Strongly correlated
systems
T. Giamarchi
http://dpmc.unige.ch/gr_giamarchi/
Details in:
TG, cond-mat/0605472 (Salerno
lectures)
TG, Quantum physics in one
dimension, Oxford (2004)
Questions
• Understood: free electrons
• Real systems : Coulomb interaction
• Properties of realistic systems
• Free electron theory works quite well
Effect of interactions
Landau Fermi liquid
Individual fermionic excitations
exist (quasiparticles)
Fermi liquid theory
• Shown perturbatively in U
• Much more general and robust
Element m*/m cc
Nb 2 1
3He 6 20
Heavy fermion 100-1000 100
Photoemission
Measures Spectral function A(k,)
4 3 2 1 0
binding energy
0°
60°
em
issio
n a
ng
le
E(k)
wave vector [Å-1]
bin
din
g e
nerg
y [eV
]
EF
-4
-3
-2
-1
0
1.51.00.50.0
zoomed
Bandmapping
[001]
[111]
[110]
X
L
W
.
..
Cu Fermi surface (Ashcroft/Mermin)
Fermi surface mapping
k(EF)
(M. Grioni)
T. Valla et al. PRL 83 2085 (1999)
Mo(110) surface
• Transistor 1956
• Supraconductivité
• Magnétorésistance
géante
(1913),1972,
1973,1987,2003
2007
Need to undestand the effects
of interactions ?
Doped SrTiO3
CMR-manganites
Organics, fullerenes,
novel compounds
largest polarization
reached by the
field effect in oxides
FM-insulator
FM-M
High-Tc cupratesSC metalAF-insulator
SC
AF-M
Silicon, GaAssemiconductor (Wigner, FQHE)
n 2D ( e / cm2 )10 1510 7 10 9 10 1310 1110 5
AF-insulator
FM-I
insulator / semiconductor / metal metal
Densities
Oxydes: do not follow band
theory
Leo Kouwenhoven et al
Quantum Dots
Nanoscopic Systems
b Moty Heiblum et al.
Fractional quantum Hall effect
I
V
two dimensional electron gas
High Tc
Bednorz Muller
Ferroelectrics, High Tc, Manganites, Oxydes,
Organics,, nanotechnologies....
Interactions: tomorrow’s
materials
Superfluid
Mott
Insulator
1 μm
Pb
Cold atoms
Atoms in a lattice
Tunnelling
Short range
interaction
Proposal: D. Jaksch et al PRL81 3108 (98)P. Zoller
Effect of interactions in CaG
M. Greiner, O. Mandel, T. Esslinger, T. W. Hansch, I. Bloch, Nature 415 39 (2002)
Cold atomic gases
• Extreme
control and
versatility
• Novel ``materials’’ to address condensed matter
issues
Quantum simulators !
Interactions
StatisticsDimensionality
ENS, ETH, LENS, Mainz, MIT,
NIST, Penn State,
Hubbard model
Reduced dimensionality ?
• Effect of interactions at their strongest
• Many realizations (CM and CA)
• Qualitatively Novel physics !
Plastic electronic1973: F.Wudl, D.Cowan, A.Garito, A.Heeger
TTF° + TCNQ°----->TTF+TCNQ-
2 TMTSF° + X- -----> TMTSF2X + e-
Charge transfer compounds and organic salts
+
+
Superconductivity ?
NO !!
Sliding CDW !!
Bechgaard salts
1979: K. Bechgaard
TMTTF : atoms S (Fabre)
TMTSF : atoms Se (Bechgaard)
TMTSF
(TMTSF)2X, X=PF6-,…. ClO4
-,.
c
a
c
b
• Salt
• Quarter filled
band
ta 3000K
tb 300K
tc 30K
Quasi 1D
Superconductivity !
1981: D. Jerome
D. Jerome, M,Ribault,J.Mazaud and K.Bechgaard (1980)
D. Jaccard et al., J. Phys. C, 13 L89 (2001)
O.M Ausslander et al., Science 298 1354 (2001)Y. Tserkovnyak et al., PRL 89 136805 (2002) Y. Tserkovnyak et al., PRB 68 125312 (2003)
Quantum wires
Cees Dekker
CARBONNANOTUBES
Nanotubes
Yao et al., Nature 402 273 (1999)
Spin chains and ladders
Ladder systemsSpin chains
Spin dimer systemsB. C. Watson et al., PRL 86 5168 (2001)
M. Klanjsek et al.,
PRL 101 137207 (2008)
B. Thielemann et al.,
arXiv:0809.0440 (2008)
Cold atoms: 1D systems
S. Hofferberth et al. Nat. Phys
(2008)
One dimension is different
• No individual excitation can exist (only
collective ones)
• Strong quantum fluctuations
Continuous symmetry
How to make a theory
60’: A. Larkin, I. Dzialoshinskii, L. Gorkov, Bichkov
E. Lieb, D. Mathis
70’: A. Luther, V.J. Emery
80’: F.D.M. Haldane
Good excitations: collective onesTomonaga Luttinger
Details in:
TG, cond-mat/0605472 (Salerno
lectures)
TG, Quantum physics in one
dimension, Oxford (2004)
So now let us start......
