Onset of nematic order in dwave superconductors Y

Outline 1. Nematic order in YBCO Broken lattice symmetry but no spin order 2.

Nematic order in YBCO V. Hinkov, D. Haug, B. Fauqué, P. Bourges, Y. Sidis,

V. Hinkov, D. Haug, B. Fauqué, P. Bourges, Y. Sidis, A. Ivanov, C. Bernhard,

M. Vojta, Y. Zhang, and S. Sachdev, Phys. Rev. Lett. 85, 4940 (2000) E.

Nematic ordering is equivalent to the appearance of subsidiary s pairing

Also consider in parallel another simpler, and previously understood case

Field theory for d. SC to d. SC+nematic transition Ising theory for nematic ordering

Field theory for d. SC to d. SC+nematic transition Yukawa coupling is strongly relevant

Field theory for d. SC to d. SC+idxy transition Yukawa coupling is strongly relevant

Expansion in number of fermion spin components Integrating out the fermions yields an effective

Expansion in number of fermion spin components Nf Integrating out the fermions yields an

Renormalization group analysis Couplings are local in the fermion action, so perform RG on

Fermion spectral functions E. -A. Kim, M. J. Lawler, P. Oreto, S. Sachdev, E.

Quasiparticle spectra from STM on BSCCO Y. Kohsaka, C. Taylor, P. Wahl, A. Schmidt,

Scanning tunneling microscopy of BSCCO Good fit with J. W. Alldredge, Jinho Lee, K.

Neutron Scattering-LSCO K Brillouin zone Vignolle et al. , Nature Phys. 07 Christensen et

J. Chang, Ch. Niedermayer, R. Gilardi, N. B. Christensen, H. M. Ronnow, Mc. Morrow,

SDW order parameters K 2 K 1 Brillouin zone

SDW field theory Most general theory invariant under spin rotation, square lattice space group,

SDW field theory x-translations y-translations lattice rotations spin rotations

SDW field theory M. De Prato, A. Pelissetto, and E. Vicari Phys. Rev. B

SDW field theory spin rotations x-translations y-translations lattice rotations

A. Pelissetto, S. Sachdev and E. Vicari, ar. Xiv: 0802. 0199

Coupling of quasiparticles to SDW order Wavevector mismatch suggests SDW order and nodal quasiparticles

Coupling of quasiparticles to SDW order No “Yukawa” coupling

Coupling of quasiparticles to SDW order Possible higher order coupling ~ 2

Coupling of quasiparticles to SDW order Higher - order couplings allowed by symmetry: Energy-energy

Coupling of quasiparticles to SDW order Higher - order couplings allowed by symmetry: Nematic

Coupling of quasiparticles to SDW order Higher - order couplings allowed by symmetry: Spiral

Coupling of quasiparticles to SDW order Scaling dimensions of these couplings: A. Pelissetto, S.

Coupling of quasiparticles to SDW order Coupling of nematic order is nearly marginal: Quantum-critical

Photoemission spectra of La 2 -x. Srx. Cu. O 4 EDC x=0. 145 MDC

Photoemission spectra of La 2 -x. Srx. Cu. O 4 x=0. 145 J. Chang,

Conclusions 1. Theories for damping of nodal quasiparticles in cuprates 2. Nematic theory for

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Onset of nematic order in dwave superconductors Y. Huh and S. Sachdev, ar. Xiv: 0806. 002. E-A. Kim, M. Lawler, P. Oreto, S. Sachdev, E. Fradkin and S. Kivelson, Phys. Rev. B 77, 184154 (2008). Yejin Huh Harvard A. Pelissetto, S. Sachdev, and E. Vicari, ar. Xiv: 0802. 0199.

Outline 1. Nematic order in YBCO Broken lattice symmetry but no spin order 2. Theory of the onset of nematic order in a d-wave superconductor Infinite anisotropy fixed point 3. SDW order in LSCO Emergent O(4) symmetry 4. Nodal quasiparticles at the O(4) critical point Unique selection of quasiparticle coupling to (composite) nematic order

Nematic order in YBCO V. Hinkov, D. Haug, B. Fauqué, P. Bourges, Y. Sidis, A. Ivanov, C. Bernhard, C. T. Lin, and B. Keimer , Science 319, 597 (2008)

V. Hinkov, D. Haug, B. Fauqué, P. Bourges, Y. Sidis, A. Ivanov, C. Bernhard, C. T. Lin, and B. Keimer , Science 319, 597 (2008) Nematic order in YBCO

M. Vojta, Y. Zhang, and S. Sachdev, Phys. Rev. Lett. 85, 4940 (2000) E. -A. Kim, M. J. Lawler, P. Oreto, S. Sachdev, E. Fradkin, S. A. Kivelson, ar. Xiv: 0705. 4099

Nematic ordering is equivalent to the appearance of subsidiary s pairing

Also consider in parallel another simpler, and previously understood case

Field theory for d. SC to d. SC+nematic transition Ising theory for nematic ordering M. Vojta, Y. Zhang, and S. Sachdev, Physical Review Letters 85, 4940 (2000)

Field theory for d. SC to d. SC+nematic transition Ising theory for nematic ordering Free nodal quasiparticles M. Vojta, Y. Zhang, and S. Sachdev, Physical Review Letters 85, 4940 (2000)

Field theory for d. SC to d. SC+nematic transition Yukawa coupling is strongly relevant RG analysis close to 3 dimensions yields runaway flow to strong coupling M. Vojta, Y. Zhang, and S. Sachdev, Physical Review Letters 85, 4940 (2000)

