Electronic Liquid Crystal Phases in Strongly Correlated Systems
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1 Electronic Liquid Crystal Phases in Strongly Correlated Systems Eduardo Fradkin University of Illinois at Urbana-Champaign Talk at the workshop Large Fluctuations and Collective Phenomena in Disordered Systems, Institute for Condensed Matter Theory, University of Illinois, Urbana, May 17-19, 2011
2 Collaborators S. Kivelson, V. Emery, E. Berg, D. Barci, E.-A. Kim, M. Lawler, T. Lubensky, V. Oganesyan, K. Sun, C. Wu, S. C. Zhang, J. Eisenstein, A. Kapitulnik, A. Mackenzie, J. Tranquada S. Kivelson, E. Fradkin and V. J. Emery, Electronic Liquid Crystal Phases of a Doped Mott Insulator, Nature 393, 550 (1998) S. Kivelson, I. Bindloss, E. Fradkin, V. Oganesyan, J. Tranquada, A. Kapitulnik and C.Howald, How to detect fluctuating stripes in high tempertature superconductors, Reviews of Modern Physics 75, 1201 (2003) E. Fradkin, Lectures at the Les Houches Summer School on ``Modern theories of correlated electron systems'', Les Houches, France (2009), to appear in the Proceedings of Les Houches; arxiv: E. Fradkin, S. Kivelson, M. Lawler, J. Eisenstein, and A. Mackenzie, Nematic Fermi Fluids in Condensed Matter Physics, Annual Reviews of Condensed Matter Physics 1, 153 (2010) E. Berg, E. Fradkin, S. Kivelson, and J. Tranquada, Striped Superconductors: How the cuprates intertwine spin, charge, and superconducting orders, New Journal of Physics 11, (2009).
3 Electronic Liquid Crystal phases in doped Mott insulators
4 Electronic Liquid Crystal phases in doped Mott insulators Doping a Mott insulator leads to a system with a tendency to phase separation frustrated by strong correlations
5 Electronic Liquid Crystal phases in doped Mott insulators Doping a Mott insulator leads to a system with a tendency to phase separation frustrated by strong correlations The result are electronic liquid crystal phases: crystals, stripes (smectic), nematic and fluids, with different degrees of breakdown of translation and rotational symmetry
6 Electronic Liquid Crystal phases in doped Mott insulators Doping a Mott insulator leads to a system with a tendency to phase separation frustrated by strong correlations The result are electronic liquid crystal phases: crystals, stripes (smectic), nematic and fluids, with different degrees of breakdown of translation and rotational symmetry In lattice systems these symmetries are discrete
7 Electronic Liquid Crystal phases in doped Mott insulators Doping a Mott insulator leads to a system with a tendency to phase separation frustrated by strong correlations The result are electronic liquid crystal phases: crystals, stripes (smectic), nematic and fluids, with different degrees of breakdown of translation and rotational symmetry In lattice systems these symmetries are discrete In addition to their charge and spin orders, these phases may also be superconducting
8 How Liquid Crystals got an or Soft Quantum Matter
9 Conducting Liquid Crystal Phases and HTSC
10 Conducting Liquid Crystal Phases and HTSC Stripes: unidirectional charge ordered states
