Tuesday, September 6, 2011

Hi, this is Andy O. The talk today was by David Tong, based on two papers with his student Joao Laia, 1108.1381 and 1108.2216.

The topic was a Dirac fermion in AdS4, dual to some gauge-invariant fermionic operator in a 2+1d CFT. Let's call the fermion

with mass m. A given solution of the Dirac equation behaves near the boundary as

Here we're using units in which the AdS radius is one, and we're being sloppy with the indices: the bulk fermion has four complex components, while A and D each have two. For generic m, we interpret the coefficient of the non-normalizable term, A, as the source for the dual operator and the coefficient of the normalizable term, D, as the vacuum expectation value (vev), or one-point function, which via functional differentiation gives the two-point function. Let

denote the dual operator, of dimension

In AdS, the symmetries fix the two-point function up to normalization.

For special values of m, namely when m < 1/2, both terms are normalizable, hence we have a choice whether to call A and D the source and vev, or vice versa. The story is similar to that of scalars in AdS. The two choices correspond to different boundary conditions at the AdS boundary, Dirichlet versus Neumann, sometimes called "standard quantization" and "alternative quantization," respectively. These two choices correspond to different CFT's, which have the same dynamics but slightly different operator content. Let's call these CFT1 and CFT2. In the former A is the source, while in the latter D is the source. CFT2 has a special relevant operator,

of dimension 3 - 2m < 3. If we start with CFT2, we can perturb by this double-trace operator and flow at low energy to CFT1. The two CFT's have different central charge: each CFT has an order N^2 central charge, but they differ by an amount of order one, i.e. N^0.

Actually, I was confused for a while during the talk. We're discussing a 2+1d CFT, so how can we define central charge? For the purposes of this discussion, the scaling of the free energy in a thermal equilibrium state should suffice. I guess we could also use the partition function of the Euclidean theory on a three-sphere, but that's heading outside my comfort zone.

Anyway, David's talk was studying in more detail how the RG flow between CFT1 and CFT2 occurs. He had two main points.

The first main point was that in fact TWO boundary terms are invovled in the flow from CFT2 to CFT1. The boundary action is

where we must impose

to guarantee that we don't fix BOTH A and D at the boundary. Clearly the g term breaks parity. The two fixed points correspond to

David and Joao then studied the holographic Wilsonian renormalization group flows for the fermions, in other words, they repeated the analysis of 1010.1264 and 1010.4036 for fermions. A key point here is that only FREE fields were studied in truly explicit detail in 1010.1264 and 1010.4036, and David and Joao studied FREE fermions. (For those of you looking for projects: try and think of interesting systems involving free fields in AdS where you could do holographic Wilsonian RG. I can think of a few.) To my knowledge the only explicit examples involving interacting fields (in the published literature) appear in 1107.5318 (and see also 1009.3094).

The prodcedure is as follows. Introduce a cutoff surface and ask how these boundary terms change as we vary the position of the cutoff surface (so that the low-energy correlators remain unchanged). They found a holographic beta function

Actually, the above is for small g. For larger g, the right-hand side changes sign. The exact solution for g is

where J is the source for the double-trace operator. Clearly g goes to zero in the UV (small cutoff), reaches a maximum somewhere, and then returns to zero far in the infrared.

The story is thus as follows. We start in the UV with CFT2, where f = -1 and g=0. We deform by the two boundary terms written above. g will flow as written above, and f is fixed by f^2 = 1- g^2. Eventually we reach CFT1, where f = +1 and again g=0.

A number of things are odd about such an RG flow. Notice that the g term breaks parity. In CFT2, parity is preserved (g=0). Along the flow, parity is broken. At the fixed point, CFT1, g=0 again and parity is restored. That is really weird. During the talk, I was wondering, if parity is broken and the field theory has gauge fields, why wouldn't a Chern-Simons term be generated? How can parity be restored at the IR fixed point? Right after I thought that, Shamit Kachru asked that very question, which was a confidence booster for me (I was on the right track!). Nobody really had an explanation, except the usual one: that's what holography tells you. Apparently parity is an accidental symmetry in the IR.

Another odd feature is that the f term does not correspond to a double-trace term. Instead, the g term does. That's obvious if you think about parity. In the field theory, the double-trace term looks like a Dirac mass in 2+1d, which breaks parity. The f term preserves parity while the g term breaks parity, hence the g term must correspond to the double-trace operator. (David explained that to me later.)

Another odd thing is that the RG flow is exactly the same for Dirac or Majorana fermions in either AdS4 or AdS5. From what David said, and what is in his papers, that ``universal'' result arises differently in each case. Some fundamental principle, which imposes that result, may be lurking in the background.

