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

2. 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

3. Basic Concepts
AdS/CFT as a tool.

4. Gauge/String duality or AdS/CFT
Solvable Toy Model(s) of
non-equilibrium
strong coupling dynamics.
“large Nc N=4 SYM”
“Finite temperature field theory
= Gravity with Black Hole”

5. Toy model needed, since:
Strong Coupling prevents perturbation
theory from being applicable.
Real Time dynamics challenging for the lattice

6. 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.
lmicr. = 1/(g2 T) at weak coupling,
lmicr. = 1/T at strong coupling!
Effective theory is more effective at strong coupling!

7. Hydrodynamics for global currents
Conserved charge density:
Conservation law:
Constitutive relation:
Diffusion Constant

8. Matching: Kubo Formulas.
(from Arnold, Moore & Yaffe)
Diffusion
constant
Current-current
2-pt function.
Need correlation
functions in microscopic theory.

9. 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?

10. Solvable Toy model can
test formalism (you don’t need AdS/CFT but it surely helps): 2nd 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

11. Defect Conformal Field Theories
Basic Concepts
Defect Conformal Field Theories

12. Defects and Interfaces.
Defects are everywhere y
We can study the same
in N=4 SYM.
Janus-Solution x

13. 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

14. 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
d-dim
CFT
(d+1)-dim
CFT
SO(d,2)
SO(d,2)
SO(d+1,2)

15. 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

16. NRCFTs and the Unitary Fermi Gas (UFG).
Basic Concepts
NRCFTs and the Unitary Fermi Gas (UFG).

17. Free Fermions: At low temperatures fermions behave very
different than classical
particles.

18. Dilute, interacting Fermi gas:
If an interacting Fermi gas is dilute, the only
interaction that matter are 2-body collisions.
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

19. 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.

20. The BEC/BCS crossover. BCS BEC
How do the properties depend on the one free paramter a?
BCS
BEC
In atomic systems a can be dialed via Feshbach resonance

21. The BEC/BCS crossover. Can one shed light on this system via AdS/CFT?
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?

22. Universality in the UFG
ξ is a pure number

23. Goals for this talk: Use holography (or AdS/CFT) to study
conformal defects in non-relativistic settings!

24. Application of NRdCFTs:

25. 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.

26. Holographic Perspecitve on Nonrelativistic dCFT
Correlation Functions in a relativistic CFT.

27. 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?

28. 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?

29. ? CFT Correlators First example: 1-pt-function (“expectation value”)
(Osborn and Petkos)
First example: 1-pt-function (“expectation value”)
For this talk we will focus on scalar operators. ?

30. ? CFT Correlators First example: 1-pt-function (“expectation value”)
(Osborn and Petkos)
First example: 1-pt-function (“expectation value”) ?
Scaling:
Δ: Dimension of the operator

31. CFT Correlators First example: 1-pt-function (“expectation value”)
(Osborn and Petkos)
First example: 1-pt-function (“expectation value”)
Scaling:
But x can not appear by translation invariance!

32. CFT Correlators First example: 1-pt-function (“expectation value”)
(Osborn and Petkos)
First example: 1-pt-function (“expectation value”)
All 1-pt functions vanish in a CFT.
Exception: Identity operator. Dimension 0.

33. CFT Correlators 2nd example: 2-pt-function:
(Osborn and Petkos)
2nd example: 2-pt-function:
|x-y| is translation and rotation invariant!

34. CFT Correlators 3-pt-function:
(Osborn and Petkos)
3-pt-function:
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!

35. OPE: OPE allows to reduce all higher point functions to
2-pt functions. 2-pt functions + OPE contain the full
dynamical information in the CFT.

36. Holographic Perspecitve on Nonrelativistic dCFT
Relativistic defects and the dOPE

37. Correlation functions in a dCFT.
Presence of planar interfaces
preserves rotations and
translations in x plane. y
y unconstrained by
translations and rotations!
But: x

38. dCFT Correlators 1-pt-function (“expectation value”)
(McAvity and Osborn)
1-pt-function (“expectation value”)
Scalar operators can have non-trivial (position
dependent) expectation values (e.g. <F2> in Janus).

39. dCFT Correlators 2-pt-function: where: (McAvity and Osborn)
free function!
(undetermined by
symmetry)
Do we really need to know all these to specify
the dCFT? Is there an analog of the OPE?

40. The dOPE Since the defect localized operators are independent
(McAvity and Osborn)
Defect localized
operator
Ambient space
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!

