Transcript Slide 1
A holographic perspective on non relativistic defects.
Andreas Karch (University of Washington, Seattle) based on work with Piotr Surowka and Ethan Thompson Talk at IPMU, March 12 2009
Outline Introduction of the basic concepts: AdS/CFT as a tool.
Conformal field theories and defects.
Fermions at unitarity.
Outline of the main research talk: Relativistic defects: dOPE and holography Non-Relativistic Conformal Defects Lessons for NR holography and Future Directions
Basic Concepts
AdS/CFT as a tool.
Gauge/String duality or AdS/CFT Solvable Toy Model(s) of non-equilibrium strong coupling dynamics.
“large N c N=4 SYM” “Finite temperature field theory = Gravity with Black Hole”
Toy model needed, since: Strong Coupling prevents perturbation theory from being applicable.
Real Time dynamics challenging for the lattice
Effective Theory: hydrodynamics Nevertheless: Low energy effective theory works!
We are looking for a description that is valid for: long wavelength: Expand in powers of derivatives.
l micr.
l micr.
= 1/(g 2 T) at weak coupling, = 1/T at strong coupling!
Effective theory is more effective at strong coupling!
Hydrodynamics for global currents Conserved charge density : Conservation law: Constitutive relation : Diffusion Constant
Matching: Kubo Formulas.
(from Arnold, Moore & Yaffe) Diffusion constant Need correlation functions in microscopic theory.
Current-current 2-pt function .
Do we need a toy model?
Hydrodynamics determines the non-equilibrium dynamics up to a few transport coefficients which in principle are determined by matching to microscopic physics but in practice have to be taken from experiment.
What good does it do to have a toy model where one can do that matching?
Solvable Toy model can test formalism (you don’t need AdS/CFT but it surely helps): 2 nd order hydro give quantitative guidance (what are typical/limiting values?): η/s, Nernst, relaxation time indentify new dynamical mechanisms at strong coupling: energy loss, jets
Basic Concepts
Defect Conformal Field Theories
y Defects and Interfaces.
Defects are everywhere We can study the same in N=4 SYM.
Janus-Solution x
Defects and Interfaces Clean systems are easier to study, but often all the interesting physics comes from the defects: Josephson Junctions Edge Currents Localization D-branes Typically defects have internal structure, but from far away every interface looks like a step: Scale invariance or Conformal defects, dCFT
CFTs and dCFTs: Long distance physics of the bulk material (“the ambient space”) described by a CFT, includes rescaling --- but a dCFT is not the same as CFT.
(d+1)-dim dCFT with d-dim defect SO(d,2)
t x
x
t y
y
d-dim CFT SO(d,2) (d+1)-dim CFT SO(d+1,2)
Solvable Toy model again can Be a guide in developing formalism to describe dCFTs – for standard dCFTs the formalism predates AdS/CFT. But holography helped (at least us) to generalize this formalism to NRdCFTs (to be defined).
Give quantitative guidance – should be interesting in CM applications
Basic Concepts
NRCFTs and the Unitary Fermi Gas (UFG).
Free Fermions: At low temperatures fermions behave very different than classical particles.
Dilute, interacting Fermi gas: If an interacting Fermi gas is dilute , the only interaction that matter are 2-body collisions .
V
i
j
If the range of the interactions is finite , at long distances they are essentially delta functions .
Completely universal description for many systems (neutron stars, fermionic atoms in trap, …) One free parameter: scattering length a
Pairing due to fermion interactions: Despite simplicity, the interactions lead to complex dynamics.
Fermions on opposite sides of the Fermi surface can pair up and condense.
The BEC/BCS crossover.
How do the properties depend on the one free paramter a?
BCS BEC In atomic systems a can be dialed via Feshbach resonance
The BEC/BCS crossover.
How do the properties depend on the one free paramter a?
No scale.
Strongly interacting Universal “Unitary Fermi Gas”
Can one shed light on this system via AdS/CFT?
Universality in the UFG ξ is a pure number
Goals for this talk: Use holography (or AdS/CFT) to study conformal defects in non-relativistic settings!
