F-Theorem ande Expansion
Igor Klebanov

Talk at YKIS Conference
Quantum Matter, Spacetime andInformation
June 15, 2016

Talk mostly based on• IK, S. Pufu, B. Safdi, arXiv:1105.4598• IK, S. Pufu, B. Safdi, S. Sachdev, arXiv:1112.5342• S. Giombi, IK, arXiv:1409.1937• L. Fei, S. Giombi, IK, G. Tarnopolsky,arXiv:1507.01960• S. Giombi, IK, G. Tarnopolsky, arXiv:1508.06354• S. Giombi, G. Tarnopolsky, IK, arXiv:1602.01076

The c-theorem• A deep problem in QFT is how to define aquantity which decreases along RG flows andis stationary at fixed points.
• In two dimensions this problem wasbeautifully solved by Alexander
Zamolodchikov who, using the two-point
function of the stress-energy tensor, found thec-function which satisfies these properties.

• At RG fixed points the c-function coincides with theVirasoro central charge, which is also the Weylanomaly• Determines the thermal free energy.
• Determines the EE of a segment of size r Holzhey, Larsen, Wilczek• cIR < cUV follows from boost invariance and SSA Casini, Huerta• The central charge can also be found using the 2-d CFT
on the sphere of radius R:
F=-log Z = -c/3 log R

The a-theorem• In d=4 there are two Weyl anomaly coefficients• One of them, called a, is proportional to the 4-d
Euler density. It can be extracted from theEuclidean path integral on the 4-d sphere:
F=-log Z = a log R• Cardy conjectured that the a-coefficientdecreases along any RG flow.
• A proof was provided a few years ago. Komargodski,
Schwimmer

The F-theorem• How do we extend these successes to odd
dimensions, where there are no anomalies?
• In d=3 there are many CFTs, some of themdescribing critical points in statisticalmechanics and condensed matter physics.
• The free energy on the 3-sphere• In a CFT, F is a well-defined, regulator
independent quantity (there are no Weyl
invariant counter terms).
• F-theorem: FIR < FUV Jafferis, IK, Pufu, Safdi

The Entanglement Connection• -F is the universal entanglement entropy
across a circle of radius R in any (2+1)-d CFT.
Casini, Huerta, Myers• Can be tested with Ryu-Takayanagi method.
• Using the language of EE, the F-theorem wasformulated and its proof was found. Myers, Sinha;
Casini, Huerta• The interpolating function used in the proof is
the Renormalized Entanglement Entropy (REE)
Liu, Mezei

Calculating F• The simplest CFT’s involve free conformal
scalar and fermion fields. Adding mass terms
makes such a theory flow to a theory with nomassless degrees of freedom in the IR whereF=0.
• For consistency with the F-theorem, the Fvalues for free massless fields should bepositive.

Conformal Scalar on Sd• In any dimension• The eigenvalues and degeneracies are• Using zeta-function regularization in d=3,• An integral representation valid in continuous
dimension d:• Near even d, it has simple poles whose
coefficients are the a-anomalies.
• For example, inSphere free energy in continuous d• A natural quantity to consider is Giombi, IK• In odd d, this reduces to IK, Pufu, Safdi• In even d, -log Z has a pole in dimensional
regularization whose coefficient is the Weyl aanomaly. The multiplication byremoves it.
•smoothly interpolates between a-anomaly
coefficients in even and ``F-values” in odd d.

Free conformal scalar in continuous d• For a free conformal scalar on Sd• This is positive for all d and smoothly interpolates between aand F

Generalized F-theorem in continuous d?
• Based on the known F- and a-theorems, it is natural to
ask whether

holds in continuous dimension d.
• We have calculated in various examples of CFTs thatcan be defined in continuous dimension, including
double-trace flows in large N CFTs and perturbative
Wilson-Fisher fixed points in the epsilon-expansion.
• In all unitary examples that we considered, we findthatindeed decreases under RG flow. For nonunitary fixed points, the inequality
does nothave to hold.

