Non equilibrium dynamics in strongly correlated systems

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CONDENSEDMATTER THEORYSECTOR

Non Equilibrium Dynami s

in Strongly Correlated Systems

Thesis submitted for the Degree of Do tor Phylosophiæ

Advisor: Candidate:

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I Introdu tion 7

1 In and Out of Equilibrium 9

2 Non-Equilibrium Physi s in Correlated Systems 13

2.1 QuantumTransportat Nanos ale . . . 14

2.2 Ultra-Cold Atomi Gases inOpti alLatti es . . . 18

2.2.1 NonEquilibrium Experiments . . . 19

2.3 Time-Resolved Spe tros opieson CorrelatedMaterials . . . . 23

2.3.1 Time Evolution of theEle troni Stru ture a ross an Insulator to Metaltransition . . . 26

3 Theoreti al Motivations 27

3.1 A Glimpse onManyBodyTheoryof Non Equilibrium States 28

3.2 QuantumImpuritiesOut of Equilibrium . . . 29

3.2.1 AndersonImpurity Outof Equilibrium . . . 30

3.2.2 Vibrationalee tsinnon equilibriumtransport . . . . 31

3.2.3 Real-TimeDynami s afterLo alQuantum Quen h . . 31

3.3 QuantumQuen hesinIsolatedSystems . . . 33

3.3.1 GlobalQuantum Quen hes . . . 33

3.3.2 Quen h Dynami sinthe Fermioni Hubbard Model . 35

3.4 Plan oftheThesis . . . 37

II Dynami s in Quantum Impurity Models 39

4 Real-Time Diagrammati Monte Carlo 41

4.1 Non Equilibrium Dynami s inQIM . . . 42

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4

Chapter

1

In and Out of Equilibrium

Nonequilibrium phenomenaareubiquitous in Nature andthey appear under

verydierent avours. At the sametime, asa result of a huge

phenomenol-ogy, a simple denition of this on ept is learly a deli ate issue, whi h

ne erssarilyrequires further lari ations. What non-equilibriumstands for,

atleast fromthe perspe tive of the present work, isthe rst question we

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The on ept of thermodynami equilibrium is one of the buildingblo k

of modern statisti al physi s. It deals with properties of ma ros opi

sys-tems whi h are isolated or weakly oupled to an environment and an be

summarizedasfollows:

Ama ros opi systemissaid tobeinthermal equilibriumwhen

(i) its state (physi al properties) is dened interms of aunique

set of intensive and extensive variables whi h do not hange

withtime and(ii)no urrentsof harges asso iated to onserved

quantities (parti les, energy,...) owthroughit.

Ifonethinkfor amomentaboutthisdenitionthenheimmediatelyrealizes

thatsu ha on eptismoreanex eptionthanaruleineverydaylife. Indeed,

non equilibrium ee ts are extremely ommon in many dierent physi al

situations of the greatest simpli ity, the ow of ele tri urrent through a

metalli ondu tor being just a trivial example. This is even more true if

we think that the very basi idea of performing experiments on materials

and ompounds amounts to a t with some external eld on an otherwise

equilibriumsystemand tomonitor itsresponseto theappliedperturbation.

Asaresultone ouldbetemptedto askwhy,inspiteoftherestri tiveness of

the above denition, the assumption of thermal equilibrium has been (and

a tually still is) so powerful and useful to des ribe physi al properties of

ma ros opi systems. Apossibleanswerto thisquestionshouldbebasedon

two observations.

From one side,asoftenhappens inphysi s,what mattersaretheorders

ofmagnitudeoftipi altimes alesasystemneedtorea haquasi-equilibrium

state ompared to the time s ales on whi h observation takes pla e. As a

onsequen e,withagooddegreeofapproximation,ama ros opi system an

be onsidered asinthermalequilibriumifall fastpro esseshavetaken pla e

whilethe slowestonestillhavetoo ur. Clearlythedistin tionbetweenfast

andslow dependson the observation time thatis onsidered [111,114,56℄.

A se ond key observation omes from a basi result in the statisti al

theoryofmanyparti le systemswhi hgoesundertheapparently inno uous

nameoflinearresponsetheory. Itsimplystatesthatasmallexternal

pertur-bation anonlyprobesmallu tuationsaround equilibrium. Hen e, aslong

asappliedeldsareweakenough,thesystem anbe onsideredasinthermal

equilibrium for all pra ti ally purposes. This major result is of paramount

importan e in onne ting theoryto experiments, sin e itprovides a wayto

omputeexperimentalrelevant quantitiessu hassus eptibilitiesintermsof

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CHAPTER 1. IN AND OUT OF EQUILIBRIUM

11

Therearehoweverphysi alsituationsinwhi htherelevantee ts annot

bea ountedforbyassumingthatthesystemis losetoanequilibriumstate.

Wewill referto two main lasses ofnonequilibrium phenomenathroughout

this work. The rst one an be realizedbyatta hingthe systemof interest

toexternal sour eswhi hallowit tosustainstationary urrentsprovidinga

dissipation me hanism whi h prevent indenite heating. In this ase, after

sometransient,thesystemwillrea hastationarystatewherephysi al

prop-erties do show time-translational invariant. However, due to the external

for ing a nite urrent ows a ross the systems and thesteady state is not

anequilibrium state.

A se ond lass of non equilibrium phenomena on erns expli itly time

dependent situations. These an be realized, for example, by exposing the

systemtotimedependentexternaleldsortotimedependentvariationofits

Hamiltonianparameters. Thesimplestexampleone animaginewithinthis

lass on erns the relaxationof anhighly ex itedstate toward equilibrium.

In the next two hapter we will present experimental and theoreti al

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Chapter

2

Non Equilibrium Physi s in

Strongly Correlated Systems

Re ent years have seen an enourmous progress in preparing, ontrolling and

probing quantum systems in non-equilibrium regime. Experimental

break-throughs inthe eldof nanos aletransport,ultra oldatomi gasesandtime

resolved pump probe spe tros opies on solid state materials triggered a huge

interest on the eld of strongly orrelated systems out of equilibrium. In

this hapter we will briey review some of these re ent a hievements whi h

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14

2.1. QUANTUM TRANSPORT AT NANOSCALE

Figure 2.1: Single Mole ule Transistor built with

C

60

mole ule using ele -tromigration te hnique. AFMimage from [152℄.

2.1 Quantum Transport at Nanos ale

Re ent advan esinnanote hnology have made itpossibleto onta t

mi ro-s opi quantumobje tstolargemetalli reservoirs[179,37℄. Singlemole ules

orevenarti ialatoms,so alledquantumdots,havebeen onta ted,

open-ing a routetowards promising nanoele troni devi es. Beside their obvious

relevan eforte hnologi alappli ations,inviewofbuildingamole ular-based

ele troni s over oming the famous Moore's Law, these experimental

break-throughs have triggered an enormous s ienti interest around the eld of

quantumtransport. Manyquestionsthatuptore entyearsseemtobepurely

spe ulative (if not ompletely meaningless) su h as measuring the

ondu -tan e of a single atom, may be now experimentally addressed in a more of

less ontrolledsetup [172℄.

From an experimental point of view, many dierent te hniques an be

employed dependingwhether the obje tto onta t is a lithographi ally

de-signed quantum dot, a magneti adatom on a metalli surfa e or a single

mole ule bridging two metalli ele trodes. In the rst ase the oupling

between the reservoirs and the arti ial atom obtained by quantum

on-nement of the two dimensional ele tron gas,is through tunnelingbarriers.

In thelatter two ases the usual setup typi ally involves respe tively

s an-ning tunneling mi ros ope (STM) te hniques, for magneti adatoms, and

breakjun tion/ele tromigration te hniquesfor single mole uletransistors.

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CHAPTER 2. NON-EQUILIBRIUM PHYSICS IN CORRELATED SYSTEMS

15

port phenomena in a ompletely dierent regime with respe t to

onven-tionalsolid-statematerials[129℄. Whileele tron-ele tronintera tionsinbulk

ma ros opi ondu tors are in general e iently s reened and other

phys-i al me hanisms are responsible for the relaxation pro esses, the behavior

of those strongly- orrelated nanodevi es is greatly ae ted by a large

(un-s reened)Coulombrepulsionexperien edbylo alizedele tronssittingonthe

dot/mole ule. Noti e that despite the dis rete set of levels of the isolated

quantumsystem, whi hwouldsuggestan exa t treatment, hybridization to

thereservoirsmakesthe problemextremely hallenging. Inaddition, dueto

thesize of the tunneling rate whi h may be ome omparable or even lower

with respe t to lo al energy s ales (su h as the ele tron-ele tron repulsion

or the energy ofatomi displa ements) atsu iently lowtemperaturesnon

trivialmanybodyphenomena an emerge.