Labelling the particles
1D: unique way
of labelling
(x) varies slowly
q » 0 q » 20
CDW
: superfluid phase
Quantum
fluctuations
All short distance properties: form of
the operators. , : smooth fields
Hamiltonian
Luttinger liquid theory
Low energy effective description
All effects described by two parameters:
u and K (Luttinger liquid parameters)
Power law correlation functions
Luttinger parameters
u: velocity of collective excitations
K: dimensionless parameter
U0 1
K1 1
Tonks gas
Luttinger parameters
Correlations
1/2
Difference with ``hydrodynamics’’
(quasicondensates)
à = Ã0 Exp[i µ(x,t)]
S = s [ ( )2 + (x )2]
but !!!
½ = ½0
Condensate ?
No true condensate
(K 0)
Finite temperature
Conformal theory
b
c
Spins
Powerlaw correlation functions
Non universal exponents K(h,J)
1D : excitations are more like fermions, not bosons !!
Fermions
Right (+kF) and left (-kF) particles
Observed ?
Hint in spin systems
Tennent et al. PRB 52 13368 (1995)
H. J. Schulz PRB 34 6372 (1986)
Lineshape:
FT of power law better
than two
gaussian/Lorentzian peaks
Organic conductors
c
b
A. Schwartz et al. PRB 58
1261 (1998)First observation of LL !!
¾(!) » !º
TG Physica B 230 975
(1997)
Nanotubes
Z. Yao et al. Nature 402
273 (1999)
Tonks limit
U = 1 : spinless fermions
Not for n(k) : F B
B. Paredes et al Nature (2004)
T. Kinoshita et al. Science (2004)
M. Kohl et al. PRL (2004)
Interferences
s0L h Ãy(r) Ã(0) i
but K large !! ½ » ½0 hydrodynamic !!
S. Hofferberth et al. Nat. Phys
(2008)
Some aspects of LL but......
Exponent is an adjustable parameter
Universality ? Different correlation
functions ?
Variation with a control parameter ?
can one do a quantitative test ????
BPCB-HPIPB. C. Watson et al., PRL 86 5168 (2001)
M. Klanjsek et al.,
PRL 101 137207 (2008)
B. Thielemann et al.,
PRB 79, 020408(R) (2009)
A. Tsvelik (BNL)
R. Chitra (Jussieu)
E. Orignac (ENS-Lyon)
R. Citro (Salerno U.)
P. Bouillot (Geneva)
C. Kollath (Polytechnique)
M. Zvonarev (Geneva)
Collaborations:M. Klanjsek +
group C. Berthier
(Grenoble)
B. Thielemann +
group C. Ruegg
(LCN/PSI)
+LANL Group
D. Poilblanc ,
S Capponi (Toulouse)
A. Laüchli (Dresden)
B. Normand
Spin Ladders
J
J’
Jr
3 scales :
Jr : rungs
J : legs
J’ : interladder exchange
Triplons:
hard core bosons
1D : spinless fermions
Effect of magnetic field
E
H
S
T
Nature of the transition ?
Properties of massless phase ?
Magnetic field: like a chemical potential (hc1, hc2)
(TG and A. M. Tsvelik PRB 59 11398 (1999))
h
m
Gapped
hc1 hc2
A way to study interacting spinless fermions !
hc1 » Jr ; hc2 » Jr + J
All microscopic parameters are known !
Expected phase diagram
T
H
hc1 hc2
LL
3D
Thermal
gapped
J
J’
How to compute
• Analytical calculations (Luttinger liquid +
BA)
• Numerical calculations (DMRG+ MC)
DMRG
• Zero temperature DMRG
(LL parameters)
• Zero temperature DMRG
Mean field theory
+ (-1)i hx
Finite temperature DMRG
Specific heat Spin chain
P. Bouillot M. Zvonarev
C. Kollath
Magnetization
M. Klanjsek et al., PRL 101 137207 (2008)
Fixes:
Jr = 12.9 K
J = 3.6 K
Specific heat
Ch. Rüegg et al.,
PRL 101, 247202 (2008)
Peak : end of LLLuttinger liquid: Cv / ° T
Specific heat
Ch. Rüegg et al., PRL
101, 247202 (2008)
Can one control LL ?