Weakly Relevant Flows• A special class of RG flows is obtained by perturbing a CFT
by a slightly relevant operator O(x) with dimension D=d-e(e<<1)• Working in conformal perturbation theory, one finds theb-function• Here, where• There is a perturbative IR fixed point,, at• To compute the change in F from UV to IR, we conformally
map to the sphere Sd and obtain• I2 and I3 are the 2-point and 3-point integrals on Sd (Cardy)• The final result for the change ofis:• Setting g=g* ,• In a unitary CFT,is positive and C is real, so we findagreement with the generalized F-theorem in all d:• This generalizes to all d previous computations in odd d(Klebanov, Pufu, Safdi) and even d (Komargodski)

eE of EE• The fact thatis a smooth function of dimension suggests
that, in the spirit of the Wilson-Fisher e expansion, it is a
useful tool to estimate the value of F for interacting CFTs.
• Consider the 3d Ising model, and more generally the O(N)
Wilson-Fisher CFTs in d=3.
• They are strongly coupled CFTs in d=3, but they have aperturbative description in d=4-e.
• We will compute the sphere free energy perturbatively andextrapolate the result to e=1 to estimate the value of F.
• This gives an e expansion of the universal Entanglement
Entropy across a circle.

The O(N) models in d=4-e• The e-expansion works well for operator dimensions:• The 1/N expansion is worse for the small N models,such as Ising. Wilson, Kogut

The Wilson-Fisher fixed points in curved space• To renormalize the theory in curved space in d=4-e, one
starts with the bare action (Brown-Collins ’80; Hathrell ‘82)

• Divergences in the free energy are removed by expressing
all bare couplings in terms of renormalized ones• Each renormalized coupling l, a, b,… then acquires a nontrivial beta function bl, ba, bb,…
• The renormalized free energy is a finite function of therenormalized couplings and renormalization scale m thatsatisfies the Callan-Symanzik equation• The conformally invariant IR fixed point is obtained by setting
to zero all beta functions in d=4-e• The sphere free-energy at the IR fixed point in d=4-e
is then a R-independent quantity which is a function of e only

F for the O(N) scalar theory in d=4-e• We performed a perturbative calculation of F to order l5(Fei, Giombi, IK, Tarnopolsky)• The poles in the above diagrams fix the curvature beta
functions to the needed order. At the IR fixed point, we get
the final result for:

F for the 3d Ising model• Extracting precise estimates from the e-expansion requires
some resummation technique. A simple approach is to use
Pade approximants• For the Ising model (N=1), we expect to be a smoothfunction of d, such that near d=4 it reproduces theperturbative e-expansion we computed, and in d=2 itreproduces the exact central charge of the 2d Ising CFT:
c=1/2.
• The accuracy of the Pade approximants can be improved if
we impose the exact value c=1/2 (which in terms ofcorresponds to = p/12) as a boundary condition at d=2

• Using the constrained Pade approximant method, we get• Consistent with the F-theorem.
• The value of F for 3d Ising is very close to the free fieldvalue!
• A similar result was found for cT in the conformalbootstrap approach El-Showk et al.

• The dimension ofis 0.518… which is only 3.6% above
the free field value.

• The large N expansion for the d=3 O(N) model is Pufu, Safdi, IK

The first correction to Fcrit/Ffree is 24% for N=1.
• In the e expansion the first correction is only 1% for e=1:

• The e expansion gives a much more accurate approximation methodfor small N than the large N expansion.
• We have also used our results and Pade approximants to find F for thecritical O(N) models in d=3 for N>1.
They are also slightly below the corresponding free field values.

QED3• Consider QED in d=3 with massless fermions• Hereare Nf 4-component spinors, and are (three of)the usual 4x4 Dirac matrices.
• Writing, the action can be stated equivalently
in terms of 2Nf two-component spinors , .
• The model has SU(2Nf) global symmetry.
• This is often called a “chiral” symmetry. A parity invariant
mass term
would explicitly break SU(2Nf) down
to SU(Nf)xSU(Nf)xU(1).

