Superconductivity, broken symmetry and the Higgs mechanism in

Superconductivity, broken symmetry and the
Higgs mechanism in condensed matter
Dirk van der Marel
Université de Genève
Università degli Studi di Pavia, 14 november 2013
The Making of Mass
Detection of the Higgs particle at LHC
Symmetry breaking and the Higgs mechanism
in a Superconductor
Leggett’s exciton and Other Collective Modes
Università degli Studi di Pavia, 14 november 2013
Let There Be Mass
•  Mass causes:
-Inertia
-Gravitational field
•  Einstein noticed that you can’t
tell the difference between
these two effects
•  But Einstein didn’t explain what
actually causes mass
Painting by John Philipp (Berlin 1929)
Università degli Studi di Pavia, 14 november 2013
1964: Higgs, Guralnik, Hagen, Kibble, Brout, and Englert explained what causes mass
François Englert and Peter W. Higgs
Nobel Prize for Physics 2013
As Higgs phrased it, "this phenomenon is just the relativistic analog of the plasmon
phenomenon to which Anderson has drawn attention: that the scalar zero-mass
excitations of a superconducting neutral Fermi gas become longitudinal plasmon
modes of finite mass when the gas is charged."
P. W. Higgs, Phys. Rev. Lett. 13 (1964) 508.
F. Englert, R. Brout, Phys. Rev. Lett. 13 (1964) 321.
G. S. Guralnik, C. R. Hagen, and T. B. Kibble, Phys. Rev. Lett. 13 (1964) 585.
P. W. Anderson, Phys. Rev. 112, 1900 (1958).
Università degli Studi di Pavia, 14 november 2013
g
A1
A2
A3
υeL
eL eR
g’
B
Ge
g
g’
φ
Electro-Weak Lagrangian
L=

  2 1 

1 
− ∂µ Aν − ∂ν Aµ + gAµ × Aν − ∂µ Bν − ∂ν Bµ
4
4
(
)
(
)
2
Gauge Fields
______
$ ν '$
$ ν '

'

ig
ig'
− && e ))&∂µ − Aµ ⋅ σ L + Bµ )&& e ))
2
2 (% e (
% e (%
Lepton -Field +
L-G Coupling
2
2
2

Higgs-Field +
2
$
'

