Time-Dependent Kondo Model with a Factorized Initial State

with a Fa torized Initial State

Von der Mathematis h-Naturwissens haftli hen Fakultät

der UniversitätAugsburg

zur Erlangung eines Doktorgradesder Naturwissens haften

genehmigteDissertation

von

M.S. DmitryLobaskin

aus

Zweitguta hter: Prof. Dr. Th. Kopp

CONTENTS

1. Introdu tion

. . . .

1

1.1 NonEquilibrium Kondo Ee t . . . 1

1.2 Theoreti al Motivation . . . 4

1.3 ExperimentalMotivation . . . 7

1.4 Goals of This Work . . . 12

2. Time-dependent Kondo Model

. . . .

17

2.1 Equilibrium Kondo Problem . . . 17

2.2 Timedependent Kondo Hamiltonian . . . 21

3. Non-Equilibrium Kondo Model at the Toulouse point

. . . .

27

3.1 Diagonalization of the Ee tive Hamiltonian . . . 31

3.2 Magnetization . . . 34

3.3 Spin-spin Correlation Fun tion. . . 36

3.4 Con lusion . . . 44

4. NonEquilibrium Kondo Model in the Kondo Limit

. . . .

45

4.1 FlowEquation Approa h . . . 46

4.2 Ee tiveHamiltonian. . . 49

4.3 Numeri al Results. . . 53

4.4 Analyti al Estimations of the Asymptoti s . . . 54

5. Violation of the Flu tuation-Dissipation Theorem in the

Non-Equilibrium Kondo Model

. . . .

61

5.1 Flu tuation-DissipationTheorem . . . 61

5.2 FDT Violation inNonequilibrium . . . 64

5.3 Toulouse Point . . . 66

5.4 Kondo Limit. . . 74

5.5 Con lusion . . . 75

6. Quasiparti le Spe tral Fun tion

. . . .

79

6.1 NonequilibriumSpe tralDensity . . . 79

6.2 Results of the Fermi LiquidTheory . . . 81

6.3 Buildup of the Kondo Resonan e . . . 82

6.4 Con lusion . . . 85

7. Summary

. . . .

87

Appendix 93 A. Details of nonregular sum al ulations

. . . .

95

Publi ations

. . . .

105

1. INTRODUCTION

1.1 NonEquilibrium Kondo Ee t

Sin e early1970s, the mi roele troni industry has been rapidly progressing

fol-lowing the Moore's Lawto doublepro essing power every18 months. Formany

yearsthishas beensatisedlargelybys alingdevi estoeven smallerdimensions.

Extrapolating the Moore's law into the next 10-15 years, it be omes apparent

that the s aling will run into physi al limits: limits of material used, but also

fundamental limits arising from the laws of quantum me hani s whi h be ome

in reasinglyimportant at very small length s ales. Of ourse, the ultimate goal

thenisto onstru ta fun tionaldevi e apableoftransportingasingle ele tron.

In1985D.AverinandK.Likharev(Averinand Likharev1986)proposedtheidea

of a new three-terminal devi e alled a single-ele tron tunneling (SET)

transis-tor. Two years later Theodore Fulton and Gerald Dolan at Bell Labs in the US

fabri atedsu h adevi e and demonstrated howit operates.

Asof August232004,StanfordUniversityhasbeenableto onstru tatransistor

fromsingle-walled arbon nanotubesand organi mole ules. These single-walled

arbon nanotubes are basi ally a rolled up sheet of arbon atoms. They have

a omplished reatingthis transistormaking ittwonanometerswide andable to

maintain urrent three nanometers inlength.

Unlikeeld-ee ttransistors, single-ele trondevi es arebasedonanintrinsi ally

quantum phenomenon: the tunnel ee t. This is observed when two metalli

ele trodes are separated by an insulating barrier about 1 nm thi k - in other

words, just 10 atoms in a row. Ele trons at the Fermi energy an "tunnel"

through the insulator, even though in lassi al terms their energy would be too

lowto over omethe potential barrier.

Due to in redibly small sizes of these devi es all kinds of nonequilibrium

quantum orrelatedpro esses start play the role. Even smallexternal

perturba-tions omparingtothatinthebulkma ros opi samplesmaydriveSETtransistor

far from equilibrium. Small sizes and low temperatures ne essary for operating

su h a system giverise towide variety of nonequilibrium quantum manybody

orrelations.

single-ele tron ee ts will repla e onventional ir uits based on s aled-down versions

ofeld-ee t transistors. Onlyonethingis ertain: ifthe pa eofminiaturization

ontinues unabated, the quantum properties of ele trons will be ome ru ial in

determining the design of ele troni devi es beforethe end of the next de ade.

Fast development of nanoele troni s is already su ient motivation to study

nonequilibrium quantum manybody physi s whi h however is not onned to

it. A tually, nonequilibriummanybody physi s is one of the most fas inating

and hallengingtopi sinmodernphysi s,duetoboththewidevarietyofobserved

phenomena and di ulty in theoreti aldes ription.

Typi alsituationofnonequilibriumin ludesthesystemsubje tedtosome

exter-nalgenerally timedependentperturbation. Ifthe perturbation isinnitesimally

smalland,therefore,thestateofthesystemisalmosttheequilibriumone,the

lin-earresponsedes riptionofanexternallydriven systemdynami sisvalid. Within

this methoddeviationsof allquantities fromtheir equilibriumvalues analways

be expressed via their u tuations in equilibrium. In parti ular, the so alled

u tuationdissipationtheorem isfullledinthis regime. It onne ts generalized

sus eptibility of the system to some external perturbation with the equilibrium

orrelation fun tions.

Unfortunately,the linearresponseformalismis onlyvalidwhilethe perturbation

does not drive the system far from equilibrium. For example, during measuring

of transport properties ( ondu tivity, thermoresistivity, et .) an applied bias

voltage ausesanonequilibrium urrenttoowthrough asample. After ertain

relaxationtimeastationary urrent establishesand systemsare saidtobeinthe

steady state. Although su h a state is timeindependent and thus seems to be

equilibrium, this statement is false sin e it is a highly exited state with respe t

to a thermodynami ally equilibrium one. Another ommon example is a non

adiabati ally applied external timedependent ele tromagneti eld. It an be

a mono hromati perturbation if we are interested in measurement of a Fourier

omponent of a spe i quantity related to response to su h a perturbation. It

an be an external onstraint hanged at some moment, as well. In latter ase,

for example, the equilibrium state of the system might dier signi antly from

the initial one and an be even nonanalyti ally related to parameters of the

originalproblem. Forsu ha asestandardtimeindependentperturbationtheory

doesnot work. Itsnonequilibriumextension(Keldysh diagrammati te hni ) is

appli able inthe ase of a small,weakly oupledperturbation,when summation

up tosome niteorder of the perturbationtheory issu ienttoget ana urate

result. Forthe strong ouplingproblemthe seriesofdiagramsare divergent, and

the method is only appli able if one an manage to sum up all diagramswhi h

ontributemost. Even if it ispossiblethe result is oftenapproximate and might

miss some importantfeatures of a true solution.

di ulties of the theoreti al des ription together with the pra ti al importan e

innanoele troni s is the nonequilibrium Kondo ee t. Study of this one of the

paradigmmodel of the ondensed matter physi s will inevitably help to a hieve

mu h better understanding of the basi s of ele troni properties of nanos aled

stru tures, in ludingmany-body, olle tiveand spin ee ts.

