Numerical evidence for marginal scaling at the integer quantum Hall transition
NNumerical evidence for marginal scaling at the integer quantum Hall transition
Elizabeth J. Dresselhaus, Bj¨orn Sbierski, and Ilya A. Gruzberg Department of Physics, University of California, Berkeley, California 94720, USA Ohio State University, Department of Physics, 191 West Woodruff Ave, Columbus OH, 43210, USA
The integer quantum Hall transition (IQHT) is one of the most mysterious members of the familyof Anderson transitions. Since the 1980s, the scaling flow close to the critical fixed point in theparameter plane spanned by the longitudinal and Hall conductivities has been studied vigorouslyboth by experiments and with numerical simulations. Despite all efforts, it is notoriously difficultto pin down the precise values of critical exponents, which seem to vary with model details and thuschallenge the principle of universality. Recently, M. Zirnbauer [Nucl. Phys. B , 458 (2019)] hasconjectured a conformal field theory for the transition, in which linear terms in the beta-functionsvanish, leading to a very slow flow in the fixed point’s vicinity which we term marginal scaling. Inthis work, we provide numerical evidence for such a scenario by using extensive simulations of variousnetwork models of the IQHT at unprecedented length scales. At criticality, we confirm the marginalscaling of the longitudinal conductivity towards the conjectured fixed-point value σ ∗ xx = 2 /π . Awayfrom criticality we describe a mechanism that could account for the emergence of an effective criticalexponents ν eff , which is necessarily dependent on the parameters of the model. We confirm this ideaby exact numerical determination of ν eff in suitably chosen models. I. INTRODUCTION
A two-dimensional electron gas subject to a strong per-pendicular magnetic field exhibits the integer quantumHall effect. It is usually described in a non-interactingapproximation where the number of filled Landau levelsdetermines the (dimensionless) quantized Hall conductiv-ity σ xy . Disorder is essential as it broadens the otherwiseflat Landau bands and localizes eigenstates on a scale ξ ,so that beyond this scale the longitudinal conductivityvanishes, σ xx = 0. This holds except when the energy E (or field) is tuned to a critical value E c where ξ di-verges. The associated integer quantum Hall transition(IQHT) belongs to the family of Anderson transitions and is believed to be governed by a conformally-invariantfixed point in the parameter space that includes σ xx and σ xy .For a long time the commonly accepted paradigm ofthe IQHT fixed point was that of a conventional criticalpoint with renormalization group (RG) beta functionswhose expansions in the vicinity of the fixed point con-tain linear terms. In this case the RG flow equations forthe deviations of the longitudinal and Hall conductivitiesfrom their fixed-point values δ − ≡ σ xx − σ ∗ xx , δ + ≡ σ xy − σ ∗ xy , (1)take the form dδ − dl = −| y | δ − + ..., dδ + dl = ν − δ + + ... . (2)Here, l = ln( L/L ), σ ∗ xy = 1 / σ ∗ xx (cid:39) .
6, where thelatter is not known precisely. The ellipses denote higherorder terms in δ ± that are usually neglected close to thefixed point δ ± = 0.The critical exponents ν > y < ξ ( E ) ∼ | E − E c | − ν . In light of the notorious difficulty with analytical approaches to the IQHT, thisrelation is at the heart of a long history of numericalfinite-size scaling studies, mostly employing the Chalker-Coddington (CC) network model. These works re-port ν = 2 . .
62 but the leading irrelevant exponent y (cid:39) . Zhu et al. reported a slightlydifferent but incompatible value ν = 2 . .
