Geometry of random potentials: Induction of 2D gravity in Quantum Hall plateau transitions
Riccardo Conti, Hrant Topchyan, Roberto Tateo, Ara Sedrakyan
GGeometry of random potentials: Induction of 2D gravity in Quantum Hall plateautransitions
Riccardo Conti, Hrant Topchyan, Roberto Tateo, and Ara Sedrakyan Grupo de F´ısica Matem´atica da Universidade de Lisboa,Av. Prof. Gama Pinto 2, 1649-003 Lisboa, Portugal. Alikhanyan National Laboratory, Yerevan Physics Institute, Armenia Dipartimento di Fisica, Universit`a di Torino, and INFN,Sezione di Torino, Via P. Giuria 1, I-10125 Torino, Italy (ΩDated: August 28, 2020)Integer Quantum Hall plateau transitions are usually modeled by a system of non-interactingelectrons moving in a random potential. The physics of the most relevant degrees of freedom, theedge states, is captured by a recently-proposed random network model, in which randomness isinduced by a parameter-dependent modification of a regular network. In this paper we formulatea specific map from random potentials onto 2D discrete surfaces, which indicates that 2D gravityemerges in all quantum phase transitions characterized by the presence of edge states in a disorderedenvironment. We also establish a connection between the parameter in the network model and theFermi energy in the random potential.
Introduction.
The investigation of plateau transitions inthe Quantum Hall Effect (QHE) continues to be one ofthe most exciting research topic in modern condensedmatter physics. Much of the current interest is motivatedby the emergence of a similar type of physics in the con-text of topological insulators. The Quantum Hall plateautransition is in fact an example of a metal-insulator tran-sition (see [1] for a review) with the plateau region be-tween the Landau Levels (LLs) corresponding to the in-sulating phase where all the states are localized due tothe external magnetic field. The transition is a disorder-induced localization/delocalization transition of Ander-son type, characterized by a divergent localization length ξ at the critical point with localization index ν .In this paper we focus on QHE plateau transitions, whichare usually modelled by a system of non-interactingelectrons moving in a 2D random potential (RP) V ( r )characterized by a white-noise Gaussian distribution.Throughout the paper, we shall consider RPs with a fi-nite correlation length generated by N Gaussian sources,i.e. V ( r ) = N (cid:88) i =1 W i exp (cid:18) − | r − r i | σ (cid:19) , (1)where σ is the variance, r i is the position vector of the i − th source and the coefficients { W i } Ni =1 are randomlychosen in [ − W, W ], with W ∈ R . In such RP landscape,electrons are localized [2] and fill the Fermi sea which ac-tually consists of a collection of lakes with characteristicsize l , as displayed in FIG. 1. The delocalization of elec-trons is triggered by the presence of an external magneticfield B which does not change the total energy of the sys-tem but rearranges the eigenstates and forms LLs. Themechanism of delocalization can be intuitively explainedusing the following semi-classical picture. At large B ,the electrons in a LL are strongly localized and their FIG. 1: RP generated by N = 2500 Gaussian sources on atorus. Points mark maxima (red), minima (blue) and saddlepoints (green). The plane denotes the Fermi level. state corresponds to an orbital motion with small radius r ∼ /B . At the boundary of a lake, the orbital motioncombines with the reflection due to the potential givingrise to a precession motion along equipotential lines (cy-cloid). The electrons localized along the boundaries ofthe lakes form the so-called edge states. When an edgeelectron approaches a saddle point that is sufficientlyclose to the boundaries of two neighbour lakes, quantumtunnelling of the particle from one boundary to the otherbecomes sizable (see FIG. 2). Therefore, edge electronsmay either jump from one lake to another with a finiteprobability t , or continue to move along the boundaryof the original lake with probability r , with t + r = 1.The presence of such quantum scattering nodes at sad-dle points enables electrons to reach arbitrary distanceswith a finite probability and is at the origin of the lo- a r X i v : . [ c ond - m a t . d i s - nn ] A ug FIG. 2: Neighborhood of a saddle point (green dot) separatingtwo lakes (blue areas) in a RP. The cycloid represents themotion of edge states along the boundary of a lake. Theparameters r and t denote the reflection and transmissionprobabilities, respectively, while B is the magnetic field. calization/delocalization transition. Taking inspirationfrom this semi-classical picture, J. Chalker and P. Cod-dington (CC) [3] formulated a network model of quantumscattering nodes based on a regular lattice that is meantto provide an effective description of the physics of edgestates (the only relevant degrees of freedom in plateautransitions). Its generalization on a Kagome lattice wasproposed in [4] and a similar network model for the SpinQuantum Hall Effect (SQHE) was studied in [5, 6]. Nu-merical investigations of the localization length ξ at thecritical point, i.e. ξ ∼ ( t − t c ) − ν with t c = 1 / √
2, resultedin ν = 2 . ± .
