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arXiv:0806.0002 · Physical Review B

Renormalization group theory of nematic ordering in d -wave superconductors

Yejin Huh, Subir Sachdev A

Published Aug 13, 2008• 111 citations
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Abstract

We examine the quantum theory of the spontaneous breaking of lattice rotation symmetry in $d$-wave superconductors on the square lattice. This is described by a field theory of an Ising nematic order parameter coupled to the gapless fermionic quasiparticles. We determine the structure of the renormalization group to all orders in a $1/{N}_{f}$ expansion, where ${N}_{f}$ is the number of fermion spin components. Asymptotically exact results are obtained for the quantum critical theory in which, as in the large ${N}_{f}$ theory, the nematic order has a large anomalous dimension, and the fermion spectral functions are highly anisotropic.

DOI
10.1103/physrevb.78.064512

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The ddd-wave high-temperature cuprate superconductor may exhibit a nematic ordering transition at zero temperature. Near the quantum critical point, the coupling between gapless nodal quasiparticles and nematic order parameter fluctuation can result in unusual behaviors, such as extreme anisotropy of fermion velocities. We study the disorder effects on the nematic quantum critical behavior and especially on the flow of fermion velocities. The disorders that couple to nodal quasiparticles are divided into three types: random mass, random gauge field, and random chemical potential. A renormalization-group analysis shows that random mass and random gauge field are both irrelevant and, thus, do not change the fixed point of extreme velocity anisotropy. However, the marginal interaction due to random chemical potential destroys this fixed point and makes the nematic phase transition unstable.
We present a theory for the onset of spin-density-wave order in the superconducting ground state of the cuprates. We compute the scaling dimensions of allowed perturbations of a "relativistic" fixed point with O4 x O(3) symmetry, including those associated with the fermionic nodal Bogoliubov quasiparticles. Analyses of up to six loops show that all perturbations with square lattice symmetry are likely irrelevant. We demonstrate that the fermion spectral functions are primarily damped by the coupling to fluctuations of a composite field with Ising nematic order. A number of other experimental implications are also discussed.
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We present a renormalization-group theory for the onset of Ising-nematic order in a Fermi liquid in two spatial dimensions. This is a quantum phase transition, driven by electron interactions, which spontaneously reduces the point-group symmetry from square to rectangular. The critical point is described by an infinite set of 2+12+12+1-dimensional local field theories, labeled by points on the Fermi surface. Each field theory contains a real scalar field representing the Ising order parameter and fermionic fields representing a time-reversed pair of patches on the Fermi surface. We demonstrate that the field theories obey compatibility constraints required by our redundant representation of the underlying degrees of freedom. Scaling forms for the response functions are proposed and supported by computations up to three loops. Extensions of our results to other transitions of two-dimensional Fermi liquids with broken point-group and/or time-reversal symmetry are noted. Our results extend also to the problem of a Fermi surface coupled to a U(1) gauge field.
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