Inthis respe tthe experimentalobservation oftheKondo Ee t, oneof

the hallmark of strong orrelation phenomena, in a quantum dot oupled

to a metalli lead [72, 36℄ reated huge ex itement. Many theoreti al and

experimental investigations followed, whi h have made the eld of

trans-portthrough orrelated nanostru turesa privilegedarenawhere strong

or-relation phenomena an be experimentally probed with an high degree of

tunability and, more interestingly, in novel physi al regimes. Inparti ular,

indu ingnonequilibriumee tsinthesenano-devi esisrathernatural,on e

thedi ulttaskof onta tingthemole ulehasbeena omplished. The

sim-plest example one an think of isto apply a d -biasvoltage between sour e

and drain ele trodes to indu e a urrent owing a ross the onta t. This

drives the system through a transient regime, toward a steady state whi h

although being stationary - hen e time-translational invariant- features a

niteamount of urrent throughit. Su han highlyex ited stateofthe

ou-pledsystem(mole ule+ele trodes)isreferredtoasanon-equilibriumsteady

state (NeqSS).It isworth noti ingthat, by measuringthe

I

− V

hara ter-isti or the dierential ondu tan e

∂I/∂V

,experiments an dire tlyprobe

physi al properties ofsu h nonequilibrium state.

In thefollowing we brieysket h some re ent experimental results that

are parti ularly relevant for the subje t of this work, sin e they highlight

beautiful interplay between orrelations and non equilibrium ee ts. The

reader hasto be areful however, sin e many other interesting non

equilib-rium phenomena an be investigated in those systems. Indeed beside the

aseofd transport,whi histhelikelysimplestone animagineand

pra ti- allyrealize,thehigh degreeof ontrol oered bythese nano-devi es allows

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in-16

2.1. QUANTUM TRANSPORT AT NANOSCALE

Figure2.2: Experimental datafor transportthroughasemi ondu tor

quan-tum dot, from [182℄. Dierential ondu tan e as a fun tion of the

sour e-drainvoltage

V

sd

for dierent temperaturesranging from

T = 15mK

(thi k bla k) to

T = 900mK

(thi k red). Noti ethe Kondo anomaly at low bias,

where the unitarity limit isalmost approa hed.

rapidlymoving inthe dire tionof time-resolvedte hniques todete t harge

transportby ounting individualele trons whiletunnelinga ross orrelated

nanostru turessu h assemi ondu ting quantum dots[184,165,78℄.

Coulomb Blo kade and Kondo Ee t Out of Equilibrium

Signaturesofele tron-ele tron orrelationsintransportthroughnano-devi es

learly appearinthe

I(V )

hara teristi and inthedierential ondu tan e

G(V ) = ∂I/∂V

,whi h areverysensitiveprobesof lo almanybodyphysi s [119℄. Thesimplestexampleisthe CoulombBlo kade ee t,whi hhasbeen

observed ina numberof experimentswithquantumdots or singlemole ule

transistors,see[138℄. Itresultsfromthelarge hargingenergy

E

C

onehasto paytoaddanextraele trononthedot/mole ule,duetoCoulombrepulsion.

It tipi ally appears as sharp peaks in the zero bias ondu tan e as a

fun -tionofthe gatevoltageor equivalently asa gapin the

I(V )

urve,meaning

that a nite bias voltage of order

E

C

is needed to make harge transport possible. The most striking ee t of many body orrelations o urs at low

enough temperaturesandfor gatevoltagessu hthatthedota ommodates

anoddnumberofele trons. Then,uponde reasingtheappliedbiasvoltage,

the ondu tan e rosses over from the low- ondu tan e Coulomb blo kade

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CHAPTER 2. NON-EQUILIBRIUM PHYSICS IN CORRELATED SYSTEMS

17

Figure 2.3: Experimental data for transport through single

C

60

mole ule, from [137℄. The urrent-voltage urve show at low bias the

hara teris-ti Coulomb Blo kade plateau. At larger bias signatures of ex itations of

mole ular phononsarevisibleassteps at regularlydispla ed voltages.

perature goestozero, ree tingthe omplete s reeningofthelo almoment

by the ondu tion ele trons of the nearby Fermi Sea. While the ee t of

temperature on this many-body state is well known sin e the early

experi-mentson metalli alloys, its ompetitionwithan external sour e drainbias

isagenuinenon-equilibrium ee t whi h hasbeen possibleto measureonly

mu h more re ently thanksto nano-s ale devi es [182,76℄.

Phonons Ee ts in Mole ular Transistors

A relevant issue in transport through devi es that are built with single

mole ules is the role played by the internal vibrational degrees of freedom

during the transport pro ess. Indeed due to the urrent owing a ross the

system, ele trons are repeatedly added and removed from the mole ule, a

pro ess whi h may result into novel physi al ee ts, su h as a hange of

shape or position of the mole ule itself with respe t to theleads. In

addi-tion,duetothenitebiasapplieda rossthejun tiontheinternalvibrational

degreesoffreedom anbealsodrivenoutofequilibriumduetotheir oupling

withele troni degrees offreedom.

In this respe t pioneering transport measurements on vibrating single

mole ule transistors have been performed [137℄ where the ondu tan e of

a mole ular jun tion made by a

C

60

mole ule onne ted to gold ele trodes hasbeen measured. Beside theCoulomb blo kade plateau,the

I

− V

urve

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18

2.2. ULTRA-COLD ATOMIC GASES IN OPTICAL LATTICES

Figure2.4: A artoonpi tureoftwodimensional(top)andthreedimensional

(bottom)opti allatti es. From[17℄.

shows hara teristi steps at regularly displa ed voltagesdue to ex itations

ofmole ular phonons, a pe uliar nonequilibrium ee t.

2.2 Ultra-Cold Atomi Gases in Opti al Latti es

Re ent advan es in the eld of ultra old atoms have allowed to engineer

ma ros opi quantummany-bodysystemswithtunableintera tions and

al-most perfe t isolation from the environment [17℄. This has been possible

thanksto a series of experimental breakthroughs whi h start withthe

real-izationofnovel oolingme hanisms,allowingtoobservedilutegasesmadeby

bosoni and fermioni atoms at extremely lowtemperatures

1

. Heregenuine

quantumee ts dueto spin-statisti sbe omerelevantand phenomenasu h

asBose-Einstein ondensation [7,21,40℄or Fermi Degenera y [42,169,180℄

have been observed.

Theseresultstriggeredalargebodyofresear hwhi hmainlyfo usedon

ma ros opi quantum oheren ephenomenawhi h hara terizebothbosoni

and fermioni ondensates. However, due to the very diluted regime (

n

≃

10

14 cm

−3

), ee tive intera tions in these systems are often weak enough thatanee tivesingleparti ledes riptionissu ientandinteresting many

bodyee tsaremissed. Thereforeanhugeex itementhasbeengeneratedby

theexperimental a hievement of two major steps inthe dire tion toward a

1

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CHAPTER 2. NON-EQUILIBRIUM PHYSICS IN CORRELATED SYSTEMS

19

strong-intera tion regime inultra old quantumgases. The rstone wasthe

developmentofFesba hresonan ete hniques[35,93℄totunetheintera tion

strength of individual atoms by means of an external magneti eld. This

allowed to in rease the ee tive s attering length far beyond the average

interparti le spa ing, thus turning a weakly intera ting diluted gas into a

strongly intera tingone.

A se ond important step was the possibility to onne old atoms into

ongurations of redu ed dimensionality or to load them in periodi al

lat-ti es built with laser light elds, known as opti al latti es [74℄. Here

dipo-lar for es lo alize the atoms in the minima (or maxima) of the stationary

andmono hromati ele tromagneti potential,thus generatingsituationsin

whi h theee ts ofthe intera tions areenhan ed. Interestinglythe

param-eters of the light eld dire tly ae t the properties of the latti e, the half

wavelength beingthelatti espa ingwhiletheintensityoftheradiationeld

ontrolling the depth of the potential, hen e thehopping strengthand the

value oftwo parti le intera tions.

