A Quantum Quench of the Sachdev-Ye-KitaevModel 17... · 2017-11-14 · A Quantum Quench of the...
Transcript of A Quantum Quench of the Sachdev-Ye-KitaevModel 17... · 2017-11-14 · A Quantum Quench of the...
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?