TheequilibriumKondoee t,aswehavementioned,istheparadigmmodelofthe

ondensedmatterphysi s. Itdes ribestheintera tionbetweenmagneti impurity

embedded into a metal and the ondu tion band Fermi sea. For the

antiferro-magneti oupling, attemperatures lowerthanthe so alledKondotemperature

T

K

ondu tionele trons tendtos reenthe impurityspin. Sin eitisnot possible to form a bound state from the impurity spin and single ondu tion ele tron,

intera tion leads toa ompli atedmanybody s attering state s reening loud

made of ondu tion ele trons. For the impurity spin equal to one half ground

state of the system be omes a singlet. Its energy is proportional to the Kondo

temperatureanddepends nonanalyti allyonthe strengthof the ouplingwhi h

demonstrates breakdown of the onventional perturbation theory. The Kondo

ee t manifests itself in many thermodynami and transport properties su h as

spe i heat, magneti sus eptibility, ondu tivity, et . Be ause of formation of

aquasiboundedstateof ondu tionele tronsinthe vi inityoftheimpuritysite,

thedensity of statesgetsenhan ement (so alledKondoresonan e) atthe Fermi

level. This, in its turn, provides additional ontribution to above mentioned

properties with the weight proportionaltothe impurity on entration.

In experiment during the measurement of transportproperties an applied nite

bias voltage tries to destru t the Kondo loud. On e the voltage is high enough

the physi sis nolonger des ribed by the equilibriumKondo ee t together with

thelinearresponsetheory. Thisisunimportantforabulkmeasurements. Dueto

thema ros opi sizeofasampleone annotrea hhighenoughele tri eldsinside

togofarfromequilibrium. Thatiswhyoneisalwaysinthelinearresponseregime

there. On the otherhand, in miniaturedevi es su h elds an be easilyrea hed

evenforsmallenoughbiasvoltages(Fran es hi,Hanson,vanderWiel,Elzerman,

Wijpkema, Fujisawa, Taru ha and Kouwenhoven 2002). Similar problem arises

whilethe timedependent ele tromagneti eld is appliedtothe sample(Kogan,

Amasha and Kastner 2004). All su h perturbationsare extremely relevantfor a

smallsizeddevi es.

Anotherimportantexample is the equilibrationafterimposing (or, equivalently,

removing) nonadiabati ally a onstraint to the system. One parti ular aspe t

is the general problem of an initial preparation of a system and its subsequent

relaxationtowards equilibriumwithanexternalreservoir. Itisa ommon ase in

many experiments that a measurable observable is initially de oupled from the

otherdegrees offreedom. Su hafa torizedinitialstatemightbeevenorthogonal

systems willbea speed of the equilibrationof observables of interest.

In the nonequilibrium Kondo ee t language su h a problem takes pla e when

the impurityspinisinitiallyde oupledfromtheFermisea andthenatsometime

the ouplingisswit hed on. Whereasinitialstate of ondu tionbandele tronsis

the unperturbed Fermi sea, the swit hingon the ouplingleads tothe formation

of the Kondo loud ompli ated manybody orrelated state. The speed of its

buildup denes orresponding relaxation rate of thermodynami and transport

properties.

Nextnonequilibriumproblemistheequilibrationofthe initiallyfrozen impurity

spin (e.g. byastrongmagneti eld)afterthe onstraintisreleased. Inthis ase

the initialstate of the ondu tion band is a polarized potentials attering state

be ause of absen e of spin ip pro esses.

Inthe followingthesis wearefo usedonabovementionednonequilibriumtypes

of the initialpreparation, namely:

I) the impurity spin is initially de oupled from the Fermi sea and then at some

time the oupling isswit hed on;

II)the impurityspinisfrozenforsometime(e.g. by astrongmagneti eld) and

at a ertain time onstraintis removed.

Aspe ial aseofinterestsare al ulationofvariousthermodynami andtransport

propertiesoftheimpurityspindegreeoffreedom,andparti ularlytheirrelaxation

laws.

In the next two se tions we will dis uss why this problem is theoreti ally and

experimentallyrelevant.

1.2 Theoreti al Motivation

Theoreti al study of the nonequilibrium Kondo ee t has a long history. After

applying spe ial transformation Kondo problem an be mapped on the spin

boson model  also one of the paradigm modelof the ondensed matter physi s

whi h des ribes dissipative quantum me hani s of the twolevel system. In this

formulationtheproblemwasalready onsideredbyLeggett,Chakravarty,Dorsey,

Fisher, Garg and Zwerger (1987). In spinboson language the nonequilibrium

prepared initial state orresponds to the parti le originally situated at one of

levels(for Kondo modelit means denite impurity spin proje tion) and then at

some time it starts its dynami s with hopping between levels. Using so alled

nonintera tingblipapproximation Leggett et al. (1987)have derived results for

the levelo upation(magnetizationinthe Kondomodellanguage)exa tlyatthe

and approximately in the whole parameter spa e. Their main result is that

o upationof the level(magnetization) goestozeroexponentiallyfast withtime

P (t))

def

_{= hS}

z

(t)i ∼ exp



−

_{character. time scale}

t



.

(1.1)

Thisresult was onrmed later by Lesageand Saleur (1998),whohave proven it

asymptoti allyusing the formfa tor approa h. However, there are no exa t or

numeri al results giving behavior of the magnetization in the whole parameter

rangeat alltime s ales.

Anotheropen questionisbehaviorof the impurityspinspin orrelationfun tion

C(t, t

′

₎

dened as

C(t, t

′

)

def

_{= hS}

z

(t)S

z

(t

′

)i .

(1.2) This generaltwotime denition is suitablein and out of equilibriumsituations.

Times

t

and

t

′

are the times of the rst and the se ond measurements of the

impurityspin. Whilethesystem isinequilibriumthetimetranslationinvarian e

ispresentandall orrelationfun tionsdependonlyonthetimedieren ebetween

two measurements

τ = t − t

′

. In our ase there is one spe i point on the time

axisthe timeofswit hingonthe oupling. Thus, timetranslationinvarian eis

violated immediatelyand timedependent orrelation fun tions depend on both

times expli itly. One may only expe t that when both times are su iently far

fromthe swit hing time orrelation fun tions shouldbe ome almost equilibrium

ones. Howfast thispro ess happens dependsonthe expli it system setup andis

one the most important questionsof the nonequilibrium physi s. In ase of the

Kondomodelinequilibrium,withtheFermiliquidtheoryone anshowalgebrai

longtimede ay of this fun tion

C(t, t

′

_{) = C(t − t}

′

_{) ∝ 1/(t − t}

′

)

2

.

(1.3) On the other hand,

C(t, 0) =

1

2

P (t)

(1.4)

and it should de ay exponentially out of equilibrium a ording to (1.1).

Parti -ular interest is how su h an exponential de ay rosses into algebrai longtime

behavior when the system observables evolve toward their equilibrium values.

Onepossibles enariowas suggestedin the workby Nordlander, Pustilnik, Meir,

Wingreen, and Langreth (1999). Using as an example the height of the Kondo

resonan e atthe Fermienergy, they have introdu ed the on ept of the ee tive

temperature laiming that the relaxation of system observables atzero

tempera-tureoutofequilibriumhappens asifitwere equilibrium timeindependent

behav-iorbut at somenite ee tive temperature whi h, initsturn,istimedependent.

Ifabovestatementistrue andappli abletotheotherobservables,one on lusion

orrelation fun tion in experiment sin e at any nite temperaturethe orrelator

de ays exponentially. This is one of the most important issue to be solved. It is

also tightly onne ted with the u tuationdissipation theorem sin e imaginary

and real parts of the Fourier transform of (1.2) are onne ted by this relation.

If it is violated then the on ept of the ee tive temperature serves exa tly as

a measure of its violation. Thus, one may ask the following questions: whether

the u tuationdissipation theorem isfullled ornot; how todene the ee tive

temperatureifitisviolatedinspe iednonequilibrium ase;towhatextentthis

denition is appli able. Finally,one is alsointerested in al ulation of transport

properties, su h as ondu tivity, whi hare mosteasily measured in experiment.

