Intertwined Orders in High Temperature Superconductors

Transcription

1 Intertwined Orders in High Temperature Superconductors Eduardo Fradkin University of Illinois at Urbana-Champaign Lectures at the CIFAR Summer School 2015 May 5, 2015 Vancouver, British Columbia, Canada 1

2 Collaborators S. Kivelson, E. Berg, A. Jaefari, R. Soto-Garrido, G. Y. Cho, D. Barci, E.-A. Kim, M. Lawler, V. Oganesyan, J. Tranquada S. Kivelson, E. Fradkin and V. J. Emery, Electronic Liquid Crystal Phases of a Doped Mott Insulator, Nature 393, 550 (1998) E. Berg, E. Fradkin, S. Kivelson, and J. Tranquada, Striped Superconductors: How the cuprates intertwine spin, charge, and superconducting orders, New. J. Phys. 11, (2009). E. Berg, E. Fradkin, and Steven A. Kivelson, Charge 4e superconductivity from pair density wave order in certain high temperature superconductors, Nat. Phys. 5, 830 (2009). E. Berg, E. Fradkin, and Steven A. Kivelson, Pair Density Wave correlations in the Kondo- Heisenberg Model, Phys. Rev. Lett. 105, (2010). A. Jaefari and E. Fradkin, Pair-Density-Wave Superconducting Order in Two-Leg Ladders, Phys. Rev. B 85, (2012). R. Soto-Garrido and E. Fradkin, Pair-Density-Wave Superconducting States and Electronic Liquid Crystal Phases, Phys. Rev. B 89, (2014). G. Y. Cho, R. Soto-Garrido, and E. Fradkin, Topological Pair-Density-Wave Superconducting States, Phys. Rev. Lett. 113, (2015). R. Soto-Garrido, G. Y. Cho and E. Fradkin, Quasi-One-Dimensional Pair-Density-Wave Superconducting States, ArXiv: E. Fradkin, S. A. Kivelson and J. M. Tranquada, arxiv:

3 Outline The Problem of High Temperature Superconductors The Competing Orders Scenario Intertwined Orders Electronic Liquid Crystal Phases and Frustrated Phase Separation The Role of Quasi-two-dimensionality Pair-Density-Wave Order: an Intertwined Order Outlook 3

4 Why is High Temperature Superconductivity different High Tc superconductors are quasi-two dimensional materials The normal phase is not a good metal: the electronic quasiparticles are not well defined: Γ ω for ω EF The mean-free-path is smaller that the distance between electrons, l n -1/2 (violation of the Ioffe-Rigel criterion) The quasiparticles of the SC state are not the leftover of the quasiparticles of the normal state The dominant interactions are repulsive, not attractive They arise from doping (i.e. changing the number of mobile electrons) a strongly correlated antiferromagnetic Mott insulator A host of other ordered (or almost ordered) states are also seen 4

5 Electronic Liquid Crystal Phases and HTSC High Tc superconductors arise from doping an antiferromagnetic Mott insulator The added holes disrupt the correlations of the antiferromagnet: attractive interaction and phase separation Coulomb repulsion and kinetic energy frustrate phase separation The result is a host of generally inhomogeneous and anisotropic phases, with and without superconductivity: electronic liquid crystal phases (Kivelson, Fradkin, Emery, 1998) Crystals, nematics, smectics (stripe), uniform phases 5

6 Electronic Liquid Crystal Phases of Doped Mott Insulators S. A. Kivelson, E. Fradkin and V. J. Emery (1998) Isotropic (Disordered) Crystal Nematic Temperature Nematic Crystal Superconducting Smectic Isotropic Smectic C 3 hω C 2 C 1 Break translation and rotational invariance to varying degrees Smectic (stripe): breaks translational symmetry in one direction Nematic: uniform and anisotropic metallic or SC phase; breaks the point group symmetry How are they related to SC? 6

7 How Liquid Crystals got an ~ or Soft Quantum Matter

8 Conducting Liquid Crystal Phases and HTSC Charge stripes: unidirectional charge order. The order parameter for wave vector K is the Fourier component ρk Spin stripes: unidirectional incommensurate spin order. The order parameter for wave vector Q is the Fourier component SQ Charge Nematic: a uniform metallic (or superconducting) state with anisotropic transport. The order parameter is a spin singlet traceless symmetric tensor (director) Qij Spin triplet nematic: Qij is a spin-triplet traceless symmetric tensor Superconductivity: the order parameter is the pair field condensate Δ Pair-Density-Waves: striped superconducting states. The order parameters are the Fourier components Δ±Q of the local pair field. Is charge order a friend or a foe of high Tc superconductivity?

