bosonization: a primer - lancaster university
TRANSCRIPT
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Bosonization: a primer
T. Giamarchi
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References on 1D Physics• Emery, V. J. (1979). Highly conducting one dimensional solids, pp. 247. Plenum.
• Solyom, J. (1979). Adv. Phys., 28, 209.
• Schulz, H. J. (1995). Les Houches LXI pp. 533. Elsevier.
• Voit, J. (1995). Rep. Prog. Phys., 58, 977.
• von Delft, J. and Schoeller, H. (1998). Ann. Phys., 7, 225.
• Schonhammer, K. (2002). J. Phys. C, 14, 12783.
• Senechal, D. (2003). CRM Series in Mathematical Physics, Springer. cond-mat/9908262.
• Gogolin, A. O., Nersesyan, A. A., and Tsvelik, A. M. (1999). Bosonization and Strongly Correlated Systems. Cambridge University Press, Cambridge.
• Giamarchi, T. (2004) Quantum Physics in one dimension. Oxford University Press, Oxford.
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1d Systems
Organicconductors
Quantum wires
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Nanotubes
But also: josephson junctionsladdersEdge states …..
Cold Bosons
N0 ~ 10 to 103
atoms
T. Stoferle et al. PRL 92 130403 (2004)
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Fermi liquid : crash course
• Individual fermionic excitations exist (as for free electrons)
n(k)
k
Z
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D=1• No individual excitation can exist (onlycollective ones)
• Strong quantum fluctuations (``no’’ordered state or mean field possible)
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How to study
• Exact methods (Bethe Ansatz)
spectrum; limited to very special models• Numerics
special models, size limitations, quantities specific to models
``Exact’’
Exact
• Low energy methods methods
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How to solve
• Identify the low energy excitations
• Bosonization method
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Particle hole excitations
)()(),( kqkqkE εε −+=
+kF-kF
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E(q)
q2kF
D>1:
continuum
E(q)
q2kF
D=1:
Well definedexcitations
qvqE F=)(
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Gory details
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ppp
F bbpvH *
0∑
≠
=
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More convenient to use
)()]()([)()]()([)(
xxxxxxx
LR
LR
Π=−=Θ∇+−=Φ∇
πρρπρρπ
]))(())([(2
22 xxvdxH F Φ∇+Π= ∫ ππ
])),(()),(([2
221 ττπ
ττ xvxdxdS xFvF
∂+∂= ∫• Phonon Hamiltonian
• Link with 2d stat mech
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Interactions2))(()'()( xxx Φ∇≈ρρ
]))(())(([2
22 xKuxuKdxH Φ∇+Π= ∫ π
π• u velocity of sound
• K dimensionless parameter,K< 1 : repulsiveK>1 : attractive
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Properties• Only collective excitations
• Thermodynamics
uvKL
uTC F
V =⎟⎠⎞
⎜⎝⎛= 0/
3κκπ
• Looks like a Fermi liquid for q~0
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Fermion operator)()](),([ xexp R
ipxRR ψψρ −=
)(xρ )(xΦ
x x
)]()([)()(
2)()( xxri
xirk
rxi
ydyi
eexeexF
x
Θ−Φ−ΘΠ
==∫
≈ ∞−
παψψ
π
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Correlation functions
)(22
**
1)(
))()())(()(()(
xxki
LRLR
Fex
xxxxx
Φ++Φ∇−
=
++=
πρ
ψψψψρ
K
F xxk
xx
2
2
1)2cos(1)0()( ⎟⎠⎞
⎜⎝⎛+=ρρ
Non universal decay of correlationfunctions
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)(2** )()()( xiLRSU exxxO Θ≈= ψψ
K
SUSU xOxO
2/11)0()( ⎟⎠⎞
⎜⎝⎛=
Single particle excitations
τψψ
ψψ τ
FRR
xiArgKK
RR
vxxK
ex
x
−==
⎟⎠⎞
⎜⎝⎛=
−+
1)0()(1
1)0()(
*
)/(][
21
*
1
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No Landau Quasiparticlesn(k)
n(k)1][
21 1 −+ −
− KKFkk
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Finite temperatureConformal theory
β
χ
K
r
21⎟⎠⎞
⎜⎝⎛
βξβπ
/)2( xxK
ee −−=
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Phase diagram2)( ),(),( −+ ≈= ∫ ηωτ ωχτχτωχ xedxdq qxi
Most divergent fluctuations
K<1 K>1
CDW SU
-V
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System with spinSame treatment