The Kondo problemis the strong ouplingproblemand itisalready di ultby

itself, without any timedependen e. With nonequilibrium initial onditions it

be omes a time-dependent strong- oupling model for whi h no general methods

exist.

Results obtained by various approximate analyti te hni s, su h as time

dependent non rossing approximation (Nordlander et al. 1999, Nordlander,

Wingreen, Meirand Langreth2000, Plihal,Langreth andNordlander 2000)have

onrmed existen e of time s ale

t

K

∝ ~/k

B

T

K

(1.5)

related to the Kondo temperature

T

K

relevant for the buildup of the Kondo resonan e. Another line of atta k of this problem is to use dierent numeri al

methods. Suitable approa hes whi h have been already applied to the

equi-librium Kondo model are the numeri al renormalization group (Costi 1997),

densitymatrixrenormalizationgroup(S hollwoe k2005),quantumMonteCarlo

(Egger2004). Inprin iple,thesete hni s anbeappliedun hangedforthe

al u-lationofthehighlyex itedinitialstaterelaxation. However, te hni allyitisvery

di ult sin e the number of eigenstates is limited by al ulation time whereas

oneneedsaveryhighenergyresolutionwhilesear hingforthelongtime

dynam-i s. Implementationofthese methodstootherproblems su hasnonequilibrium

steady state with applied voltage bias is even mu h harder sin e all methods

need to be generalized and are not dire tly appli able in their equilibrium

ver-sions. There are few exa t results in addition to the Leggett's result for the

magnetizationwhi h an serve asa ben hmark for adjusting of numeri al

meth-ods. S hiller et al in a number of works (S hiller and Hers held 2000, S hiller

and Hers held 1995, S hiller and Hers held 1998, Majumdar, S hiller and

Hers held 1998) got exa t solution for the nonequilibriumKondo model with

the nite voltage bias at some spe i point of the parameter spa e. Yet, there

isno ompletepi ture ofthese phenomenaas wellasany exa t results regarding

the rossover from nonequilibrium to equilibrium behavior in this one of the

Figure 1.1: Pi ture of a lateral quantum dot used by Folk et al. (1996). Gate and

bias voltages provide ontinuous ontrol of this devi e, whi h is not possible in bulk

materials.

1.3 Experimental Motivation

Equilibrium Kondo ee t in bulk materials is known for a long time (de Haas,

de Boerand vanden Berg1934). Howevernonequilibriumrealizationswith the

impurityspininitiallyde oupledfromthe ondu tionbandareobviouslynot

pos-sible in a bulk material. Another experimental di ulty is to avoid intera tion

betweenimpuritiessin eoneusuallyneedssubstantialimpurity on entrationfor

dete ting singleimpurity ee ts. A tually, all theories of the Kondo ee t

de-pendonseveralparameterswhosevaluesare notindependentlyand ontinuously

tunableinbulk materials.

NonequilibriumKondoproblemattra tedmu hinterestre entlywhenmany

ex-perimental groups su eeded in fabri ating various low-dimensional

nanometer-sizedsystems usually alled 'quantum dots'(Kastner 1992). This namerefers to

the quantum onnement in all three spatial dimensions. Typi al quantum dot

is made by forming a two-dimensional ele tron gas in the interfa e region of a

semi ondu tor heterostru ture (usually GaAs)and applying an ele trostati

po-tentialtometal gatesto onne ele tronstoasmallregionintheinterfa eplane.

Changing ouplings of the entral region ("dot") to leads and passing urrent

throughthe systemone measuresvarioustransportproperties. Thereare a

num-ber of other miniaturedevi es su h as metalli nanoparti les, fullerenes, arbon

nanotubes whose measured properties exhibit unexpe tedly similar behavior to

that of the quantum dots. This suggests that quantum dots are generi systems

toinvestigate oherent quantum devi es. A typi al exampleof aquantum dot is

There are several te hniques to produ e quantum dots. A typi al example on

the Fig.1.1ismade by mole ular-beam epitaxywhenalayerof AlGaAsis grown

on the top of a layer of GaAs. Ele trons in the interfa e between two layers

form a two-dimensional ele tron gas, be ause of the onnement in the verti al

dire tion. Metalgates(lighterregions)are reatedbyele tron-beamlithography.

Applying a negative bias to the top metal gate restri ts ele tron motion to a

small region ("dot"). Dot is oupled to the 2D ele tron gas by two adjustable

point onta ts. Gatevoltages

V

g1

and

V

g2

ontrolthe shapeofthe quantumwell. Transportproperties ofthe system are measuredbyapplying avoltage

V

sd

(sfor sour e and d for drain)and the ondu tan e of the dot an bemeasured via

G =

I

V

sd

(1.6)

Typi alvaluesfor quantumdots produ edinexperimentsarethe following:

on-nedele trons are

∼ 50 − 100

nmbelowthesurfa e; ee tivemassof anele tron in GaAs is

m

∗

_{= 0.067m}

e

; typi al sheet density

n

s

∼ 4 × 10

11

_cm

−2

, whi h gives

for aFermi wavelength

λ

F

= (2π/n

s

)

1/2

_{∼ 40}

nm (twoorders ofmagnitude larger

than ina metal) and Fermi energy

E

F

= 14

meV; mobility of ele trons of order

µ

e

∼ 10

4

−10

6

cm

2

/V,givingtypi almeanfreepathof

l = v

F

m

∗

_µ

e

/e ∼ 0.1−10µm

. In the dots with ee tive area

A ∼ 0.3µm

motion of dot ele trons is, therefore, ballisti . To observe quantum oheren e ee t in dots that are weakly oupled

to leads, so alled " losed"dots,

G ≪ e

2

/h

(1.7)

one should have a temperature whi h are smaller than a mean single-parti le

level spa ing

δE

inside the quantum well. Taking above numbers one gets for the spa ing

δE = π~

2

_/m

∗

_{A ∼ 11µeV}

whi h an be resolved at temperatures of

∼ 100

mK(

8.6µeV

). Nowadays,the temperature anbemadeeven loweroforder

20 − 25

mK.

Most interesting phenomenon observed in quantum dots is harge quantization

inthe dotwhi his alledCoulombBlo kade(Giaeverand Zeller1968,Kulikand

Shekhter 1975, Ben-Ja oband Gefen 1985, Likharevand Zorin1985, Averin and

Likharev1986,FultonandDolan1987,S ott-Thomas, Field,Kastner, Smithand

Antonadis1989). It alreadytakespla eat temperatures lowerthan the harging

energy of the dot

k

B

T ≪ E

C

= e

2

_/2C

, where

C

is lassi al apa itan e of the harged region. This is the energy one needs to pay to add an extra ele tron to

the dot. At su h temperatures the tunneling through the dot is blo ked by the

Coulombrepulsion. Changinggatevoltage

V

g

one an ompensatethatrepulsion and ausea tivationlesstransportthrough thedot whenthe lledenergylevelof

the dot rosses the Fermi energy of the atta hed leads (see Fig.1.2).

At temperatures lowerthan single-parti le level spa ing

Figure 1.2: Explanation of the Coulomb Blo kade phenomenon. When the gate

voltage makes

|Ni

and

|N + 1i

states equally favorable, it auses a tivationless transport of the ele tri harge through the dot. Tuning the gate voltage we an

swit hbetween variousregimes when the topmostlevel iszero, single or double

Figure 1.3: Elasti o-tunnelingwith a ipof spin.

one has totakeintoa ountwhether the top-mostlevelissinglyordoubly

o u-pied. When the top-mostlevelissingle-parti leo upied the dothas amagneti

moment and the level is spin degenerate. Ground state of the dot is des ribed

by a ertain spin proje tion and elasti transfer of the ele trons from one lead

to another lead is a ompanied by a spin ip (see Fig.1.3). Resulting ee tive

modelis the Kondo modeland itis validfor allenergies belowthe threshold

δE

(see review by Glazmanand Pustilnik (2003), forexample).

