Disordered Holographic Systems II: Marginal Relevance of Imperfection

E. Abrahams, P. W. Anderson, D. C. Licciardello and T. V. Ramakrishnan, Phys. Rev. Lett. 42

1979

Earlier work this paper cites.

For a review, see P. A. Lee and T. V. Ramakrishnan, Rev. Mod. Phys. 57

1985

Cited alongside, same era.

A. Adams and S. Yaida, arXiv:1102.2892 [hep-th]

Original

Cited in the paper.

One way to arrive at g M ​ N ( 2 ) ​ ( 𝐤 ) g^{(2)}_{MN}\left({\bf k}\right) for 𝐤 ≠ 0 {\bf k}\neq 0 is to first work in a coordinate system where g z ​ z ( 2 ) = 0 g^{(2)}_{zz}=0 , g μ ​ μ ( 2 ) = 0 g^{(2)}_{\mu\mu}=0 , and ( 𝐤 ) i ​ g i ​ l ( 2 ) ​ ( δ l ​ j − ( 𝐤 ) l ​ ( 𝐤 ) j 𝐤 2 ) = 0 \left({\bf k}\right)_{i}g^{(2)}_{il}\left(\delta_{lj}-\frac{\left({\bf k}\right)_{l}\left({\bf k}\right)_{j}}{{\bf k}^{2}}\right)=0 and then transform back to the radial coordinate

Cited in the paper.

Note that A ( 3 ) A^{(3)} is essentially a product of g μ ​ ν ( 2 ) g^{(2)}_{\mu\nu} and A μ ( 1 ) A^{(1)}_{\mu}

Cited in the paper.

A quick path to get [ T z ( 4 ) ​ z ] d . a . \left[T^{(4)z}_{\ \ \ \ z}\right]_{\rm d.a.} is to use T M M = 0 T^{M}_{\ \ \ M}=0 for d = 2 + 1 d=2+1 and the energy-momentum conservation ∇ M T z M = 0 \nabla_{M}T^{M}_{\ \ \ z}=0 , together with the regularity at the horizon. Incidentally, the same logic implies that [ T z ( 2 ) ​ z ] d . a . = 0 \left[T^{(2)z}_{\ \ \ \ z}\right]_{\rm d.a.}=0

Cited in the paper.

One may be concerned with the fact that the U ⁡ ( 1 ) U(1) symmetry of the holographic CFT, whose current is coupled to the random potential, is global on the field theory side. Although we do not fully justify it, this may be a red herring: at least, for electronic systems, one can often proceed without explicitly taking U ⁡ ( 1 ) U(1) gauge field into account. For holographic systems, the literature on the issue of gauging U ⁡ ( 1 ) U(1) symmetry is surprisingly scarce and it calls for careful development

Cited in the paper.

For reviews with a view toward condensed matter physics, see S. A. Hartnoll, Class. Quant. Grav. 26

Original

2010

Later among the works it cites.