AQuantumQuenchoftheSachdev-Ye-Kitaev Model

JuliaSteinbergHarvardUniversity

arXiv:1703.07793[cond-mat.str-el]

Chaos,Topology,andDualitiesinCondensedMatterTheoryUIUCNovember4,2017

AndreasEberleinHarvardUniversity

ValentinKasperHarvardUniversity

Subir SachdevHarvardUniversityPerimeterInstituteofTheoreticalPhysics

Collaborators

Quantummatterwithoutquasiparticles

•Wanttostudypropertiesofsystemswithoutquasiparticles

•First:whatisaquasiparticle?

•Longlivedadditiveexcitationwithsamequantumnumbersasfreeparticle

•Howdoweidentifysystemswithoutquasiparticles?

•Fastestrelaxation

•Nolonglivedexcitationsinanybasis

•“Toofast”:cannotstudylongtimebehaviorwithconventionaltechniques

⌧eq � C ~kBT

, T ! 0

TheSYKmodel:asolvablesystemwithoutquasiparticles

• ModelofNflavorsofMajorana fermionswithinfiniterangeq-bodyinteractions

• SolvableinlargeNlimit

• Maximallychaotic

• Disorderaverage→melondiagrams,onlykeepone

• Fullpropagator

H = (i)q2

X

1i1<i2<...<iqN

ji1i2...iq i1 i2 ... iq hj2i1...iq i =J2(q � 1)!

Nq�1.

= +

Strongcouplingbehavior

• For→“Emergentreparameterization symmetry

• Spontaneouslyandexplicitlybroken→ Schwarzian action

• SYKis“dual”tonearly

�J � 1

↔

iGR(t) = C(J,�)

1

� sinh ⇡t�

!2�

✓(t) � =1

q

AdS2

Escher“Heavenandhell”

Quantumquenchesandthermalization

•QuenchofSYK→ probenonequilibrium dynamicswithoutquasiparticles

•Possiblyreachthermalstate

•Whatisathermalstate?

•Systemisitsownheatbathforsubsystems

•Steadystate,observablesreachthermalvalues

• ForSYK,ourworkingdefinitionofthermalization:

2-pointfunctionobeysKMS,Tfromenergyconservation

BA

Quenchprocedure

• StartwithSYKmodelwithqandpq interactions

• Turnoffpq terminstantaneously

• TrackevolutionofGreen’sfunction

• DoesSYKGreensfunctionthermalize?

• Howlongdoesittake(scalingdependenceonT)?

• Whatisthebestwaytodothis?

Green’sFunctionsontheClosed-Time-Contour

• Outofequilibrium,muststudyfullevolutionalongcontour

• TwoGreensfunctions:and

• Usetoform2by2matrix

• Dyson(matrix)equationfromdisorderaverage:

⌃(t1, t2) =X

i

iqiJ2qiG(t1, t2)

qi�1

G�10 (t1, t2)�G�1(t1, t2) = ⌃(t1, t2)

G>(t1, t2) ⌘ G(t�1 , t+2 ) G<(t1, t2) ⌘ G(t+1 , t

�2 )

TheKadanoff-Baym equations

• Howdowestudy2-ptfunctionwithnotimetranslation?

• Kadanoff-Baym equationdirectlyfromDysonequation

• Gotorealtimeplanegettwointegro-differentialequations:

• ForMajorana fermionshavecondition:

• Alwaystrue→everythingfrom!

G�10 ⌦G> =

�⌃R ⌦G> + ⌃> ⌦GA

�G> ⌦G�1

0 =�GR ⌦ ⌃> +G> ⌦ ⌃A

�

G>(t1, t2) = �G<(t2, t1)

G>(t1, t2) = �G<(t2, t1)

CausalStructure

• EvolutionfromintegralstructureofKadanoff Baym equations

• Rewriteeverythingintermsof

• Stepfunctions→limitsofintegration

• Forpoint→ integrate“rectangleregion”

• Pre-quench

• Postquench

• Passthroughotherquadrants→causaleffect

G>(t1, t2) = �G<(t2, t1)

G>(t1, t2) = �G<(t2, t1)

t2

t1G>(t1, t2) = �G<(t2, t1)

t1 0, t2 0

t1 � 0, t2 � 0

Numerics model

• ConsidertheSYK+random matrixmodel(p=1/2q=4)

• Onlypq:integrable

• Bothterms:“Fermiliquid”

• Onlyq:“strangemetal”

• Useandtospecifyquench

• SolvefullKadanoff-Baym equationsnumerically

• UseMajorana condition→ solvefor

H(t) = iX

i<j

j2,ijf(t) i j �X

i<j<k<l

j4,ijklg(t) i j k l

⌧�1eq ⇠ T

⌧�1eq ⇠ T 2

H(t) = iX

i<j

j2,ijf(t) i j �X

i<j<k<l

j4,ijklg(t) i j k lH(t) = iX

i<j

j2,ijf(t) i j �X

i<j<k<l

j4,ijklg(t) i j k l

G>(t1, t2) = �G<(t2, t1)

Procedure

• SolveDysonequationself-consistentlyforthermalinitialstate

• UseasBCforquench

• Immediatelypostquench,notimetranslationinvariance,define

• Absolutetime:

• Relativetime:

• Nearequilibrium,variesslowlywithlookatlowfrequencybehavior

• Wignertransform

• Alsodefine

T =t1 + t2

2t = t1 � t2

f(t1, t2) ! f(T ,!)

