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arXiv:0905.4532 · Physical Review B

Low-energy effective theory of Fermi surface coupled with U(1) gauge field in 2 + 1 dimensions

Sung-Sik Lee

Published Oct 1, 2009• 360 citations
arXivHTMLPDF

Abstract

We study the low-energy effective theory for a non-Fermi-liquid state in $2+1$ dimensions, where a transverse U(1) gauge field is coupled with a patch of Fermi surface with $N$ flavors of fermion in the large $N$ limit. In the low-energy limit, quantum corrections are classified according to the genus of the two-dimensional surface on which Feynman diagrams can be drawn without a crossing in a double line representation and all planar diagrams are important in the leading order. The emerging theory has the similar structure to the four-dimensional SU(N) gauge theory in the large $N$ limit. Because of strong quantum fluctuations caused by the abundant low-energy excitations near the Fermi surface, low-energy fermions remain strongly coupled even in the large $N$ limit. As a result, there are infinitely many quantum corrections that contribute to the leading frequency dependence of the Green's function of fermion on the Fermi surface. On the contrary, the boson self-energy is not modified beyond the one-loop level and the theory is stable in the large $N$ limit. The nonperturbative nature of the theory also shows up in correlation functions of gauge-invariant operators.

DOI
10.1103/physrevb.80.165102

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We study the low-energy effective theory for a non-Fermi-liquid state in 2+12+12+1 dimensions, where a transverse U(1) gauge field is coupled with a patch of Fermi surface with NNN flavors of fermion in the large NNN limit. In the low-energy limit, quantum corrections are classified according to the genus of the two-dimensional surface on which Feynman diagrams can be drawn without a crossing in a double line representation and all planar diagrams are important in the leading order. The emerging theory has the similar structure to the four-dimensional SU(N) gauge theory in the large NNN limit. Because of strong quantum fluctuations caused by the abundant low-energy excitations near the Fermi surface, low-energy fermions remain strongly coupled even in the large NNN limit. As a result, there are infinitely many quantum corrections that contribute to the leading frequency dependence of the Green's function of fermion on the Fermi surface. On the contrary, the boson self-energy is not modified beyond the one-loop level and the theory is stable in the large NNN limit. The nonperturbative nature of the theory also shows up in correlation functions of gauge-invariant operators.
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In metals, low-energy effective theories are characterized by a set of coupling functions. Among them, the angle-dependent Fermi momentum specifies the size and shape of Fermi surface. Since the Fermi momentum grows incessantly under the renormalization group flow, a metallic fixed point is defined only modulo a rescaling of Fermi momentum. In this paper, we discuss the physical consequences of this projective nature of fixed points for non-Fermi liquids with hot Fermi surfaces. The first is the absence of a unique dynamical critical exponent that dictates the relative scaling between energy and momentum. The second is mismatches between the scaling dimensions of couplings and their relevancy. Nonetheless, each projective fixed point is characterized by a few marginal and relevant coupling functions, and the notion of universality survives. We illustrate our findings by charting the space of projective fixed points and extracting their universal properties for the Ising-nematic quantum critical metal beyond the patch theory. To control the theory, we use the dimensional regularization scheme that tunes the co-dimension of Fermi surface. Near the upper critical dimension, two exactly marginal coupling functions span the space of stable projective fixed points: functions that specify the shape of the Fermi surface and the angle-dependent Fermi velocity. All other coupling functions, including the Landau functions and the universal pairing interaction, are fixed by those two marginal functions. With decreasing dimensions, the forward scattering remains irrelevant while the pairing interaction becomes relevant near two dimensions. In two dimensions, it is expected that the universal superconducting fluctuations lower the symmetry of the non-Fermi liquid realized above the superconducting transition temperatures from the loop U(1) group to a proper subgroup.
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We study critical Fermi surfaces in generic dimensions arising from coupling finite-density fermions with transverse gauge fields by applying a dimensional regularization scheme developed previously [Phys. Rev. B 92, 035141 (2015)]. We consider the cases of U (1) and U (1) × U (1) transverse gauge couplings and extract the nature of the renormalization group (RG) flow fixed points as well as the critical scalings.Our analysis allows us to treat a critical Fermi surface of a generic dimension m perturbatively in an expansion parameter = (2m)/(m + 1).One of our key results is that although the two-loop corrections do not alter the existence of an RG flow fixed line for certain U (1) × U (1) theories, which was identified earlier for m = 1 at one-loop order, the third-order diagrams do.However, this fixed line feature is also obtained for m > 1, where the answer is one-loop exact due to UV/IR mixing.