Corner Multifractality for Reflex Angles and Conformal Invariance at 2D Anderson Metal-Insulator Transition with Spin-Orbit Scattering
Hideaki Obuse, Arvind R. Subramaniam, Akira Furusaki, Ilya A. Gruzberg, Andreas W. W. Ludwig
aa r X i v : . [ c ond - m a t . d i s - nn ] M a r Corner Multifractality for Reflex Angles and Conformal Invariance at 2DAnderson Metal-Insulator Transition with Spin-Orbit Scattering
H. Obuse a , A. R. Subramaniam b , A. Furusaki a , I. A. Gruzberg b , A. W. W. Ludwig c a Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan b James Franck Institute, University of Chicago, 5640 South Ellis Avenue, Chicago, Illinois 60637, USA c Physics Department, University of California, Santa Barbara, California 93106, USA
Abstract
We investigate boundary multifractality of critical wave functions at the Anderson metal-insulator transition in two-dimensional disordered non-interacting electron systems with spin-orbit scattering. We show numerically that multifractalexponents at a corner with an opening angle θ = 3 π/ θ < π to θ > π , and gives further supporting evidence for conformal invariance at criticality. We also propose a refinementof the validity of the symmetry relation of A. D. Mirlin et al. , Phys. Rev. Lett. (2006) 046803, for corners. Key words: conformal invariance, Anderson transition, multifractality, spin-orbit interaction
PACS:
Anderson metal-insulator transitions are continu-ous phase transitions driven by disorder. Examples oflocalization-delocalization (Anderson) transitions oc-curring in two-dimensions (2D) include non-interactingelectronic systems with spin-orbit scattering (‘sym-plectic symmetry class’), with sublattice symmetry, orin strong magnetic fields (quantum Hall effect).Recently, we have reported numerical evidence forthe presence of conformal invariance at the 2D Ander-son transition in the symplectic symmetry class [1]. Tothat end, we have considered multifractal properties ofcritical wave functions near boundaries of disorderedsamples of finite size, and verified numerically thatthe multifractal exponents of critical wave functionsat corners with opening angle θ (corner multifractal-ity) are related, through simple relations derived fromconformal invariance, to the exponents computed near straight edges (surface multifractality). In Ref. [1] wehave discussed corner multifractality at wedges withangles θ < π only, both acute ( θ = π/
4) and obtuse( θ = 3 π/ θ = 3 π/ θ > π ).Following Refs. [1,2], we define bulk, surface, andcorner multifractality from the scaling of moments ofwave functions ψ ( r ) in bulk (b), surface (s), and corner( θ ) regions, L d x | ψ ( r ) | q ∼ L − τ x q , (x = θ, s , b) , (1)where d x is the spatial dimension of each region ( d b =2, d s = 1, and d θ = 0). The overbar represents theensemble (disorder) average and the simultaneous spa-tial average over a region x surrounding the point r . Preprint submitted to Physica E 30 October 2018 he exponents τ b q , τ s q , and τ θq are the bulk, surface,and corner multifractal exponents, respectively. Fromthe multifractal exponents we extract non-vanishinganomalous dimensions ∆ x q ,∆ x q = τ x q − q + d x . (2)The multifractal singularity spectra f x ( α ) are obtainedfrom τ x q by Legendre transformation, f x ( α x ) = α x q − τ x q , α x = dτ x q dq . (3)As explained in Ref. [1], under the assumption thatthe q -th moment | ψ ( r ) | q is represented by a primaryoperator in an underlying conformal field theory [3],one can derive, using the conformal mapping w = z θ/π ,the