Due toextremeexibilityin ontrolofthe quantum dot one an realizeallkinds

of the nonequilibrium onditions we have dis ussed before. Starting with the

applying onstant bias voltage and nishing with the applied timedependent

external ele tromagneti elds.

Onepossibleexperimentforrealizationofthenonequilibriumsituationwhenthe

spinisinitiallyde oupledfromthe ondu tionbandwasproposedby Nordlander

et al. (1999). Quantum dot with a singly o upied topmost level well below

the Fermienergy ofthe leads anbe onsideredasee tivelyde oupled fromthe

Fermi seaof the ondu tionbandele trons. Indeed,if the impuritylevelismu h

belowtheFermienergythenthe ouplingbetweenthe impurityspinand

ondu -tion ele trons istoo smalland the spin isuns reened. One an alsothinkabout

the Kondo temperaturemu hsmaller than the a tual temperature of the

0

200

400

600

800

t [1/

Γ

_dot

]

0.0

0.1

0.2

0.3

0.4

0.5

ρ

dot

(

ε

F

,t)

scaled equilibrium

nonequilibrium

00000

00000

00000

00000

00000

00000

00000

00000

00000

11111

11111

11111

11111

11111

11111

11111

11111

11111

00000

00000

00000

00000

00000

00000

00000

00000

00000

11111

11111

11111

11111

11111

11111

11111

11111

11111

000000000000000

111111111111111

ε

dot

(t)

V (t)

_g

Figure 1.4: Quantum dots experiments with time-dependent gate voltages:

Time-dependent buildup of many-body orrelations (gurefrom Nordlander et al. (1999))

levelup and swit h onthe Kondo regimewhen

T

K

≫ T

. Proposed experimentis plottedinFig.1.4. Measuringmagnetizationand magneti sus eptibilityone an

he k deviationsof this quantities fromtheir equilibriumanalogues.

Themain obsta letoobservethis phenomenon inexperiment isthe smallnessof

the time s ale

t

K

. For a typi al Kondo temperature in quantum dots of order

0.01 − 1

K we get the Kondo time s ale

t

K

whi h orresponds to the frequen ies fromseveral gigahertzes up toterahertz. Hen e, to nd indi ations of des ribed

non-equilibrium Kondo ee t the time of swit hing on the oupling should be

mu h smaller than this time s ale. The lower is the Kondo temperature, the

longeristherelaxationofthedotspin and, onsequently,the easieristomeasure

it. On the other hand, the Kondo temperature must be mu h higher than the

a tualtemperatureof the experiment forsystem stilltobeinthe Kondo regime.

We believe that su h anexperiment an be made inthe nearestfuture.

Experimental importan e of su h an investigation is tightly onne ted with an

attempt to make a fun tional qubit, a twolevel system, the building blo k of

a possible quantum omputer realization made of SET transistors. The most

promising andidates are the quantum dots and the Josephson jun tions. In

dissipation. Thus, to build a qubit apable of performing tens of thousands

operationsbeforeenvironmentalde oheren e setsoneshoulda hievearelaxation

time whi h is mu h longer than the operation time. Although the oupling to

the environment isusually weak and, therefore,des ribed by thedierent model

alled the spinboson modelsolutionfor the nonequilibriumKondo regimeof a

strongly s reened impurityisa very importantben hmark inallfurther study of

these systems sin e both models an be exa tlymapped on ea h other.

1.4 Goals of This Work

Summingup all open questions we an formulate goals of the underlined thesis.

The main subje t is to study the nonequilibrium Kondo ee t for two dierent

initial preparations I) when the impurity spin is initially de oupled from the

ondu tion band and II) when it is xed by the external magneti eld at zero

temperature.

Weaim at

1 Findingfull rossover between the exponential nonequilibriumde ay and

the equilibrium algebrai longtime behavior of the spinspin orrelation

fun tions

•

atthe exa tly solvable Toulousepoint

•

inthe physi ally relevantKondo limit

2 Che king the fulllmentof the u tuationdissipation theorem for the real

and imaginary parts of the spinspin orrelation fun tion in the quantum

limit.

3 Deriving analyti al asymptoti results for the magnetization

P (t)

at large and smalltimes and extend these results for allintermediate time s ales.

4 Conne tingrelaxationofthethermodynami propertieswiththebehaviorof

the transport properties (e.g. ondu tivity)by al ulationof the

quasipar-ti le spe tral fun tion, whi h is dire tly related with the nonequilibrium

urrent through a quantum dot.

The entralproblemisto hoose appropriatemethodswhi hallowustosolveall

posed issues.

We use bosonization and refermionization te hniques to al ulate the zero

tem-peraturespinspin orrelationfun tionoftheKondomodelattheToulousepoint.

model be omes equivalent to the nonintera ting resonant level model whi h is

quadrati and therefore an be solved exa tly. We nd equations of motion for

all observables in the Heisenberg pi ture and average afterwards with respe t

to the nonequilibrium initial states I) and II) whi h are timeindependent in

thisrepresentation. Although al ulationis lengthy itisstraightforward and the

nal solution for the orrelation fun tions

P (t)

and

C(t, t

′

₎

an be given in the

losedform. Themainresultfollowingfromthissolutionisthatrelaxationo urs

exponentially fast with a time s ale set by the inverse Kondo temperature and

independentlyonthein ipientstateI)orII)ofthesystem. Resultfor

P (t)

isthe onrmationoftheLeggettresultbutobtained byanothermethod. However,

re-sult for

C(t, t

′

₎

is ompletelynew and wasnot known before. Closed expressions

letus tra e the rossover from the exponentialde ay to the algebrai longtime

behaviorat allintermediate time s ales.

Remarkable on lusion whi h follows immediatelyfromthe exa t solutionis the

inappli ability of the ee tive temperature on ept to al ulation of orrelation

fun tions in the manner proposed by Nordlander et al. (1999). From exa t

an-alyti al expressions it is lear that the dieren e between nonequilibrium and

equilibriumspinspin orrelators vanishes exponentiallyfor

t

w

≫ t

K

and one re-alizes the algebrai longtimede ay immediatelyat both initialpreparations I)

and II) on ethe time of the rst measurement is larger thanzero.

Sin e the Toulouse point is a very spe ial point of the parameter spa e, it is

interestinginwhi hextentthese resultsaregeneri forthe wholeproblem. Away

from this point the Kondo model is no longer des ribed by a nonintera ting

Hamiltonianand another approa h isneeded. For this purpose we use the

ow-equation method developed by Wegner (1994) and applied later to a number of

problemsin ludingtheequilibriumKondoee taswell(Kehrein2001,Hofstetter

and Kehrein 2001, Kehrein and Mielke 1996, Kehrein and Mielke 1997). This

approa h is the ontrolled approximation whi h gives results with a very high

a ura y for the Kondo model in the whole parameter region. Flow equations

be omeexa tattheToulousepoint,hen e, onne ting ontinuouslyresultsatall

oupling strengths with our exa t analyti al expressions. Main result one gets

afterapplyingthe owequation approa histhe onrmationof the exponential

relaxation at large times. At shorter times omparing with

t

K

the relaxation is even faster, whi h is due to unrenormalized oupling onstants at large energies

that dominate the shorttime behavior.