50 obtainedfrom scaling the total number of conducting states inboth lattice and continuum models projected to the low-est Landau level. Even larger deviations were reportedin a structurally disordered version of the CC model where exponents as low as ν ≈ . ν ≈ . Such disor-dered Dirac fermions were conjectured before to be in theIQHT universality class. To sum up, the IQHT sets itself apart from other An-derson transitions in two ways: (i) A significant appar-ent variability of numerical estimates of ν across differentmodels assumed to be in the same universality class; (ii)A very small (and possibly vanishing ) leading ir-relevant exponent | y | .Although (i) might be rationalized by finite size effectsor the occurrence of novel universality classes, there isa more radical and intriguing alternative explanation:What if the conventional paradigm of a critical fixedpoint with linear beta functions does not apply to theIQHT? In Ref. [21], Zirnbauer proposed a solvable confor-mal field theory for the IQHT, which, in striking contrastto other Anderson transitions, comes without relevant orirrelevant perturbations. All physically allowed pertur-bations turn out to be marginal , implying ν − = 0 and y = 0. The theory moreover predicts the concrete value σ ∗ xx = 2 /π ≈ . a r X i v : . [ c ond - m a t . d i s - nn ] J a n earlier developments. In this work, we explore the consequences of such amarginal scaling scenario on the level of the flow equa-tions and present numerical evidence for its validity. Weset the stage by discussing the form of the sub-leadingterms on the right-hand sides of the flow equations (2)once the linear terms vanish (Sec. II). Along the criticalline δ + = 0, the equation for δ − can be solved analyti-cally. The result is a logarithmically slow flow of σ xx ( L )towards its fixed point value σ ∗ xx = 2 /π , governed by asingle universal number.Using simulations of the well-established CC net-work model and a much less studied two-channelgeneralization, (described in Sec. III), we confirmthis scaling prediction in Sec. IV. Unlike all previouslystudied models, as system size increases towards thethermodynamic limit, the two-channel network modelapproaches the fixed-point conductivity from above, σ xx ( L ) > σ ∗ xx . We also show that the proposed scaling isconsistently found for a disordered 2d Dirac model. Tuning away from criticality in Sec. V, we demonstratehow the marginal flow equations can mimic relevant scal-ing with an effective exponent ν eff , offering a new per-spective on the variability of numerically determined ν discussed above. If this mechanism is indeed realized atthe IQHT, why is the so-far-observed variation of ν eff onlyin the few percent range? Do models with a drasticallydifferent value of ν eff exist? To answer these questions,we first show that ν eff is controlled by the longitudinalconductivity σ xx in the fixed point’s vicinity, which isnumerically close in all standard models for which high-accuracy estimates of ν have been obtained. Crucially,as stated above, the two channel network model is anexception and indeed realizes ν eff (cid:39) II. MARGINAL FLOW EQUATIONS
We now explore the consequences of the Zirnbauer con-jecture for the flow equations (2) and assume σ ∗ xx =2 /π ≈ . δ ± have to be taken into account. Based on symmetries ofthe Pruisken field theory (periodicity in σ xy and be-havior under reversal of the magnetic field), Khmelnit-skii argued that dδ − dl must be even in δ + and dδ + dl mustbe odd, and proposed a global flow diagram. To respectthe topology of the flow diagram we require that the flowis always away from the axis δ + = 0 for both signs of δ − (no term δ + δ − in dδ + dl ). Likewise, to get a fixed pointwhich is stable along the δ + = 0 axis, no term δ − can + = + b = | b | FIG. 1. Flow diagram based on Eqs. (3) and (4). The flowis depicted for the rescaled RG variables ∆ ± (see axis labels)and parameters A = 0 . B = 8 introduced in Sec. V. Thefixed point at (0 ,
0) is denoted by a dot. appear in dδ − dl . Thus, we arrive at the RG equations dδ − dl = b δ − + b δ + ..., (3) dδ + dl = b δ − δ + + b δ + ... . (4)The expected phenomenology of the IQHT requires b , < b , >
0. For a certain choice of these pa-rameters discussed in Sec. V below, the flow is depictedin Fig. 1. We note that similar flow equations (with a dif-ferent b -term of the form b δ − δ ) have been suggestedrecently. We emphasize that, from an applied point of view, σ ∗ xx = 2 /π and the parameters b , , , in Eqs. (3) and (4)take over the role of ν and y in defining the IQHT univer-sality class. In Sec. IV we determine | b | = 45 . ± . III. NETWORK MODELS
For numerical simulations of the IQHT in subsequentsections we rely on network models originally introducedbased on semiclassical arguments and widely applieddue to their numerical efficiency. The standard CCmodel [abbreviated CC1, see Fig. 2(a)] is defined on acheckerboard lattice with inequivalent sites A,B (dots) atwhich the incoming chiral states on the links (arrows) arescattered quantum mechanically into two possible outgo-ing states with scattering amplitudes r = 1 √ e − x , t = 1 √ e x , (5)controlled by the model’s single parameter x , which en-codes the probability for right vs. left turns. The disor-der is realized by U(1)-phases exp( iφ j ) with φ j ∈ [0 , π )associated randomly to each link j . AB AB (a) CC1 (b) CC2 C=0C=1C=2(c) CC2 tt r-r rr t-t
FIG. 2. Network models employed in this work. The stan-dard CC model [abbreviated CC1, panel (a)] is defined ona checkerboard lattice with inequivalent sites A,B (dots) atwhich the incoming chiral states on the links (arrows) arescattered quantum mechanically into two possible outgoingstates with scattering amplitudes ± r, ± t , controlled by theparameter x , see Eq. (5). The disorder is introduced by ran-dom U(1) phases on each link. Panel (b) depicts the two-layergeneralization of the CC1 termed CC2, which features layer-preserving node scattering parametrized by the tuple ( x a , x b ).The disorder, which causes both inter- and intra-layer scatter-ing is modeled by random U(2) matrices mixing states on eachlink (boxes). Panel (c) shows the schematic phase diagram ofthe CC2 in the x a - x b plane following Ref. [25]. The dottedline indicates the (ensemble) symmetry under layer exchange x a ↔ x b , the dashed line represents the symmetry of the bulkphase diagram under ( x a , x b ) → ( − x a , − x b ) and C denotesthe number of edge states for a finite system with a specificchoice of boundary termination. The colored dots correspondto the critical parameter values ( x a , x b ) = ( − . ,
