QED3 Theory of High Temperature Superconductors • What is the ...

QED 3 Theory of
High Temperature Superconductors
Zlatko Tešanović
The Johns Hopkins University
August 2002
LT 23, August 20-27, 2002, Hiroshima, JAPAN
Collaborators: Prof. Marcel Franz (UBC), O. Vafek and A. Melikyan (JHU)
• What is the Problem in HTS? An “Inverted” View
• Protectorate of the Pairing Pseudogap
• Vortices, Quasiparticles and Topological Frustration
• QED 3 Theory of HTS – From d-wave Superconductor
to Antiferromagnet via Strange Metal

This talk is based on:
∗∗ M. Franz and ZT, Phys. Rev. Lett. 87, 257003 (2001).
∗∗ ZT, O. Vafek and M. Franz, Phys. Rev. B 65, 180511(R) (2002). †
∗∗ M. Franz, ZT and O. Vafek, cond-mat/0203333.
∗∗ O. Vafek, A. Melikyan, MF and ZT, Phys. Rev. B 63, 134509 (2001).
∗∗ M. Franz and ZT, Phys. Rev. Lett. 84, 554 (2000).
∗∗ ZT, Phys. Rev. B 59, 6449 (1999) (particularly Appendix A).
∗∗ I. F. Herbut and ZT, Phys. Rev. Lett. 76, 4588 (1996).
∗∗ other related papers available at www.pha.jhu.edu/people/faculty/zbt.html
† see also I. F. Herbut, Phys. Rev. Lett. 88, 047006 (2002).
Other significant papers on pseudogap and SC fluctuations:
∗∗ J. Emery and S. A. Kivelson, Nature 374, 434 (1995).
∗∗ L. Balents, M. P. A. Fisher and C. Nayak, Phys. Rev. B 60, 1654 (1999).
∗∗ A. K. Nguyen and A. Sudbø, Phys. Rev. B 60, 15307 (1999).
∗∗ M. Franz and A. J. Millis, Phys. Rev. B 58, 14572 (1998).
∗∗ H. J. Kwon and A. T. Dorsey, Phys. Rev. B 59, 6438 (1999).
∗∗ L. Marinelli, B. I. Halperin and S. Simon, Phys. Rev. B 62, 3448 (2000).

What is The Problem in high T c superconductors?
• Superconducting state appears d x 2 −y2 “BCS-like”. Low energy
fermionic quasiparticles seem to have only “weak” interactions.
• In contrast, the “normal” state appears strange and non-Fermi
liquid-like. Its fermionic excitations are strongly correlated.
“Inverted” solution:
• Today, everything seems to be a high temperature superconductor
(cuprates, C 60 ’s, MgB 2 ,. . . ). The superconducting ground
state is as common as normal metal.
• Traditional paradigm: we must understand the normal state before
we can understand the superconductor (Anderson, Pines,. . . ).
We turn this paradigm around. The “inverted paradigm” states:
we must understand the superconductor before we can understand
the adjacent “normal” state.
• We construct a theory in which the correlated BCS-BdG superconductor
serves as the reference ground state.
⇒
The key
questions: (i) how does such a superconductor become nonsuperconducting?
(ii) what are the states adjacent to such a
superconductor, i.e.
what are its “failure modes” (Laughlin,
LT23)? (iii) what is the theory that assumes the role of Fermi
liquid in the traditional paradigm?

How Does a Superconductor Become Normal?
In superconductors electrons form Cooper pairs: 〈c † ↑(r)c † ↓(r)〉 ∼
∆(r). The order parameter ∆(r) is a complex wave function with
amplitude |∆| and phase exp(iϕ(r)). In the superconducting state
Cooper pairs are “Bose condensed” and 〈∆〉 ≠ 0.
Superconducting vortex is a topological
defect in ϕ (see arrows).
If one goes around the vortex,
ϕ winds by 2π (−2π around
antivortex). The amplitude is
uniform except in the core where
|∆| → 0.
If vortices proliferate, through thermal or quantum fluctuations,
it is this strong phase winding that ultimately destroys superconducting
order: lim |r|→∞ 〈∆(r)∆ ∗ (0)〉 → 0 , 〈∆〉 = 0

