We study the influence of short-range electron-electron interactions on scaling behavior near the integer quantum Hall plateau transitions. Short-range interactions are known to be irrelevant at the renormalization group fixed point which represents the transition in the noninteracting system. We find, nevertheless, that transport properties change discontinuously when interactions are introduced. Most importantly, in the thermodynamic limit the conductivity at finite temperature is zero without interactions, but nonzero in the presence of arbitrarily weak interactions. In addition, scaling as a function of frequency $\ensuremath{\omega}$ and temperature T is determined by the scaling variable $\ensuremath{\omega}{/T}^{p}$ (where p is the exponent for the temperature dependence of the inelastic scattering rate) and not by $\ensuremath{\omega}/T,$ as it would be at a conventional quantum phase transition described by an interacting fixed point. We express the inelastic exponent p and the thermal exponent
zT in terms of the scaling dimension $\ensuremath{-}\ensuremath{\alpha}<0$ of the interaction strength and the dynamical exponent z (which has the value
z=2), obtaining $p=1+2\ensuremath{\alpha}/z$ and
zT=2/p.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
lB 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≃0.21 in the screened regime and
κ≃0.41 in the unscreened regime; combining this with current scaling reveals that screening changes
z from
≃1 in the unscreened regime to
≃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.
We study dynamical scaling in the quantum-critical fan of the pseudogap-metal to Fermi-liquid transition of the two-dimensional Hubbard model. Using a four-patch dynamical cluster approximation with the numerical renormalization group as a cluster impurity solver, we access real-frequency dynamics over several decades at arbitrary temperatures. Close to the critical doping, the local spin and cluster-current susceptibility spectra exhibit
x=ω/T scaling of the form
χ′′(ω,T)∼tanh(x/2), and the cluster contribution to the optical conductivity obeys
Tσcl′(ω,T)∼tanh(x/2)/x, implying a
1/T cluster dc conductivity. In the scaling regime, the vertex contribution to the cluster optical response is much larger than the bubble contribution. We further find evidence for a marginal-Fermi-liquid nodal self-energy. This, together with the
1/T vertex contribution to the conductivity, implies strange-metal optical transport in the quantum critical region. Our results describe several qualitative aspects of several experimental observations.
We analyze optical conductivity of a clean two-dimensional electron system in a Fermi liquid regime near a
T=0 Ising-nematic quantum critical point (QCP) and extrapolate the results to a QCP. We employ direct perturbation theory up to the two-loop order to elucidate how the Fermi surface's geometry (convex vs concave) and fermionic dispersion (parabolic vs nonparabolic) affect the scaling of the optical conductivity $\ensuremath{\sigma}(\ensuremath{\omega})$ with frequency $\ensuremath{\omega}$ and correlation length $\ensuremath{\xi}$. We find that for a convex Fermi surface the leading terms in the optical conductivity cancel out, leaving a subleading contribution $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{2}{\ensuremath{\xi}}^{4}\mathcal{L}$, where
L=const for a parabolic dispersion and $\mathcal{L}\ensuremath{\propto}ln\ensuremath{\omega}{\ensuremath{\xi}}^{3}$ in a generic case. For a concave Fermi surface, the leading terms do not cancel, and $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\xi}}^{2}$. We extrapolate these results to a QCP and obtain $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{2/3}$ for a convex Fermi surface and $\ensuremath{\sigma}(\ensuremath{\omega})\ensuremath{\propto}1/{\ensuremath{\omega}}^{2/3}$ for a concave Fermi surface.
The hyperscaling property implies that spatially isotropic critical quantum states in
d spatial dimensions have a specific heat, which scales with temperature as
Td/z, and an optical conductivity, which scales with frequency as ${\ensuremath{\omega}}^{(d\ensuremath{-}2)/z}$ for $\ensuremath{\omega}\ensuremath{\gg}T$, where
z is the dynamic critical exponent. We examine the spin density wave critical fixed point of metals in
d=2 found by Sur and Lee [Phys. Rev. B 91, 125136 (2015)] in an expansion in $\ensuremath{\epsilon}=3\ensuremath{-}d$. We find that the contributions of the ``hot spots'' on the Fermi surface to the optical conductivity and specific heat obey hyperscaling (up to logarithms), and agree with the results of the large
N analysis of the optical conductivity by Hartnoll et al. [ Phys. Rev. B 84, 125115 (2011)]. With a small bare velocity of the boson associated with the spin density wave order, there is an intermediate energy regime where hyperscaling is violated with $d\ensuremath{\rightarrow}{d}_{t}$, where
dt=1 is the number of dimensions transverse to the Fermi surface. We also present a Boltzmann equation analysis which indicates that the hot-spot contribution to the dc conductivity has the same scaling as the optical conductivity, with
T replacing $\ensuremath{\omega}$.
We investigate the frequency and temperature dependence of the low-energy electron dynamics in a Landau Fermi liquid with a local self-energy. We show that the frequency and temperature dependencies of the optical conductivity obey universal scaling forms, for which explicit analytical expressions are obtained. For the optical conductivity and the associated memory function, we obtain a number of surprising features that differ qualitatively from the Drude model and are universal characteristics of a Fermi liquid. Different physical regimes of scaling are identified, with marked non-Drude features in the regime where $\ensuremath{\hbar}\ensuremath{\omega}\ensuremath{\sim}{k}_{\text{B}}T$. These analytical results for the optical conductivity are compared to numerical calculations for the doped Hubbard model within dynamical mean-field theory. For the ``universal'' low-energy electrodynamics, we obtain perfect agreement between numerical calculations and analytical scaling laws. Both results show that the optical conductivity displays a non-Drude ``foot,'' which could be easily mistaken as a signature of breakdown of the Fermi liquid, while it actually is a striking signature of its applicability. The aforementioned scaling laws provide a quantitative tool for the experimental identification and analysis of the Fermi-liquid state using optical spectroscopy, and a powerful method for the identification of alternative states of matter, when applicable.