Theresultingset-upmaybeseenasanidealizedversionofa onventional

solidstatesystemwherespin-fullele trons,herefermioni atomswith

dier-ent hypernestates,feelthe periodi potentialofioni latti e. Equivalently,

ultra old atoms inopti allatti es an be onsidered asthesimplest

experi-mentalrealizationofpopularlatti emodelsofintera tingquantumparti les.

The ombinationoftheseexperimentalresultslargelyextendedtherangeof

physi s whi h isa essiblewithultra old atoms,openingthewayto study

strongly orrelated systems in a ompletely tunable set-up. An important

stepinthisdire tionhasbeen theexperimentalrealizationofaMott

Insula-tormadebybosoni [74℄andfermioni atoms [96,167℄. Itisworthnoti ing,

however,thatthemajor bottlene ktoward theexperimentala hievementof

exoti many body phases is still represented by the issue of ooling

me h-anisms. Indeed the energy s ales ontrolling the physi s of those systems

areso small thatrea hingtemperature for theonset of antiferromagnetism

or super ondu tivity has been so far elusive. This is parti ularly true for

fermions,whi h aremore di ultto ooldowndueto Pauli prin iple whi h

largelyredu es their s attering amplitudeinthes-wave hannel.

2.2.1 Non Equilibrium Experiments

Oneofthemainfeatureofexperimentswith oldatomi gases,whi hlargely

dierfrom onventionalsolid-statesetup,isthepossibilityto hange

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20

2.2. ULTRA-COLD ATOMIC GASES IN OPTICAL LATTICES

Figure2.5: Timeofight measurements[75℄ofthemomentum distribution

ofultra- old

Rb

bosoni atoms,takenat dierenttimes

t

afterthe hange of

thepotential depth. From left to right: (a) 0

µ

s, (b)100

µ

s, ( ) 150

µ

s, (d)

250

µ

s,(e)350

µ

s,(f)400

µ

s,(g)550

µ

s. Noti ethe ollapse andrevivalof

thesuperuid oheren e peak.

theaforementionedlowenergys alesinthesesystems(typi ally

∼

kHz)and

thealmostperfe tisolationfrom the environment resultinto relatively long

times ales omparedto solidstatematerials(seenext se tion). Thismakes

those systemsthe natural laboratory where theunitary quantum dynami s

following anexternalperturbation an beprobedinreal-time. Inthe

follow-ingwewillbrieysurveysomekeynote re ent experimentsthattriggered an

enormousinterestontheeldofnonequilibriumdynami s ofisolatedmany

bodysystems.

Collapse and Revival Os illations

Therstseminalexperimentprobingthenonequilibriumdynami sof

ultra- oldbosoni atomsinopti allatti eshasbeenrealizedin[75℄. Hereasudden

hange of thestrengthof the latti e depthhasbeen performed, driving the

system from a weakly intera ting superuid regime to the Mott Insulator.

Hen ethe systemis holdwiththenal valueof latti e depthfor a variable

time

t

, after whi h time of ights measurements are taken to probe the

evolutionof momentumdistribution, see gure2.5.

As we an see from panel a), the initial state shows oherent peaks in

the interferen e pattern whi h are hara teristi features of the superuid

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sys-CHAPTER 2. NON-EQUILIBRIUM PHYSICS IN CORRELATED SYSTEMS

21

Figure2.6: Left Panel: artoon pi ture of the lassi al Newton's radle and

itsquantumanalogue. TheinitialBEC issplitted intwowavepa kets whi h

os illateout ofphasewithperiod

τ

. RightPanel: Absorptionimagesduring

the rstos illation y le. From[106℄.

enough, however, after waiting further time the oheren e is restored as

learly shown in the last panel g. A tually, the non equilibrium dynami s

of the system shows a full series of ollapse and revival os illations of the

superuidphase oheren e,whi h eventually fadeout at longer times ales.

While a qualitative explanation of the observed periodi pattern has been

given in termsof simple mean eldarguments [197℄a full understandingof

thenonequilibriumdynami sinintera tingbosoni (andfermioni )systems

isstill asubje tofintense resear ha tivity.

Quantum Newton's Cradle

Ase ondground-breakingexperimentprobingthedynami sofbosoni atoms

hasbeen performedmore re ently [106℄. Herea BEC of

Rb

atoms was on-nedbymeansofa strongtwodimensionalopti allatti einto anee tively

one-dimensional geometry. Then, by applying along the axial dire tion a

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22

2.2. ULTRA-COLD ATOMIC GASES IN OPTICAL LATTICES

Figure 2.7: Experimental data from [177℄ showing time evolution of

dou-bly o upied sites ina strongly orrelated system made by ultra old 40

K

fermioni atoms.

is splitted in two oherent wave-pa kets with momentum

±2~k

that

re ol-lide periodi ally after a time

t

col

∼ π/ω

0

, see artoon in gure 2.6. The remarkableexperimentalobservation wasthateven afterseveralhundredof

os illationstheinitial non-equilibriummomentumdistributiondidnotrelax

toanewequilibriumone onsistentwithgivenma ros opi onstraintssu h

asenergy andparti le onservation.

Thisexperimentalresulttriggeredmanytheoreti alinvestigationsonthe

nonequilibriumdynami sofintegrableandnon-integrableintera ting

quan-tumsystems.

Lifetime of Doublons

Non equilibrium experiment involving ultra old fermioni atoms in the

strongly orrelated regime have been reported only mu h more re ently. A

re ent work addresses thede ayof doubly o upied latti e sites (doublons)

inthe fermioni single band Hubbard model. In order to prepare a largely

outofequilibriuminitial statethesystemwasexposedtolatti emodulation

whi h reated anex essofdoublons. After themodulationthesystemislet

evolve freely at the initial latti e depth and intera tion strength for up to

4

s. Asshown ingure2.7theexperimental results showan elasti de ay of doublons spanning two-order of magnitude. The de ay rate of this highly

ex itedstateisfoundtobe,withinagoodagreement,exponentiallylargein

theratio

U/J

,where

U

istheHubbardrepulsionand

J

thehoppingintegral.

Thisresult onrmswhatrstlynoti edin[155℄. Indeedthede ayofhighly

ex ited stateson a latti e withbandwidth

D

is inhibited by energy

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CHAPTER 2. NON-EQUILIBRIUM PHYSICS IN CORRELATED SYSTEMS

23

Figure2.8: S hemati set-up of atimeresolvedpumpprobe experiment. In

thisapproa h, anultrashort laserpulse issplitinto twoportions,a stronger

beam(pump)isusedtoex itethesamplegeneratinganon-equilibriumstate,

whilea weakerbeam (probe)is usedto monitor thepump-indu ed hanges

inphysi al properties ofthesystem.

pro ess is required,with

n

∼ E/D

. In a systemwithtwo-body intera tions

su h a pro ess is expe ted to have an exponentially small rate, thus to be

long-lived.

2.3 Time-ResolvedSpe tros opies onCorrelated

Ma-terials

In the previous se tions we have des ribed arti ially engineered systems

inwhi h ele tron-ele tron intera tions an have dramati ee ts on the

ob-served physi al properties. This is not only true in arti ial systems but

also,andperhaps moreinterestingly, inrealsolidstate materials.

In this respe t relevant examples are provided by a ertain lass of

transition-metal oxides with partially lled d-shells, or by many mole ular

solids with large separations between neighboring mole ules. Here the

en-ergygaindue to ele tronwavefun tion delo alization an be lowerthan the

energy asso iated to the lo al ele tron-ele tron Coulomb repulsion. When

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un-24

2.3. TIME-RESOLVED SPECTROSCOPIES ON CORRELATED MATERIALS

sites and the system is a so alled Mott Insulator (MI). Su h a strongly

orrelated state ofmatter represents a remarkable example of thefailure of

onventional band theory. A ording to it, due to the partially lled band,

aMottInsulatorshould indeedbehave like ametal, whileitfeatures harge

gapped ex itations due to the large Coulomb repulsion. This is typi ally

missed in standard ele troni stru tures al ulations whi h a ount for

in-tera tions only at meaneldlevel.

Fromthephysi alpointofviewitisworthnoti ingthatmanynontrivial

phenomenaareknowntoo urinstrongly orrelatedmaterialswhi hareon

the verge ofbe omingMottinsulators, high-temperature super ondu tivity

beingoneofthemost strikingexample[112℄. Itishen enotsurprisingthat

manydierentexperimental te hniqueshavebeendeveloped,sin e theearly

dis overy of super ondu tivity in opper-based oxides[128℄, to hara terize

physi al propertiesof thesematerials withan in reasing resolutioneitherin

energy [38℄orinreal-spa e [139℄.

