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arXiv:1507.05962 · Physical Review B

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

Aavishkar A. Patel, Philipp Strack, Subir Sachdev A

Published Oct 6, 2015• 43 citations
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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 ${\ensuremath{\omega}}^{(d\ensuremath{-}2)/z}$ 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 ${d}_{t}=1$ 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 $\ensuremath{\omega}$.

DOI
10.1103/physrevb.92.165105

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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 \ensuremathω(d\ensuremath−2)/z{\ensuremath{\omega}}^{(d\ensuremath{-}2)/z}\ensuremathω(d\ensuremath−2)/z for \ensuremathω\ensuremath≫T\ensuremath{\omega}\ensuremath{\gg}T\ensuremathω\ensuremath≫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 \ensuremathϵ=3\ensuremath−d\ensuremath{\epsilon}=3\ensuremath{-}d\ensuremathϵ=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→dtd\ensuremath{\rightarrow}{d}_{t}d\ensuremath→dt​, 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 \ensuremathω\ensuremath{\omega}\ensuremathω.
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 ddd 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 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 $\ensuremath{\theta}=1$ in d=2d=2d=2. We expect that similar results apply to Fermi surfaces coupled to gauge fields in d=2d=2d=2.
We study the frequency dependence of the electron self-energy and the optical conductivity in a recently developed field theory of the spin-density-wave quantum phase transition in two-dimensional metals. We focus on the interplay between the Fermi surface ``hot spots'' and the remainder of the ``cold'' Fermi surface. Scattering of electrons off the fluctuations of the spin-density-wave order parameter, $\ensuremath{\varphi}$, is strongest at the hot spots; we compute the conductivity due to this scattering in a rainbow approximation. We point out the importance of composite operators, built of products of the primary electron or $\ensuremath{\varphi}$ fields: These have important effects also away from the hot spots. The simplest composite operator, ${\ensuremath{\varphi}}^{2}$, leads to deviations from Landau Fermi-liquid behavior on the entire Fermi surface. We also find an intermediate frequency window in which the cold electrons lose their quasiparticle form due to effectively one-dimensional scattering processes. The latter processes are part of umklapp scattering, which leads to singular contributions to the optical conductivity at the lowest frequencies at zero temperature.
We present a field-theoretic renormalization group analysis of Abanov and Chubukov's model of the spin density wave transition in two dimensional metals. We identify the independent field scale and coupling constant renormalizations in a local field theory and argue that the damping constant of spin density wave fluctuations tracks the renormalization of the local couplings. The divergences at two-loop order overdetermine the renormalization constants and are shown to be consistent with our renormalization scheme. We describe the physical consequences of our renormalization-group equations, including the breakdown of Fermi liquid behavior near the ``hot spots'' on the Fermi surface. In particular, we find that the dynamical critical exponent zzz receives corrections to its mean-field value z=2z=2z=2. At higher orders in the loop expansion, we find infrared singularities similar to those found by Lee [Phys. Rev. B 80, 165102 (2009)] for the problem of a Fermi surface coupled to a gauge field. A treatment of these singularities implies that an expansion in 1∕N1∕N1∕N (where NNN is the number of fermion flavors) fails for the present problem. We also discuss the renormalization of the pairing vertex and find an enhancement which scales as logarithm squared of the energy scale. A similar enhancement is also found for a modulated bond order which is locally an Ising-nematic order.
The theory for the onset of spin density wave order in a metal in two dimensions flows to strong coupling, with strong interactions not only at the ``hot spots,'' but on the entire Fermi surface. We advocate the computation of dc transport in a regime where there is rapid relaxation to local equilibrium around the Fermi surface by processes which conserve total momentum. The dc resistivity is then controlled by weaker perturbations which do not conserve momentum. We consider variations in the local position of the quantum-critical point, induced by long-wavelength disorder, and find a contribution to the resistivity which is linear in temperature (up to logarithmic corrections) at low temperature. Scattering of fermions between hot spots, by short-wavelength disorder, leads to a residual resistivity and a correction which is linear in temperature.
We revisit the problem of two-dimensional metals in the vicinity of a quantum phase transition to incommensurate Q=2kF\mathbf{Q}=2{k}_{F}Q=2kF​ charge-density-wave order, where the order-parameter wave vector Q\mathbf{Q}Q connects two hot spots on the Fermi surface with parallel tangents. Earlier theoretical works argued that such critical points are potentially unstable, if the Fermi surface at the hot spots is not sufficiently flat. Here we perform a controlled, perturbative renormalization-group analysis and find a stable fixed point corresponding to a continuous quantum phase transition, which exhibits a strong dynamical nesting of the Fermi surface at the hot spots. We derive scaling forms of correlation functions at the critical point and discuss potential implications for experiments with transition-metal dichalcogenides and rare-earth tellurides.
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.