1. Pairing of critical Fermi-surface states
Max Metlitski
Kavli Institute for Theoretical Physics
XVIIth International Conference on Recent Progress in Many-Body Theories, Rostock, Germany, September 11, 2013
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2. Collaborators Subir Sachdev Senthil Todadri David Mross
(Harvard) (MIT) (MIT Caltech)
Erez Berg (Weizmann Inst.)
Sean Hartnoll (Stanford)
Diego Hofman
(Stanford/SLAC)
Rahul Nandkishore (Princeton)
M.M, S. Sachdev, Phys. Rev. B 82, , 2010 M.M., S. Sachdev, Phys. Rev. B 82, , 2010
M.M., D. Mross, S. Sachdev and T. Senthil - forthcoming

3. Critical Fermi surface states
Are there states with
- a sharp Fermi-surface - no Landau quasiparticles
1d: Luttinger liquids:
d > 1 ?

4. Some candidate critical Fermi surface states
Phase transitions in metals
Spinon Fermi-surface state of Mott-insulators
Halperin-Lee-Read state of Quantum Hall system at ν=1/2
Sr3Ru2O7
S.A. Grigera, R. S. Perry, A. J. Schofield et al (2001)
Field(T)
Watanabe et al (2012)
R. Willet et al (1987)

5. Phase transitions in metals
Order parameter:
Non-Fermi- liquid
Fermi liquid
Fermi liquid
pnictides, electron-doped cuprates, heavy-fermion compounds, organics, Sr3Ru2O7*

6. “Strange Metal” Resistivity
Fermi-liquid:
Strange metal:
S.Kasahara et al (2010)
Nd2-xCexCuO4
La2-xSrxCuO4
Sr3Ru2O7
Field (T)
N.P. Fournier, P. Armitage and R.L. Greene (2010)
R.A. Cooper et al (2009)
S.A. Grigera, R. S. Perry, A. J. Schofield et al (2001)

7. Phase transitions in metals: pairing instability
Non-Fermi- liquid SC
Fermi liquid
Fermi liquid

8. Phase transitions in metals (d = 2)
ferromagnet:
nematic:
charge-density wave:
spin-density wave:

9. under 90 degree rotations,
Ising-Nematic QCP
Breaking of rotational symmetry of the lattice, translational symmetry preserved
Introduce an Ising order parameter:
Transition out of a metallic state (Pomeranchuk instability)
under 90 degree rotations,

10. Some candidate critical Fermi surface states
Phase transitions in metals
Spinon Fermi-surface state of Mott-insulators
Composite-fermion liquid of Quantum Hall system at ν=1/2

11. Spinon Fermi-surface Exotic Mott insulators with no symmetry breaking.
Low energy excitations:
fermionic quasiparticles forming a Fermi-surface
- an emergent U(1) gauge-field
- gauge field fluctuations render the Fermi-surface critical
Numerical evidence for the presence of this state in the triangular lattice Hubbard model at intermediate U/t.

12. Spinon Fermi-surface
Experimental candidate: organic triangular lattice insulator EtMe3Sb[Pd(dmit)2]2
S. Yamashita et. al., (2011)
M. Yamashita et. al., (2011)

13. Some candidate critical Fermi surface states
Phase transitions in metals
Spinon Fermi-surface state of Mott-insulators
Composite-fermion liquid of Quantum Hall system at ν=1/2

14. Half-filled Landau level
R. Willet et. al. (87)

15. Half-filled Landau level: Composite Fermion Liquid
B. I. Halperin, P. A. Lee, N. Read (1993)
Form composite fermions by attaching two flux-quanta to each electron
Composite fermions see no net magnetic field and form a Fermi-surface
A U(1) Chern-Simons gauge field is used to attach flux:
Fluctuations of make the Fermi-surface of the composite fermion critical.