• In d=4 the electric charge is dimensionless and has positive
beta function, i.e. QED is free in the IR• In d=3, e2 has dimension of mass, so the theory is free in theUV, but it can have non-trivial behavior in the IR.
• In the UV we have the free Maxwell theory and a collectionof free fermions. This free UV theory is not conformallyinvariant. Its sphere free energy has an explicit radiusdependence due to the Maxwell term IK, Pufu, Safdi, Sachdev• In the IR, it is believed that theory flows to an interacting
conformal phase for Nf > Ncrit

QED3 at large Nf• Integrating out the fermions, the one-loop vacuum
polarization diagram

yields an induced kinetic term Appelquist, Pisarski• At low momenta, this induced term dominates over theMaxwell term, and one gets an interacting CFT wherehas dimension 2 instead of the UV dimension 3/2.
• Scaling dimensions and correlation functions of otheroperators can be computed in 1/Nf expansion.

F at large Nf• At large Nf, the free energy on S3 in the IR CFT can also becomputed in the 1/Nf expansion. The first non-trivial
correction comes from the determinant of the induced
kinetic operator for the gauge field IK, Pufu, Sachdev, Safdi• In the UV limit, the free energy FUV diverges due to thelog(R) dependence of the Maxwell term
• Thus, there is no contradiction with F-theorem FUV>FIR,
despite the fact that the log(Nf) term in Fconf can growwithout bound for large Nf

QED3 at finite Nf and symmetry breaking• For sufficiently large Nf, this interacting fixed point is expected
to be stable, because there are no relevant operatorspreserving SU(2Nf) and parity.
• As we lower Nf, a widely discussed scenario is that for Nf lessor equal than some critical value Ncrit , the model displays
spontaneous symmetry breaking according to the pattern• This symmetry breaking is due to the condensation of thefermion bilinear
which breaks SU(2Nf) but preserves parity Pisarski; Appelquist et al• If SSB occurs, then the IR theory consists of the 2Nf2 NambuGoldstone bosons and an extra massless scalar (dual photon)

QED3• At Nf = Ncrit a quartic fermion operator (invariant underSU(2Nf) and parity) can become relevant in the IR, andrender the IR fixed point unstable towards the symmetry
breaking Di Pietro et al; Kaveh, Herbut; Braun et al• We would like to use the F-theorem to provide a constraint
on the value of Ncrit (a similar F-theorem approach to Ncrit
was considered earlier by Grover)• Since we are interested in finite Nf physics, we use theepsilon expansion and the dimensional continuation ofto estimate the value of F in QED3

QEDd in the epsilon expansion• The dimensional continuation of the model is defined in sucha way thatare 4-component spinors in all d, i.e. thegamma matrices are a formal set of 4x4 matrices wherethe vector index is continued to d-dimensions• In this way, the usual QED in 4d with Nf massless Diracfermions is connected to QED3 with 2Nf two-componentfermions.
• The even number of 3-d Dirac flavors avoids the “parityanomaly.” A. N. RedlichQEDd in the epsilon expansion• In d=4-e , the coupling e has dimension (4-d)/2=e/2. So
one finds the beta function• Thus, similarly to the Wilson-Fisher fixed point of theO(N) scalar field theory, we get a perturbative IR stable
fixed point at• This fixed point can be studied by usual perturbativemethods for any Nf , for instance scaling dimensions ofsome local operators can be determined as series
expansions in eF of QEDd in the e-expansion• As in the O(N) model discussed earlier, the calculation
requires careful renormalization of the model in curved
space in d=4-e• Building on previous results for the renormalization of QED
on Sd (Adler; Drummond, Shore; Hathrell; Jack, Osborn;…) we have performed
the calculation of F at the IR fixed point working to ordere4 ~ e2 in perturbation theory• At the IR fixed point Giombi, IK, Tarnopolsky• The term 31p/90 is from the a-anomaly coefficient of the d=4
Maxwell field.
• A new feature is the non-analytic term ~ log(Nf/e). Thisoriginates from the free Maxwell contribution, whichcontains the term log(e2R4-d)• The R dependence drops out at e=e* as a result of delicate
cancellations between the free Maxwell term and terms due
to interactions. Consistent with the expected conformal
invariance in the IR.