Nextissuewhi hweaddressisthe fulllmentoftheu tuationdissipation

theo-remwhi h onne ts real(equilibriumspin u tuations)and imaginary(magneti

sus eptibility)partsof thespinspin orrelationfun tioninequilibrium. Wend

inour ase that the u tuationdissipationtheorem ismaximallyviolatedat

in-termediate time s ales of order the inverse Kondo temperature. At this point

the u tuationdissipationtheorem violation,as it isusually done inthe widely

investigated ontextof glassysystems (seeFisherandHertz (1991)forareview).

The ee tivetemperaturebe omes oforder the Kondotemperaturedue to

heat-ingof the ondu tion bandele trons by the formationof theKondosinglet. The

system then relaxes towards equilibrium and the u tuationdissipation

theo-rem be omesfullled exponentially fast atlarger times. Againthis relaxation is

given inthe analyti alformatthe Toulouse pointand extended away fromitby

using the owequation method. Although our denition of the ee tive

tem-perature is not unique itsbehavior qualitativelygrabs the a tual thermi ee ts

whi hhappen duringthe formationof the Kondo loud and an be measured in

experiment. These observations ould be relevant for designing time-dependent

fun tional nanostru tures with time-dependent gate potentialssin e they give a

quantitativeinsightintohowlongoneneedstowaitafterswit hingforthesystem

to returnto ee tively zero temperature.

Inexperimentone typi allymeasures ondu tan e ofagivensample. Aquantum

dot suddenly shifted into the Kondo regime exhibits an in rease in the

ondu -tan e due to a forming of the Kondo resonan e at the Fermi energy. Thus, an

interesting quantity to evaluate isthe non-equilibriumdensity of states whi h is

proportional to su h an in rease. Unfortunately, the dire t translation of this

problemintothe languageoftheee tivemodels,whi harethemainingredients

of our exa t Toulousepoint al ulationand approximateowequation solution,

looksimpossible. Ex itationsof the ee tive models are manyele tron soliton

like stru tures and are not suitable obje ts for evaluation of the singleparti le

quantities (like density of states needed for al ulation of ondu tan e). It is

very hardtoreexpress themeven atthe exa tly solvableToulouse point. Thus,

anotherapproa h isimplemented. For al ulation of the spe traldensity, weuse

the resultofthe mi ros opi Fermiliquidtheory whi hstatesthat inequilibrium

the density of states atthe impuritysite

ρ

imp

(ǫ)

, inthe vi inityof the Fermi en-ergy, is proportional to the quasiparti le density of states of the ee tive model

ρ

eff

(ǫ)

(see Hewson (1993))

ρ

imp

(ǫ) = zρ

eff

(ǫ),

(1.9) where

z

is known as a wavefun tion renormalizationfa tor. In our approa h we assume that

z

fa tor remainstime-independent in the non-equilibriumsituation or, at least, it remains onstant on the s ale of order of

t

K

= 1/T

K

. Next step is to al ulate the spe tral fun tion for the ee tive model. Resulting spe tral

densities exhibit exponential relaxation toward their equilibrium values and are

in agreement with results obtained by non rossing approximation (Nordlander

etal.1999)whi hhoweverisnotappli ableattemperatures mu hlowerthan

T

K

. Energy dependen e of the nonequilibriumspe tralfun tions demonstratesvery

pe uliar Friedeltype os illations with the frequen y equal to the time distan e

All the mentioned results are published in a sequen e of papers (Lobaskin and

Kehrein 2005a, Lobaskinand Kehrein 2005b,Lobaskin and Kehrein 2005 ).

The dissertationis organized as follows:

In the next hapter wedis uss the Kondo modelin equilibriumand out of

equi-libriumanddes ribe indetailsboth typesof thenonequilibriumsituations

on-sidered. With the help of bosonization and refermionization te hni s we derive

ee tivetimedependent Hamiltonianneeded fora further study.

In hapter IIIwe give fulldetails ofexa t al ulationsof the magnetization

P (t)

andthespinspin orrelationfun tions

C(t, t

′

₎

attheToulousepointoftheKondo

model out of equilibrium. We show that results are independent on the type of

the initialpreparation.

In the next hapter details of the owequation approa h applied to this model

aredis ussed,andnumeri alresultsfor orrelationfun tionsare givenalongwith

analyti al asymptoti s of these fun tions in small and large time limits. Again

we onrm independen y of the full solution on the original state I) or II). We

alsodis uss importantdieren es between the full owequation solution of the

problemand analyti alToulousepointresult.

ChapterV ontainsdis ussionofthe u tuationdissipationtheorem violationin

thequantum ase

T = 0

out ofequilibrium. Con ept oftheee tivetemperature isintrodu ed todene the measure of the u tuationdissipation theorem

viola-tion. These generaltheoreti alresultsare appliedtothe nonequilibriumKondo

ee t.

In hapterVIthe resultsforthenonequilibriumdensityof statesoftheee tive

model are presented, whi h is dire tly related with experimentally measurable

ondu tan e of a sample.

2. TIME-DEPENDENT KONDO MODEL

2.1 Equilibrium Kondo Problem

Re ognitionofthefundamentalroleofthemagneti impuritiesinuen eon

trans-portandthermodynami propertiesof metalsgot itsstartfromthe famouswork

ofJ. Kondo(Kondo 1969). It washimwho explainedthe temperatureminimum

intheele tri resistivityobserved earlierinseveral experimentswithdoped

met-als, phenomenon whi h puzzled people for 30 years. In most metals resistivity

is mainly aused by s attering of ondu tion ele trons over latti e phonons and

de reasesveryrapidlywith temperatureas

T

5

,whereasinexperimentson

Au

by de Haas et al. (1934) and on dilute alloys of

F e

in a series of

Nb − Mo

alloys ashost metalsby Sara hik, Corenzvitand Longinotti(1964) resistivity dropped

with temperature down to some nite value and unexpe tedly in reased with

further temperature lowering (see Fig.2.1).

Basis for Kondo al ulations was provided in earlier ontributions by J.Friedel

(Friedel 1952, Friedel 1958, Friedel 1956, Blandin and Friedel 1959) and

P.W.Anderson (Anderson 1961). In the former work an important on ept of

the "virtual bound states", almost lo alized states due to a resonant s attering

attheimpuritysite,wasintrodu ed. Ifalo alimpuritypotentialisnotattra tive

enoughtoform aboundstate in 3D itstill may lo alizeele trons inthe vi inity

of an impurity for su iently long time. Provided su h a resonan e is situated

near the Fermi energy,it enhan es impurity ontributions to allthermodynami

properties of metal. In the latter work by Anderson a dierent approa h was

formulatedwithin the famous modelwhi h nowbears hisname

H

Anderson

=

X

σ

ǫ

d

n

dσ

+ Un

d↑

n

d↓

+

X

kσ

ǫ

k

c

†

_kσ

c

kσ

+

X

kσ

(V

k

c

†

_dσ

c

kσ

+ V

k

∗

c

†

kσ

c

dσ

).

