Hyperscaling at the spin density wave quantum critical point in two dimensional metals

TL;DR

This paper investigates hyperscaling properties at the spin density wave quantum critical point in 2D metals, showing that hot spot contributions obey hyperscaling up to logarithms and identifying regimes where hyperscaling is violated.

Contribution

It demonstrates that hot spot contributions in 2D SDW quantum critical metals follow hyperscaling laws and explores conditions leading to hyperscaling violation.

Findings

Hot spot contributions obey hyperscaling up to logarithms.

Agreement with large N analysis of optical conductivity.

Identification of an intermediate regime with hyperscaling violation.

Abstract

The hyperscaling property implies that spatially isotropic critical quantum states in $d$ spatial dimensions have a specific heat which scales with temperature as $T^{d/z}$, and an optical conductivity which scales with frequency as $\omega^{(d-2)/z}$ for $\omega \gg T$, 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 $\epsilon = 3-d$. 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 $d \rightarrow d_t$, where $d_t = 1$ 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 $\omega$.