relation between the surface and corner multifrac-tal spectra f x ( α x q ), α θq − πθ ( α s q − , f θ ( α θq ) = πθ ˆ f s ( α s q ) − ˜ . (4)The validity of these relations provides direct evidencefor conformal invariance at a 2D Anderson transitionand for the primary nature of the operator.In Ref. [1] we have shown that the probability dis-tribution of ln | ψ ( r ) | becomes broader, as the openingangle θ is reduced. This implies that the distributionis narrower at a corner with larger θ . We may thusexpect that multifractal exponents can be more accu-rately calculated for corners with reflex angles than forcorners with angles θ < π . We can then estimate thesurface f s ( α s q ) by taking the numerical data for α θq and f θ ( α θq ) obtained for θ > π as input into Eq. (4). More-over, we can relate multifractal spectra of corners withdifferent angles θ and θ ′ ( θ < θ ′ ), yielding α θq − θ ′ θ ( α θ ′ q − , f θ ( α θq ) = θ ′ θ f θ ′ ( α θ ′ q ) . (5)As we pointed out in Ref. [1], Eqs. (4) and (5) arevalid only if all occurring α x q >
0, because α x q is non-negative for normalized wave functions. Thus, whenthe prefactor θ ′ /θ is larger than one (and hence 0 α θq < α θ ′ q ), the first of Eq. (5) cannot be used for q >q θ , where q θ is a solution to α θq = 0 in Eq. (5). (We donot know if q θ is finite for θ > π .) Taking this physicalconstraint into account, we find the following relationbetween anomalous dimensions for corners ( θ < θ ′ ),∆ θq = θ ′ θ ∆ θ ′ q , q q θ ,θ ′ θ ∆ θ ′ q θ − q − q θ ) , q > q θ . (6) If we set θ = π in Eq. (6), we obtain a relation betweenanomalous dimensions at a surface ( θ = π ) and a cor-ner with a reflex angle ( θ ′ > π ). In Ref. [1] we also dis-cussed the symmetry relation of Ref. [4], ∆ x q = ∆ x1 − q ,and its application to corners x = θ . Here we propose,as a refinement of that discussion, that this symmetryrelation (i) is valid for corners of any angle θ including θ = π (straight boundaries), but only in the range of q satisfying 1 − q θ q q θ , corresponding precisely [1,4]to the range 0 α θq
4, and (ii) makes no statementsabout ∆ θq for values of q outside of this range. [The de-pendence on q of ∆ θq is linear for q > q θ (correspondingto the termination of the multifractal spectrum [5]),whilst it may, in general, continue to be non-linear for q < − q θ , even [1,4] when α θq > θ = 3 π/ ψ having energy eigenvalue closest to acritical point E c = 1 . W c = 5 . L isvaried through L = 24 , , · · · ,
120 and the number ofdisordered samples is 6 × for each L . We set w = 2 -2.0-1.00.01.0 0.0 1.0 2.0 3.0 4.0 f () w/2 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)yyyyyyyyyyyyyyyyyy(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)yyy yyy(cid:0)(cid:0)(cid:0) (cid:0)(cid:0)(cid:0)yyy yyy L /
2L w
Fig. 1. (color online) Multifractal spectra f ( α ) for cornerswith θ = 3 π/ θ = π/ q for the corner with θ = 3 π/
2. The solid and short--dashed curves represent the theoretical prediction fromEq. (4), where f π/ ( α π/ q ) and f s ( α s q ) are used as input,respectively. The dashed curve is calculated from Eq. (5)with f π/ ( α π/ q ) used as input. Inset: L-form geometrywith 3 L / θ = 3 π/ w / .20.40.60.8 -2 -1 0 1 2 3 4 ∆ xq / [ q ( - q )] q Fig. 2. (color online) The exponents ∆ x q / [ q (1 − q )] for cornerwith θ = 3 π/ θ = π/ of the corner region shown in Fig. 1. Multifractal spec-tra are computed in the same way as in Ref. [1].In Fig. 1, we show multifractal spectra f ( α ) of cor-ners with θ = 3 π/
2, together with those of cornerswith θ = π/
