Finite Size Scaling in Quantum Hallography
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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.
