Bosonization: a primer

T. Giamarchi

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.

1d Systems

Organicconductors

Quantum wires

Nanotubes

But also: josephson junctionsladdersEdge states …..

Cold Bosons

N0 ~ 10 to 103

atoms

T. Stoferle et al. PRL 92 130403 (2004)

Fermi liquid : crash course

• Individual fermionic excitations exist (as for free electrons)

n(k)

k

Z

D=1• No individual excitation can exist (onlycollective ones)

• Strong quantum fluctuations (``no’’ordered state or mean field possible)

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

How to solve

• Identify the low energy excitations

• Bosonization method

Particle hole excitations

)()(),( kqkqkE εε −+=

+kF-kF

E(q)

q2kF

D>1:

continuum

E(q)

q2kF

D=1:

Well definedexcitations

qvqE F=)(

Gory details

ppp

F bbpvH *

0∑

≠

=

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

Interactions2))(()'()( xxx Φ∇≈ρρ

]))(())(([2

22 xKuxuKdxH Φ∇+Π= ∫ π

π• u velocity of sound

• K dimensionless parameter,K< 1 : repulsiveK>1 : attractive

Properties• Only collective excitations

• Thermodynamics

uvKL

uTC F

V =⎟⎠⎞

⎜⎝⎛= 0/

3κκπ

• Looks like a Fermi liquid for q~0

Fermion operator)()](),([ xexp R

ipxRR ψψρ −=

)(xρ )(xΦ

x x

)]()([)()(

2)()( xxri

xirk

rxi

ydyi

eexeexF

x

Θ−Φ−ΘΠ

==∫

≈ ∞−

παψψ

π

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

)(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

No Landau Quasiparticlesn(k)

n(k)1][

21 1 −+ −

− KKFkk

Finite temperatureConformal theory

β

χ

K

r

21⎟⎠⎞

⎜⎝⎛

βξβπ

/)2( xxK

ee −−=

Phase diagram2)( ),(),( −+ ≈= ∫ ηωτ ωχτχτωχ xedxdq qxi

Most divergent fluctuations

K<1 K>1

CDW SU

-V

System with spinSame treatment

More convenient

↓↓↑↑ Φ∇→Φ∇→ ρρ

)(2

1)(2

1↓↑↓↑ −=+= ρρσρρρ

σρ HHHHHkin +=+= ↓↑

)(

))((int

σσρρ

σρσρρρ

−=

−+== ∑ ↓↑

U

UUHi

σρ HHH +=

),( ρρ Ku Charge excitations

),( σσ Ku Spin excitations

Charge-spin separation

spinonholon

Correlation functions

( ) ρσ KKxFx

xkSxS ++= 11 )2cos()0()( 2

•Perturbation (small U)

π

π

σσ

ρρ

σσρρ

//

//

UvKu

UvKu

vKuKu

F

F

F

−=

+=

==

ρK

σK

SDW

CDW SU singlet

SU triplet

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

J. Voit

Luttinger liquid concept

• How much is perturbative

• Nothing provided the correct u,K are used

• Low energy properties: Luttingerliquid (fermions, bosons, spins…)

General Derivation (Haldane)

Interacting Bosons

No condensate

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

How to compute (u,K)

• Perturbation• Exact solutions (Bethe Ansatz): thermodynamics

cuLELEuKDKuuCV

∝∞−=→→

/))()((/κ

• Numerics

H.J. Schulz PRL 64 2831 (90)

Conclusions

• Good control of d=1 massless phases

• Massive phases

• Luttinger Liquid: crucial startingpoint to study perturbations (lattice, disorder, etc.)

Examples

Lattice

Mott insulator:φ is locked

HZ

dxV x ½ xZ

dxV x ei p ¼½ x¡ Á x

Coupled one dimensional bosons

Cold Bosons

N0 ~ 10 to 103

atomsT. Stoferle et al. PRL 92 130403 (2004)

Interchain hopping

Wants to order θ on each chain

Competes with Mott potential that wants to order φ

“Mott” potential: localizes atoms

AF Ho, M.A. Cazalilla, and TG, PRL 92 130403 (2004)

R’

RJosephson coupling: delocalizes atoms

Mott vs. Josephson

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

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

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

ExperimentsT. Stoferle et al. PRL 92 130403 (2004)

Large quantum depletion

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

Bosonized representation

])),(()),(([2

221 τφτφπ

ττ xux

KdxdS xu ∂+∂= ∫

SZ

dxd¿V x ½ x; ¿Z

dxd¿V x ei Á x;¿

DisorderedElastic System