(2.1)

This Hamiltonian des ribes the ondu tion band ele trons with a dispersion

ǫ

k

intera ting with the lo alized impurity orbitalat anenergy

ǫ

d

, whi hhas an on-site intera tion

U

. Coulomb repulsion

U

between lo alizedele trons is essential toexplain originationof the lo alized magneti moments in host metal. Indeed,

onsideringthe ase

V

k

= 0

, whi his reasonableassumptionfora further pertur-bationtheoryexpansionsin ethe hybridizationparameterisof order

0.1 − 0.7

eV whereasCoulombrepulsion isinrangeof several eV,we immediatelysee thatfor

Figure 2.1: Resistan e minima for Fe in a series of Mo-Nb alloys (taken from

Sara hik etal. (1964))

ǫ

d

< ǫ

F

and

ǫ

d

+ U > ǫ

F

the most favorable ongurationis the singly o upied one. Additionof anextra ele tron orremovingone osts extraenergy,hen e,the

ground state is two-fold degenerate. Taking into a ount doubly-o upied and

empty states asan exited states inlowest order in

V

k

and repla ing

S

+

= c

†

_d↑

c

_d↓

,

S

z

=

1

2

(n

d↑

− n

d↓

)

(2.2) whi h always hold in the

n

d

= 1

subspa e, one an derive from (2.1) the well-known s-dmodelor, simply,the Kondo model(S hrieer and Wol 1966)

H

Kondo

=

X

kσ

ǫ

k

c

†

kσ

c

kσ

+

X

kk

′

J

kk

′

(S

+

c

†

k↓

c

k

′

_↑

+S

−

c

†

k↑

c

k

′

_↓

+S

_z

(c

†

k↑

c

k

′

_↑

−c

†

k↓

c

k

′

_↓

))

(2.3)

with an ee tiveex hange oupling given by

J

kk

′

= V

_k

∗

V

_k

′



1

U + ǫ

d

− ǫ

k

′

+

1

ǫ

k

− ǫ

d



(2.4)

together with a potentials attering term

X

kk

′

_σσ

′

where

K

kk

′

=

V

_k

∗

V

k

′

2



1

ǫ

k

− ǫ

d

−

1

U + ǫ

d

− ǫ

k

′



.

(2.6)

Above orresponden e is valid for ondu tion band energies below the threshold

(levelspa ing

U

)whi h anbeexpressedby

|ǫ

k

| ≪ |ǫ

d

−ǫ

F

|

and

|ǫ

k

| ≪ |U+ǫ

d

−ǫ

F

|

. Kondo,in hisseminal paper (Kondo 1969), onsidered a lo almagneti moment

with spin

S =

1

2

oupled via a onstant ex hange intera tion

J

with the on-du tionband ele trons. He has shown inthe third order perturbation theory in

oupling

J

the appearan e of

ln T

term inthe resistivity al ulation

R(T ) = aT

5

+ c

imp

R

0

− c

imp

R

1

ln(k

B

T /D),

(2.7)

where

c

imp

is impurity on entration,

R

0

,

R

1

- some onstants,

D

- bandwidth. That al ulation explained the observed resistan e minimum, though obviously

brokedown atsmalltemperatures.

Ahuge numberoftheoreti alworksdevoted tothisproblemappearedin60s and

early 70s (Suhl 1965, Suhl and Wong 1967, Nagaoka 1965, Abrikosov 1965,

Zit-tartzandMüller-Hartmann1974,BloomeldandHamann1967,Hamann1967)).

People su eeded inidentifying anenergy s ale

k

B

T

K

∼ De

−

1

2J ρ0

(2.8)

alled the Kondo temperature, beyond whi h standard perturbation approa hes

fail. Here

D

is the bandwidth,

ρ

0

is the density of states at the Fermi energy. Non-analyti dependen e of

T

K

on Kondo oupling demanded non-perturbative methods to determine system properties at low temperatures

T < T

K

. In 60s Anderson and o-workers introdu ed "poor-man" s aling approa h (Anderson

1967, Anderson and Yuval 1969, Anderson, Yuval and Hamann 1970, Anderson

andYuval1970)basedon ontinuousredu tionofbandwidth

D

ofthe ondu tion ele trons. Eliminationofhighenergyex itationsleadstoastrongrenormalization

of the ee tive oupling between animpuritymomentand ondu tion ele trons,

andeven be omes divergentatlowenergies

T ≪ T

K

. Natural on lusionfollows that su h a s aling pi ture orresponds to the ground state with the innite

oupling between an impurity and the ondu tion band forming a singlet state

(Mattis 1967,Nagaoka 1967).

In 1974 Wilson developed numeri al renormalization group approa h (NRG)

(Wilson1974, Wilson 1975) whi h enabledhimtoderive anee tive low-energy

Hamiltonian of the problem and onrmed existen e of the singlet state for

S = 1/2

Kondomodel. Wilson gotnumeri alresultsforimpurity ontributionto thermodynami and transportproperties whi h were also onrmed by Nozières

(1974) using phenomenologi al Fermi-liquid pi ture of low-lying energy

γ

imp

inthe spe i heat

C

imp

C

imp

= γ

imp

T

(2.9)

to impurity magneti sus eptibility

χ

imp

atzero frequen y dened as

χ

imp

(ω + iδ) = −(gµ

B

)

2

hhd

†

d; d

†

dii

ω+iδ

,

(2.10)

wheredoublebra kets standforauto orrelator. Sommerfeldratio(forthe Kondo

modelit issometimes alledWilson ratio)then equals to

R =

χ

imp

/χ

c

γ

imp

/γ

c

=

4π

2

_k

2

B

3(gµ

B

)

2

χ

imp

γ

imp

= 2

(2.11)

with

c

meaning ondu tion ele tron values without the impurity. In non-intera ting ase, for

U = 0

and onstant hybridization fun tion

V

k

, this ratio is

1

or

4

depending on the magneti sus eptibility denition: if the denition is lo almeaningthe magneti eld a ts onlyon the impurity site then this ratio is

4

;if we use the globalmagneti elda ting onboth the ondu tionband aswell as the impurity site then the Wilson ratio is

1

.

Both the NRG andthe Fermi-liquidapproa h give the samevaluesfor

T

2

oe- ient inthe ondu tivity

σ

imp

(T ) = σ

0

1 +

π

4

_w

2

16



T

T

K



2

+ O(T

4

)

!

(2.12)

where

w = 0.4128

so- alledWilsonnumberand

σ

0

= ne

2

_πρ

0

(0)/2mc

imp

 ondu -tivityatthe"unitarity"limitwhenthe

s

wavephaseshiftofs atteringele trons atthe Fermilevelisequalto

π/2

. Later Yamada(1975) derived the mi ros opi Fermi-liquidtheory of the Kondo modelfromthe AndersonHamiltonian(2.1).

Another sour e of theoreti al insight ame from an exa t analyti al solution of

the s-d model using Bethe ansatz (Bethe 1931) applied by Andrei (1980) and

Wiegmann (1980). They have su eeded in al ulating thermodynami

prop-erties of the Kondo model, providing the same results to Wilson's NRG. This

approa h works for the linear dispersion Kondo model with the innite

band-width. Exa t al ulation of the ex itation spe trum gives all thermodynami

properties as wellas analyti al expressions for the Kondo temperature and

Wil-son number

w

. The method an be generalized to provide exa t results for the ground state and thermodynami s of

S >

1

2

magneti impurity models, whi h are dierent from the usual spin

1/2

problem, sin e in the ground state an im-purity spin is unders reened and equal to

S −

1

2

. N-fold degenerate problem was parti ularly interesting for experimentalists who have already been

than two. Sin e the Bethe ansatz annot be used for al ulation of

dynami- al response fun tions dire tly measured in experiment, a number of

approxi-mate te hniques was invented, treating

1/N

as a small parameter. Results of these theories be ome asymptoti ally exa t in the limit

N → ∞

(mean eld point). Many very good approximations around this limit an be obtained

by dierent methods: perturbation theory (Ramakrishnan and Sur 1982,

Ra-suland Hewson 1984, Bi kers 1987, Brandt, Keiter and Liu 1985), Fermi-liquid

theories (S hlottmann 1983, Yoshimori 1976, Newns and Hewson 1980), non

rossing approximation (Kuromoto and Müller-Hartmann 1985, Kuromoto and

Kojima 1984, Bi kers, Cox and Wilkins 1987), slave-boson and mean eld

the-ory (Newns and Read 1987, Read and Newns 1983a, Read and Newns 1983b),

variational

1/N

expansions (Gunnarsson and S hönhammer 1983a, Gunnarsson andS hönhammer1983b),bosonizationand onformaleldtheories(Ae kand

Ludwig1991a,Ae k and Ludwig1991b,Ae k andLudwig1991 , Ae k and

Ludwig 1993, Ae k and Ludwig 1994, Ae k and Ludwig 1992, Ae k and

Ludwig 1991d). All these methods were su essfully applied to the

investiga-tionof the bulkKondo ee t manifestedin dierent properties of rareearth and

a tinide ompounds.