Application of NRdCFTs:
Nishida’s claim: BEC/BCS crossover realize “ s-wave ” superconductor (order parameter of the condesate has spin 0) Many interesting material exhibit “p-wave” condensates.
Can one study this in trapped fermionic gases?
Conventional Wisdom: No. p-wave Feshbach resonance unstable.
Nishida: Yes. 3d fermionic gas gives rise to p-wave condensate of 2d fermions localized on a defect.
Holographic Perspecitve on Nonrelativistic dCFT
Correlation Functions in a relativistic CFT.
Conformal Field Theories.
Conformal field theories give a universal description of the low energy physics close to critical points. Many different microscopic theories at long distances give rise to the same CFT.
Even more general: What constraints does conformal invariance give on observables?
Observables of CFT: CFT = no scale = no particles = no S-matrix Observables: Correlation functions.
Want to understand: What are the constraints on correlation functions?
How to organize the information about all the different correlation functions?
CFT Correlators (Osborn and Petkos) First example: 1-pt-function (“expectation value”) For this talk we will focus on scalar operators .
O
(
x
) ?
CFT Correlators (Osborn and Petkos) First example: 1-pt-function (“expectation value”)
O
(
x
) ?
Scaling:
x
x O
O
Δ: Dimension of the operator
CFT Correlators (Osborn and Petkos) First example: 1-pt-function (“expectation value”) Scaling:
x
x O
O O
(
x
)
C O x
But x can not appear by translation invariance!
CFT Correlators (Osborn and Petkos) First example: 1-pt-function (“expectation value”)
O
(
x
) 0 All 1-pt functions vanish in a CFT.
Exception: Identity operator . Dimension 0.
1
1
CFT Correlators (Osborn and Petkos) 2 nd example: 2-pt-function: |x-y| is translation and rotation invariant!
O
(
x
)
O
(
x
' )
C OO
|
x
x
|'
CFT Correlators 3-pt-function: (Osborn and Petkos) Do we need to know the full structure of these increasingly more complicate n-pt-functions to completely specify the dynamics of a CFT? NO!
OPE:
O
1 (
x
)
O
2 (
x
' )
n
|
x
x n C
|' 12 1 2
n O n
(
x
' ) OPE allows to reduce all higher point functions to 2-pt functions. 2-pt functions + OPE contain the full dynamical information in the CFT.
Holographic Perspecitve on Nonrelativistic dCFT
Relativistic defects and the dOPE
y Correlation functions in a dCFT.
x
Presence of planar interfaces preserves rotations and translations in x plane.
y unconstrained by translations and rotations!
But:
t x
x
t y
y
dCFT Correlators (McAvity and Osborn) 1-pt-function (“expectation value”)
O
(
x
)
C O y
Scalar operators can have non-trivial (position dependent) expectation values (e.g.
dCFT Correlators (McAvity and Osborn) 2-pt-function: where: Do we really need to know all these to specify the dCFT? Is there an analog of the OPE?
free function!
(undetermined by symmetry)
The dOPE (McAvity and Osborn) Ambient space operator Defect localized operator Since the defect localized operators are independent of y they form a standard representation of a standard (non-defect) CFT in one lower dimension!
The dOPE (McAvity and Osborn) Upshot: dOPE takes correlation function in (d+1)-dim dCFT (with d-dim defect) into correlation function of a standard d-dim CFT.
What about vev? Didn’t we say dCFT allows vev, but CFT does not?
The dOPE (McAvity and Osborn)
n
0 Except:
O
B O
1
y
1
1
dOPE: an example Simple example: the “no-braner” • • • take a standard CFT in d+1 dimensions declare y=0 to be a defect without changing theory boundary conditions on ambient fields at defect: X(y=0 + )=X(y=0 ) and X’(y=0 + )=X’(y=0 ) Can I use the dOPE in this case to reduce a (d+1) dimensional CFT to a d-dimensional CFT?
dOPE: an example dOPE for the “no-braner” is just standard Taylor expansion!!
