Quantum physics

in one dimension

T. Giamarchi

http://dqmp.unige.ch/giamarchi/

N. Kestin A. Kantian

S. Furuya

E. CoiraL. Foini

C. Berthod

N. Kamar

A. Tokuno

S. Tapias

Plan of the lectures

Lecture 1: basic concepts for interacting

quantum systems

Lecture 2: : basic concepts for interacting

quantum systems

Lecture 3: One dimensional systems; Concept of

Tomonaga-Luttinger liquid.

Lecture 4: Experimental realizations

Lecture 5: Beyond Tomonaga-Luttinger liquids

References for Lecture 1&2

Lectures Les houches (Singapore) 2009, TG

arXiv:1007.1030

Lectures Les Houches 2010, A. Georges + TG

arXiv:1308.2684

Basic course on many-body physics

http://dqmp.unige.ch/giamarchi/local/people/

thierry.giamarchi/pdf/many-body.pdf

Effect of interactions

Landau Fermi liquid

Individual fermionic excitations

exist (quasiparticles)

*m m

Fermi liquid theory

• Shown perturbatively in U

• Much more general and robust

Element m*/m c/c0

Nb 2 1

3He 6 20

Heavy fermion 100 100

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

Finding new functionalities

/physics

(Y. Tokura, Japan)

Ferroelectrics, High Tc, Manganites,

Oxydes, Organics,, nanomaterials....

10 23 particules + Interactions:

Crucial fundamental problem

``non Fermi liquids’’

tomorrow’s materials

1 μm

Reduced dimensionality

Future electronic

Need to worry about reduced dimensionality

Physics at the edge

LaO/StO interface

(JM Triscone et al.)

Presence of edge

(B. I. Halperin)

Quantum hall effect

Topological insulators….

Superconductivity

between insulators…

One dimension is specially

interesting

• No individual excitation can exist (only

collective ones)

• Strong quantum fluctuations

Difficult to order| | ie

Three urban legends about 1D

It is a toy model to understand higher

dimensional systems.

It does not exist in nature ! This is only for

theorists !

Everything is understood there anyway !

References

TG, arXiv/0605472 (Salerno lectures)

TG, Quantum physics in one

dimension, Oxford (2004)

M. Cazalilla et al.,

Rev. Mod. Phys.83 1405 (2011)

TG, Int J. Mod. Phys. B 26 1244004 (2012)

What does “1D” means in the real

(3D ?) world

22

2 2

2

yx

y

y

kkE

m m

nk

L

Fine …..

But does it exist ?

TG, Int J. Mod. Phys. B 26 1244004 (2012)

Organic conductors

c

b

D. Jaccard et al., J. Phys. C, 13 L89 (2001)

Cees Dekker

CARBON NANOTUBES

Quantum Wires

O.M Ausslander et al., Science 298 1354 (2001)

Spin chains and laddersB. C. Watson et al., PRL 86 5168 (2001)

M. Klanjsek et al.,

PRL 101 137207 (2008)

B. Thielemann et al.,

PRB 79, 020408® 2009

Cold atoms

Control on the dimension

I. Bloch, Nat. Phys 1, 23 (2005)

Typical problem (e.g. Bosons)

• Continuum:

• Lattice:

How to treat ?

``Standard’’ many body theory

Exact Solutions (Bethe ansatz)

Field theories

(bosonization, CFT)

Numerics

(DMRG, MC, etc.)

Luttinger liquid physics

Labelling the particles

1D: unique way

of labelling

(x) varies slowly

CDW

~ 0q 0~ 2q

: superfluid phase

Quantum

fluctuations

U0 1

K1 1

Luttinger liquid concept

• How much is perturbative ?

