Edge state critical behavior of the integer quantum Hall transition
Martin Puschmann, Philipp Cain, Michael Schreiber, Thomas Vojta
EEPJ manuscript No. (will be inserted by the editor)
Edge state critical behavior of the integerquantum Hall transition
Martin Puschmann , Philipp Cain , Michael Schreiber , and Thomas Vojta Department of Physics, Missouri University of Science and Technology, Rolla, Missouri65409, USA Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany
Abstract.
The integer quantum Hall effect features a paradigmaticquantum phase transition. Despite decades of work, experimental, nu-merical, and analytical studies have yet to agree on a unified under-standing of the critical behavior. Based on a numerical Green functionapproach, we consider the quantum Hall transition in a microscopicmodel of non-interacting disordered electrons on a simple square lat-tice. In a strip geometry, topologically induced edge states extend alongthe system rim and undergo localization-delocalization transitions asfunction of energy. We investigate the boundary critical behavior inthe lowest Landau band and compare it with a recent tight-bindingapproach to the bulk critical behavior [Phys. Rev. B 99, 121301(R)(2019)] as well as other recent studies of the quantum Hall transitionwith both open and periodic boundary conditions.
Applying a strong perpendicular magnetic field on a two-dimensional free-electron gasleads to highly degenerate eigen energies E n = ( n +1 / (cid:126) ω , the Landau levels. Here, n is a non-negative integer and ω is the cyclotron frequency, ω = eB/m . Disorder liftsthe degeneracy and broadens the Landau levels into Landau bands (LBs), leadingto extended states in the band center E c that separate two localized phases. Theinteger quantum Hall (IQH) transition is characterized by a power-law divergenceof the localization lengths ξ ∼ | E − E c | − ν at the critical energy E c . The value ofthe localization-length exponent ν is not settled despite a large body of work in theliterature. There are deviations between experimental and theoretical reports as wellas between several numerical approaches [1,2,3,4].We recently analyzed the IQH transition in a microscopic tight-binding model ofnon-interacting electrons on a square lattice using the topology of an infinite cylinder[5]. By means of a careful scaling analysis, we obtained ν = 2 . ν ≈ . a r X i v : . [ c ond - m a t . d i s - nn ] A p r Will be inserted by the editor3.415 3.420 3.425 3.4300.20.40.60.81.01.21.41.6 E Γ L ry − − r − α L − y − α E × / / / / r E < E c E = E c E > E c ≤ -4 -3 -2 -1 0 1log N ( Nρ i / P i ρ i ) Fig. 1.
IQH transition in the lowest LB for flux Φ = 1 /
10 and disorder W = 0 .
5. Left:Local density of states ρ i (visualized by color) for a strip of width L = 32 for an edge state( E = 3 . E = 3 . E = 3 . N = 10 ) is shown. Right: Dimensionless Lyapunov exponent Γ ( E, L ) asfunction of E for several L . The statistical errors are well below the symbol size. The solidlines are third-order polynomial fits. The inset shows an analysis of the crossing energy E × according to Eq. (3) with y = 0 .
