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arXiv:1410.6811 · Physical Review B

Aspects of renormalization in finite-density field theory

A. Liam Fitzpatrick, Gonzalo Torroba, Huajia Wang

Published May 26, 2015• 27 citations
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Abstract

We study the renormalization of the Fermi surface coupled to a massless boson near three spatial dimensions. For this, we set up a Wilsonian RG with independent decimation procedures for bosons and fermions, where the four-fermion interaction ``Landau parameters'' run already at tree level. Our explicit one-loop analysis resolves previously found obstacles in the renormalization of finite-density field theory, including logarithmic divergences in nonlocal interactions and the appearance of multilogarithms. The key aspects of the RG are the above tree-level running, and a UV-IR mixing between virtual bosons and fermions at the quantum level, which is responsible for the renormalization of the Fermi velocity. We apply this approach to the renormalization of $2{k}_{F}$ singularities, and to Fermi surface instabilities in a companion paper, showing how multilogarithms are properly renormalized. We end with some comments on the renormalization of finite-density field theory with the inclusion of Landau damping of the boson.

DOI
10.1103/physrevb.91.195135

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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 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.
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.
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