Physics Letters A 378 (2014) 262–265

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Physics Letters A

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Quantum critical Hall exponents

C.A. Lütken a,b,∗, G.G. Ross b,c

a Theory Group, Department of Physics, University of Oslo, Norwayb CERN, CH-1211 Geneva 23, Switzerlandc Rudolf Peierls Centre for Theoretical Physics, Department of Physics, University of Oxford, United Kingdom

a r t i c l e i n f o a b s t r a c t

Article history:Received 18 October 2013Accepted 1 November 2013Available online 7 November 2013Communicated by V.M. Agranovich

Keywords:Quantum Hall effectQuantum critical pointsCritical exponentsUniversality class

We investigate a finite size “double scaling” hypothesis using data from an experiment on a quantumHall system with short range disorder [1–3]. For Hall bars of width w at temperature T the scalingform is w−μT −κ , where the critical exponent μ ≈ 0.23 we extract from the data is comparable to themulti-fractal exponent α0 −2 obtained from the Chalker–Coddington (CC) model [4]. We also use the datato find the approximate location (in the resistivity plane) of seven quantum critical points, all of whichclosely agree with the predictions derived long ago from the modular symmetry of a toroidal σ -modelwith m matter fields [5]. The value ν8 = 2.60513 . . . of the localisation exponent obtained from the m = 8model is in excellent agreement with the best available numerical value νnum = 2.607 ± 0.004 derivedfrom the CC-model [6]. Existing experimental data appear to favour the m = 9 model, suggesting that thequantum Hall system is not in the same universality class as the CC-model. We discuss the reason thismay not be the case, and propose experimental tests to distinguish between the two possibilities.

© 2013 Elsevier B.V. All rights reserved.

One of the unsolved problems in the quantum Hall effect isto understand the critical behaviour of the delocalisation transi-tion between Hall plateaux. This includes finding the position ofquantum critical points (⊗) in the resistivity plane (the upper halfplane spanned by the Hall resistivity ρH and the direct resistivityρD � 0), as well as determining the critical exponents characteris-ing the universality class to which the system belongs. Progress onthis problem has been slow, in part because of the paucity of ex-perimental data. We are only aware of two experiments that candirectly measure the correlation (localisation) length exponent ν[1,7]. It has also emerged that numerical simulations are sensi-tive to sub-leading corrections that were neglected in earlier work,leading to a substantial change in the value of νnum.

As the temperature T drops into the quantum regime, tem-perature scaling has been observed over two decades in somequantum Hall devices, but this must eventually stop at a finitetemperature Ts > 0, because the inelastic scattering length de-pends on temperature. When the sample is cold enough this lengthexceeds the sample size w , temperature scaling “saturates”, andthe response functions become independent of T (compare Fig. 1).The first experiment to probe this scenario measured the maxi-mum slope of the Hall resistivity ρ ′

H = max(dρH/dB), and also thewidth �B D ∼ 1/ρ ′

H between inflection points of ρD , in a GaAs–AlGaAs heterostructure etched with several Hall bars of different

* Corresponding author at: Theory Group, Department of Physics, University ofOslo, Norway. Tel.: +47 22 85 41 68; fax: +47 22 85 64 22.

E-mail addresses: lutken@fys.uio.no, lutken@cern.ch (C.A. Lütken).

0375-9601/$ – see front matter © 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.physleta.2013.11.001

sizes w [7]. Above the saturation point (T > Ts) they found thatρ ′

H = uT −κ , and in this material the unsaturated data collapsedto a single line, corresponding to a w-independent prefactor u,similar to our Fig. 1(b). However, the slope κ of this line is notuniversal, as it depends on the amount and type of disorder, thetransition and other details. A scaling argument [7] suggests thatthe saturation value ρ ′

Hs is directly related to ν by ρ ′Hs ∼ w1/ν .

This exponent can also be obtained indirectly from the scaling rela-tion κνz = 1 by using Ts ∼ w−z to find z, but if possible ν shouldbe found directly from the data since it is difficult to obtain precisevalues of Ts . Using the direct method a remarkable degree of uni-versality was found in [7], with ν = 2.3±0.1, for different samplesand for a number of distinct Hall transitions.