2.2 Timedependent Kondo Hamiltonian

Westart fromthe KondoHamiltonianwhi h des ribesthe ouplingbetween the

ondu tionband Fermi sea and the impurityorbital inthe lo almomentregime

H =

X

k,α

ǫ

k

c

†

kα

c

kα

+

X

i

J

i

(t)

X

α,β

c

†

_0α

S

_i

σ

_i

αβ

c

_0β

.

(2.13) Here

c

†

0α

,

c

0α

isthelo alizedele tronorbitalattheimpuritysitewhi haredened as

c

†

_0α

def

=

X

k

c

†

_k

√

L

,

c

0α

def

=

X

k

c

k

√

L

.

(2.14)

We allowfor anisotropi ouplings

J

i

= (J

⊥

(t), J

⊥

(t), J

k

(t)).

(2.15) Su hageneralizationisessentialfortheexa tanalyti alsolutionattheso alled

Toulouse point. At all ouplings equal to ea h other (isotropi oupling) we get

the originalKondo Hamiltonian(2.3). Dispersion relation islinear

ǫ

k

= ~v

F

k,

(2.16)

whi h is also ne essary for the exa t solution. However, it is not needed for the

generally set

v

F

= ~ = 1

. Sometimes we will restore these onstants to get the right dimension of anevaluatedquantity.

A ording tothe introdu tion hapter we onsider two typesof nonequilibrium

preparations:

I) Theimpurityspin isxed forallnegativetimesby alargemagneti eld term

h(t)S

z

that is swit hed oat

t = 0

:

h(t) ≫ T

K

for

t < 0

and

h(t) = 0

for

t ≥ 0

H =

X

k,α

ǫ

k

c

†

kα

c

kα

+

X

i

J

i

(t)

X

α,β

c

†

0α

S

i

σ

αβ

i

c

0β

+ h(t)S

z

.

(2.17)

This ase orresponds to the absen e of spin-ip pro esses at negative times,

although impurity is oupledto the ondu tion band:

J

⊥

(t) = 0, J

k

(t) = J

k

> 0

for

t < 0

and

J

i

= J

i

> 0

for

t ≥ 0

H(t < 0) =

X

k,α

ǫ

k

c

†

_kα

c

_kα

+ J

k

X

α,β

c

†

_0α

S

_z

σ

_z

αβ

c

_0β

(2.18) and

H(t > 0) =

X

k,α

ǫ

k

c

†

_kα

c

_kα

+

X

i

J

i

X

α,β

c

†

_0α

S

_i

σ

_i

αβ

c

_0β

.

(2.19)

II) The impurity spin is ompletelyde oupled from the bath degrees of freedom

for allnegative times:

J

i

(t) = 0

for

t < 0

H(t < 0) =

X

k,α

ǫ

k

c

†

kα

c

kα

.

(2.20)

SimilartosituationI)weassumea ertainproje tionofspin

hS

z

(t ≤ 0)i = +1/2

, whi h is realized by a innitesimallysmallmagneti eld. Then the oupling is

swit hedonat

t = 0

andbe omesnite

J

i

(t) = J

i

> 0

and timeindependentfor

t ≥ 0

H(t > 0) =

X

k,α

ǫ

k

c

†

kα

c

kα

+

X

i

J

i

X

α,β

c

†

_0α

S

_i

σ

_i

αβ

c

_0β

.

(2.21)

This situationisrathermoreinteresting than situationI) sin eit an berealized

infuture quantum dot experiments wherethe quantumdot issuddenly swit hed

into the Kondo regime by applying a timedependent gate voltage (Nordlander

et al.1999).

Bosonization of the Kondo Hamiltonian

Forour purposesit ismore onvenient totreat thisHamiltonianinitsbosonized

•

the model be omes quadrati at the Toulouse point;

•

inthe owequation approa h the Toulousepoint orresponds to the xed pointof the Hamiltonianow.

We will follow general bosonization rules reviewed in work by von Delft and

S hoeller (1998).

One introdu es bosoni spin-density modes

σ =

_p

1

2|p|

X

q

(c

†

_p+q↑

c

_q↑

_{− c}

†

_p+q↓

c

_q↓

) ,

(2.22)

with the ommutator

[σ(−q), σ(q

′

)] = δ

qq

′

L

2π

= δ

qq

′

1

∆

L

(2.23) for

q, q

′

_{> 0}

. Here

L

is the system size,

∆

L

is the momentum spa ing. The bosoni eld isdened ina standard way

Φ(x) = −∆

L

i

X

q6=0

√

_q

q

e

−iqx−a|q|/2

_σ(q)

(2.24)

and obeys the following ommutationrelation

[Φ(x), ∂

y

Φ(y)] = −2πiδ(x − y).

(2.25)

The parameter

a > 0

generates the UV regularization of our model and an be interpreted as the bandwidth. The type of regularization does not play any

role, while weare on erned inthe universal properties of our modelat energies

|E| ≪ a

−1

. We will dis uss the ut-o pro edure later in Se tion4.4 where we

linkUV parameter with the at band width.

Charge density modes in the Hamiltonian (2.13) de ouple ompletely and one

onlyhas to look atthe spin-density part

H = H

0

+ H

k

+ H

⊥

.

(2.26)

For a linear dispersion relation of the ondu tion band the kineti part of the

Hamiltonian

H

0

an be expressed via density modes as (Kronig1935)

H

0

= ∆

L

X

q>0

qσ(q)σ(−q) =

1

2

Z

_dx

2π

: (∂

x

Φ(x))

2

: ,

(2.27)

H

_k

isthelongitudinalintera tionbetweenimpurityspinandthe ondu tionband

H

_k

_{= −}

_√

J

k

22π

∂

x

Φ(0)S

z

_.

(2.28)

H

_⊥

is the transverse part

H

_⊥

=

J

⊥

4πa



e

i

√

2Φ(0)

S

−

+ e

−i

√

2Φ(0)

S

+



.

(2.29)

The longitudinal spin oupling an beeliminated by aunitary transformation

U = exp [iµS

z

Φ(0)]

(2.30)

with an appropriate hoi e of

µ

µ =

√

J

k

22π

.

(2.31)

Applying this transformation to the ondu tion band part

H

0

, one generates exa tly minusse ond term in(2.26), up toirrelevant onstant

UH

0

U

†

= e

iµS

z

_Φ(0)



1

2

Z

_dx

2π

: (∂

x

Φ(x))

2

:



e

−iµS

z

Φ(0)

=

= H

0

+

1

2

iµS

z

Z

dx

2π



Φ(0), (∂

x

Φ(0))

2



=

= H

0

+ µS

z

∂

x

Φ(0).

(2.32)

In both ases for

t > 0

we arrive atafter the bosonization pro edure

˜

H

I,II

(t > 0)

def

= U

†

H

I,II

U = H

0

+ g

0



V (λ

0

, 0)S

−

+ V (−λ

0

, 0)S

+



(2.33)

with the oupling onstant

g

0

=

J

_⊥

4πa

(2.34)

and s aling dimension

λ

0

=

√

2 −

√

J

k

22π

.

(2.35)

Here vertex operatorshave been introdu ed

V (λ, x) ≡ exp [iλΦ(x)]

.

