Existence and construction of Galilean invariant z≠2 theories

Abstract

We prove a no-go theorem for the construction of a Galilean boost invariant and z ≠ 2 anisotropic scale invariant field theory with a finite dimensional basis of fields. Two point correlators in such theories, we show, grow unboundedly with spatial separation. Correlators of theories with an infinite dimensional basis of fields, for example, labeled by a continuous parameter, do not necessarily exhibit this bad behavior. Hence, such theories behave effectively as if in one extra dimension. Embedding the symmetry algebra into the conformal algebra of one higher dimension also reveals the existence of an internal continuous parameter. Consideration of isometries shows that the nonrelativistic holographic picture assumes a canonical form, where the bulk gravitational theory lives in a space-time with one extra dimension. This can be contrasted with the original proposal by Balasubramanian and McGreevy, and by Son, where the metric of a (d þ 2)-dimensional space-time is proposed to be dual of a d-dimensional field theory. We provide explicit examples of theories living at fixed point with anisotropic scaling exponent z ¼ 2l lþ1 , l ∈ Z.

Non-boost invariant fluid dynamics

“…In order to compute transport coefficients using (4.17), we will drop the restriction to stationary configurations -that is to say, we relax the requirement that β µ is Killing. This will lead to an action for the hydrostatic non-dissipative transport coefficients 10 This is related to the discussion below equation (3.7) in the following way. As we will show, the energy-momentum tensor obtained by varying the geometric variables in the action that follows from the hydrostatic partition function without the condition that β µ is Killing, is equal to the HS part of the energy-momentum tensor as defined in equation (3.11).…”

“…where we have used the relation (2.24). On flat space (2.19), where u µ = (1, v i ), the energymomentum tensor -and by extension the tensor η µνρσ -may be further decomposed as, 9) where the flat space tensors κ jk , η jkl , κ i jk are given by 10) which means that…”

Non-equilibrium steady states in quantum critical systems with Lifshitz scaling

“…In addition, for z = 2, the Schrödinger group (consisting of the Bargmann group, enhanced by the addition of the dilation operator D), can have an additional generator, Ĉ, corresponding to special conformal transformations. Finally, as shown in [11] and [25], it is only for this particular value of z that one can have a Galilean boost invariant fluid with Lifshitz scaling symmetry with a discrete Hamiltonian and number operator spectrum.…”

Report on scipost_202005_00012v4

“…multiply vector structures, while, finally, the coefficient t multiplies a single tensor structure. 9) where the flat space tensors κ jk , η jkl , κ ijk are given by 10) which means that…”