Field theory for d. SC to d. SC+idxy transition Yukawa coupling is strongly relevant RG analysis close to 3 dimensions yields a relativistically invariant fixed point with all velocities equal M. Vojta, Y. Zhang, and S. Sachdev, Physical Review Letters 85, 4940 (2000)

Expansion in number of fermion spin components Integrating out the fermions yields an effective action for the nematic order parameter Nf

Expansion in number of fermion spin components Nf Integrating out the fermions yields an effective action for the nematic order parameter E. -A. Kim, M. J. Lawler, P. Oreto, S. Sachdev, E. Fradkin, S. A. Kivelson, ar. Xiv: 0705. 4099

Expansion in number of fermion spin components Integrating out the fermions yields an effective action for the nematic order parameter Y. Huh and S. Sachdev, ar. Xiv: 0806. 0002 Nf

Renormalization group analysis Couplings are local in the fermion action, so perform RG on fermion self energy Y. Huh and S. Sachdev, ar. Xiv: 0806. 0002

Renormalization group analysis Couplings are local in the fermion action, so perform RG on fermion self energy

Renormalization group analysis

Fermion spectral functions E. -A. Kim, M. J. Lawler, P. Oreto, S. Sachdev, E. Fradkin, S. A. Kivelson, ar. Xiv: 0705. 4099

Quasiparticle spectra from STM on BSCCO Y. Kohsaka, C. Taylor, P. Wahl, A. Schmidt, Jhinhwan Lee, K. Fujita, J. Alldredge, Jinho Lee, K. Mc. Elroy, H. Eisaki, S. Uchida, D. -H. Lee, & J. C. Davis, preprint

Scanning tunneling microscopy of BSCCO Good fit with J. W. Alldredge, Jinho Lee, K. Mc. Elroy, M. Wang, K. Fujita, Y. Kohsaka, C. Taylor, H. Eisaki, S. Uchida, P. J. Hirschfeld, and J. C. Davis Nature Physics 4, 319 (2008)

Neutron Scattering-LSCO K Brillouin zone Vignolle et al. , Nature Phys. 07 Christensen et al. , PRL 04 Hayden et al. , Nature 04 Tranquada et al. , Nature 04

J. Chang, Ch. Niedermayer, R. Gilardi, N. B. Christensen, H. M. Ronnow, Mc. Morrow, M. Ay, J. Stahn, O. Sobolev, A. Hiess, S. Pailhes, C. Baines, Momono, M. Oda, M. Ido, and J. Mesot, ar. Xiv: 0712. 2181 Phase diagram predicted by E. Demler, S. Sachdev, and Y. Zhang, Phys. Rev. Lett. 87, 067202 (2001). SC D. F. N. to SC+SDW quantum critical point

SDW order parameters K 2 K 1 Brillouin zone

SDW field theory Most general theory invariant under spin rotation, square lattice space group, and time-reversal symmetries

SDW field theory x-translations y-translations lattice rotations spin rotations

SDW field theory M. De Prato, A. Pelissetto, and E. Vicari Phys. Rev. B 74, 144507 (2006).

SDW field theory spin rotations x-translations y-translations lattice rotations

A. Pelissetto, S. Sachdev and E. Vicari, ar. Xiv: 0802. 0199

Coupling of quasiparticles to SDW order Wavevector mismatch suggests SDW order and nodal quasiparticles are

Coupling of quasiparticles to SDW order No “Yukawa” coupling

Coupling of quasiparticles to SDW order Possible higher order coupling ~ 2

Coupling of quasiparticles to SDW order Higher - order couplings allowed by symmetry: Energy-energy coupling A. Pelissetto, S. Sachdev and E. Vicari, ar. Xiv: 0802. 0199

Coupling of quasiparticles to SDW order Higher - order couplings allowed by symmetry: Nematic coupling A. Pelissetto, S. Sachdev and E. Vicari, ar. Xiv: 0802. 0199

Coupling of quasiparticles to SDW order Higher - order couplings allowed by symmetry: Spiral spin order coupling A. Pelissetto, S. Sachdev and E. Vicari, ar. Xiv: 0802. 0199

Coupling of quasiparticles to SDW order Scaling dimensions of these couplings: A. Pelissetto, S. Sachdev and E. Vicari, ar. Xiv: 0802. 0199

Coupling of quasiparticles to SDW order Coupling of nematic order is nearly marginal: Quantum-critical features appear in fermion spectrum via coupling to nematic fluctuations of spin density wave order. A. Pelissetto, S. Sachdev and E. Vicari, ar. Xiv: 0802. 0199

Photoemission spectra of La 2 -x. Srx. Cu. O 4 EDC x=0. 145 MDC J. Chang, M. Shi, S. Pailhes, M. Maansson, T. Claesson, O. Tjernberg, A. Bendounan, L. Patthey, N. Momono, M. Oda, M. Ido, C. Mudry, and J. Mesot, ar. Xiv: 0708. 2782

Photoemission spectra of La 2 -x. Srx. Cu. O 4 x=0. 145 J. Chang, M. Shi, S. Pailhes, M. Maansson, T. Claesson, O. Tjernberg, A. Bendounan, L. Patthey, N. Momono, M. Oda, M. Ido, C. Mudry, and J. Mesot, ar. Xiv: 0708. 2782

Conclusions 1. Theories for damping of nodal quasiparticles in cuprates 2. Nematic theory for has a fixed point with = 0 which is approached logarithmically. The theory is expressed as an expansion in 3. Exact results for a strongly interacting fixed point with large fermion anomalous dimensions - leads to “Fermi arc” spectra at low temperatures. 4. Nematic order also emerges naturally from theory of SDW ordering, as the most efficient source of quasiparticle damping.