1
ig
ig'
µ t
λ
t
− &∂µ − Aµ ⋅ σ + Bµ )ϕ + (ϕ ϕ ) − (ϕ ϕ ) H-G Coupling
2%
2
2 (
2
4
+ ______ $
/
'
-$ ν e ' & ϕ + )
)
−Ge ,&&
eR − H.c.0
Lepton -Higgs Coupling
)
0
-% e (L &% ϕ )(
.
1
Università degli Studi di Pavia, 14 november 2013
Spontaneous Symmetry Breaking
Mexican hat
Set a ball on the tip of a Mexican Hat.
The ball decides “spontaneously”
where to fall. There is no influence on
the ball’s path of choice.
The trough of the sombrero represents
Higgs field lowest energy states.
Higgs field amplitude
Università degli Studi di Pavia, 14 november 2013
υeL
W α A1+ iA2
W* α A1- iA2
<φ0>
Z α gA3+g’B
e
<φ0 > = | µ/λ |
mW= <φ0 > g / 2
mZ= <φ0 > (g2+g’2)1/2 / 2
electron mass = Ge<φ0 >
A α g’A3-gB
Symmetry breaking
Higgs: <φ>=φ0
Gauge-fields, Aj, B : E = pc
Coupling of φ and A1, A2, A3, B:
mW c2= 80 GeV mZ c2= 91 GeV
Università degli Studi di Pavia, 14 november 2013
1.
1.  The universe is full of an
invisible field called the
Higgs field
2.  When matter moves, the
Higgs field “drags” on it –
a bit like a celebrity being
slowed down by admirers.
3.  A particle’s mass is
determined by how
strongly it interacts with
the Higgs field.
↓
2.
↓
3.
Università degli Studi di Pavia, 14 november 2013
The Making of Mass
à Detection of the Higgs particle at LHC
Symmetry breaking and the Higgs mechanism
in a Superconductor
Leggett’s exciton and Other Collective Modes
Università degli Studi di Pavia, 14 november 2013
THE LARGE HADRON COLLIDER (CERN)
proton proton collider
14 TeV
27 km de circonférence
1 collision every 25 ns
ATLAS
-Higgs particle
-CP violation
-Quark-Gluon Plasma
CMS
Università degli Studi di Pavia, 14 november 2013
THE LARGE HADRON COLLIDER (CERN)
Assembly of the CMS detector
15
The ATLAS dectector
ATLAS-LHC-CERN Collaboration,
Phys. Lett. B 716, 1 (2012).
CMS-LHC-CERN Collaboration,
Phys. Lett. B 716, 30 (2012).
Symmetry breaking
Higgs: <φ>=φ0
Amplitude Fluctuations around φ0
Higgs: mHc2 = 125 GeV
Università degli Studi di Pavia, 14 november 2013
Università degli Studi di Pavia, 14 november 2013
The Making of Mass
Detection of the Higgs particle at LHC
à Symmetry breaking and the Higgs
mechanism in a Superconductor
Leggett’s exciton and Other Collective Modes
Università degli Studi di Pavia, 14 november 2013
Back to 1911
mercury
H. Kamerling Onnes
Bardeen, Cooper, Schrieffer
If electrons attract each other…..
HOT
COLD
….they form Cooper pairs when T < TBCS
Center of mass momentum of j’th Cooper pair: Kj
Università degli Studi di Pavia, 14 november 2013
which, once formed, condense immediately in a
Bose-Einstein condensate of Cooper-pairs:
T < TBCS << TBEC
N
Ky
Kx
Center of mass momentum of j’th Cooper pair: Kj = 0
Bose-Einstein condensate of Cooper-pairs
Quantum Phenomena
Magnetic Flux Penetrating a Superconductor
Integer Multiples of h/2e
Vortices
Vortices interact with each other à vortex lattice (Abrikosov )
Università degli Studi di Pavia, 14 november 2013
Spontaneous symmetry breaking: Δ≠0
Δ gap amplitude
P. W. Anderson, Phys. Rev. 112, 1900 (1958):
Collective modes of Δ, phase and density
Amplitudon mass: mH c2 ≅ 2Δ
Università degli Studi di Pavia, 14 november 2013
Δ gap amplitude
NbSe2
Università degli Studi di Pavia, 14 november 2013
Photon
ωout
Photon
ωin
Raman Spectroscopy
=
Photon Energy Loss Spectroscopy
Superconductor
Università degli Studi di Pavia, 14 november 2013