88 and α = 1 / . r . the critical energy are localized and extended, respectively, rendering a localization-delocalization transition in the boundary behavior. We study this transition using therecursive Green function method and determine the boundary critical behavior. Wefind a localization-length exponent ν = 2 . We consider a tight-binding model of non-interacting electrons moving on a squarelattice of N × L sites, given by the Hamiltonian matrix H = H II H II H . . .. . . . . . II H N with H x = u x, e iϕ x e − iϕ x u x, e iϕ x e − iϕ x u x, . . .. . . . . . e iϕ x e − iϕ x u x,L , (1)expressed in a Wannier basis. Geometrically, the lattice is a stack of N layers H x of L sites each. H and H x have block-tridiagonal and tridiagonal forms, respectively,representing open boundaries (obc) in the x and y directions. The disorder is imple-mented via independent random potentials u x,y , drawn from a uniform distribution inthe interval [ − W/ , W/ W characterizes the disorder strength. The hopping termshave unit magnitude, and the uniform out-of-plane magnetic field B is representedvia direction-dependent Peierls phases [9,10]. The hopping in the y direction suffersa complex phase shift ϕ x = 2 πΦx whereas the bonds in the x direction, representingcouplings between consecutive layers, do not have phase shifts. This leads to the off-diagonal identity matrices I in H . Φ = Bl /Φ denotes the magnetic flux through aunit cell (of size l ) in multiples of the flux quantum Φ = h /e . ill be inserted by the editor 3 In the clean case, W = 0, the interplay of the lattice periodicity and the Peierlsphases leads to feature-rich Landau-level formation as function of flux Φ , known asthe Hofstadter butterfly [11,12]. In our previous work [5], we discussed the implica-tions of the butterfly structure for the observation of universal properties of the IQHtransition. We then analyzed the bulk IQH transition for Φ = 1 / / / /
10, 1 /
5, 1 /
4, and 1 / Φ (cid:46) /
10, where our data collapse when the system size L is expressed inmultiples of the magnetic length L B = 1 / √ πΦ . In the current work, we examine theboundary transitions for the same set of system parameters.We employ the recursive Green function method [13,14,15] to characterize thebehavior of the electronic states. It recursively computes the Green function G ( E ) =lim η → [( E + iη ) I − H ] − at energy E . I is the identity matrix and η shifts the energyinto the complex plane to avoid singularities. Based on a quasi-one-dimensional latticewith N (cid:29) L , the smallest positive Lyapunov exponent, γ ( E, L, Φ, W ) = lim N →∞ N ln | G N N | , (2)describes the exponential decay of the Green function between the 1st and N th layers.For the current system, the matrix G N N = G · G · G . . . G NNN can be written asproduct of the diagonal blocks G xxx = (cid:2) ( E + iη ) I − H x − G x − x − ,x − (cid:3) − . We approxi-mate the limit η → η to a small nonzero value, η = 10 − . We use thedimensionless Lyapunov exponent Γ ≡ (cid:104) γ (cid:105) L for the scaling analysis. (cid:104) γ (cid:105) representsthe ensemble average of 50 strips of size L × with width L up to 512. For Φ = 1 / L up to 768. Using the recursive Green function method, we create Γ ( E, Φ, L ) data sets in theenergetic vicinity of the transition in the lowest LB for several Φ . The right panelof Fig. 1 shows the data for Φ = 1 /
10. We first perform a simple scaling analysis.To this end, we describe the E dependence of Γ (for each L and Φ ) by a third-orderpolynomial. For each Φ , we identify E c using the crossings of the Γ vs. E curves for twodifferent L with ratio r , Γ ( E × , L ) = Γ ( E × , L/r ). The crossings can be extrapolatedto infinite L using the scaling ansatz Γ ( E, L ) = Γ c + Γ r ( E − E c ) L /ν + Γ i L − y withrelevant (r) and irrelevant (i) correction terms, which implies E × ( L, r ) = E c + Γ i ( r y − Γ r (1 − r − /ν ) L − /ν − y . (3)The inset of Fig. 1 shows this extrapolation for Φ = 1 /
10; we use ν = 2 . y = 0 . r collapse and the largest number of crossings followEq. (3), leading to E obcc = 3 . . Unfortunately, this extrapolation depends onthe (a priori unknown) value of y . We can exclude higher values, y (cid:38) . E × vs. L curves develop a pronounced S-shape, which would imply that at leastthree correction-to-scaling terms are important. However, we cannot strictly excludesmaller y values (even though the range of crossings that follow (3) becomes smallerwith decreasing y ). For y = 0 .
4, the resulting critical energy, 3 . We consider fits as reasonable when the mean squared deviation approximates the data’sstandard deviation. Unless noted otherwise, the given uncertainties of the critical estimatesrepresent statistical standard deviations with respect to individual fits. Will be inserted by the editor0.00 0.10 0.20 0.30 0.400.40.60.81.0 (
L/L B ) − . Γ ( E c ) / / / / / / / Φ − − − L/L B Γ ( E c ) Γ ( L / L B ) − / . Fig. 2.
Lyapunov exponent Γ and its slope Γ (cid:48) = ∂Γ/∂E at criticality, E pbcc , as function ofthe effective length L/L B for several Φ . Errors are below the symbol size. Lines are guide tothe eye only. The inset shows Γ (cid:48) scaled using the relevant exponent ν = 2 .