Re ently a large interest hasbeen generated bythepossibilityto

inves-tigate orrelated materials using modern time-resolved spe tros opies with

femtose ond resolution, te hniques whi h have been mainly borrowed from

therealmof atoms andmole ules [196℄.

Pump-probespe tros opyisthesimplestexperimentalte hniqueusedto

study ultrafast dynami s insolids and mole ules. Insu h experiments, one

rst shoots an ultrafast (typi ally 10-100 fs) pumping pulse on the sample

todriveitsele troni systemoutoftheequilibriumstate. Thenafterabrief

time delay (

∆t

) of typi ally tens of femtose onds to tens of pi ose onds, a

probing pulse of either photons or ele trons is sent in to probe the sample

transient state. Byvarying

∆t

,one an studythepro ess bywhi hthe

sys-temrelaxesba ktotheequilibriumstate,thusa quiringtherelateddynami

information. Most onventional set-ups use opti al probes to measure, for

example, the hanges in theopti al onstants (su h as ree tivity or

trans-mission)asa fun tionof timedelay between thearrivalof pumpand probe

pulses. This yields information about therelaxation of ele troni states in

thesample[67℄.

It is worth mentioning that the very idea of re ording transient

phe-nomenabyshiningshort pulsesfollowed bya se ondone at xeddelaywas

developed long-time ago, already before the end of XIX entury [109℄.

Al-thoughthe on eptualframework forpump-probe spe tros opywassettled,

it takes the whole entury to develop proper te hniques to generates and

dete t fastenough pulses. In thisrespe tthe resolution inthetimedomain

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CHAPTER 2. NON-EQUILIBRIUM PHYSICS IN CORRELATED SYSTEMS

25

Figure 2.9: Time-Resolved ARPES measurements [141℄ of the evolution of

ele troni stru ture of 1T-TaS2 after an infrared opti al ex itation. Right

panelshowstheinstantaneous ollapseoftheintensityoftheHubbardband

and the slower subsequent re overy. Left panel shows the dynami s of the

spe tralfun tion whi h rstly shows a transfer of spe tral weight from the

Hubbardband tothe Fermileveland thenaslower depletion.

ti al te hniques pushes the resolution power of pump-probe te hniques six

orderofmagnitudeup,fromnanose ondtofemtose ondregime. Thislargely

widens the range of physi al pro esses that an be addressed by means of

thesete hniques. Inaddition,sin e itisnowpossibleto generateanddete t

femtose ond pulses a ross thewhole ele tromagneti spe trum, novel

time-resolved spe tros opi te hniques have been proposed in addition to more

onventional opti al probes. Among them we mention ultrafast ele tron

dira tion [29℄ and photoemission spe tros opy [141℄. As we briey

men-tioned,a ommon feature of these dynami alte hniques is that thesystem

investigated is no longer in stri t thermodynami equilibrium. The

mate-rial under study may be either in an ex ited state whose de ay into other

degreesof freedomisbeingprobed,yielding informationunavailableto

on-ventionaltime-averagedfrequen ydomainspe tros opies,orinametastable

statewithfundamentally dierent physi al properties. Manyinteresting

ex-periments have beenre ently performedon orrelated materialsusing

time-resolved spe tros opies, see for example [66, 34℄ for interesting works on

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26

2.3. TIME-RESOLVED SPECTROSCOPIES ON CORRELATED MATERIALS

Figure2.10: Cartoonpi tureofthetransientdynami sofele troni stru ture

of1T-TaS2 afterphoto-indu ed ex itation.

2.3.1 Time Evolution of the Ele troni Stru ture a ross an

Insulator to Metal transition

Dierently frommetalli ompounds,Mottinsulatorsfeaturesalargegapin

thespe trumofsingleparti leex itations. Chemi aldopingisapossibleand

widelyusedroute toindu e aninsulatorto metaltransition. Analternative

path isoered bytheso alledphoto-dopingasithasbeen shownin[141℄.

Here an infrared femtose ond laser pulse ex itesthe insulating phase of

1T-TaS2,amaterialwhi hshowsaMotttransitiondrivenbyele troni

or-relations. Hen ethetransient ele troni stru ture ismeasureddire tlywith

time-resolvedphotoele tronspe tros opybyphotoemittingvalen eele trons

by a time-delayed ultra-violet laser pulse. In gure 2.9 we report ARPES data at dierent time delays. The results show that the opti al ex itation

indu es anultrafast transformation to asemi- ondu ting transient phase.

This is shown by the instantaneous ollapse of the ele troni band gap

(right panel)andbythemajortransferofspe tralweight fromtheHubbard

band to the originally gapped region lose to theFermi level. This pro ess

o urs onarather shorttimes ale,

t < 100f s

. Thenthehotele tron

distri-bution de ays rapidly within a fewhundredfemtose ond and theband gap

is subsequently re-established. Noti e however that the peak of the

Hub-bard band is slightly shifted toward the Fermi level due to photoindu ed

doping. Inaddition, thehotele trons laun ha nu lear motionofthelatti e

atoms whi h vibrate oherently after the instantaneous ex itation. Sin e

these vibrations do not alter the band gap, the material remains

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Chapter

3

Theoreti al Challenges in

Correlated Systems Out of

Equilibrium

The experimental advan es we have des ribed in previous hapter oer the

han e to probe strongly orrelated systems in a ompletely novel regime,

where the ombinationof intera tionsand nonequilibrium ee ts mayresult

inmanynontrivial phenomena. Atheoreti al des riptionof thisnew regime

poses serious hallenges, both froma on eptual and a methodologi al point

of view. In this hapter we briey review the main theoreti al issues behind

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28

3.1. A GLIMPSE ON MANY BODY THEORY OF NON EQUILIBRIUM STATES

3.1 A Glimpse on Many Body Theory of Non

Equi-librium States

Strongly orrelated ele tronsystems and their ri h physi al properties

rep-resent a major hallengefor theoreti al ondensed matterphysi ssin e last

thirtyyears. Theyusuallyes ape anysingle parti ledes ription andare

of-tenasso iatedtointrinsi allynonperturbativeee ts. Themain goalofthe

theoreti al resear h has been so far mainly fo used on understanding low

temperature equilibrium properties of these systems, whi h is the regime

typi allyprobedin ondensed matterexperiments. Heregroundstate

orre-lations and low-lying ex ited states ompletely dominate the physi s. This

eort resultedinto a largevarietyof analyti aland numeri al methods able

to opewithele tron-ele tron intera tion beyond perturbation theory.

However, as experiments start probing physi al properties far beyond

the linear response regime, the interest on orrelated phenomena in out of

equilibriumrapidly startsto grow.

Initsoriginalformulationthemanybodytheoryofnonequilibriumstates

dates ba kto early worksbyS hwinger [170℄, Keldysh [105℄, Kadano and

Baym[97℄inthemidsixties. Intheirseminal ontributionstheseauthors

set-tledthe propertheoreti alframework todes ribe systemswhi h areevolved

underthe a tion ofexternal eldsor whi h are driven into non equilibrium

steadystatesbyexternal for ing. Withoutentering intoo mu h details,for

whi h we refer the reader to existing monographs [147, 98℄, we an try to

point out themain dieren ebetween equilibrium(zerotemperature)many

body theory and its non-equilibrium ounterpart. It lies in the fa t that,

on e the systems is pushed out of equilibrium, the nature of the quantum

state onwhi hphysi al propertiesare omputed isnot deneda priori, but

is determined by the dynami s itself. This point is entral in the whole

formulation and ree tsthe fa tthat we would like to des ribe phenomena

su h asthe ow of urrents where the system, even ifstationary, ould not

be inits ground state but inan arbitrarilyex ited state. Thisis at the

op-positeofstandard manybodytheorybasedontheso alledGell-MannLow

theorem [64℄, whi h assume that, provided the intera tion is swit hed on

adiabati allyintheinnite past, the systemwill remaininits ground state

even after innite time. While this statement is generally true for ground

statesit anbeviolated forarbitrarystates[110℄,whi hevolveafterinnite

time into a ombination of ex ited states. From a dierent perspe tive we

ould saythat while inthermal equilibrium thegeneral goal is to solve the

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CHAPTER 3. THEORETICAL MOTIVATIONS

29

Althoughextremely valuable andgeneral, thisapproa hasitstands annot

provide too mu h insights into the strongly intera ting regime where

per-turbative approa hesusually fail. Thisiseven moretrue for what on erns

the real-time dynami s. Indeed it is well known [13℄ that bare

perturba-tiontheoryinreal-timeisusually plaguedbyse ular terms whi h make the

limit of long-times/small-intera tions tri ky to handle. Partialsummations

of ertain lass of diagrams are not su ient to ensure a stable long-time

limitandonehastoresorttomoresophisti atedapproa hes,seeforexample

[68, 63℄ Among them we mention inparti ular the ow equation approa h

[81,82℄whi hprovidesareliableanalyti altooltostudythenonequilibrium

real-timedynami s inthe weak oupling regime.