16. Theory of phase transitions in metals
Theory of an order parameter interacting with the Fermi surface
Difficult problem, due in part to an absence of a full RG program
Long history: - J. A. Hertz, PRB (1976), A. J. Millis (1993) - Parallel development in the context of spinon Fermi-surface concluded class with is solvable in the limit P. A. Lee, N. Nagaosa (1992), J. Polchinski (1993), B. Altshuler, L. Ioffe, A. Millis (1994)
- Similar conclusion for class with
Ar. Abanov and A. V. Chubkov (2000), Ar. Abanov, A. V. Chubukov, J. Schmalian (2003) - Problem declared open again after work of S. S. Lee (2009).
M.M. and S. Sachdev (2010) D. Mross, J. McGreevy, H. Liu and T. Senthil (2010) M.M., D. Mross, S. Sachdev and T. Senthil (to appear)

17. Theory of the Ising-nematic transition
Theory of the order parameter interacting with the Fermi surface
Fermions:
Bosons:
Interaction:

18. Theory at RPA level RPA:
Low energy dynamics controlled by the Landau-damping:
J. A. Hertz, PRB (1976)

19. Feedback on the fermions
Non Fermi-liquid ( ! )!
Reason: singular forward scattering at small angle
P. A. Lee, N. Nagaosa (1992); J. Polchinski (1993)

20. How to scale?
Order parameter
Fermions
R. Shankar
K. Wilson

21. Two-patch regime Most singular kinematic regime: two-patch
J. Polchinski (1993); B. Altshuler, L. Ioffe, A. Millis (1994).

22. B. Altshuler, L. Ioffe, A. Millis (1994).
Why two-patch regime?
Order parameter fluctuations are very soft:
Cannot effectively absorb a Landau-damped bosonic mode.
B. Altshuler, L. Ioffe, A. Millis (1994).

23. B. Altshuler, L. Ioffe, A. Millis (1994).
Why two-patch regime?
Order parameter fluctuations are very soft:
Cannot effectively absorb a Landau-damped bosonic mode unless its momentum is nearly tangent to the Fermi surface.
This “conspiracy” between fermions and bosons is special to 2+1 dimensions.
B. Altshuler, L. Ioffe, A. Millis (1994).

24. Two-patch theory
For each expand the fermion fields about two opposite points on the Fermi surface, and
Key assumption: can neglect coupling between patches.

25. Two-patch theory
Have an infinite set of 2+1 dimensional theories labeled by points on the Fermi-surface ( )
Key assumption: can neglect coupling between patches.

26. Two dimensions are unique
Strongly coupled local field theory
Bosons and fermions enter on the same footing
No local description
Effective theory weakly coupled

27. Anisotropic scaling Fermion kinetic term dictates:
A symmetry of the theory guarantees that this scaling is exact.
Fermi surface curvature does not renormalize.
J. Polchinski (1993); B. Altshuler, L. Ioffe, A. Millis (1994).

28. Two-patch scaling Critical Fermi surface Fermi-liquid does not flow
J. Polchinski (1993); B. Altshuler, L. Ioffe, A. Millis (1994), S. S. Lee (2008).

29. Scaling properties Symmetries constrain the RG properties severely
Only two anomalous dimensions
Bosons:
Fermions:
- fermion anomalous dimension
- dynamical critical exponent
M.M and S. Sachdev (2010)

30. Scaling at criticality: bosons
Simple Landau-damped frequency dependence consistent with scaling form
Static behaviour

31. Scaling at criticality: fermions
“Fermionic dynamical exponent” is half the “bosonic dynamical exponent”
Static behaviour:
Dynamic behaviour:
Tunneling density of states:
Consistent with general theory of continuous Fermi surface disappearance
M.M and S. Sachdev (2010)
T. Senthil (2008)

32. Problem No expansion parameter. Theory strongly coupled.
Direct large- expansion fails.
A more sophisticated genus expansion also fails. Unknown if limit exists.
Uncontrolled three loop calculations
S. S. Lee (2009)
M.M. and S. Sachdev (2010)

33. Problem Uncontrolled three loop calculations give
Contrary to the previous belief that no qualitatively new physics beyond one loop.
M.M. and S. Sachdev (2010)

34. How to control the expansion?
Goes back to work on the Halperin-Lee-Read state in QHE
Some control appears when
More control if
B. I. Halperin, P. A. Lee, N. Read (1994)
C. Nayak and F. Wilczek (1994)
- fixed
D. Mross, J. McGreevy, H. Liu and T. Senthil (2010)
at three loop order.

