Presentazione

Introduction Phenomenology Model Transport properties Results Conclusion
Gauge approach to pairing and superconductivity
in high Tc cuprates
Autore: Filippo Bovo
Relatore: Prof. Pieralberto Marchetti
UNIVERSIT`A DEGLI STUDI DI PADOVA
FACOLT`A DI SCIENZE MM. FF. NN.
DIPARTIMENTO DI FISICA “GALILEO GALILEI”
13 Luglio 2011
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates

1 Introduction
2 Phenomenology
3 Model
4 Transport properties
5 Results
6 Conclusion

Crystal structure
Typical structure: Copper-Oxygen planes
↓
+ Charge reservoir
(doping)
O, 2px
O, 2pyCu, 3dx -y2 2
Hole, 3dx -y2 2
Hole-doping removes
this electron allowing
holes to move

Phase diagram
Phases:
FL: Fermi Liquid;
SM: Strange Metal;
PG: Pseudogap;
AF: Antiferromagnetic;
SC: Superconducting.
δ: doping;
T: temperature [K].

Low-energy model
Anderson’s idea of ”doping a Mott insulator” (1987)
↓
t-J model: low-energy physics of CuO2-planes
+
Gutzwiller projector PG

Low-energy model
Anderson’s idea of ”doping a Mott insulator” (1987)
↓
t-J model: low-energy physics of CuO2-planes
+
Gutzwiller projector PG
↓
Ht−J = PG
<ij>
[−t(ˆc†
iαˆcjα + H.c) + JˆSi · ˆSj ]PG

Spin-charge decomposition
electron −→ spinon(boson) + holon(fermion):
ˆciα ≡ ˆsiα
ˆh†
i

ˆciα ≡ ˆsiα
ˆh†
i
⇓
PG → ˆni = ˆc†
iαˆciα = 0, 1 →
ˆh†
i
ˆhi = 0, 1 Automatic
α ˆs†
iαˆsiα = 1 Constraint

ˆciα ≡ ˆsiα
ˆh†
i
⇓
PG → ˆni = ˆc†
iαˆciα = 0, 1 →
ˆh†
i
ˆhi = 0, 1 Automatic
α ˆs†
iαˆsiα = 1 Constraint
+
Invariance under:
ˆhi → ˆhi eiϕ
ˆsiα → ˆsiαeiϕ
,
ϕ
U(1) local phase
⇒
Aµ
U(1) gauge ﬁeld

Improved mean-field approximation
Optimization of the spinon configurations and mean-field treatment:
ˆhj eiφh(j), fermion
ˆzj eiφs (j), boson
←−
The product
is still
a fermion

Improved mean-field approximation
Optimization of the spinon configurations and mean-field treatment:
ˆhj eiφh(j), fermion
ˆzj eiφs (j), boson
←−
The product
is still
a fermion
⇒ φh(j) → Charge vortex φs(j) → Spin vortex
PG SM
↓
π-flux
↓
0-flux

Low-energy effective action
Complete low-energy effective action =
Spinon effective action:
Non-linear σ-model with mass gap ms ∼ |δ ln δ| minimally coupled to Aµ
1
Magnetic Brillouin zone

+
Holon eﬀective action:
PG: ⇒
(Formally) relativistic spinless fermion with small
half-circle Fermi surface (∼ δ) centered in
(±π/2, ±π/2) in MBZ1
, minimally coupled to Aµ
1

+
Holon eﬀective action:
PG: ⇒
(Formally) relativistic spinless fermion with small
half-circle Fermi surface (∼ δ) centered in
(±π/2, ±π/2) in MBZ1
, minimally coupled to Aµ
SM: ⇒
Non-relativistic spinless fermion with big circular
Fermi surface (∼ 1 − δ) centered in (±π, ±π) in
MBZ, minimally coupled to Aµ
1

Phase diagram of the model
Underdoped Overdoped
0.150.03 0.04 0.250
MI
SC
PG
SM
400
T (K)
δ
Parent compound
coherence
240
h
s
s
Attraction scheme:
h

0.150.03 0.04 0.250
MI
SC
PG
SM
400
T (K)
δ
Parent compound
coherence
240
h
s
s
Attraction scheme:
h
A
A

0.150.03 0.04 0.250
MI
SC
PG
SM
400
T (K)
δ
Parent compound
coherence
240
h
s
s
Indirect spinon potential
Attraction scheme:
h
A
A
s
s

Within this model we studied, as original contribution, how the
formation of holon pairs contributes to transport properties.

Formation of holon pairs ⇒
energy-dependent (normalized)
holon density of states n(ω)
↓
Starting point: Kubo formula for holon conductivity
σ(ω, T) ∝ 1
Γtr (ω,T)−iω , Γtr (ω, T) transport scattering rate:

Formation of holon pairs ⇒
energy-dependent (normalized)
holon density of states n(ω)
↓
Starting point: Kubo formula for holon conductivity
σ(ω, T) ∝ 1
Γtr (ω,T)−iω , Γtr (ω, T) transport scattering rate:
ΓTr (Ω, T) ∝
∞
0
dω˜I2
χTr (ω){n(Ω−ω)fem(Ω, ω, T)+n(Ω+ω)fab(Ω, ω, T)}
• ˜I2χTr : interaction spectral density (momentum averaged);
• fem, fab: holon probabilities of emission and absorption of the
gauge ﬁeld.

Holon conductivity
Dressed holon propagator GR
h (ω, k) → n(ω) ∝ dk
(2π)2 GR
h (ω, k).
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
Ω0.0
0.5
1.0
1.5
2.0
n
δ
T
↑
The interaction between phase ﬂuctuations of the paired holons
gets stronger as temperature decreases →
→ interpolation between FL and SC behaviours.

Real part of optical conductivity, SM
Hole re-composition through the Ioﬀe-Larking rule for complex
conductivities:
1
σ
=
1
σh
+
1
σs
Theory2 vs Experiment:
2
ω in unities of t 0.4eV

Real part of optical conductivity, PG
Hole re-composition through the Ioﬀe-Larking rule for complex
conductivities:
1
σ
=
1
σh
+
1
σs
3
ω in unities of t 0.4eV

Transport scattering rate, SM
Ioﬀe-Larkin rule ⇒ Γtr = Γtr;h + Γtr;s
Γ
4
T=300,270,250,230K. ω and Γtr in unities of t 0.4eV

Resistivity, SM
Ioﬀe-Larkin rule ⇒ ρtr ρtr;h + ρtr;s
5
δ = 0.10, 0.15, 0.20. ω and T in unities of t 0.4eV

Resistivity, PG
Ioﬀe-Larkin rule ⇒ ρtr ρtr;h + ρtr;s
6
δ = 0.03, 0.05. ω and T in unities of t 0.4eV

Conclusions
Good qualitative comparison between theoretical and experimental
results
⇓
The technique used to take into account the formation of holon
pairs is correct
+
Further step toward the validity of the model

Further developments
Mapping of ∂2ρ/∂T2 to be compared with Ando et al. (Phys.
Rev. Lett., 93(26):267001, 2004) experiment:
Computation of the reﬂectivity to be compared with Giannetti et
al. (Nat Commun, 2:353, 06 2011) experiment:

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