Thishasmotivatedahugeeortfordevelopingmethodsthattreatstrong

orrelationsandnonequilibriumee tsonequalfooting. Thiseorthasbeen

triggered by few main theoreti al issues whi h we will briey review here,

sin e theyhave aspe ialinterest fromtheperspe tive of thepresent work.

3.2 NonEquilibrium Physi s in Quantum Impurity

Models

Several intriguing theoreti al questions arise from the possibility to ouple

smallintera ting quantumsystems to externalreservoirs and to drivethem

outofequilibriumbyapplyingafor ingeld,su hasad voltagebias. Asa

resultastrongly orrelatednonequilibriumsteadystate(NeqSS)isrea hed,

after waiting some transient time, with a nite urrent owing a ross the

system. A natural question on erns howto des ribe su h a state when

in-tera tions arenot weak and in parti ular what is the ee t of de oheren e

anddissipation-two genuinenon equilibriumee ts- onthephysi al

prop-ertiesof the system. Also,an interesting question on erns thetypi al time

s aleswhi h ontrol the onset ofa NeqSS.

QuantumImpurity(QI)modelsrepresentthenaturalframeworktostudy

quantumtransportthroughnano onta ts. These onsistofasmallquantum

system with few intera ting degrees of freedom, the impurity, tunnel

ou-pledto a reservoirof fermioni ex itations. Worthily, whilethe equilibrium

physi s of these nutshell strongly orrelated systems an be studied witha

wide rangeofpowerful numeri al andanaliti altools,their non equilibrium

dynami s is still hallenging. The reason for this gap is mainly due to the

fa tthatmost of the theoreti al tools whi h has been developed inthelast

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30

3.2. QUANTUM IMPURITIES OUT OF EQUILIBRIUM

plied to the out of equilibrium ase. This has triggered a large amount of

theoreti al works. Manyinteresting issues have been addressed inthe

on-text of quantum impurities innon equilibrium steady states. Without sake

of ompleteness, we brieysummarizesome of themhere.

3.2.1 Anderson Impurity Out of Equilibrium

Atheoreti al paradigmfor orrelated quantumtransportisprovided bythe

so alledAndersonImpuritymodelrstlyintrodu edin[8℄toexplainthelow

temperaturebehaviourofmagneti impuritiesembeddedinmetalli hosts. It

ontainsthebasi physi softheKondoee t[192,133℄,namelythe omplete

s reening of the impurity spin by ondu tion ele trons as the temperature

is lowered to zero, and itwaslater re ognized [131,69,2℄to beresponsible

for thezero bias anomaly observed in the low temperature ondu tan e of

semi ondu ting quantumdots.

Non equilibrium ee ts on this orrelated lo al many body state have

been studied theoreti ally very intensively in last years, due to their dire t

relevan eto experiments onnanodevi es.

In parti ular, the ee t of a d voltage bias has been rstly addressed

using (weak- oupling) perturbative renormalization group methods, see for

example [99, 100, 153, 103, 44, 59℄, that provide a sensible approximation

in theregime

eV

≫ T

K

. The general out ome of these studies is that the ow of the urrent a ross the impurity indu es de oheren e thus

utting-othe hara teristi logarithmi singularities asso iated withKondo ee t.

Also the role of a magneti eld has been addressed in the same regime

[154, 60, 168℄. Moreover, the behaviour of dierential ondu tan e at low

bias voltage

eV

≪ T

K

has been also obtained using Fermi Liquid Theory [100,135℄.

However, as these works fo us mainly on the asymptoti regimes of

lowand highvoltages, thetheoreti al des riptionof thewholebias-indu ed

rossoverfromlow ondu tan e upto theunitarylimit is stilla hallenging

open problem. Inadditionmany hara teristi nonequilibriumfeaturesdue

to thebias voltage, su h as for example its ee t on the impurity spe tral

fun tion,hasnotbeenyetfully laried. Finally,wealsomentionthat

theo-reti alapproa hesabletodealwithmore ompli atedimpuritymodels(su h

asthosewithmore impurities or orbitals)would be highlydesirable.

Re ently a new interest has been triggered by the developments of non

perturbative methods for quantum impurities out of equilibrium. We

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CHAPTER 3. THEORETICAL MOTIVATIONS

31

appliedtostudyquantumtransportthrougha orrelatedAndersonImpurity

inthe non linear regime we mention thes attering state extension of NRG

[3℄, the time-dependent Density Matrix Renormalization Group [107, 88℄

(DMRG), and the ISPI method [186, 46℄. In addition, a novel numeri al

method based on Diagrammati Monte Carlo [183℄ has been re ently

pro-posedbya numberof authors[127,162,191℄. Thiswill bethemainsubje t

ofthe rst partofthis thesis andwerefer thereader to hapters 4 and5. 3.2.2 Vibrational ee ts in non equilibriumtransport

Theoreti alinvestigationsofvibrationalee tsinquantumtransportthrough

singlemole ules,startingwiththepioneeringworks[70,193℄,keepattra ting

alargeinterestinthe ommunity[122,61℄. Theinterplaybetween ele troni

andvibroni degrees offreedominmole ular ondu torsandthelarge

num-berofenergys alesthatareinvolved resultinnontrivialbehaviorsthatare

generally di ult to grasp within a unied theoreti al framework. Several

interesting issueshave been re ently addressed.

One on erns,forexample,thepossiblesignaturesofele tron-vibron

ou-plingintransportpropertiessu hasdierential ondu tan e[51℄,shotnoise

[87℄ andmore generallyfull ounting statisti s[166,10℄. Thesemayappear

asso alledvoltage-indu ed singularities[53℄,arising whenbiasvoltagehits

the vibrationalfrequen y ofthemole ule, or asside bandsinthenitebias

dierential ondu tan e [57℄.

A relevant issuein thepro ess where ele trons owa ross themole ule

is the role played by phonon distribution fun tion whi h may be or not

thermalized with the fermioni reservoirs [54, 122℄. In the latter ase we

ouldexpe tthebiastoplaytheroleofee tivetemperatureforthevibrioni

ex itations.

Morere entlyalsothetransientnonequilibriumdynami shasbeen

stud-ied,usinglowest orderKeldyshperturbationtheoryandamean-eldstrong

oupling approa h [151℄. It is worth noti ingthat thedynami al behaviour

of these single mole ule devi es may display intriguing ee ts espe ially in

the strong ele tron-vibron oupling regime where the jun tion may be on

theverge of a bistable behaviour [121℄. In this respe t a proper treatment

of quantum u tuations is also ru ial to orre tly reprodu e the physi s

[122,123℄.

3.2.3 Real-Time Dynami s after Lo al Quantum Quen h

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32

3.2. QUANTUM IMPURITIES OUT OF EQUILIBRIUM

suddenperturbation,aso alled quantumquen h.

In the ontext of impuritymodels this problem has a long history that

goes ba k to the seminal works by Nozières and De Domini is on the

X-rayedgesingularity[134℄, passingthrough thefamous Anderson and Yuval

approa h to the Kondo model[195℄.

More re ently, this problem stimulated new interest [132, 113, 5℄, due

totheexperimental progressesinnanote hnology, whi h madeitpossibleto

onta tmi ros opi quantumobje ts withmetalli ele trodes,thusrealizing

quantum impuritymodelsina fullytunable set-up [71℄.