35. ε-expansion
Theory described by a single dimensionless coupling constant:
- stable fixed-point:
C. Nayak and F. Wilczek (1994)

36. Response functions Same as before,
Except non-locality of the action constrains
at three loop order.
D. Mross, J. McGreevy, H. Liu and T. Senthil (2010)

37. How to control the expansion?
Perform more conventional scaling dictated by the fermion kinetic term:
Fermion-gauge field interactions are at tree level:
Theory described by a single dimensionless coupling constant:
- irrelevant
- relevant
- marginal

38. ε-expansion Quantum corrections to scaling
C. Nayak and F. Wilczek (1994)

39. ε-expansion
- interactions marginally irrelevant. “Marginal Fermi-liquid.”
C. Nayak and F. Wilczek (1994)

40. Beyond one-loop Two expansions have been suggested:
potential difficulty:
systematic counting of powers of needs to be done!
cannot be applied to the case. -
- fixed, “perturbative” expansion
C. Nayak and F. Wilczek (1994) -
- fixed,
- expansion.
D. Mross, J. McGreevy, H. Liu and T. Senthil (2010)

41. Pairing of critical Fermi surfaces
A regular Fermi-liquid is unstable to arbitrarily weak attraction in the BCS channel.
How about a critical Fermi surface?

42. Pairing instability of the nematic transition
Nematic fluctuations lead to attraction in the BCS channel
Fundamental problem: as one approaches the critical point the pairing glue becomes strong, but the quasiparticles are destroyed
Who wins?
Non-Fermi-liquid SC
Fermi liquid
Fermi liquid

43. Pairing instabilities for spin/charge liquids
Magnetic fluctuations mediate a long-range repulsion
Can a short range attractive interaction compete with this?
What is the critical interaction strength?

44. Pairing in a spin-liquid
U(1) spin-liquid spinon Fermi surface
- spin-liquid
Excitations:
gapped spinons gapped Abrikosov vortices (visons)
gapless spinons gapless gauge (magnetic field) fluctuations

45. Pairing in a half-filled Landau level
Paired state (e.g. Moore-Read)
Composite fermion liquid
Excitations:
gapped fermions gapped vortices (charge )
gapless composite fermions gapless gauge (density) fluctuations
Exact diagonalisation data may be interpreted as a transition/crossover
H.E.Rezayi and F.D.M. Haldane (2000), G. Moller, A. Wojs and N.R.Moller (2011)

46. Pairing singularities
Scattering amplitude in the BCS channel at
In the regime , the one loop diagram is enhanced by
irrelevant in two-patch theory marginal in Fermi-liquid theory

47. Conceptual difficulties with two-patch RG
Low-energy states on the Fermi-surface cannot be integrated out

48. Conceptual difficulties with two-patch RG
Treatment of the pairing instability requires a marriage of two RG’s:
does not flow

49. D. Son’s RG procedure Keep interpatch couplings!
D. T. Son, Phys. Rev. D 59, (1999).

50. Perturbations Forward-scattering BCS scattering
Only two types of momentum conserving processes keep fermions on the FS
Forward-scattering
BCS scattering

51. Fermi-liquid RG

52. D. Son’s RG
Generation of inter-patch couplings:
Generates an RG flow:

53. Combined RG
- (from intra-patch theory)

54. Pairing: Ising-nematic transition
Always flows to (transition unstable to pairing)
Pairing preempts the Non-Fermi-liquid physics
whenever calculation controlled ( )
Fermi liquid
Non-Fermi-liquid SC

55. Pairing: Ising-nematic transition
Always flows to (transition unstable to pairing)
Pairing preempts the Non-Fermi-liquid physics
whenever expansion controlled
Fermi liquid SC

56. Pairing: Ising-nematic transition
- similar to dense QCD in 3+1d (D. T. Son, (1999)).
E. A. Yuzbashyan, unpublished (A. Chubukov, private communication)

57. Pairing: gauge field, ε =0•
Single fixed point (CFL)
Pairing gap onsets as
M.M., D. Mross, S. Sachdev, T. Senthil, forthcoming

58. Pairing: gauge field, ε >0
Gap onsets as
Pairing transition
Critical FS
Paired state
M.M., D. Mross, S. Sachdev, T. Senthil, forthcoming

59. Eliashberg approximation
All results for can be reproduced by summing rainbow graphs in the Eliashberg approximation.

60. Open questions Pairing transition Critical FS Paired state
Can we make Son’s RG more systematic?
- explore anologies with problems in particle physics
Are fermion and gauge field propagators different at the two fixed points?
Properties of the paired state: - is the “superconductor” type I or type II - how does the vortex mass vanish at the pairing transition?

61. Conclusion
Progress in understanding phase transitions in metals.
Controlled description of a superconducting instability of the QCP.
Implications for other critical Fermi-surface state (spinon Fermi-surface, Halperin-Lee-Read state of QHE).
Thank you!