Effective Field Theories for Doped
Antiferromagnets
Uwe-Jens Wiese
Bern University
Oberwölz, September 12, 2008
Collaboration: C. Brügger, C. Hofmann, F. Kämpfer, M. Moser, M. Pepe
Outline
Cuprates
Correspondences between QCD and Antiferromagnetism
Models for Doped Antiferromagnets
Effective Field Theory for Magnons
Effective Field Theory for Magnons and Holes
Magnon-mediated Two-Hole Bound States
Spiral Phases of the Staggered Magnetization
Chiral Spiral and Pion Condensation
Conclusions
Outline
Cuprates
Correspondences between QCD and Antiferromagnetism
Models for Doped Antiferromagnets
Effective Field Theory for Magnons
Effective Field Theory for Magnons and Holes
Magnon-mediated Two-Hole Bound States
Spiral Phases of the Staggered Magnetization
Chiral Spiral and Pion Condensation
Conclusions
Antiferromagnetic precursors of high-Tc superconductors
LaCuO
YBaCuO
Properties of cuprates
Temperature-dependence of resistivity
Outline
Cuprates
Correspondences between QCD and Antiferromagnetism
Models for Doped Antiferromagnets
Effective Field Theory for Magnons
Effective Field Theory for Magnons and Holes
Magnon-mediated Two-Hole Bound States
Spiral Phases of the Staggered Magnetization
Chiral Spiral and Pion Condensation
Conclusions
Phase diagrams of QCD and of doped antiferromagnets
QCD phase diagram
Phase diagram of cuprates
Quark-gluon plasma
T
strange metal
T
hadronic
phase
color superconductor
chemical potential
antiferromagnetic
phase
high-temperature
superconductor
hole concentration
Correspondences between QCD and Antiferromagnetism
broken phase
global symmetry
symmetry group G
unbroken subgroup H
Goldstone boson
Goldstone field in G /H
order parameter
coupling strength
propagation speed
conserved charge
charged particle
long-range force
dense phase
microscopic description
effective description
of Goldstone bosons
effective description
of charged fields
QCD
hadronic vacuum
chiral symmetry
SU(2)L ⊗ SU(2)R
SU(2)L=R
pion
U(x) ∈ SU(2)
chiral condensate
pion decay constant Fπ
velocity of light
baryon number U(1)B
nucleon or antinucleon
pion exchange
nuclear or quark matter
lattice QCD
chiral perturbation
theory
baryon chiral
perturbation theory
Antiferromagnetism
antiferromagnetic phase
spin rotations
SU(2)s
U(1)s
magnon
~e (x) ∈ S 2
staggered magnetization
spin stiffness ρs
spin-wave velocity c
electric charge U(1)Q
electron or hole
magnon exchange
high-Tc superconductor
Hubbard or t-J model
magnon effective
theory
effective theory
presented here
Outline
Cuprates
Correspondences between QCD and Antiferromagnetism
Models for Doped Antiferromagnets
Effective Field Theory for Magnons
Effective Field Theory for Magnons and Holes