Twokindsofquen hes anbe onsideredinthis ontext,dependingonthe

amount of energy thatis inje ted into the system,also referredasthework

doneduringthe quen h. Global quantumquen hesareparti ularly relevant

for transportthrough orrelated nanostru tures,where a net urrent owis

for ed by suddenly swit hingon e.g. a d bias voltage. Sin e the swit hed

perturbationisextensive,thesystemisdriven into anon-equilibriumsteady

state at long times [44℄. Conversely, lo al quantum quen hes amount to

suddenly hange the impurityHamiltonian. Thesekindsofquen hes anbe

realizedinanopti alabsorptionexperiment, assuggestedin[181℄and more

re ently in[90℄, and the resulting non-equilibrium dynami s anbe tra ked

inreal-time usingpump-probe te hniquesor, inreal-frequen ies,measuring

theabsorptionlineshape. Furthermore,lo alquen hesareinterestingasthey

arethesimplestexamplesofnon-equilibrium pro esseswhosestatisti smay

shownon trivialu tuations [171℄.

In the ontext of the Anderson Impurity Modelthe real-time dynami s

hasbeen studied using td-NRG[5℄. Thisstudy reveals thepresen e of

sep-arate time s ales for harge and spin ex itations, the rst being ontrolled

by the hybridization width while thelatter being long lived and ontrolled

by the Kondo temperature

T

K

. We will ome ba kon this problemin next hapters.

In the ontext of Kondo Model many theoreti al works addressed the

real-timedynami safteralo alperturbationusinganalyti alandnumeri al

approa hes[6,143,102℄.

Anintriguingproblem,whi hwillbepartiallyaddressedinnext hapters,

is related to the interplay between thequen h and thelow energy spe tral

properties of the bath. Indeed, as it is well known, the equilibrium xed

point stru ture of the AIM is very sensitive to the low energy properties

of the bath [194, 73℄ and this may result in non trivial behaviors for what

on erns the non equilibrium dynami s. This point has some onne tion

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CHAPTER 3. THEORETICAL MOTIVATIONS

33

3.3 Non Equilibrium Dynami s of Correlated Bulk

Systems

Ase ondmainlineoftheoreti alresear honstrongly orrelated physi sout

of equilibrium has been triggered by the opportunity to probe relaxation

dynami s ofhighly ex ited statesinma ros opi quantum systems. This is

oered by re ent developments in ultra old atoms and pump-probe

spe -tros opies on orrelated materials that we have briey reviewed in hapter

2.

Asoppositetoprevious asesofquantumimpuritieswheremostofthe

in-terestwastriggeredbytheexisten eofastrongly orrelatednon-equilibrium

steadystate due to the reservoirs, here the main fo us is on the relaxation

dynami saftertheex itation,hen eontransientdynami al phenomena. As

a onsequen eoftheir ri h equilibriumlowtemperature phasediagram,

fea-turing many ompeting phases all very lose in energy, strongly orrelated

ele trons are expe ted to display intriguing dynami al behaviors when an

externalperturbationdrive themawayfromthermalequilibrium. Thismay

in lude, for example, the trapping into long-lived metastable states whi h

maydier ompletely fromtheir lowenergy equilibrium ounterpart [155℄.

It is worth mentioning that a theoreti al modeling of non equilibrium

experimentswehavedes ribedin hapter2,bothforwhat on ernsultra old atoms and parti ularly strongly orrelated materials, may be very di ult.

Forthisreasontheoreti alinvestigationsmostlyfo usedonprototypi alnon

equilibriumproblemswiththeideaofgainingfurtherinsightsonthephysi s

oftime dependent strongly orrelated phenomena.

3.3.1 Global Quantum Quen hes

From a theoreti al point of view the simplest way to push the system out

ofequilibrium isthrougha so alledquantum quen h [25℄. Herethesystem

is rstly (i) prepared in the many-body ground state

|Ψ

i

of some initial Hamiltonian

H

i

whi h is then(ii)suddenly hanged to

H

f

6= H

i

. Asa on-sequen e of this instantaneous hange the initial state

|Ψ

i

turns to be an highlyex itedstate of the nalHamiltonian, whi h willdrive thedynami s

for later times. It is worth to noti e that the perturbation indu ed by the

hange of Hamiltonian is in general extensive. We an therefore say that

quantum quen hes provide the simplest proto ol to indu e a global

ex ita-tion into the system and to monitor how the energy is distributed among

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34

3.3. QUANTUM QUENCHES IN ISOLATED SYSTEMS

These primarily address the onset of thermalization at long time s ales or

the possible realization ofnon thermalsteady states.

Before we pro eed a word is in order to further larify the meaning of

the so alledthermalizationdebate [27,126,124℄.

Sin e the quantum dynami al evolution of isolated systems is unitary,

noentropy produ tion isultimately possible. Hen e apure state,su h that

Tr

[ρ

2 i

] = 1

,will neverrelax stri tly speakingtoward a thermal state, whi h isby onstru tion a mixedone Tr

[ρ

2 therm

] < 1

. However, aslong assuitable observables are onsidered, their quantum dynami s an rea h long-time

steady states whi h (i) are robust against hanges in the initial onditions

and(ii)whosepropertiesresemblethoseobtainedwithinanitetemperature

Gibbs ensemble at the same intensive energy and parti le density. From a

broaderperspe tive,wenoti ethattheissueofthermalizationinintera ting

quantum systems hasbeen rstly addressedin ontexts whi h arefar from

ondensed matterphysi s. In parti ular,in onne tion withquantum haos

roughlyade adeago[43,176℄andwithhigh-energy physi s and osmology,

more re ently [14, 15, 68℄. In this respe t, the experimental developments

withultra oldgaseshavebroughtfreshnewideasandtriggeredtheattention

ofthe ondensed matter ommunity[149℄.

The re ent literature on quantum quen hes in intera ting bosoni and

fermioni systems is by now very broad, see for example the re ent topi al

reviews[50,12,31℄

An interesting issuewhi h hasbeen widelydis ussed and is still matter

ofs ienti debateistherole playedbyintegrabilityintheissueof

thermal-ization. Thegeneralexpe tationisthatintegrablesystemsfailtothermalize

due to the extensive number of onserved quantities whi h forbid loosing

memory ofthe initial ondition. Whilethese expe tationshave been

gener-ally onrmed [30,94,47,150,148℄,the spe i me hanismsfor su ha la k

ofthermalization isstill underdebate [16,156,157℄.

For generi non integrable systems the physi al expe tation is that

re-laxationto thermalequilibriumwill take pla eaftersome(eventuallylarge)

transient times ale. Howeverfewresultsareavailable inthis ase, sin ethe

problemofsolvingreal-timedynami sofnon-integrableintera tingquantum

systemsisextremely hallenging. Thisleavesstillopenthequestionwhether

andin whi h wayrelaxationto a thermalsteadystate takespla e.

Forlatti eonedimensionalsystemsresultshavebeenobtainedmainly

us-ingtime-dependentDMRGandLan zosalgorithm. Inthebosoni ase[108,

158℄ the numeri al results show that while for small intera tion quen hes

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CHAPTER 3. THEORETICAL MOTIVATIONS

35

thermalonewasrea hed,thusmakingdi ultto on ludewhether

thermal-ization o urs on a mu h longer times ale or not. Even more surprisingly,

theresults forone dimensionalspinlesslatti efermions [116℄bothinthe

in-tegrableandnon-integrable aseshowthatupto thelongesta essibletime

s alethedynami s relaxtoastationarystatewhi hdiersfromthethermal

one.

For strongly orrelated ele trons in more than one dimension only few

results are available on erning the dynami s after an intera tion quen h

inthe Fermioni Hubbard Model. In the following we will dis uss inmore

detailstheseworkswhi hwill playanimportantrole fromthepointof view

ofthis work.

3.3.2 Quen h Dynami s in the Fermioni Hubbard Model

The singleband Hubbardmodel [91,79,101℄represent one of the simplest

yetnontrivialmodelsen odingthephysi sofstrong orrelations,namelythe

ompetitionbetween ele troni wave fun tiondelo alizationdue to hopping

t

and harge lo alization due to large Coulomb repulsion

U

. Out of this ompetition, on the verge of a Mott metal-insulator transition, many non

trivialphenomena mayarise. While theoreti alinvestigations onits ground

stateproperties ontinuesin ethirtyyears,thenonequilibriumdynami sof

this paradigm strongly orrelated model has been started only mu h more

re ently. The dynami s of Fermi system after a sudden swit h-on of the

Hubbard intera tion has been studied rstly in [125, 126℄ using the

ow-equationapproa h.

