Electronic quasiparticles and competing orders in the cuprate superconductors

Electronic quasiparticles and competing orders in the cuprate superconductors Andrea Pelissetto Rome Subir Sachdev Ettore Vicari Pisa Yejin Huh Harvard Harvard

Gapless nodal quasiparticles in d-wave superconductors Brillouin zone

Photoemission spectra of La2-xSrxCuO4 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, arxiv:0708.2782

Photoemission spectra of La2-xSrxCuO4 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, arxiv:0708.2782

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

T c T QUANTUM T X superconducting CRITICAL d x 2-y2 superconductor 0 state X r c r Needed: Quantum critical point with nodal quasiparticles part of the critical theory

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

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

T=0 SC to SC+SDW quantum critical point E. Demler, S. Sachdev, and Y. Zhang, Phys. Rev. Lett. 87, 067202 (2001).

J. Chang, Ch. Niedermayer, R. Gilardi, N.B. Christensen, H.M. Ronnow, D.F. McMorrow, M. Ay, J. Stahn, O. Sobolev, A. Hiess, S. Pailhes, C. Baines, N. Momono, M. Oda, M. Ido, and J. Mesot, arxiv:0712.2181 SC to SC+SDW quantum critical point

SDW order parameters K2 K1 Brillouin zone S i (r, τ) = Re [ e ik 1 r Φ 1i (r, τ)+e ik 2 r Φ 2i (r, τ) ]. ( 2π )( 1 ) ( 2π )( 1 ) K 1 = a 2 ϑ, 1 2, K 2 = a 2, 1 2 ϑ,

SDW field theory L Φ = τ Φ 1 2 + v 2 1 x Φ 1 2 + v 2 2 y Φ 1 2 + τ Φ 2 2 + v 2 2 x Φ 2 2 + v 2 1 y Φ 2 2 + r( Φ 1 2 + Φ 2 2 ) + u 1 2 ( Φ 1 4 + Φ 2 4 )+ u 2 2 ( Φ2 1 2 + Φ 2 2 2 ) +w 1 Φ 1 2 Φ 2 2 + w 2 Φ 1 Φ 2 2 + w 3 Φ 1 Φ 2 2 Most general theory invariant under spin rotation, square lattice space group, and time-reversal symmetries T x T y R I T Φ 1i e iϑ Φ 1i Φ 1i Φ 2i Φ 1i Φ 1i Φ 2i Φ 2i e iϑ Φ 2i Φ 1i Φ 2i Φ 2i

SDW field theory L Φ = τ Φ 1 2 + v 2 1 x Φ 1 2 + v 2 2 y Φ 1 2 + τ Φ 2 2 + v 2 2 x Φ 2 2 + v 2 1 y Φ 2 2 + r( Φ 1 2 + Φ 2 2 ) + u 1 2 ( Φ 1 4 + Φ 2 4 )+ u 2 2 ( Φ2 1 2 + Φ 2 2 2 ) +w 1 Φ 1 2 Φ 2 2 + w 2 Φ 1 Φ 2 2 + w 3 Φ 1 Φ 2 2 Symmetries: U(1) U(1) Z 4 O(3) x-translations y-translations lattice rotations spin rotations

SDW field theory Stable fixed point in a 6-loop RG analysis: w 1 = u 1 u 2, w 2 = w 3 = u 2, v 1 = v 2 O(4) O(3) invariant theory for ϕ ai, with a =1... 4 an O(4) index, and i =1... 3 an O(3) index, and Φ 1i = ϕ 1i + iϕ 2i, Φ 2i = ϕ 3i + iϕ 4i. M. De Prato, A. Pelissetto, and E. Vicari Phys. Rev. B 74, 144507 (2006).

SDW field theory Symmetries: U(1) U(1) Z 4 O(3) spin x-translations rotations y-translations lattice rotations

SDW field theory Symmetries: O(4) O(3) spin x-translations rotations y-translations lattice rotations

Properties of O(4) O(3) fixed point: The following 9 order parameters have divergent fluctuations at the spin-density wave ordering transition with the same exponent γ: The real nematic order parameter, φ ( i Φ1i 2 Φ 2i 2) which measures breaking of Z 4 symmetry Charge density waves at 2K 1 and 2K 2 : i Φ2 1i and i Φ2 2i Charge density waves at K 1 ± K 2 : i Φ 1iΦ 2i and i Φ 1i Φ 2i At the quantum critical point, the susceptibilities of these orders all diverge as χ T γ, with γ = { 0.90(36) MZM, 6 loops 0.80(54) d = 3MS, 5 loops A. Pelissetto, S. Sachdev and E. Vicari, arxiv:0802.0199

Brillouin zone L Ψ = Ψ 1 ( τ i v F 2 ( x + y )τ z i v 2 ( x + y )τ x ) Ψ 1 +Ψ 2 ( τ i v F 2 ( x + y )τ z i v 2 ( x + y )τ x ) Ψ 2.

