)Combining Eqs. (3.7) and (B17), we recover thequadratic multifractality for τ ( q ) quoted in the text,Eqs. (2.3) and (3.15). We have used dimensional regular-ization to obtain the result in Eq. (B17). Although theClifford algebra becomes formally infinite upon dimen-sional continuation to d = 2 − ǫ , this causes no problemsfor the renormalization of the q th LDOS moment becausethe latter is already an eigenoperator. We omit detailsof the (lengthy) two-loop calculation in this paper. ∗ Electronic address: [email protected] L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. ,106803 (2007); J. E. Moore and L. Balents, Phys. Rev. B , 121306(R) (2007); X.-L. Qi, T. L. Hughes, and S.-C.Zhang, ibid. , 195424 (2008); R. Roy, ibid. , 195322 (2009). M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. , 3045(2010). X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. , 1057(2011). A. J. Niemi and G. W. Semenoff, Phys. Rev. Lett. , 2077(1983); G. W. Semenoff, ibid. , 2449 (1984). A. N. Redlich, Phys. Rev. Lett. , 18 (1984); R. Jackiw,Phys. Rev. D , 2375 (1984). F. D. M. Haldane, Phys. Rev. Lett. , 2015 (1988). A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Lud-wig, Phys. Rev. B , 195125 (2008); New J. Phys. ,065010 (2010). P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. ,287 (1985). J. H. Bardarson, J. Tworzydlo, P.W. Brouwer, and C.W.J. Beenakker, Phys. Rev. Lett. , 106801 (2007); K. No-mura, M. Koshino, and S. Ryu, ibid. , 146806 (2007). S. Ryu, C. Mudry, H. Obuse, and A. Furusaki, Phys. Rev.Lett. , 116601 (2007); P. M. Ostrovsky, I. V. Gornyi,and A. D. Mirlin, ibid. , 256801 (2007). L. Capriotti, D. J. Scalapino, and R. D. Sedgewick, Phys.Rev. B , 014508 (2003). P. Roushan, J. Seo, C. V. Parker, Y. S. Hor, D. Hsieh,D. Qian, A. Richardella, M. Z. Hasan, R. J. Cava, and A.Yazdani, Nature , 1106 (2009); T. Zhang, P. Cheng,X. Chen, J.-F. Jia, X. Ma, K. He, L. Wang, H. Zhang,X. Dai, Z. Fang, X. Xie, and Q.-K. Xue, Phys. Rev. Lett. , 266803 (2009); Z. Alpichshev, J. G. Analytis, J.-H.Chu, I. R. Fisher, Y. L. Chen, Z. X. Shen, A. Fang, andA. Kapitulnik, ibid. , 016401 (2010). H. Beidenkopf, P. Roushan, J. Seo, L. Gorman, I. Drozdov,Y. S. Hor, R. J. Cava, and A. Yazdani, Nat. Phys. , 939(2011). H.-M. Guo and M. Franz, Phys. Rev. B , 041102(R)(2010). X. Zhou, C. Fang, W.-F. Tsai, and J. P. Hu, Phys. Rev. B , 245317 (2009); W.-C. Lee, C. Wu, D. P. Arovas, andS.-C. Zhang, ibid. , 245439 (2009). Q. Liu, C.-X. Liu, C. Xu, X.-L. Qi, and S.-C. Zhang,Phys. Rev. Lett. , 156603 (2009); R. R. Biswas andA. V. Balatsky, Phys. Rev. B , 233405 (2010); A. M.Black-Schaffer and A. V. Balatsky, arXiv:1110.5149 (un-published). G. Paladin and A. Vulpiani, Phys. Rep. , 147 (1987). For reviews, see B. Huckestein, Rev. Mod. Phys. , 357(1995); M. Janssen, Int. J. Mod. Phys. B , 943 (1994). F. Evers and A. D. Mirlin, Rev. Mod. Phys. , 1355(2008). F. Wegner, Z. Phys. B , 204 (1980); D. H¨of and F. Weg-ner, Nucl. Phys. B , 561 (1986); F. Wegner, ibid. ,193 (1987); , 210 (1987). A. M. M. Pruisken, Phys. Rev. B , 416 (1985). B. L. Altshuler, V. E. Kravtsov, and I. V. Lerner, PismaZh. Eksp. Teor. Fiz. , 342 (1986) [JETP Lett. , 441(1986)]; Zh. Eksp. Teor. Fiz. , 2276 (1986) [Sov. Phys.JETP , 1352 (1986)]; Phys. Lett. A , 488 (1989);in Mesoscopic Phenomena in Solids , edited by B. L. Alt-shuler, P. A. Lee, and R. A. Webb (North-Holland, Ams-terdam, 1991), Vol. 449. V. I. Fal’ko and K. B. Efetov, Phys. Rev. B , 17413(1995). A. W. W. Ludwig, M. P. A. Fisher, R. Shankar, and G.Grinstein, Phys. Rev. B , 7526 (1994). A. Schuessler, P. M. Ostrovsky, I. V. Gornyi, and A. D.Mirlin, Phys. Rev. B , 075405 (2009). W. Pook and M. Janssen, Z. Phys. B , 295 (1991). R. Klesse and M. Metlzer, Europhys. Lett. , 229 (1995);Int. J. Mod. Phys. C , 577 (1999); F. Evers, A. Milden- berger, and A. D. Mirlin, Phys. Rev. , 241303(R) (2001). H. Obuse, A. R. Subramaniam, A. Furusaki, I. A.Gruzberg, and A. W. W. Ludwig, Phys. Rev. Lett. ,116802 (2008); F. Evers, A. Mildenberger, and A. D. Mir-lin, ibid. , 116803 (2008). K. Nomura, S. Ryu, M. Koshino, C. Mudry, and A. Fu-rusaki, Phys. Rev. Lett. , 246806 (2008). H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C.Zhang, Nat. Phys. , 438 (2009). F. Meier, L. Zhou, J. Wiebe, and R. Wiesendanger, Science , 82 (2008). K. Nomura and A. H. MacDonald, Phys. Rev. Lett. ,256602 (2006). See, e.g., J. G. Checkelsky, Y. S. Hor, M.