• Nothing (Haldane):

provided the correct u,K are used

• Low energy properties: Luttinger liquid

(fermions, bosons, spins…)

Correlations

1/2

S(q,!) J.S. Caux et al PRA 74 031605 (2006)

Finite temperature

Conformal theory

b

c

K

r

21

bb

/)2(

x

xK

ee

Fermions

Right (+kF) and left (-kF) particles

Spins

Powerlaw correlation functions

Non universal exponents K(h,J)

Use boson or fermions mapping

( 1) cos(2 )i i iS e e

1(-1) cos(2 )z iS

Calculation of Luttinger

parameters

• Trick: use thermodynamics and BA or numerics

• Compressibility: u/K

• Response to a twist in boundary: u K

• Specific heat : T/u

• Etc.

Tonks limit

U = 1 : spinless fermions

Not for n(k) : F B

Free fermions: 2

1( ) (0) cos(2 )Fx k x

x

K=1

Note: 1/2

† 1( ) (0)B Bx

x

Luttinger parameters

M. Klanjsek et al., PRL 101 137207 (2008)

NMR relaxation rate:

Tests of Luttinger liquids

Organic conductors

c

b

A. Schwartz et al. PRB 58 1261 (1998)

First observation of LL/powerlaw !!

TG PRB (91) :

Physica B 230 (1996)

Z. Yao et al. Nature 402

273 (1999)

Cold atoms

Bosons (continuum)

Optical lattices (dilute)

B. Paredes et al., Nature 429 277 (2004)

†( ) ( ) (0)ikxn k dxe x

I am your worst nightmare

Confining potential / Trap

3 2 2

0

1( )

2H d r r r

• No homogeneous phase !

Atom chips

K large (42)

S. Hofferberth et al. Nat. Phys 4

489 (2008)†

0( ) (0)

L

dr r

Magnetic insulators

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)

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)

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’ =c 1D(Tc)

M. Klanjsek et al., PRL 101 137207 (2008)

R. Chitra, TG PRB 55 5816 (97); TG, AM Tsvelik PRB 59 11398 (99)

M. Klanjsek et al.,

PRL 101 137207 (2008)

NMR

D. Schmidiger et al. PRL 108 167201 (12):

K. Yu et al. arxiv/1406.6876 (14)

†

, ,q qS S

Spin ladders

Fractionalization of excitations

1 = ½ + ½

E(k) = cos(k1) + cos(k2)

k = k1 + k2

Neutron scattering

B. Thieleman et al. PRL 102, 107204 (2009)

P. Bouillot et al. PRB 83, 054407 (2011)

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 »

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)

A. Kleine, C. Kollath et al. PRA 77 013607 (2008); NJP 10 045025 (2008)

Proposal for cold atoms (Rb)

End of the story ...... ??????

Quantitative agreement with the

predictions of the Luttinger liquid

NO! Luttinger liquid plays the

same role than Fermi liquid did

To boldly go where no theorist

has gone before….

Beyond Luttinger liquids

• 1D additional perturbation:

Lattice (Mott transition), disorder (Bose glass) etc.

Multicomponents, mixtures, …..

• Out of equilibrium situations

A. Mitra, TG, Phys. Rev. Lett. 107, 150602 (2011)

E. Dalla Torre, E. Demler,TG, E. Altman, Nat. Phys. 6

806 (2010); PRB 85 184302 (2012)

• Impurities, polarons in 1D systems:

• Ladders and magnetic fields:E. Orignac, TG Phys. Rev. B 64, 144515 (2001)

M. Atala et al., arxiv/1402.0819

• Dimensional crossover 1d – 2d/3d:AF Ho, M.A. Cazalilla, TG, PRL 92 130403

(2004);NJP 8 158 (2006).

Mott transition

Lattice: Mott transition

Costs U

Quantum phase transition

SU MI U

Superfluid

Mott

Insulator

Mott transition and cold atoms

Lase

r

Lase

r

[D. Jaksch et al. PRL 81 (1998)]

[M Greiner et al. Nature, 415 (2002)]

Superfluid to Mott insulator

transition in a 3D optical

lattice

How to treat?

