intertwined orders in high temperature superconductors...

Intertwined Orders in High Temperature Superconductors

Eduardo FradkinUniversity of Illinois at Urbana-Champaign

Seminar at the Department of Physics of Cornell University, Ithaca NY April 17 2012

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Collaborators

S. Kivelson, V. Emery, E. Berg, A. Jaefari, 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. Fradkin, S. Kivelson, M. Lawler, J. Eisenstein, and A. Mackenzie, Nematic Fermi Fluids in Condensed Matter Physics, Annu. Rev. Condens. Matter Phys. 1, 153 (2010)•E. Berg, E. Fradkin, S. Kivelson, and J. Tranquada, Striped Superconductors: How the cuprates intertwine spin, charge, and superconducting orders, New. J. Phys. 11, 115009 (2009).•E. Berg, E. Fradkin, and S. Kivelson, Charge 4e superconductivity from pair density wave order in certain high temperature superconductors, Nature Physics 5, 830 (2009).•E. Berg, E. Fradkin and S. Kivelson, Pair Density Wave correlations in the Kondo-Heisenberg Model, Phys. Rev. B 83,100509(R) (2011).•A. Jaefari and E. Fradkin, Pair Density Wave Superconducting Order in Two-Leg Ladders, Phys. Rev. B 85, 035104 (2012).

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Outline

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Outline

• The Competing Orders Scenario

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Outline

• The Competing Orders Scenario• Intertwined Orders

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Outline

• The Competing Orders Scenario• Intertwined Orders• Relation with the concept of Electronic Liquid

Crystal Phases

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Outline


Crystal Phases

• The Role of Quasi-two-dimensionality

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Outline


Crystal Phases

• The Role of Quasi-two-dimensionality• Pair-Density-Wave Order as an Intertwined Order

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Outline


Crystal Phases

• The Role of Quasi-two-dimensionality• Pair-Density-Wave Order as an Intertwined Order• Theoretical Evidence of PDW Order in Strongly

Correlated Systems

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Orders in High Temperature Superconductors

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• There is by now abundant evidence of several types of order present in high Tc superconductors, namely

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• spin and charge stripe order LBCO, YBCO (RIXS, INS)

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• nematic order in BSCCO (STM), YBCO (INS+Nernst)

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• d-wave uniform superconductivity

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• d-wave uniform superconductivity• evidence for striped superconductivity (pair-density-

wave) in LBCO (transport) and LSCO (INS + Josephson resonance)

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• d-wave uniform superconductivity• evidence for 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

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The YBCO Phase Diagram

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• Tc vs x has a plateau with an anomaly at “1/8” (Bonn & Hardy)

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• INS finds nematic order for y~6.45 with Tc~150 K

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• INS finds nematic order for y~6.45 with Tc~150 K• NQR measurements at high fields finds a charge

stripe signature in agreement with quantum oscillations with a Tc~150 K

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• Nernst experiments find nematic order in the pseudogap regime, with Tc~150 K

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• Nernst experiments find nematic order in the pseudogap regime, with Tc~150 K

• RIXS find static stripe charge order near x=1/8 with a Tc~150 K (unpublished)

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Pseudogap

T

antif

erro

mag

net nematic

charge stripe

x

d wave superconductor

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Phase Diagram of LBCO

LBCO

SO

0.100 0.125 0.1500

10

20

30

40

50

60

70

CO

SC

LTO

Tc

TLT

SC

LTT

Tem

per

ature

(K

)

hole doping (x)

M. Hücker et al (2009)7Monday, May 7, 12

The Competing Orders Scenario

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• In the Landau Theory of Phase Transitions with several order parameters typically one order will be strongest and the others will be suppressed

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• This means that the critical temperatures for the different phases are in general quite different

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• Regimes in which orders have comparable strength are exceptional

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• Regimes in which orders have comparable strength are exceptional

• The exception to this rule are systems close to a multi-critical point. This is an exceptional situation that requires fine tuning

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Why are the orders of comparable strength?