62 for a regular lattice [7–9, 11, 12] and ν = 2 . ± .
046 for the Kagome lattice [4]. Both thesevalues are not compatible with the experimental value ν = 2 . ± .
06 measured for plateau transitions in theIQHE [13, 14]. A possible solution to fix the discrep-ancy was put forward in [15, 16] by considering randomnetworks (RNs), which should better account for the dis-order present in a RP. The numerical estimate obtainedin this framework ν = 2 . ± .
017 [15, 16] confirmsindeed a very good agreement with the experimental re-sult.In [15, 16] it was also argued that the randomness ofRNs leads to the appearance of 2D quantum gravity, inthe continuum limit. The primary objective of this paperis to show that 2D gravity is indeed emerging from theRPs framework, establishing a precise correspondencebetween RNs and RPs. Notice that quantum gravityis also involved in the understanding of Fractional QHE[17, 18] revealing the physics of Laughlin wave-function.In that context, the interaction between fermions is re-sponsible for the emergence of gravity in the bulk. In-stead, in the present paper gravity is related to the 1+1dimensional edge states, which originates from the RP.
Network models with geometric disorder.
Let us brieflyreview the construction of RNs carried on in [15, 16]. Themain idea is to generate randomness in a regular CC net-work making an extreme replacement, which consists in “opening” a scattering node in the horizontal (vertical)direction with probability p ( p ) setting t = 0 ( t = 1)(see FIG. 3), or leaving it unchanged with probability1 − p − p . In the following, we shall set p n = p = p to maintain statistical isotropy [15, 16]. Since in the RPpicture the scattering node simulates a saddle point andthe four squares surrounding it corresponds to an alter-nate sequence of maxima and minima (see FIG. 3), afterthe extreme replacement the scattering node becomes anhexagon containing a maximum (minimum) and two ad-jacent triangles each containing a minimum (maximum),as depicted in FIG. 3. Thus, starting from a regular FIG. 3: Top: “opening” of a scattering node in the horizontaland vertical directions. Bottom: result of the extreme re-placement on the network. Red, blue and green points markmaxima, minima and saddle points in the corresponding RPframework.
CC network where all the faces are quadrangles and ran-domly making the extreme replacement with probabil-ity p n , a tiling of the plane with n − gons is obtained.In [16] it was shown that in this type of RNs the lo-calization index has a non-trivial dependence on the re-placement probability p n , with a line of critical pointsfor p n ∈ [0 , / p n within the RP model. As we shall see, p n is connected with the height or Fermi energy level of edgestates in the RP landscape. Random potentials and discrete surfaces.