3) (brown)and ( x a , x b ) = (0 . , . The two-layer (or two-channel) generalization of theCC1, termed CC2, features two parallel chiral chan-nels per link, see Fig. 2(b). Without loss of general-ity, the scattering at the node is layer preserving andparametrized by the tuple ( x a , x b ) as above. The disor-der, which causes both inter- and intra-layer scattering,is modeled by Haar-random U(2) matrices acting on co-moving states on the links (boxes). This model has beenintroduced in Refs. [24, 25], where the phase diagramreproduced schematically in Fig. 2(c) was reported.The bulk phase diagram of the CC N , N = 1 , x to one in the − x ensemble. Together with the factthat the CC N generically has N + 1 topological distinctphases with C = 0 , , .., N edge states, this fixes thecritical point of the CC1 to x = 0. The above mirrorsymmetry also gives rise to the dashed symmetry line inthe phase diagram for the CC2. The dotted symmetryline for the CC2 phase diagram in Fig. 2(c) arises froma statistical layer-exchange symmetry x a ↔ x b .An important practical complication for the CC2 andany even N is that the positions of critical lines are notfixed by any symmetry argument but have to be foundnumerically. We defer the description of our numericalapproach to this task to Sec. VI. On the other hand, dueto the two-dimensional parameter space, the CC2 offersthe possibility to tune along the critical line, a feature of paramount importance to our study that is absent in theCC1.The critical properties of the CC2, so far assumed tobe in the IQHT universality class at all points along thecritical line, are not known with great accuracy due tolarge localization lengths and the aforementioned uncer-tainty about the critical ( x a , x b ). In Ref. [25], the authorssettled for a modified model with an ad-hoc weakeningof inter-layer scattering and reported ν = 2 .
45 from astudy of quasi-1d Lyapunov exponents at a certain pointon the critical line. No error bars were given. Our resultsfor the CC2 with full interlayer scattering presented inSec. VI are significantly different.All observables defined and computed in subsequentsections are based on the steady-state, four-terminal scat-tering matrix S of large network models. Each termi-nal refers to the union of incoming and outgoing linksalong one of the four sides of a rectangular-shaped net-work. To numerically obtain S for systems of linear size L up to order 1000 efficiently, we use an iterative patchingapproach, concatenating the scattering matrices of fourrectangular subsystems with size L × aL into the scatter-ing matrix of a single system of size 2 L × aL , where a isthe aspect ratio. Note that unlike transfer matrix mul-tiplication, the scattering matrix concatenation does notrequire further numerical stabilization. The main com-putational bottleneck limiting system sizes is the largememory required to store the iteratively obtained matri-ces S . A useful feature of this iterative approach is thatsystems of exponentially different sizes are generated ina single run. IV. MARGINAL SCALING AT CRITICALITY
In this section, we focus on the critical line δ + = 0.In this case, the right-hand side of Eq. (4) vanishes whileEq. (3) becomes dδ − dl = b δ − , and we neglect higher orderterms. This can be solved as δ − ( L ) = δ − ( L ) (cid:113) | b | δ − ( L ) ln ( L/L ) . (6)Using the network models introduced in the previ-ous section, we numerically demonstrate the validity ofEq. (6) and determine the value of b . While for CC1the critical point occurs at x = 0, for the CC2 we tune to( x a , x b ) = ( − . ,
3) [brown dot in Fig. 2(c)] which willbe shown to be on the critical line below in Sec. VI. Weuse wide slabs of length L , aspect ratio a (cid:38) σ xx = g xx /a from thedisorder averaged two-terminal Landauer conductance g xx with the terminals in the transverse direction short-circuited to realize a torus-geometry (periodic boundaryconditions). Unlike in square samples, the histograms ofthe conductivities σ xx for an ensemble of disordered, wide( a (cid:38)
4) slabs are Gaussian in shape (data not shown).The data on the top panel in Fig. 3 shows a mono-tonously increasing (decreasing) flow of σ xx ( L ) for CC1(CC2) with the conjectured critical conductivity σ ∗ xx =2 /π consistently placed between the two data sets. Weremark that to the best of our knowledge, this is the firstnumerical observation of a decreasing σ xx ( L ) at a IQHT,hypothesized long ago in the proposed flow diagram ofKhmelnitskii. For a quantitative test of marginal scaling, in Fig. 3(bottom) we plot the quantity | b ( L ) | ≡ δ − − ( L ) − δ − − ( L )2 ln ( L/L ) , (7)which, according to Eq. (6), should be length-independent and identified with the universal number | b | . The only freedom is the choice of the initial sys-tem length L which must be large enough such that | δ − ( L ) | is sufficiently small to apply the expansion of theflow equations in Sec. II. Indeed, if we consider the CC1data points for which | δ − ( L ) | ≤ .
085 (grey region, datapoints connected by solid lines) we confirm that | b ( L ) | inEq. (7) is practically a constant over almost two decadesin L/L . A least-squares fit yields (purple line) | b | = 45 . ± . . (8)For the CC2 model at ( x a , x b ) = ( − . ,
3) (brown),there are only two data points in the grey region, whichare nevertheless consistent with Eq. (8). Sliding along theCC2 critical line to the point ( x a , x b ) = (0 . , . σ xx ( L = 1920) (cid:39) .
