Dynamical scaling analysis is theoretically performed for the ac (optical) Hall conductivity ${\ensuremath{\sigma}}_{xy}({\ensuremath{\epsilon}}_{F},\ensuremath{\omega})$ as a function of Fermi energy ${\ensuremath{\epsilon}}_{F}$ and frequency $\ensuremath{\omega}$ for the two-dimensional electron gas and for graphene. In both systems, results based on exact diagonalization show that ${\ensuremath{\sigma}}_{xy}({\ensuremath{\epsilon}}_{F},\ensuremath{\omega})$ displays a well-defined dynamical scaling, for which the dynamical critical exponent as well as the localization exponent are obtained. A crossover from the dc-like behavior to the ac regime is identified. The dynamical scaling analysis has enabled us to quantify the plateau in the ac Hall conductivity previously obtained and to predict that the plateaux structure in ac is robust enough to be observed in the terahertz regime.Dynamical scaling analysis is theoretically performed for the ac (optical) Hall conductivity ${\ensuremath{\sigma}}_{xy}({\ensuremath{\epsilon}}_{F},\ensuremath{\omega})$ as a function of Fermi energy ${\ensuremath{\epsilon}}_{F}$ and frequency $\ensuremath{\omega}$ for the two-dimensional electron gas and for graphene. In both systems, results based on exact diagonalization show that ${\ensuremath{\sigma}}_{xy}({\ensuremath{\epsilon}}_{F},\ensuremath{\omega})$ displays a well-defined dynamical scaling, for which the dynamical critical exponent as well as the localization exponent are obtained. A crossover from the dc-like behavior to the ac regime is identified. The dynamical scaling analysis has enabled us to quantify the plateau in the ac Hall conductivity previously obtained and to predict that the plateaux structure in ac is robust enough to be observed in the terahertz regime.

arXiv:1004.4778Physical Review BAug 9, 201014 citations

Screening-controlled dynamical criticality in the quantum Hall regime

Tanima Chanda, Simrandeep Kaur, Anantbir Virk +5 more

At continuous electronic phase transitions, Coulomb interactions can modify the relation between length, energy, and temperature, but experimentally disentangling their effects on spatial versus dynamical criticality has remained difficult, since finite-temperature scaling alone measures only the combined exponent $κ= 1/(zγ)$ . Here, we introduce two advances that resolve this limitation. First, by combining temperature scaling with independent current scaling, we separately extract the dynamical exponent $z$ and the localization-length exponent $γ$ at the quantum Hall plateau transition -- rather than inferring one from an assumed value of the other. Second, using dual-graphite-gated graphene devices in which the effective Coulomb interaction range is tuned geometrically by the ratio of the magnetic length $l_B$ to the graphite-gate distance $d$ , we track this separation across both screened and unscreened interaction regimes within the same device platform. Temperature scaling gives $κ\simeq 0.21$ in the screened regime and $κ\simeq 0.41$ in the unscreened regime; combining this with current scaling reveals that screening changes $z$ from $\simeq 1$ in the unscreened regime to $\simeq 2$ in the screened regime. In contrast, $γ$ remains close to $2.4$ throughout. Our results establish that gate-controlled screening selectively modifies the interaction-dependent dynamical sector of the quantum Hall transition, leaving the localization-length exponent $γ$ unchanged within experimental uncertainty. More broadly, this work establishes geometric screening as a versatile tool for controlling interactions and disentangling interaction and disorder effects in correlated two-dimensional systems, including fractional quantum Hall states, moiré materials, and other strongly localized electronic phases.At continuous electronic phase transitions, Coulomb interactions can modify the relation between length, energy, and temperature, but experimentally disentangling their effects on spatial versus dynamical criticality has remained difficult, since finite-temperature scaling alone measures only the combined exponent $κ= 1/(zγ)$. Here, we introduce two advances that resolve this limitation. First, by combining temperature scaling with independent current scaling, we separately extract the dynamical exponent $z$ and the localization-length exponent $γ$ at the quantum Hall plateau transition -- rather than inferring one from an assumed value of the other. Second, using dual-graphite-gated graphene devices in which the effective Coulomb interaction range is tuned geometrically by the ratio of the magnetic length $l_B$ to the graphite-gate distance $d$, we track this separation across both screened and unscreened interaction regimes within the same device platform. Temperature scaling gives $κ\simeq 0.21$ in the screened regime and $κ\simeq 0.41$ in the unscreened regime; combining this with current scaling reveals that screening changes $z$ from $\simeq 1$ in the unscreened regime to $\simeq 2$ in the screened regime. In contrast, $γ$ remains close to $2.4$ throughout. Our results establish that gate-controlled screening selectively modifies the interaction-dependent dynamical sector of the quantum Hall transition, leaving the localization-length exponent $γ$ unchanged within experimental uncertainty. More broadly, this work establishes geometric screening as a versatile tool for controlling interactions and disentangling interaction and disorder effects in correlated two-dimensional systems, including fractional quantum Hall states, moiré materials, and other strongly localized electronic phases.