The RP (1) cor-responds to a 2D smooth surface characterized by N max maxima, N min minima, N sp saddle points (see FIG. 1)and with Euler characteristics χ = N min + N max − N sp , (2)according to Morse theory [19]. Connecting maxima andminima following the gradient of V ( r ) gives a uniquequadrangulation of the surface, namely a 2D discrete sur-face S made of v = N max + N min vertices, e edges and f = N sp quadrangular faces (see FIG. 4). Denoting by n i the connectivity of the i − th vertex, i.e. the num-ber of edges connected to it, the Euler characteristics χ = v − e + f of S can be expressed as2 πχ = v (cid:88) i =1 R ( n i ) , R ( n ) = π − n ) , (3)where, according to Gauss-Bonnet theorem, R ( n ) can beinterpreted as the discrete Gaussian curvature associatedto each vertex of S . Formula (3) follows immediatelyfrom the fact that e = 2 f = (cid:80) vi =1 n i , from which χ = v − e + f = v − f = (cid:80) vi =1 (4 − n i ).Notice that by construction each face of S contains ex-actly one saddle point. Therefore, connecting saddlepoints belonging to nearest neighbor faces of S resultsin a dual 2D discrete surface S ∗ , made of v ∗ = f ver-tices with connectivity 4, e ∗ edges and f ∗ = v n − gonalfaces, where n is the connectivity of the vertex of S ly-ing within each face of S ∗ (see FIG. 4). By duality, each n − gonal face of S ∗ carries a discrete Gaussian curvature R ( n ) and brings a local contribution to 2 πχ as describedin eq. (3). Following this procedure, to each RP is asso-ciated a 2D random discrete surface S or equivalently S ∗ ,which ultimately corresponds to a network model whereeither the connectivity of the sites or the number of sidesof the faces carries the discrete Gaussian curvature of thesurface. In the following, we will denote by S or S ∗ boththe discrete surface or the network associated to it. FIG. 4: Topography of a RP generated by N = 900 Gaussiansources placed on a torus. Points mark maxima (red), minima(blue) and saddle points (green). White and black lines arethe edges of S and S ∗ , respectively. Random potentials vs. Random networks.
To establishcontact between RPs and RNs, let us give a concreteexample. Consider a RP generated by N = L Gaus-sian sources evenly distributed on a regular square latticeof size L with unit spacing and doubly periodic bound-ary conditions, i.e. a torus. Let r i = ( x i , y i ) = ( i mod ( L ) , (cid:100) i/L (cid:101) ) be the position of the i − th source on FIG. 5: Networks associated to the truncated discrete sur-faces S c and S ∗ c . White and black lines are the links of S c and S ∗ c , respectively, while the areas highlighted in light bluecorrespond to the regions under the Fermi level. the lattice, where (cid:100)∗(cid:101) is the ceiling function. Then, theRP at the generic point r = ( x, y ) ∈ [0 , L − × [0 , L − V ( r ) = N (cid:88) i =1 (cid:88) n ∈ Z W i exp (cid:18) − | r − r i + n L | σ (cid:19) , (4)where the range of the summation index n = ( n x , n y ) isrestricted to {− , , } × {− , , } in the numerical sim-ulation. Eq. (2) implies that the critical points of (4) aresuch that N max + N min = N sp since χ = 0 for a torus.In FIG. 6, the distributions of critical points per unitheight of the potential are reported for a statistical sam-ple consisting of m = 45 simulations on square latticesof size L = 300 with W = 1 /
10 and σ = √
2. Since atfinite W and σ the potential V ( r ) is bounded, the distri-butions are defined on a finite support, also in the limit L → ∞ . However, in the case under consideration, theyare well approximated by Gaussian distributions with ex-pectation values µ max = − µ min = 0 . µ sp = 0 andstandard deviations σ max = σ min = σ sp = 0 . S or equivalently S ∗ can be uniquely as-sociated to (4) (see FIG. 4). The introduction of a Fermilevel c ∈ R induces a truncation of both S and S ∗ since afraction of the vertices stays inside the Fermi lakes. Theresulting truncated surface S c or S ∗ c is such that the dis-connected portions of the original surface lying under theFermi level, which may contain an arbitrary number ofvertices, can be replaced with single isolated vertices asdisplayed in FIG. 5. This operation is indeed physicallymeaningful since the scattering of edge states is not af-fected by the bulk of Fermi lakes. Therefore, varying theFermi level produces a flow within the space of discretesurfaces parametrized by c . FIG. 6: Number of maxima (∆ N max ), minima (∆ N min ) andsaddle points (∆ N sp ) in the height range [ h, h + ∆ h ], with∆ h = 1 / A of the lattice. The statis-tical sample consists of m = 45 simulations on square latticesof size L = 300, i.e. A = mL , with W = 1 /
