Discrete Time Evolution and Baxter's Q-operator - - PowerPoint PPT Presentation

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Discrete Time Evolution and Baxter's Q-operator - - PowerPoint PPT Presentation

Mar 30, 2024 162 likes •337 views

Discrete Time Evolution and Baxter's Q-operator Christian.Korff@glasgow.ac.uk ( :( ( : a un nu nu 1) Cylindric Macdonald functions and a deformed Verlinde algebra, CMP 318 (2013) 173-246 2) From quantum Bcklund transformations to TQFT,

slide-1
SLIDE 1 Christian.Korff@glasgow.ac.uk 1) Cylindric Macdonald functions and a deformed Verlinde algebra, CMP 318 (2013) 173-246 2) From quantum Bäcklund transformations to TQFT, JPA 49 (2016) 104001 Discrete Time Evolution and Baxter's Q-operator RAQIS 16 a ( nu ( nu : :( un
slide-2
SLIDE 2 Road map : quantisation
  • f
the Ablowitt
  • Ladik
chain Classical integrate system [Abeowik . Ladik ' 76 ] Quantum integrate system zxj = { H ,4j } , 2t4j*= { Hits 't } #= .

jz(Bkjt÷BB*

  • znj
) µ
  • tgttsttjnttitjtt
'
  • 2lnHY*4D )
[ kueishiqgi Poisson algebra
  • f
  • boson
algebra [ . , . ] =
  • it
{ . , . }t0th2 ) { Yi ,Xj* } = Sij C l
  • 4g*4j )
[ pi,Pj*I= Sijll
  • of )
a- Rpi ) quantisation { 4. . ,4j}={Yi*,Xg*3=o £ → it ,
  • < q=
etc < 1 4*→±5 integrals of motion ' Bethe algebra ' =

ZDTQFT

Ljcu ) =

his

. "

u%* )

Ljlui = ( B , " u&* ) → monodromy matrix , no monodromy matrix , YB
  • algebra
spectral invariants Baxter 's commuting transfer matrices
slide-3
SLIDE 3 Ablowitt . Ladik chain : separation of time flow Equations of motion 2- Xj = Yj+ , -24J + Yj . ,
  • Xj*4j ( 4g ; ,
+ Yj . , ) { 2.4¥
  • YE
, tarts 's
  • YE
't4j*4jHjEt Yit ' ) Decomposition
  • f
Hamiltonian into left
  • and
right movers H = Hr + Hit Ho , Hr = ,? 4j*Xj+ , , Hi ,?4j4j*t , , { He ,Hr}=o ' Auxiliary time flow ' at ,4j=
  • { HL
,4j } = 4g , ( l
  • 4g*4j)
Since all 3 flows commute , we can consider them separately .
slide-4
SLIDE 4 Datboux matrices : Dj+ , ( u ,v ) Lj ( u ) = [j ( u ) Djcu ,v , , discrete zero . curvature equation det Dj C v.v )=o and Djlu ,0 ) = ( ff ) v ) Bcicklund transform : (Xj ,4j* ) / ( Tlj ,Ij* ) , not , Ij=YjCot) 1 Canonical map which preserves the Poisson structure { . ) . } 2 Commutativity : BCV , ) ° Blvd = B ( vz )°B( v , ) [ Veselov ' 91 ] ' time ' discretisalion : Yj ¥1210 ' = ( 1
  • 4j*4j
Cot ) ) Yj . , Cot ) [ Sun 's 1997 ]
  • t
What is the quantum analogue of this evolution eqn ?
slide-5
SLIDE 5
  • f
  • boson
took space periodic boundary conditions
  • Vacuum
: plo >=o
  • mot
. boson stale : im >=H*zM,o , :^ . !^ . . : : 9- mz mz m , Mn n multi . particle stale : li >=01miH ) > partition d=( d , , ... ,

,\n)=(

Im'2m? . .nmn ) i= , P*jlX>=( 1-qmjtl ) lm , , ... ,mj+ , , ... .mn > , Example shown above : n=io X= ( 10,10 , 817,7 , 5,5 ,5 , 4,4 , 3,313,3 , -2,111,1 ) fj It >= Im , , ... ,mj
  • 1
, ... ,mn ) K = 2+1+2+3 +2+4+1+3 = 18
  • →
Canonical quantisation of the Poisson algebra : An
  • [ pi,pj*]=Sija-9.111
. pitpi )