arXiv:2607.09244Jul 10, 20260 citations

Scaling to Non-Scaling Transition of Quantum Hall Systems

X. R. Wang, Gang Xiong, S. D. Wang +2 more

Bulletin of the American Physical SocietyMar 1, 20040 citations

Quantum geometric localization length and localization criticality in an ideally flat Chern band

Xu-Cheng Wang, Yang Qi

We propose that the localization length in an isolated, ideally flat Chern band is set by quantum geometry. We explore the corresponding localization transition and its critical scaling by applying transfer matrix calculations in the maximally localized hybrid Wannier basis, whose spatial spread is exactly characterized by a quantum geometric length. Remarkably, upon tuning the quantum metric of the Chern band, we observe a crossover from a universal regime controlled by the Dirac fixed point to a non-universal regime with continuously varying critical exponents. Within the universal regime, the localization length exhibits a pronounced linear dependence on the quantum geometric length, supporting its quantum geometric nature. These findings provide a novel quantum geometric perspective on the localization in quantum Hall systems such as twisted moiré superlattices, and shed new light on the long-standing controversy over the criticality of the integer quantum Hall transition.We propose that the localization length in an isolated, ideally flat Chern band is set by quantum geometry. We explore the corresponding localization transition and its critical scaling by applying transfer matrix calculations in the maximally localized hybrid Wannier basis, whose spatial spread is exactly characterized by a quantum geometric length. Remarkably, upon tuning the quantum metric of the Chern band, we observe a crossover from a universal regime controlled by the Dirac fixed point to a non-universal regime with continuously varying critical exponents. Within the universal regime, the localization length exhibits a pronounced linear dependence on the quantum geometric length, supporting its quantum geometric nature. These findings provide a novel quantum geometric perspective on the localization in quantum Hall systems such as twisted moiré superlattices, and shed new light on the long-standing controversy over the criticality of the integer quantum Hall transition.

arXiv:2608.08042Aug 8, 20260 citations

Supersymmetry and localization in the quantum Hall effect

Jané Kondev, John Bradley Marston

arXiv:cond-mat/9612223Nuclear Physics BJul 1, 199741 citations

κ= 1/(zγ)

. Here, we introduce two advances that resolve this limitation. First, by combining temperature scaling with independent current scaling, we separately extract the dynamical exponent

z

and the localization-length exponent

γ

at the quantum Hall plateau transition -- rather than inferring one from an assumed value of the other. Second, using dual-graphite-gated graphene devices in which the effective Coulomb interaction range is tuned geometrically by the ratio of the magnetic length

l_B

to the graphite-gate distance

d

, we track this separation across both screened and unscreened interaction regimes within the same device platform. Temperature scaling gives

κ\simeq 0.21

in the screened regime and

κ\simeq 0.41

in the unscreened regime; combining this with current scaling reveals that screening changes

z

from

\simeq 1

in the unscreened regime to

\simeq 2

in the screened regime. In contrast,

γ

remains close to

2.4

throughout. Our results establish that gate-controlled screening selectively modifies the interaction-dependent dynamical sector of the quantum Hall transition, leaving the localization-length exponent

γ

unchanged within experimental uncertainty. More broadly, this work establishes geometric screening as a versatile tool for controlling interactions and disentangling interaction and disorder effects in correlated two-dimensional systems, including fractional quantum Hall states, moiré materials, and other strongly localized electronic phases.