I also talked to Tassos Petkou after the talk (we share an office), and he said he actually encountered the marginal deformation earlier, in hep-th/0304217, in a very different context: supersymmetric higher-spin theories in AdS. He had an N=1 multiplet in the bulk, which included a conformally-coupled scalar and a massless fermion. Playing the usual games with the scalar's boundary conditions (standard vs. alternative), and requiring supersymmetry to fix the fermion's boundary conditions, led to very similar RG flows. Actually, they are even more mysterious, and tantalizing: with the scalars present the fixed points become more lively, since at each fix point one sector, scalar or fermion, is free while the other is interacting. (For more details, I recommend talking to Tassos!)

David's second main point was about deformations that break the Lorentz group to a subgroup including rotations and scale transformations, but not boosts and special conformal transformations. Explicitly, he considered double-trace deformations by the operators

If we start in the UV from CFT2 and deform by one of these, we arrive at a "non-relativistic" fixed point CFT (NRCFT). In the bulk, these are described by funny boundary conditions where we fix individual components of A and D, namely we fix A1 and D2 (these are the sources) or D1 and A2. At these fixed points parity and charge conjugation are broken. We can reach CFT1 again by an appropriate double-trace deformation, so these two fixed points are sort of "intermediate" between CFT1 and CFT2.

The most interesting lesson of the talk (for me) was the following. The NRCFT's include a charged marginal operator, made of mixed-up components, something like

We can deform the NRCFT's by operators of this type. According to David, we obtain an S^2 of fixed points! All of those fixed points have broken parity and charge conjugation. We can start in the UV with CFT2, flow to any point on that S^2, and then flow down to CFT1. These charged marginal operators are thus ``Goldstone operators.''

According to David, these statements generalize quite broadly. The general rule is: if you break any bulk symmetry G to a subgroup H only via boundary conditions, you will always obtain in the field theory a marginal, double-trace operator, producing a manifold G/H along the RG flow. David later emphasized to me that while such a statement may be true in the large-N/supergravity limit, 1/N (or quantum) corrections may change it. I think he credited Witten with the original statement of the rule, but I'm not certain.

David also discussed in detail the form of the retarded Green's function for the fermionic operator in the NRCFT's. They all have a pole at exactly zero frequency, for any momentum! More precisely, they have the form

where the stuff depends on frequency and momentum. He called that a ``flat band.'' The spectral function has a ridge (a continuum of poles) at zero frequency, for all momenta. The kinetic energy is irrelevant. He was quick to add that finite temperature and density can suppress and dampen the pole at small momentum, and that 1/N effects may kill it completely, although I didn't entirely understand that.

I'll end with some open questions. A lot of the above physics that seems simple in the bulk sounds very weird in the field theory, especially the restoration of parity in the IR. Can we understand that better from a purely field theory point of view? Can similar games be played with other higher-spin fields (spin 1 and 3/2 come to mind)? Can any of the above can be usefully exploited for applications?


Friday, September 9, 2011 talk by Max Metlitski on Fermi quantum criticality.

I am Tarun Grover and I would be blogging about a very interesting talk presented by Max Metlitski on fermionic quantum criticality.

The problem that Max talked about concerns gapless bosonic modes coupled to a Fermi surface. The importance of this class of problems can be gauged from the fact that there exists a large set of experiments where one expects such a scenario to arise, and many of these phenomena cannot be understood within the framework of Landau Fermi liquid theory even qualitatively. Max gave a brief overview of the experimental situation involving such materials such as copper based high-Tc materials and certain quasi two-D organic systems (just to make this list even more diverse, one might add to this list recently discovered "dimensionality tuned" heavy-fermionic layered systems, see http://meetings.aps.org/Meeting/MAR11/Event/137776). Even more, Max argued that a large set of these materials have certain striking similarities:

1. The non-Fermi liquid behavior often arises near the phase transition between two phases, often in the "quantum-critical regime" (though in the context of cuprates, Max mentioned a caveat that the NFL behavior may extend even beyond the critical regime and in fact may comprise a whole new phase.).

2. The resistivity in the NFL regime is often linear in temperature T over a wide range of temperatures (going as low as one possibly can without encountering a superconducting instability) as compared to T^2 in FL at low temperatures.

So there is an ample evidence that something beyond Landau Fermi liquid paradigm is desperately needed to describe states that may have Fermi surfaces but perhaps do not have Fermi liquid quasiparticle. As it was mentioned in the talk, that in 1-D we already have such an example, the Luttinger liquid. Going beyond 1-D is what makes this problem rather challenging.