More convenient
↓↓↑↑ Φ∇→Φ∇→ ρρ
)(2
1)(2
1↓↑↓↑ −=+= ρρσρρρ
σρ HHHHHkin +=+= ↓↑
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)(
))((int
σσρρ
σρσρρρ
−=
−+== ∑ ↓↑
U
UUHi
σρ HHH +=
),( ρρ Ku Charge excitations
),( σσ Ku Spin excitations
Charge-spin separation
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spinonholon
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Correlation functions
( ) ρσ KKxFx
xkSxS ++= 11 )2cos()0()( 2
•Perturbation (small U)
π
π
σσ
ρρ
σσρρ
//
//
UvKu
UvKu
vKuKu
F
F
F
−=
+=
==
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ρK
σK
SDW
CDW SU singlet
SU triplet
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Spin sector more complicated (gap)
∫∫
Φ+
Φ∇+Π=
))(8cos(
]))(())(([2
22
xdxg
xKuxuKdxH π
π
Anomalous correlation functions
12
21][
41 1
)(−
−+
≈
−≈−
ρ
ρρ
χ Kk
KKF
T
kkkn
F
photoemission
NMR
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J. Voit
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Luttinger liquid concept
• How much is perturbative
• Nothing provided the correct u,K are used
• Low energy properties: Luttingerliquid (fermions, bosons, spins…)
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General Derivation (Haldane)
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Interacting Bosons
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No condensate
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ρ
ρ
ρρ
K
F
K
F
xxkA
xxkA
xx
4
2
1
12
1)4cos(
1)2cos(1)0()(
⎟⎠⎞
⎜⎝⎛+
⎟⎠⎞
⎜⎝⎛+=
+
Hubbard
2/1
10
=∞=
==
ρ
ρ
KU
KU
Fermions
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How to compute (u,K)
• Perturbation• Exact solutions (Bethe Ansatz): thermodynamics
cuLELEuKDKuuCV
∝∞−=→→
/))()((/κ
• Numerics
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H.J. Schulz PRL 64 2831 (90)
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Conclusions
• Good control of d=1 massless phases
• Massive phases
• Luttinger Liquid: crucial startingpoint to study perturbations (lattice, disorder, etc.)
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Examples
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Lattice
Mott insulator:φ is locked
HZ
dxV x ½ xZ
dxV x ei p ¼½ x¡ Á x
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Coupled one dimensional bosons
Cold Bosons
N0 ~ 10 to 103
atomsT. Stoferle et al. PRL 92 130403 (2004)
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Interchain hopping
Wants to order θ on each chain
Competes with Mott potential that wants to order φ
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“Mott” potential: localizes atoms
AF Ho, M.A. Cazalilla, and TG, PRL 92 130403 (2004)
R’
RJosephson coupling: delocalizes atoms
Mott vs. Josephson
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Phase diagram
10-6
10-5
10-4
10-3
10-2
10-1
1 1.5 2 2.5
J/µ
K
1D MI
( γ = + ∞) ( γ = 3.5) ( γ = 2)( γ = 8)
Anisotropic 3D SF (BEC)
Deconfin
ement
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Mesoscopic effects
• Charging enery of a tube EC = ~ π vs/K L
Array of atomic‘quantum dots’
10 -3 10 -2
-0.4
-0.2
0
0.2
0.4
J/E C µ/
EC
SF2D MI
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10-6
10-5
10-4
10-3
10-2
10-1
1 1.5 2 2.5
J/µ
K
1D MI2D MI
Anisotropic 3D SF (BEC)
( γ = + ∞) ( γ = 3.5) ( γ = 2)( γ = 8)
Phase diagram for finite tubes
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ExperimentsT. Stoferle et al. PRL 92 130403 (2004)
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Large quantum depletion
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Disorder, no interactions
])()([])[(
)()(*****
0
0
RLLRLLRR xxdxxdxH
xxdxVHH
ψψξψψξψψψψη
ρ
∫∫∫
++++=
+=
Localization length is mean free path
Compressible system locre ξψ /~ −
Backscattering gives localization
)(log~)( 22 ωωωσ
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Bosonized representation
])),(()),(([2
221 τφτφπ
ττ xux
KdxdS xu ∂+∂= ∫
SZ
dxd¿V x ½ x; ¿Z
dxd¿V x ei Á x;¿
DisorderedElastic System