Compute u(h) and K(h) from (Jr,J,h)
No adjustable parameters !!
Get all (several) correlation functions
Allows to test for Luttinger Liquid !
Luttinger parameters
M. Klanjsek et al., PRL 101 137207 (2008)
Red : Ladder (DMRG)
Green: Strong coupling (Jr 1 ) (BA)
Correlation functions
NMR relaxation rate:
Tc to ordered phase: 1/J’ = Â1D(Tc)
M. Klanjsek et al., PRL 101 137207 (2008)
R. Chitra, TG PRB 55 5816 (97); TG, AM Tsvelik PRB 59 11398 (99)
Ordered phase
Order parameter at T=0
M. Klanjsek et al.,
PRL 101 137207 (2008)
NMR
Neutrons
J’ = 27 mKB. Thielemann et al.,
PRB 79, 020408(R) (2009)
Systems with « spins »
Luttinger liquid
Same treatment
More convenient
1 1( ) ( )
2 2
kinH H H H H
int ( )( )
( )
i
H U U
U
H H H
( , )u K Charge excitations
( , )u K Spin excitations
Charge-spin separation
Spin
Spin-Charge Separation
Charge
Spinon Holon
Spin
Spin-Charge Separation
higher D ?
Charge
Energy increases with spin-charge separation
Confinement of spin-charge: « quasiparticle »
Can one observe spin-charge
separation ?
• Condensed matter: difficult
• One serious experiment: Yacoby et al.
O.M Ausslander et al., Science
298 1354 (2001)
Y. Tserkovnyak et al., PRL 89
136805 (2002)
Y. Tserkovnyak et al., PRB 68
125312 (2003)
Bosons with “spins”
[L.E. Sadler et. al., Nature 443, 164 (2006)]
Spin 1 system[J.M. McGuirk et. al., PRL 89, 090402 (2002)]
[J.M. Higbie et. al., PRL 95, 050401 (2005)]
87Rb atoms, F=1 states
|F=1,mF=-1i and |F=2, mF = 1i “spin” ½ system
[M.Erhard et. al., PRA 69, 032705 (2004)] [A. Widera et. al., PRL 92, 160406 (2004)]
Proposal
• 87Rb |F=2,mF=-1i and |F=2,mF=1i good
potential candidates to observe spin-charge
separation
• No problem of temperature ( fermions)
• Phase separation
A. Kleine, C. Kollath et al. PRA 77 013607 (2008);
NJP 10 045025 (2008)
(Partial) Conclusions
Highly non trivial 1D physics due to interactions
Luttinger liquid theory provides a framework to
study this physics
Testable in many different microscopic systems
(organics, nantubes, quantum wires, cold atoms,
spin systs)
Luttinger liquid provides a firm footing to go
beyond
Luttinger liquids as a cornerstone
to study additional effects
Effect of a lattice:
Mott transition
How to treat?
• Incommensurate: Q 2 0
• Commensutate: Q 2 0
Competition
2 21[ ( ( , )) ( ( , )) ]2
xu
dxdS x u x
K
Beresinskii-
Kosterlitz-Thouless
transition
K=2
Mott insulator:
is locked
Density is fixed
Gap in the excitation spectrumn(k)
T. Kuhner et al. PRB 61
12474 (2000) TG, Physica B 230 975 (97)
Important for 1D
J 0 ``trivial’’ MI
J À V but K < 2
MI !!
T. Stoferle et al.PRL 92 130403 (2004)
Cold atoms
Effect of disorder
Disorder: Bose Glass
TG + H. J. Schulz EPL 3 1287 (1987); PRB 37 325 (1988)
``Two’’ fourier components of disorder
Backward scattering (q » 2 0)
Responsible for localization (Bose Glass)
Pinning of a CDW of bosons
Tonks limit: Anderson localization of free
fermions
Bose glass phase
1D : TG + H. J. Schulz PRB 37 325 (1988)
Superfluid – Localized (Bose glass) transition
for K < 3/2
BKT like transition
Higher dimensions: M.P.A. Fisher et al. PRB 40 546 (1989)
Bose glass also exists (scaling theory)
continuous transition
Phase diagram
K
Db
1 3/2
Bose Glass
Superfluid
Important points
Weak disorder enough : localized for
K < 3/2 even if V ¿
Quantum effect: destructive interferences
Short correlations of disorder: rf » d
Probing: tilting of lattice (``transport’’)
Numerics
S. Rapsch, U.
Schollwoeck, W.
