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arXiv:1605.00657 · Physical review. B./Physical review. B

Hyperscaling violation at the Ising-nematic quantum critical point in two-dimensional metals

Andreas Eberlein, Ipsita Mandal, Subir Sachdev A

Published Jul 25, 2016• 54 citations
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Abstract

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 $d=2$.

DOI
10.1103/physrevb.94.045133

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Violation of hyperscaling at the Ising-nematic quantum critical point in a two-dimensional metal

Andreas Eberlein, Ipsita Mandal, Subir Sachdev A

Bulletin of the American Physical Society•Mar 16, 2016•0 citations
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 \ensuremathθ\ensuremath{\theta}\ensuremathθ, so that in a spatially isotropic state in ddd spatial dimensions, the specific heat scales with temperature as T(d\ensuremath−\ensuremathθ)/z{T}^{(d\ensuremath{-}\ensuremath{\theta})/z}T(d\ensuremath−\ensuremathθ)/z, and the optical conductivity scales with frequency as \ensuremathω(d\ensuremath−\ensuremathθ\ensuremath−2)/z{\ensuremath{\omega}}^{(d\ensuremath{-}\ensuremath{\theta}\ensuremath{-}2)/z}\ensuremathω(d\ensuremath−\ensuremathθ\ensuremath−2)/z for \ensuremathω\ensuremath≫T\ensuremath{\omega}\ensuremath{\gg}T\ensuremathω\ensuremath≫T, where zzz 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 \ensuremathθ=1\ensuremath{\theta}=1\ensuremathθ=1 in d=2d=2d=2. We expect that similar results apply to Fermi surfaces coupled to gauge fields in d=2d=2d=2.
The hyperscaling property implies that spatially isotropic critical quantum states in ddd spatial dimensions have a specific heat, which scales with temperature as Td/z{T}^{d/z}Td/z, and an optical conductivity, which scales with frequency as ${\ensuremath{\omega}}^{(d\ensuremath{-}2)/z}$ for $\ensuremath{\omega}\ensuremath{\gg}T$, where zzz is the dynamic critical exponent. We examine the spin density wave critical fixed point of metals in d=2d=2d=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 NNN 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{d}_{t}=1dt​=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 TTT replacing $\ensuremath{\omega}$.
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 sss 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=2d=2d=2 by an expansion below d=5/2d=5/2d=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 (TTT) as ${T}^{\ensuremath{-}2/z}$, where zzz is the dynamic critical exponent for fermionic excitations dispersing normal to the Fermi surface.
We study the optical conductivity of a pristine two-dimensional electron system near an Ising-nematic quantum critical point. We discuss the relation between the frequency scaling of the conductivity and the shape of the Fermi surface, namely, whether it is isotropic, convex, or concave. We confirm the cancellation of the leading order terms in the optical conductivity for the cases of isotropic and convex Fermi surfaces and show that the remaining contribution scales as ${|\ensuremath{\omega}|}^{2/3}$ at T=0T=0T=0. On the contrary, the leading term, $\ensuremath{\propto}{|\ensuremath{\omega}|}^{\ensuremath{-}2/3}$, survives for a concave FS. We also address the frequency dependence of the optical conductivity near the convex-to-concave transition. Explicit calculations are carried out for the Fermi-liquid regime using the modified (but equivalent to the original) version of the Kubo formula, while the quantum-critical regime is accessed by employing the space-time scaling of the Z=3Z=3Z=3 critical theory.
We study finite-temperature properties of metals close to an Ising-nematic quantum critical point in two spatial dimensions. In particular we show that at any finite temperature there is a regime where order parameter fluctuations are characterized by a dynamical critical exponent z=2z=2z=2, in contrast to z=3z=3z=3 found at zero temperature. Our results are based on a simple Eliashberg-type approach, which gives rise to a boson self-energy proportional to $\mathrm{\ensuremath{\Omega}}/\ensuremath{\gamma}(T)$ at small momenta, where $\ensuremath{\gamma}(T)$ is the temperature dependent fermion scattering rate. These findings might shed some light on recent Monte Carlo simulations at finite temperature, where results consistent with z=2z=2z=2 were found.
We analyze optical conductivity of a clean two-dimensional electron system in a Fermi liquid regime near a T=0T=0T=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\mathcal{L}=\mathrm{const}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.