78 way outside the scal-ing region (data not shown). This is consistent with theobservation of very large localization length in the CC2model made in Refs. [24, 25]. Sliding along the criticalline in the other direction (towards larger x b ) did notlower σ xx appreciably.To further test the prediction σ ∗ xx = 2 /π , we re-peated the above analysis with ad-hoc variations of σ ∗ xx by +0 .
01 ( − . | b ( L ) | , see upward (downward) triangles. Theresulting values for CC1 (blue triangles) exhibit a con-siderable dependence on L . Significantly, for CC2 (browntriangles) the resulting values shift in the opposite waycompared with CC1. Thus, the expected universality of b in the marginal flow scenario holds only for the pre-cise value σ ∗ xx = 2 /π , lending additional support for theprediction of Ref. [21].Notice that the numerically determined conductivity σ xx = g xx /a is expected to coincide with the theoreticalprediction σ ∗ xx = 2 /π only in the double limit a → ∞ and L → ∞ . Therefore, it is important to keep the aspectratio a as large as is practically possible. An insufficientaspect ratio leads to a drop of the quantity | b ( L ) | inEq. (7) at large length scales: compare the data for a = 4(cyan) and a = 6 (blue) in Fig. 3. However, at small L , | b ( L ) | is surprisingly affected only slightly by the changefrom a = 6 to a = 4.Finally, in Fig. 3 (top) we also include data for theCC1 at small, finite x = 0 .
02 (green crosses) which show L xx = g xx / a xx = 2/CC1, x = 0, a = 6CC1, x = 0, a = 4CC1, x = 0.02CC2, ( x a , x b ) = ( 0.483, 3)2d Dirac, E = 0.8, W = 2 L / L [ ( L ) ( L )] / [ l n ( L / L )] CC1, x = 0, a = 6 : L = 44CC1, x = 0, a = 4 : L = 40CC2, ( x a , x b ) = ( 0.483, 3) : L = 11522d Dirac, E = 0.8, W = 2 : L = 20 FIG. 3. Numerical demonstration for marginal scaling of δ − = σ xx − σ ∗ xx at criticality δ + = 0. The top panel shows thedisorder averaged conductivity σ xx (dots) obtained from theLandauer conductance of wide slabs with aspect ratio a (cid:38) σ ∗ xx (greybox) connected by solid lines follow the scaling prediction inEq. (7) with | b | = 45 . ± .
22 (purple line), see bottom panel.The upward (downward) triangles denote the CC1 and CC2data for ad-hoc variations of σ ∗ xx by +0 .
01 ( − . L -dependence, while theCC2 data move away from the purple line in the oppositeway compared with CC1. Data for the CC1 at small finite x = 0 .
02 is also included in the top panel to show that theconstant b in Eq. (3) is negative. an initial increase toward σ ∗ xx for small L , but then curveaway from the fixed-point conductivity at large L . Thisconfirms the negative sign of the constant b in Eq. (3). l + A=0.1, B=80 5 10 15 l l = 0) = 0.35 ( l = 0) = 0.37 6 5 4 3ln + ( l = 0)51015 l c l n eff = 2.55 eff = 3.34 FIG. 4. Numerical solution of the rescaled flow equations(10) and (11) with the parameters A = 0 . B = 8,∆ − (0) = − .