Early on in the talk, a distinction was made between problems where the bosonic modes coupled to the Fermi surface live at zero momenta (such as ferromagnetic or nemtic flcutuations) Vs those where they live at non-zero momenta (such as charge or spin density wave). Max mentioned that the former are relatively easier to tackle within the formalism to be described later on in the talk.

Next, Max described a few theoretical routes that lead to Fermi surfaces coupled to gapless bosons. One rather old example is that of half-filled Landau level where fermions are coupled to an internal U(1) gauge field (the same gauge field that leads to Chern-Simons term in the action). In this case the fermions are actually composite fermions that are obtained by binding two flux quanta to the electrons. Two another contexts where one obtains a similar looking (but different) low energy action are the problem of a metallic Fermi surface coupled to a Q = 0 order parameter (such as ferromagnetic or nematic fluctuations) and the case of a "spinon fermi surface" coupled to an internal gauge field. Here spinons refer to spin-1/2 neutral particles that formally arise when one expresses a spin operator using Schwinger fermions. The spinons take on their own life when the internal gauge field they are coupled to is in the deconfined phase (the gauge field arises because of the redundancy in the definition of spin operator).

One route to understand conventional Landau Fermi liquid theory is via renormalization group (using the work done in early 90's by R. Shankar and J. Polchinski), and it is a natural question if there is an RG approach to understand the theory of fermions coupled to gapless bosons. To set the stage for RG treatment of critical Fermi surface, Max gave a brief overview of RG for Fermi liquids. Here, one important point to note is that the low energy excitations in a Fermi liquid lie along a surface (as compared to a point e.g. for a gapless Goldstone mode) and therefore the RG thinning out of degrees of freedom needs to be accomplished along this whole surface. Since energy of particle-hole excitations scales differently parallel and perpendicular to the Fermi surface, this would have important consequences for the RG treatment of critical states as one would later see in the talk. Let us also note down the results that comes out of such an RG analysis of Fermi liquids: 1. The only relevant channels for four-fermion interaction are forward scattering and BCS scattering. 2. Forward scattering is marginal at all orders while BCS scattering is relevant (irrelevant) at second order for attractive (repulsive) interaction. 3. The fixed point theory can be solved exactly and (surprise!) the quasiparticles are infinitely long-lived. As Max mentioned, it would be interesing to consider perhaps the effect of irrelvant operators and derive the correct result for the self-energy (omega^2 log(omega) in D=2).

Having laid the ground, Max next moved to the main topic of his talk, the RG treatment for the critical Fermi states. Among other terms, the action consists of a transverse part of a dynamical boson field coupled to a fermion along the Fermi surface. A simple RPA calculation shows that the boson would be damped (this is called Landau damping in literature) and it's propagator would have a term |omega|/|q|. This has important consequences, since this means that the flcutuations of the boson are rather soft (z = 3) and if one calculates the feedback of these soft bosons on the fermions, one finds that for |omega| >> |k|^3, the fermion self-energy has a term |omega|^(2/3), that is, at this level of treatment, it is a non-Fermi liquid! One finds that it is the singular nature of forward scattering that has killed the Landau quasiparticles. Having gained some confidence that a non-Fermi liquid may be within sight, Max next described an RG treatment for this state. It turns out that for many purposes one can consider an approximate RG treatment where only patches on the opposite sides of the Fermi surface are coupled together. One interesting feature of this "two-patch theory" is that momenta along ('kx') and perpendicular ('ky') to the Fermi surface scale differently as alluded above. This is not very surprising since the energy cost for a particle hole excitation along and perpendicular to the Fermi surface scale differently. This anisotropic scaling also leads to an interesting "shift symmetry" between the momenta 'kx' and 'ky' that is akin to the usual Galilean symmetry between 'x' and 't'. In the RG treatment, one also needs to decide how to rescale the time and Max explained that it is most convenient to scale it so that the fermion gauge coupling is kept invariant under the RG transformations. This leads to a theory with dynamical exponent z = 3. Like any RG treatment, one needs a small parameter in the theory to make any sense of the RG, and one would think that having a large number of flavors of the fermion field might help. In fact, that is how the problem was addressed in late 80's and 90's and it was declared to be a closed problem with the above mentioned RPA results to be the answer in the large-N limit. However, as recently shown by Sung-Sik Lee, this does not quite work (for the reason Max would explain in part-2 of the talk) and one still ends up with a strongly coupled theory even at large N. In the next talk, Max would describe approaches that would help us get out this apparent quagmire, using the recent work done by Harvard and MIT groups.