Magnon-mediated Two-Hole Bound States
Spiral Phases of the Staggered Magnetization
Chiral Spiral and Pion Condensation
Conclusions
The Hubbard model on the square lattice
H = −t
X
(cx† cy
+
cy† cx )
+U
X
(cx† cx
2
− 1) ,
cx =
x
hxy i
cx↑
cx↓
U(1)Q and SU(2)s symmetries
Q=
X
(cx† cx − 1),
~S =
X
x
cx†
x
~σ
cx ,
2
[H, Q] = [H, ~S] = 0
The t-J model
H=P
−t
X
(cx† cy
+
cy† cx )
+J
hxy i
X
~Sx · ~Sy P
hxy i
reduces to the Heisenberg model at half-filling
H=J
X
hxy i
~Sx · ~Sy
Outline
Cuprates
Correspondences between QCD and Antiferromagnetism
Models for Doped Antiferromagnets
Effective Field Theory for Magnons
Effective Field Theory for Magnons and Holes
Magnon-mediated Two-Hole Bound States
Spiral Phases of the Staggered Magnetization
Chiral Spiral and Pion Condensation
Conclusions
Effective Goldstone boson field in SU(2)/U(1) = S 2
~e (x) = (e1 (x), e2 (x), e3 (x)),
~e (x)2 = 1
Transformation rules of magnon fields
~e (x)0 = R~e (x),
SU(2)s :
Di :
Di
~e (x) = −~e (x),
O:
O
~e (x) = ~e (Ox),
Ox = (−x2 , x1 , t),
R:
R
~e (x) = ~e (Rx),
Rx = (x1 , −x2 , t),
T :
T
~e (x) = −~e (Tx),
Tx = (x1 , x2 , −t)
Low-energy effective action for magnons
Z
S[~e ] =
ρs
d x dt
2
2
1
∂i ~e · ∂i ~e + 2 ∂t~e · ∂t~e
c
Fit to predictions in the ε-regime of magnon chiral
perturbation theory with βc ≈ L, l = (βc/L)1/3
M2s L2 β
χs =
3
2ρs
χu = 2
3c
2048
(
1 c e
1
1+
β1 (l) +
3 ρs Ll
3
c
ρs Ll
c
ρs Ll
)
2
β1 (l) + 3β2 (l)
Q/J = 0.1
Q/J = 0.5
Q/J = 1
Q/J = 2
Q/J = 3
Q/J = 4
64
1024
2
2 )
1
βe2 (l) − βe1 (l)2 − 6ψ(l)
3
128
Q/J = 0.1
Q/J = 0.5
Q/J = 1
Q/J = 2
Q/J = 3
Q/J = 4
2
χSJa
4096
c
1+2
β1 (l) +
ρs Ll
< Wt >
8192
(
32
16
8
512
4
256
32
64
32
L/a
Ms = 0.3074(4)/a2 ,
64
L/a
ρs = 0.186(2)J,
c = 1.68(2)Ja
Nonlinear realization of the SU(2)s symmetry
u(x)~e (x) · ~σ u(x)† = σ3 ,
u11 (x) ≥ 0
Under SU(2)s the diagonalizing field u(x) transforms as
u(x)0 = h(x)u(x)g † , u11 (x)0 ≥ 0,
exp(iα(x))
0
h(x) = exp(iα(x)σ3 ) =
∈ U(1)s
0
exp(−iα(x))
The composite vector field
vµ (x) = u(x)∂µ u(x)† = ivµa (x)σa ,
vµ± (x) = vµ1 (x) ∓ ivµ2 (x)
transforms as
vµ3 (x)0 = vµ3 (x) − ∂µ α(x),
vµ± (x)0 = exp(±2iα(x))vµ± (x)
Outline
Cuprates
Correspondences between QCD and Antiferromagnetism
Models for Doped Antiferromagnets
Effective Field Theory for Magnons
Effective Field Theory for Magnons and Holes
Magnon-mediated Two-Hole Bound States
Spiral Phases of the Staggered Magnetization
Chiral Spiral and Pion Condensation
Conclusions
Hole dispersion in the t-J model
p2
π
a
π
π
a
π /2
0
p1
-π /2
-π -π
-π /2
π /2
0
π
Hole pockets centered at lattice momenta
kα =
π π
,
,
2a 2a
k α0 = −k α ,
kβ =
π
π
,
,−
2a 2a
0
k β = −k β
Hole fields
0
1 kf
f
kf
ψ+
(x) = √ ψ+
(x) − ψ+
