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arXiv:1406.3029 · Physical Review B

Quantum critical metals in 4−ε dimensions

Gonzalo Torroba, Huajia Wang

Published Oct 31, 2014• 44 citations
arXivHTMLPDF

Abstract

We study the quantum theory of a Fermi surface coupled to a gapless boson scalar in $D=4\ensuremath{-}\ensuremath{\epsilon}$ space-time dimensions as a simple model for non-Fermi liquids (NFL) near a quantum phase transition. Our analysis takes into account the full backreaction from Landau damping of the boson, and obtains an RG flow that proceeds through three distinct stages. Above the scale of Landau damping, the Fermi velocity flows to zero, while the coupling evolves according to its classical dimension. Once damping becomes important, its backreaction leads to a crossover regime where dynamic and static damping effects compete and the fermion self-energy does not respect scaling. Below this crossover and having tuned the boson to criticality, the theory flows to a $z=3$ scalar interacting with an NFL. We finally analyze the IR phases of the theory with arbitrary number of flavors ${N}_{c}$. When ${N}_{c}$ is small, the superconducting dome covers the NFL behavior; strikingly, for moderately large ${N}_{c}$, we find that NFL effects become important first, before the onset of superconductivity. A generic prediction of the theory is that the Fermi velocity and quasiparticle residue vanish with a power law ${\ensuremath{\omega}}^{\ensuremath{\epsilon}}$ as the fixed point is approached. These features may be useful for understanding some of the phenomenology of high-${T}_{c}$ materials in a systematic $\ensuremath{\epsilon}$ expansion.

DOI
10.1103/physrevb.90.165144

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We present a reduced dimension theoretical framework for studying quantum field theories at finite density, providing a tractable model for investigating non-Fermi liquid (NFL) behavior near quantum phase transitions. Our approach departs from the standard paradigm by placing bosons and fermions in different spatial dimensions: bosonic fields reside in a (d+1)(d+1)(d+1)-dimensional bulk, while fermionic fields are confined on a ddd-dimensional boundary. This dimensional separation significantly simplifies the renormalization group (RG) analysis of gapless boson-fermion coupling. We demonstrate that the tree-level boson exchange contributions, which typically exhibit logarithmic divergences, become finite in our reduced-dimension scheme. Furthermore, the log⁡2\log^2log2 and log⁡3\log^3log3 divergences that characterize 1-loop four-fermion interactions in conventional treatments are reduced to logarithmic divergences within this framework, substantially improving the convergence properties of the perturbative expansion and allowing controlled theoretical analysis of NFL physics.
We revisit the interplay between superconductivity and quantum criticality when thermal effects from virtual static bosons are included. These contributions, which arise from an effective theory compactified on the thermal circle, strongly affect field-theoretic predictions even at small temperatures. We argue that they are ubiquitous in a wide variety of models of non-Fermi liquid behavior, and generically produce a parametric suppression of superconducting instabilities. We apply these ideas to non-Fermi liquids in d=2d=2d=2 space dimensions, obtained by coupling a Fermi surface to a Landau-damped soft boson. Extending previous methods developed for $d=3\ensuremath{-}\ensuremath{\epsilon}$ dimensions, we determine the dynamics and phase diagram. It features a naked quantum critical point, separated by a continuous infinite order transition from a superconducting phase with strong non-Fermi liquid corrections. We also highlight the relevance of these effects for (numerical) experiments on non-Fermi liquids.
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