The resulting evolution, evaluated up to se ond order in the

intera -tion, shows a full a sequen e of transient regimes. For short time s ales

1

,

0 < t

≪ 1/U

2

,one observes a fastredu tion of Fermi surfa e dis ontinuity

Z

whi h an be understood as formation of quasiparti les from the initial non intera ting state. At time s ales of order

t

1 ≃ 1/U

2

the quasiparti le

distribution fun tion has relaxed to a quasi steady state whi h makes the

systemresembling a zero-temperature Fermi Liquidbut witha

hara teris-ti mismat hinthequasiparti leweight

1 −Z

N EQ

= 2(1

−Z

EQ

)

. Noti ethat su ha prethermal regimeis stablefor times ales

1/U

2 _{< t < 1/U}

4

. Inthis

timewindowex itationenergyisalreadyrelaxedtoequilibrium, hen efrom

an energeti point of view the system an be onsidered as thermal, while

the distribution fun tion is trapped into a metastable onguration. This

hasbeen attributed to phase spa e onstraints hara teristi of intera ting

1

Hereafterwextheunitofenergiesandtimesbysettingthenon-intera tingdensity

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36

3.3. QUANTUM QUENCHES IN ISOLATED SYSTEMS

Fermisystems. Full thermalizationeventually o ursonamu h longertime

s ales,

t

2 ≃ 1/U

4

, due to residual quasiparti le intera tions. Thishas been

des ribed phenomenologi ally within a quantum Boltzmann equation. The

twostage relaxationthathasbeen des ribed withinowequationapproa h

isstri tlyspeakingrestri tedtosmallintera tionquen hes. Forlargervalues

oftheintera tiononemayexpe tthetwotimes ales

t

1 , t

2

tostillbepresent, although lessseparated.

This hasbeen indeed onrmed by solving thequen h dynami s for the

Hubbard model using non-equilibrium extension [58℄ of Dynami al Mean

FieldTheory[65℄(DMFT).DMFThasemergedinthelastde adesasavery

powerful non-perturbative approa h to strongly orrelated ele tron systems

inthermalequilibrium. Thismethodmapsthefulllatti emodelof

intera t-ingfermions onto a Quantum Impuritymodel oupledto a self- onsistently

determinedexternalbath. Inageneralnonequilibriumsituation,whereone

isinterestedinstudyingrelaxationdynami sinreal-time,thebath annotbe

assumedto be inthermalequilibrium. Usinga re ently developed real-time

Diagrammati Monte Carlo algorithm [191℄ the dynami s after an

intera -tionquen hstartingfromazerotemperatureFermiseawasinvestigated[48℄.

Theresultsshowthatbothforweakandstrongintera tionquen hesthe

sys-temistrappedintoquasi-stationarystatesonintermediatetimeswhi hdelay

thermalization to mu h longer time s ales, far beyond the longest time so

far a essiblewithreal-time QMC.Theweak oupling prethermal regimeis

onsistent with ow equation analysis and appears as a plateau in the

dy-nami sof quasiparti le weight. As opposite thestrong oupling prethermal

regime originates from the exponentially long-time s ales asso iated to the

de ay of double o upations [155℄and results ina hara teristi pattern of

ollapseand revivalos illations.

In addition, DMFT results show eviden e for a fast thermalization

o - urringataspe ialvalueofthenalintera tion, in orresponden eofwhi h

thequasiparti leweightrelaxestozero onsistently withtherelatively

high-temperature

T

⋆

xed by the initial energy. Aslater analysis has onrmed thatnotonlyone-timeobservablesbutalso orrelationsfun tionarethermal

for thisspe ialvalue ofquen h[49℄.

The appearan e of su h a sharp rossover at

U

dyn

c

in thedynami al be-havior issurprising ifthought intermsof theequilibrium phasediagram of

HubbardModelwithin DMFT. Indeed this would appearat a mu h higher

temperature

T

⋆

than the riti alMott endingpoint. Itsultimate origin and itsrelationwiththeequilibriumMottmetal-insulatortransitionhasnotbeen

(37)

CHAPTER 3. THEORETICAL MOTIVATIONS

37

3.4 Plan of the Thesis

In light of previous dis ussions we an on lude that understanding the

physi s of intera ting quantum systems out of thermal equilibrium

repre-sentsone ofthe mostintriguing openproblem inmodern ondensed matter

physi s.

In this perspe tive, the aim of present thesis is to address some of the

issuesarisinginthiseldbymeansofnovelanalyti alandnumeri almethods

able to deal with strong ele tron-ele tron orrelations and non equilibrium

ee ts.

In the rst part of this work we onsider quantum impurity models as

paradigmati examples of strongly orrelated nanodevi es. We develop a

general approa h to dealwith their real-time non equilibrium quantum

dy-nami sandapplythisReal-TimeDiagrammati MonteCarlomethodtotwo

relevantphysi al examples. Thispartofthethesisisorganizedasfollows. In

hapter4weintrodu ethemethod,whi hisbasedonaDiagrammati Monte Carlosampling ofthereal-timeperturbationtheoryinthehybridization

be-tween the impurity andthebath. In hapter5 we present two appli ations, namely we study dynami s after a lo al quantum quen h in the Anderson

Impurity Model and non linear transport through a simple Holstein model

ofmole ular ondu tor.

In the se ond part we move our attention to bulk ma ros opi systems

andtothenonequilibriumdynami s indu edbyglobal quantumquen h. In

this ontext weproposea non perturbative approa hto quantumdynami s

ofstrongly orrelated ele tronsystemsbasedonatimedependent extension

of the Gutzwiller wave fun tion. In hapter 6 we present a general formu-lationof the timedependent variational method and onsider as a relevant

(38)

(39)

Dynami s in Quantum

(40)

(41)

Chapter

4

Diagrammati Monte Carlo on the

Keldysh Contour

Inthis hapterweintrodu etheReal-TimeDiagrammati MonteCarlomethod

we have proposed to study non equilibrium dynami s in orrelated

quan-tum impurity models. We formulate the problem on the full three bran hes

Kadano-Baym-Keldysh ontour whi h allows us to deal with an arbitrary,

evenintera ting,initialdensitymatrix. Wedis usstwomainnonequilibrium

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42

4.1. NON EQUILIBRIUM DYNAMICS IN QIM

4.1 Non Equilibrium Dynami s in Quantum

Impu-rity Models

The aim of this se tion is to set-up the proper framework to study non

equilibrium real-time dynami s in quantum impurity (QI) models. To this

purpose, we onsider a set of dis rete ele troni levels, the impurity, with

reation operator

c

†

a

labeled by an integer

a = 1, . . .

N

whi h may in lude both spinandorbitaldegrees offreedom. Theselevelsare oupledto oneor

more baths of free fermions with momentum

k

and reation operator

f

†

k a

. Thegeneri Hamiltonian ofa QImodelreads

H

−

=

X

k a

ε

−

_k

f

_{k a}

†

f

_{k a}

+

_H

−

_loc

h

c

†

_a

, c

a

i

+

X

k a

V

_{k a}

−

f

_{k a}

†

c

a

+ h.c.

,

(4.1)

where the rst term des ribes the ontinuum of fermioni ex itations, the

lo al Hamiltonian

H

−

loc

h

c

†

a

, c

a

i

generally ontains many-body intera tions

for ele tronson the impurity,whilethe last term isthehybridization whi h

ouples the impurity and the bath and it is assumed here, for the sake of

simpli ity,to bediagonal inthe

a

index.

Sin e we are interested instudying non equilibrium dynami s of model

(4.1), we have to spe ify an initial ondition as well as a proto ol to drive this systemout of equilibrium. Following general ideas of non equilibrium

manybodytheory[147,97℄,we imagine toprepare thesystemat

t = 0

ina thermalstateof

H

−

,namelywe hoosetheBoltzmanndistributionasinitial densitymatrix

ρ(t = 0) = ρ

eq

≡

e

−βH

−

Z

,

Z =

Tr

e

−βH

−

_,

(4.2)

and then,for

t > 0

,let thesystemevolve underthe a tionof a new

Hamil-tonian

H (t) = H

−

+

V (t) ,

t > 0

(4.3) Choosingtheinitial densitymatrixasthethermalonegivesa essto the

re-sponseofa orrelated quantum impurity modelto externalelds. For what

on erns the driving proto ol, namely the nature of the external

perturba-tion, therearea tuallyseveralways topush aquantum impuritymodelout

of equilibrium. In this work we shall fo us on the simplest one, namely a

quantum quen h experiment, but the method allows to address even more

general time dependent out of equilibriumproblems. Ina quantum quen h,

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CHAPTER 4. REAL-TIME DIAGRAMMATIC MONTE CARLO

43

Hamiltonian(

H

−

inthe aseofourinterest)andthen,for

t > 0

,tosuddenly hange some of its parameters letting evolve the system under the unitary

a tion of a new Hamiltonian

H

+

. Su h a proto ol therefore represents the simplestexample of time-dependent problem where the variation intime is

step-like

H (t) = H

−

+ θ (t) δ

H ,

δ

H = H

+

− H

−

.