(x) ,
2
0
1 kf
f
kf
ψ−
(x) = √ ψ−
(x) + ψ−
(x)
2
Transformation rules of fermion fields
SU(2)s :
f
f
ψ±
(x)0 = exp(±iα(x))ψ±
(x),
U(1)Q :
Q
f
f
ψ±
(x) = exp(iω)ψ±
(x),
Di :
Di
f
f
ψ±
(x) = ∓ exp(ikif a) exp(∓iϕ(x))ψ∓
(x),
O:
O
β
α
ψ±
(x) = ∓ψ±
(Ox),
O
R:
R
β
α
ψ±
(x) = ψ±
(Rx),
R
β
α
ψ±
(x) = ψ±
(Ox),
β
α
ψ±
(x) = ψ±
(Rx)
Leading terms in the effective Lagrangian for holes
L =
X h
f
f
Mψsf † ψsf + ψsf † Dt ψsf + Λ ψsf † v1s ψ−s
+ σf ψsf † v2s ψ−s
f =α,β
s=+,−
+
i
1
1
f†
f
f†
f
f†
f
D
ψ
D
ψ
+
σ
D
ψ
D
ψ
+
D
ψ
D
ψ
1
2
2
1
i
i
f
s
s
s
s
s
s
2M 0
2M 00
Covariant derivative coupling to composite magnon gauge field
f
f
Dµ ψ ±
(x) = ∂µ ± ivµ3 (x) ψ±
(x)
Outline
Cuprates
Correspondences between QCD and Antiferromagnetism
Models for Doped Antiferromagnets
Effective Field Theory for Magnons
Effective Field Theory for Magnons and Holes
Magnon-mediated Two-Hole Bound States
Spiral Phases of the Staggered Magnetization
Chiral Spiral and Pion Condensation
Conclusions
Magnon exchange
p~+
′
p~−
f−
f+
~q
f˜−
f˜+
′
p~+
p~−
One-magnon exchange potentials
V αα (~r ) = γ
sin(2ϕ)
,
r2
V αβ (~r ) = V βα (~r ) = γ
sin(2ϕ)
,
r2
Λ2
γ=
2πρs
V ββ (~r ) = −γ
cos(2ϕ)
,
r2
Two-hole Schrödinger equation for an αβ pair
− M1 0 ∆ V αβ (~r )
V αβ (~r ) − M1 0 ∆
Ψ1 (~r )
Ψ2 (~r )
=E
Ψ1 (~r )
Ψ2 (~r )
Making the ansatz
Ψ1 (~r ) ± Ψ2 (~r ) = R(r )χ± (ϕ)
for the angular part of the wave function one obtains
−
d 2 χ± (ϕ)
± M 0 γ cos(2ϕ)χ± (ϕ) = −λχ± (ϕ)
dϕ2
1
0.5
0
-0.5
-1
-3
-2
-1
0
j
1
2
3
The radial Schrödinger equation
d 2 R(r ) 1 dR(r )
λ
−
+
− 2 R(r ) = M 0 ER(r )
2
dr
r dr
r
is solved by the Bessel function
R(r ) = AKν
p
M 0 |En |r ,
√
ν=i λ
with the energy eigenvalue (for large n)
√
En ∼ −(M 0 r02 )−1 exp(−2πn/ λ)
Two-hole bound state of an αα pair
1
0.5
0
-0.5
-1
-1.5
-2
-3
-2
-1
0
1
2
3
ϕ
e
Angular wave function
Probability density
Outline
Cuprates
Correspondences between QCD and Antiferromagnetism
Models for Doped Antiferromagnets
Effective Field Theory for Magnons
Effective Field Theory for Magnons and Holes
Magnon-mediated Two-Hole Bound States
Spiral Phases of the Staggered Magnetization
Chiral Spiral and Pion Condensation
Conclusions
Spiral configurations
vi3 (x)0 = vi3 (x) − ∂i α(x) = ci3 ,
vi± (x)0 = vi± (x) exp(±2iα(x)) = ci
have a non-zero magnetic energy density
m =
ρs
∂i ~e (x) · ∂i ~e (x) = 2ρs vi+ (x)vi− (x) = 2ρs (c12 + c22 )
2
Single-particle Hamiltonian
H f (~p ) =
M+
pi2
2M 0
+ σf
p1 p2
M 00
Λ(c1 + σf c2 )
Λ(c1 + σf c2 )
M+
pi2
2M 0
+ σf
p1 p2
M 00
yields energy values
E±f (~p ) = M +
pi2
p1 p2
+ σf
± Λ|c1 + σf c2 |
0
2M
M 00
Total density of fermions
β
β
α
α
n = n+
+ n−
+ n+
+ n−
Total energy density of holes
h = α+ + α− + β+ + β−
!