9 Strong Coupling vs Weak Coupling Perspectives Smectic or stripe phase Fermi Liquid Via melting stripes Via Pomeranchuk Instability Nematic phases 9

10 The Competing Orders Scenario Landau Theory of Phase Transitions with several order parameters: one order is strongest and the others are suppressed Tc s for the different phases are quite different Regimes in which orders have similar Tc s are exceptional and require fine tuning to a (generally complex) multicritical point Although multicritical points do exist they are rare and are not a generic feature of all materials! Phase diagrams with generically comparable Tc s require a physical explanation without fine tuning 10

11 Multicriticality and Fine Tuning Example: SC and CDW competing orders F = s 2 r 2 + r sc u sc 4 + v 2 cdw 2 + apple cdw 2 r cdw 2 + r cdw 2 cdw 2 + u cdw cdw 4 If usc=ucdw=2v and the stiffnesses are equal and the MF Tc s are also equal, then this system has an U(2)=O(4) symmetry Perturbations that break the symmetry to O(2) x O(2) are strongly relevant and the Tc s split rapidly, e.g. N=4 and D=3 T SC T CDW / r SC r CDW 1/, '

12 Typical Phase Diagram of Competing Orders T One order typically dominates The Tc s are typically very different Small (or non-existent) coexistence regions No domes e.g. 1T-TaS2 CDW CDW + SC SC v 12

13 Why are the orders of comparable strength? This is difficult to understand in terms of the competing order scenario This fact suggests that all the observed orders may have a common physical origin and are intertwined Strong hint: electronic inhomogeneity. STM sees stripe and nematic local order in exquisite detail in BSCCO on a broad range of temperatures, voltage and field This phenomenology is natural in doped Mott insulators: frustrated phase separation Electronic liquid crystal phases have also been seen heavy fermions and iron superconductors 13

14 Intertwined orders in two-leg ladder arrays Jaefari, Lal and Fradkin (2010); Fradkin Kivelson and Tranquada (2014) Generalized Hubbard-Heisenberg models on 2-leg ladders: broad range of parameters with a spin gap Δs and d- wave SC power law correlations Only two relevant inter-ladder interactions: coupling between the d- wave SC order parameters and coupling between the CDW order parameters The relative relevance of these couplings changes with the charge Luttinger parameter Kc (tuned by doping) The Tc s for SC and CDW are comparable and O(1) Multicritical point P at a special value of parameters (Kc =1) where the couplings are equally relevant but with a strongly suppressed Tc s T T sc SC Luttinger Liquid Luther-Emery Liquid P SC + CDW CDW 1/2 2 1 T cdw K c Phase diagram with SC and CDW orders in quasi-1d strongly correlated systems with a spin gap. P is a multicritical point 14

15 Orders in High Temperature Superconductors There is by now abundant evidence of several types of order present in high T c superconductors, namely spin and charge stripe order LBCO, YBCO (RIXS, INS) nematic order in BSCCO (STM), YBCO (INS+Nernst) d-wave superconductivity striped superconductivity (pair-density-wave) in LBCO (transport) and LSCO (INS + Josephson resonance) When these orders appear they do so in full strength, i.e. the observed critical temperatures are comparable in magnitude 15

16 70 60 Phase Diagram of La2-xBaxCuO4 LTO LBCO Temperature (K) SC T LT LTT CO SO SC T c hole doping (x) 16 M. Hücker et al (2009)

17

18 Anisotropic Transport Below the Charge Ordering transition Li et al, 2007

19 The 2D Resistive State and 2D Superconductivity Li et al, 2007

20 Layer decoupling in LSCO in magnetic fields Schafgans et al (PRL, 2010)

21 Phase Diagram of the High Tc Superconductor YBa2Cu3 O6+y T" A" N" T" I" F" E" R" R" O" M" A" G" N" E" T" T N" Spin;" glass" MESOSCALE" ELECTRONIC" """""PHASE" SEPARATION" T*" INCIPIENT" "CHARGE" ""DENSITY" """"WAVE" T c" SUPERCONDUCTIVITY" y" 21

22 Charge Order in Underdoped YBCO Phase diagram of YBCO from thermal conductivity and resistivity in magnetic fields (Taillefer et al 2013) 22

23 Charge Order in the pseudogap phase of YBCO Sound velocity anomalies in YBCO D. LeBoeuf et al (2012) Hard X-ray diffraction J. Chang et al (2012) 23

24 Stripe and Nematic Order in BSCCO R map from STM spectra in underdoped superconducting BSCCO K. Fujita et al (2014) 24

