Pairing of critical Fermi-surface states Max Metlitski Kavli Institute for Theoretical Physics XVIIth International Conference on Recent Progress in Many-Body.

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

Collaborators

Subir Sachdev Senthil Todadri David Mross (Harvard) (MIT) (MIT Caltech)

Diego Hofman(Stanford/SLAC)

Erez Berg(Weizmann Inst.)

Sean Hartnoll(Stanford)

Rahul Nandkishore(Princeton)

Critical Fermi surface states

• Are there states with

- a sharp Fermi-surface- no Landau quasiparticles

• 1d: Luttinger liquids:

• d > 1 ?

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)

Phase transitions in metals

Fermi liquid

• Order parameter:

Non-Fermi-liquid

Fermi liquid

•pnictides, electron-doped cuprates, heavy-fermion compounds, organics, Sr3Ru2O7*

“Strange Metal” Resistivity

Nd2-xCexCuO4 La2-xSrxCuO4 Sr3Ru2O7

Field (T)

Fermi-liquid:

Strange metal:

N.P. Fournier, P. Armitage

and R.L. Greene (2010)

S.A. Grigera, R. S. Perry,

A. J. Schofield et al (2001)

R.A. Cooper et al (2009)

S.Kasahara et al (2010)

Phase transitions in metals: pairing instability

Fermi liquid

Non-Fermi-liquid

Fermi liquidSC

Phase transitions in metals (d = 2)

ferromagnet:

nematic:

charge-density wave:

spin-density wave:

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,

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)

Theory of the Ising-nematic transition

• Theory of the order parameter interacting with the Fermi surface

Fermions:

Bosons:

Interaction:

Theory at RPA level

• RPA:

• Low energy dynamics controlled by the Landau-damping:

J. A. Hertz, PRB (1976)

Feedback on the fermions

• Non Fermi-liquid ( ! )!

• Reason: singular forward scattering at small angle

P. A. Lee, N. Nagaosa (1992); J. Polchinski (1993)

How to scale?

Order parameter Fermions

K. Wilson R. Shankar

• Most singular kinematic regime: two-patch

Two-patch regime

J. Polchinski (1993); B. Altshuler, L. Ioffe, A. Millis (1994).

• For each expand the fermion fields about two opposite points on the Fermi surface, and .

• Key assumption: can neglect coupling between patches.

Two-patch theory

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).

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)

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)

Problem

• Uncontrolled three loop calculations give

• Contrary to the previous belief that no qualitatively new physics beyondone loop.

M.M. and S. Sachdev (2010)

How to control the expansion?

• Goes back to work on the Halperin-Lee-Read state in QHE

• Some control appears when

• More control if

C. Nayak and F. Wilczek (1994)

B. I. Halperin, P. A. Lee, N. Read (1994)

- fixed

at three loop order.

D. Mross, J. McGreevy, H. Liu and T. Senthil (2010)

• A regular Fermi-liquid is unstable to arbitrarily weak attraction in the BCS channel.

• How about a critical Fermi surface?

Pairing of critical Fermi surfaces

• 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?

Pairing instability of the nematic transition

Fermi liquid

Non-Fermi-liquid

Fermi liquidSC

• Low-energy states on the Fermi-surface cannot be integrated out

Conceptual difficulties with two-patch RG

•

• Treatment of the pairing instability requires a marriage of two RG’s:

Conceptual difficulties with two-patch RG

does not flow

D. Son’s RG procedure

• Keep interpatch couplings!

D. T. Son, Phys. Rev. D 59, 094019 (1999).

Perturbations

• Only two types of momentum conserving processes keep fermions on the FS

Forward-scattering BCS scattering

Fermi-liquid RG

D. Son’s RG

• Generation of inter-patch couplings:

• Generates an RG flow:

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

Fermi liquidSC

Pairing: Ising-nematic transition

• Always flows to (transition unstable to pairing)

• Pairing preempts the Non-Fermi-liquid physics

whenever expansion controlled

SC

Fermi liquid

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!