Schwinger Model• In d=2, the IR behavior of QED with 2Nf two-component
fermions is that of the multi-flavor Schwinger model, which
has a description in terms of the level 1 SU(2Nf) WZWmodel. This has c=2Nf-1, or =p(2Nf-1)/6. This expectationis also supported by a large Nf calculation of F in
• Therefore we use a “two-sided” Pade approximant with thisd=2 boundary condition in order to better estimate thevalue in d=3.

• The plot of the Pade resummed e-expansion evaluated at
e=1, compared to the d=3 large Nf expansion result showsthat they are very close already at Nf ~ 3

F-theorem and Ncrit• Assume that the mechanism for SSB is that of a quartic
fermion operator becoming relevant at some Nf=Ncrit• For Nf slightly above Ncrit, we have a slightly irrelevant
operator, and by conformal perturbation theory we expect
a nearby UV fixed point, which we may call QED3*• A commonly discussed scenario is that as Nf approaches
Ncrit, the two fixed points QED3 and QED3* merge andannihilate at Nf=Ncrit Herbut, Kaveh; Braun et al; Kaplan et al.

F-theorem and Ncrit• For, the conformal phase no longer exists, but theRG flow originating in the UV can “hover” near the complex
fixed points before running away to large quartic coupling andpresumably towards SSB. During this “hovering” F can be madeparametrically close to Fconf(Ncrit)• The F-theorem then requires Fconf(Ncrit) > FSB, where FSB is that of
2Nf2 +1 massless scalars

F-theorem and Ncrit• QED3 is in the SU(2Nf) invariant conformal phase forNf>4.4• For Nf=1 the SB seems to be disallowed again (justbarely!). QED3 is conformal? What about Nf=1/2?

CJ and CT• CJ determines the universal charge or spin
conductivity.
• CT enters in many contexts including theentanglement entropy. It is one of the natural
measures of degrees of freedom in a CFT.
• In d=2 satisfies the Zamolodchikov C-theorem,
but there are counterexamples in d>2.

CJ and CT in Conformal QED• Here the 1/N corrections are calculated using
the induced photon propagator.
• To find CJ we calculated Giombi, Tarnopolsky, IK• The electron mass anomalous dimension is• The calculation of CT requires more diagrams
becauseHuh and Strack• With an analytic regulator we find• Agrees with the 4-e expansion for QED.
• In d=2 agrees with the exact result for multiflavor Schwinger model• In d=3 we findAnother Estimate for SB?
• The far UV theory of free fermions anddecoupled Maxwell field is not conformal.
Define CT via the 2-point function of• Then• For the SB phase with Nambu-Goldstonebosons• IF we assumethenConclusion and Discussion• We studied the dimensional continuation of the sphere freeenergy and provided some evidence for a “generalized Ftheorem” in continuous d, interpolating between F-theorems
in odd d and a-theorems in even d.
• The quantity that appears to decrease under RG flow is•a smooth function of d, and its e-expansion is a useful tool
for finding F of CFTs in the physical integer dimension.

Conclusion and Discussion• For the critical 3d Ising model, using the e-expansion we
found that F is only 2-3% below that of the free conformal
scalar.
• Can this be compared with a direct numerical calculationof F, or of the disk Entanglement Entropy, for the 3d Ising
CFT?
• For the QED3 with Nf massless fermions, we havecomputed F using the e-expansion.
• Used F-theorem to argue that the theory is conformal forNf>4.4.
• Is QED3 conformal again for Nf=1?
• What about Nf=1/2 (a single charged 2-component
fermion)?