T =t1 + t2

2

GK(t1, t2) = G>(t1, t2) +G<(t1, t2)

ThermalstatefromtheKMScondition

• “Thermal”2-ptfunctionobeysKMS

• KMS→FDT

• “EffectiveinverseT”:

• Startwiththermalstate

• Rightafterquenchoutofequilibrium

• ,variesslowly→“thermal”

−2.5 0 2.5

ω/J4,f

−1

0

1

iGK(T

,ω)/A(T

,ω)

T J4 = −50T J4 = 0T J4 = 50

J2,i = 0.5, J2,f = 0, J4,i = J4,f = 1, Ti = 0.04J4

iGK(T ,!)

A(T ,!)= tanh

✓�(T )!

2

◆

�(T )

T ! 1 �(T )

Effectivetemperature

• fromfitofFDTrelation

• Relaxesexponentially

• Checkthroughoutquench

• Determines

• Dependsononlythrough −50 0 50

T

0.05

0.075

0.1

Teff

Ti = 0.04J4Ti = 0.08J4

J2,i = 0.0625, J2,f = 0, J4,i = J4,f = 1

Teff

hHi = Ef

Teff

J2 hHi = Ef

Relaxationrate

• Knowfinaltemperaturefromenergy

conservation

• Howlongdoesittaketoreachfinal

temperaturefor?

• Exponentialrate

• Highertemperature,controlledby

�J � 1

� / T

� / T

Thelargeqlimit

• Considerlargeqinteraction(afterLargeN)

• Expand

• exponentialformofselfenergy

• DerivativesofKBeqns→ Lorentzian-Liouville eqn

• Exactsolutionforp=1/2,or2

J 2(t) = qJ2(t)21�q , J 2p (t) = qJ2

p (t)21�pq

q ! 1

G>(t1, t2) = �i h (t1) (t2)i = � i

2

1 +

1

qg(t1, t2) + . . .

�

@2

@t1@t2g(t1, t2) = 2J (t1)J (t2)e

g(t1,t2) + 2Jp(t1)Jp(t2)epg(t1,t2)

G>(t1, t2) = �i h (t1) (t2)i = � i

2

1 +

1

qg(t1, t2) + . . .

�

QuenchRegions

• Generalsolutioninallregions

• Majorana condition→

• Forequilibriumsolution

• StructureofintegralsinKBequationsshowthisisalwaystrue

• Needtosolvein5regions

g(t, t) = 0

g(t2, t1) = [g(t1, t2)]⇤

g(t1, t2) = ln

"�h

0

1(t1)h0

2(t2)

J 2(h1(t1)� h2(t2))2

#

t1 0, t2 0

Quenchtimeplane@

@t1g(t1, t2) = 2

Z t2

�1dt3 J (t1)J (t3)e

g(t1,t3)�Z t1

�1dt3 J (t1)J (t3)

heg(t1,t3) + eg(t3,t1)

i

+2

Z t2

�1dt3Jp(t1)Jp(t3)e

pg(t1,t3) �Z t1

�1dt3Jp(t1)Jp(t3)

hepg(t1,t3) + epg(t3,t1)

i

@

@t2g(t1, t2) = 2

Z t1

�1dt3 J (t3)J (t2)e

g(t3,t2)�Z t2

�1dt3 J (t3)J (t2)

heg(t3,t2) + eg(t2,t3)

i

+2

Z t1

�1dt3Jp(t3)Jp(t2)e

pg(t3,t2) �Z t2

�1dt3Jp(t3)Jp(t2)

hepg(t3,t2) + epg(t2,t3)

i

BoundaryConditions

• Need3BCs

• Get,andfrom,,,

• ToomanyBCs

• SL(2,C)invariance:

• Constraint:

• Resultindependentofchoicesfor,,

• SL(2,R)invariance→ “gauge”choice!

h(t) ! a h(t)+bc h(t)+d

ad� bc = 1

hA1(0) hA2(0) h0A1(0) h0

B2(0)hB2(0)hB1(�1) gC(t)

hB2(0)hB1(�1) h0B2(0)

Post-QuenchSolution

• Chooseansatz

• Findsolutionforp=1/2

• FromKMS:

• Onlydependsonrelativetime:instantthermalization!

gA(t1, t2) = ln

��2

4J 2 sinh2(�(t1 � t2)/2 + i✓)

�(1)

hA1(t) =ae�t + c

ce�t + d, hA2(t) =

ae�2i✓e�t + b

ce�2i✓e�t + d

�f =2(⇡ � 2✓)

�

� = 2J sin ✓ e�4i✓ =(b� dhB1(�1))(a⇤ � c⇤h⇤

B1(�1))

(b⇤ � d⇤h⇤B1(�1))(a� chB1(�1))

RelationtotheSchwarzian Action

• SYKdescribedbySchwarzian for

• TakefromKBequations

• issolutiontoSchwarzian EOM

• Thermalization connectedtoreparameterization modes

• Schwarzian shouldalsoexhibitinstantaneousthermalization

[h0(t)]2h0000(t) + 3 [h00(t)]

3 � 4h0(t)h00(t)h000(t) = 0

�J � 1

L[h(t)] = h000(t)

h0(t)� 3

2

✓h00(t)

h0(t)

◆2

h(t) ! a h(t)+bc h(t)+d

h(t) ! a h(t)+bc h(t)+d

FinalRemarks

• Lowenergylimit,ratelinearinTasexpected

• Alsodependsonq,whatdo1/q2 correctionslooklike?

• 2-pointfunctioninstantlythermalizes,butotherquantitiesdonot

• Whichquantitiesthermalizeandonwhattimescale(somedonot)?

• WhendoesthelargeNlimitbreakdown?

• Whatotherconsequencesdoesthishaveingravity?