• Incommensurate: Q 2 0

• Commensutate: Q 2 0

cos(2 ( ) )H dx x x

cos(2 ( ))H dx x

0 cos( ) ( )H dxV Qx x

0(2 2 ( ))

0 0cos( )i x x

H dxV Qx e

TG, Physica B

230 975 (97)

Competition

2 210 [ ( ( , )) ( ( , )) ]

2xu

dxdS x u x

K

Beresinskii-

Kosterlitz-Thouless

transition at K=2

String order

parameter

0 0 cos(2 ( ))LS V dxd x

Mott insulator:

is locked

Density is fixed

Gap in the excitation spectrum

n(k)T. Kuhner et al. PRB 61 12474 (2000)

TG, Physica B

230 975 (97)

/( ) rG r e

E. Haller et al. Nature 466 597 (2010)

Disorder and interaction effects

Dirty bosons

TG + H. J. Schulz EPL 3 1287 (1987); PRB 37 325 (1988)

``Two’’ fourier components of disorder

Bose glass phase

U

[interactions]

Db

0

Uc

Bose GlassSuperfluid

TG + H. J. Schulz EPL 3 1287 (87); PRB 37 325 (1988);

M.P.A. Fisher et al. PRB 40 546 (1989)

SU-BG transition in d=1

Universal exponent at the SU-BG transition !

Db

0 K

Two loop RG:

Z. Ristivojevic, A. Petkovic, P. Le Doussal, TG PRL (2012)

Various phases

• Mott insulator: incompressible; h i = 0

0dN

d

• Superfluid: compressible; h i 0

0dN

d

• Bose Glass : compressible; h i = 0

0dN

d

TG, P. le Doussal PRB 53 15206 (96); T. Nattermann et al. PRL 91

056603 (03); E. Altman et al PRB 81 174528 (10),…….

Other potentials: Biperiodics

G. Roux et al. PRA 78 023628 (2008);

T. Roscilde, Phys. Rev. A 77, 063605 2008;

X. Deng et al PRA 78, 013625 (2008);

Effect of interactions?

Same as ``true’’ disorder ?

J. Vidal, D. Mouhanna, TG PRL 83 3908 (1999);

PRB 65 014201 (2001)

0 0 1 1( ) cos( ) V cos( )V x V Q x Q x

• U= 0

Aubry-André model

• Localization transition

Hunt for the Bose glass

Numerics (disorder)

S. Rapsch, U. Schollwoeck,

W. Zwerger EPL 46 559

(1999);

G. Batrouni et al. PRL 65

1765 (90);

N. V. Prokof'ev and B. V.

Svistunov, PRL 80 4355

(96);

N. Prokofev et al. PRL 92

015703 (04);

O. Nohadani et al. PRL 95,

227201 (05)

K. G. Balabanyan et al. PRL

95, 055701 (05);

L. Pollet et al. PRL 103,

140402 (2009)

…..

Old Experiments

Josephson junction arrays

Disordered superconducting films(D.B. Haviland et al, PRL 62 2180 (89))

Helium in porous media

New remarkable systems

Cold atoms

New superconducting films

Spin dimers

Comptes Rendus AS

Cold atomic gases

Speckle (Palaiseau,

Florence, Urbana)

Biperiodic lattices

(Florence)

Mixtures

(Stony Brook)

Quasi-periodic

Quasiperiodic (1D):

J. E. Lye, et al., PRA 75, 061603R 2007.

L. Fallani, et al. PRL 98, 130404 2007

Speckle

Pasienski et al. Nat. Phys 6 677 (2010)

Control of Interactions and

Disorder

C. D Errico, E. Lucioni, L. Tanzi, L. Gori, G. Roux,

I. P. McCulloch, TG, M. Inguscio, G. Modugno,

PRL 113, 095301 (2014)

L. Gori, T. Barthel, A. Kumar, E. Lucioni, L. Tanzi,

M. Inguscio, G. Modugno, TG, C. D'Errico, G. Roux

PRA 93, 033650 (2016)

• Reentrant phase diagram (consistent with theory)

• Disordered insulator (``Bose glass’’)

• But one can improve on : temperature, Mott

insulator, trap

Absence of equilibrium: MBL

Artificial Gauge fields

Meissner effect in bosonic

ladders

E. Orignac, TG, PRB 64 144515 (2001)

A dl

Orbital currents (``Meissner’’ effect)

Field ``Hc1’’: appearance of vortices

Observed in cold atoms (I. Bloch’s group)

Interactions: devil’s staircase

Plateau at ½ should be visible

Fermionic ladders

Many works (Nersesyan, Lehur, Roux, ……)

Experimental realisations ?