-H. Liu, D.-X.Qu, R. J. Cava, and N. P. Ong, Phys. Rev. Lett. ,246601 (2009); Z. Ren, A. A. Taskin, S. Sasaki, K. Segawa,and Y. Ando, Phys. Rev. B , 241306(R) (2010); J. G.Checkelsky, Y. S. Hor, R. J. Cava, and N. P. Ong, Phys.Rev. Lett. , 196801 (2011). S. Adam, E. H. Hwang, V. Galitski, and S. Das Sarma,Proc. Natl. Acad. Sci. USA , 18392 (2007); V. V.Cheianov, V. I. Falko, B. L. Altshuler, and I. L. Aleiner,Phys. Rev. Lett. , 176801 (2007). For a recent review, see e.g. S. Das Sarma, S. Adam, E. H.Hwang, and E. Rossi, Rev. Mod. Phys. , 407 (2011). L. A. Ponomarenko, A. A. Zhukov, R. Jalil, S. V. Morozov,K. S. Novoselov, V. V. Cheianov, V. I. Fal’ko, K. Watan-abe, T. Taniguchi, A. K. Geim, and R. V. Gorbachev, Nat.Phys. , 958 (2011). Y. L. Chen, J.-H. Chu, J. G. Analytis, Z. K. Liu, K.Igarashi, H.-H. Kuo, X. L. Qi, S. K. Mo, R. G. Moore,D. H. Lu, M. Hashimoto, T. Sasagawa, S. C. Zhang, I. R.Fisher, Z. Hussain, Z. X. Shen, Science , 659 (2010). T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia,and B. I. Shraiman, Phys. Rev. A , 1141 (1986). I. L. Aleiner, B. L. Altshuler, and M. E. Gershenson, WavesRandom Media , 201 (1999). M. L. Mehta,
Random matrices , 3rd ed. (Academic Press,Amsterdam, 2004). C. C. Chamon, C. Mudry, and X.-G. Wen, Phys. Rev. Lett. , 4194 (1996). H. E. Castillo, C. C. Chamon, E. Fradkin, P. M. Goldbart,and C. Mudry, Phys. Rev. B , 10668 (1997). M. S. Foster, S. Ryu, and A. W. W. Ludwig, Phys. Rev.B , 075101 (2009). B. L. Altshuler and V. N. Prigodin, JETP Lett. , 687(1987); JETP , 198 (1989). I. V. Lerner, Phys. Lett. A , 253 (1988). D. V. Khveshchenko, Phys. Rev. B , 241406(R) (2007). W. Richter, H. K¨ohler, C. R. Becker, Phys. Status SolidiB , 619 (1977). M. R. Zirnbauer, J. Math. Phys. , 4986 (1996); A. Alt-land and M. R. Zirnbauer, Phys. Rev. B , 1142 (1997);P. Heinzner, A. Huck Leberry, and M. R. Zirnbauer, Com-mun. Math. Phys. , 725 (2005). D. Bernard and A. LeClair, J. Phys. A , 2555 (2002). K. Efetov,
Supersymmetry in Disorder and Chaos (Cam-bridge University Press, Cambridge, England, 1999). M. L. Horbach and G. Schoen, Ann. Phys. (Leipzig) , 51(1993). B. Duplantier and A. W. W. Ludwig, Phys. Rev. Lett. ,247 (1991). As discussed in the paragraphs following Eq. (3.6) inSec. III A, a magnetic impurity will typically induce a scalar potential V ( r ) deformation, in addition to mass andvector potential point exchanges for out-of-plane and in-plane polarization components, respectively. P. M. Ostrovsky, I. V. Gornyi, and A. D. Mirlin, Phys.Rev. B , 235443 (2006). J. Li, R.-L. Chu, J. K. Jain, and S.-Q. Shen, Phys. Rev.Lett. , 136806 (2009); H.-M. Guo, G. Rosenberg, G.Refael, and M. Franz, ibid. , 216601 (2010); C. Weeks,J. Hu, J. Alicea, M. Franz, and R. Wu, Phys. Rev. X ,021001 (2011). P. W. Anderson, Phys. Rev. , 1492 (1958). R. Abou-Chacra, P. W. Anderson, and D. J. Thouless, J.Phys. C , 1734 (1973). More precisely, strings with the same value of n R − n L mix, where n R ( n L ) denotes the total number of barredand unbarred R ( L ) labels, and n R + n L = 2 q . Stringswith different values of n R − n L transform with different U (1) charges under spatial rotations in the xy plane, and cannot mix. For composite LDOS fluctuations, it is necessary to re-strict the eigenspace of Eq. (B9) to rotationally invariant( J z = 0), “parity”-invariant states. Here, parity denotessimultaneous invariance under spatial x - and y -reflectionsin the plane of the system. In the chiral decomposition ofEq. (B1), parity-invariant operators are symmetric underthe exchange R ↔ L , ¯ R ↔ ¯ L . See, e.g., D. J. Amit,
Field Theory, the RenormalizationGroup, and Critical Phenomena , 2nd ed. (World Scientific,Singapore, 1984). M. S. Foster and I. L. Aleiner, Phys. Rev. B , 195413(2008). A. Bondi, G. Curci, G. Paffuti, and P. Rossi, Ann. Phys. , 268 (1990); J. F. Bennett and J. A. Gracey, Nucl.Phys. B563