35 (blue) and − . + (0) = 10 − ... − are chosen as detailed in the bottompanel. Each flow is stopped at l = l c defined as ∆ + ( l c ) = 1(dashed line in top panel). The relation between the initial∆ + (0) and l c shown in the bottom panel approximately fol-lows l c ∼ − ν eff ln ∆ + (0) + const (solid lines) with ν eff depen-dent on ∆ − (0). V. MIMICRY OF RELEVANT SCALING FROMMARGINAL RG FLOW
In this section, we demonstrate how the marginal flowequations in Sec. II can give rise to an apparent conven-tional scaling of the localization length, ξ ∼ | δ + ( L ) | − ν eff .To reduce the number of constants in the flow equationsof Sec. II, we define rescaled variables ∆ ± ( l ) and (un-known) universal numbers A , B :∆ − ( l ) ≡ (cid:112) | b | δ − ( l ) , ∆ + ( l ) ≡ (cid:112) b δ + ( l ) ,A = (cid:112) | b | | b | b , B = b | b | . (9)In terms of these, and neglecting higher-order terms, theRG Eqs. (3) and (4) become d ∆ − dl = − ∆ − − A ∆ , (10) d ∆ + dl = B ∆ − ∆ + + ∆ . (11)As usual, the localization length ξ is defined via ln ξ ∼ l c with l c the RG cutoff time given by ∆ + ( l c ) = 1. The qualitative behavior of the solutions of the flowequations (10) and (11) very close the fixed point canbe obtained by neglecting the second terms on the right-hand side. Then ∆ − ( l ) flows logarithmically slow towardzero (see Eq. (6) above), and the factor B ∆ − ( l ) can beapproximately treated as a constant in front of ∆ + ( l ), asin Eq. (2), resulting in ν eff ∼ [ B ∆ − (0)] − . (12)This is the mimicry of the conventional critical scaling asdefined above.Eventually, RG trajectories leave the vicinity of thefixed point and ∆ + ( l ) grows large enough such that thesecond terms on the right-hand side in (10) and (11) sig-nificantly alter the global flow. The condition for themimicry of the conventional scaling is that the RG tra-jectory spends a long time sufficiently close to the fixedpoint where the powers of ∆ + ( l ) higher than the linearcan be neglected, and ∆ − ( l ) does not vary appreciablyacross this range of l . This is easily achieved for A (cid:28) B (cid:29) | ∆ − (0) | (cid:28)
1. Inthis case we still expect that ν eff ∼ [ B ∆ − (0)] − . Thisexpectation is confirmed by the exact solution of the sys-tem (10) and (11) in the case A = 0, presented in theAppendix, where Eq. (12) is shown to hold for B (cid:29) − (0) are not much smallerthan unity in magnitude. Employing ∆ − (0) = − . A = 0 . B = 8 for the uni-versal parameters, Fig. 4 shows a numerical solution, seeblue lines. A linear approximation to the data in thebottom panel using ln ξ ∼ − ν eff ln ∆ + (0) + const yields ν eff (cid:39) .
55. The red lines denote results when the ini-tial value ∆ − (0) is slightly changed to − .
3, in this case ν eff (cid:39) .
34 emerges. Both values are beyond the accuracyof the estimate (12), since B = 8 is not large enough.Larger values of B would reduce the curvature in Fig. 4(bottom) and thus better approximate conventional scal-ing, but also push towards smaller values of ν eff . Thenumerical value of A is of minor importance for ν eff , as isdemonstrated by the exact solution for A = 0. The sameis true for the precise value of the constant used in thedefinition of l c above, as long as it is of the order of unity.This is clear from the steep slope (and the eventual diver-gence) of ∆ + ( l ) close to l c , see Fig. 4 (top). In addition,we have confirmed (results not shown) that the mimicrymechanism is qualitatively unchanged for the alternativeflow equations proposed in Ref. [28], indicating that theprecise nature of the higher-order terms does not play asignificant role.We stress, however, that for all known numerical mod-els of the IQHT, the quantitative validity of the abovetruncation of the flow equations leading to (10) and (11),is questionable. The reason is the aforementioned largevalue of | ∆ − | = √ b | σ xx − /π | (cid:39) . ± fromtheir starting values. These effects and their quantitativeinfluence on the mimicry of conventional scaling are leftfor future work.In summary, we have shown how the conjecturedmarginal flow equations (3) and (4) can approximatelymimic conventional scaling with an effective critical ex-ponent ν eff . There are two qualitative conclusions thatcould serve as hallmark signatures of the marginal scal-ing scenario: (i) The dependence of the effective criticalexponent ν eff on δ − ( L ) and through it on the chosenmodel and its parameter values, c.f. Fig. 4. Althoughthe relation is likely more complicated than the simpleestimate ν eff ∼ δ − − (0), we should expect sizeable varia-tions of ν eff between models for the IQHT if they real-ize different δ − (0). (ii) The relation ξ ∼ | δ + (0) | − ν eff isonly approximately fulfilled. Indeed, Fig. 4 (bottom) re-veals a small residual curvature, that can be captured bythe introduction of a δ + (0)-dependent critical exponent.Anticipating the relation δ + (0) ∼ δ x ≡ x − x c for thenetwork models, we thus use the ansatz ξ = λ | x − x c | − ν eff ( δ x ) . (13)Here, the non-universal parameter λ has the dimensionof length. In the next section, we investigate both signa-tures (i) and (ii) with exact numerical simuations of thenetwork models. While we confirm a model and param-eter dependent critical exponent ν eff , we can only put anupper bound on a putative δ x dependence of ν eff ( δ x ) inthe CC1. VI. NUMERICAL DEMONSTRATION OFVARIABLE ν eff IN MODELS OF THE IQHT