O n
(
x
) ˆ
n
n y O
(
x
,
n B O n y
0 ) 1 /
n
!
Indeed Taylor expansion in y reduces one dimension, but for each operator we get an infinite tower of its y derivatives
Defects, Interfaces and Boundaries (Janus) (D3/D5) (D-brane) Defect = same ambient theory on both sides, new DOFs on interface.
Interface = different ambient theories on both sides, potentially new DOFs on interface.
Boundary = ambient space ends at boundary, potentially new DOFs localized on boundary.
Defect is special case of interface, interface is special case of boundary
The folding trick Any dCFT or iCFT can be written as a bCFT.
They are special cases in the sense that the ambient theory of this bCFT has two decoupled systems that only interact via boundary conditions.
Our dOPE results generalize to the bOPE
Entanglement entropy for defects (Azeyanagi, AK, Takayanagi, Thompson) There are however some new questions one can ask in a dCFT that can not be asked in a bCFT: EE in (1+1)d bCFT: Must be true in dCFT for
symmetric
region. But asymmetric region has non-universal EE.
Holographic Perspecitve on Nonrelativistic dCFT
Relativistic defects: dOPE and holography (Aharony, DeWolfe, Freedman, AK)
AdS/CFT with defects.
What is the dual geometry to a (d+1) dim dCFT?
CFT: dCFT: Conformal symmetry Isometry SO(d+1,2) SO(d,2) AdS d+2
ds
2
e
2
A
(
r
) 2
ds AdS d
1
dr
2 What information is encoded in A(r)?
Holographic “No-braner” Holographic Defect :
ds
2
e
2
A
(
r
)
ds
2
AdS d
1
dr
2 Can we write pure AdS in this way (after all we can interpret the defect-less N=4 as a dCFT)?
AdS 5 AdS 4
Slicings of AdS Poincare Patch
General defect geometry the asymptotic geometry is still AdS 5 the geometry close to defect is modified
ds
2
e
2
A
(
r
) 2
ds AdS d
1
dr
2
e A
(
r
)
r
cosh(
r
)
Examples of holographic defects D5 brane probe (AK, Randall) Defect-localized fundamental flavor.
Janus Solution (Bak, Hirano, Gutperle) Jump in the dielectric constant (iCFT).
holographic dOPE.
Holographic Defect :
ds
2
e
2
A
(
r
)
ds
2
AdS d
1
dr
2 Operator/state map: dOPE = mode expansion :
O
dual to defect localized operators
dOPE = mode expansion.
Eigenvalues of radial equation = dimension of defect localized operators appearing in dOPE Coefficients with which a given operator contributes from normalized eigenfunction.
A(r) gives the “potential” of the mode Schrodinger equation
“A(r) encodes the dOPE”.
Mass of the bulk scalar
Holographic Perspecitve on Nonrelativistic dCFT
Non-relativistic holography.
Non-relativistic scaling symmetry.
Non-relativistic conformal group includes the standard Gallilean boosts , translations, rotations + scale invariance (+ one special conformal) Relativistic Scaling
t x
x
t
free particle Non-relativistic Scaling
t x
x
2
t
realization
i
What spacetime has this isometry?
NRConformal symmetry NR conformal group in d+1 dimensions Isometry Sch d+3 (Son; Balasubramanian & McGreevy)
Sch d+3
Sch d+3 AdS d+3 in lightcone coordinates . The space transverse to the lightlike direction v exhibits non-relativistic dynamics in one dimension less. Similarly, NRCFT algebra in d+1 is subalgebra of relativistic CFT algebra in d+2.
x
x t z
2
t
z
v is needed to make Gallilean boosts a symmetry.
Sch d+3 This is invariant all by itself.
What is the role of v?
Non-relativistic holography needs an extra lightlike direction in the bulk. Typically taken to be compact.
Φ d+3 ~ e iMv introduces a new conserved quantum number in the bulk. What is this in the field theory?
Particle number!
The role of v.