Zwerger EPL 46
559 (1999)
Experiments with interactions !
L. Fallani et al. PRL 98, 130404 (2007)
Quasiperiodics
V(x) = Cos(Q1 x) + Cos(Q2 x)
[Harper potential]
Commensurate (Mott transition): K=2 ??
Disorder (Bose Glass): K=3/2 ??
Other ??
Quasiperiodic
J. Vidal, D. Mouhanna, TG PRL 83 3908 (1999); PRB 65 014201 (2001)
G. Roux et al. PRA 78, 023628 2008
Expansion
Beyond Luttinger
liquids
• Coupling between 1D chains (deconfinement
transition)
[M. A. Cazalilla et al. N. J. Phys. 8 158 (2006) ]
• Coupling of the LL to an external bath
[A. M. Lobos et al. arxiv/0906.5570; E.G. Dalla
Torre et al. (2009)]
• Systems with non linear spectrum (e.g. ! » k2 –
ferromagnet)
[M. B. Zvonarev et al. PRL 99, 240404 (2007)]
Coupled chains: deconfinement
and higher dimensions
Spin systems
E
k
Weakly coupled 1D ladders
J’
Quasi-1D ; Luttinger liquid approach
¹ = h – hc1
Close to QCP:
Dimensional crossover
Always 3D close to QCP !
E
kJ’
¹ = h – hc1
J’ > ¹:
3D
behavior
Bose Einstein Condensation
(TG and A. M. Tsvelik PRB 59 11398 (1999))
triplon = hard core boson
h hc dilute limit: « free » bosons
BEC vs BEC
Cold atoms Dimers/Spins
TG, Ch. Rüegg,
O. Tchernyshyov,
Nat. Phys. 4 198 (08)
Crossover LL (``fermions’’) –
BEC (``bosons’’)
E. Orignac, R. Citro, TG, PRB 75, 140403R 2007
NMR,
1/T1
Local spin
correlations
Bosons
[87Rb]
A. F. Ho, M. A. Cazalilla, TG PRL 92 130405 (2003)
M. A. Cazalilla, A. F. Ho, TG, NJP 8 158 (2006)
T. Stoferle et al.
PRL 92 130403 (2004)
1D physics (Luttinger Liquids)
Cold atoms
1D Mott insulator
10-6
10-5
10-4
10-3
10-2
10-1
1 1.5 2 2.5
J/
K
1D MI2D MI
Anisotropic 3D SF (BEC)
( ) ( ) ( )( )
Phase diagram
Array of atomic
‘quantum dots’
transverse
hopping
repulsion
T. Stoferle et al.PRL 92 130403 (2004)
Experiments
Fermions
D. Jaccard et al., J. Phys. C, 13 L89 (2001)
Deconfinement
0 5 10 15 20 25
60
80
100
200
400
600 TMTTF2PF
6
a
T*
IC
SDW
C
SDWSpin-Peierls
c
Act
ivat
ion
en
ergy,
T* (
K)
Pressure (kbar)
P. Auban-Senzier, D. Jérome, C. Carcel and J.M. Fabre J de Physique IV, (2004)
TG Chemical
Review 104 5037
(2004)
Coupled 1D chains
C. Berthod, TG, S. Biermann, A. Georges, PRL 97 136401 (06)
Coupled tubes with Spin gap
b
M.A. Cazalilla, A. F. Ho, TG,
PRL 95 226402 (2005); arxiv
0604525
AF
Order
1
2
Triplet superconductivity
(repulsive interactions)
Non luttinger liquids
Itinerant ferromagnet
M. B. Zvonarev, V. V. Cheianov, TG, PRL 99 240404 (2007)
One dimensional two bosonic species
G?(x; t) ' t¡®"¯ ln
Ãt
tF
!+
it¹h
2m¤
#¡1=2exp
(im¤x2
2t¹h¡ 4i¯m¤ ln(t=tF)
)
Trapped/open
regimes
Light cone of
spinless bosons
Transverse spin excitations
Conclusions
Tour of one dimensional physics
Luttinger liquid theory provides a framework to
study this physics, and to go beyond
Many effects: lattice, disorder, long range forces,
competition between orders, out of equilibrium
physics, .......
Many experimental realizations: organics, spin
systems, nanowire, nanotubes, adatoms,
FRQHE, Joseph. junctions, cold atoms, ......