RG on a Fermi surface Evelyn Tang (Dated: May 7, 2010) We review the renormalization group (RG) method for interacting fermions, building o® typical Á4 theory to account for the presence of a Fermi surface. A few geometries and the general procedure are briefly discussed. The method is illustrated with a model of two Fermi circles with di®erent spin components and momenta, which can describe experiments with interacting fermions. Gen- eral results obtained from this method, such as implications for Fermi-liquid theory and the BCS instability, are summarized. A. Introduction tion is then Z Y Y ¹ S0 The RG has been useful in treating interactions { both Z = dÃi(!; k)dÃi(!; k)e (1) conceptually and as a method for removing high energy i jkj<¤ degrees of freedom. This is because critical behavior is X Z ¤ d Z 1 d k ¹ ¡1 assumed to be arise from the low energy degrees of free- S0 = d! Ãi(!; k)(¡G (!))Ãi(!; k) (2¼)d dom of the system. As a concept it arose in the 1950s and i ¡¤ ¡1 a decade later was applied to a lattice model by Kadano®, and consolidated into its modern form with Wilson. In where G(!) is the Green's function, ¤ is the cuto® mo- condensed matter physics it has since been applied to a mentum and i labels each fermion. All other terms in variety systems and has been of great success in describ- the action will be taken as perturbations to the Gaussian ing critical phenomena. action S0. In this paper we assume familiarity with Á4 theory for bosons, and review the modi¯cations necessary to apply RG to fermions. We discuss dealing with a Fermi surface A. With a Fermi Surface for a few dimensions and simple geometries, and outline the general procedure. Next we illustrate the method by applying it to two colliding Fermi surfaces, i.e. two In one-dimension the Fermi surface is simply two points, so the RG is very similar to that in quantum ¯eld surfaces displaced by momentum 2KD and with di®erent spin. The results are compared with those from the more theory where we shrink our phase space down to a point typical case of two stationary surfaces. Lastly, we discuss under renormalization. In two-dimensions the circular the broader relevance of the RG and the scope of this Fermi surface allows for a set of transitions that obey technique. momentum conservation rules. Constraining ourselves to excitations at the Fermi surface, allowed transitions must have de¯nite angular relations. In three-dimensions the physics is similar, only that the allowed angles have an additional degree of freedom and thus the allowed pa- rameter space forms a line instead of a point. The non- I. RG FOR FERMIONS rotationally invariant case di®ers from the rotationally invariant one where the coupling functions depend only The RG relies on the idea of scale invariance, in that on relative di®erences in scattering angle, while in the distances or energies can be successively rescaled with- former they are now functions of the angles themselves. out loss of underlying physics, rather it allows the exam- For nested Fermi surfaces, an additional scattering by the ination of the critical behavior of the system at certain nested momentum vector is possible. ¯xed points. The basic ideas and methods of Á4 scalar With a Gaussian action, we can use Wick's theorem to ¯eld theory can be applied to fermions, however anti- give us all possible pairings then integrate over the high- commutation rules and the presence of a Fermi surface energy modes from ¤=s to ¤. Expanding the coupling demand a di®erent treatment [1]. To this end Grassmann functions in momentum and frequency, many couplings variables are introduced, which describe fermionic coher- are found to be marginal at the tree-level renormaliza- ent states. Using the path-integral approach, the parti- tion and so we calculate those to the order of one-loop. tion function at inverse temperature ¯ can be mapped Additional loops are supressed by an order of ¤=KF and to a system of d spatial dimensions and width ¯ in the hence we have a solution as ¤=KF ! 0; analogous to a imaginary time direction. Variables are written in the 1=N expansion. As an extension from the cases discussed Fourier bases of frequency and momentum which give us in [1], we now apply the RG to two Fermi surfaces with an additional integral over all frequency values (at zero di®erent spin and momenta. to illustrate in detail the temperature ! ! 