2, and of the surface region [1]. The peakposition α x0 of f π/ ( α π/ q ) is α π/ = 2 . ± . α π/ = 2 . ± . f π/ ( α ) is smaller than that of f s ( α s q ). Thisis consistent with Eq. (4) at θ = 3 π/
2. Figure 1 clearlyshows that f π/ ( α ) computed directly for the cor-ner with θ = 3 π/ f s ( α ) as input,which verifies Eq. (4) derived from conformal invari-ance. We have also calculated the surface f s ( α s q ) andcorner f π/ ( α π/ q ) from Eqs. (4) and (5), respectively,using f π/ ( α π/ q ) as input into these equations. Thisallows us to estimate f s ( α s q ) near α s ≈
0, providing anestimate for q s = df s ( α s = 0) /dα s . The theoretical pre-dictions (solid and dashed curves) are in good agree-ment with the numerical data (triangles and squares)for f s ( α s q ) and f π/ ( α π/ q ), respectively.Figure 2 shows the anomalous dimensions ∆ x q forcorners with θ = 3 π/ θ = π/
2, and the surfaceregion, which are numerically calculated from the scal-ing | ψ ( r ) | q / ( | ψ ( r ) | ) q ∼ L − ∆ x q . The solid and dashedcurves represent the theoretical prediction, Eq. (6),from the conformal mapping using ∆ π/ q as inputs.The data points for ∆ s q (triangles) agree with the solidcurve for | q | < ∼ .
5, while those for ∆ π/ q (squares) areclose to the dashed curve only near q ≈
0. The datapoints for ∆ π/ q satisfy, within error bars, the symme-try relation [4] ∆ π/ q = ∆ π/ − q in the vicinity of q = 1 /
2, indicating good numerical accuracy. This opensthe possibility that one can use corner multifractalityfor θ > π to obtain, with the help of Eqs. (4) and (6),more accurate estimates for multifractal properties ata straight surface and corners with θ < π .We briefly comment on multifractality of a wholesample with boundaries. We have found in Refs. [1,2]that corner multifractality may dominate multifractal-ity of a whole system, even in the thermodynamic limit,for large values of | q | if τ θq < τ b q , τ s q . Here we pointout that this cannot happen with corners of reflex an-gles ( θ > π ). The proof goes as follows. We first notethat, from Eqs. (2) and (6), the difference of corner andsurface multifractal exponents is given by τ θq − τ s q =1 + ∆ s q ( π/θ −
1) as long as α s q > α θq > α s q for θ > π ). Thus, when π < θ < π , the inequality τ θq > τ s q holds if ∆ s q
2. Secondly, since τ x q is a con-vex function of q with the constraints τ x0 = − d x and τ x1 = 2 − d x (recalling [1] µ = 0, and thus ∆ x1 = 0),we find ∆ x q | q − / | > / < ∆ x q < < q <
1. We thus conclude that τ θq > τ s q when α s q >
0. Finally, when τ θq − τ s q is positive for α s q > q s < ∞ and in theregime q > q s where α s q s = 0, because τ s q is then con-stant for q > q s (and dτ θq /dq = α θq > α s q ). Hence, con-tributions from corner multifractality at θ > π cannotbe larger than contributions from surface multifractal-ity. The numerical results shown in Fig. 1 are consis-tent with the above general argument.In summary, we have investigated corner multifrac-tality for the reflex angle θ = 3 π/ [1] H. Obuse, A. R. Subramaniam, A. Furusaki, I. A.Gruzberg, and A. W. W. Ludwig, Phys. Rev. Lett. (2006) 126802.[3] A. A. Belavin, A. M. Polyakov, and A. B.Zamolodchikov, Nucl. Phys. B (1984) 333.[4] A. D. Mirlin, Y. V. Fyodorov, A. Mildenberger, and F.Evers, Phys. Rev. Lett. (2006) 046803.[5] F. Evers and A. D. Mirlin, arXiv:0707.4378(unpublished).[6] Y. Asada, K. Slevin, and T. Ohtsuki, Phys. Rev. Lett. (2002) 256601; Phys. Rev. B (2004) 035115.[7] T. Nakayama and K. Yakubo, Phys. Rep. (2001)239.(2001)239.