11 Conducting Liquid Crystal Phases and HTSC Stripes: unidirectional charge ordered states Nematic: a uniform metallic or superconducting state with anisotropic transport
12 Conducting Liquid Crystal Phases and HTSC Stripes: unidirectional charge ordered states Nematic: a uniform metallic or superconducting state with anisotropic transport Stripe and nematic ordered states in HTSC: static stripes in LBCO and LNSCO x~1/8, in LSCO and YBCO in magnetic fields. Nematic order in the pseudogap regime of YBCO: INS ~6.45 and anisotropy in Nernst measurements
13 Conducting Liquid Crystal Phases and HTSC Stripes: unidirectional charge ordered states Nematic: a uniform metallic or superconducting state with anisotropic transport Stripe and nematic ordered states in HTSC: static stripes in LBCO and LNSCO x~1/8, in LSCO and YBCO in magnetic fields. Nematic order in the pseudogap regime of YBCO: INS ~6.45 and anisotropy in Nernst measurements Fluctuating stripes, in superconducting LSCO, LBCO away from 1/8, and underdoped YBCO
14 Conducting Liquid Crystal Phases and HTSC Stripes: unidirectional charge ordered states Nematic: a uniform metallic or superconducting state with anisotropic transport Stripe and nematic ordered states in HTSC: static stripes in LBCO and LNSCO x~1/8, in LSCO and YBCO in magnetic fields. Nematic order in the pseudogap regime of YBCO: INS ~6.45 and anisotropy in Nernst measurements Fluctuating stripes, in superconducting LSCO, LBCO away from 1/8, and underdoped YBCO The high energy electronic states seen in BSCCO by STM/STS have local nematic order
15 Conducting Liquid Crystal Phases and HTSC Stripes: unidirectional charge ordered states Nematic: a uniform metallic or superconducting state with anisotropic transport Stripe and nematic ordered states in HTSC: static stripes in LBCO and LNSCO x~1/8, in LSCO and YBCO in magnetic fields. Nematic order in the pseudogap regime of YBCO: INS ~6.45 and anisotropy in Nernst measurements Fluctuating stripes, in superconducting LSCO, LBCO away from 1/8, and underdoped YBCO The high energy electronic states seen in BSCCO by STM/STS have local nematic order Is charge order a friend or a foe of high T c
16 Nematic Order Sr3Ru2O7, Mackenzie et al, DEG in large magnetic fields J. Eisenstein et al, 1998 BSCCO, Davis et al, 2010
17 Stripe Order in HTSC La 1.86 Sr 0.14 Cu Zn O 4 La 1.85 Sr 0.15 CuO I 7K (arb. units) (a) E = 1.5 mev (c) E = 2 mev Intensity (arb. units) I 7K -I 80K (arb. units) (b) E = 0 (d) E = (0.5+h,0.5,0) T = 1.5 K T = 50 K (0.5+h,0.5,0) Intensity (arb. units) Fluctuating Stripes in LSCO Tranquada et al, 1995 Static Charge Stripes in LBCO Abbamonte et al 2004
18 The case of La1-x Bax CuO4
19 The case of La1-x Bax CuO4 LBCO, the original HTSC, is known to exhibit low energy stripe fluctuations in its superconducting state
20 The case of La1-x Bax CuO4 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
21 The case of La1-x Bax CuO4 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 Experimental evidence for superconducting layer decoupling in LBCO at x=1/8
22 The case of La1-x Bax CuO4 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 Experimental evidence for superconducting layer decoupling in LBCO at x=1/8 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!