1). The unperturbed partition func- method briefly outlined above. 2 FIG. 2: A geometrical representation of scattering momenta FIG. 1: Spin " fermions move to the right with momentum that form solutions to Eq. 8. Case I and II are labelled in the diagram, while case III is when the two shells coalesce to sit KD and spin # ones move to the left with ¡KD. In [3] for example, this describes the collision of two atomic clouds. atop each other. Figure taken from [1]. II. TWO COLLIDING FERMI GASES we can expand the coupling function ¹ ¹(k; !) = ¹00 + ¹10k + ¹01i! + ::: (4) Here we use the method briefly outlined above to study two interacting Fermi gases that have di®erent spin and The constant piece is a relevant perturbation that causes momenta. This gives us two Fermi surfaces each with the Fermi sea to adjust to a change in the chemical poten- radius KF and separated by 2KD, see Fig. 1. There are tial, while the following two terms are marginal interac- several examples of this in two-dimensional electron gas tions that modify terms already present in the action. As or cold-atom systems. none of this changes the physics and all other terms are In the former an applied magnetic ¯eld can change irrelevant, we turn to the e®ect of a quartic perturbation: Z the momentum of a lower quantum well with respect to 1 an upper well [2], corresponding to a displacement of its ±S = ù(4)ù(3)Ã(2)Ã(1)u(4; 3; 2; 1) (5) 4 2!2! Fermi circle. For cold-atom systems, in recent work by K!θ ¹ ¹ Sommer et. al [3] two strongly interacting atomic clouds where Ã(i) = Ã(Ki;!i; θi) etc., collide into each other, mix and form molecules . We use 2 3 spin up " and down # as isospin labels for the degrees of Z Y3 Z 2¼ Z ¤ Z 1 dθj dkj d!j freedom relevant to the sysem, e.g. in [3] the gases are = 4 5 θ(¤ ¡ jk j)(6) 2¼ 2¼ 2¼ 4 Li-6 atoms in distinct hyper¯ne states. The calculation K!θ j 0 ¡¤ ¡1 is done for the 2D case as in 3D only the angular phase space is increased while the physics does not change. The step function in Eq. 6 results from evaluating the momentum conserving ±-function in the quartic interac- tion (Eq. 5), ±(K1 + K2 ¡ K3 ¡ K4) (momentum is A. Tree-level calculation conserved mod2¼ in a lattice problem). As all four momenta are required to lie within an an- nulus of thickness 2¤ around the Fermi circle (see Fig. As in Fig. 1, a fermion labelled i in this scheme carries 2), the step function ensures that after choosing the momentum ¯rst three momenta, the fourth is also constrained to lie Ki,σ = σKd + (KF + ki)­i (2) within the cuto® ¤. Expanding u(4; 3; 2; 1) similar to what we did for ¹ in where ­i is a unit vector from the center of its Fermi Eq. 4, we ¯nd that the constant term u0 is marginal circle to Ki,σ. Near the ground state, we can linearize and other terms are irrelevant. Further, the only allowed the dispersion relation near the Fermi surface as values of u0 are ²(Ki) = E(Ki) ¡ ¹ = vki (3) u0 = u"""" ; u#### ; u"#"# = u#"#" = ¡u#""# = ¡u"##" (7) where v is the Fermi velocity. This gives G¡1 = vk ¡ i!. d as other couplings are forbidden by the Pauli principle We write d k = KF kdkdθ in Eq. 1 for each surface in the and Wick's theorem. Here we take a coupling strength unperturbed action and proceed to consider the e®ects of independent of spin. interactions. For a quadratic perturbation written as For each coupling listed above, the KD's cancel in the Z Z Z X ¤ dk 2¼ dθ 1 d! momentum ±-function which reduces to ±S = ¹(k!)ù (θ!k)à (θ!k) 2 2¼ 2¼ 2¼ i i i ¡¤ 0 ¡1 ±(­1 + ­2 ¡ ­3 ¡ ­4) (8) 3 that we call F (θ1 ¡ θ2 ´ θ12) for forward scattering. The second is in Case III that we call V (θ13). These functions only depend on the di®erence between two angles due to rotational invariance. As they are both marginal at this level, we continue to the one-loop graphs. FIG. 3: Diagrams that contribute to one-loop corrections. B. One-loop corrections The ¯rst is the tadpole graph that renormalizes the quadratic term, while the next three renormalize the quartic term. Fig- ures taken from [1]. The tadpole graph (Fig. 3) contributes to the quadratic term and acts as a shift in the chemical po- tential. However, we would like to maintain a ¯xed KF when we take ¤=KF ! 0 after an in¯nite amount of as this corresponds to a ¯xed density of the system, so we renormalization. This is when all vectors lie on the Fermi solve for a term ±¹ in the chemical potential that repro- surface at the ¯xed point.