T < T c
T > T c
τ
τ
x
y
x
y
Thermal/quantum proliferation of vortex loops (unbound vortexantivortex
pairs) is the mechanism of superconductor-to-normal transition
(Kosterlitz-Thouless). Vortex fluctuations are particularly enhanced
in copper oxides due to quasi 2D structure, high T c AND
Important: in correlated dSC
large ∆ x 2 −y 2 but small ρ s
in the underdoped regime
⇒ small penalty for
twisting the phase: ρ s (∇ϕ) 2
Vortices in a doped Mott Insulator:
PA Lee, Nagaosa
Franz & ZT
DH Lee & Han
Ogata

T
Vortex Fluctuations in Underdoped Cuprates
T
Nernst
T
AF
QED 3
T c
dSC
*
T < T ∗ – Strong SC Fluctuations
Experimental Evidence:
Uemura (Columbia),
Ong (Princeton),
Orenstein (Berkeley),
Fischer (Geneva),
Oda & Ido (Hokkaido),
Below T ∗ (∼ 100-200 K) there are strong fluctuations of Cooper
pairs. These fluctuations involve both the amplitude and the phase
of the SC order parameter. The phase fluctuations are dominant
while the amplitude fluctuations are frozen below T ∗ . The true SC
order sets in only at T c ≪ T ∗ . The strongest evidence for large
vortex fluctuations in the pseudogap state (above T c and below
T Nernst ) comes from Nernst effect experiments by Ong et al.
x
Phase fluctuations:
Smooth (“spin waves”)
Singular (vortices)
⇒ Vortices are responsible for the SC transition at T c
and are dominant fluctuations below T Nernst

Evidence for the pairing pseudogap, Oda, Ido et al. (Hokkaido)

HTS are unique not only because they have such high T c and strong
vortex-antivortex fluctuations. They are also d-wave superconductors,
in contrast to conventional s-wave superconductors. ⇒
Low energy effective theory must contain FERMIONS !! This is in
contrast to He-type (BOSON ONLY) models of SC fluctuations
+
+
+ +
s-wave superconductors: The gap function
∆(k) does not change sign on the Fermi
surface. The quasiparticle spectrum is
gapped: E(k) = ± √ (ε(k) − µ) 2 + |∆| 2
-
+
+
-
d-wave superconductors: The gap function
∆(k) changes sign on the Fermi
surface. The quasiparticle spectrum is
gapless near the nodes, where ∆(k) → 0
E(k) = ± √ (ε(k) − µ) 2 + |∆(k)| 2 ∼ ±|k|
⇒ Massless Dirac Fermions !! ⇐

Superconductor
“Normal”
-
+
-
+
τ
+ -
τ
+ -
x
y
x
y
In correlated dSC we must consider two types of degrees of freedom:
fluctuating vortices and nodal quasiparticles (vortices in k-space),
and their mutual interactions generated by the pairing term in H eff :
H t−J ⇒ H eff ⇒ ∆(r)c † ↑(r)c † ↓(r) + (· · ·)
where ∆(r) = ∆ 0 exp(iϕ(r)). The physical coupling should be
through ∂ϕ, due to the global U(1) gauge invariance. But, to
extract ∂ϕ, we must take a “square-root” of exp(iϕ(r)).

Some Notorious Square Roots
√
a2 + b 2 = c ⇒ Pythagoras’ Theorem
( √ 5 − 1)/2 ⇒ Penrose Tiles
√ −1 = i ⇒ Complex Analysis
√ˆp2 + m 2 = α · ˆp + βm ⇒ Dirac equation, S = 1 2
√
f(z) ⇒ Riemann sheet
What is the square-root of exp(iϕ(r)) appearing in exp(iϕ)c † ↑c † ↓ ?
exp(iϕ(r)/2) looks nice but it introduces half-vortices and leads to
branch cuts and non-single valued wavefunctions.
Is there another way of taking a “square-root” of exp(iϕ(r)) which
does not lead to branch cuts and keeps (hc/2e) vortices whole?