Photon
ωout
Photon
ωin
2 K, SC
8 K, N
0.0
2.5 5.0
Superconductor
NbSe2
EXP: R. Sooryakumar, M. V. Klein, PRL. 45 (1980) 660.
7.5
-1
Energy Loss (cm )
ωin - ωout (meV)
THE: P. B. Littlewood & C. M. Varma, PRL. 47 (1981) 811.
29
Δ≠0
Amplitudon mass in NbSe2: mH c2 = 2 meV
Università degli Studi di Pavia, 14 november 2013
P. W. Anderson, Phys. Rev. 112, 1900 (1958)
Light particles in vacuum (photons)
c = 3Ÿ108 m/s
Light particles inside a material (polaritons)
v = c / ε1/2
ε = dielectric permeability
Dirty metal
Longitudinal: ωp,L = 0
Transverse: ωp,T= pv
Superconductor with superfluid density ns
Longitudinal: ωp,L = (4πe2 ns /me)1/2
Transverse: ωp,T = (p2v2 + ωp2)1/2
Plasmons (longitudinal)
&
Plasma-polaritons (transverse)
25
(meV)
15
ωEp
20
10
Transverse
Longitudinal
5
0
0
250
500
750
1000
-1
p / h (cm )
1250
1500
High Tc cuprates
C
J
C
J
C
J
Ca
Cu
ϕi-1 , Ni-1
O
Bi
Sr
Cu
ϕi , N i
Inter Layer Plasmon
J
1
2
2&
H = ( φi − φi+1 ) +
( N i − N i+1 ) (
2
2C
' ⇒ ω p = JC
(
"#φi , N j $% = iδi, j
)
Università degli Studi di Pavia, 14 november 2013
Inter Layer Plasmon of Bi2Sr2CaCu2O8
ω = 0.19 meV
K. Kadowaki et al., PRB 56, 5617 (1997)
t=
2 ε
(1+ ε )
iω d
e
2
ε /c
r=
1− ε
1+ ε
Università degli Studi di Pavia, 14 november 2013
−1
"
%
ω p2
−1
Longitudinal modes - Energy Loss function: L (ω ) = Im
= Im $$
−1''
ε (ω )
ω
ω
+
i
/
τ
(
)
#
&
La1.85Sr0.15CuO4
Tc= 30 K
2
ω p,S
T < Tc ;
= Δ(T ) τ
ω p2
Undamped superfluid
L(ω)
L (ω ) ∝ δ (ω − ω p,S )
T > Tc ; ω pτ << 1
Overdamped
ω p2τ
L (ω ) ≅
ω
J. H. Kim, DvdM et al.,
Physica C 247, 297 (1995).
0
5
10
15
ω (meV)
Università degli Studi di Pavia, 14 november 2013
20
25
Transverse modes: Polariton dispersion
8K
30 K
33 K
25
Eω (meV)
20
La1.85Sr0.15CuO4
Tc= 30 K
15
10
0
( )
p=c ω ε ω
−1
5
0
250
500
750
1000
1250
1500
-1
p / h (cm )
D. van der Marel, J. of Superconductivity: Incorporating Novel Magnetism, 559, 17 (2004) .
Symmetry breaking
BCS: Δ≠0
Coupling of Δ and A : Photon mass in La1.85Sr0.15CuO4 = 6 meV
Università degli Studi di Pavia, 14 november 2013
−1
"
%
ω p2
$
Energy Loss function: L (ω ) = Im $
−1''
# ω (ω + i / τ ) &
ω pτ > 1 : Underdamped plasmon
Electric field
Current
Bi2Sr2CaCu2O8
Inelastic
Electron
Scattering
L(ω)
Optical
C. N. Presura, Ph D thesis, Rijksuniversiteit Groningen, (2003).
S Nakai et al., Physica Scripta 41, 596 (1990).
(eV)
0.90
Intra Layer Plasmon of Bi2Sr2CaCu2O8
Tc2
Tc
0.89
ωp
T (K)
H. J. A. Molegraaf et al., Science 295, 2239 (2002)
The Making of Mass
Detection of the Higgs particle at LHC
Symmetry breaking and the Higgs
mechanism in a Superconductor
àLeggett’s exciton and Other Collective
Modes
Università degli Studi di Pavia, 14 november 2013
Leggett excitons in two-band superconductors
A.J. Leggett, Progr Theor. Phys. 36, 901 (1966)
ϕ1 , N 1
ϕ2 , N 2
Josephson coupling: Inertia
Finite compressibility : Restoring potential
J
1
2
2&
φ
−
φ
+
N
−
N
( 1 2)
( 1 2) (
J
2
2κ 0
' ⇒ ωL =
κ0
(
"#φi , N j $% = iδi, j
)
H=
Università degli Studi di Pavia, 14 november 2013
Transverse Optical Plasmons in Bilayer Superconductors
DvdM & A. Tsvetkov, Czech. J. Phys. 46, 3165 (1996)
SmLaCuO4
CuO2
LaO
LaO
CuO2
C1
ω p,1 = J1C1
J1
Sm
O2
Sm
C2
ω p,2 = J 2C2
J2
CuO2
LaO
LaO
CuO2
C1
ω p,1 = J1C1
J1
Sm
O2
Sm
C2
ω p,2 = J 2C2
J2
CuO2
LaO
LaO
CuO2
C1
ω p,1 = J1C1
J1
Transverse optical plasmons & Leggett excitons
DvdM & A. Tsvetkov, PRB 64, 024530 (2001)