23 70 60 Phase Diagram of LBCO LTO LBCO Temperature (K) SC T LT LTT CO SO SC T c hole doping (x) M. Hücker et al (2009)
24 Li et al (2007): Dynamical Layer Decoupling in LBCO
25 Li et al (2007): Dynamical Layer Decoupling in LBCO ARPES: anti-nodal d-wave SC gap is large and unsuppressed at 1/8
26 Li et al (2007): Dynamical Layer Decoupling in LBCO ARPES: anti-nodal d-wave SC gap is large and unsuppressed at 1/8 Static charge stripe order for T< T charge=54 K
27 Li et al (2007): Dynamical Layer Decoupling in LBCO ARPES: anti-nodal d-wave SC gap is large and unsuppressed at 1/8 Static charge stripe order for T< T charge=54 K Static Stripe Spin order T < T spin= 42K
28 Li et al (2007): Dynamical Layer Decoupling in LBCO ARPES: anti-nodal d-wave SC gap is large and unsuppressed at 1/8 Static charge stripe order for T< T charge=54 K Static Stripe Spin order T < T spin= 42K ρab drops rapidly to zero from Tspin to TKT
29 Li et al (2007): Dynamical Layer Decoupling in LBCO ARPES: anti-nodal d-wave SC gap is large and unsuppressed at 1/8 Static charge stripe order for T< T charge=54 K Static Stripe Spin order T < T spin= 42K ρab drops rapidly to zero from Tspin to TKT ρab shows KT behavior for Tspin > T > TKT
30 Li et al (2007): Dynamical Layer Decoupling in LBCO ARPES: anti-nodal d-wave SC gap is large and unsuppressed at 1/8 Static charge stripe order for T< T charge=54 K Static Stripe Spin order T < T spin= 42K ρab drops rapidly to zero from Tspin to TKT ρab shows KT behavior for Tspin > T > TKT ρc as T for T>T ** 35K
31 Li et al (2007): Dynamical Layer Decoupling in LBCO ARPES: anti-nodal d-wave SC gap is large and unsuppressed at 1/8 Static charge stripe order for T< T charge=54 K Static Stripe Spin order T < T spin= 42K ρab drops rapidly to zero from Tspin to TKT ρab shows KT behavior for Tspin > T > TKT ρc as T for T>T ** 35K ρc 0 as T T3D= 10K
32 Li et al (2007): Dynamical Layer Decoupling in LBCO ARPES: anti-nodal d-wave SC gap is large and unsuppressed at 1/8 Static charge stripe order for T< T charge=54 K Static Stripe Spin order T < T spin= 42K ρab drops rapidly to zero from Tspin to TKT ρab shows KT behavior for Tspin > T > TKT ρc as T for T>T ** 35K ρc 0 as T T3D= 10K ρc / ρab for TKT > T > T3D
33 Li et al (2007): Dynamical Layer Decoupling in LBCO ARPES: anti-nodal d-wave SC gap is large and unsuppressed at 1/8 Static charge stripe order for T< T charge=54 K Static Stripe Spin order T < T spin= 42K ρab drops rapidly to zero from Tspin to TKT ρab shows KT behavior for Tspin > T > TKT ρc as T for T>T ** 35K ρc 0 as T T3D= 10K ρc / ρab for TKT > T > T3D Meissner state only below T c= 4K
34 Anisotropic Transport Below the Charge Ordering transition Li et al, 2007
35 How Do We Understand This Remarkable Effects?
36 How Do We Understand This Remarkable Effects? Broad temperature range, T3D < T < T2D with 2D superconductivity but not in 3D, as if there is not interlayer Josephson coupling
37 How Do We Understand This Remarkable Effects? Broad temperature range, T3D < T < T2D with 2D superconductivity but not in 3D, as if there is not interlayer Josephson coupling In this regime there is both striped charge and spin order
38 How Do We Understand This Remarkable Effects? Broad temperature range, T3D < T < T2D with 2D superconductivity but not in 3D, as if there is not interlayer Josephson coupling 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.
39 A Striped Textured Superconducting Phase
40 A Striped Textured Superconducting Phase The stripe state in the LTT crystal structure has two planes in the unit cell.
41 A Striped Textured Superconducting Phase The stripe state in the LTT crystal structure has two planes in the unit cell. Stripes in the 2nd neighbor planes are shifted by half a period to minimize the Coulomb interaction: 4 planes per unit cell
42 A Striped Textured Superconducting Phase The stripe state in the LTT crystal structure has two planes in the unit cell. Stripes in the 2nd neighbor planes are shifted by half a period to minimize the Coulomb interaction: 4 planes per unit cell The AFM spin order suffers a π phase shift accross the charge stripe which has period 4
43 A Striped Textured Superconducting Phase The stripe state in the LTT crystal structure has two planes in the unit cell. Stripes in the 2nd neighbor planes are shifted by half a period to minimize the Coulomb interaction: 4 planes per unit cell The AFM spin order suffers a π phase shift accross the charge stripe which has period 4 We propose that the superconducting order is also striped and also suffers a π phase shift.