Unconventional superfluidity in

fermionic ladders

S. Uchino, A. Tokuno, TG,

PRA 89 023623 (2014)

Zero flux: singlet superconductivity

(d-wave U >0 , s-wave U < 0)

Triplet superconductivity with purely

local interactions !

Other/related route:

State dependent hopping

S. Uchino, TG,

PRA 91 013604 (2015)

Challenge: t_up = - t_down

Shaking or artificial gauge fields

Triplet

superconductivity

Non-LL and impurity problems

M. B. Zvonarev, V. V. Cheianov, TG,

PRL 99 240404 (2007)

A. Kantian, U. Schollwöck, and T. Giamarchi

PRL 113, 070601 (2014)

Interacting 1D system (Luttinger liquid):

e.g. bosons -- ``spin up’’

Add one particle of ``spin down’’, or flip a spin

Example: two component bose-Hubbard model

( , ) ( , ) (0,0)G x t S x t S

Trapped/open

regimes

Light cone of

spinless bosons

M. B. Zvonarev, V. V. Cheianov, TG, PRL 99 240404

(2007)

Experiments: Cambridge, Florence,

Innsbruck, Munich, Bonn, …

Theories: Akhanjee, Tserkovnyak, Matveev,

Furusaki, Kamenev, Glazman, Gangardt,

Lamacraft, ……

Driven impurity

Diffusive impurity

J. Catani et al. PRA 85, 023623 (2012)

T. Fukuhara, A. Kantian, M. Endres, M. Cheneau,

P. Schauss, S. Hild, D. Bellem, U. Schollwock, TG,

C. Gross, I. Bloch, S. Kuhr , Nat. Phys. (2013)

Coherent spin excitation

Heisenberg ferromagnet

Ferromagnetic Heisenberg

Coherent propagation of a magnon

Heisenberg model : ex 1·i i

i

H J S S 2

ex

4JJ

U

Impurity in a LL (weak lattice)

Polaronic effect

Driven impurity vs diffusion

• Normal transport

• Einstein relation: = D

Coupled 1D chains:

deconfinememt

“Mott” potential: localizes atoms

Mott vs. Josephson

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

NJP 8 158 (2006).

R’

RJosephson coupling: delocalizes atoms

Higgs “amplitude” mode

AF Ho, M.A. Cazalilla, TG,NJP 8 158 (2006).

c

b

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 e

ner

gy,

T* (

K)

Pressure (kbar)

P. Auban-Senzier, D. Jérome, C. Carcel and J.M. Fabre J de Physique IV, (2004)

A. Pashkin, M. Dressel, M. Hanfland, C. A. Kuntscher, PRB 81 125109 (2010)

TG Chemical

Review 104 5037

(2004)

Cold atoms

AF Ho, M.A. Cazalilla, TG, PRL 95, 226402 (2005);

Arxiv/0604525

Triplet

superconductor

(repulsive

interactions)

Out of equilibrium Luttinger

liquids

Quenches and thermalization in LL

A. Mitra, TG,

Phys. Rev. Lett. 107, 150602

(2011)

Phys. Rev. B 85, 075117 (2012)

J. Lancaster, TG, A Mitra,

Phys. Rev. B 84, 075143 (2011)

Thermalization for the low energy part

Out of equilibrium sine-Gordon

E. Dalla Torre, E. Demler,TG, E. Altman, Nat. Phys. 6 806

(2010); PRB 85 184302 (2012)

LL + external noise

And many other problems

and works……

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, gauge fields,…..

Many recent excellent realization of 1D/Q-1D

experimental systems