A"Chiral"Majorana"Sachdev2Ye2Kitaev Model

Biao"LianPrinceton"Center"for"Theoretical"Science,"Princeton"

University

BL,"Shivaji."L."Sondhi,"Zhenbin Yang,"2019"(to"appear"1905.xxxxx)

Shou'Cheng,Zhang,Memorial,Workshop,May,4,,2019

Graduation*2017.06

Cabo*San*Lucas*2017.11Stanford*dish,*2017.12.31

The$school$of$Athens$- Raphael

A"Chiral"Majorana"Sachdev2Ye2Kitaev Model

Biao"LianPrinceton"Center"for"Theoretical"Science,"Princeton"

University

BL,"Shivaji."L."Sondhi,"Zhenbin Yang,"2019"(to"appear"1905.xxxxx)

Shou'Cheng,Zhang,Memorial,Workshop,May,4,,2019

Outline

Overview:(Majorana(fermion(&(SYK

SYK(generalized(to(chiral(Majorana(fermions

Conclusions

Majorana fermion

Majorana'fermion is#a#fermion#being#its#own#antiparticle:

! " = !$ " , ! " , ! "& = ' " − "& .

The#Majorana'equation'with#real#Gamma#matrices#*+:

,*+-+ − . ! = 0Ettore#Majorana

• Chiral#Majorana#fermion#in#1+1d#(massless&&&unidirectional)

• Majorana#zero#mode#(MZM)#in#0+1d

E

k

Chiral#Majorana#fermionExist&in&condensed&matter&systems&(FQH,&0 + ,0 TSC,&chiral&spin&systems,&etc) Moore,'Read'1991,'Read,'Green'

2000,'Kitaev 2000,'2006

My work with Shoucheng on chiral Majorana

! + #! TSC$&$chiral$Majorana$fermion$device$proposal$(Wang$et#al.$2015)

Chiral$Majorana$fermion$quantum$gates$(Lian$et#al.$2018)

The original SYK model in 0+1d

Sachdev,)Ye)1993,)Kitaev KITP)talk)2015,)Polchinski,)Rosenhaus)2016,)Maldacena,)Stanford)2016,)Kitaev,)Suh)2018)

Black&Hole&(near&extremal)

AdS2

relevant

!"

0+1d&Majorana&Fermions

#"$%&

The$SYK$model$(NMajorana$fermions$with$random$interactions):

• Asymptopic*conformal*symmetry*(for%! ≫ #$ ≫ 1)%under%& → &( =*(&) ,%implying%duality%to%AdS2 spacetime.

Green’s'function''- & ≈ |&|01/3

Effective'action'(Schwarzian)'''

• Ground*state*entropy*45 ∼ 7(8)!,%implying%spin'glass'states and%duality%to%Near'Extremal'Black'Hole with%area%%9 ∼ !:;1 .

The original SYK model

OTOC & maximal chaos

• Out$of$time$order,correlation,(OTOC)

large+N SYK+model:

!" # !$ 0 !" # !$ 0 ∼ 1 − )*(,-,./)

Maximal(Lyapunov(exponent((1 → 345

Classical+Lyapunov+exponent+&+commutator:

)*, ∼ 67 #67 0

3= 9 0 , 7 # 3

= 9 0 7 # 9 0 7 # − 9 0 7 # 7 # 9 0 + ⋯

Maldacena,*Shenker,*Stanford*2016

Higher dimension generalizations of SYK

• Chain&or&array&of&SYK&clusters&(higher&dimension)

• Nonchiral&1+1d&fermions&(large&N Thirring model):&&interaction*irrelevant

• Modified&nonchiral&fermions,&boson&models

• …&…

Gu,%Qi,%Stanford%2016,%Jian,%Yao%2017

SYK%cluster%chain%(Gu,%Qi,%Stanford%2016)

Berkooz,%Narayan,%Rozali,%Simon%2017

Turiaci,%Verlinde 2017,%Murugan,%Stanford,%Witten%2017,Blake,%Davison,%Sachdev%2017,%…%…

Outline

Overview:(Majorana(fermion(&(SYK

SYK(generalized(to(chiral(Majorana(fermions

Conclusions

! + #! TSCN"copies

E

k

Chiral SYK model

Chiral)Majorana)fermion

Minimal)generalization)of)SYK:)))to"chiral"Majorana"fermions?

• A"travelling"version"of"SYK?

• Chaos"of"topologically"protected"chiral"states

Chiral SYK model

The$Chiral$SYK$model$of$! flavors$of$chiral'Majorana'fermions:

"#"$"%"& has$scaling(dimension(2(& conformal(spin(2,$so$the$interaction$'#$%& is

• Exactly'marginal

• Lorentz'symmetry'breaking'(allowed(in(condensed(matter)

Point splitting regularization

Regularization of#operator#products#! ", $ %(", $):

• The$point$splitting$direction$matters$(Lorentz$symmetry$broken)

Condensed$matter:$splitting$in$$ direction$(lattice$cutoff),

(SO(N)1 KacCMoody$algebra)

Integrable via$bosonization:

For$! = 4 flavors$of$fermions,$the$model$reduces$to$the$chiral'Luttingerliquid'with'spin3charge'separation:

Where$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$,$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$.