44 A Striped Textured Superconducting Phase The stripe state in the LTT crystal structure has two planes in the unit cell. Stripes in the 2nd neighbor planes are shifted by half a period to minimize the Coulomb interaction: 4 planes per unit cell The AFM spin order suffers a π phase shift accross the charge stripe which has period 4 We propose that the superconducting order is also striped and also suffers a π phase shift. The superconductivity resides in the spin gap regions and there is a π phase shift in the SC order across the AFM regions
45 Period 4 Striped Superconducting State E. Berg et al, 2007 (x) x
46 Period 4 Striped Superconducting State E. Berg et al, 2007 (x) x This state has intertwined striped charge, spin and superconducting orders.
47 Period 4 Striped Superconducting State E. Berg et al, 2007 (x) x 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)
48 How does this state solve the puzzle?
49 How does this state solve the puzzle? If this order is perfect, the Josephson coupling between neighboring planes cancels exactly due to the symmetry
50 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 J1 and J2 between planes two and three layers apart also cancel by symmetry.
51 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 J1 and J2 between planes two and three layers apart also cancel by symmetry. 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
52 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 J1 and J2 between planes two and three layers apart also cancel by symmetry. 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
53 Landau-Ginzburg Theory of the striped SC: Order Parameters
54 Landau-Ginzburg Theory of the striped SC: Order Parameters Striped SC: Δ(r)=ΔQ(r) e i 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)
55 Landau-Ginzburg Theory of the striped SC: Order Parameters Striped SC: Δ(r)=ΔQ(r) e i 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) Nematic: detects breaking of rotational symmetry: N, a real neutral pseudo-scalar order parameter
56 Landau-Ginzburg Theory of the striped SC: Order Parameters Striped SC: Δ(r)=ΔQ(r) e i 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) Nematic: detects breaking of rotational symmetry: N, a real neutral pseudo-scalar order parameter Charge stripe: ρk, unidirectional charge stripe with wave vector K
57 Landau-Ginzburg Theory of the striped SC: Order Parameters Striped SC: Δ(r)=ΔQ(r) e i 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) Nematic: detects breaking of rotational symmetry: N, a real neutral pseudo-scalar order parameter Charge stripe: ρk, unidirectional charge stripe with wave vector K Spin stripe order parameter: S Q, a neutral complex spin vector order parameter, K=2Q
58 Ginzburg-Landau Free Energy Functional
59 Ginzburg-Landau Free Energy Functional F=F2+ F3 + F4 +...
60 Ginzburg-Landau Free Energy Functional F=F2+ F3 + F The quadratic and quartic terms are standard
61 Ginzburg-Landau Free Energy Functional F=F2+ F3 + F The quadratic and quartic terms are standard F3= γs ρk * SQ. SQ + π/2 rotation + c.c.
62 Ginzburg-Landau Free Energy Functional F=F2+ F3 + F The quadratic and quartic terms are standard F3= γs ρk * SQ. SQ + π/2 rotation + c.c. +γδ ρk * Δ-Q * ΔQ + π/2 rotation + c.c.
63 Ginzburg-Landau Free Energy Functional F=F2+ F3 + F The quadratic and quartic terms are standard 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.