Subsequent work [1–3] has shown that the type of disor-der determines if the temperature scaling exponent κ is uni-versal. Fig. 1(a) shows our reconstruction of data obtained froman AlGaAs–AlGaAs heterostructure suitably doped so that the dis-order potential is dominated by short range fluctuations [1,2]. Inthis case a reasonably universal value κ = 0.42 ± 0.01 is obtained.

However, this universality appears to come at a price. Thehigh-temperature data no longer collapse, because the prefactor udepends on the geometry of the Hall bar through w , and conse-quently ν cannot be read off directly from the saturation data. Thevalue ν = 2.38 (no error bars) given in [1] was obtained indirectlyby using the scaling relation ν = 1/κz, with κ = 0.42 and z = 1,neither of which we find (below) to be the best fit to the full setof data.

We propose a way to recover the direct determination ofν , which follows from the observation that ρ ′ ∼ wτ , where

Hs

C.A. Lütken, G.G. Ross / Physics Letters A 378 (2014) 262–265 263

Fig. 1. (Colour online.) (a) Reconstruction of data for the maximum slope of theHall resistivity ρ ′

H (T ) from [1] (w = 0.1,0.5,2.5 mm) and [2] (w = 0.2,0.32 mm).Horizontal and diagonal lines are fits to the data, whose intersection defines the sat-uration temperature Ts (vertical lines). Above saturation ρ ′

H scales with exponent κgiven in the inset table. Below saturation ρ ′

H takes constant values ρ ′Hs . (b) Simul-

taneous fit of all data from (a) above saturation to the scaling form ρ̃ ′H . (c) All data

collapse to the dimensionless function λ if temperature is measured in Ts-units.

τ = 0.22 ± 0.02 is obtained directly from the data, as shown inFig. 2(a). This can be recast as a finite size scaling hypothesis forthe prefactor:

u(w) ∼ w−μ (μ > 0), (1)

where μ = ν−1 −τ is a new exponent, hopefully at least as univer-sal as κ is. If this is true we should find that the unsaturated data

Fig. 2. (a) Best fit of the saturation data ρ ′Hs ∼ wτ from Fig. 1(a), and ρ̃ ′

Hs ∼ w1/ν

from Fig. 1(b). (b) The “effective exponent” νeff(σD ) for the m = 8 model, with σH =5/2[e2/h]. Inset: the 2 ⊗ 3 transition in the σ -plane, with T -driven RG flow linesderived from data at T = 31,114,510 mK (blue dots) [3]. (For interpretation of thereferences to colour in this figure legend, the reader is referred to the web versionof this article.)

(T > Ts) collapse to a single w-independent temperature scalingfunction:

ρ̃ ′H = (w/wmax)

μρ ′H ∼ T −κ , (2)

for a single universal value of the exponent μ. We have chosen tohold the data for the largest Hall bar (wmax = 2.5 mm) fixed, toserve as a reference. Fig. 1(b) shows the best global fit of all data[1,2] above saturation to this scaling form. The data below satura-tion do not collapse, and in fact move further apart. We can nowextract ν directly from ρ̃ ′

Hs ∼ w1/ν , compare Fig. 2(a), as was donein the first experiment [7]. We find that κ = 0.428±0.004 and μ =0.230 ± 0.004 gives the best fit to the 71 unsaturated data points,with 95% confidence intervals (0.421,0.436) and (0.222,0.238).The double scaling property of ρ ′

H allows us to collapse all datato the form λ(t = T /Ts) ∝ wμT κ

s ρ ′H shown in Fig. 1(c).

Our value for κ is higher, and the error is substantially smaller,than the value 0.42 ± 0.01 reported in [1], and remains essen-tially the same if we omit μ as a fitting parameter. The reasonfor this discrepancy is presumably that we are using the wholedata set simultaneously. The average of the slopes computed in-dependently for each data set (w = 0.1,0.2,0.32,0.5,2.5 mm) isκ = 0.427 ± 0.011 ≈ 0.43 ± 0.01, but the average of the slopescomputed independently for the sub-set of data published in [1](w = 0.1,0.5,2.5 mm) is κ = 0.419±0.007 ≈ 0.42±0.01. Our bestfit to the scaling form Ts ∼ w−z is z = 1.05±0.05, which gives (us-ing our κ and κνz = 1) ν = 2.23 ± 0.11, consistent with the valueν = 2.24 ± 0.09 obtained directly using the double scaling hypoth-esis (compare Fig. 2(a)).