FT Singular Gauge Transformation †
Eliminate the phase ϕ(r, τ) from the pairing term ˆ∆:
c † ↑ → exp (iϕ A)c † ↑ , c† ↓ → exp (iϕ B)c † ↓ ,
where ϕ A +ϕ B → ϕ ; exp(iϕ) = exp(iϕ A ) exp(iϕ B )
ϕ A(B) (r, τ) is the singular part of the phase due to A(B) vortex
defects. A and B are a convenient but arbitrary division of vortex
defects into two equivalent groups.
+
-
A
The figure shows a convenient
choice of sets A and B for
+
-
+
+
- -
B
fluctuating vortices. (↑, ↓)
symmetry is restored after
summation over all vortex
configurations.
† M. Franz and ZT, Phys. Rev. Lett. 84, 554 (2000); O. Vafek, A. Melikyan, M.
Franz and ZT, Phys. Rev. B 63, 134509 (2001)

Quasiparticle-Vortex Interactions I
An (hc/2e) vortex is seen by a quasiparticle as a confined state
of a U(1) “doppleron” and a Z 2 “berryon”:
2e
vortex
qp (e)
Doppler shift:
ω → ω ′ = ω − ⃗v s (⃗r) · ⃗k
⃗v s (⃗r) - superfluid velocity
Volovik (’93)
• However, Doppler shift provides only “half” of the phase:
2e
vortex
qp (e)
Berry phase † :
supplies additional
∆ϕ = π ; Berry gauge
field describes topological
frustration experienced by
BdG quasiparticles in
presence of vortices
† M. Franz and ZT, Phys. Rev. Lett. 84, 554 (2000)

Quasiparticle-Vortex Interactions II
We call new, singularly gauge transformed, quasiparticles Topological
Fermions. Interactions of TFs with vortices are described by
the BdG Hamiltonian for d-wave superconductor:
⎛
⎜
⎝
⎞
ˆD
ˆD − 1
2m (ˆp − mvB s ) 2 ⎟
+ ɛ F
1
2m (ˆp + mvA s ) 2 − ɛ F
⎠ ,
ˆD = ∆ 0
p 2 [ˆp x + m
F 2 (vA sx − vsx)][ˆp B y + m 2 (vA sy − vsy)]
B
and vs ζ = 1 m (¯h∇φ ζ − e cA), ζ = A, B.
• Doppler gauge field v µ = 1 2 (∂ µϕ A +∂ µ ϕ B ) couples to “charge”
of topological fermions (µ = x, y, τ)
(Meissner coupling).
• Berry gauge field a µ = 1 2 (∂ µϕ A − ∂ µ ϕ B ) couples to “spin” of
topological fermions (µ = x, y, τ) (minimal coupling). ⇒
⇒
Doppler gauge field is massive due to Meissner effect of
TFs. Berry gauge field must remain massless since d-wave SC is
spin singlet. The only way to make a µ massive is to bind vortices
into finite loops – this is what happens in the superconducting state.

QED 3 in Cuprates
M. Franz and ZT, PRL 87, 257003 (2001)
The low energy quasiparticles are located at the four nodal points
of the d-wave gap function denoted as (1, ¯1) and (2, ¯2). Linearizing
the spectrum near the nodes leads to the effective Lagrangian:
L QED =
∑
α=1,¯1
+ ∑
α=2,¯2
Ψ † α[D τ − iv F D x σ 3 − iv ∆ D y σ 1 ]Ψ α
Ψ † α[D τ − iv F D y σ 3 − iv ∆ D x σ 1 ]Ψ α + L 0 , (1)
Ψ † α is two-component spinor, α is node index, D µ = ∂ µ + ia µ and:
L 0 [a µ ] = K τ (∂ × a) 2 τ/(2π 2 ) + K d (∂ × a) 2 d/(2π 2 ) . (2)
Key parameters of the theory are v F , v ∆ , N and K τ,d ∝ ξ sc (x, T ) .
Effective Lagrangian (1,2) for low energy quasiparticles interacting
with fluctuating unbound vortices (〈∆〉 = 0)
Anisotropic Euclidean quantum electrodynamics of massless
Dirac fermions in (2+1)-dimensions (QED 3 ).
⇒

Electron (Quasiparticle) Propagator
G(x, x ′ ) ≈ 〈exp(i ∫ x ′
x a µ ds µ (Γ))[ ˜Ψ(x) ˜Ψ † (x ′ )] 11 〉, where x = (r, τ).
• The real electron propagator can be computed from the TF propagator
in a gauge in which 〈exp(i ∫ x ′
x ds µ a µ )〉 = 1 (Franz and ZT):
Real electron propagator (Franz and ZT):
⇒ G α (p) = p/
p 2−η
⇐
η = 16/3π 2 N ∼ = 0.27 (N = 2) is the anomalous dimension of real
electrons (η ≪ 1). The Fermi liquid description of quasiparticles
breaks down ⇒ Non-Fermi liquid in the pseudogap state !!
Phase Vortex Loops Berryons Nodal Fermions
dSC Bound Massive Free – Coherent
Pseudogap Unbound Massless Interacting – Incoherent