We now tune our numerical models away from theircritical points x = x c to study the divergence of thelocalization length close to criticality which we assumeto be described by Eq. (13). We adopt the scattering-matrix based observable Λ initially proposed by Fulga et al. , Ref. [30], and recently employed to study scalingfor the 2d Dirac model. This observable is qualitativelysimilar to the scaling variable δ + = σ xy − /
2, that is,it changes sign at the critical point. This property pro-vides a simple and precise method to determine x c formodels like the CC2 where the critical point is not fixedby symmetry.In this section, we fix the aspect ratio to a = 1 andwrap the L × L system in one direction, called the trans-verse direction, to form a cylinder. To define Λ( x, L ) forthe cylinder, we attach a lead extended over the full widthof the system to one of the open ends of the cylinder.Then we consider the reflection matrix r ( φ ) of the lead asa function of the phase φ of twisted boundary conditionsin the transverse direction, or equivalently the value ofan Ahronov-Bohm flux piercing the cylinder. For a givendisorder realization and smoothly tuned x , the criticalpoint x = x c occurs when there exists a φ such that r ( φ ) has a zero eigenvalue and thus det r ( φ ) = 0. This followsdirectly from the definition of a reflection-matrix basedtopological invariant in the unitary symmetry class. To obtain Λ, we generalize to boundary conditionscharacterized not only by a phase but also by a complexnumber z ∈ C . In terms of scattering states ψ i,j definedon all links of the network, the generalized boundary con-dition reads: ψ i,j = L − = z ψ i,j =0 for all i = 0 , , ..., L − i, j ). Thisadditional freedom allows for solutions z of det r ( z ) = 0to exist even away from criticality x (cid:54) = x c . However, gen-eralized zeros of this kind occur away from the unit circle, | z | (cid:54) = 1. To measure the distance to criticality, considerlog | z | for the z closest to the unit circle. This quan-tity indeed changes sign at x = x c and features a Gaus-sian histogram in the ensemble of disorder realizations. Finally, the scaling observable is defined as Λ = ln | z | where the overline denotes disorder average.We proceed with the conventional single-parameterscaling hypothesis stating that a dimensionless observ-able like Λ depends not on the system size L and thedimensionless parameter x separately, but as Λ( x, L ) =Λ( L/ξ ( x )), with the localization length ξ from Eq. (13).Requiring that Λ( x, L ) is analytic in δ x ≡ x − x c andusing the property Λ( x = x c , L ) = 0, we expandΛ( x, L ) = α δ x (cid:16) Lλ (cid:17) ν − ( δ x ) + α δ x (cid:16) Lλ (cid:17) ν − ( δ x ) + ... (14)We now turn to the demonstration of a sizeable model-and parameter dependence of ν eff , point (i) above. Weanticipate the δ x -dependence of ν eff [point (ii)] to bea comparatively small effect which we neglect for nowand revisit towards the end of this section. In Fig. 5,we report Λ for CC1 and CC2 network models of size L = 100 , , , , a = 1 and sev-eral x around x c . While x c = 0 for the CC1 by symme-try, for the CC2 we focus on two line-cuts in the phasediagram of Fig. 2(c), ( x a , x b ) = ( x,
3) and ( x, x ), respec-tively. We take the range of x -values small enough so thatthe higher order terms in Eq. (14) do not contribute, seeFig. 5. For each L , we perform a fit of Λ( x, L ) linear in x (dotted line) and extract its zero-crossing. Remarkably,these crossing points agree for all L within an accuracybetter than ∆ x c = 0 . x c isreported in Fig. 5 and was used for the study of the crit-ical longitudinal conductivity in Sec. IV.The CC2 is known to have localization lengths largecompared to CC1, reflected by a larger λ . In our study,this leads to drastically smaller slopes for the CC2 whencompared to the CC1 at the same L , see Fig. 5. In prac-tice, this requires hundreds of thousands of disorder re-alizations to achieve an acceptable ratio between datapoint separation and error bars.To extract ν eff , we employ the scaling predictionΛ( x, L ) /δ x ∼ L /ν eff , valid for small enough δ x and L ,c.f. Eq. (14). We approximate the left hand side by theslopes of the linear fits mentioned above, and plot the x ( x , L ) CC1 x c = 0.0 x CC2: (x,3) x c = 0.483 x CC2: (x,x) x c = 0.227 L l n ( x ( L ) | x = + c o n s t . CC1 eff = 2.6 ± 0.02CC2: (x,3) eff = 3.42 ± 0.04CC2: (x,x) eff = 3.9 ± 0.05
FIG. 5. The left three panels show the scaling variable Λ( x, L ) for x crossing the critical point x c and square systems of size L = 100 , , , , − ∂ x Λ( x, L ) obtained from linear fits to each individual systemsize L . They follow power-laws ∼ L /ν eff with various ν eff indicated in the legend. The data points are offset in the verticaldirection for clarity. slopes in the right panel of Fig. 5 where we extract ν eff .For the CC1, our result ν eff = 2 . ν eff = 3 . ν eff = 3 . − . ,
3) and (0 . , . ν eff in Eq. (13) shoulddepend on δ x . We stress that a check of this predictionrequires an analysis of numerical data at several fixed values of x , which is usually not attempted in the existingliterature. Such an analysis crucially relies on our abilityto represent Λ( x, L ) as the expansion (14) with a non-trivial exponent ν eff ( δ x ).We focus on the CC1 for its numerical convenience,exactly known x c = 0 and odd-in- δ x expansion [im-plying α = 0 in Eq. (14)], and select the data for x = 0 . , . , .
