Understanding optical conductivity data in the optimally doped cuprates in the framework of quantum criticality requires a strongly coupled quantum critical metal which violates hyperscaling. In the simplest scaling framework, hyperscaling violation can be characterized by a single nonzero exponent $\ensuremath{\theta}$, so that in a spatially isotropic state in $d$ spatial dimensions, the specific heat scales with temperature as ${T}^{(d\ensuremath{-}\ensuremath{\theta})/z}$, and the optical conductivity scales with frequency as ${\ensuremath{\omega}}^{(d\ensuremath{-}\ensuremath{\theta}\ensuremath{-}2)/z}$ for $\ensuremath{\omega}\ensuremath{\gg}T$, where $z$ is the dynamic critical exponent defined by the scaling of the fermion response function transverse to the Fermi surface. We study the Ising-nematic critical point, using the controlled dimensional regularization method proposed by Dalidovich and Lee [Phys. Rev. B 88, 245106 (2013)]. We find that hyperscaling is violated, with $\ensuremath{\theta}=1$ in $d=2$ . We expect that similar results apply to Fermi surfaces coupled to gauge fields in $d=2$ .Understanding optical conductivity data in the optimally doped cuprates in the framework of quantum criticality requires a strongly coupled quantum critical metal which violates hyperscaling. In the simplest scaling framework, hyperscaling violation can be characterized by a single nonzero exponent $\ensuremath{\theta}$, so that in a spatially isotropic state in $d$ spatial dimensions, the specific heat scales with temperature as ${T}^{(d\ensuremath{-}\ensuremath{\theta})/z}$, and the optical conductivity scales with frequency as ${\ensuremath{\omega}}^{(d\ensuremath{-}\ensuremath{\theta}\ensuremath{-}2)/z}$ for $\ensuremath{\omega}\ensuremath{\gg}T$, where $z$ is the dynamic critical exponent defined by the scaling of the fermion response function transverse to the Fermi surface. We study the Ising-nematic critical point, using the controlled dimensional regularization method proposed by Dalidovich and Lee [Phys. Rev. B 88, 245106 (2013)]. We find that hyperscaling is violated, with $\ensuremath{\theta}=1$ in $d=2$. We expect that similar results apply to Fermi surfaces coupled to gauge fields in $d=2$.

arXiv:1605.00657Physical review. B./Physical review. BJul 25, 201654 citations

Shear viscosity at the Ising-nematic quantum critical point in two-dimensional metals

Andreas Eberlein, Aavishkar A. Patel, Subir Sachdev A

In an isotropic strongly interacting quantum liquid without quasiparticles, general scaling arguments imply that the dimensionless ratio $({k}_{B}/\ensuremath{\hbar})\phantom{\rule{0.16em}{0ex}}\ensuremath{\eta}/s$, where $\ensuremath{\eta}$ is the shear viscosity and $s$ is the entropy density, is a universal number. We compute the shear viscosity of the Ising-nematic critical point of metals in spatial dimension $d=2$ by an expansion below $d=5/2$ . The anisotropy associated with directions parallel and normal to the Fermi surface leads to a violation of the scaling expectations: $\ensuremath{\eta}$ scales in the same manner as a chiral conductivity, and the ratio $\ensuremath{\eta}/s$ diverges at low temperature ( $T$ ) as ${T}^{\ensuremath{-}2/z}$, where $z$ is the dynamic critical exponent for fermionic excitations dispersing normal to the Fermi surface.In an isotropic strongly interacting quantum liquid without quasiparticles, general scaling arguments imply that the dimensionless ratio $({k}_{B}/\ensuremath{\hbar})\phantom{\rule{0.16em}{0ex}}\ensuremath{\eta}/s$, where $\ensuremath{\eta}$ is the shear viscosity and $s$ is the entropy density, is a universal number. We compute the shear viscosity of the Ising-nematic critical point of metals in spatial dimension $d=2$ by an expansion below $d=5/2$. The anisotropy associated with directions parallel and normal to the Fermi surface leads to a violation of the scaling expectations: $\ensuremath{\eta}$ scales in the same manner as a chiral conductivity, and the ratio $\ensuremath{\eta}/s$ diverges at low temperature ($T$) as ${T}^{\ensuremath{-}2/z}$, where $z$ is the dynamic critical exponent for fermionic excitations dispersing normal to the Fermi surface.

arXiv:1607.03894Physical review. B./Physical review. BFeb 15, 201726 citations

d

spatial dimensions, the specific heat scales with temperature as ${T}^{(d\ensuremath{-}\ensuremath{\theta})/z}$, and the optical conductivity scales with frequency as ${\ensuremath{\omega}}^{(d\ensuremath{-}\ensuremath{\theta}\ensuremath{-}2)/z}$ for $\ensuremath{\omega}\ensuremath{\gg}T$, where

z

is the dynamic critical exponent defined by the scaling of the fermion response function transverse to the Fermi surface. We study the Ising-nematic critical point, using the controlled dimensional regularization method proposed by Dalidovich and Lee [Phys. Rev. B 88, 245106 (2013)]. We find that hyperscaling is violated, with $\ensuremath{\theta}=1$ in

d=2

. We expect that similar results apply to Fermi surfaces coupled to gauge fields in

d=2

s

is the entropy density, is a universal number. We compute the shear viscosity of the Ising-nematic critical point of metals in spatial dimension

d=2

by an expansion below

d=5/2

. The anisotropy associated with directions parallel and normal to the Fermi surface leads to a violation of the scaling expectations: $\ensuremath{\eta}$ scales in the same manner as a chiral conductivity, and the ratio $\ensuremath{\eta}/s$ diverges at low temperature (

T

) as ${T}^{\ensuremath{-}2/z}$, where

z

is the dynamic critical exponent for fermionic excitations dispersing normal to the Fermi surface.