25 PDW SC state in LBCO LBCO, the original HTSC, is known to exhibit low energy stripe fluctuations in its superconducting state It has a very low Tc near x=1/8 where it shows static stripe order in the LTT crystal structure Superconducting layer decoupling in LBCO at x=1/8 (Li et al) Layer decoupling, long range charge and spin stripe order and superconductivity: a novel striped superconducting state, a Pair Density Wave, in which charge, spin, and superconducting orders are intertwined! This can only happen if there is a special symmetry of the superconductor in the striped state that frustrates the c-axis Josephson coupling. 25

26 Period 4 Striped Superconducting State E. Berg et al, 2007 (x) x This state has intertwined striped charge, spin and superconducting orders. 26 A state of this type was found in variational Monte Carlo (Ogata et al 2004) and MFT (Poilblanc et al 2007)

27 How does this state solve the puzzle? If this order is perfect, the Josephson coupling between neighboring planes cancels exactly due to the symmetry The Josephson couplings between planes two and three layers apart also cancel by symmetry. The first non-vanishing coupling occurs at four spacings which is negligibly small Defects and/or discommensurations gives rise to small Josephson coupling between neighboring planes 27

28 Landau-Ginzburg Theory of the PDW: Order Parameters Unidirectional PDW: Δ(r)=ΔQ(r) e i Q.r + Δ-Q(r) e -iq.r, complex charge 2e singlet pair condensate with wave vector, (an LO state at zero magnetic field) Two complex SC order parameters: ΔQ(r) and Δ-Q(r) Two amplitude and two phase fields Nematic: real neutral pseudo-scalar order parameter N Charge stripe: ρk, unidirectional charge stripe with wave vector K Spin stripe order parameter: SQ, charge neutral complex spin vector order parameter 28

29 Ginzburg-Landau Free Energy F=F2+ F3 + F4 + The quadratic and quartic terms are standard Interesting cubic terms allowed if K=2Q F 3 = K Q Q +c.c. Landau theory: these orders will not develop with comparable strength unless all the parameters are fine-tuned to a multicritical point with very large symmetry! Composite order parameters K Q Q 4e Q Q Q 0 Q 29

30 Some Consequences of the GL theory The symmetry of the term coupling charge and spin order parameters requires the condition K= 2Q PDW SC order implies charge stripe order with 1/2 the period, and nematic order PDW state has pockets with a finite DOS of of Bogoliubov quasiparticles Striped SC order (PDW) uniform charge 4e SC order F 3=g4 [Δ4 * (ΔQ Δ-Q+ rotation)+ c.c.] PDW quartet condensate! The PDW state couples strongly to disorder: fragile 30

31 Coexisting uniform and striped SC order PDW order ΔQ and uniform SC order Δ0 F3,u=ϒ Δ Δ0 * ρq Δ-Q+ρ-Q ΔQ+g ρ ρ-2q ρq 2 +rotation +c.c. If Δ0 0 and ΔQ 0 there is a ρq component of the charge order! The small uniform component Δ0 removes the sensitivity to quenched disorder of the PDW state 31

32 Topological Excitations of the PDW SC E. Berg. E. Fradkin and S. A. Kivelson Strongly layered system: 2D thermal phase fluctuations play a key role Unidirectional PDW SC Two phase fields: the SC phase θ+ and the CDW phase ϕ H=(ρs ( θ+) 2 + κ ( ϕ) 2 )/2 SC vortex with Δθ+ = 2π and Δϕ=0 Bound state of a 1/2 vortex and a CDW dislocation Δθ+ = π, Δϕ= 2π Double CDW dislocation, Δθ+ = 0, Δϕ= 4π All three topological defects have logarithmic interactions 32

33 Topological Excitations of the PDW π Half-vortex and a Dislocation Double Dislocation 33

34 Thermal melting of the PDW state Three paths for thermal melting of the PDW state by a generalized Kosterlitz-Thouless mechanism Three types of topological excitations: (1,0) (SC vortex), (0,1) (double dislocation), (±1/2, ±1/2) (1/2 vortex, single dislocation bound pair) Scaling dimensions: p,q=π(ρsc p 2 +κcdw q 2 )/T If p,q=2 the defects proliferate (there is a free energy gain) Resulting phases: uniform SC, PDW, Charge 4e SC, CDW, and normal (Ising nematic) 34

35 Schematic Thermal Phase Diagram Normal T n T Nematic P P CDW 4e SC PDW T d Striped SC κ/ρ pdw Tc s are comparable without fine tuning 35