64 Ginzburg-Landau Free Energy Functional F=F2+ F3 + F The quadratic and quartic terms are standard 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)
65 Ginzburg-Landau Free Energy Functional F=F2+ F3 + F The quadratic and quartic terms are standard 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)
66 Some Consequences of the GL theory
67 Some Consequences of the GL theory The symmetry of the term coupling charge and spin order parameters requires the condition K= 2Q
68 Some Consequences of the GL theory The symmetry of the term coupling charge and spin order parameters requires the condition K= 2Q Striped SC order implies charge stripe order with 1/2 the period, and of nematic order
69 Some Consequences of the GL theory The symmetry of the term coupling charge and spin order parameters requires the condition K= 2Q Striped SC order implies charge stripe order with 1/2 the period, and of nematic order 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
70 Some Consequences of the GL theory The symmetry of the term coupling charge and spin order parameters requires the condition K= 2Q Striped SC order implies charge stripe order with 1/2 the period, and of nematic order 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 Coupling to a charge 4e SC order parameter Δ4
71 Some Consequences of the GL theory The symmetry of the term coupling charge and spin order parameters requires the condition K= 2Q Striped SC order implies charge stripe order with 1/2 the period, and of nematic order 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 Coupling to a charge 4e SC order parameter Δ4 F 3=g4 [Δ4 * (ΔQ Δ-Q+ rotation)+ c.c.]
72 Some Consequences of the GL theory The symmetry of the term coupling charge and spin order parameters requires the condition K= 2Q Striped SC order implies charge stripe order with 1/2 the period, and of nematic order 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 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
73 Coexisting uniform and striped SC order
74 Coexisting uniform and striped SC order PDW order ΔQ and uniform SC order Δ0
75 Coexisting uniform and striped SC order PDW order ΔQ and uniform SC order Δ0 F3,u=ϒΔ Δ0 * ρq Δ-Q+ρ-Q ΔQ+gρ ρ-2q ρq 2 +rotation +c.c.
76 Coexisting uniform and striped SC order PDW order ΔQ and uniform SC order Δ0 F3,u=ϒΔ Δ0 * ρq Δ-Q+ρ-Q ΔQ+gρ ρ-2q ρq 2 +rotation +c.c. If Δ0 0 and ΔQ 0 there is a ρq component of the charge order!
77 Coexisting uniform and striped SC order PDW order ΔQ and uniform SC order Δ0 F3,u=ϒΔ Δ0 * ρq Δ-Q+ρ-Q ΔQ+gρ 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
78 Topological Excitations of the Striped SC
79 Topological Excitations of the Striped SC ρ(r)= ρk cos [K r+ Φ(r)]
80 Topological Excitations of the Striped SC 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)]
81 Topological Excitations of the Striped SC 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)]
82 Topological Excitations of the Striped SC 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
83 Topological Excitations of the Striped SC 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
84 Topological Excitations of the Striped SC 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+
85
86 Topological Excitations of the Striped SC
87 Topological Excitations of the Striped SC SC vortex with Δθ+ = 2π and Δϕ=0
88 Topological Excitations of the Striped SC SC vortex with Δθ+ = 2π and Δϕ=0 Bound state of a 1/2 vortex and a dislocation
89 Topological Excitations of the Striped SC SC vortex with Δθ+ = 2π and Δϕ=0 Bound state of a 1/2 vortex and a dislocation Δθ+ = π, Δϕ= 2π
90 Topological Excitations of the Striped SC SC vortex with Δθ+ = 2π and Δϕ=0 Bound state of a 1/2 vortex and a dislocation Δθ+ = π, Δϕ= 2π Double dislocation, Δθ+ = 0, Δϕ= 4π
91 Topological Excitations of the Striped SC SC vortex with Δθ+ = 2π and Δϕ=0 Bound state of a 1/2 vortex and a dislocation Δθ+ = π, Δϕ= 2π Double dislocation, Δθ+ = 0, Δϕ= 4π All three topological defects have logarithmic interactions
92 Half-vortex and a Dislocation
93
94 Thermal melting of the PDW state
95 Thermal melting of the PDW state Three paths for thermal melting of the PDW state
96 Thermal melting of the PDW state 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)
97 Thermal melting of the PDW state 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) Scaling dimensions: p,q=π(ρsc p 2 +κcdw q 2 )/T=2 (for marginality)
98 Thermal melting of the PDW state 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) Scaling dimensions: p,q=π(ρsc p 2 +κcdw q 2 )/T=2 (for marginality) Phases: PDW, Charge 4e SC, CDW, and normal (Ising nematic)
99 Schematic Phase Diagram
100
101 Effects of Disorder
102 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
103 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
104 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.
