This document summarizes Filippo Bovo's thesis on a gauge theory approach to pairing and superconductivity in high-temperature cuprate superconductors. The thesis: 1) Develops a t-J model and uses a spin-charge decomposition to derive a low-energy effective action, 2) Studies how the formation of holon pairs contributes to transport properties within this model using a Kubo formula approach, and 3) Finds good agreement between the model's predictions for the real part of the optical conductivity in the strange metal and pseudogap phases and experimental data.

1. 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

2. Introduction Phenomenology Model Transport properties Results Conclusion
1 Introduction
2 Phenomenology
3 Model
4 Transport properties
5 Results
6 Conclusion
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates

3. Introduction Phenomenology Model Transport properties Results 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
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates

4. Introduction Phenomenology Model Transport properties Results Conclusion
Phase diagram
Phases:
FL: Fermi Liquid;
SM: Strange Metal;
PG: Pseudogap;
AF: Antiferromagnetic;
SC: Superconducting.
δ: doping;
T: temperature [K].
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates

5. Introduction Phenomenology Model Transport properties Results Conclusion
Low-energy model
Anderson’s idea of ”doping a Mott insulator” (1987)
↓
t-J model: low-energy physics of CuO2-planes
+
Gutzwiller projector PG
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates

6. Introduction Phenomenology Model Transport properties Results Conclusion
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
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates

7. Introduction Phenomenology Model Transport properties Results Conclusion
Spin-charge decomposition
electron −→ spinon(boson) + holon(fermion):
ˆciα ≡ ˆsiα
ˆh†
i
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates

8. Introduction Phenomenology Model Transport properties Results Conclusion
Spin-charge decomposition
electron −→ spinon(boson) + holon(fermion):
ˆciα ≡ ˆsiα
ˆh†
i
⇓
PG → ˆni = ˆc†
iαˆciα = 0, 1 →
ˆh†
i
ˆhi = 0, 1 Automatic
α ˆs†
iαˆsiα = 1 Constraint
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates

9. Introduction Phenomenology Model Transport properties Results Conclusion
Spin-charge decomposition
electron −→ spinon(boson) + holon(fermion):
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
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates

10. Introduction Phenomenology Model Transport properties Results Conclusion
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
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates

11. Introduction Phenomenology Model Transport properties Results Conclusion
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
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates

12. Introduction Phenomenology Model Transport properties Results Conclusion
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
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates

13. Introduction Phenomenology Model Transport properties Results Conclusion
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
+
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
Magnetic Brillouin zone
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates

14. Introduction Phenomenology Model Transport properties Results Conclusion
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
+
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
Magnetic Brillouin zone
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates

15. Introduction Phenomenology Model Transport properties Results Conclusion
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
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates

16. Introduction Phenomenology Model Transport properties Results Conclusion
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
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates

17. Introduction Phenomenology Model Transport properties Results Conclusion
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
A
A
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates

18. Introduction Phenomenology Model Transport properties Results Conclusion
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
Indirect spinon potential
Attraction scheme:
h
A
A
s
s
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates

19. Introduction Phenomenology Model Transport properties Results Conclusion
Within this model we studied, as original contribution, how the
formation of holon pairs contributes to transport properties.
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates

20. Introduction Phenomenology Model Transport properties Results Conclusion
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:
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates

21. Introduction Phenomenology Model Transport properties Results Conclusion
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:
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.
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates

22. Introduction Phenomenology Model Transport properties Results Conclusion
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.
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates

23. Introduction Phenomenology Model Transport properties Results Conclusion
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
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates

24. Introduction Phenomenology Model Transport properties Results Conclusion
Real part of optical conductivity, PG
Hole re-composition through the Ioffe-Larking rule for complex
conductivities:
1
σ
=
1
σh
+
1
σs
Theory3 vs Experiment:
3
ω in unities of t 0.4eV
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates

25. Introduction Phenomenology Model Transport properties Results Conclusion
Transport scattering rate, SM
Ioffe-Larkin rule ⇒ Γtr = Γtr;h + Γtr;s
Theory4 vs Experiment:
Γ
4
T=300,270,250,230K. ω and Γtr in unities of t 0.4eV
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates

26. Introduction Phenomenology Model Transport properties Results Conclusion
Resistivity, SM
Ioffe-Larkin rule ⇒ ρtr ρtr;h + ρtr;s
Theory5 vs Experiment:
5
δ = 0.10, 0.15, 0.20. ω and T in unities of t 0.4eV
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates

27. Introduction Phenomenology Model Transport properties Results Conclusion
Resistivity, PG
Ioffe-Larkin rule ⇒ ρtr ρtr;h + ρtr;s
Theory6 vs Experiment:
6
δ = 0.03, 0.05. ω and T in unities of t 0.4eV
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates

28. Introduction Phenomenology Model Transport properties Results Conclusion
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
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates

29. Introduction Phenomenology Model Transport properties Results Conclusion
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:
Autore: Filippo Bovo - Relatore: Prof. Pieralberto Marchetti Gauge approach to superconductivity in high Tc cuprates