41. The dOPE Upshot: dOPE takes correlation function
(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?

42. The dOPE
(McAvity and Osborn)
Except:

43. 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?

44. dOPE: an example dOPE for the “no-braner” is just standard Taylor
expansion!!
Indeed Taylor expansion in y
reduces one dimension, but for
each operator we get an
infinite tower of its y derivatives

45. Defects, Interfaces and Boundaries
(D3/D5)
(Janus)
(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

46. 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

47. 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.

48. Holographic Perspecitve on Nonrelativistic dCFT
Relativistic defects: dOPE and holography
(Aharony, DeWolfe, Freedman, AK)

49. AdS/CFT with defects. What is the dual geometry to a (d+1) dim dCFT?
Conformal symmetry
Isometry
CFT:
SO(d+1,2)
AdSd+2
SO(d,2)
dCFT:
What information is encoded in A(r)?

50. Holographic “No-braner”
Holographic Defect:
Can we write pure AdS in this way (after all we
can interpret the defect-less N=4 as a dCFT)?
AdS5
AdS4

51. Slicings of AdS Poincare Patch

52. General defect geometry
the asymptotic geometry
is still AdS5
the geometry close to
defect is modified

53. Examples of holographic defects
D5 brane probe
Janus Solution
(AK, Randall)
(Bak, Hirano, Gutperle)
Defect-localized
fundamental flavor.
Jump in the dielectric
constant (iCFT).

54. holographic dOPE. Holographic Defect: Operator/state map:
dOPE = mode expansion:
dual to defect localized operators

55. 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
Mass of the
bulk scalar
“A(r) encodes the dOPE”.

56. Holographic Perspecitve on Nonrelativistic dCFT
Non-relativistic holography.

57. Non-relativistic scaling symmetry.
Non-relativistic conformal group includes the
standard Gallilean boosts, translations, rotations
+ scale invariance (+ one special conformal)
Relativistic
Scaling
free particle
realization
Non-relativistic
Scaling

58. What spacetime has this isometry?
NRConformal symmetry
Isometry
NR conformal group
in d+1 dimensions
Schd+3
(Son; Balasubramanian & McGreevy)

59. Schd+3

60. Schd+3 AdSd+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.
v is needed to make Gallilean
boosts a symmetry.

61. Schd+3
This is invariant all by itself.

62. What is the role of v? Non-relativistic holography needs an
extra lightlike direction in the bulk. Typically
taken to be compact.
Φd+3 ~ eiMv introduces a new conserved quantum
number in the bulk. What is this in the field theory?
Particle number!

63. The role of v. Like states in QM, Operators in a NRCFT are
classified by particle number.
holography:
One bulk field is dual to an infinite tower of
operators (one for each M).

64. Correlation Functions.
2-pt function:
Very different in form compared to relativistic case.
Still determined by Symmetry.
Depends on both, dimension and particle number.

65. 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 Schd+3 but just AdSd+3 in lightcone
coordinates.

66. Example with Sch5 To get not just AdS5 in LCC but Sch5
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 dt2/z4 term
(as well as a non-trivial NS B-field).

67. Holographic Perspecitve on Nonrelativistic dCFT
Put it all together: Non-relativistic Conformal Defects.

68. What we learned so far:
Relativistic
Non-Relativistic
(no defect) ?
(defect)
x, t no longer on equal footing. But still neither
can appear on the rhs by translation invariance!

69. What we learned so far:
Relativistic
Non-Relativistic
(no defect)
(defect)
x, t no longer on equal footing. But still neither
can appear on the rhs by translation invariance!

70. The NRdOPE. Same for the dOPE. Its structure determined
by translations and scaling of y.

71. 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.

72. The NRdOPE and holography.
Relativistic:
A(r) encodes the dOPE via mode expansion
So presumably: Non-Relativistic:
A(r) encodes the NRdOPE via mode expansion

73. Not so fast! and independently preserve NR conformal group!
Symmetry alone allows for:
“no braner” = pure Schd+3: A=B=cosh(r)

74. The NR holographic defect
If A(r) encodes the NRdOPE, what is B(r)?
Look at mode decomposition:

75. The radial equation: Operators dual to one and the same field but with
the eigenvalues
we want
as in relativistic mode
expansion (with d+2 instead d+1)
the 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

76. 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.

77. An Example: Showed: NMT takes CFT into NRCFT in one less
dimension. Same works for dCFTs.
NMT of Janus:

78. 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….

79. Conclusions: While NRCFTs are very different from CFTs the
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.