6, emphasizingthat ν ≈ . Φ (cid:46) / perfectly with our value E pbcc = 3 . E c is more accurate and robust [5]. Within the standard picture ofthe IQH effect, the critical energies for open and periodic boundaries should coincidein the thermodynamic limit because chiral edge states cannot Anderson localize dueto the absence of back scattering. This suggests that the above estimate y ≈ . Φ ; Fig. 2 shows theresulting Γ ( E pbcc , Φ, L ) and their slopes Γ (cid:48) ( E pbcc , Φ, L ) at bulk criticality E pbcc . Thedata for Φ = 1 / / Φ , whose dataasymptotically collapse as function of L/L B . As in the case of the cylinder geometry[5], we thus consider systems with Φ (cid:46) /
10 to be in the universal regime. If we use E obcc instead of E pbcc in Fig. 2, the data collapse is of significant lower quality.In the following, we use the data for Φ = 1 /
10 for which we have better statisticsand larger sizes to extract estimates of the critical exponents and amplitudes. Weperform fits at both E obcc and E pbcc to capture errors stemming from the uncertaintiesof E c . For E obcc , power-law corrections Γ ( E c , L ) = Γ c (1+ aL − y ) lead to reasonable fitsfor L ≥
32, yielding Γ c = 0 . y = 0 . E pbcc , the simple power-lawdescription is limited to a smaller L range. We obtain Γ c = 0 . y = 0 . Γ c = 0 . y = 0 . L ≥
128 and L ≥ Γ (cid:48) to get estimates for the relevant exponent ν . For both E c estimates, we get good-quality fits Γ (cid:48) ( E c , L ) = Γ (cid:48) c L /ν even without irrelevantscaling corrections for L ≥ ν = 2 . E obcc and ν = 2 . E pbcc . For a wider range, L ≥
32, power-law corrections to scaling need to beincluded, Γ (cid:48) ( E c , L ) = Γ (cid:48) c (1 + aL − y ) L /ν . This yields ν = 2 . y = 1 . ν = 2 . y = 1 . E obcc and E pbcc , respectively.In addition to the simple scaling analysis, we also perform fits of sophisticatedscaling functions Γ ( x r L /ν , x i L − y ), expanded in terms of relevant and irrelevant scal-ing field, x r L /ν and x i L − y [16]. We consider a large collection of such fits based onvarious subsets of the data and different fit expansions. The results of these fits showfluctuations similar to the results presented above. Hence, whereas the compact fitsgive robust estimates of ν , they do not give a reliable estimate of y , systematicallyaffecting E c , and Γ c . ill be inserted by the editor 5 Table 1.
Critical parameters obtained by means of the tight-binding lattice (TBL) and theCC network model (CCNM) for systems in topology of a cylinder (pbc) and a strip (obc).TBL, pbc [5] CCNM, pbc [2] TBL, obc (current) CCNM, obc [17] ν y (cid:46) . Γ c In summary, we have investigated the IQH transition in the lowest Landau band in astrip geometry with open boundary conditions for a microscopic model of electrons. Incontrast to cylindrical systems, edge states lead to a transition between an extendedand a localized phase. Table 1 compares the critical parameters of the IQH transitionfor the tight-binding model and the CC model for both cylinder and strip geometries.Interestingly, literature values of the irrelevant exponent y seem to have a strongdependence on the geometry. Whereas y is very small in cylinders, it is significantlyhigher ( y (cid:38)
1) for strips. Does this imply that strong but shorter-ranged boundarycorrections are dominant at the current system sizes whereas longer-ranged bulk cor-rections dominate asymptotically, or do bulk corrections vanish in strip geometry? Inthe current model, the estimate of y is strongly correlated with the critical energy;a straightforward analysis yields a critical energy marginally different from the bulkvalue as well as a larger y . However, assuming the bulk critical value to be valid, weobserve a significant better agreement of Γ c with the result of the open-boundary CCmodel investigation.The main message of the present paper is, however, that the estimate of thelocalization length exponent ν is very robust. Combining statistical and systematicerrors, we estimate ν = 2 . ν = 2 . ν ≈ . This work was supported by the NSF under GrantNos. DMR-1506152 and DMR-1828489.
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