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• This is difficult to understand in terms of the competing order scenario

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• This fact suggests that all the observed orders may have a common physical origin and are intertwined

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• A 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

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• This phenomenology is natural in doped Mott insulators: frustrated phase separation

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• This phenomenology is natural in doped Mott insulators: frustrated phase separation

• Electronic liquid crystal phases now also seen heavy fermions and iron superconductors

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How Do We Understand This Remarkable Effects?

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• Broad temperature range, T3D < T < T2D with 2D superconductivity but not in 3D, as if there is not interlayer Josephson coupling

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• In this regime there is both striped charge and spin order

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• In this regime there is both striped charge and spin order

• This can only happen if there is a special symmetry of the superconductor in the striped state that leads to an almost complete cancellation of the c-axis Josephson coupling.

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The case of La1-x Bax CuO4

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• LBCO, the original HTSC, is known to exhibit low energy stripe fluctuations in its superconducting state

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• It has a very low Tc near x=1/8 where it shows static stripe order in the LTT crystal structure

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• Experimental evidence for superconducting layer decoupling in LBCO at x=1/8

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• Clearest example of the interconnection between charge and spin order with superconductivity

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• Clearest example of the interconnection between charge and spin order with superconductivity

• 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!

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Period 4 Striped Superconducting State

• This state has intertwined striped charge, spin and superconducting orders.

• A state of this type was found in variational Monte Carlo (Ogata et al 2004) and MFT (Poilblanc et al 2007)

x)(

x

E. Berg et al, 2007

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How does this state solve the puzzle?

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• If this order is perfect, the Josephson coupling between neighboring planes cancels exactly due to the symmetry of the periodic array of π textures

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• The Josephson couplings J1 and J2 between planes two and three layers apart also cancel by symmetry.

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• The first non-vanishing coupling J3 occurs at four spacings. It is quite small and it is responsible for the non-zero but very low Tc

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• The first non-vanishing coupling J3 occurs at four spacings. It is quite small and it is responsible for the non-zero but very low Tc

• Defects and/or discommensurations gives rise to small Josephson coupling J0 neighboring planes

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Landau-Ginzburg Theory of the striped SC: Order Parameters

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• Striped SC: Δ(r)=ΔQ(r) ei Q.r+ Δ-Q(r) e-iQ.r , complex charge 2e singlet pair condensate with wave vector, (i.e. an FFLO type state at zero magnetic field)

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• Nematic: detects breaking of rotational symmetry: N, a real neutral pseudo-scalar order parameter

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• Charge stripe: ρK, unidirectional charge stripe with wave vector K

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• Charge stripe: ρK, unidirectional charge stripe with wave vector K

• Spin stripe order parameter: SQ, a neutral complex spin vector order parameter, K=2Q

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Ginzburg-Landau Free Energy Functional

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•F=F2+ F3 + F4 + ...

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•F=F2+ F3 + F4 + ...

•The quadratic and quartic terms are standard

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•F=F2+ F3 + F4 + ...


•F3= γs ρK* SQ . SQ + π/2 rotation + c.c.

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•F=F2+ F3 + F4 + ...


•F3= γs ρK* SQ . SQ + π/2 rotation + c.c.+γΔ ρK* Δ-Q* ΔQ + π/2 rotation + c.c.

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•F=F2+ F3 + F4 + ...


•F3= γs ρK* SQ . SQ + π/2 rotation + c.c.+γΔ ρK* Δ-Q* ΔQ + π/2 rotation + c.c.+gΔ N (ΔQ* ΔQ+ Δ-Q* Δ-Q -π/2 rotation) + c.c.

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•F=F2+ F3 + F4 + ...


•F3= γs ρK* SQ . SQ + π/2 rotation + c.c.+γΔ ρK* Δ-Q* ΔQ + π/2 rotation + c.c.+gΔ N (ΔQ* ΔQ+ Δ-Q* Δ-Q -π/2 rotation) + c.c.+gs N (SQ* . SQ -π/2 rotation)

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•F=F2+ F3 + F4 + ...