36 Effects of Disorder The striped SC order is very sensitive to disorder: disorder pinned charge density wave coupling to the phase of the striped SC SC gauge glass with zero resistance but no Meissner effect in 3D Disorder induces dislocation defects in the stripe order Due to the coupling between stripe order and SC, ±π flux vortices are induced at the dislocation core. Strict layer decoupling only allows for a magnetic coupling between randomly distributed ±π flux vortices Novel glassy physics and fractional flux the charge 4e SC order is unaffected by the Bragg glass of the pinned CDW

37 Phase Sensitive Experiments I=J2 sin(2δθ)

38 Microscopic Theories of the PDW In BCS (i.e. in FFLO) a PDW state is found if the band structure has special features, e.g. two bands with opposite spins (Zeeman coupling) and satisfying a nesting condition Otherwise the PDW instability only arises at a finite (and typically large) coupling (see Kampf et al, 2009) BCS is a weak coupling theory that does not reliably predict finite coupling condensed phases Intertwined orders necessarily require at least intermediate coupling physics Spin triplet nematic order can produce PDW phases (Soto-Garrido and EF (2014) This type of order appears naturally in the 1D Kondo-Heisenberg chain (Berg, Fradkin and Kivelson, 2010) and in two-leg ladders (Jaefari and Fradkin, 2012) 38

39 Evidence for PDW Order in the t-j model Corboz, Rice and Troyer did numerical simulations of the 2D t-j model using a large-scale tensor network approach (ipeps) As the degree of quantum entanglement increases, the uniform d wave SC, the PDW (`anti-phase ) state, and d wave SC coexisting with in-phase stripes have (essentially) the same ground state energy Effective degeneracy over a broad range of doping and for intermediate values of J/t Ordering wave vectors and stripe filling fraction vary with x and J/t VMC (Ogata et al 2004) and MFT (Poilblanc et al 2007) found a similar degeneracy P. A. Lee on an RVB PDW state (arxiv:1401:0519) PDW orders result from strong correlation physics 39

40 The Kondo-Heisenberg chain Hamiltonian t 1DEG J K Heisenberg Chain J H H KH = t X j c j c X j+1, +h.c. + J H S j S j+1 + J K X S j s(x j ) j j where s(x j )= 1 2 X c j, c j, =",# JK is the Kondo coupling and JH is the Heisenberg coupling 40

41 Earlier Results on the KH chain Sikkema et al (1997) used DMRG and found a broad regime with a spin gap In the weak coupling regime, K<<t, J, K>0 is marginally relevant and flows to a spin gap phase (numerics) Spin gap phase with staggered correlations found in the Toulouse limit (with J=0) (Zachar, Emery, Kivelson, 1999) and for K<< t, J (Zachar and Tsvelik, 2001) Possible Ordering Wave vectors: 2kF T =2kF+π/b, with 2kF=π<n> Gapless neutral (CDW) mode Generalized Luttinger theorem of Yamanaka et al (1997) with 2kF the size of the large Fermi surface Gapless charge 2e mode (η pairing composite order parameter) Search for composite order parameters (Coleman et al, 1998) 41

42 Possible Order Parameters h i b (n) =h c n, c n+1, + c n+1, c n, i (n)+cos(qn + ) PDW (n) 4e(n) =hc n, c n, c n+1, c n+1, i PDW: amplitude ΔQ of a pair condensate with wave vector Q It is a composite operator of the triplet SC Δ of the 1DEG and the Néel order parameter SQ of the spin chain ΔQ =Δ SQ The charge 4e order parameter is a 4-fermion condensate which is stable in the absence of zero momentum pair condensate We carried out DMRG simulations on chains of up to N=256 Applied a pairing field and a Zeeman field at the edge bond and edge spin (respectively) to determine which orders are favored (S. White et al) These composite order parameters show power-law decay into the bulk All fermion bilinears decay exponentially into the bulk 42

43 Results from DMRG 43

44 Expectation Values of the Order Parameters as a function of distance from the left edge 44

45 Extended Hubbard-Heisenberg model on a 2-leg ladder A. Jaefari and E. Fradkin, PRB (2012) ε t t E F j j +1 t t k a k b k H = t X i,j, c i,j+1, c i,j, +h.c. t? X j, c 1,j, c 2,j, +h.c. + U X X X n j," n j,# + V? n 1,j n 2,j + V k n i,j n i,j+1 j j i,j X X + J? S 1,j S 2,j + J k S i,j S i,j+1 j i,j bonding and anti-bonding bands cb,a=(c2±c1)/ 2 DMRG: spin gap for a substantial range of parameters (Noack et al, 1997) RG analysis (continuum model): uniform SC phase (Wu et al, 2003) 45