;

is
slide-6
SLIDE 6 Quantum Backlund transform → Baxter 's Q . operator [ Pasquier
  • Gaudin
1992 ] ( Toda chain ) ( Bj ,pg*li→ ( fj ,fg* ) fj = Qcnpj Qcvi ' fj* =Qcvspjtaa , .it#n : find Q Define Qcv ) as the transfer matrix
  • f
an exactly solvable vertex model :
  • f
  • insertion
: insert a particles into a pile
  • f
b- c particles inside a gravitational potential with
  • f
= 5 PE , E energy to lift
  • ne
particle .
  • •I
c
  • →
  • b
4
  • '
  • at
:::←s¥tI÷=i Iq ' '' ' . [ da ] ,

a##µ

1 insertion q 4+3+3+1 = ql 'll D= at b
  • c
a ,b , C , d E 2120 particle picture Boltzmann weight vertex configuration
slide-7
SLIDE 7 Lattice configurations &
  • f
  • Whittaker
polynomials Let

Me

Match . , ,×n( No ) and set air pipit , INPUT

µ=(

In 'zm . .cn . nmn " ) t¥ µ=⇐po*Mioa,Mn . . .in?mpnTi0

Mom

, i. .

Mn

. , ( )

Mio

M , ,

Mhm

, , . . . m

,o

Define A
  • then
✓ , it th . ' ( of )Mio(9- )Mi , " . l F )Min . , yMzo

Mz

,Mzz

Mz

}

Mzo

vmmmomn

. "

Man

mm

. the Matrix elements
  • f

mimi

... min Zlv )= Em

Tam

' ,

vM=

IT vomij . 1 xiijs d=( 1mi . :( mjmh ' ' ) are the partition function . OUTPUT ( cylindrical ) skew
  • f
  • Whittaker
functions HIM [ CK ' 13 ] Open boundaries

Mio=o

: C

KIZCVIIM

> =P

. ,µ,(

v ;q , 0 ) periodic boundaries Mio >0

:<

KIZCVIIM

> = Ed ZDP

.io/ailv;q..o

)
slide-8
SLIDE 8 ~ Quantum Backlund transform pjcv ) = Qlv ) PjQ(v5 ' where Qcv ) is the row
  • to
  • row
transfer matrix
  • f
the
  • f
. insertion model . htm [ CK ' 16 ] discrete quantum " time flow "

±

YI =( 1-

pjfgcvhfjncvi

k*MjPI*=

Pitt , (

Bitfjm

  • 1)
Discrete time evolution [ Qlu ),Q*lv ) ]=O § ; = Q*cvi§jQ*cv ) §j
  • pj=
  • v(
1- FYP ; )Pj+ , functional equation p ; (
  • t
) =

Q*fioH" Qciot

) pj Qliotl

"Q*fioH

, ucotii Ucots 't

*

  • 5
' Ulot ) time evolution operator &l9toEfI= (l
  • Fyttiotlp
; )pj+ , ta
  • fjtciotspjcot
, )p , . ,(
  • t
, for time step 't
slide-9
SLIDE 9 Multivariate Biicklund transforms & TQFT fusion matrices BTCV, )
  • ...
  • Btcvn
. ,) m> ZCV ) = QCV , ) " .

QCK !

= ? Qx Pxcv ;
  • f
, 0 ) J 4 Fusion matrix
  • f
  • Whittaker function
TIM [ CK ' 13 ] Fusion matrices
  • f
ZD TQFT for fixed particle number k . v.

Naujukcqt

< vl Qxlm > = ¢n§ ' pair of pants ' 2- cobordism R in Recurrence relations for fusion coefficients a- qmim " ) NYjs*kw ' = a- qm . " " ) NEW '±a .qmit ' ' '' 's Nrfdtjuhtka . qmic " )a . qmt ' ' "sNrPBjtjYL "
  • f
→o : nisei '= New '±NrB" think .FI ' " such )h
  • WZW
fusion ring ' phase model ' [ CK . , C . Stoppel Adv . Math . 2009 ]
slide-10
SLIDE 10 ZD TQFT ± Symmetric Frobenius algebras [ Aliyah

19881

Functor Z : Zcob → Vec # IF
  • vector

spaces

with dimV< co Z C On )=V ZCO )=V* s ' multiplication m : ZCSYOZCS 's → ZCS 's Z(&÷j§)E Horn (

ZCSYOZCS

's , ZCS ' ) ) ( commutative ) aoxbt ab invariant c.isiZCsyoxzC5s-EZl@1eHomCZCsY0ZCss.ZC . ) ) bilinear form IF cab ,c > = < a ,bc > unit element e :F→z( s ' ) ZCO ) E Horn ( IF , ZCS ' ) ) 1 H I TQFT partition function