03 from Fig. 5, left panel. In Fig. 6we show that the anticipated scaling relationln Λ(
L, x ) x = ln α + 1 ν eff ( x ) ln Lλ (15)holds with ν eff ( x ) values that agree within error bars forall chosen x and are, moreover, consistent with the valueof ν eff obtained above using the slope of Λ extracted frommultiple x . Decreasing x further was found not to be suit-able due to enhanced statistical error bars, while larger x would require including higher-order corrections in Eq.(14) and the corresponding modifications to Eq. (15).In summary, using a variation of x by a factor of three,we were not able to positively identify the proposed δ x dependence of ν eff anticipated in the marginal scaling sce-nario. However, this should not be taken as a serious ar-gument against the possible validity of the latter, as thecurvature of the data in the bottom panel of Fig. 4 sen-sitively depends on the parameters in the flow equations.Detecting a small curvature corresponding to a weak x -dependence of ν eff might very well require varying x by L ( x , L ) / x eff ( x = 0.03) = 2.6 ± 0.01 eff ( x = 0.02) = 2.59 ± 0.02 eff ( x = 0.01) = 2.61 ± 0.03 FIG. 6. Scaling plot for Λ( x, L ) /x at fixed x computed fromthe CC1 at aspect ratio a = 1 for L = 100 , , , , x = 0 . , . , .
03 agree within error bars and thepower law fit [Eq. (15)] (dashed lines) yields compatible valuesfor ν eff ( x ) for all three x . one or two orders of magnitude, which is currently be-yond the capability of our numerical approach. VII. CONCLUSION AND OUTLOOK
In this work, we scrutinized the consequences of Zirn-bauer’s marginal scaling scenario recently conjecturedfor the IQHT. We proposed and analyzed a beyond-linear-order expansion of the RG beta functions for thelongitudinal and Hall conductivities taking into accountthe known topology of the flow diagram. At criticality,our numerical simulations of the one- and two-channelnetwork models confirmed that the resulting RG equa-tion indeed describes the flow of the critical longitudinalconductivity with the system size L . The RG flow de-pends on a single universal number | b | which we deter-mined numerically to be | b | = 45 . ± .
22. For futurework, it is important to find this parameter | b | via ananalytical calculation within the conjectured conformalfield theory and confirm the surprisingly large valuefound numerically.We stress that consistency with the critical marginalscaling defined by Eqs. (6) and (8) is a necessary condi-tion for any model to be in the IQHT universality class.To further support such universality beyond the two net-work models studied so far, in Fig. 3, we present theconductivity data for a 2d Dirac model with disorder inthe unitary class at a finite energy E = 0 . W = 2, see Ref. [19] for details of themodel definition and numerical method. Although therange of system sizes is not as large as for the networkmodels, the Dirac data is also consistent with the univer-sal marginal scaling. We expect the universality to holdfor the wide-slab conductivity in the variety of numeri-cal models studied as realizations of the IQHT but notconsidered in this work.Moving away from the critical point, we showed by aproof-of-principle numerical solution of the conjecturedflow equations how an effective localization length expo-nent ν eff can arise from marginal scaling where formally ν − = 0. The effective exponent depends on the criticalconductivity at short distances, albeit the detailed func-tional relation probably requires an account of higher-order terms in the flow equations, which is left for futurework. Interestingly, it appears that the selection of mod-els for the IQHT studied in the literature so far conspiredto have very close critical conductivities and, accordingly,only moderate (less than 10%) variations of ν eff from thevalue ν CC1 = 2 . ν eff , models with drasti-cally increased tuning capabilities are required. One ex-ample is the 2d Dirac model, where energy and dis-order strength can be varied independently. As shownrecently, ν eff (cid:39) .
33 was found at half-filling, E = 0,and a certain disorder strength, the largest deviationfrom ν CC1 = 2 . ν CC2 between 3 and 4 and thus takes acentral role in providing strong evidence of the validityof the marginal scaling picture.We emphasize that according to the conjectured flowequations, a larger absolute value of δ − ( L ) = σ xx ( L ) − /π , i.e. a larger distance of short-length longitudinalconductivity to the fixed point value 2 /π should lead to asmaller ν eff . Naively, comparing our results for the CC1and CC2 shows the opposite trend. This can be rational-ized by the intrinsic difficulty in defining the length scale L beyond which a beta-function description of σ ( L ) be-comes possible at all. And even for L (cid:38) L , an expansionof the beta functions beyond the order considered abovemight be required.Very recently, the IQHT was also studied in the frame- work of dissipation-induced topological states and theresult ν (cid:39) ACKNOWLEDGMENTS
We acknowledge useful discussions with John Chalker,Ferdinand Evers, Matthew Foster, Igor Gornyi and Mar-tin Zirnbauer. Computations were performed at the OhioSupercomputer Center and the Lawrencium cluster atLawrence Berkeley National Lab. E.J.D was supportedby the NSF Graduate Research Fellowship Program, NSFDGE 1752814. B.S. acknowledges financial support bythe German National Academy of Sciences Leopoldinathrough grant LPDS 2018-12.