δ (ω )
ωT
ε1(ω)
ε1(ω)
ωε2(ω)
ωε2(ω)
ωT
L(ω)
ωp,1
0
ε (ω ) = ε s
ω p,1
2
2
2
2
ω
−
ω
ω
−
ω
(
p,1 ) (
p,2 )
ω 2 (ω 2 − ωT2 + i0 + )
ω p,2
ω
L(ω)
ωp,2
2
2
; ω p,1
< ωT2 < ω p,2
0.5
1.0
1.5
2.0
ω (meV)
D.Dulic et al. PRL 86, 4144 (2001)
Transverse optical plasmons & Leggett excitons
DvdM & A. Tsvetkov, PRB 64, 024530 (2001)
4
4
Plasmons
Polaritons
ωp (meV)
3
3
2
ωp,2 m2
2
ωT
1
1
ωp,1 m1
0
0.6
0.4
0.2
-1
pz/h (A )
0.0
20
40
-1
p// / h (cm )
60
0
Leggett excitons in two-band superconductors
Symmetry breaking
BCS: Δ1≠0 : Δ2≠0
2 Higgs bosons & 1 Leggett exciton
Coupling of Δ1 , Δ2 and A :
Two photon flavors in SmLaCuO4 :
m1c2 = 0.8 meV
m2c2 = 1.6 meV
Università degli Studi di Pavia, 14 november 2013
Collective Modes in Superconductors
Fluctuation of
Appellation
Parity
Spin
No. of
gaps
No. of
bands
Probe
Observed ?
|Δ|
Higgs mode
even
0
1
1
Raman
y
ϕ
Josephson
Plasmon
Resonance
odd
0
1
1
EELS/
Optics
y
ϕ
Josephson
Polariton
odd
0
1
1
Optics
y
ϕu− ϕd
Bound
Paramagnon
even
1
1
1
INS
y
Δs / Δ p
BardasisSchrieffer
odd
1
2
1
INS
n
Δd / Δ s
BardasisSchrieffer
even
0
2
1
Raman
y
ϕ1g − ϕ2g
Leggett Exciton
even
0
1
2
Raman
y
ϕ1g − ϕ2u
Leggett Exciton
/ Transverse
Optical JPR
odd
0
1
2
Optics
y
E. S. Reich, Nature 495, 22 (2013)
PRL 100, 205701 (2008)
PRB 88, 060508(R) (2013)
PRL 111, 057002 (2013)
arXiv:1108.5207
49
Summary"
"
The mass of elementary particles is generated by the Higgs
mechanism"
"
Unifying principles connect observations on elementary particles and
phenomena in condensed matter. "
"
Photons in a superconducting material acquire a non-zero mass due to
their coupling to the superconducting condensate. "
"
Under certain conditions a collective mode is observed, similar to the
Higgs particle. "
"
These and many other phenomena in elementary particles and
condensed matter are marvelous by the beauty and simplicity of the
principles uniting them, despite conditions and energy scales being so
vastly different."
"
V. L. Ginzburg, L. D. Landau, Zh. Eksp. Teor. Fiz. 20 (1950) 1064.
J. Bardeen, L. N. Cooper, J. R. Schrieffer, Phys. Rev. 108 (1957) 1175.
P. W. Anderson, Phys. Rev. 112, 1900 (1958).
Y. Nambu, Phys. Rev. Lett. 4 (1960) 380.
J. Goldstone, Nuovo Cimento 19 (1961) 154.
J. Goldstone, A. Salam, S. Weinberg, Phys. Rev. 127 (1962) 965.
J. Schwinger, Phys. Rev. 125 (1962) 397.
P. W. Anderson, Phys. Rev. 130 (1963) 439.
P. W. Higgs, Phys. Lett. 12 (1964) 132.
P. W. Higgs, Phys. Rev. Lett. 13 (1964) 508.
F. Englert, R. Brout, Phys. Rev. Lett. 13 (1964) 321.
G. S. Guralnik, C. R. Hagen, and T. B. Kibble, Phys. Rev. Lett. 13 (1964) 585.
A. J. Leggett, Prog. Theor. Phys. 36 (1966) 901.
S. Weinberg, Phys. Rev. Lett. 19 (1967) 1261.
G. ’t Hooft, Nucl. Phys. B 35 (1971) 167.
R. Sooryakumar, M. V. Klein, Phys. Rev. Lett. 45 (1980) 660.
P. B. Littlewood, C. M. Varma, Phys. Rev. Lett. 47 (1981) 811.
A. Bardasis and J. R. Schrieffer, Phys. Rev. 121 (1960) 1050 .
D. van der Marel, Phys. Rev. B 51 (1995) 1147.
K. Kadowaki, I. Kakeyam, M. B. Gaifullin, et al., Phys. Rev. B 56 (1997), 5617
D. van der Marel and A. A. Tsvetkov, Phys. Rev. B 64 (2001) 024530.
D. van der Marel, J. of Superconductivity: Incorporating Novel Magnetism, 17 (2004) 559.
ATLAS-Collaboration,Phys. Lett. B 716 (2012) 1.
CMS-Collaboration, Phys. Lett. B 716 (2012) 30.
Università degli Studi di Pavia, 14 november 2013