Coupling of quasiparticles to SDW order π Ψ h P Ψ 2 Ψ 1 0 Ψ h K Φ α Φ α Ψ h Ψ 1 K Ψ 2 π π Ψ h 0 π Wavevector mismatch suggests SDW order and nodal quasiparticles are decoupled

Coupling of quasiparticles to SDW order π Ψ h P Ψ 2 Ψ 1 0 Ψ h K Φ α Φ α Ψ h Ψ 1 K Ψ 2 π π Ψ h 0 π No Yukawa coupling ΦΨ Ψ

Coupling of quasiparticles to SDW order π Ψ h P Ψ 2 Ψ 1 0 Ψ h K Φ α Φ α Ψ h Ψ 1 K Ψ 2 π π Ψ h 0 π Possible higher order coupling ~ ΦΨ 2 Ψ

Coupling of quasiparticles to SDW order Higher - order couplings allowed by symmetry: L 1 = λ 1 ( Φ1 2 + Φ 2 2) ( Ψ 1 τ z Ψ 1 + Ψ 2 τ z Ψ 2 ) ( ) ( ) [ Energy-energy coupling A. Pelissetto, S. Sachdev and E. Vicari, arxiv:0802.0199

Coupling of quasiparticles to SDW order Higher - order couplings allowed by symmetry: L ( ) ( ) ( L 2 = λ 2 Φ1 2 Φ 2 2) ( ) Ψ 1 τ x Ψ 1 + Ψ 2 τ x Ψ 2 [ L ( Nematic ) ( coupling A. Pelissetto, S. Sachdev and E. Vicari, arxiv:0802.0199

Coupling of quasiparticles to SDW order L ( ) ( ) Higher - order couplings allowed by symmetry: ( ) ( ) [ L 3 = ɛ ijk ( Φ 1j Φ 1k + Φ ) ( ) 2jΦ 2k λ 3 Ψ 2 τ x σ i Ψ 2 + λ 3Ψ 1 τ z σ i Ψ 1 + ( Φ 1jΦ 1k Φ 2jΦ 2k ) ( λ 3 Ψ 1 τ x σ i Ψ 1 λ 3Ψ 2 τ z σ i Ψ 2 )]. Spiral spin order coupling A. Pelissetto, S. Sachdev and E. Vicari, arxiv:0802.0199

Coupling of quasiparticles to SDW order Scaling dimensions of these couplings: dim[λ 1 ] = 1 ν 2= 0.9(2) dim[λ 2 ] = dim[λ 3, λ 3] = { (γ 1) 0.05(18) MZM, 6 loops = 2 0.10(27) d = 3MS, 5 loops { 0.84(8) MZM, 6 loops 0.76(8) d = 3MS, 5 loops A. Pelissetto, S. Sachdev and E. Vicari, arxiv: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, arxiv:0802.0199

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)

STM studies of the underdoped superconductor Ca 1.90 Na 0.10 CuO 2 Cl 2 Bi 2.2 Sr 1.8 Ca 0.8 Dy 0.2 Cu 2 O y

di/dv Spectra Ca 1.90 Na 0.10 CuO 2 Cl 2 Bi 2.2 Sr 1.8 Ca 0.8 Dy 0.2 Cu 2 O y Intense Tunneling-Asymmetry (TA) variation are highly similar Y. Kohsaka et al. Science 315, 1380 (2007)

TA Contrast is at oxygen site (Cu-O-Cu bond-centered) R map (150 mv) Ca 1.88 Na 0.12 CuO 2 Cl 2, 4 K 4a 0 12 nm Y. Kohsaka et al. Science 315, 1380 (2007)

TA Contrast is at oxygen site (Cu-O-Cu bond-centered) R map (150 mv) Ca 1.88 Na 0.12 CuO 2 Cl 2, 4 K 4a 0 12 nm Broken lattice translation and rotation symmetries: valence bond supersolid or supernematic S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991). S. A. Kivelson, E. Fradkin, V. J. Emery, Nature 393, 550 (1998). M. Vojta and S. Sachdev, Phys. Rev. Lett. 83, 3916 (1999).