•F3= γs ρK* SQ . SQ + π/2 rotation + c.c.+γΔ ρK* Δ-Q* ΔQ + π/2 rotation + c.c.+gΔ N (ΔQ* ΔQ+ Δ-Q* Δ-Q -π/2 rotation) + c.c.+gs N (SQ* . SQ -π/2 rotation)+gc N (ρK* ρK -π/2 rotation)

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Some Consequences of the GL theory

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• The symmetry of the term coupling charge and spin order parameters requires the condition K= 2Q

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• Striped SC order implies charge stripe order with 1/2 the period, and of nematic order

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• Charge stripe order with wave vector 2Q and/or nematic order favors stripe superconducting order which may or may not occur depending on the coefficients in the quadratic part

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• Coupling to a charge 4e SC order parameter Δ4

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• Coupling to a charge 4e SC order parameter Δ4• F’3=g4 [Δ4* (ΔQ Δ-Q+ rotation)+ c.c.]

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• Coupling to a charge 4e SC order parameter Δ4• F’3=g4 [Δ4* (ΔQ Δ-Q+ rotation)+ c.c.]• Striped SC order (PDW) ⇒ uniform charge 4e SC order

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Coexisting uniform and striped SC order

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• PDW order ΔQ and uniform SC order Δ0

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• PDW order ΔQ and uniform SC order Δ0 • F3,u=ϒΔ Δ0* ρQ Δ-Q+ρ-Q ΔQ+gρ ρ-2Q ρQ2 +rotation

+c.c.

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+c.c.

• If Δ0≠0 and ΔQ≠0 ⇒ there is a ρQ component of the charge order!

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+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

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Topological Excitations of the Striped SC

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• ρ(r)=|ρK| cos [K r+ Φ(r)]

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• ρ(r)=|ρK| cos [K r+ Φ(r)]• Δ(r)=|ΔQ| exp[i Q r + i θQ(r)]+|Δ-Q| exp[-i Q r + i θ-Q(r)]

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• ρ(r)=|ρK| cos [K r+ Φ(r)]• Δ(r)=|ΔQ| exp[i Q r + i θQ(r)]+|Δ-Q| exp[-i Q r + i θ-Q(r)]• F3,ϒ=2ϒΔ |ρK ΔQ Δ-Q| cos[2 θ-(r)-Φ(r)]

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• ρ(r)=|ρK| cos [K r+ Φ(r)]• Δ(r)=|ΔQ| exp[i Q r + i θQ(r)]+|Δ-Q| exp[-i Q r + i θ-Q(r)]• F3,ϒ=2ϒΔ |ρK ΔQ Δ-Q| cos[2 θ-(r)-Φ(r)]• θ±Q(r)=[θ+(r) ± θ-(r)]/2

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• ρ(r)=|ρK| cos [K r+ Φ(r)]• Δ(r)=|ΔQ| exp[i Q r + i θQ(r)]+|Δ-Q| exp[-i Q r + i θ-Q(r)]• F3,ϒ=2ϒΔ |ρK ΔQ Δ-Q| cos[2 θ-(r)-Φ(r)]• θ±Q(r)=[θ+(r) ± θ-(r)]/2• θ±Q single valued mod 2π ⇒ θ± defined mod π

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• ρ(r)=|ρK| cos [K r+ Φ(r)]• Δ(r)=|ΔQ| exp[i Q r + i θQ(r)]+|Δ-Q| exp[-i Q r + i θ-Q(r)]• F3,ϒ=2ϒΔ |ρK ΔQ Δ-Q| cos[2 θ-(r)-Φ(r)]• θ±Q(r)=[θ+(r) ± θ-(r)]/2• θ±Q single valued mod 2π ⇒ θ± defined mod π

• ϕ and θ- are locked ⇒ topological defects of ϕ and θ+

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• SC vortex with Δθ+ = 2π and Δϕ=0

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• SC vortex with Δθ+ = 2π and Δϕ=0• Bound state of a 1/2 vortex and a dislocation

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• SC vortex with Δθ+ = 2π and Δϕ=0• Bound state of a 1/2 vortex and a dislocationΔθ+ = π, Δϕ= 2π

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• Double dislocation, Δθ+ = 0, Δϕ= 4π

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• Double dislocation, Δθ+ = 0, Δϕ= 4π

• All three topological defects have logarithmic interactions

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Half-vortex and a Dislocation