ARPES in Cuprates
ARPES measures the spectral function of electrons:
A(k, ω) = 1 π Im [G α(k, ω)] 11 ∝ Im
p/
[v 2 Fk 2 x + v 2 ∆k 2 y − ω 2 ] 1−(η/2)
EDC
MDC
ω ω
−1.5 −1.0 −0.5 0.0 0.5
ω, v F
k x
0.0 0.5 1.0
v F
k x
Algebraic Fermi liquid: Spectral function exhibits Luttinger-like
behavior in the “normal” (pseudogap) state – Σ ′′ α(k, ω) finite only
inside the cone defined by ω 2 > vFk 2 x 2 + v∆k 2 y
2
M. Franz and ZT, PRL 87, 257003 (2001)
Coherent d-wave quasiparticles (η = 0) appear only in the true
phase-ordered superconducting state (ZX Shen, LT23)

BdG Chiral Symmetry and HTS Phase Diagram I
2
1
1
Q Q
22 11
Q
12
2
In L QED we can group four nodal
fermions, (1 ↔ ¯1), (2 ↔ ¯2), into two
four-component massless Dirac fermions
⇒ N = 2 QED 3 ⇒
L QED = ¯ψ n c µ,n γ µ D µ ψ n + L 0 [a µ ] + (· · ·) ,
n labels (1, ¯1) and (2, ¯2) pairs of nodes, c τ,n = 1, c x,1 = c y,2 = v F ,
c x,2 = c y,1 = v ∆ , and γ 0 = σ 3 ⊗σ 3 , γ 1 = −σ 3 ⊗σ 1 , γ 2 = −σ 3 ⊗σ 2 ,
{γ µ , γ ν } = 2δ µν .
• L QED has a U(2) global symmetry for each pair of nodes
generated by 1 ⊗ 1, γ 3 , −iγ 5 and 1 2 [γ 3, γ 5 ], where γ 3 = σ 1 ⊗ 1 and
γ 5 = iσ 2 ⊗1 anticommute with all γ µ ’s. This BdG chiral symmetry
is broken by two “mass” terms:
m ch ¯ψn ψ n and m PT ¯ψn
1
2
[γ 3 , γ 5 ]ψ n

BdG Chiral Symmetry and HTS Phase Diagram II †
T
AF/
SDW
(CSB)
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¤¡¤¡¤¡¤
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£¡£¡£¡£
¤¡¤¡¤¡¤
¤¡¤¡¤¡¤
£¡£¡£¡£
T*
QED
3
¡ ¢¡¢¡¢
¡ ¢¡¢¡¢
¢¡¢¡¢ ¡
T c
(a)
CSB
(b)
CSB
dSC
FL
dSC
dSC
x
We begin with
m ch = m PT = 0.
However, m ch ≠ 0 can be
generated spontaneously
(but not m PT ≠ 0 !!) ⇒
Chiral Symmetry
Breaking (CSB) ⇒
Three Phases of our QED 3 :
• Superconductor: a µ is massive, ψ n is massless.
• Chiral Symmetric QED 3 : this phase exhibits non-Fermi liquid
behavior and is the reference symmetric state of the pseudogap
regime – This Algebraic Fermi liquid plays the role of the “Fermi
liquid theory” for the pseudogap state. Both a µ and ψ n are massless.
• CSB QED 3 : finite m ch is dynamically generated through spontaneous
chiral symmetry breaking (CSB) and ψ n gains mass. a µ
remains massless and confines spinons (topological fermions).
† ZT, O. Vafek and M. Franz, PRB 65, 180511(R) (2002) (cond-mat/0110253)

