[Paper Review] Finite Size Scaling in Quantum Hallography

Allan Bayntun, C. P. Burgess|arXiv (Cornell University)|Dec 16, 2011

Black Holes and Theoretical Physics39 references3 citations

TL;DR

This paper extends the AdS/QHE model to incorporate finite-size effects in quantum Hall systems, showing that the observed transition from power-law temperature dependence to temperature-independent Hall resistance at low temperatures arises naturally from gravity duals with a brane tension that shifts the crossover scale. It predicts $ T_s \propto 1/L $ or $ T_s \propto 1/L^5 $, reconciling the measured $ p = 0.42 \pm 0.01 $ with $ z = 5 $ in the infrared while allowing $ z = 1 $ in the ultraviolet, and identifies a first-order transition analogous to the Chandrasekhar limit in stellar collapse.

ABSTRACT

At low temperatures observations of the Hall resistance for Quantum Hall systems at the interface between two Hall plateaux reveal a power-law behaviour, dR_xy/dB ~ T^(-p) (with p = 0.42 +/- 0.01); changing at still smaller temperatures, T < T_s, to a temperature-independent value. Experiments also show that the transition temperature varies with sample size, L, according to T_s ~ 1/L. These experiments pose a potential challenge to the holographic AdS/QHE model recently proposed in arXiv:1008.1917. This proposal, which was motivated by the natural way AdS/CFT methods capture the emergent duality symmetries exhibited by quantum Hall systems, successfully describes the scaling exponent p by relating it to an infrared dynamical exponent z with p = 2/z. For a broad class of models z is robustly shown to be z = 5 in the regime relevant to the experiments (though becoming z = 1 further in the ultraviolet). By incorporating finite-size effects into these models we show that they reproduce a transition to a temperature-independent regime, predicting a transition temperature satisfying T_s ~ 1/L or ~ 1/L^5 in two separate regions of parameter space, even though z = 5 governs the temperature dependence of the conductivity in both cases. The possibility of a deviation from naive z = 5 scaling arises because the brane tension introduces a new scale, which alters where the transition between UV and IR scaling occurs, in an L-dependent way. The AdS/CFT calculation indicates the two regimes of temperature scaling are separated by a first-order transition, suggesting new possibilities for testing the picture experimentally. Remarkably, in this interpretation the gravity dual of the transition from temperature scaling to temperature-independent resistance is related to the Chandrashekar transition from a star to a black hole with increasing mass.

Motivation & Objective

• To reconcile experimental observations of a temperature-independent Hall resistance at low temperatures with the AdS/QHE model’s prediction of $ p = 0.42 \pm 0.01 $.
• To explain the sample-size dependence of the crossover temperature $ T_s \propto 1/L $ within the holographic framework.
• To resolve the apparent conflict between $ z = 5 $ (from $ p = 2/z $) and experimental evidence for $ z = 1 $ in the same energy regime.
• To show that finite-size effects, mediated by brane tension, alter the UV/IR crossover scale in a system-size-dependent way.
• To identify the gravity dual of the transition as a first-order phase transition analogous to the Chandrasekhar limit in stellar collapse.

Proposed method

• Incorporates finite-size effects into the AdS/QHE model by introducing a brane tension that modifies the effective action and alters the location of the UV/IR crossover.
• Uses the AdS/CFT correspondence to compute the conductivity $ \sigma_{\theta\theta} $ in a black hole geometry with finite system size $ L $, deriving two distinct scaling regimes based on brane tension.
• Derives the temperature dependence of the conductivity in both the high- and low-tension limits, showing identical $ T^{-2/5} $ scaling in both cases.
• Evaluates the free energy of the stellar and black hole phases using on-shell gravity actions, with normalization ensuring a meaningful comparison at finite temperature.
• Identifies a first-order phase transition between the stellar (non-black hole) and black hole phases, with the transition temperature scaling as $ T_s \propto 1/L $ or $ T_s \propto 1/L^5 $ depending on the parameter regime.
• Relates the gravity dual of the quantum Hall transition to the Chandrasekhar limit, where increasing mass triggers collapse into a black hole.

Experimental results

Research questions

• RQ1 How does finite size affect the scaling of the Hall resistance in holographic quantum Hall systems?
• RQ2 Can the observed $ T_s \propto 1/L $ dependence of the crossover temperature be reproduced in the AdS/QHE framework?
• RQ3 Why does the measured $ p = 0.42 \pm 0.01 $ correspond to $ z = 5 $, while other experiments suggest $ z = 1 $, and is this consistent?
• RQ4 What role does brane tension play in shifting the UV/IR crossover scale in holographic models of quantum Hall systems?
• RQ5 Is the transition from temperature-dependent to temperature-independent resistance in the quantum Hall effect dual to a gravitational phase transition, such as the Chandrasekhar limit?

Key findings

• The model reproduces the experimentally observed $ T_s \propto 1/L $ scaling by introducing a brane tension that shifts the UV/IR crossover scale in an $ L $-dependent manner.
• Despite $ z = 5 $ governing the infrared scaling, the model predicts $ T_s \propto 1/L^5 $ in one parameter regime, while $ T_s \propto 1/L $ in another, both consistent with $ z = 5 $ in the relevant regime.
• The conductivity exhibits the same $ T^{-2/5} $ scaling in both finite-size regimes, confirming that the temperature dependence is robustly governed by $ z = 5 $.
• The transition between the power-law and temperature-independent regimes is identified as a first-order phase transition in the gravity dual, analogous to the Chandrasekhar limit in stellar collapse.
• The free energy difference between the stellar and black hole phases is computed using on-shell gravity actions, confirming the phase transition is thermodynamically favorable under appropriate normalization.
• The model resolves the apparent conflict between $ z = 5 $ (from $ p = 2/z $) and $ z = 1 $ (from other experiments) by showing that $ z = 1 $ arises in the UV, while $ z = 5 $ governs the IR regime where $ p $ is measured.