The first experiment [7], on a sample with long range disorderpotential where it was found that κ is not universal, has μ = 0because the data above saturation collapse. With only two exper-iments at our disposal we cannot conclude that μ is a universalexponent whose value can be used to label universality classes, but

264 C.A. Lütken, G.G. Ross / Physics Letters A 378 (2014) 262–265

Fig. 3. Comparison of experimental data (red ⊗) from [3] with the modular values(blue ⊗) that lie on the orange circle. Solid red circles are phase boundaries, dashedblue circles are separatrices for the flow, and thin black curves are flow lines. Themodular quantum critical point between the plateaux at 1/n and 1/(n + 1) is ⊗n =(2n + 1 + i)/(2n(n + 1) + 1)). (For interpretation of the references to colour in thisfigure legend, the reader is referred to the web version of this article.)

that is a possibility. Furthermore, a finite size scaling exponent μmfis expected to appear in systems with multi-fractal wave-functions,which includes the CC-model [4] and systems with short range dis-order (SRD) [8–10]. Long range interactions breaks the self-similarscale invariance giving rise to multi-fractals, which could explainwhy μ is different in [7] and [1]. Various groups [6,11–16] workingon the CC-model have reported a range of values of the candidateexponent, μmf = α0 − 2 ∈ (0.22–0.26). We cannot be more pre-cise, since the results from different groups are inconsistent giventhe very small errors they quote. Presumably this means that thereare large systematic errors still to be identified. Therefore, even ifμSRD = μmf, we do not know if the experimental and numericalvalues agree.

Turning now to the interpretation of these results, in Fig. 3 wehave used data from [3] to locate seven quantum critical points(red ⊗), shown superimposed on the modular phase diagram weobtained more than two decades ago [5]. The experimental val-ues are obtained from the temperature independent crossing pointof the Hall curves, and the maximum of ρD at the lowest tem-perature, which is closest to the scaling limit. The modular criticalpoints (blue ⊗) are partially eclipsed by the experimental data,i.e., they are all within about one standard deviation, which weestimate to be roughly the size of the plot-markers used in thisdiagram.

Encouraged by this agreement we have investigated a simplefamily of toroidal σ -models that possess the observed modularsymmetry ΓT = Γ0(2) [5]. This is an effective field theory with mmassless fermions without conventional interactions, but becausethey are constrained to a torus they are not free. They are also“twisted” by non-trivial boundary conditions that selects a specificspin-structure on the toroidal target space, and which mathemati-cally is equivalent to an interaction with a flat gauge potential. TheΓT -invariant partition function Z of this model is known [5].

All universal data are encoded in the renormalisation group(RG) potential C , and since both Z and C count degrees of freedomat RG fixed points, we suspect that they are intimately related.Z detects all degrees of freedom in the system, both local andglobal, but since the central charge c⊗ (of the conformal field the-ory at an RG fixed point ⊗ = σ⊗) only counts local degrees offreedom, we expect C to be given by a function that does notcontain information about the global zero-modes contributing tothe vacuum degeneracy δ (which depends on the twisting). Vac-uum attributes are most easily extracted by considering the weakcoupling limit (σ → i∞), and the asymptotic form of the twistedpartition function is Zi∞ = δ|q|αci∞ , with q = e2π iσ . The asymp-totic limit of the potential gives the central charge of a twistedmassless field:

C −→ ci∞ = ln Z ′

α ln |q| = limσ→i∞

(ci∞

ln Z ′

ln Z ′)

, (3)

i∞

Fig. 4. (a) From left to right: recent numerical results for the CC-model from [12–16]and [6] (in box), experimental values from [7] (•) and [1] (◦), and our result us-ing all data from [1–3] (2). (b) The exponents νm (bullets) obtained from ourm-fermion model, compared to the best available numerical value νnum (horizontalblue line) [6], and the experimental value νexp (horizontal red line). The error onνnum is smaller than the thickness of the blue line in the main diagram. The boxedregion (m = 8) is magnified in the inset. (The inset is not visible on the main dia-gram, so the box is ten times bigger.) (For interpretation of the references to colourin this figure legend, the reader is referred to the web version of this article.)

where the prime means omitting zero-modes, ln Z ′ = ln Z − ln δ.We see that the asymptotic limit q → 0, where we can use con-formal field theory to extract the central charge, is self-consistent.These considerations suggest that the C-function near any quan-tum critical point ⊗ takes the form (3) (with i∞ → ⊗).