Z(€YFfE¥c

:

:#

) genus g Surface
slide-11
SLIDE 11 ZD TQFT
  • perator
version
  • f
  • bosons

ZCQ

) E Vec . , F=Z[ 9- it ' ' ] Bethe algebra Bmkc End ( Fr , )

si

p÷j§)eHom(

Zcssozcs

's ,Zcs 's )

Q×Qµ=FNdi%9*

÷
  • f ]
IF 1- Z ( @ )eHom( ZCSYOZCS 's ,ZC . ) ) < Qx

,Qµ>=£xµ*Fh[

mica ) ]q ! invariant bilinear form ×= imizmz ... nmn Z ( O ) E Horn ( IF , ZCS ' ) ) Qcn , ... ,n ) Qx = Qx unit element ZC

EXETER

:

:#

) TQFT partition function Tr (

,§Qx§*)9

" genus g surface
slide-12
SLIDE 12 ZD TQFT
  • perator
version
  • f
  • bosons

ZCQ

) E Vec . , E=Z[ 9- it ' ' ] Bethe algebra Bmkc End ( Fr , )

:

( &÷j§)eHom(

Zcssozcs

's ,Zcs 's ) Q×Qµ= ? Nd

'µY9*

÷
  • f ]
IF 1- Z ( @ )eHom( ZCSYOZCS 's , ZC . ) ) < Qx

,Qµ>=£xµ*Fh[

mica ) ]q ! invariant bilinear form ×= |m , zmz ... nmn Z ( O ) E Horn ( IF , ZCS ' ) ) Qcn , ... ,n ) Qx = Qx unit element ZC

EXETER

:

:#

) TQFT partition function Tr (

,§Qx§*)9

" genus g surface AIM : describe the discrete time dynamics in terms
  • f
the TQFT
slide-13
SLIDE 13 Two Q . operators Q±cv)=§,ovrQ±r Qtr . Epn .PE#EHR*Fi*an ,

as

. cnzz.pitanHRI.ch#Mlpnanggaea* Qtr ( Fla. . " ( of Ian ( of la , . . ( of Ian T±M0 Functional identities 0(u1= GZNU " N=#ofq
  • bosons
CK ' 13,16 Tcu )Q+Cul= Qtcuqijtocu )QHuq ' ) , TQ+ equation Qtutlu )= QTuq3+o(uq2)QTuq2) QT equation ) Qtcu ) Qtuqtj
  • unq2N Qtcuq's
QTU ) = I ' quantum Wronskian ' The functional relations also imply the quantum analogue of BH . )°Blk)=BWoBw , ) Cor [ Qtr ,TsI=[ Qtr ,Q's

]=[Q±r,Q±s]=O

. ' Bethe algebra ' C
  • f
  • boson
algebra commutative non . commutative
slide-14
SLIDE 14 Because we are dealing with non . commutative variables in the quantum case , the equation defining the Darboux transformation is now replaced with the Yang
  • Baxter eqn
:

Dfduiv

) Lytu )

Esjlvl

=

Ltsjlv

) Ly . lu )

Dfzluiv

) This allows
  • ne
to define Q± in a similar way as Tcu ) = Tr Lncul . :L , lu ) Q± . operators for the
  • f
boson model [ CK 2013 , 2016 ] M 1 * M Eons = ( trim MIMI " ) mm , , .

ija

' = ( vmqm

'z±

' ' fight) mm . , . Current
  • perators
( formal power series in v with coefficients in
  • f
  • boson algebra)

Qttv

) =

Trench

. . E Cr ) = Four QE " explicitly known
slide-15
SLIDE 15 Omitted from the discussion D Combinatorial approach to compute Ny ,u ( q ) e z [ of 1 f. Hall polynomial Recall skew Macdonald functions : Px ,µ( x ; of it ) = ? ftp.vlq.tl Pvlx ;qH → cylindnc
  • f
  • Whittaker functions
Px,d,µ( x ;q , 0 ) = ? Nstuvlq ) Pv ( x ; q , o ) D N ,Iu (OIEZ ,o are the such )r
  • WZW
  • fusion
coefficients k=#
  • ff
  • bosons
TQFT g=o I Ko ( E ) , E lens or category of Ue such ) tilting modules with e = el "k+h ( QFT : Chern . Simons ) → Geometric interpretation at
  • f
to ?