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Double Dislocation

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Thermal melting of the PDW state

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• Three paths for thermal melting of the PDW state

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• Three paths for thermal melting of the PDW state• Three types of topological excitations: (1,0) (SC

vortex), (0,1) (double dislocation), (±1/2, ±1/2) (1/2 vortex, single dislocation bound pair)

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• Scaling dimensions: ∆p,q=π(ρsc p2+κcdw q2)/T=2 (for marginality)

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• Scaling dimensions: ∆p,q=π(ρsc p2+κcdw q2)/T=2 (for marginality)

• Phases: PDW, Charge 4e SC, CDW, and normal (Ising nematic)

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Schematic Phase Diagram

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Intertwined Orders!

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Intertwined Orders!

• LBCO exhibits intertwined orders!

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Intertwined Orders!

• LBCO exhibits intertwined orders!• The orders melt in different sequences, they appear

essentially with similar strength

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Intertwined Orders!



• In quasi 2D systems it is natural to get complex phase diagrams with comparable critical temperatures!

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Intertwined Orders!



• 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

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Microscopic Theories of the PDW

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• 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

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• Otherwise the PDW instability only arises at a finite (and typically large) coupling (see Kampf et al, 2009)

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• BCS is a weak coupling theory that does not reliably predict finite coupling condensed phases

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• Intertwined orders necessarily require at least intermediate coupling physics

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• Intertwined orders necessarily require at least intermediate coupling physics

• We will show that it appears naturally in the 1D Kondo-Heisenberg chain and in two-leg ladders

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The Kondo-Heisenberg chain Hamiltonian

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Earlier Results on the KH chain

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• Sikkema et al (1997) used DMRG and found a broad regime with a spin gap

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• Sikkema et al (1997) used DMRG and found a broad regime with a spin gap

• In the weak coupling regime, K

Possible Order Parameters

⇥b(n) = hhc†n,�c

†n+1,⇥ + c

†n+1,�cn,⇥

ii ⇠ ¯⇥(n) + cos(Qn + �)⇥PDW (n)

�4e(n) = hc†n,�c†n,⇥c

†n+1,�c

†n+1,⇥i

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• The PDW condensate is the amplitude of a pair condensate with wave vector Q


†n+1,⇥ + c

†n+1,�cn,⇥

ii ⇠ ¯⇥(n) + cos(Qn + �)⇥PDW (n)

�4e(n) = hc†n,�c†n,⇥c

†n+1,�c

†n+1,⇥i

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• The charge 4e order parameter is a 4-fermion condensate which is stable in the absence of zero momentum pair condensate


†n+1,⇥ + c

†n+1,�cn,⇥

ii ⇠ ¯⇥(n) + cos(Qn + �)⇥PDW (n)

�4e(n) = hc†n,�c†n,⇥c

†n+1,�c

†n+1,⇥i

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• We carried out DMRG simulations on chains of up to N=256


†n+1,⇥ + c

†n+1,�cn,⇥

ii ⇠ ¯⇥(n) + cos(Qn + �)⇥PDW (n)

�4e(n) = hc†n,�c†n,⇥c

†n+1,�c

†n+1,⇥i

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• 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)


†n+1,⇥ + c

†n+1,�cn,⇥

ii ⇠ ¯⇥(n) + cos(Qn + �)⇥PDW (n)

�4e(n) = hc†n,�c†n,⇥c

†n+1,�c

†n+1,⇥i

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• 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)

• Computed the expectation value of these order parameters (as well as CDW order parameters) as a function of distance from the edge


†n+1,⇥ + c

†n+1,�cn,⇥

ii ⇠ ¯⇥(n) + cos(Qn + �)⇥PDW (n)

�4e(n) = hc†n,�c†n,⇥c

†n+1,�c

†n+1,⇥i

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Results from DMRG

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

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The Two-leg Ladder System

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Role of Nematic Fluctuations in PDW Melting?