Like states in QM, Operators in a NRCFT are classified by particle number.
holography:
N O
M
One bulk field is dual to an infinite tower of operators (one for each M).
Correlation Functions.
2-pt function: • • • Very different in form compared to relativistic case.
Still determined by Symmetry.
Depends on both, dimension and particle number.
Examples: Simple procedure to get NRCFTs with gravity dual out of CFTs with gravity dual: Start with (d+2)-dim CFT with gravity dual Compactify light-like direction (DLCQ) Obtain (d+1)-dim NRCFT
Doesn’t give Sch d+3 coordinates.
but just AdS d+3 in lightcone
Example with Sch 5 To get not just AdS 5 in LCC but Sch 5 Start with (d+2)-dim CFT with gravity dual Compactify light-like direction (DLCQ) Impose twisted boundary conditions on the circle; all fields must transform by a global U(1) shift.
Obtain (d+1)-dim NRCFT On the gravity side: NMT. Generates dt 2 /z 4 term (as well as a non-trivial NS B-field).
Holographic Perspecitve on Nonrelativistic dCFT
Put it all together: Non-relativistic Conformal Defects.
What we learned so far:
Relativistic
O
(
x
)
O
(
x
' )
C OO
|
x
x
|'
O
(
x
,
t
) 0 (no defect)
O
(
x
,
y
,
t
)
C O y
(defect)
Non-Relativistic ?
x, t no longer on equal footing. But still neither can appear on the rhs by translation invariance!
What we learned so far:
Relativistic
O
(
x
)
O
(
x
' )
C OO
|
x
x
|'
Non-Relativistic
O
(
x
,
t
) 0 (no defect)
O
(
x
,
y
,
t
)
C O y
(defect)
O
(
x
,
t
) 0
O
(
x
,
y
,
t
)
C O y
x, t no longer on equal footing. But still neither can appear on the rhs by translation invariance!
The NRdOPE.
Same for the dOPE. Its structure determined by translations and scaling of y.
NRdOPE vs dOPE.
The reduction of correlation functions of ambient operators to correlation functions of defect localized operators via the dOPE is identical in the NRdCFT and the dCFT case.
The correlators of the defect localized operators then take the standard R/NR form respectively.
The NRdOPE and holography.
Relativistic:
ds
2
e
2
A
(
r
)
ds
2
AdS d
1
dr
2 A(r) encodes the dOPE via mode expansion
So presumably: Non-Relativistic:
ds
2
e
2
A
(
r
) 2
ds Sch d
2
dr
2 A(r) encodes the NRdOPE via mode expansion
Not so fast!
and independently preserve NR conformal group!
Symmetry alone allows for: “no braner” = pure Sch d+3 : A=B=cosh(r)
The NR holographic defect If A(r) encodes the NRdOPE, what is B(r)? Look at mode decomposition:
The radial equation: the eigenvalues as in relativistic mode we want expansion (with d+2 instead d+1) t he M-dependent potential term is modified unless A=B.
Unless A=B for different values of M the radial wave equation has different potential terms !
Operators dual to one and the same field but with different values of M have completely different dOPEs
Upshot of holographic dOPE.
The one free function in the relativistic holographic defect allows us to encode one tower of modes and modefunctions via A(r).
In the non-relativistic holographic defect we need two free functions as we don’t only want to encode one dOPE for one operator in the mode expansion, but an infinite tower of dOPEs (each with an tower of modes), one for each M.
An Example: Showed: NMT takes CFT into NRCFT in one less dimension. Same works for dCFTs.
NMT of Janus:
Are there more?
From every dCFT we get NRdCFTs via Melvin twists. But there should be more NRdCFTs then this! So far we haven’t found any other analytic solutions….
Conclusions: Field Theory: While NRCFTs are very different from CFTs the dOPE that allows to reduce a dCFT to a (one lower dimensional) CFT is identical in both. Should be a useful tool in any study of NRdCFTs.
AdS/CFT: In non-relativistic holography the operator/field map really has to be understood as mapping one bulk field to an infinite tower of boundary operators.