105 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. Halo of uniform SC order at half vortex cores
106 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. Halo of uniform SC order at half vortex cores Strict layer decoupling only allows for a magnetic coupling between randomly distributed ±π flux vortices
107 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. Halo of uniform SC order at half vortex cores Strict layer decoupling only allows for a magnetic coupling between randomly distributed ±π flux vortices Novel glassy physics and fractional flux
108 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. Halo of uniform SC order at half vortex cores 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
109 Role of Nematic Fluctuations in PDW Melting?
110 Role of Nematic Fluctuations in PDW Melting? If nematic fluctuations become strong, orders that break translational symmetry become progressively suppressed
111 Role of Nematic Fluctuations in PDW Melting? If nematic fluctuations become strong, orders that break translational symmetry become progressively suppressed In the absence of a lattice in 2D smectic order is not possible (at T>0) (Toner & Nelson, 1980)
112 Role of Nematic Fluctuations in PDW Melting? If nematic fluctuations become strong, orders that break translational symmetry become progressively suppressed In the absence of a lattice in 2D smectic order is not possible (at T>0) (Toner & Nelson, 1980) Coupling to the lattice breaks continuous rotational invariance to the point group of the lattice
113 Role of Nematic Fluctuations in PDW Melting? If nematic fluctuations become strong, orders that break translational symmetry become progressively suppressed In the absence of a lattice in 2D smectic order is not possible (at T>0) (Toner & Nelson, 1980) Coupling to the lattice breaks continuous rotational invariance to the point group of the lattice For a square lattice the point group is C4 and the nematicisotropic transition is 2D Ising
114 Role of Nematic Fluctuations in PDW Melting? If nematic fluctuations become strong, orders that break translational symmetry become progressively suppressed In the absence of a lattice in 2D smectic order is not possible (at T>0) (Toner & Nelson, 1980) Coupling to the lattice breaks continuous rotational invariance to the point group of the lattice For a square lattice the point group is C4 and the nematicisotropic 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
115
116 PDW melting as a McMillan-de Gennes theory The electronic nematic order parameter is an antisymmetric traceless tensor N ij = N(ˆn iˆn j δ ij /2) Its presence amounts to a local change of the geometry, the local metric, seen by the charge 4e SC order parameter 4e the CDW order parameter ρ K and the PDW order ±Q Metric tensor: g ij = δ ij + λn ij Novel derivative couplings in the free energy F d = dx 2 detg g ij D cdw i ρ K D cdw j ρ K + +(Di sc ±Q ) D sc j ±Q + D 4e i 4e D 4e j 4e Di sc Di cdw Di 4e = i i2ea i ± iq δn i = i + i2q δn i = i i4ea i
117 Deep in the PDW phase the amplitude of the SC, CDW and nematic order parameters are fixed but their phases are not F = d 2 ρs x 2 gij ( i θ 2eA i )( j θ 2eA j ) + κ 2 gij ( i ϕ + Qδn i )( j ϕ + Qδn j ) + K 1 ( δn) 2 + K 3 ( δn) 2 + h δn 2 ρ s = ±Q 2 + 4e 2 κ = ±Q 2 + ρ K 2 Effective CDW stiffness reduced by gapped nematic fluctuations κ eff = κ 1+κQ 2 /h
118 Deep in the charge 4e nematic superconducting phase F = d 2 x K α 2 + ρ s θ 2 + ρ sn 2 (n θ)2 α and θ are the orientational nematic fluctuations and SC phase fluctuations 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 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
119 Conclusions 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 Several predictions can be tested experimentally 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 ladders.
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