QED 3 Unified Theory (QUT) of HTS †
2
1
1
Q Q
22 11
Q
12
2
In m ch ¯ψn ψ n the “mass” m ch acts as
order parameter for bilinears 〈 ¯ψ n ψ n 〉
of topological fermions ψ n . If v F ≠ v ∆ ,
the chiral manifold of these bilinears
has a U(1)×U(1) degeneracy for each
pair of nodes.
What are the states in the BdG chiral manifold?
〈c † ↑αc ↑ᾱ − c † ↓αc ↓ᾱ 〉 + h.c. (cos SDW)
i〈c † ↑αc ↑ᾱ − c † ↑ᾱc ↑α 〉 + (↑→↓)
(sin SDW)
This is an incommensurate SDW at wavevectors ±Q 1¯1 and ±Q 2¯2.
It is the dominant state as x → 0 due to umklapp processes.
⇒ QUT predicts antiferromagnetism at low doping
⇒ dSC → “strange metal” → SDW (AF) ⇐
⇐
† ZT, O. Vafek and M. Franz, Phys. Rev. B 65, 180511(R) (2002) (condmat/0110253);
I. F. Herbut, Phys. Rev. Lett. 88, 047006 (2002).

QED 3 Unified Theory (QUT) of HTS
SDW is a 2D incommensurate antiferromagnetic
insulator. In QED 3 theory
its spinon excitations are confined.
This SDW is a progenitor of Neel (Mott-
-Hubbard) antiferromagnet at half-filling.
The two join smoothly as x → 0.
BdG chiral symmetry
breaking leads to the
opening of a small gap
in the nodes of the large
d x 2 −y2 pairing pseudogap.
2∆ SDW

Spin and charge ordering near a vortex within QED 3 theory:
30
a spin magnetization
30
b spin magnetization FT
M. Franz, D. Sheehy and ZT,
PRL 88, 257005 (2002)
25
25
20
20
a charge density
b charge density FT
15
15
30
10
5
0
0 5 10 15 20 25 30
10
5
0
0 5 10 15 20 25 30
0.68
0.66
0.64
0.62
20
30
25
20
15
10
30
25
c staggered magnetization
d staggered magnetization
10
20
10
5
0
30 0 5 10 15 20 25 30
20
15
10
0.5
0.25
0
0.25
0.5
20
30
30
25
20
c integrated LDOS
30
25
20
d integrated LDOS FT
5
0
0 5 10 15 20 25 30
10
20
30
10
15
10
15
10
5
5
0
0 5 10 15 20 25 30
0
0 5 10 15 20 25 30
Coexistence of SDW/CDW and dSC orders:
Incommensurate SDW at Q SDW = π(1 ± δ SDW , 1 ± δ SDW ) with δ SDW = 1 4
.
Incommensurate CDW at Q CDW = π(±δ CDW , 0), π(0, ±δ CDW ) with δ CDW = 1 2 .
CDW pattern consistent with Hoffman et al.

QED 3 Unified Theory (QUT) of HTS
Other states in [U(1)×U(1)] 2 (v F ≠ v ∆ ) BdG chiral manifold:
i〈ψ ↑α ψ ↓α − ψ ↑ᾱ ψ ↓ᾱ 〉 + h.c.
(dipSC)
This is a d+ip phase-incoherent superconductor. It is written in
terms of topological fermions ψ σα (r, τ) instead of the original nodal
quasiparticles c σα (r, τ). This state breaks parity but preserves time
reversal, translational invariance and superconducting U(1) symmetries.
All continuous chiral rotations between SDW and dipSC are
also included as well as “diagonal stripes” – combinations of
(1, ¯1) SCDW and (2, ¯2) dipSC.
When v F = v ∆ the BdG chiral manifold is enlarged to U(2)×U(2):
1
√
2
〈c † ↑1c ↑2 + c † ↑¯2 c ↑¯1 + h.c.〉 + (↑→↓)
(CDW)
1
√
2
〈ψ ↑1 ψ ↓2 + ψ ↑¯2ψ ↓¯1 + h.c.〉 + (↑↔↓) (SCDW) ⇒

QED 3 Unified Theory (QUT) of HTS
These are superpositions of one-dimensional p-h and p-p states,
an incommensurate CDW accompanied by a non-uniform phaseincoherent
superconductor (SCDW) at wavevectors ±Q 12 and ±Q¯2¯1
⇒
QUT “stripes”
i〈ψ ↑1 ψ ↓1 + ψ ↑¯1ψ ↓¯1 + h.c.〉 + (1 → 2)
(disSC)
This is a phase-incoherent d+is superconductor.
Large U(2)×U(2) BdG chiral manifold of nearly degenerate
states provides means to unify the phenomenology
of cuprates within the universal framework of QED 3 theory
and emerges as a likely culprit behind complexity of
the HTS phase diagram.