The $ = % case

free'boson'action

Define&!± = (!↑ ± !↓)/ 2 ,&the&action

where&&&&&&&&&&&&&&&&&&&&&&&&&(spin&&&charge&velocities,&* exactly&marginal)

+ = , & spin charge separation

physical)bound)of)interaction:

! = # two-point function & OTOC

The$fermion$$+point$function$for$% = 4 can$be$obtained$using$Wick’s'theorem.

• 2+point$function$'( ), + = ,- ⟨/0 ), + /0 0,0 ⟩ at$temperature$34,:

• OTOC$ℱ6 ), + = /0 ) /7 0 /0 ) /7 0 :

Large N case

Leading(1/N(expansion((melon)(diagrams

Two$point)function)!" #, % = '( ⟨*+, −./, % +, 0,0 ⟩ satisfies)the)large)N

Schwinger*Dyson/(SD)/equation:

• Scaling(invariance((2 exactly(marginal)(

34, 5 → 7(34, 5),))))! .34, 5 → 7:'!(.34, 5),))))Σ(.34, 5) → 7Σ(.34, 5)

indispensable

Comparison: original SYK & chiral SYK

SD#equation#of#original#1d#SYK: SD#equation#of#2d#chiral#SYK:

relevant indispensable

• Scaling,invariance,(! exactly,marginal)

• No,reparameterization,(conformal),symmetry

• Reparameterization,symmetry,under," → $(")(for,'! ≫ 1)

Zero temperature solution

Scaling(invariant(ansatz for(the(SD#equation#solution:

SD#equation#!

By#solving#!(#) exactly,#one#finds#the#zero;temperature#solution:

Zero temperature solution

Physical)bound)of)interaction)(exactly)marginal):

• Large)! has)the)same)2;point)function)as)! = # !"(SO(N)"symmetry?)

• $(−'(, *) factorizes"like"two)1d)SYKs)along",±( − '* directions".

Spectral)weight:

Nonzero temperature solution

Scaling(invariant(ansatz(fails(due(to(the(scale(set(by(temperature.

A#guess for#the#large#! solution:##the(same(as(! = 4 &(double(SYK

Verified#to#satisfy#the#Schwinger<Dyson equation:

Thermal quantities

• Energy'density:

! Thermal'entropy'density''''''''''''''''''''''''''''''''''''''''''''''''''''''.

• Energy'current:

! Thermal'Hall'conductivity'quantized

(topological'invariant)∇"#

$ℰ&

Kane,'Fisher'1997

Physical)picture:""energy"spectrum"""""""""""""""""""""""""""""",

!

!± :"partition"function"of"N free"chiral"Majorana"with"velocity"#± .

Ground state entropy density

The"partition"function:

Calculating+! (large+N):

! Ground'state'entropy'density''$% = % .

OTOC, chaos

Ladder&diagram&(leading&contribution)

Regularized+OTOC+40point+function

Growing+!ℱ as+eigenfunction+(eigenvalue+1)+of+one$rung(kernel:

• Eigenfunction:,,product,of,that,of,two,1d,SYK:

eigenvalue:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,= 1

Lyapunov,exponent

By#defining###################################################the#kernel#factorizes#into#two#1d#SYK#kernel#in##$ &#−#& :

OTOC, chaos

maximum

OTOC$growing$piece$$(! < # < ! ln&):

where

OTOC, chaos

Saddle&approximation

Saddle&approximation:&'ℱ dominated&by&either&

• saddle&point&of&&&&&&&&&&&&&&&&&&&&&&&&,

or

• pole&of&&&&&&&&&&&&&&&&&&&&&&.

Gu,%Kitaev 2019

Tilted butterfly cone

Growing(piece

(t direction(Lyapunov(reaching(chaos(bound)

Velocity(dependent(Lyapunov(exponent

The$model$is$also$integrable for$! = 5 &$! = 6 via$bosonization.

By#suitable#SO(N)#basis#rotation####################,#one#can#simplify#the#model:

• ! = 5 :##equivalent#to#! = 4 (2#free#chiral#bosons)#plus#a#free#chiral#Majorana.

• ! = 6 :##equivalent#to#3#free#chiral#bosons

Integrable: & ≤ (

! ≥ # & thermalization

The$model$becomes$increasingly$complicated$for$$ ≥ 7,$and$chaotic$in$the$large$N limit.

Conjecture:$$transition)from)integrable to)chaotic at)certain $ = $'

Difference$between$integrable and$chaotic:$$

no$thermalization$$$$$$vs. thermalization

Relation to generic Fermi surface

Fermi&surface

Analogy(between(chiral'SYK and(a'piece'of'interacting'Fermi'surface'(Fermi'liquid):

! = #$%&:(Chiral(fermion

'∥:((Large(N

Patel,'Sachdev'2017

Conclusions

• The$large$N chiral$Majorana$SYK$model$has$exactly$marginal$&$Lorentz$breaking$4>fermion$interactions.

• The$2>point$function$for$large$N is$exactly$solvable,$and$has$the$same$form$as$that$of$! = 4.$$(SO(N)$symmetry$constrain?)

• The$ground$state$entropy$is$zero,$while$the$OTOC$approaches$maximal$chaotic$as$$ → 2' .$The$butterfly$cone$is$tilted.

• Transition$from$integrable$to$chaotic$as$! increases.

• Effective$action:$Schwarzian or$not?

• Relation$to$Fermi$liquid?$Effects$of$spatial$disorders?

Shoucheng,*may*the*force*be*with*you!