Geometrically disordered network models, quenched quantum gravity, and critical behavior at quantum Hall plateau transitions

Abstract

Recent results for the critical exponent of the localization length at the integer quantum Hall transition differ considerably between experimental (νexp ≈ 2.38) and numerical (νCC ≈ 2.6) values obtained in simulations of the Chalker-Coddington (CC) network model. The difference is at least partially due to effects of the electron-electron interaction present in experiments. Here we propose a mechanism that changes the value of ν even within the single-particle picture. We revisit the arguments leading to the CC model and consider more general networks with structural disorder. Numerical simulations of the new model lead to the value ν ≈ 2.37. We argue that in a continuum limit the structurally disordered model maps to free Dirac fermions coupled to various random potentials (similar to the CC model) but also to quenched two-dimensional quantum gravity. This explains the possible reason for the considerable difference between critical exponents for the CC model and the structurally disordered model. We extend our results to network models in other symmetry classes.

Random network models with variable disorder of geometry

“…Hence, the average over randomness of the saddle points leads to the average over all configurations of the curved space [41,42], with yet to be determined functional measure. The field, which is characterizing different surfaces and by use of which one can ensure reparametrization invariance of the And, as it appeared [40], by taking equal probability 1/3 for each of the three nodes (complete reflection, complete transmission, regular scattering), the localization length index becomes ν = 2.37 ± 0.011, very close to the experimental value. In passing, we like to remark that recently the problem of the fractional quantum Hall effect on arbitrary gravitational background has attracted considerable interest [43][44][45] .…”

“…2). To overcome this difficulty, following [40] we take for every open node either t or r to be equal to ε…”

Integer quantum Hall transition on a tight-binding lattice

“…Our results suggest that the discrepancy between the experimental and theoretical values of ν cannot be attributed to a too regular structure of the semiclassical CC network model [48]. Why did the modified CC network [29], the Chern number calculation [31], and the study of the IQH transition in the presence of δ impurities [30] lead to different critical behavior with smaller ν? The latter two investigation used linear system sizes up to about 100L B , much smaller than our largest sizes of more than 600L B .…”

“…We then present our results and compare them with those of Refs. [29] (1) in the clean case (W = 0). For Φ = 0 and 1, the graph shows a single band from E = −4 to 4.…”

Many-body localization in Landau-level subbands

“…The realization that the Hall conductance is robust to impurities[1] paved the way for our understanding of the integer quantum Hall plateau transition as a problem of electron localization in a system with a diverging localization length protected by a topological invariant [2][3][4]. Even after decades of research, several aspects remain unclear, including the precise value of the localization length critical exponent [5][6][7], the agreement between single-particle numerics and experimental measurements [8][9][10], and the nature of the effective field theory at the critical point [11,12].The last decade has also seen burgeoning interest in many-body localization (MBL) [13][14][15][16][17][18][19][20], a generalization of Anderson localization to highly excited eigenstates of interacting many-body systems. However, the two fields have remained largely separated.…”