Stability range of various phases
κ1
κ2
κ3
κ4
κi
0
0
1
Homogeneous phase
Zero degree spiral
2
MeffΛ2/2πρs
Inhomogeneous phase
Compressibilities are given by
Λ2
π
Λ2
M 0 M 00
2π
−
, κ2 =
−
, Meff = p
Meff 2ρs
Meff 4ρs
M 00 2 − M 0 2
1
Meff Λ2
π
2π
1−
, κ4 =
κ3 =
3Meff
2 6πρs − Meff Λ2
2Meff
κ1 =
Outline
Cuprates
Correspondences between QCD and Antiferromagnetism
Models for Doped Antiferromagnets
Effective Field Theory for Magnons
Effective Field Theory for Magnons and Holes
Magnon-mediated Two-Hole Bound States
Spiral Phases of the Staggered Magnetization
Chiral Spiral and Pion Condensation
Conclusions
Chiral spiral (with D. Arnold and G. Colangelo)
U = cos α1 + i sin α [cos(ki xi )τ1 + sin(ki xi )τ2 ] = u 2 ,
1 †
1
†
2
τ3 ,
u ∂i u + u∂i u = iki sin α −
vi =
2
2
k
1 †
i
ai =
u ∂i u − u∂i u † = sin 2α τ2
2i
2
Energy density of a spiral configuration
SU(2)
Fπ2 2 2
hψψi
k 2 cos2 α
k sin α +
cos α(mu + md ) + MN +
n
2
2
8MN
gA |k| sin α
(3π 2 )5/3 5/3
5/3
(n+ − n− ) +
n
+
n
, n = n+ + n−
−
2
10π 2 MN +
ε =
+
Critical baryon density for pion condensation
nc ≈ (0.1GeV)3 ≈ nnuclear matter
Outline
Cuprates
Correspondences between QCD and Antiferromagnetism
Models for Doped Antiferromagnets
Effective Field Theory for Magnons
Effective Field Theory for Magnons and Holes
Magnon-mediated Two-Hole Bound States
Spiral Phases of the Staggered Magnetization
Chiral Spiral and Pion Condensation
Conclusions
Conclusions
• There are intriguing analogies between antiferromagnets and QCD.
• Doped antiferromagnets are described by a systematic low-energy
effective field theory analogous to chiral perturbation theory.
• One-magnon exchange leads to two-hole bound states analogous to the
deuteron.
• Spiral phases in doped antiferromagnets are analogous to pion
condensation in nuclear matter.
• Electron-doped systems as well as systems with other lattice geometries
have been studied with similar techniques.
Some references
B. I. Shraiman and E. D. Siggia, Phys. Rev. Lett. 60 (1988) 740
F. Kämpfer, M. Moser, and U.-J. Wiese, Nucl. Phys. B729 (2005) 317
C. Brügger, F. Kämpfer, M. Moser, M. Pepe, and U.-J. Wiese, Phys. Rev. B74 (2006) 224432
C. Brügger, F. Kämpfer, M. Pepe, and U.-J. Wiese, Eur. Phys. J. B53 (2006) 433
C. Brügger, C. P. Hofmann, F. Kämpfer, M. Pepe, and U.-J. Wiese, Phys. Rev. B75 (2007) 014421