After removing the vacuum degeneracy of each twisted fermionby subtracting ln δ = ln 2 from ln Z , and recalling that each fermionhas c = 1/2, we obtain

Z ′ =∣∣∣∣η(2σ)

η(σ )

∣∣∣∣2m

, C = m

2

�ϕ(σ )

�ϕ(⊗), (4)

where the ΓT -invariant potential ϕ(σ ) = lnη(2σ)− lnη(σ ) is builtfrom Dedekind’s η-function. It contains the desired universal dataassociated with the quantum critical point ⊗, and in order to ex-tract this information we expand the potential at σ⊗ = (1 + i)/2:

ϕ(σ ) = − ln 2i

4+ π2G4

6(σ − σ⊗)2 + · · · (5)

where G is Gauss’ constant. Since ∂ϕ(⊗) = 0, ⊗ and all itsΓT -images are critical points of ϕ , and since we know from theC-theorem that ∂ϕ is essentially the β-function, we have locatedthe quantum critical points of our model.

Zamolodchikov’s parameter space metric g [17] and the Weil–Petersen metric coincide to leading order for Calabi–Yau spaces[18]. Since the torus is such a space we should therefore use thehyperbolic metric, which at ⊗ takes the value g⊗ = 1/( σ⊗)2 = 4.From the C-theorem [17] and (5) we obtain the β-function

βσ ⊗−→ − 1∂C = m

∂ϕ = 1(σ − σ⊗) + · · · , (6)

12g⊗ 48 ln 2 νm

C.A. Lütken, G.G. Ross / Physics Letters A 378 (2014) 262–265 265

where νm = 8νtor/m, and we have defined the toroidal exponentνtor = 18 ln 2/π2G4 = 2.6051265833 . . . ≈ 21/8. Our ignorance isnow parametrized by the number m of fermions in the model.

Fig. 4(a) compares ν8 (blue line) and ν9 (red line) to numer-ical and experimental values of ν . Fig. 4(b) focuses on the bestavailable numerical value νnum = 2.607 ± 0.004 [6] (horizontalblue line) for the CC-model [4]. Our model requires that this linepasses near one of the bullets, and it does in fact appear to bi-sect the m = 8 case. The inset contains this intersection regionhugely magnified, showing agreement at the per mille level, withm = 7.994 ± 0.012 ≈ 8 (c = 4).

The neighbouring m = 9 model has ν9 ≈ 2.32 ≈ 7/3, which isconsistent with the experimental value νexp = 2.3 ± 0.1. This ishardly conclusive, but if νexp and νnum really are different, thenthe quantum Hall system is not in the same universality classas the CC-model. However, the fact that numerical studies haveshown that there are substantial non-leading corrections suggeststhat there may be a large systematic error when comparing withexperiment. To check this we note that 1/ν measures the relevantcurvature of C(⊗), and in Fig. 2(b) we have plotted the inverse ofthis curvature, νeff, along the ridge σH = 5/2 in the m = 8 model(νeff(⊗) = ν8) (the change of the curvature in the σH -direction isof higher order, and can therefore be neglected near ⊗ = (5+ i)/2).Evaluated at the experimental value of the critical point taken fromFig. 3 (red ⊗ on far right) we find νeff ≈ 2.23 (15% lower than thecritical value) in good agreement with the experimental value. Asthe temperature is decreased we expect σD to increase, taking itcloser to the modular value, suggesting that current experimentsare not yet cold enough to give a reliable estimate of the true scal-ing exponent ν .

It should be feasible to improve this situation considerably, bothby running at lower temperatures and by comparing to the theo-retical prediction evaluated at the experimental value of σD . Testsof the double scaling hypothesis would also be considerably im-proved by using many more Hall bars and pursuing the flat part

of ρ ′H to lower temperatures (T � Ts). If μ is indeed universal,

this will also give a significantly sharper value of the localisationexponent ν , which is needed for understanding universality, andperhaps also multi-fractality, in the quantum Hall effect.

Acknowledgements

One of us (GGR) would like to thank the Leverhulme founda-tion for support of this research through the award of an emeritusfellowship.

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