(2.36) After applying the unitary transformationfor negative timesin ase I) one gets

˜

In ase II)

˜

H

II

(t < 0) = H

0

+

J

_k

√

22π

∂

x

Φ(0)S

z

_,

(2.38)

sin e the unitary transformation

U

, in spite of an eling longitudinal oupling term forpositivetimes, generates it for negative times.

Forafuturereferen ewealsogivethevalueofthes alingdimensionwithrestored

dimension onstants

λ

0

=

√

2 −

√

J

k

22π~v

F

.

(2.39)

3. NON-EQUILIBRIUM KONDO MODEL AT THE

TOULOUSE POINT

At the Toulouse point, whi h isdened by the unity s aling dimension

λ

0

λ

0

=

√

2 −

√

J

k

22π

= 1,

(3.1)

vertex operators (2.36) obey fermioni ommutation relations (von Delft and

S hoeller 1998):

{V (1, x), V (−1, x

′

)} = 2πaδ(x − x

′

)

(3.2) orfor the normalordered version

{: V (1, x) :, : V (−1, x

′

) :} = Lδ(x − x

′

).

(3.3)

Bothversions are onne ted with ea hother by

: V (1, x) :=



L

2πa



1

2

V (1, x)

(3.4)

Thenone may introdu e standard fermioni operators a ording to

Ψ(x)

def

= : V (1, x) : .

(3.5)

Thus, transformed Hamiltonian

H = U

˜

†

_HU

an be refermionized (see Leggett

etal.(1987))

˜

H(t > 0) =

X

k

kΨ

†

_k

Ψ

_k

+

X

k

V

k



Ψ

†

_k

d + d

†

Ψ

_k



.

(3.6)

where the hybridization onstant

V

k

is onne ted with the ex hange oupling onstant

J

⊥

via

V

k

= g

0



2πa

L



1

2

=

J

⊥

4πa



2πa

L



1

2

.

(3.7)

Herethespinlessfermions

Ψ(x)

orrespond tonon-trivialsolitontypeex itations builtfromthe bosoni spindensity waves(see the denition (2.36)of the vertex

operators). Fermioni impurity orbital

d

†

,

d

is onne ted with the original spin degree of freedom by the identities

S

z

= d

†

d −

1

2

,

(3.8)

S

+

= d

†

,

S

−

= d .

(3.9)

Eq.(3.6)inthesenotationsissimplytheresonantlevelmodelwiththe

hybridiza-tion fun tion

∆(ǫ)

def

= π

X

k

V

_k

2

_{δ(ǫ − k).}

(3.10)

This is a well-known modelwith a lotof results already obtained (S hlottmann

1982).

Summarizing,forpositivetimesmodelsI)andII)arerepresented bytheresonant

level model with the hybridizationfun tion (3.10). For negative times model I)

is des ribed only by the free ondu tion band part (2.37), be ause the large

magneti eld suppresses all ex hange pro esses between the impurity orbital

and the ondu tionband, leavingin(2.26)only thelongitudinal oupling,whi h

is, nally, removed by the unitary transformation

U

. In model II) the spin is de oupled from the ondu tion band for negative times: this leads to a time

dependent potential s attering term (again due to the polaron transformation

U

), a ording to(2.38). All onditions an be summarized givingfor both ases a Hamiltonianof the followingstru ture:

H =

P

_k

kΨ

†

_k

Ψ

_k

+















P

kk

′

g

kk

′

Ψ

†

_k

Ψ

_k

′

(d

†

d − 1/2) , t < 0

P

k

V

_k

(Ψ

†

_k

d + d

†

_Ψ

k

)

, t > 0

(3.11)

with the non-zero hybridizationfun tion (3.10)for positivetimes

∆(ǫ; t ≥ 0) = ∆ =

J

2

⊥

16πa~v

F

=

T

K

πw

.

(3.12)

Couplings

J

i

s ales as

L

with the system size be ause of the denition (2.14), thus, one sometimes uses res aled ouplings

J

′

=

J

L

(3.13)

whi h have dimension of energy. With this redenition and using the standard

expression for the at band density of states at the Fermi energy

ρ

0

=

L

2π~v

F

we an rewrite (3.12) as

∆ =

π(ρ

0

J

′

₎

2

4

~

_v

_F

a

(3.15)

and linkthe Kondo temperature with the transverse oupling

J

⊥

T

K

=

π

2

_w(ρ

0

J

′

)

2

4

~

_v

_F

a

(3.16)

The lastterm inthe produ t has dimension of energy and isproportionalto the

highest energy dieren e or simply to the band width. Above denition of the

Kondotemperatureatthe Toulousepointis ne essary toobtain rightdimension

onstants for onne tion our results with results of Leggett et al. (1987) and

Lesageand Saleur(1998)

Potentials attering ouplingis onstant and equals to

g

kk

′

= 0

(3.17)

for ase I) and

g

kk

′

=

√

2 − 1

ρ

0

(3.18)

for ase II).

Correlation fun tions

Correlationfun tions, whi h we are interested in,are magnetization

P (t)

def

_{= hS}

z

(t)i = hd

†

(t)d(t)i − 1/2

(3.19)

and spin-spin orrelator

C(t

w

+ τ, t

w

)

def

= hS

z

(t

w

+ τ )S

z

(t

w

)i

(3.20)

where

h. . .i

means the average with respe t to either state I) or II). Last or-relation fun tion has real and imaginary parts. Cal ulation of both parts is of

parti ular interest, sin e they have dierent physi al meaning. The real part

des ribesequilibrium u tuationsof the spin

Re C(t

w

+ τ, t

w

) =

C

_{S

z

,S

z

}

(t

w

+ τ, t

w

)

def

=

1

whereas, the imaginarypart is proportional toa response fun tion

Im C(t

w

+ τ, t

w

) =

C

[S

z

,S

z

]

(t

w

+ τ, t

w

)

def

=

1

2

h[S

z

(t

w

+ τ ), S

z

(t

w

)]i.

(3.22) In equilibriumboth parts are onne ted by the u tuation-dissipation theorem,

whi hin our situationdoes not hold (Lobaskin and Kehrein 2005b).

All averages are taken with respe t to the initial state, whi h is equilibrium

ground state of the Hamiltonian

H

˜

I,II

(t < 0)

. The most intriguing in the follow-ing study is that su h a state is orthogonal with respe t to the ground state of

the Hamiltonianof the resonant level modelinthe thermodynami limit.

More-over, these states have dierent energies. Therefore, possibility to investigate

the problem in the S hrödinger pi ture is questionable. Our approa h suggests

simply to avoid this di ulty by translating our problem into the language of

the Heisenberg pi ture. Alloperatorsthena quiretimedependen eleavingwave

fun tions time independent.

In the Heisenberg pi ture

In order to evaluate the nonequilibrium spin dynami s we use the quadrati

form of

H

˜

I,II

(3.11) to solve the Heisenberg equations of motion for

d(t)

and

d

†

_(t)

. DiagonalizingHamiltoniansby aunitary transformation

A

†

H

˜

I,II

A

(3.23)

for positive times we arriveto

A

†

H

˜

I,II

(t > 0)A =

X

ε

εa

†

ε

a

ε

,

(3.24)

where the onstant has been dropped. Dynami s of the operators

a

ε

indiagonal basis is trivial

a

ε

(t) = a

ε

(0)e

−iεt

(3.25) and

a

†

ε

(t) = a

†

ε

(0)e

iεt

.

(3.26) Only thing we need to do is to onne t expe tation values of the produ t of

several su h operators at time

t = 0

with expe tation values of the produ t of original operators

d

and

Ψ

k

. New operators

a

ε

and

a

†

ε

are linked with the old ones via

a

ε

=

X

n=k,d

A

εn

a

n

=

X

k

A

εk

Ψ

k

+ A

εd

d.

(3.27)