10 and σ = √ S ∗ c (red squares) for various values of theparameters p n and p c which minimize the SSE. Observe that the truncation of the discrete surfacescaused by the Fermi level corresponds to the removalof sites in the associated networks and, consequently, tothe emergence of polygonal faces with a larger numberof sides (see FIG. 5). The net effect of this process isvery reminiscent of that induced in the CC network bythe surgery performed in [15, 16] leading to RNs. Forthis reason, we expect the replacement probability p n ofRNs to be somehow related to the Fermi level in RPs.For the purposes of comparing these two models, it isfirst necessary to restore particle-hole duality in the RPframework since RNs, which tend to Dirac fermion mod-els in the continuum limit, possess it. To this aim, therange of energies accessible to fermions in the RP is re- FIG. 8: Correspondence between the replacement probabil-ity p n and p c obtained searching for the best match betweenthe two curvature distributions. The inset plot gives the es-timated SSE as a function of p c . stricted from [ c, + ∞ [ to the symmetric interval [ −| c | , | c | ]and the complementary interval [ −∞ , −| c | ] ∪ [ | c | , + ∞ [ islabelled as “non-valid” region.Secondly, notice that the replacement probability p n inthe RN framework is equivalent to half the ratio betweenthe number of removed scattering nodes, i.e. saddlepoints, and the total number of them. This observationsuggests that half of the ratio between the number of sad-dle points in the non-valid region and the total number ofthem, denoted by p c , is the most appropriate parameterof the RP to be put in relation to p n . In this respect,the observable taken into consideration is the distribu-tion of discrete Gaussian curvatures R of the polygonstiling both the RN and the dual network S ∗ c to varyingof p n and p c , respectively. Denoting by n n ( R ) and n c ( R )the number of polygons with curvature R in the RN andin S ∗ c , respectively, the defining criterion for the associ-ation between p n and p c is the minimization of the sumof squared errors, SSE = (cid:80) R ∈ N ( n n ( R ) − n c ( R )) . InFIG. 7 distributions of curvatures in both the RN and S ∗ c are compared for some values of p n and p c that min-imize the SSE. The statistical samples consist of morethan 50 RN simulations on a 100 × p n ∈ [0 , /
2] and 45 RP simulations on a squarelattice of size L = 300 for each value of c ∈ [0 , / p n ↔ p c , as reported in FIG.8. Conclusions.
There are strong evidences that the field-theory description of plateau transitions corresponds toa model of fermions interacting with random gauge andscalar potentials and also with structurally-disordered ge-ometry. Indicating that, in the scaling limit, localizationtransitions of this type are correctly described by matterfields coupled to 2D quantum gravity. Starting from arandom potential model, we have explicitly constructeda map onto the 2D disordered graphs S c and S ∗ c depend-ing on the Fermi-level. Thus, observing the appearanceof the basic ingredient of random network models [15, 16]for Quantum Hall plateau transitions and giving an inter-pretation of the replacement probability in term of theFermi energy. S c and S ∗ c , being quadrangulations and n − gon tiling’s of the plane, have a straightforward inter-pretation as discrete random surfaces, explicitly showingthe emergence of 2D gravity. As discussed also in [15],the notion of functional measure of random surfaces re-mains an open problem. From the current analysis, itappears that the distribution of Gaussian curvatures onthe random surface associated with the random poten-tial coincides with the corresponding distribution in therandom network model, suggesting that the functionalmeasure of random surfaces can be defined in terms ofthe measure of random potentials. In conclusion, we re-vealed a deep link between random potentials in Ander-son localization problem and 2D curved surfaces, wherethe edge states responsible for plateau transitions live. Acknowledgments.