(4.4) Thesuddenquen hinje tsenergy into the systemand leadsto a relaxation

dynami stowardsanewsteadystate,providedtheperturbation

δ

H

isnota

onserved quantityof

H

−

. The maintask isthereforeto ompute quantum averages withthe fulldensitymatrix

ρ(t)

evolved inreal-time

hO(t)i = T r [ρ (t) O ] = T r

h

ρ

eq

U

†

(t)

O U (t)

i

.

(4.5)

where the tra e has to be taken over the bath and the impurity degrees

of freedom, while

U (t)

and

U

†

_(t)

are, respe tively, the unitary operator

generatingthedynami sand itshermitian onjugate. Inthespe i aseof

a time independent Hamiltonian, as we have for

t > 0

see Eq. (4.4), these operators read

U (t) = e

−i H

+

t

_U

†

_{(t) = e}

i

H

+

t

_.

(4.6)

Topro eedfurther,itis onvenient tospe ifythenatureoftheperturbation

δ

H

indu ed by the quantum quen h. To keep the dis ussionas general as possible we writetheHamiltonian ofthe systemfor

t > 0

as

H

+

=

X

k a

ε

+

_k

f

_{k a}

†

f

_{k a}

+

_H

+

_loc

h

c

†

_a

, c

a

i

+

X

k a

V

_{k a}

+

f

_{k a}

†

c

a

+ h.c.

,

(4.7)

namely we allow for an abrupt hange of all the parameters entering in

the Hamiltonian (4.1), insu h a way thatdierent kind of non equilibrium phenomena an be treated within the present approa h. Throughout this

hapter we will mainlyrefer to two dierent setup.

Ina rstproto oltheimpurityisprepared inequilibriumwiththebath

thenadynami sisindu edbyasudden hangeofanyimpurityenergys ales.

These anbe,forexample,thelo alele tron-ele tronintera tionorthe

posi-tionoftheimpurityenergylevel. Alternatively,one animaginetoa tonthe

impurity-bath hybridization,namelytostartfromade oupledimpurityand

suddenly swit h-on the oupling with the reservoir. Interestingly enough,

thislattersetup anbeusedtostudytransient urrentsowingthroughthe

quantumimpurity,provided two reservoirsheldat dierent hemi al

(44)

44

4.2. HYBRIDIZATION EXPANSION

we plot a artoon of the dierent non equilibrium proto ols we are mostly

interestedin. On ewe have spe ied the stru tureof theHamiltonian after

Γ

L

R

Γ

L

R

Γ

µ − µ = 0

_L

R

L

QI

R

µ

L

R

t = 0

L

QI

R

µ

L

R

t > 0

µ

Γ

L

R

Γ

µ − µ = 0

_L

R

L

QI

R

µ

L

R

t = 0

L

QI

R

t > 0

1)

2)

µ

L

µ

R

Figure 4.1: The two non equilibrium set-up we will onsider to study

real-time dynami sintheQuantum ImpurityModels. Inset-up1 we prepareat

time

t = 0

the impurityinequilibriumwiththebathsat temperature

T

and

hemi al potentials

µ

L

= µ

R

. Thenattime

t > 0

someenergys aleentering the impurity Hamiltonian is suddenly hanged. In set-up 2 we prepare at

time

t = 0

the impurity de oupled from theleads whi h arein equilibrium

at temperature

T

and hemi al potentials

µ

L

6= µ

R

. Thenat time

t > 0

the ouplings

Γ

α

between then impurity and then leads are suddenly swit hed on.

thequen h,we anperformthehybridizationexpansioninformalanalogyto

whathasbeendonepreviously intheequilibriumimaginary-time ase[187℄,

with however important dieren es ree ting the genuine non-equilibrium

nature of the problem. This is will be des ribed in detail in the next few

se tions.

4.2 HybridizationExpansionon the

Kadano-Baym-Keldysh Contour

In order to study the non equilibrium real-time dynami s of quantum

im-purity models starting from a generi initial density matrix, we formulate

thediagrammati monte arlo algorithm (diagMC), inits hybridization

ex-pansionversion, onthe Kadano-Baym-Keldysh ontourmade bytheusual

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CHAPTER 4. REAL-TIME DIAGRAMMATIC MONTE CARLO

45

quantumaveragesgiven inequation(4.5). Topro eedfurther,we introdu e a dynami al time-dependent partition fun tion for the QI model whi h is

dened as

Z (t, β) ≡

Tr

h

e

−βH

0 _U

†

_{(t) U (t)}

i

.

(4.8)

We note that this quantity does not a tually depend on time

t

sin e, by

onstru tion, theevolutionis unitary, nevertheless itrepresents thenatural

quantitytoderivethehybridizationexpansion. Asitwillappearmore learly

lateron,

Z (t, β)

anbeseenadynami algeneratingfun tionaloftheMonte

Carlo weights needed to ompute any quantum average in real-time. The

basisofany ontinuous-time diagMCalgorithmistheexpressionofevolution

operators as time-ordered exponentials. For the real-time operator and its

hermitian onjugate weget

U (t) =

T

exp

−i

Z

t

0 dt

_H

+

(t)

(4.9)

U

†

(t) =

T

¯

exp

i

Z

t

0 dt

H

+

(t)

(4.10) whereT(

¯

T)isthetimeordering(anti-timeordering)operator,whosea tion

is order a string of real-time fermioni operators a ording to their time

arguments, pla ing to the left theoperatorsat later (earlier) times,withan

overallplusorminussigna ording,respe tively,totheparityofthenumber

of fermioni ex hanges needed to move the string from the original to the

nal ordering. Using the well known properties of the equilibrium density

matrix (4.2) we an write also the Boltzmann weight asan imaginary time evolutionalongthe path

Non equilibrium dynamics in strongly correlated systems

CONTENTS

CONTENTS

Chapter

1

CHAPTER 1. IN AND OUT OF EQUILIBRIUM

Chapter

2

2.1. QUANTUM TRANSPORT AT NANOSCALE

C

60

CHAPTER 2. NON-EQUILIBRIUM PHYSICS IN CORRELATED SYSTEMS

I

− V

∂I/∂V

2.1. QUANTUM TRANSPORT AT NANOSCALE

V

sd

T = 15mK

T = 900mK

I(V )

G(V ) = ∂I/∂V

E

C

I(V )

E

C

CHAPTER 2. NON-EQUILIBRIUM PHYSICS IN CORRELATED SYSTEMS

C

60

C

60

I

− V

2.2. ULTRA-COLD ATOMIC GASES IN OPTICAL LATTICES

n

≃

10

14

cm

−3

CHAPTER 2. NON-EQUILIBRIUM PHYSICS IN CORRELATED SYSTEMS

2.2. ULTRA-COLD ATOMIC GASES IN OPTICAL LATTICES

Rb

t

µ

µ

µ

µ

µ

µ

µ

∼

t

sys-CHAPTER 2. NON-EQUILIBRIUM PHYSICS IN CORRELATED SYSTEMS

τ

Rb

2.2. ULTRA-COLD ATOMIC GASES IN OPTICAL LATTICES

K

±2~k

t

col

∼ π/ω

0

4

U/J

U

J

D

CHAPTER 2. NON-EQUILIBRIUM PHYSICS IN CORRELATED SYSTEMS

n

∼ E/D

2.3. TIME-RESOLVED SPECTROSCOPIES ON CORRELATED MATERIALS

∆t

∆t

CHAPTER 2. NON-EQUILIBRIUM PHYSICS IN CORRELATED SYSTEMS

2.3. TIME-RESOLVED SPECTROSCOPIES ON CORRELATED MATERIALS

t < 100f s

Chapter

3

_{< t < 1/U}