Appendix A: Exact solution of the flow equationswith A = 0 When we neglect the second term in the right-handside of Eq. (10), the flow equations become d ∆ − dl = − ∆ − , d ∆ + dl = B ∆ − ∆ + + ∆ . (A1)This system is exactly solvable. The first equation (A1)is solved in the same way as in Section IV:∆ − ( l ) = ∆ − (0) (cid:113) − (0) l . (A2)Then the second equation (A1) becomes a linear equationin terms of the variable X ( l ) ≡ ∆ − ( l ): dXdl = − B ∆ − ( l ) X − . (A3)This equation has the solution X ( l ) = e − F ( l ) (cid:18) X − (cid:90) l e F ( l (cid:48) ) dl (cid:48) (cid:19) ,X ≡ ∆ − (0) , F ( l ) = 2 B (cid:90) l ∆ − ( l (cid:48) ) dl (cid:48) . (A4)Performing the integrals, we obtain F ( l ) = B ln[1 + 2∆ − (0) l ] = 2 B ln ∆ − (0)∆ − ( l ) , (cid:90) l e F ( l (cid:48) ) dl (cid:48) = [1 + 2∆ − (0) l ] B +1 − B + 1)∆ − (0) , (A5)which gives the explicit solution1∆ ( l ) = [1 + 2∆ − (0) l ] − B ∆ (0) − [1 + 2∆ − (0) l ] − [1 + 2∆ − (0) l ] − B ( B + 1)∆ − (0) . (A6)Rewritten in terms of ∆ − ( l ) this equation describes the integral curves of the system (A1) in the ∆ + –∆ − plane:∆ (0)∆ ( l ) = (cid:16) ∆ − (0)∆ − ( l ) (cid:17) − B − ∆ (0)( B + 1)∆ − (0) (cid:20) ∆ − (0)∆ − ( l ) − (cid:16) ∆ − (0)∆ − ( l ) (cid:17) − B (cid:21) . (A7) + ( l = 0)5678910 l n A = 0, B = 60, (0) = 0.3 eff = 0.22, B (0) = 0.19 A = 0, B = 20, (0) = 0.3 eff = 0.85, B (0) = 0.56 FIG. 7. Analytical solution for ln ˜ ξ defined in the main text.This figure is similar to the bottom panel in Fig. 4. As every-where in this appendix A = 0, while we chose B = 20 (green)and B = 60 (brown) for the two data sets. The dots representof ln ˜ ξ computed from Eq. (A9) for the same values of ∆ + (0)as in the bottom panel in Fig. 4, and ∆ − (0) = − . Notice that the resulting flow is even in ∆ − ( l ), which isa consequence of setting A = 0. This property is lost forthe full system (10), (11).The RG flow described by Eq. (A6) begins at l = 0with a very large left-hand side ∆ − (0) (cid:29)
1. As l grows,the first term on the right-hand side decreases, while thesecond (negative) term increases in magnitude. When wereach the localization length, l c = ln ξ , the left-hand sidebecomes one, and the resulting equation can be numeri-cally solved for ln ξ . However, for our current analysis itis better to adopt a slightly different definition of ln ξ asthe RG time for which ∆ + (˜ l c ) = ∞ . With this modifica-tion the terms on the right-hand side cancel each otherwhen l = ˜ l c . This gives an equation for ln ˜ ξ :[1 + 2∆ − (0) ln ˜ ξ ] B +1 − B + 1) ∆ − (0)∆ (0) , (A8) with the solutionln ˜ ξ = 12∆ − (0) (cid:18)(cid:104) ( B + 1) ∆ − (0)∆ (0) + 1 (cid:105) B +1 − (cid:19) . (A9)This quantity is shown in Fig. 7 for B = 20 and B = 60,together with linear fits resulting in effective exponents ν eff whose values are given in the legend.Let us estimate ν eff analytically. If we start the flowsufficiently close to the critical line, ∆ + (0) (cid:28)
1, the firstterm in the square brackets in Eq. (A9) dominates, andwe get approximatelyln ˜ ξ ≈ [( B + 1)∆ − (0)] B +1 − (0) [∆ + (0)] − B +1 − − (0) . (A10)Now comes the mimicry. If B (cid:29)
1, the factor with ∆ + (0)has a very small exponent, and can be approximated by[∆ + (0)] − B +1 = exp (cid:16) − B + 1 ln ∆ + (0) (cid:17) ≈ − B + 1 ln ∆ + (0) . (A11)Notice that this approximation only works for | ln ∆ + (0) | (cid:28) B , that is, sufficiently far from criti-cality. When legitimate, this approximation leads toa linear relation between ln ˜ ξ and ln ∆ + (0) with thecoefficient ν eff = [( B + 1)∆ − (0)] − BB +1 ≈ [ B ∆ − (0)] − . (A12)This analytical estimate is given in the legend of Fig. 7for the same values of B and ∆ − (0) as the actual valuesof ln ˜ ξ . We see that the agreement between the analyticalestimates and the results of linear fits becomes reasonableonly for unrealistically large values of B ( B (cid:38)
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