46 At certain electron densities the bonding band b has a rational filling, e.g. 1/2, 14/ etc., where Umklapp operators can open a charge gap in the spectrum of the b band The state breaks spontaneously translation invariance (except at 1/2 filling) and a charge gap opens in the b band The b band spin degrees of freedom at this stage behave as an effective spin chain Low-energy electron tunneling between the bands is suppressed by the charge gap The charge degrees mode of the a band is gapless and decouples (at low energies) The only (marginally) relevant coupling is the spin current interaction which drives the system into a state with a full spin gap Only one gapless charge mode. All spin degrees of freedom are gapped ( C1S0 ) All fermion bilinears have exponentially decaying correlations The physical order parameters are composite operators PDW: ΔQa=Δa SQa (Qa=2ka); CDW: ρk=sqa SQb (K=2ka+2kb) 46

47 Effective Field Theory At low energies we can expand the lattice fermions with momenta close to their Fermi wave vector ki (I=a, b) into left and right moving chiral Dirac fermions c I, (j) =e ik I x(j) R I, (x)+e ik I x(j) L I, (x) I, (x) = RI, L I, (x) (x) H 0 = v F Z dx I, (x) 3 ( i)@ x I, (x) The interactions involve local operators written in terms of the charge and spin densities and currents of the two bands, e.g. the spin current couplings H int = g s1 (J b J b + J a J a )+g s2 J b J a J = 1 2 X, =",# R R + L L We will use bosonization and the renormalization group to solve this problem 47

48 Bosonization Rules In 1D the fluctuations of the Fermi sea are particles and holes that move with the same speed they can be represented as the fluctuations of a Bose field density : J 0 =R R + L L 7! 1 x current : J 1 =R R L L 7! 1 0 (x, t) =@ t (x, t) [ (x), (y)] = i (x y) Hamiltonian: x R x L) 7! (@ x ) 2 chiral bosons : R = 1 2 ( #), L = 1 2 ( + #) Fermions are solitons (or kinks) of the Bose field Fermion-Boson mapping: R(x) 7! 1 p 2 a : e i2p R (x) : L(x) 7! 1 p 2 a : e i2p L (x) : 48

49 Effective Field Theory and RG Spin-Charge separation: I, = I,c ± I,s (I = a, b) The charge degrees of freedom of the b band, φb,c, are gapped The charge degrees of freedom of the a band, φa,c, are gapless The spin degrees of freedom of both bands, φa,s and φb,s, remain in play Define φ±,s=(φb,s± φa,s)/ 2 and θ±,s=(θb,s± θa,s)/ 2 H s = v s± 2 apple K s± (@ x # s± ) K s± (@ x s± ) 2 + cos p 4 s+ 2( a) 2 h g s1 cos p 4 s + g s2 cos p 4 #s i RG: φs+ and θs+ are gapped and Ks± 1 (SU(2) invariance) 49

50 2 Leg ladder with kf,b=π/2 Effective Hamiltonian for φs-: H s = v apple s K s (@ x # s ) (@ x s ) 2 2 K s h p p + gs1 cos 4 s + gs2 cos 4 #s i Since Ks- 1, the scaling dimensions of both perturbations equals 1 For gs1 * >0 and gs2 * >0, gs1 * 0 and gs2 * 0: Luttinger liquid (C1 S1) It can be shown (by refermionization) that this system is equivalent to two massive Majorana fermions (equivalent to two Ising models) On model is critical for gs1 * =gs2 * For gs1 * =0 and gs2 * - : PDW fixed point For gs2 * =0 and gs1 * - : a and b bands decouple, uniform SC 50

51 Phase Diagram g s1 = I g s1 = g s2 < 0 Ising critical g s2 = g s2 LL phase II g s1 For kf,b=π/2 we find three phases a LL phase a PDW phase an uniform SC The PDW-SC phase transition is in the Ising universality class The other transitions are KT We recently found that the PDW and SC phases are topological with protected Majorana zeromodes (Soto-Garrido, et al, 2015) 51

52 Conclusion: Intertwined Orders! HTSC (and LBCO in particular) exhibit intertwined orders! The orders melt in different sequences, they appear essentially with similar strength In quasi 2D systems it is natural to get complex phase diagrams with comparable critical temperatures! The structure of the phase diagrams owes as much to strong correlation physics as to quasi two dimensionality Intertwined orders appear in other HTSC (YBCO) The PDW originates from strong correlation physics Challenge: construct a general microscopic theory of intertwined orders! 52