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• If nematic fluctuations become strong, orders that break translational symmetry become progressively suppressed

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• In the absence of a lattice in 2D smectic order is not possible (at T>0) (Toner & Nelson, 1980)

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• Coupling to the lattice breaks continuous rotational invariance to the point group of the lattice

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• For a square lattice the point group is C4 and the nematic-isotropic transition is 2D Ising

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• For a square lattice the point group is C4 and the nematic-isotropic transition is 2D Ising

• As the coupling to the lattice is weakened the structure of the phase diagram changes and the nematic transition is pushed to lower temperatures

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PDW melting as a McMillan-de Gennes theory

• The electronic nematic order parameter is an antisymmetric traceless tensor

• Its presence amounts to a local change of the geometry, the local metric, seen by the charge 4e SC order parameter the CDW order parameter and the PDW order

• Metric tensor: • Novel derivative couplings in the free energy

gij = �ij + ⇥Nij

Fd =Z

dx2p

detg gijn

�

Dcdwi �K��

Dcdwj �K�

+

+ (Dsci �±Q)� �Dscj �±Q

�

+�

D4ei �4e��

D4ej �4e�

o

�4e⇢K �±Q

Dsci = ⇥i � i2eAi ± iQ �niDcdwi = ⇥i + i2Q �niD4ei = ⇥i � i4eAi

Nij = N(n̂in̂j � �ij/2)

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• Deep in the PDW phase the amplitude of the SC, CDW and nematic order parameters are fixed but their phases are not

F =Z

d2xn⌅s

2gij (⌃i⇥ � 2eAi)(⌃j⇥ � 2eAj)

+⇤

2gij (⌃i⇧ + Q�ni)(⌃j⇧ + Q�nj)

+ K1 (⇤ · �n)2 + K3 (⇤⇥ �n)2 + h|�n|2o

⇢s = |�±Q|2 + |�4e|2 = |�±Q|2 + |⇢K |2

Effective CDW stiffness reduced by gapped nematic fluctuations

�e� =�

1 + �Q2/h

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Monday, May 7, 12

Deep in the charge 4e nematic superconducting phase

F =Z

d2x

⇢K|��|2 + ⇤s|�⇥|2 +

⇤sN

2(n ·�⇥)2

�

40

↵ ✓

Monday, May 7, 12


• and are the orientational nematic fluctuations and SC phase fluctuations

F =Z

d2x

⇢K|��|2 + ⇤s|�⇥|2 +

⇤sN

2(n ·�⇥)2

�

40

↵ ✓

Monday, May 7, 12



• in the absence of orientational pinning SC (half) vortices and nematic disclinations attract each other due to the fluctuations of the local geometry (nematic fluctuations): the only topological excitations are disclinations bound to half vortices

F =Z

d2x

⇢K|��|2 + ⇤s|�⇥|2 +

⇤sN

2(n ·�⇥)2

�

40

↵ ✓

Monday, May 7, 12




• If the nematic fluctuations are gapped by lattice effects there is a crossover to the pinned case discussed before

F =Z

d2x

⇢K|��|2 + ⇤s|�⇥|2 +

⇤sN

2(n ·�⇥)2

�

40

↵ ✓

Monday, May 7, 12




• If the nematic fluctuations are gapped by lattice effects there is a crossover to the pinned case discussed before

• Fermionic SC quasiparticles couple to the pair field which is uncondensed in the charge 4e SC. This leads to a “pseudogap” in the quasiparticle spectrum

F =Z

d2x

⇢K|��|2 + ⇤s|�⇥|2 +

⇤sN

2(n ·�⇥)2

�

40

↵ ✓

Monday, May 7, 12

Conclusions and Questions

• The static stripe order seen in LBCO and other LSCO related materials can be understood in terms of the PDW, a state in which charge, spin and superconducting orders are intertwined rather than competing

• This state competes with the uniform d-wave state• Nematic and charge stripe order is seen in the pseudogap

regime of YBCO. How is it related to superconductivity?

• The observed nematic order is a state with “fluctuating stripe order”, i.e. it is a state with melted stripe order. Is the nematic state a melted PDW?

• Is the PDW peculiar to LBCO? If so why? If it is generic, why?

• A microscopic theory of the PDW is needed. This state is not accessible by a weak coupling BCS approach. Strong evidence in 1d Kondo-Heisenberg chain and in ladders.

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intertwined orders in high temperature superconductors...

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