T T c T n Quantum Critical φ/v 1/λ c 1/λ 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, arxiv:0705.4099

d x 2 y 2 superconductor + nematic order d x 2 y 2 superconductor r c r

d x 2 y 2 ± s superconductor d x 2 y 2 superconductor r c r Order parameter - s pairing amplitude φ Also φ Φ 1 2 Φ 2 2

Field theory for dsc to dsc+nematic transition S 0 φ = [ d 2 1 xdτ 2 ( τ φ) 2 + c2 2 ( φ)2 + r 2 φ2 + u 0 24 φ4] ; [ ( )] Ising theory for nematic ordering S Ψ = d 2 k (2π) 2 T ω n Ψ 1a ( iω n + v F k x τ z + v k y τ x ) Ψ 1a + d 2 k (2π) 2 T ω n Ψ 2a ( iω n + v F k y τ z + v k x τ x ) Ψ 2a. Free nodal quasiparticles [ ] M. Vojta, Y. Zhang, and S. Sachdev, Physical Review Letters 85, 4940 (2000)

[ Field theory for dsc to dsc+nematic transition ] S Ψφ = d 2 xdτ [ λ 0 φ (Ψ 1a τ x Ψ 1a + Ψ 2a τ x Ψ 2a )], Yukawa coupling is now permitted and 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)

Expansion in number of fermion spin components Nf Integrating out the fermions yields an effective action for the nematic order parameter S φ = N f v v F Γ [ + N f 2 λ 0 φ(x, τ); v v F d 2 xdτ + irrelevant terms ] ( ) rφ 2 (x, τ) where Γ is a non-local and non-analytic functional of φ. The theory has only 2 couplings constants: r and v /v F.

Expansion in number of fermion spin components Nf Integrating out the fermions yields an effective action for the nematic order parameter S φ = N f 2 k,ω + λ2 0 8v F v φ(k, ω) 2 [r ( ω 2 + vf 2 k2 x ω 2 + vf 2 k2 x + v 2 k2 y +(x y) )] +higher order terms which cannot be neglected E.-A. Kim, M. J. Lawler, P. Oreto, S. Sachdev, E. Fradkin, S.A. Kivelson, arxiv:0705.4099

Renormalization group analysis Couplings are local in the fermion action, so perform RG on fermion self energy The 1/N f expansion has only one coupling constant at criticality: v /v F. The RG has the structure: dynamic critical exponent : z = 1 + 1 F 1 (v /v F ) N f fermion anomalous dimension : η f = 1 F 2 (v /v F ) N f RG flow equation : d(v /v F ) dl = 1 N f F 3 (v /v F ) where we have computed the functions F 1,2,3 (v /v F ).

Renormalization group analysis The RG flow is to v /v F 0 with d(v /v F ) dl = 4 π 2 N f (v /v F ) 2 ln ( ) 0.46987 (v /v F ) This implies that at the critical point, as T 0 we have the asymptotic result v v F = π2 N f 4 ln ( Λ T ) ln [ 1 0.1904 N f ln ( Λ T ) ], and a more precise result is obtained by numerically integrating the RG equation.

Renormalization group analysis In the limit v /v F 0, the effective action becomes S φ = N f v v F Γ [ + N f 2 λ 0 φ(x, τ); v v F d 2 xdτ + irrelevant terms ] ( ) rφ 2 (x, τ)

Renormalization group analysis In the limit v /v F 0, the effective action becomes S φ N v [terms that have at most a divergence ] which is a power of ln(v F /v ) The theory is controlled by the expansion parameter v /N f, and so results are asymptotically exact even for N f = 2.

Fermion spectral functions φ fluctuations broaden the fermion spectral functions except in a wedge near the nodal points E.-A. Kim, M. J. Lawler, P. Oreto, S. Sachdev, E. Fradkin, S.A. Kivelson, arxiv:0705.4099

Conclusions 1. Theories for damping of nodal quasiparticles in cuprates 2. SDW theory for LSCO yields (nearly) quantum critical spectral functions with arbitrary values of v /v F 3. Nematic theory for YBCO/BSCCO has a fixed point with v /v F = 0 which is approached logarithmically. The theory is expressed as an expansion in v /v F 4. Theories yield Fermi arc spectra.